MULTIO
Kaisa M. Miettinen
Kluwer's INTERNATIONAL SER E, 

INTERNATIONAL SERIES IN
OPERATIONS RESEARCH & MANAGEMENT SCIENCE
Frederick S. Hillier, Series Editor
Department of Engineering-Economic Systems and
Operations Research
Stanford Univ¢rsity
Stanford, Califomia
Saigal, Romesh
Nagumey, Anna/Zhaag, Ding
PROJECTED DYNAMICAL SYSTEMS AND VARIATIONAL LVEQUAL1TIES WITH
APPLICATIONS
Padberg, Manfred/Rij al, Minendra P.
LOCATIO3L SCHEDUL[NG, DESIGN AND [NTEGER
PROGRAMMING
Vanderbei, Robert J.
MILITARY OPERATIONS RESEARCI4: Quaniitative Dacision Maklng
Gai, Tomas / Greenberg, Harvey J.
ADVANCES IN SENSIT1VITY ANAL¥SIS AND pARAMETPdC PROGRAMMING
Prabhu, N.U.
FOUNDATIONS OF QUEUEING THEORY
ENTROpy OPTIM1ZdTIONAND MATHEMAT1CAL PROGRAMM1NG
Ho, Teck-Hua / Tang, Christopher S.
PRODUCT VARIETY MANAGEM3IT
EI-Tah, Muhanmaad/Stidham, Jr., Shaler
SAMPLE-PATHANALYSIS OF QUEUEING SySTEMS
NONLINEAR MUL TIOBJECTIVE
OPTIMIZA TION
by
Kaisa Miettinen, PhD
University of Jyvaskyla
k
Kluwer Academic Publishers
Boslon/London/Dor dr echl

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Massachusetts 02061
Prinled on acid-free paper.
Prhated in t United States of America
Anna-Liisa and Kauko
To my parents
with love and respect

CONTENTS
PREFACE
ACKNOWLEDGEMENTS
NOTATION AND SYMBOLS
PtI TERMINOLOGY AND THEORY
INTRODUCTION
CONCEPTS
2.1. Problem Setting and General Notation
2.1.1. Multiobjective Optimization Problem
2.1,2. Background Concepts
2.2. pareto Optimality
2,3, Dccisiot Maker
2,4, Ranges of the pareto Optimal Set
2.4.1. Idem Objective Vector
2.4.2, Nadlr Objective Vector
2.4.3. Related Topics
2.5. Weak Pareto Optimality
2.6, Value Function
2.7, Eflïciency
2,8, From One Sohtion to Anothcr
2,8.1, Trade-Offs
2.8.2. Marginal Rate of Substitution
2.9, Proper pareto Optlmality
2.10, Pareto Optimality Tests with Existence Results
xxi
3
5
5
5
6
23
25
26
27
29
33

viii
3.3.
3.4.
Part II
THEORETICAL BACKGROUND
3.1. Differentlable Optimallty Conditions
3,1.1, First Order Conditiov3
3.1.2. Second-Order Conditions
3,1,3. Conditions for Proper Pareto Optimallty
3.2. Nondifferentiable Optlmallty Conditions
3.2.1. First-Order Conditions
3.2.2, Second Order Conditions
More Optimality Conditions
Sensitivlty Analysls and Duality
METHODS
INTRODUCTION
NO-PREFERENCE METHODS
2,1, Method of the Global Criterion
2,1,1. Diffcrent Metrics
2,1,2. Theorctlcal lesults
2.1.3. Concludlng Remarks
2,2. Multlobjective Proximal Bundle Method
2.2.1. ]ntroduction
2,2.2. MPB Algorit}m
2.2.3. Theoretical Isults
2,2,4, Concluding Remar ks
A POSTERIORI METHODS
3.1. Weighting Method
3.1.1. Tbeoretical Resuls
3.1.2. Applications and Extensios
3.1.3. Weighting Method  m A Priori Method
3.1.4. Concluding Rcmarks
3.2.  Constraint Mcthod
3.2.1. Theoretical Pcsults on Weak md Pareto Optlmality
3.2.2. Connections witb the Weighting Method
3.2.3. Theoretical Results on Proper pareto Optinmlity
3.2.4. Connectior with Tade Off Rates
3.2.5. Applications and Extensios
3.2.6. Concluding Rcnmrks
3.3, Hybrid Method
37
37
37
42
43
45
47
52
54
56
67
67
69
73
75
75
77
78
78
82
83
84
85
85
88
89
92
94
95
96
3.4. Method of Weighted Metrics
3.4,1. Introduction
3.4.2. Theoretical Reslllts
3.4.3. Comments
3.4.4, Connections with Trade Off Rates
3.4.5. Variants of the Weighted Tchebycheff Problem
3.4.6. Connections wlth Global Trade-Offs
3.4.7. Applications and Extensions
3,4.8. Concluding Remarks
3.5. Achievement Scalarizing lmction Approch
3.5.1, htroduction
3.5.2. Theoretlcal Results
3.5.3. Conments
3.5,4. Concluding Remarks
3.6. 0ther A Posteriori Methods
h PRIORI METHODS
4.1. Value ]mction Method
4,1,1. htroduction
4.1,2. Contaient s
4.1.3. Concluding Renecks
4,2. Lexicograplfic 0rdcring
4.2.1, htroduction
4.2.2, Conmmnts
4.2.3, Concluding Remarks
4.3. Goal Progranming
4,3,1. Introduction
4.3.2, Different Approaches
97
97
98
99
100
}00
103
106
106
107
107
108
120
120
122
126
127
129
136
136
137
140
143
146
146
148

5.3. Sequential Proxy OptLnalzation Technique
5,3.1. Introduction
5.3.2. SPOT Algorithm
5.3.3. Conunents
5.3.4. Applications and Extensiotm
5.3.5, Concludng Pmmarks
5.4. Tchebycheff Method
5.4.I. Introduction
5.4.2. Tchebycheff Algorirhm
5,4,3. Comments
5.4,4. Concluding Remarks
5.5, Step Method
5.5.1. Introduction
5.5.2. STEM Algorithm
5.5.3. Comments
5,5.4. Conctudiag Remarks
5.6. Reference Point Method
5.6.1. Introduction
5.6.2. Reference Poút Algorithm
5.6.3. Comments
5.6.4. Inlplement atlol
5.6.5. Applications and Extensions
5.6.6. Concluding Remarks
5.7. GUESS Method
5.7.1. Introduction
5.7.2. GUESS Algoritiml
5.7.3. Comments
5,7,4. Concludhtg Remarks
5.8. Satisficlag Trade-Off Method
5.8.1, Introduction
5.8.2. STOM Aigorlthm
5.8.3. Commelts
5.8.4. Imptement ation
5.8,5. Applications and Exenslons
5.8,6. Concludng Remarks
5.9. Light Beam Search
5.9.1, Introduction
5.9.2. Ligbt Beam Algorithm
5.9.3. Comments
5,9,4. Concludbg Remarks
5.10. Reference Direction Approach
5.10,1. Introduction
5.10.2. Reference Direction Approach Algorithm
5.10.3. Comments
5,10.4. Conciuding Remarks
149
149
151
152
153
153
154
I54
158
159
160
161
161
162
163
164
164
165
165
I67
167
169
17o
170
17I
173
173
174
174
174
176
177
178
178
179
179
180
182
183
184
184
185
187
189
5.11, Reference Direction Method
5.11.1. Introduction
5.I1,2, ID Algorithm
5.11.3. Conunents
5.11.4, ConcludLng Remarks
5.12. NIMBUS Method
5.12.1, Introduction
5,12.2. Vector Subproblem
5.12.3, Scalar Subproblem
5,12,4. NIMBUS Algorithm
5.12.5. Optimality Results
5.12.6. Comparison of the Two Versions
5.12.7. Comments
5.12.8. Implementations
5.12.9. Applications
5.12.10. Concluding Remarks
5.13. Other Interative Methods
5,13.1. Methods Based on Goal Programming
5,13.2. Methods Bsed on Weigited Metrics
5.13.3. Methods Bsed on Reference Points
5.13.4. Methods Bsed on Miscellanoous Ides
Part III RELATED ISSUES
COMPARING METHODS
1.1. Comparative Table of Interactive Methods Presented
1.2. Comparlsons AvaltabIe in the Literature
1.2.1. Introduction
1.2.2. NonLnterative Tests
1.2.3. Intcrative Tests with Human Decision Makers
1.2,4. Interactive Tests with Value Functlons
1,2,5. Comparisons Bsed on Intuirlon
1.3. Selecting a Method
1.3,1. Ceneral Guidelines
1.3,2. Method Selection Too/s
1.3.3. Declslon Tee
2. SOFTWARE
2.1. Introduction
2,2. Review
GRAPHICAL ILLUSTRATION
3.1. Introduction
3.2. I]lustrating the Pareto Optimal Set
199
190
192
193
193
195
195
197
198
198
201
203
205
205
206
207
208
208
209
210
217
218
219
220
221
221
225
226
227
227
228
229
233
233
235
239
239
240

3.3.8. General Remarks
4. FUTURE DIRECTIONS
5. EPILOGUE
240
240
242
243
243
244
245
246
247
251
255
257
293
PREFACE
Lffe inevitably involves decision nmldng, choices and searching for compro-
nfises. It is only natural to want ail of these to be s good s possible, in other
words, optimal. The di6ïculty here lles in the (at le,st partial) conflict between
out various objectives ad goals. Most everyday decisions ad compronfises are
ruade on the bsis of intuition, comtnon sense, chance or ail of these. However,
there are ares where nmthematlcal modelling and prograrnnfing are needed
such s engineerlng and economics. Here, the problems to be solved vary from
desiginng spacecraft, bridges, robots or camera lenses to blendin 6 sausages,
planning and pricing production systems Or managing pollution problems in
envronmental control. May phenomena are of a notflinear nature, whic]l is
why we need tools for nonlinear prOgranmang capable of handling several coin
flicting or inconmlensurable objectives. In tlfis case methods of traditlonal
slngle objective optinUzation arc not enough; we need new ways of thhaldng,
new concepts, and new methods notfllnear mtfltiobjective optimizatlon.
Problems with multiple objectives and criterin are generally known s mul-
tiple critea optimizatïon or multiple criteria decision-making (MCDM) prob-
lems. The area of multiple criteria declsinn making hs developed rapidly, s
the statistics collected in Stcuer et al. (1996) demonstrate. For cxample, by the
year 1994, a nunaber of 144 conferences had been held and over 200 books and
proceedlngs volumes had appeared ou the toplç. Morcover, some 1216 refereed
journal articles were published between 1987 and 1992.
The MCDM field is so extensive that there is good reason to clssify prob-
lems on the bmis of thinr characteristlcs. Tbey ca be dlvlded into two distinct
types (in accordance with MacCrinmmn (1973)). Dependlng on the properties
of the feslble solutions, we distin6uish multinttribute decislon analysis zatd
multiobjective optimizatlon, ha multiatO5bute decision analysis, the set of fea-
slble alternatives is discrete pïedeteïnfined and finite. Specific examples arc
tbe selectlott of the locations of power plants and dumping sites Or the pur-
chase of cars and bouses. In multiobjective optimization problems, the fesible
alternatives are hot explicltly known in advance. An infinite numbeï of them
cxists and they are represented by declsion varinbles restrlcted by constraint
functions. These problems can be called conrinuous. In these cses, one lins to
generate the alternatives belote they can be valuated.
As far s multinttïibute declsion analysls is concerned, we refeï to the mono-
graphs by Hwang and Yoon (1981) and Keeney and Raiffa (1976). More reg

erellces, together with 17 major illethods in the area accolllpadied by sbcnple
exanlples, can be round in the latter monograph. A more recent sunImary of
the methodology is given ñ Yoon and Hwang (1995). A brief hist orical accouzt,
including the bsic ides behind both multlobjectlve optimization and multiat
trlbute declsion analysls together wlth suggestions for further reading, can be
round in Dyer et al. (1992) and Zionts (1992), (The latter also handies mult
attribute utility theory and negotition.) In addition, a review of the research
in both of these problem classes accompanied by future directions appears La
Korhonen et al. (1992a). [t contalns short descriptions of many concepts and
ares in multiple criteria optñniztion and declsion making not included here.
In this book we concentrate solely on continuous multiobjectlve optimiza-
tion. Thls does hot mean that some of the noet hods presented cannot be applied
to multiattñbute decislon analysis. Neveïtheless, most of the methods have
been designed odiy for one or otbeï of the problem types, exploiting certain
speclal characteristlcs.
The importance of multiobjcctive optimiztion can be seen iX'oto the large
variety of pplictions presented in the literature. Some ide of its posslbitities
can be gained from the fact that over 500 papers describing different ppllca-
tions (between tire years 1955 and 1986) are llsted in Wtdte (1990). They cover,
for examplc, pïoblemS concerning agriculture, banking the health service, en-
ergy, industry, water and wildiife.
Even though we hvo restïicted OUrselves to handlig odiy multiobjective
optimization pïoblems, it nonetheless remains a broad area of ïesearch and we
are therefre tbliged to omit several topics to be able to give a udiform prcsen-
tation. We shall restrlct the treatment to deteïn/nistic problems, Nevertheless,
a few words and futher references are in order in relation to problems invol
ing uncertalnties. These can be divided into stochstic and fuzzy problens. In
stochstic programnñng itis usually ssnned that uncertainty is due toa lack
of infornmtion about prevailing states, and that this uncertainty only cOncerns
the occurrence of the states and hot the defidition of the states, results or crb
teria themselves. A problem contalning random vaïibles s coeftlcients on a
certain probabflity space is called a stochstic progranmñng problem (treated,
for exantple, in the moographs of Guddat et al. (1985) and Stancu-Minsian
(1984)), When declsion making takes place in an eavironment where the goals,
constïalnts and consequences tf possible actions are ntt precisely known, it
is called decision making in fuzzy environments (handled, for example, in the
proceedlngs of Kacprzyk and Orlovski (1987)). izzy coeftlcients may also be
iavolved in the problem formulation. Both stochstic and fuzzy multiobjective
optlmization (for linear problems) are dealt with and compared in the pioceed-
ings of Sltwinskl and Teghem (1990), Let us stress once again that here we
assume the problems tobe deteïmidistic; tht is, the outcome of any fesible
solution is known for certain.
Solvlng problems with several conflictbag objectives tsuaily requires the
participation of a human declsion nker who can express prefereace relations
betweea alternative solutions and who continues from the point where math-
enatlcal tools end. Here we assume ttmt  single decision maker is involved.
With several decision makers, the whole question of problem settiag is very dif-
ferent, In addition to the mat hematical side of the solution process, t here is also
the spect of negotitlon and consensus strivhg between the decision makers.
The number of decision makers affects the means of approaching and solving
the problem sigaificantly. A sunmary of group decision makñg is given in the
monograph of Hwang and Lin (1987). Here we settle for one decision maker,
A mnnber of specific problem types requlres special handling (hot included
here), Amogg these are problemS in which the fesible solutions must have inte-
ger values or 0-1 values, multiobjective trjectory optimiztion problems, wheïe
the multiple objectives have multiple observation points, multlobjective net
works or transportation networks and maltiobjective dynamic programmlng.
Here we shall hot go into these ares but adhere to standard methods,
Thus far we have outlined out interest here s being in deter ministic contin-
uous multiobjective optinfization with a sitgle decision maker. Thls definition
focuslng malnly on linear problems is Steuer (1986). However, the methodol-
Hwang and Msud (1979) (currently out of print). One more fact to notice is

the origin and the acbievements lu1 this fleld of research rom 1776 te 1960
are widely treated in Stadler (1979). A brief sulmnary of the history is also
glven in Gal and Hanne (1997). There it is demonstrated that multiobjectlve
optimizatinn hs its foundations iu utility theory and economlcs, gaine theory,
theory, and nonlinear progranmaing.
Let us mention seine farther readlngs. The monographs of Chankong and
Haimes (1983b), Cohon (1978), Hwang and Masud (1979), Osyczka (1984),
Sawaragi et al. (1985), Steuer (1986) and Yu (1985) provide an extensive
overview of the area of multiohjective optimization. Further noteworthy mono-
graphs on the topic are those of Rietveld (1980), Vincke (1992) and Zeleny
(1974, 1982). A slgniflcant part of Vincke (1992) deals, however, with multiat
tribute declsion analysis. The hehavioural spects of multiobjective optlmiz
tion are mostly treated in Pdnguest (1992), wheres the theoretical spects are
extensively handled in the monographs by Jahn (1986a) and Luc (1989).
As far s this book is concerncd, the contents are divided into three parts.
Part I provides the theoretical hackground. Chapter 1 leads into the topic and
Chapter 2 presents important notatlon concepts and definitinns in mu]tlob-
jective optimlzatinn with seine i[[ustratlve figures. Various theoretical spects
appcar in Chapter 3. For example, analogous optimality conditions for dif
ferentiahle and nondifferentiable problems are considered. A solid, conceptual
bsis and foundation for the remalnder of the hook is lald. Throughout the
hook we keep te problems involvlng only finite-dlmensional Euctidean spaces.
(Dauer and Staoee (1986) provide a Survey on multiobjective optimization in
bffinite dimensional spaces 0
The methodology is handied in Part II. Methods are divided into four classes
in Chapter 1 accordng te the role of a (single) decisinn maker in the solution
process. The state ofthe art iu method development is portrayed hy descrihing
a numher of different methods accompanled by thelr theoretical hackground in
Chapters 2 te 5. For ese of comparison, ail the methods are presented using a
unifurm notation. The good and the weak properties ofthe methods are aise in-
troduced with references te ex%ensions and applications. The ctss of interactlve
methods in Chapter 5 contalns most ofthe methods, and it is the most exten
sively handled. Linear problems and methods are onty occslonally touched
on. Ia addition te descrihing solution methods, we iultroduce seine hnplemen-
tatlons. In connectlon with every method descñbed, seine authorS colmnents
appear in the concludlng remarks. Seine of the methods are depicted lu1 more
detall and seine only mentinned. Appropriate references te the ]iterature are
always included.
Part III is Related Issues. Afer the presentation of a set of different se-
intion methods, seine comparison is approprlate in Chapter 1. Naturally, no
ahsointe order of superlority tan he givea, but seine points can he ralsed. A
tahle comparing seine of the features of the interactlve methods descrihed is
incinded. In addition, we present hrief summarles of seine of the comparlSons
preface xvii
available ha the literature. Moreover» we suggest seine outllnes regarding the
important question of selecting an appropriate method. Method selection itself
is a problem wlth multiple objectives. Nevertheless, in additlo te considering
seine siglhïcant factors, we present a decision tree te aid selection. T]ñs tree
contains ail the interactive met/aods previnusly described in seine detail. Itis
bsed on sonoe of the fandamental assumptions underlying the methods and
different ways of exchanging infurmation hetween the method and its user.
Compared with the plethoïa of methods, only a relatively few computer
implementations are widely kmwn and avallahte. However, seine implementa-
tional spects arc touched on and seine software mentioned in Chapter 2.
As computers and monitors have developed, graphical illustration hs in-
cresed in importance and hs aise become esier te produce. Hence graphical
illustration of alternative solutions togetheï with retated matters are featured
in Chapter 3. The potential and restrlctlons of graphics are treated and sonm
clarifying figares are enclosed.
We conclude with comments on fUture dixectinns in Chapter 4 and an epl-
logue in Chapter 5.
This book is intended both for researchers and students in the ares such s
(apptied) mathematics, engineering, econonfics, operations research and man-
agement science; it is meant hoth for professionals and practitioners in many
different ficlds of application. For heginners, this hook provldes an introduc-
tion te the theory and methodology of nonlinear multiohjectivc optitr5zation.
For otber readerS, it effets an extensive rerence te many related results and
methods. Obviousty it is net possihle in a single hook te include ait the aspects
atd methods of notflinear muttiohjective optimlzatlon. However, the intentioI
hs been te provtde a consistent sumnary using a uniform notation leading
te further references. The unifurm style of presentation may help in selectlng
an approprinte method for the problem te be solved. Itis hoped the extensive
bibIiograpby will be of value te researchers.
The hook glves su6ïcient theoretical background te allow those interested te
fullow the deñvation of the featured methods. Howover, the theoretical tïeat-
ment in Chapter 3 of Part I, for example is net essential for the continua-
tion. For both theoretically and practicalIy orieated readers, the algorithms
are descrihed in a consistent manner with seine implementatlonal remarks and
software information aise presented. Because, however, thls is net an actual
textbook, no exerclses or ilinstrative examp]es ]lave bee incinded.

ACKNOWLED GEMENTS
I wlsh to express my gratitude to several indivlduals. First, I mLtst thattk
Professor Peldia Neittaanm/iki, who orl6inalIy proposed multlobjectlve optl-
nllzatio to me as a research subject. I also want to take tIUs oppotunity
to express a speclal thank you to Professor Pekka Korhonen whose ideas and
contacts helped me la publlshing this book.
I mn ladebted to lectureïs Michael Freeman, Arl Lehtonen and Jukka Peldia
Santanen as well as to my collea6ue Marko M/ikeli for their efforts in readln 6
the manuscrlpt and su6gcstlag improvements. In addition, I wish to thank
Marko for availiag me freely of tus expertise in nondifforentiable optinUzatlon.
I mn grateful to my collea6uc Thno Mlanikk5 for tus helpful TEXnlcal
tUnts, lrther, I wásh to express my appreclation to Markku KSnkkSli as well
as Tapani Tarvalnen for theh" hcIp la numerous practlcal and tecImlcal matters
in the course of the prepaa'atlon of this book, ot to mentioa Markku's efforts
wlth some of the figures. I also want to thank my students for their assistance
in the implementations and Marja-Leena Rantalalnen for some revlsions.
On this occasion a speclal vote of thanks should be extended to all those
software developers who bave glven thetr pro6rams to me for test purposes.
My thanks go to the Acdemy of Finland (grant number 22346) for financial
support and to the Universlty of Jyvskyli for providlng me with the facilities
lit which to work.
Finally, I want to truly acknowled6e the support and love of my dcar par-
ents, Atna-Liisa and Kuuko, who laid tbe foundatlons for my educatiom
Kalsa Miettinen

NOTATION AND SYMBOLS
ï(x), z
dist(x, S)
Vf(x)
R
V2fi(x)
of,(x)
U
D
P
n-dimensional Euclidean space
objective function page 5
numbeï of objective functions page 5
decision (variable) vector page 5
number of decision variables page 5
feasible region page 5
fcasible objective region page 5
objective vectoÏ page 5
achievement (scMaïizing) function page 108
2.1.11, page 9
2.1.14, page 10
2.1.14, page 10
2.3.1, page 14
2.3.1, page 14
2.4.1, page 16
2.4.2, page 16
2.6.1, page 21
2.8.4, page 27
2.8.5, page 27
2.8.6, page 28

Part I
TERMINOLOGY AND THEORY

1. INTRODUCTION
We begœel by laybg a conceptual and ttleoretical basis for tire continuation
ad rcstrict out treatmet to finite-dñalensional Euclidean spaces. First, we
present tbe deterministic, continuous problem formulation to be handled and
some geIleral notation. Ttmn we introduce several cocepts and definitions of
multiobjective optimi.atlon as well as t tcir interconnections. Tbe concepts and
terres used in the ficld of multiobjective optimization ae ot completely fixed.
The terminology used terc is occasionally slightly different from that in gen-
eral use. In some cases only one of tbe existing terres is employed. Somewbat
ditïerent definitions of cocepts are presented, for exampl% in Zionts (1989).
To deepen the ttmoretical basls, we treat optimality conditions for ditïer
entiable and nondifferentiable multiobjective optimization problems. Wc also
briefly towh on tire topics of sensltivity analysis, stability and duality.
Througtout the book, even some simple results ae proved, for the conve-
nlence of the readcr (witb possible approprlate refereccs), in order to lay firm
corcrstones for tbe cotimatlon. However, to keep tire text to a reasonable
length, some proos tave been omltted if they can directly be found as suct
elsewtmre. In those cases appropriate refereces in tbe llterature are indlcated.
Multiobjective optlmlzation problems are usually solved by scalarizatioL
Scalarization means that te problem is converted into a siagle (scalar) or a
fmmily of single objective optlmization problems, bi this way tire new problem
bas a real-valued objective function, posslbly depcndlng o some paameters.
After the multiobjective optimizatio problem has bee scalm-ized, the widely
developed theory and mettods for sigle objective optimizatio tan be used.
Eveix ttougb multiobjective optimization mchods ae presentcd in Pav II,
we emphaslze here at the autset tht the mettlods «md the ttmory of slngle
objective optlmization are prcsumed tobe lmwn.

2. CONCEPTS
This 01opter introduces the basic concepts of (nonlinea) multiobjective
optlmfization and the notations uscd b the coatinaation.
2.1. Problem Setting and General Notation
We begin by deflning the problem tobe tndled.
2.1.1. Multiobjective Optimizatlon Problem
We study a multiob3ective optimization pvblem of the form
miiïilize { fl (X), f(x), . . . , A(X)}
(2.1.1)
subject to x
where we havc k (OE 2) objective nctions fi: Rn  R. We denote the vector
of objective nctio::s by f(x) -- (f: (x), f(x),.., f(x))
able) vectors x -- (x:,x,...
(set) S, w]fich is a subset
the form of thc consin ctions formg S, but refer o S ù general.
The word minimize' means that we woet to nfinze all the objective nc
tions simult aneously. If there is no conflict between the objective functlons, then
a solution con be td where every objective nction attains its optimum. In
this c, no speciM methods e needed. To avoid such triviM ces we sume
that there docs not est a single solution that is optinml with respect to every
objective nction. This mem that the objective nctions e at let partly
confiicttg. They nmy ao be bconmensurable (i.e., in dirent units).
In tle followlng, we denote the image of the feible on by Z (= f(S))
and call ita feasïble objective gion. Itis a subset of the objective spaoE R  . The
elements of Z are cMled objective (ction) vectors or cteon vtors and
denoted by f(x) or z -- (z, z,..., z) T, where zi = f(x) for M1 i -- 1,..., k are
objective (nction) values or ctemon values. The words in the parentheses
above e usuMly omitted for short.

For clarity and simplicity of the treatment we assume that ail the objective
fonctions are to be minimized. If an objective function f is tobe ma.ximized,
2.1.2. Background Concepts
First, we present some geaeral concepts and notations. We use bold face
and superscripts for vectors, for exanlple, x 1, and subscripts for components
of vectors, fOr example, x. Ail the vectors bere are assumed to be column
vectors. For two vectors x and x*  1 ', the notation XTX * denote3 tbeir
scalar pvduct and the vector inequa]Jty x < x* rtleJ that xl < x7 for ail
i -- 1...,n. Correspondingly x < x* stands for
The nonnegative orthant of R  is deuoted by R. In other words R --
{x fi R" I zi _> 0 fOr i - 1,...,n}. The Euclidean norm of a vector x
s denoted by Ilxll  (_ z ) . Tbe Euchdean distance function between a
poiut x* and a set Sis deaoted by dist(x*,S) - infxs IIx* - xii. The symbol
B (x*, 6) deuotes an open ball with a centre x" and a radius 6 > 0, B (x', 6) --
{x  R  I Il x* xii < 6}. The notation int S stands for the interior of a set S.
The vectors x , i -- 1,..., m, are linearly independent if the only welghtiug
coefiïcieats fil for which :ï fllx i -- 0 are fl -- 0, i -- 1,...,m. Tbe sure
 flx i is called a convex combïnation of the vectors x,x2,... ,x "  S if
fil > 0 for ail i and '-1 fli -- 1. The convex hull of a set S C R , denoted by
conv S, is the set of ail couvex combinations of vectors in S.
A set S C R  is a cone if çx -- (flz,...,flzm) T  S whenever x  S and
ç>0. Thenegativeofaconeis-S-{ xR lxS}.AconeSissaidto
be pointed if it satsfle3 S t3 -S -- {0}. Acone S transformed to x*  R  is
denoted by x" S-- {x  R  I x -x* +d, wbere d • -S}.
It is said that d  R  is a feasible direction emahating from x  S if there
existe * > 0 such that x +ad  S for 0 <
In some connections we asume that the foasible region is formed of iuequal
ity constraints, t}lat is, S -- {x E R  I g(x) -- (gl (x),g2(x),..., gin(x)) T < 0}.
An nequality contraint ça is said tobe active at a point x* if gj(x*) - 0,
and tbe set of active constraints at x* is denoted by J(x*) - {j
g(*) - 0}.
Different types of mLdtiobjective optimizatiou problems can be defined.
Defiaition 2.1.1. Wben ail tbe objective fuctions and tbe constraint fuc-
tions forming tbe feasible region are linear, tben tbe multiobjective optimlz
tion protflem is called linear. In brief, it is an MOLP (ntdtiobjectivc linear
progranmlhg) problem.
If at least one of tbe objective or tbe constraint functions is nonlinear, tbe
problem is callcd a nonlinear multiobjective optimization pvblem.
A large variety of solution techniques have been created as to enable the
special characteristics of MOLP problems to be taken into account. Here we
concentrate on cases where nonliaear functioas are included and tbus methods
for nordinear problems are needed. Metbods and details of MOLP problems are
meationed only iacidentally.
Before we define convex multiobjective optlmlzation problems, we briefiy
write down tbe definitions of convex fonctions and convex sets.
Definltlon 2.1.2. A fonctlo f: 1  - 1 is convex if for ail xx 2  1  is
valid tbat fi(flx I ÷ (1 - fl)x 2) < 3f(X 1 ) ÷ (1 -- 3)f(x 2) for ail 0 < ç < 1.
AsetScRisconvexifxl,x2Simplie3thatflxl÷(1 fl)x  S for
ail0 <ff< 1.
Definltlon 2.1.3. Tbe multiobjective optimizatlon problem is convex if all tbe
objective fonctions and tbe foasible region are convex.
tbe continuatioa. We sball also need related generalized concepts, quasiconvex
ferentiability. For completeness, we write down tbe deflnltions of differentiable
and coatinuously differentlable foctlons.
Definitlon 2.1.4. A function f): R  - Ris diffe)ntiable at x* if
/,(x* ÷d) /,(x') --V/,(x*)Td+ IId[I e(x',d),
wbeïe Vf(x*) is tbe gudient of f at x* and e(x*,d) - 0 as Ildll -- 0.
In addition, ri is continuously differentiable at x* if ail of its partial driva-
rives  (j = 1,...,n) that is ail the compouents of tb «rutilent are
The gïadient of f at x* can also be denotcd by Vf(x*) to emphasize that
the deriwtion is carried ont subject to x.
Now we can deflnc quasiconvex and pseudoconvex functions.
Definition 2.1.5. A fonction f: P - P is quasiconvex if f(flx  + (1 -
3)X 2) < mL [f(xl), fl(x2)] for ail 0 < fl < 1 and for ail x,x 2 G R .
Let ri be differcntiable at every x  R . Then it is pseudoconvex if for ail
x I ,x 2  R  sucb tbat Vf,(x)T(x 2 -- x ) _> 0, we bave fi(x ) _> fl (x) -
As far as the relations of quasiconvex and pseudoconvex functions are con
cerned, every pseudocouvex fnuction is also quaslconvex.
The definition of convex functions can be modified for concave functions by
replaclng '< by '>'. CorrespondingJy, tbe defiition of quasiconvex fanctions
become3 appropriate for quasiconcave functions by tbe excbange of _<' to '>
and mmx' to 'rein'. In tbe deflnition of pseudoconvex functions we replace

Part I -- 2. Concepts
functfon ri is quiconvex, ail ofits level sers {x • R' ] ri(x) <_ a} are convex
and if it is quicoacave, ail of its level sets {x • R  ] fl(x) > a} are COavex
(see, for example, Mangarian (1969, pp. 133-134)).
ç)x 2) < çfi(x 1) + (1 - ç)fi(x 2) d stctly asiconvex if f,(çx I + (I -
ç)x a) < mx[fi(x), fi(x)] r M1 0 < ç < 1 d r ail x,x   R TM, where
fï(x 1) # fï(x).
Notice that strict convexity of a nction impli convety d convexity
plies both strict quiconvexity and quiconvety. If derentiability is 
sed, oenty impli pseudooenvety which implies strict quiconvexity.
the detls of the relations. The corresponng results are vMid r concave
oencavlty and related concepts c be defined in a oenvex set S C R   well
We Mso need othor ction types. The st of these e re/ated to mon
tonicity.
Correspondlngly, the nction fi is decasing if fi(x ) OE f(x).
decreing. Mnotonicity c be tightened Up in severM ways.
x <x ra j--1,...,n imply fi(x )
Deflnltion 2.1.9. A function ri: R  " Ris strongly increasing if for x  and
xj_<x) forall j=l,...,n and x <x fOrsome I imply f(x 1) < ri(x2).
Correspondingly, the function fl is sfrongly decreasing if f(x ) >
Notice that if a function is strongly decreasing and differentiable, ail of its
partial derivatives have to be (strlctly) negative.
In the nex definition we need a subset R  of R , It is defined as
Deflnition 2.1.10. A ftmction f: pn > p is e-stvngly increaMng if for x 
xlx 2 R\{0} imply f(xl)<f(x2).
For the convenience of the tender we define twice differentiable functions
and some related concepts,
Definition 2.1.11. A function f,: R  - Ris twice-differentiable at x* if
fi(x* + d) fi(x*) : Çfï(x*)Td + dTv2fi(x*)d + ]ld]]2e(x *, d),
where V ff(x*) is the gradient, the symmetric n × n matix V2fi(x * ) is a Hessian
matrix of f at x* and (x*,d) - 0 as ]]dl] --> 0. The Hessian matrix of a
twice-differentiable fianction consists of second-ortier partial devatives 
o :, *  \
In addition, f, is twice confinuously differentiablc at x* if ail of its second-
order partial derivatives are continuous at x*.
A symmetric n × n matrix Mis ca]]ed positive de]ïnite, if xTMx > 0 for
]fi(x ) -- f(x2)] _ K]]x I x2]] for ail x,x 2 • B(x*,6).
Notice that a convex fonction fl : R' - Ris for any point x • R  locally
Lipschitzian at x.
In what fdilows, a function is called nondifferentiable if itis locally Lip-
schitzian (and hot necessarily continuously ditferentiable).

Definition 2.1.13. The multiobjective optimization problem is nondifferen
tiable if some of the objective functlons or the constraint functions forming the
feasible region are nondifferentiable.
According te RoAenmcher's Theorem (sec, e.g., Foderer (1969)), we know
that a locally Lipschitzian function, àefined in an open set, is differentiable
ahnost everywhere in that set. A set where a function f, is net differentiable
is denoted hcre by /22.. ht the sequel, ve employ the concept subdifferential
as defined in Clarke (1983). It corresponds te the gradient in the differentiable
DelhfitioŒE 2.1.14. Let the flmction ri : 1 -> 1 be locally Lipschitzian st a
point x* E 1 . The set
0f(x*) -COnv{E R [ - lira Vf(xl); xl->x
is called a subdifferential of the function f evaluated st the point x*. In addi-
tion, tbe vectors  E 0ff(x*) are called subgradïents.
Wc end with a special type of upper semidiffcrentiable function (sec Wang
(1989)).
Deflnltion 2.1.15. Let the fmction ri: R  -> R be locally Lipschitzian st a
poht x* E RL Then it is upper semidiffentiable st x* if for every d
any sequence {tj}_ wlth tj -> 0 and sequence {}, where
for eery j, we bave
limhf fi(x* + td) fi(x*) < "m
 sup(¢') c.
Special properties of nondifferentiable flmctions are htroduced in Section
3.2, h-a the context where nondifferentiability is }mdled.
Afer these general definitio and concepts we can contrme with muItiol
jective optimizatlon texminology.
2.2. Pareto Optimality
In t}5s section, we handle a crucial concept in optimization, namely op
timality. In single objective optimization problems, the main focus is on
decision variable space. In the multiobjective context we are often more inte
ested in the objective space. For one thing, itis usually of a lower dimemsion
than the decision variable space. Furt}oer, objective values are used below in
defining optimality.
Because of the contradiction and possible incommensurability of the objec-
tive functions, itis net possible te flnd a single solution tht would be optimal
for ail the objectives sbnultaneously. Multiobjective optbnization problems are
in a sense ill-deflned. There is no natural ordering in the objective spce be
cause t m only partally ordered meanng that, for example, (1,1) can b.
said te be less than (3,3) , but how te compare (1,3)  and (3,1)). Ths s
always the case when vectors are compared in real spaces (sec aise Chankong
and Hairaes (1983b, pp. 64ï7)).
Anyway, seine of the objective vectors can be eXracted fOr examintion.
Such vectors are those where none of the components can be improved without
deterioration te at lcast one of the other components, Edgeworth (1987) pre
sented ttfis definition in 1881. However, the definition is usually called Pareto
optbnality affer the lench Italian economist and sociologist Vi[fredo Pareto,
who in 1896 developed it further (sec Pareto (1964, 1971)). However, in seine
connections, like it Stadler (1988b), the terre Ed9eworth-Pato optimality L
Definition 2.2.1. A decision vector x" E Sis Parto optimal if there does net
ex]st another dcclsion vector x  S such that ri(x) <_ fi(x') for ail i -- 1,..., k
and f(x) < f(x*) for st leaSt ote index j.
An objective vectoï z"  Z is Pareto optimal if there does net exlst «mother
objective vector z E Z such that zi < z; for ail i -- 1,...,k and z < z for
st least one index j; or equivalently, z* is Pareto optimal if the declsion vector
corresponàing te itis Parcto optimal.
In Figure 2.2.1, a feasible region S C 1 3 and its image, a feasible objective
region Z C 1 , are illustrated. The fat line contains ail the Pareto optimal
objective vectors. The vector z* is an example of them.
z 2
Figure 2.2.1. The sers S and Z and the Pareto optimal objective vectors.

this result, see e.g., Censor (1977).)
a nelghbour tlood B(x*, 5) of x* sltctl ttat ttere is no x E SN B(x*, 5) fOr which
fl(x) < ri(x*) for ail i -- 1,..., k and for at least one index j is fj (x) < f (x').
(2.2.1) fl(x °)_<ri(x*) for Mli--1,...,kandf(x °) < fj(x*) for somej.
Let lts define  -- flx ° + (1 fl)x', wtiere 0 < fl < 1 is sclected suctl that
fi() < flfi(x °) + (1 - fl)f,(x*) < 3f(x*) + (1 - fl)f/(x*) -- f(x*) for every
tvo i() -- L(x*) for ail i.
Furtter, f(x*) < flfi(x °) + (1 - fl)f,(x') for every i - 1,..., k. Because
fl > 0, we can dide by il and obtaitl f,(x*) _< f(x °) for all i. According
to aSSnmption (2.2.1), we havc f2(x*) > fa(x °) for some j. Here we bave a
2.2. Pxeto Optimality 13
in Ru£z-Canales and Iuflhn-Lizana (1995). These assunlptions can be further
rel.xed according to Luc and Schaible (1997).
Theorem 2.2.4. Let the nmltiobjective optlmizatlon problem Iiave a convex
foasible region and quasiconvex objective flmctions wlth at least one strictly
quasiconvex objective function. Tben every locally Pareto optimal solution is
also globally Pareto optimal.
Proof. Let x* é S be focally Pareto optimal. Thus there exist some  > 0 and
ri(x) _< f(x*) for M1 i -- 1,..., k and for at least one index j is f(x) < f(x').
Let us assume that x* is hot globoJ]y Pareto optimal. In this cse, there
exists some other pobIt x °  S such that
(2.2.2) fi(x °) < fi(x*) for Ml i -- 1,... ,k and fj(x ) < f(x*) for some j.
Let us derme : -- flx ° + (1 - ff)x*, wtiere 0 < ç < 1 is selected such tbat
Employiug (2.2.2) and by the quasiconvexlty of ttic objective functfons,
respectively, for each index i such that fi(x °) -- fi(x*), we obtain
fi(:) _< max[f,(x°),f(x*)] -- f,(x'),
and for each index j sucl tlat f(x °) < fj(x'), we ave
fj() <_ max[f(x°), fj(x*)] -- f/(x*).
Because at least one of ttle objective hactions is strictly quasiconvex, t least
local Pareto optiraality of x*. Thus, x* is globally Pareto optimal. []
For the sake of brevity, we shall usually speak only about Pareto optimality
in tbe sequel. In practice, tiowever, we oRly bave locally Pareto optimal solu-
tloRs computationally avallable, tmless some addltinnal reqàremett, suctl as
convexity, is fulfilled.
Usually, we are interested in Pareto optimal solutions and can forger ttie
ottier feasible solutions. Exceptions to tins practice are problems wtiere one of
tte objective functions is a approximation of an unknown fluction or ttlere
arc underlying unexpressed objective flmctions involved. Ttien, ttie real Pareto
optimal set is malfliown.
According to the deflnitfon of Pareto opt[mality, noving from one Pareto
optimal solution to anotber necessitates trading off. Ttiis is oae of ttie hasic
concepts in multiobjective optimization. Let us, tlowcver, mention tbat ttie
idea of trading off can be called into question, as suggested, for exma}ple, in
Zeleny (1997). It is not pertuUps always necessary to trade off bt orde to attal
impoved results. One can argue that it tins been possible to produce tkings

both at lower cot and with higher quality. Changlng the way of approaching the
2.3. Decision Maker
and Lin (1987) md Y|l (1973)).
in Zionts (1997a, b)).
By solving a multiobjectlve opthnization problem we here mean finding a
and the reqnrements of the decision maker. Assuming such a solution exists,
it is coeed a final solution. However, as stressed in Zionts (1997a, b), it may
solutions in real problems. If this is the case, the emphasis should be on findiig
good solutions (and sometlmes, only, on flnding solutions).
We de net focus here on the problems of decision madng, whicb is a research
wlth incomplete information, validity of the problem formulation and habitual
doma[ns. The flrst of these matter is treated, for example, in Weber (1987).
Reasons for incomplete information include lack of knowledge, pressure of time,
problem bas net been formulated well exmugh. As already emphasized, non
Pareto optimal solutions may be important ff there are seine unformulated or
hidden objective functions in the mind of the decision maker or seine of the
objective fonctions are stmply proxies of the objective functions proper (see,
for example, Zionts (1997a, b)). La such cases, the pareto optimal sets of the
problem handled and the actual problem wlfich sholfld be solved, de net ce
A habitual do*nai is deflned in Yu (1991) as a set of ways of thinking, judg-
ig and respondlng, as well as the fomwledge and experience on wlfich they are
based. Yu emphasizes that in order te make effective decisions it is important
te expand and enrich tho habitual domains of the decision makers. Several ways
of carrying this out are presented in Yu (1991, 1995). Understanding, expand
ing and enrlchlag thc domains of ttdnking is aise stressed, for example, in Yu
0994) and Yu and Liu (1997).
2.4. Ranges of the Pareto Optimal Set
region S.
ideal (or peïfect) objective vector.

Deflnltion 2.4.1. The components zî of tbe ideal objective vector z* E R 
are ohtained by mhlimizlng each of the objective functlons individually subject
to the constraints, that is, by solving
minimize fdx)
subjcct to x E S,
[t is obváous that ff the ideal objective vector were feasible (that is, z* E Z),
it would be the solution of the multiobjective optimization problem (and the
Pareto optimal set wo/tld be reduced toit). This is not possible in general since
there is some confiict anaoag the objectives. Even though the ideal objective
for. lom the ideal objective vector ve obtain the lower botmds of thc Pareto
optimal set for each objective function.
The delïnition of the ideal objective vector assumes that we know the global
minima of the individual objective functioas. Guarahteelng global opt[mality
in mtmerical calculations is aot that simple. ïis must be kept in mind with
practical problems. Properties of ideal objective vectors, for example, their
uniqueness, are treated in Skulimowski (1992).
strictly domlnates, every Pareto optimal solution.
tire vector whose components are formed by
for oe i - 1,..., k, where zî is a component of the ideal objective vector and
2.4.2. Nadir Objective Vector
A payoff table is fOrmed by using ttæ decision vectors obtalned when calcu-
the bold lines.
z 2
z 2
Figure 2.4.1. Ideal objective vectors and nadlr objective vectors.
Note that the objective vectors in the rows of the payoff table are Pareto
opt[mal ff thcy are unique. In othcr words, if the indivádual objective functions
have alternative opt[ma, the obtabmd objective vector may hot be Pareto op
t[mal. Ttfis fact can weaken the approach and it can happen in linear as well
as i nonlinear problems.
It is important to note that the est[mates based on the payoff table are
not nccessarily equal to the real components of the nadir objective vector as
demonstrated, foï example, in Korhonen et al. (1997) and Weistroffer (1985).
Instead of being COrrect, the nadir objective value approximatc may be either
far too low Or too high.
The difference between the complete Pareto optimal set and the subset of
the Pareto optimal set bounded by the ideal objective vector and the upper
bounds obtalned from the payoff table in linear cases is explored in Reeves and
Rcid (1988). It is proposed that r elaxing (i.e., increasing) the approximated up-
per bounds by a relatively small tolerance should iraprove the approximation,
although it is ad hoc in ature. However, small tolerances may hot necessar
ily help because the error between the correct and tbe approx[mated nadir
objective value may be significanç.
For nonltnear problems, there is no constructive method for calculating the
nadiï objective vector. That is why wc here mention somc treatments for MOLP
problenm, lsermam and Steuer (1988) include an examination of how many of
the Pareto optinml extreme solutions of some MOLP problems are above the
upper bounds obtained from the payoff table. Three methods for determining
the exact nadir objective vector in a linear case are also suggested. None of
them is especially economical computationoey. Ln Dessouky et al. (1986), three
heuristlcs are presented fOr calculating the nadir objective vector when the

problem is [ineêx. A heuristic for MOLP problems s also described i Korhoe
nen et al. (1997). Itis demonstrated how much better êxe the approxhnatior
the heuristic can provde. Heuristics êxe usually able to improve the approxi-
mations obtalued from the payoff table even though they amy hot always find
the correct nadir objective values. Heuristics are ofeu compttationally much
less demanding than exact procedures.
Nonetheless, the payoff table my be used as a rough estiamte as long
as its robustness is kept in mind. Because of the bove described difficulty
of calculating the actual nadir objective vector, we shall usually refer to the
approximate nadir objective vector as z nad.
2.4.3. ttelated Toplcs
In nmny occasions it is advisable to rescale, that is, normalize the objective
functions so that their objective values are of approximately the saine mag-
nitude. If the ideal objective vector atd a good enough approxinmtinn to the
nadir objective vcctor are known, we can replace each objective functioa
In this case, thc range of each new objective fimction is [0,1].
Another relted possibility is to use  range equalization factor, as suggested
kn Steuer (1986). The range Ri of each objective nctlon is first estimatad by
the difference betweea the (possibly appro:dmated) nadir objective vector and
the ideal ohjective vector. Then, constants
re defined for every i - 1,...  k, and finally eacb objective flnction is maltl
A simpl alternative for noïmalizlng thc objective function values is to di-
vide each objective fuictinn by its (nolzero) ideal objective value. This has
been suggestad, for cxarnple, in Osyczka (1984, 1992). This is hot as exact as
the prvious methods but does hot neccssitte infornmtion about the nadir
objective vector.
Itis usually advisable to use nornmlized objective values only in calcu-
ltions and to display restored objective values in thc oïiginal scales to the
decision nmker. Ia this way thc dhfferent seales do hot confuse computaion
and significant objective values are offered to the decision *aakeï.
Itis possible that (some) objective functions are nbounded, for instance,
ïom below. In this case some cautioa is b order. In multiobjectivc optimiza-
tion problems this does hot aecessêxily meat that thc problem is foïmulatad
2.5. Weak Preto OptimMity 19
incorrectly. There may still exist Pêxeto optimal solutions. However, if, for la-
stance, some comportent of the ideal objective vector is unbonded and it is
replaced by a snmll but finite number, methods utilizing the ideal objective
vector nmy not De able to overcome the replacemelt.
Finally, let us look at some examples ofthe problem of optimizin a flnction
over the Pêxeto optimal set ofa multiobjective optinfization problem. This is a
more general problem than just looking for the ranges of the Pareto optinml set.
Lu Benson and Sayin (1994), the authors deal with the *naxinization ofa linear
function over the pareto optimal set of an MOLP problem. A gcneral fmction
is nfininized oveï the Pêxeto optimal se of an MOLP problem in Dauer and
Fosnugh (1995), and a convex fnctlon is optinifzed over the Pareto optinml set
of lineêx objective finctions and a convex feasiblc region by duality techiques
in Thach et al. (1996). Maxinfization of a functioa over the Pêxeto optimal set
is also considered in Horst and Thoai (1997).
2.5. Weak Pareto Optimality
h addition to Pareto optinmlity, other related concepts êxe widely used.
These are weak and pïoper Pareto optinmlity. The relationship between these
concepts is that the properly Pêxeto optinml set is a subset of the Pêxeto
optinml set which is a subset of the weakly Pêxeto optinml set.
A vector is wealdy Pêxeto optimal if there does hot exist any other vector
for whicb ail the components are better. More formally it means thc followlng:
Definltlon 2.5.1. A decislon vector x*  Sis weakIy Pareto optimal if there
does not exist another decision vector x  S such that f,(x) < f(x*) for ail
An objective vector z*  Z is wekly Pêxeto optinal if there does not
exist another objective vector z  Z such that zi < z for ail i - 1,... ,k; or
equlvalently, ff the decision vector corresponding toit is wekly Pareto optinml.
The bold line in Figure 2.5.1 represents the set of weakly Pêxcto optinal
objective vectors. The fact that the Pareto optimal set is a subset ofthe wealdy
Pêxeto optinlal set tan also be seen in tke figure. Tbe Pêxeto optimal objective
vectors êxe situated aloag the line between the dors.
Similarly to Pareto optinmlity, local weak pêxeto optinality can be defined
in addition to the global weak Pêxeto optimadty of Definition 2.5A. It must
still be kept in mind that usually only locally weakly pêxeto optinml solutions
re co*nputationMly awallable. Neverthcless, for the sake of brevity, we shall
usuady refer only to weak Pêxeto optimadty.
Let us state as a cm'iosity that if the feasible region is convex and the objec-
tive functlons êxe quasiconvex wifb at least one strictly quasiconvex function,
the set of locally Pareto optinal solutions is a s,ïbset ofthe set of wealdy pêxeto

z 1
Figure 2.5.1. Weakly Pareto optinml vectors.
optimal solutions. This result is ma mmediate corollary of Theorem 2.2.4, where
we proved that undcr the above-mentioned assumptlons ail the locally Pareto
optimal solutions are also globally Pareto optimal.
The connectedness ofthe sets of Pareto optimal and weakly Pareto optimal
solutions has hot been Adely tzeated. Yet, this is an important feature becase
it is often lseful to know how well one tan more conthtuously from one (weakly)
pareto optinml solltion to maother.
The Pareto optinml set of ma MOLP pzoblem is proved to be connected in
Steuer (1986, pp. 158, 220). It is stated in Warburton (1983), that thc Pareto
optimal set is connected in convex multiobjective optimization problems. In
addition, Warforton shows that if the feasible regio is coavex and compact
and the objective functions are quasiconvex, then the set of wealdy Pareto
optimal solutior is connected. The comaectedness ofthe Pareto optimal set is
guaranteed for a certain subclass of quasiconvex functions. A noncompact case
is also studied in Warforton (1983).
The structure, including connectedness, of the sers of weakly, poperly or
Pareto optimal solutions for nonconvex problems with two objective functions
is iavestigated in Tenhuisen and Wiecek (1996). A review of connectedness
resalts for Pareto optimallty is given in Benoist (1998). Benoist also proves
that the Pareto optinml set is coimected for continuous strictly quasiconvex
objective functions (when transformed for minimizatlon problen) defined on
a covex and compact set.
Although weakly Pareto optimal solutions are mportmat for theoretical
considerations, they are mt always useful in practice, because of the large
size of the weakly Pareto optinml set. However, they are often relevant from
a technical point of view because they are sometimes easier to generate thma
Pareto optimal poiIlts. A more restrictive concept thma Pareto optimality is
proper Pareto optinmlity (to be defined in Section
2.6. Value Function
]tis ofteR asstnned that the decision maker makes dealsioas on the basis of
ma underlying fonction of some kind. This fonction is called a value fonction.
Definltlon 2.6.1. A fonctiot U: R  - R representing the preferences ofthe
decision maker among the objective vectors is called a value function.
Let z t and z 2  Z be two different objective vectors. If U(z I ) > U(z), then
the decision maker prefers z  to z . If U(z t ) - U(z2), tlmn the decision maker
finds Che objective vectors equally desirable, that is, they are indif]erent.
]t mst be pointed out that the value fonctio is totaJly a decision maker-
dependent concept. Different decision makers may have different valte fonctions
for the same problcm.
Sometimes the terre utilty function is used instead of the value fonction.
Here we follow tire common way of referrflg to value fonctions in deterministic
problems. The terre utility functlon is reserved for stochastlc problems (hOt to
be hmadled here). See Keeney and RaLffa (1976) foï a more exteaded discussion
of both terms.
If we bad at our disposal the ñtathematical expression of the decision
makcr's value fonction, if would be easy to solve tbe multiobjective optinñz
tion problem. The value function would simply be maximized by some method
of single objective optinñzatiom The value fonction wotdd offer a total (coin
plete) ordering ofthe objective vectors. However, there are several reasoas why
this seemingly easy way is hot geaerally used in practice. The most important
decision maker.

22 Pt t 2. Concepts
nlethods, convergence results are obtalned by nmking certain assumptions for
example, quaslconcavity about the impllcit value fnctlon. In ail, we can say
that alue functloas are usually more important to the analyst than to the
decslon nmker (see Zionts (1997a, h)).
Generally, the value fnction is assumed to be strongly decreasing. This
means that the preference of the decialon maker will increase ff the value of
an objective functlon decreases whde ail the other objective values renmln un-
changed (i.e., less is preferred to more). This assumptlon is justified by Rosea-
thal (1985), who stresses that "Clearly, under the monotonlclty assumption a
rational decision nmker woaid noyer dellbertaly select a dominated poht. This
is probably the only important statement h nultlobjective optimizatlon that
can be nade without the possibillty of generating some disagreement."
Howeve L there are exceptions to this situation. Rosenthal nœentions as an
(nuximizatlon) example the deer population, where nore deer are usually pre-
ferred to fewer for aesthetlc and recreational reasons, but hot in the case when
the deer population is large enough to remove ail the forest undergowth.
The following theorem presents an important result concerning the solutions
of strongly decreasing value functions.
Tbeorem 2.6.2. Let the value function U: R  - R be strongly decreasing.
objective vctor z Ç Z such that z i <_ z for ail i = 1,...,k and z 3 < z for
at least one index j. Because U is strongly decreasing, we have U(z) > U(z*).
2,7. Efflciency 23
Itis important to realize that regardless of the existence of an tmderlying
value functlon, a general assumptlon still is that that less is pÆefered to more
by the decision nmker, that is, lower objective function values are preferred to
higher. This assumptlon is usually nade even in noethods Rot involvhg alue
functlons  any way. Thus, assuming that less is preferred to more is a more
general assumption than assumiag a strongly decreasing value functloa.
2.7. Efl]ciency
Itis possible to derme optinmlity in a multiobjective context in moïe general
ways than by Pareto or weak Pareto optimality. Let us have a poited convex
cone D defined in R a. This cone Dis called an ordeng conc and it is used to
induce a par tiai ordering on Z. Let us have two objective vectors z  and z   Z.
An objective vector z  dominatcs z: deaoted by z  /) z , if z  - z   D and
z  ¢ z , that is, z: z   D \ {0}. The saine can also be written as z  5 z  + D
and z   z , that is, z 2  z  + D \ {0} as illustïated iï Figure 2.7.1.
Figure 2.7.1. Domination iaduced by acone D.
We can now present a defmltion of ptimality based on domination, which
is an alternative to the defmitions previously given. When an orderlng cone
is sed in defining optbnaSty, thea the terre efficlency will be used in what
Defiitlon 2.7.1. Let D bea pointed convex cone. A decisioi vector x*  S
is efficient (wlth respcct to D) if there does not exist anotheï decision vectoï
x E S such that f(x) <D f(x*).
An objective vector z* E Z is efficient if there does hOt exist another objec-
tive vector z  Z such that z _<D Z*.

This definition means that a vector is efficient (nondominated) if it is not
donfinated by any other feasible vector. The definltlon above can be formulated
iii many ways. If we substltute _<D for its definitinn, we have the condition
the form 0  z* - z • D or z* - z • D \ (0} (see Corley (1980)).
Other equivalent formulations are, for instance, z* • Z is efficient ff (Z
z*) ç (-D) - ({}} (see pascoletti and Serafmi (1984) and Weldner (1988)),
(z* D \ (0})çZ - @(see Tapla and Murtagh (1989) and Wierzfficki (1986b))
or if (z* D) n Z - z* (see Chen (1984) and Jah (1987)).
Let us give an alternative formulation to Definltlon 2.7.1 using one of the
equàvalent representations.
Definition 2.7.2. Let D be a pointed convex cone. A decislon vector x* • S
is efficient (wlth respect to D) ff therc does hot exist maothcr decislon vector
x • S such that f(x*) E f(x) + D \ {0}, that is, (f(x*) D \ {0}) çZ - @.
An objective vector z* E Z is efficient if there does not exist another objec-
tlvevectorzEZsuchthatz*z+D\{O},thatis,(z* D \ {0}) çZ - 0.
DLfferent notions of efficlency are collected in Ester und T51tzsch (1986).
They provide several auxfliary problems in the inteïests of obtaining efficlcnt
solutions.
ttemark 2.7.3. The above defmltions are equivalent to Pareto optintallty ff
D - R+ (see Figtre 2.7.2).
Figure 2.7.2. Pareto optinmlity with the help of cone tt.
When Pareto optimality or efficlency is defined with the help of ordering
cones, itis trivial to verffy that Pareto optimal or efficient objective vectors
always fie on the boundary of the feasible objective region Z.
Instead of acone D which is constant for every objective vector, we can use
a point-to-set map D from Z into tt a to represent the domination structure. In
this case domination is dependent on the current objective vector. For details
of ordering cones, see Sawaragi et al. (1985, pp. 231) and Yu (1974, 1985,
pp. 163-209).
Theorem 2.6.2 ves a relatlonship between Pareto optimal solutions and
value functions. Relations can also be estabfished between efficient solutions and
value functlons. To glve an idea of them, let us consider a pseudoconcave value
fonction U. Accorditg to pseudoconcavity whenevcr VU(zt)T(z 2 z ) <_ 0, we
have U(z 2) _< U(zZ). We can now define an ordering cone as a map D(z) -
(d • pk ] V (z)Td _( 0}. This order hng cone can be uscd to determlne efficient
structure, but hot generally vice versa. See Yu (1974) for an exarnple.
Weakly efficient decision and objective vectors can he defined in a Corre-
spondlng fashion to efficient ones. If the set Z of objective vectors is ordered
by an orderlng cone D, weakly efficient vectors may be characterlzed in the
folfowing way (see Jahn (1987) and Wierzbickl (1986h)):
Definltlon 2.7.4. Let D he a polnted convex cone. A decision vector x* • S
is weakly efficient (with respect to D) if there does not exlst another declsinn
vector x • S such that f(x*) • f(x) + int D, that is, (f(x*) int D) ç Z .
An objective vector z* • Z is wealdy efficient if there does not exJst another
objectivc vector z • Z such that z*  z + int D, that is, (z* - int D) ç Z - .
efficient if (Z z*) n (-int D) -  (see Sawara et al. (1985, pp. 33 34)).
Connectedness of thc sers of wealdy efficlcnt and efficient points is studied
in Helbig (1990) whereas Luc (1989, pp. 14154) treats partlcularly weakly
efficient sets in convex problems where the ohjective functions are quasiconvex.
h addition, connectedness results for efficient points in multlobjective combi-
natorial problets are given in Ehrgott and Klamroth (1997).
In the following, we mostly sertie for treating Pareto optlmallty. Some ex-
general effic[ency and weak efficiency. Proper Pareto optimality and proper
efficiency are yet tobe introduced. To clarify their practical meaning and for
other furthcr purposes we must first, however, define trade-offs and narginal
2.8. From One Solution to Another

2.8.1. Trade-Offs
We have several concepts involved in trading off. A trade-off reflects the
ratio of change in the values of the objective functinns concerning the increment
of one ohjective function that occurs when the value of some other objective
functlon decreases. In the following definitions we have i, j -- 1,...  k, i  j.
Definitlon 2.8.1. (lrom Chankong and Haimes (1983b)) Let x ] and x 2 • S
be two decisinn vectors and let f(x I ) and f(x ) be the corresponding objective
vectors, respectively. We denote the ratio of change between the functions
and fj by
A A x x  - /(x)
where fj(x 1 ) - f(x ) ¢ 0.
Now, A,j is called a partial trade of], involving ri and f betweea
x 2 if fl(x ) - fl(x 2) for ail I - 1,..., k, I ¢ i,j. If fl(x ) ¢ fl(x ) for at least
Note that in the case of two objective functioas there is no difference be-
cision maker, (s)he can compare changes in two objective functinns at a
This is usually a more comfortable procedure thau comparing sevcral objec
objective functinns f, and fs for which the trade-off is negative. A concept
related to the trade-off is the trade-off rate.
Definitlon 2.8.2. (From Chankong and Haimes (1983b)) Let x*  S be a
declsinn vector and let d* be a feasible direction emanatlng from x*. The total
trade-of] rate at x*, involvhg f d f Mong the direction d*, is given by
If d*  a feible dectinn so that the exists a > 0 satisfylng f(x* + ad*) =
Remk 2.8.3. ff the obctlve functions e contlnuously differentlable, then
Vfj(x*)Td *'
where the denoanator differs from zero.
For continuously differentlable objective functlons we eau alternatlvely glve
the following definltion.
Definltion 2.8.4. Let the objective functions be continuously differentiable
at a decisinn vector x* • S. Then a partial trade-off rate at x*, tnvolvlng f,
and fj, is given by
"Xi = t(x. ) _ Of;(x*)
Differing from the ide& of the definitinns bove, a scalled global trade-
off is defined in Kaliszewskl and Michalowski (1995, 1997). A global trade-off
involves two objective functinns and one decisinn vector which does not have to
be Pareto optimal. It is the/argest pairwise trade-off of two objective functions
for one declsion vector. Let us consider x* • S and modify the definitinns for
mlnimization problems. We dcfine a subset of the feasible declsinn vectors in
the form
Now we eau introduce global trade-offs.
Definltion 2.8.5. (FromKaliszewskiandMichalowski(1995,1997)) Letx* •
S be a decisinn vector. We denote a global trade of] between the functinns f
and fs by
Ai=Ai(x)= sup .
x¢>/x *) fJ(x) --
If S>(x*) -- , then A(x*) -- oe for every i
A generallzed defiditlon of trade-offs in ternm of tan,gent cones, neauhg
feasible directinas, in the objective space is preseated in Henlg and Bucbauau
(1994, 1997). These generalized trade-off directions eau be used for calculat-
ing trade-off rates at every Pareto optimal point of a convex multiobjective
opt imizatinn problem.
Note that trade offs are defined mathematically and the decisinn maker
cannot affect them. If we take into consideration the opinions of the decision
maker, we can define indifference curves and marginal rates of substitution.
2.8.2. Marginal Rate of Substitution
It is sald that two feasible solutions are situated on thc same indifference
curve (or isopreference curve) if the declsion rnker finds them equally deslr-
able, that is, neither of them is preferred to the other one. This means that
indifference curves are contours of the underlylng valle function. There may
also be a 'wlder' indifference hand. In this case we do hot have any welVdefined
boundary betweea preferences, but a hand where indifference occurs. This con-
cept is studied in Passy and Levanon (1984).

Definition 2.8.6. A maryinal rate of substitution mi - m(x*) represents
the preferences of the decisioa maker ata decision vector x* E S. It is the
amount of decrement in the value ofthe objective fnction fl that compensates
the declslon maker for the one-unit increment in the value of the objective
5anctinn f, while the values of ail the other objectives remain unaltered.
Note that in the definltinn the starting and the rcsulting objective vectors lie
on the saine indifferertce curve and i, j - 1,... ,k, i 7  j.
It can be stated that the final solution of a multïobjective optinfization
problem is a Pareto optimal solution where the indifference curve is tangent to
the pareto optimal set. This tangency condition means finding an indifference
curve intersecthag the feasible objective region that is far thest to the southwcst.
This property is illustrated in Figure 2.8.1.
z I
Figttre 2.8.1. The final solution.
2.9. Proper Pareto Opthaality
temark 2.8.7. If the partial deñvatives exist, then
.j(x*) = OUff(*))/OU(f('))
- of,-/ of "
29
If the Pareto optimal set is smooth (th.t is, at every Pareto optimal point
there exists a uinque tangent), we have the following result. When one exaraines
the definition of a trade-off rate at some pohL one sees tbt it is the slope of
the tanget of the Pareto optimal set ai that point. We can also define that
when a Pareto optimal solution is a final solution, then the tangents of the
indherenee curve and the Pareto optimal set colnclde ai it, that
(2.8.1) -mi -- , for ail ï,j - 1,...,k, ï 7j.
Thus, wlth the help of the negative of the marginal rate of substitution and
the tradoff rate one can get a local lincar approximation of the indifference
curve and the Pareto optimal set, respectively.
Usually, one of the objective functions is selected as a reference function
when trade-offs and narginal rates of substitution are treated. The trade-offs
and the narginal rates of substitution are generated wlth respect to it. In the
notations above, f is the reference functinn. When co-operatlng with decisinn
makers, it is important to select the reference fnction in a meaningful way. An
important criterion in the selection is, for exarnple, that the reference fnctinn
is in fatniliar units or that it is donfinant.
2.9. Proper pareto Optimality
tions had undesirable properties (see Kuhn and Tucker (1951)). To avdid such
Definition 2.9.1. (From Geoffñon (1968)) A decision vector x*  Sis prop-
erly Pareto optimal (in the sense of Geoffion) if itis Pareto optimal and if
there is some real number M > 0 such that for each fl and each x E S sat-
isfying f;(x) < ri(x*), there exists at least one f such that f(x*) < ri(x)
and

iti,e fnction M(x) instead of a constmat (see Mishra (16) d Mshr nd
e collected  G (1986) In Cbew d Choo (1984), it is pïoved that every
be oensidered speclal ces of more generM results presentcd  We (10), In
Gulatl and Isla (1990), tt  shown that the preceding result ct bc generalized
by sumg quiconvextty of the active construits (of the form g(x) g 0)
cMly pseudoline fuactions in Kaul et al
Defition 2.9.2. (om Wlerzbickl (1980b)) A declsion vector x*  S td
(*   (o})n z = ,
whereR-(zWdlst(z, Rç) [[z[}orR-- (zR [m_, ,zi+
 î- zl  O} d Z > 0 is a predetermined scM.
2.9. Proper Preto OptimMity 31
Note that thls definition OEer8 from that of pareto optimallty 8o thst a
larger set R,  is used instead of the set R, The set of e-properly Preto optimal
solutlo is depicted in Figure 2.9.1 mad denoted by a bold line. The solutions
are obtmned by [utersectlng the feasible objective regáon wltb a blunt cone
S and the correspondlng z* E Z are -propeïly pareto optimal if (z * -R k )n Z -
cone D such that R C intDU (0}.)
Figure 2.9.1. The set of e properly Pareto optimal solutions.
trde-offs are bounded by e ztd 1/e (see Wierzblcki (1986a, b)). We return to
help of inequality constramts. In other words, S - {x  R ] g(x) --
(g(x),g(x),... ,g,(x)) T < 0}. In addition, Ml thc objective rd the con-
Definition 2.9.3. (om Kuhn and Tucker (1951)) A declslon vector x*  S
is pperly Paoeto optimal (in the see of Kuhn and cker) if it is pto
V/,(x*)d < 0

32 Pat I -- 2. Concepts
for ail i -- 1... ,k, for some j
and
Vf#(x*)Wd < 0,
Çgl(x* )Td< 0
for ail 1 satisf)5ng gt(x*) -- 0, that is, for ail ctive constralnts at x*.
An objective vector z* E Z is properly Pareto optimal if the decislon vector
correspondlng toit is properly Pareto optinlal.
Kuhn and Tuckeï also derived necessary and sufficlent conditions for proper
Pareto optimallty in Kuhn and Tucker (1951). Those conditions wifi be pre-
sented in the next section.
A comparlson of the definitions of Kuhn and Tucker mad Geoffrion is pre-
sented in Geoffrlon (1968). For example, in convex cases the definition of Kuhn
and 2hlcker implies the deflnltion of Geoifion. The reverse result is valld if
the so-called Kuhn Tucker constraint qualification (see Definition 3.1.3) is sat-
isfied. The ïelatlonships of these two definitlons are also treated, for example,
in Sawaragl et al. (1985, pp. 42-46). Several practlcal examples are glven in
Tamura and Arai (1982) to illustrate the fact that properly Pareto optimal so-
lutions according to the definltlons of Kuhn mad Tuckeï mad Geoifion (and one
more definltion by Klinger; see Klinger (1967)) ae not necessarily consistent.
Conditions under which (local) proper Pareto optimality in the sense of Kuhn
and Tucker impbes (local) proper Pareto optinmlity in the sense of Geoffrion
are given in White (1983a).
Borwein (1977) and Benson (1979a) have both defined proper efficiency
two definitions are generalized in Henlg (1982b) ushg convex ordering cones.
Definltlon 2.9.4. (From Henig (1982b)) Let D be a pointed convex cone.
A decislon vector x* E Sis pvperly efficient (in the sense of Henig) (with
f(x*) E f(x) + E \ {0}, that is, (f(x*) E \ {0}) n Z - 0 for some convex cone
E such that D\ (0} C itE.
another objective vector z  Z such that z* E z + E \ {0}, in other words,
(z* E\{O})nZ=withEasabove.
The desbrable property is valid aLso here: ff a point is properly efficient, it
we set D -- P+.
As pointed out, different definitions of proper efficiency (and proper Pareto
optimality) are not equivalent with each other but they have connections. The
ïeintionshlps between the definitions i the sense of Kuhn and Tucker Geof-
fion, Borwein, Benson and Henig are analysed in Sawarag et al. (1985, pp. 39-
44). ]r instance, Geofffion's and Benson's definitlons are equal when D -- P
(see also Benson (1983)). On the other hand, Definltlon 2.9.4 is equiralent to
Benson's definition if the orderlng cone Dis closed and its closure is pointed.
For fuïthez analysls we refer to Sawaïagi et al. (1985, pp. 394).
In HenoE (1982b), necessary and sufficient conditions for the existence of
pïoperly efficient solutions are given.
Let us finally mention that a new kind of proper efficlency, called super
efficiency, is suggested in BorweLa and Zhuang (1991, 1993).
2.10. Pareto Optlmality Tests with Existence Results
Let us have a look at how the Pareto optimallty of feaslble declsion vectors
can be tested. The procedures presented can also be used to find ma initial
Pareto optlnml salutlon fOr (interactlve) solution methods or to examine the
existence of Pareto optlnml and properly Pareto optimal solutions.
Specific results for MOLP problens are presented in Eckeï and Kouada
(1975). They are generMized fOr nonlinear problems with the help of duallty
theory in Wendell and Lee (1977). The treatment is based on ma auxiliary
problem
k
minlmize  ri(x)
(2.10.1) i-i
subject to f(x) <_ f(:) for ail i - 1,...,k,
by
is Pareto optinml and f(x*) < fi(:) for ail i - 1,..., k.

Proof. See Wendell and Lee (1977).
Theoïem 2,10.1 memas that f problem (2.10.1) has ma optlnml solution foï
some :  S, then either : is Pareto optimal or the optiml solution of (2.10.1)
Whea studyJag the (prlmal) problem (2,10.1) and its dual, a duality gap is
the optimal value of the dual problem.
Theorem 2.10.2. Let a decision vector : E S be given and assume that
¢() -- -oe. Thea some x* E Sis Pareto optimal only if there is a dua[ity gap
between the primal (2,10.1) and its dual problem at x*. If such a gap c:dsts,
the optimal solution of (2.10.1) is Pareto optimal,
Proof. See Wendell and Lee (1977).
The signhïcmace of Theorem 2.10.2 is that precluding duallty gaps the
¢() - -oe foï some   S. It can also be proved that if a multiobjectlve
optimization problem is convex and if 0(5) oe for some   S, thon no
properly pareto optiaml solutions exlst, See the details in Wendell and Lee
(1977),
Pareto optimal solutlons are also investlgated in Benson (1978), The results cma
be comblned into the following theoïem.
Theorem 2.10.3. Let a decision vector ×*  S be given. Solve the pïoblem
(2.10.2) subject o f(x) ֕ -- ff(x*) for ail i - 1,..., ,
(4) If n addition to the conditior b (3), the set (  P* [  < f(x) for
some x  S} is closed then the pareto optim] set is empty.
Proofi See Benson (1978) or Chmakong and Halmes (1983b, pp. 151 152),
Problem (2,10.2) is a pOpflar way of checking Pareto optimality mad of
generatlng Pareto optimal sohtions, Howeveï, sometlmes equa[ity constralnts
cause comptatinaal difiïculties, Therefore it is useful to note that the equalities
in (2.10.2) cma be replaced with inequalltles ri(x) +¢i < f,(x') for ail i --
1,, ., k wlthout affecting the geneïa[ity of the results presented.
Two simple tests are suggested in Brosowski and da Silva (1994) foï deteï-
nfining whetheï a given point is (locally) Païeto optimal oï hot, The tests are
not based on may scalarizing functions but [inear systems of equatlons. Theïe
are, however, severa] [imitations. The objective functions are assumed to be
coatnuously differentiable mad thelr number bas to be strictly larger than the
numbeï of varlables. Ftther, no constraints cm be included. Finally, the tests
may also rail as demonstrated in Brosowski mad da Silva (1994).
Itis proved b Sawaragl et al. (1985, p, 59) that Pareto optkal solutions
exlst to multiobjectlve optimizatlon problems wheïe all the objective functlous
are lower semlcontlmmus (more geneïal than continuity) mad the feasible ïegion
and Naumov (1981),
lems is also characteïized in Deng (1998b),
As meationed, auxillary problems (2.10.1) and (2.10.2) can be used to pro-
of weakly mad properly clficient (in the sense of Borwein) mad elficlent solutions
in the pïesence of oïdeïing cones is studied in Jalm (1986b). The existence of

efficient solutions is aLso treated in Cambini and Martein (1994) by introducffig
so-cMled quasi-D-bounded sers.
A phenomenon called complote e]ficien¢y occurs when every feasible decl
sion vector of a multiohjective optimizatinn problem is pareto optinml. Tests
are presented in Benson (1991) to check for complete efficiency in linear mad
nonllnear cases. A significmat saving of computational efforts cma be attained if
the problem is tested for complete efficiency before itis solved. If the problem is
completely efficient, no rime, effor mad special nmchlnery for generating some
or ail of the Pareto optimal solutions is needed. Anyway, no solution algorlthm
exists which first checks for complete efficiency. The fequency of completely ef
ficient probleras among multlobjective optlmizatlon problems deserves further
cial problem types, for example, when the feasible region S has no interior, as
Benson points out. Transportation problems feature in this category. Complete
efficiency is also treated in Weidner (1990).
any nonefficient solution, that is, for each x E S mad corresponding z E Z there
ex£sts ma efficient point x* mad corresponding z* such that z - z*  D, where
Dis the orderlng cone. VMidity conditions for the domination propcrty are
exanñned in Benson (1983). Thc results of Benson are corrected mad necessary
mad sufficient conditions for the domination property to hold are suppled in
Luc (1984a). The domination property mad its sufficlent conditions are also
treated in Henig (1986). Further, if is demonstïated that the existence of an
property in ffffinite-dinmnsinnal spaces mad for the sure of two sers is hmadled
in Luc (1990).
The last concept to be mentinned herc is the redundancy of ohjective func-
tions. In MOLp cases this can be understood as linear dependency. Ia other
words, an objective fnctinn is redundmat if it does hot affect the pareto optimal
set (see Gai and Leberliitg (1977)). This is net necessarily valid for nonlinear
probleras or in connectioa vAth hateractlve methods. For both of these itis
important to deffim redundmacy on the basis of coafiict between the objectives,
which is why in Agrell (1997), an objective funetion is defined as redundmat if
itis hot in confiict with any other objective functinn. Agell suggests a proba-
where the COrrelation of the objective funetinn is obseïved. Redundmacy checks
are importmat because it nmy ease the burden of the decision nmker if redun
dmtt objectives ae elininated.
3. THEORETICAL
BACKGROUND
We present a set of optlnmllty conditions for multiobjective optlmizatinn
problems. Because the conditions are different for differentiable mad nondiffet 
entiable problem, they are hmadled sepaately.
3.1. Differentiable Optimality Conditions
Optimallty conditions arc ma importmat sector in optlmizatlon. As else-
where, we restrlct the treatment also here to finitdimensional Euclidean
spaces. We considcr probleras of the form
(3.1.1)
minimize (ft(x),f(x),...,f(x)}
subject to x S-- (xPnlg(x)- (g(x),g2(x),...,gm(x)) T <_ O}.
We denote the set of active constrahats at a point x* by
J(x*) (j • (1,...,m} 19(x') -- 0}.
We assume in this section that the objective and thc constraint functloas ae
continuously differentiable. In Section 3.2 we treat nonffiffereetiable functions.
Sinxilar optimality ïesults are also handled, for example in Da Cuaha and
polak (1967), Kuhn mad Tucker (1951), Marusciac (1982), Simon (1986) and
Yu (1985, pp. 358, 49-50). In oroEr to highiight the ideas, the theorems are
here preseeted in a simplified form as compared to the general practlce. For
this reason, the proofs have been modffied.
3.1.1. First-Order Conditions
Wç begin with a necessay condition of the Fritz Jdim type.
Theorem 3.1.1. (Frtz John necessary condition for Pareto optimal*ty) Let
the objective mad the constraint functions of problem (3.1.1) be continuously
differentiable at a decision veetoï x* • S. A necessary co¢ditfon foï x* to he

Proof- See, for instance, Da Cunha and Polak (1967).
We do hot present the proof here because it is qldte extensive. The theorem
can be consldered a special case of the correspondlng theorem for nondifferen-
tlable problems, whlch is proved in Slbsectim 3.2A. For convex problems, nec-
essary optlma6ty conditions can be derived by uslng separating hyperplanes.
This is realized for example, in Zadeh (1963). A separatimi theoïem is a/so
employed in the proof of the general case in Da Cunha and Polak 0967).
Corollary 3.1.2. (Fgtz John necessary condition for weak Pareto optimafity)
The condition of Theorem 3.1.1 is also necessary for a decislon vector x* E S
to be weakly Pareto optimal.
The difference between Fritz John type ad Karush-Kuh-ucker type op
timality conditions in siagle objective opthzatifia is that the multiplier ()) of
the objective functifin is assumed to be positive in the latter case. Thls efimi
nates degeneracy since it implies that the objective function plays its impoïtant
role in the optimality coadltlons. To guaantee the posltivity of )% some reçu
larity has tobe assumed in the problem. Different regularity conditifiIs exist
and they are cdiled constint qualifications.
In the mdtlobjectlve case it is equa]ly important that ail the mditiplicrs
of the objective fimctions are hot equal to zero. Sometimes the mditlpfiers
connected to Karush Kuhn-Tucker optlmality cx)ndltions are called Karsh-
Kuhn-Tucker multipfiers. Thls concept will be used Inter.
In order to present thc Karush-Kuha-Tucker optimality conditions we must
formulate some constraint quaifilcatlou. From among several different alterna-
tives we here present the scalled Kuhn-Tucker constralnt qualification.
Definitlon 3.1.3. Let the constralnt fimctlos g of problem (3.L1) be con
tinuously differentiable at x* E S. lle problem satisfies the Kuhn-Tucker con
straint qualification at x* if for any d E R  such that Vgi(x*)Td < 0 for
ail j E J(x*), there exists a function a: [0,1]  R  which is continuously
differentlablc at 0, and some rcal scalar a > 0 such that
a(0)-x*, g(a(t))<_0 forall 0_<t_<l and a'(0)-ad.
of the alteznative. It will be needed in the proof of thc following necessary
condition.
Theorem 3.1.4. (Motzkin's theorem) Let A and C be given matrices. TheIoE
either the system of inequallties
Ax<0, Cx<0
has a solution x, or the system
ATAq-CTla--O, A_>O, AO, /a>0
has a solution (ja), but never both.
Proof. Ste, for example, Mangasarian 0969, pp. 28 29).
Now we can formulate the Kaxush Kuhn-Tucker imcessary condition for
Pareto optimality.
Theorem 3.1.5. (Kash-Kuhn Tucker necessary condition ]or Pareto opti-
mality) Let the assumptlons of Theorem 3.1.1 be satisfied by the Kulu Tucker
constralnt qualification. Theorem 3.1.1 is thea valid with the addition that
x#0.
Proof. Let x*  S be Pareto optimal. The idea of this proof is to apply
Theorem 3.1.4. For thls reason we prove that there does hot exist any d  R n
such that
(3.1.2) Tfi(x*)Td < 0 for ail i = 1,...,k, and
7.q(x*)Td  0 for ali j  J(x*).
Let us ou the contrary assume that there exists sonm d*  R" satisfying
(3.1.2). Then from the KuhmTucker const raint qualification we koEow that there
exists a function a: [0,1] - R" which is cottlnuously diffezentiable at 0 and
some real scalar a > 0 such that a(0) - x*, g(a(t)) < 0 for ail 0 _< t _< 1 and
a'(0) =
Because the fimctions ] are continuously differ entlable, we can approximate
]i(a(t)) linearly as
f,(a(t))--f¢(x*)+Vf(x*)T(a() x* ) + [[a(t) -- x*l[ç(a(t), * )
-f(:¢)+VfJx*)((t)-(o))+ll(t) (0)llv((t), (0))
where ç(a(t),a(0))  0 as lin(t) - a(0)[ I  0. As t  0 tends lin(t) - a(0)l [ to
zero and (a(0 +t) a(0))/t  a'(0) - ad*.

g _> 0), we have ïi(a(t)) < ï,(x*) for afi i --- 1,..., k for . suflfoiently snmfi t.
for j Ç J(x*) such that , )«Vfi(x*) + j,(**) jV9j(x*)  0. We obtaln
statement (1) of Theorem 3.1.1 by setting
#j = 0 md equalities (2) of Theorem 3.1.1 follow.
cia¢ (1982).
Corollaïy 3.1.6. (Ka'ush-Kuhn-Tucker necessa'y condition ]or weak Pareto
convex problems is introduced in Zhou et al. (1993).
Theorem 3.1.7. (Karush-Kuhn Tucker sufficient condition ]or optimality)
nfinlmize ri(x)
subject to g(x) (g(x),g2(x),.. ,gm(x)) T _ O,
(1) v/(*) + mv(*) = 0
(2) #9(x*)-0 foïall 3--1, ..,m.
Proof. Sec, for example, Simon (1986).
Now we can extend Theorem 3.1.7 for the multlobjective case.
Theorem 3.1.8. ( Ka'ush Kuhn-Tucker sufficient condition for Pareto opti-
mahty) Let the objective and the constralnt functions of problet (3.L1) be
(2) #jgj(x*)--O foïall j--l,...,m.
Pro of. Let the vectors * and p be such that the conditions stated are sat isfied.
We define a flmction F= tt   tt as F(x) = ç_ )f(x), where x  S.
Trivially F is convex because ail the nctlons ri are d we have A > 0.
Now om statements (1) d (2), we obtn VF(x*) + *_ p1Vg(x*) - 0
and g(x*) - 0 r ail j -- 1,... ,m. Thus, cording to Theorem 3 1.7, the
sufficlem condition for Fto attn its mimum at x* is satisfied. So F(x*) 
F(x) r ail x  S. Ia other words,
Let us sume that x* is hot Paretptimal. Ten there exists some point
= ](x) < = ](x ). T5is  a contradiction with inequity (3.1)
Note that because the multiobjectivc optiatlon problem is assumed to
be convex, Theorem 3.1.8 provides a sufficlent condition for glob Pareto op-
Theorem 3.1.9. (Kash-Kuhn Tucker sufficient conditwn for weak Pareto
optimality) The condition r Theorem 3.1.8 is suffieient for a dccision vector
x*  S to be weakly Pareto optimal for 0 < A E tt  with A  0.
Proof. The proof is a stralgbtforward modification of the proof of Theorem
3.1.8.
example, in Majumdar (1997), Marusciac (1982) and Simon (1986).

they are not only nonnegative real scalars but belong toa dual cone D*, where
D* = {A E tf  [ ATy _> 0 for all y E D}. Because of the close resemblance, we
do hot here handle optimality conditions separately for efficiency. For detafls
see, for example, Chen (1984) and Luc (1989, pp.
3.1.2. Second-Order Conditions
Second order optimallty conditions (presumlng twice contlnuously differem
tiable objective and constraint functlons) have been examlned substantially
less than fiïst order optlmaJáty conditions. Second order optimallty conditions
provlde a means of ïeducing the set of candidate solutions produced by the
first-order conditions but at the saine rime tlghteli the assmnptions set to the
ïegdiarlty of the problem.
Second order optlmality conditions for (local) Païeto optimality axe treated,
for example, ha Wan (1975). For completeness we here present examples of neoe
essaïy and suffàcient second-order optimaSty conditions followlng Wang (1991).
First we need ole more constralnt qlalificatlon, namely the ïegularity of
declslon vectors.
Delànition 3.1.10. A point x*  Sis sald to be a regular point if the gïadlents
of the active constralnts at x* are iiaeaïly independet.
Theorem 3.1.11. (Second-ortier necessavg condition for Pareto optimality)
Let the objective and the constïalnt fimctions of problem (3.1.1) be twlce con-
thauously differe1tiable ata regulaï decision vectoï x*  S. A necessary condi-
tion for x* tobe Païeto optimal is that there exist vectors 0 < A  pk, j  0,
E: / E: /" • :o
(1) V x ) + pVg(x
(2) gg(x*)-0 forall j--l,...,m
 . ))
Ca) dr(XVl(x')+V(:" dR0
foralld {0dER'lVf(x*)Td<0feralli 1,...,, Vgj(x')Td -
0 for ail j Ç J(x*)}.
Proof. Sec Wang (1991).
tions. The difference ries ha the sets of search directions.
Theorem 3.1.12. (Second-ortier sufjïcïent condition for Parelo optimality)
Let the ohjective and the constïaint functlons of pïoblem (3.1.1) be twice con
tdiuously dhfferentiable at a decision vector x*  S. A sufficlent condition foï
for which (A,/a)  (0,0) such hat
(1)  XiÇjti(x * ) +  #jÇgj(x") = 0
(2) #;g;(x*)-O forall j-1,...,m
  ))
(a) d'( X,VI(x ") +  V(" d > 0
r either Ml d e {0 ¢ d Ç R   ç/(x*)Td < 0 for alli -- l,...,k, çg;(x*) T
d 0 for aj e J(x*)} or 1 d {0 Cd e R n [ Ççj(x*)Td= 0fr lj e
J+(x*), Çgj(x*)Td  0 for MI j  J(x*)  J+(x*)}, where J+(x ) : {j 
J(x*) I m > 0}.
Proof. See Woeg (1991).
Son (1986) d more necessary d sufficieut conditions for Pcto d
weakly Pareto optimal solutions e presented ha Wang (1991).
3.1.3. Conditions for Proper pareto Optimdiity
For completetmss we also present the original necessaxy oplimality coaditlon
formulated for proper Paxeto optimality i the sense of Kuhn and Tucker (see
Definltlon 2.9.3) as stated by Kuhn and Tucker (1951). To begln with, we wïlte
down Tacker's theoïem of thc alternative, which will be utilized in tbe proof.
Theorem 3.1.13. (Tucker's theomm) Let A and Cbe given matrices. Tben
either the system of inequalities
Ax<0, Ax0, Cx<_0
bas a solution x, or tho system
AT)t+cTla 0, J>0,
hes  solution (A,/), but never both.

(i) ]ç'(') + ],ç(*) -- o
(2) g(x*)=O forall j-l,...,m.
ProoL Let x" be properly parcto optimal (in the sense of Kuhn and Tucker).
From the definition we know that no vector d
_< 0for al[i = 1,...,k, Çfi(x*)Td <Oforsomeindexj, alldÇgl(x*)Td <0for
all I E J(x*). Then, from Theorem 3.1.13 we know that thcre exist multipliers
)i > 0 for i -- 1,...,
je«(×-) pjVgj(x*) -- 0. We obtain statemelt (1) by settlng i - 0 for ail
j Ç {1,...,m}\
If g(x*) < 0 for some j, then according to thc ahovc setting  - 0 and
equalities (2) follow.
Itis proved in Geoffrion (1968) and Sawarai et al. ( 1985, p. 90), that if the
Kuhn-TUcker constralnt qualification (Definition 3.1.3) is satlsfied ata declsinn
vector x*  S, then the condition in Theorem 3.1.14 is also necessary for x* to
be properly Pareto optimal in the sensc of Geoffrinn. Finally, we write down
the sufficient condition for proper Pareto optimality.
Theorem 3.1.15. (Kuhn-Tucker sufjïcient condition for proper Pareto opti
mdiity) If problem (3A.1) is convex, then the condition in Tbcorcm 3.1.14
also sufficlent for a decision vector x*  S tobe properly Pareto optimal (in
the sease of Kuhn and Tucker).
Proof. See Sawarai et al. (1985, p. 90) or Shinfizu et al. (1997, p. 112).
Let us finally mention that necessary and snfficient conditions for propcr
Parcto optimaSty in the sense of Gcoffrion are presented in Gulatl and Islam
(1990) for pseudolinear objective (i.e., differentiabie funcinns that are both
pseudoconvex and pseudoconcavc) and quasicoavex constïaint furictinns.
3.2. Nondifferentiable Optimality Conditions
45
In this section, we no longer necessltate differentiabfiity but put forward
nondifferentiable counterparts fr the optimallty conditions presented in Sec-
tion 3.1. Usually, when the assumptinn of continuous differentlability is given
up, functlons are assumed to be locally Lipschitzinn (sec Definition 2.1.12).
Rcmember that a functlon is here calied nondifferentiable if it is locally LiI
schltzia (ad hot necessarily continuously differentiable).
In every other way the mditiobjective optlmlzation problem to be solved is
stl[1 of the form
(3.2.1)
nfinimlze (] (x), ] (x),..., ] (x)}
suhject to xS=(xER"lg(x ) (g(x),g2(x),...,g.(x))T<_O}.
We first bñefly present some pïoperties of subdierentials (see Deflnition
2.1.14) wlthout any proofs.
Theorem 3.2.1. t the functions ],: R   R, i - 1,...,k, be locly
Poof. See, r example, M£kel£ and Neittoei (12, p. 39) and Clarke
(1983, pp. 389).
1() =  1()
wheïe l(x*) C (1,..., k} denotes the set of indices i r which ](x*) - ff(x*).
Theorem 3.2.3. Let thc fimction 1,: R" R be locally Lipschltzinn at a

0  (").
]tbe [anction /ci is convex, tben the condition is also sufficlent mad tbe mini-
mam is global.
Proof. See, for exmnple, Mkel mad Neittaanmgki (1992, pp. 7(71).
Before moving on to the optimality conditions ofthe Frltz Jolm mad Karush-
Kuhn-Tucker type we should point out the following. ] a single objective fmlc-
tion is defined on a set, the coanterpar of the condition in Theorem 3.2.3
says that zero belongs to the algebralc sum of two sers formed at the point
coasidered. The sets are the subdiffereiltial of the objective [anction and the
normal cone of the feasiblc region. This result is adapted for convex maltiobjec-
tive optimizatlon problems bvolviag continuous objective fanctions mad closed
feasible regions 1 Plastria and Caxrizosa (1996). The ncccssary and sllfficiet
condition for weak Pareto optimality is that zero belots to the sum of the
union of the subdifferentlals of the objective [ancrions mad the normal cone
of the feasible ïegion. Note that the [aactions do mt have to be even locally
Lipschitzlma. Accordig to Claïke (1983, pp. 23031), the smne condition is
necessary for weak Paxeto optimality in general problems as well. We do hot
tïet these results more thoroghly hcre. [astead, we present one more resnlt
Theorem 3.2.4. (Ftz John necessary condition ]or optimality) A necessary
condition for a point x*  R" tobe a local minimum of the problem
minimize ](x)
subject to g(x)--(g(x),g(x),...,g(x))<O,
j -- 1, ... ,m, are locally Lipschitzian at x*, is that there exist multlpliers
(2) #ŒEgj(x*)-0 forall j-1,...,m.
Proof. See, for exmnple, Clarke (1983, pp. 228 230) or Kiwlel (1985c, p. 16).
Now that we bave assembled a set of tools, we are in a positioa to handle
the actual optimality codJtions. More information can be round, for example,
h Cave (1989), Doleal (1985), Minami (19801, 1981, 1983), Shimizu et
al. (1997, pp. 322-325) mad Wang (1984). The theorems aïe presented i a
simplifled form here compared to the general practice so as to emphaslze the
ideas. For thls reason, the proofs have bee modified.
3.2.1. First-Order Conditions
The first result to be presented is a necessary condition of the Fritz John
Theorem 3.2.5. (litz John necessary con&tlon for Pareto optimality) Let
the objective mad the constraint [anctions of problem (3.2.1) be locally LiI
schitzima t a poht x*  S. A necessaxy condition for the point x* tobe
which (/a) # (0,0) such that
(2) #gj(x*)-O forall j-1,...,m.
Proof. Becase itis assumed that (/a) ¢ (0,0), we cma normalize the ml-
tlpliers to sum up to one. We shall bere pïove a stïonger conditio L where
T'(x) --max[/(x) l(x*),(x) li-l,...,k, -L...,m]
(3.2.2) F(x) >_ O,
(3.2.3)
Oe conv(Oh(x*),Og(x*) l i - l,...,k, j • J(x*)}.

Employng the defmition of a convex hull, we tmow that there exist vectors
A and D of real multipfiers for which hl _> 0 for all i -- 1,..., k, #a _> 0 foi ail
j 6 J(x*) and *1 h + cd(x-) J -- 1, such that
Now we cran set  -- 0 for ail j • {1,..., m} \ J(x*). Statement (1) foilows
from this settlng.
Part (2) is triviaL If gj(x*) < O for some j, then j 6 {1, . . . , m } \ J (x*) and
we bave Pi - 0. This completes the proof. []
Delnition 3.2.6. A decision vector x* E S is caged  substationay point if
it satisfies the (necessary) optlmality condition presented ha Theorcm 3.2.5.
Theorem 3.2.5 cran also be proved by first employing a scalarization method
and then Theorem 3.2.4 for the resulting sitgle objective optimizatlon problem
(see, e.g., Doleal (1985)).
Coroll*try 3.2.7. (Fritz John necessay condition]or weak Pato optlmality)
The condition of Theorem 3.2.5 is also mcessary for a decision vector x* E S
tobe weakly Pareto optimal.
Next, we examine some constlalnt quaSfications. Itis obvious that they
dhffer from the different]able case.
Note that whea the necessary optimality conditions are derlved wlth the
help of a scalarizing 5anction, it is easy to generallze the constraint qualhqcm
tions from single objective optimizatlon to the multiobjectivc case. One slmply
assumes that both the original constrMnts and the possible dditional con-
straints satisfy a constralitt qualification. This is expressed in Doleal (1985).
The const ralilt qual]fications used there are those of cMmness and Mmagasarlan-
Fromov]tz.
The so-called Cottle constraint qualification is used in the following the
rem. Other constraint qualifications are presented, for example, in Ishizuka and
Sh]mizu (1984).
Delnitlon 3.2.8. Let the objective and he constraint lixnctions of problem
C'ottle constraint qualification at x* if eitheï g(x*) < 0 for ail j - 1,...,m or
0 ¢ con {0g/') I i(') - 0}.
Assuafing the Cottle constraint qualification, we obtain the Karush Kuhn
TUcker necessary condition for Pareto optlmality.
Theorem 3.2.9. (Karsh-Kuhn Tucker ne¢essay condition ]or Pareto opti
straint qualification. Theorem 3.2.5 is then valld wlth the addition that A  0.
Proof. The proof of Theorem 3.2.5 is heïe valld up to the observation 0 6
0F(x*) and result (3.2.3). We prove also this theorem in a stronger form,
where the multipliers sum up to one.
From the definition of F we know that
F(x*) -- 0,
We continue by fiïst assumlng that çj(x*) < 0 foi ail j - 1,,,,, m, In this case,
F(x*) > ça(x*) for ail j. Now we can apply Theorem 3,2,2 and equatlon (3.2.3)
and obtaln
0  conv {01,(*) I i - 1,...,
From the definitlon of a convex hull we tmow that there exists a vector
0 _< A  tt  of multipliers for which ,k hl - 1 (thus A # 0) such that
We obtMn the statement tobe proved (denoted by (1) in Threm 3.2.5) by
setting  - 0 for ail j - 1,...,m.
On the other hd, if there exists some index j such that gj(x*) - 0, we
denote the set of such indices by J(x*). By the CottM constrMnt quMificatlon
we know that
(3.2.4) 0 Ç COnv {Og (x*) I J e J(x*)}.
In this case, we deduce from Theorem 3.2.2 and result (3.2.3) that
0  conv{OIi(x*),Og)(x*) ]i - 1,...,k, j e J(x*)}.
Applylng the definition of a convex hull, we know that there exist multipliers
1, and by assumption (3.2.4), especlally A  0, such that
Again, we obtain the statement to be pçoved by settlng i - 0 for ail j 
Tlie proof of part (2) is the same as in Theorem 3.2.5. []

tïmality) Let problem (3.2.1) be ¢nvex. A sufficlent condition for a declsio
(1) 0 •  oS(x') +  #jogj(x')
(2) #/g(x*)--0 forall j--1,...,m.
Proof. To start with, we dcfme m addltlonal functlon F: R' - R by F(x) --
 )iSi(x) + j?  #gi (x), wheïe the multlpllers )i mad  satlsfy the above
sumptions. Becausc the functios fi and g e convex A > 0 and D  0,
thca F too is convex, and 0F(x) = _l iO/i(x) +  gOg(x) ( stated
b Theorem 3.2.1).
om sumption (1) we ow tbat 0  0F(x*), and, cordlng to Theorem
3.2.3, the point x* is a (global) minum of F. Tbis implies that foï any
x   R , especlly y x ° satisfyit g(x ) E 0, the foowlng  vid:
0  () - (*)
Employlng sumpfion (2), thc ft that g(x ) < 0 and D > 0, we obta
foranyx ° ÇS.
Let us sume that x* is hot Pct[) optimdi. Tbe thcrc cxlsts some feible
 such that 1,() < ff(x*) for MI i - 1,...,k d for at le one dex
j  1() < ](x*). Because every  w sumed tobe positive, we bavc
_ ]i(x) <  ](x*). This contïadlcs inequdilty (3.2.5) d x* is
thus Pareto optimM.
Theorem 3.2.12. ( Kaush-Kuhn - Tucker sul ïcient condition fvr weak pareto
optimallty) The condition stated in Threm 3.2.11  sufficlent for a decion
vector x* é S to be wely Peto optimM for 0 < •  R  wlth A  0.
Proof. The proof  a triviM modcation of the proof of Thcorem 3.2.11.
FhMly, we introduce one more constrnt quMification. It can only be ap-
pfied to convex problems d it will Mso be needed in Part II (Section 2.2) in
coanectioa with the multiobjcctive promal bundle methodi Itis cMled the
Slater constrnt qtMificatioa. It is dcpendent of the differentiabillty of the
functions hvdid.
Definition 3.2.13. I*t problem (3.2.1) be convex. Problem (3.2.1) satisfics
the ater constralnt alficatlon if there exts some x wlth g(x) < 0 for Ml
Threm 3.2.9 d Coroary 3.2.10 cun aow be reformulated for convex
problems suming the Slater constrnt qualification. Remember tbat convex
ity imples that fmctions e locMly Lipschitzi at y poht in the feible
Proof. Tbe proof is a trivial modification of the proof of Theorcm 3.2.9 wheIoE
we note the following, h case the set J(x*) is nonempty we deaote g(x) -
mmx[g(x) I J - 1,...,m]. Now g(x*) - gj(x*) for j • J(x*). By the Slater
constraint qualification there cxists some x ° such that
Tbus, x* cannot be the global minimum of the convex functlon g. Accordiag
to Theorem 3.2.3 we derlve
0 ¢ conv (0(*) I J  J(x*)}.
The proof of Tbeorem 3.2.9 can now be appled.
Neccssary optimality conditions for Pareto optimallty in those nmdiffeï
eitlable problems wlere the objective fimctions are fractions of covcx and

concave fnctions axe formulated in Bhatia md Dz.tta (1985). In ldition, nec-
essary Irit z John md Kaxush Kuhn-ucker type optimality conditions for weak
treated in Preda (1996).
If m ordering cone Dis used in defining eflàciency, then the optimallty
conditions are slmilar to those presented ahove, except for the multlplleis )h
(simply as in the differentiahle case). The multipliers helong to the dual cone
D* - {A  R.  ] ATy > 0 for Mly  D}. Because of the similarity, we do
suflàcient conditiOnS for eflàclency md weak eflàclency axe handled, for exmnple,
in Wmag (1984). Furthermore, in Craven (1989) md EI Ahdouni md Thihault
(1992), necessary conditions for weak eflàciency in normed spaoes and Banach
spaces, respectively, are presented. The ohjectlve md the constralnt functions
are stlil assumed to he locally Lipschltzim.
Direct counterparts of optlmallty conditions for proper Pareto optlmality
in the nondifferentiahle case. The reason is that the definltlon of Kuhn md
proper Pareto optimality in the sese of Geoffrion, when the ohjective md the
constralnt functions are compositions of convex, locally Lipschltzim fnctlons,
is formulated in Jeyakumar and Ymag (1993). This treatment natur Mly includes
ordlnary convex, locally Lipschitzian 5anctlons. The authors also present nec-
essary conditions for weak Pareto optlmallty md suflàcient conditions of their
own for Pareto optlmallty in prohlems wlth convex composlte functions. A nec-
essary md suflàclent condition for proper eflàciency (in the sense of Henig) is
derived in Henlg and Buchanan (1994, 1997) for convex prohlems.
3.2.2. Second-Order Conditions
At the end of this section we shall say a few words ahout the case where the
fnctloni involved are contlnuously differentlahle md thei gTadlents are locally
Lipschitzim. Such flnctloni are called CL-unctions. Second-order optimality
conditions for multiohjectlve prohlems wlth C'-flnctions are handled in Liu
(1991). Here we hrlefly state tlm mMn results. Firt we must introduce one
conoept accordlng to Liu.
Definitlon 3.2.15. Let the nctlon ] : S  R he a C1A-fction at the point
x*  S. The set
d) = {çi  R ] theie exists a sequence (t
is called a generalized secon&order directonal derivative of the functlon
The set O.]i(x*)(d,d) is nonempty accordlmg to Liu rond Kifek (1997).
in Liu (1991), for hoth the case where all the ohjectlvc fuctions are C
functions rond all the constraint flnctlons are twlce continUoUSly differetlahle
rond tire case where ail thc ohjectlve flmctions are twice contiuuously differen-
tiahle and all the constraint flnctions are C'l-functions. Here, we formulate
foies of the Hessima matrices and generalized second order directional deriva-
tives.
Let us again denote the set of active constraints at x*  S hy J(x*).
Theorem 3.2.16. (Secon&order necessary conditïon ]or Pato optimality)
Let the objective functions of prohlem (3.2.1) he C ' 5anctions rond its con-
0 <_ A  R  and 0 <   R" for which (A,)  (0, 0) such that
(1) v/(')+,Vg/*) =0
(2) #g(x*)-0 fora6 j-1,...,m
(3) )iV/i(x*)Td = 0, .jVgj(x*)Td : 0
for ail 0  (0 # 0  R' ] ç],(x*)d  0 r MI i = 1,...,k, çgi(x*)T0 <
0 for II j  d(x*)} =d ¢  OEI(x )(d,d).

(3)
for alld  {0  d  R' [ Çfi(x*)Td < 0 foroeli=l, .,k, Çgi(x*)''d <
0 r 1 j  J(x*)} d çl  0f,(x*)(d,d).
Proof. See Lin (1991)
Actuly, the results
polyhedral convex cone.
3.3. More Optimality Conditions
Many necessary and sufficlent conditions for weak, proper or pareto opti-
mality (or efficiency) havo been suggested in the llteratrc. They are based on
diffeïent klnds of assumptlons as to the propertles d form of the problem.
Many of them are based on a scalarizatlon of the orlgnal problem and con-
ditions are set to both the original flmctlons and the scalarization parameters
(sobre such conditions are prescnted in Part II in connectinn with the scalaïiza-
tion met hods). In thls book, we sertie for a closeï handling of the Fritz John and
the Karush Kuhn Tckeï type conditions, presented in the two earller scctions.
For the interested reader we list some other references.
Necessary conditions for proper and impropeï Pareto opt hnallty in the sense
of Kuhz and Tcker are derlved wlth the help of cones in Tamura and Aïal
(1982) Geoffrlon (1968) was the flrst to give the basic chaïacterizatinn of proi
erly Pareto optimal solutions in terms of a scalaï problem, called a weighting
problem (see Section 31 of Part II). He extended the results by a compre-
hensive theorem into necessary and sufficient conditions for local and global
proper Pareto optlmallty Geoffrlon's treatment is closely followed in Chou et
al. (1985), where properly Pareto optimal solutions are charaCterized for mu]rb
objective optimization pïoblcms with set valued functions. In addition, neces
sary and sufficient Kaïush-Kuhn-TUcker type optimality conditions for e-Pareto
optimality in convex pïoblems using the weighting method for the objectives
and exact penalty 5nctlons for the constraints are handled in Lin (1996).
In Chmakotg and Haimes (1982), the Karash-Kuhn-'/3acker optimallty con-
solutlon methods (the c-constraint method mad the jth Lagrangima problem;
see Section 3.2 of Part II) Chankong mad Halmes also propose optlmality
conditions for proper Pareto optimality (ha the sense of Geoffnon) wlth the
c-constramt method Further, in Benson mad Morin (1977), nccessary and suf-
optimal. Tbis is dolze wlth the help of the jth Lagrmaglma problem Necessary
mad suflàcient conditions for Pareto optimallty with convex and dierentible
5nctinns partly based on the ¢ constraint problem are proved in Zlobec (1984).
Necessary mad suffiment conditions for Parcto opthïlality and proper Pareto
Optlmallty are proved wlth the help of duality theory and auxiliary problem
(2.10.1) (presented in Section 210) in Wendell and Lee (1977). Howeveï, it
is stated that nonlinear pïoblems do hot generally satlsfy the conditions de-
Pareto optlmality on a point-by-point basis.
In Gulati mad Islam (1988), lineaï li'actional objective fltnctlons and general-
ized convex constïalnts are hmadled. Necessary conditions of thc Karush-Kuhn-
Necessary and sufficient coadltinns foï Pareto optimallty in problems wlth non-
linear li'actional objective functlons and nonllnear constraints are proved in Lee
(1992). In addition, necessary optimallty conditions for fractional multinbjec-
rive optimizatmn problems wlth square £oot terms are 8Jven in Egudo (1991)
In Benson (1979b), a necessary and sufficlcnt coaditinn is given for a point to
be Pareto optimal when there are two concave objective flmctions (pïoblem of
The fo[[owln 8 refereces dcal with conditions for efficiency, where the ob-
In Zubiïi (1988), necessary and sufficlent conditions are proved for weak
efficlency in Banach spaces with the help of a wclghted Lc¢-metric (sec Sec-
weakly efficient and pïoperly efficient solutions (in the sense of Borwein) in
real topolo8Jcal linear spaces are collected in Jahn (1985). Necessary and suf-
ficleut optimality conditions of the Karush Kuhn TUcker type are defived in
Hazen (1988) for cases where preferences are and are hot representable by
Let us finally brlefly mention some furtheï references handllng nondiffer
proper pareto optimality are dcrived in Bhatin and Aggarwal (1992), by the
weighting method (see Section 3.1 of Part II) and Dinl derlvatives. The func-

Optimality conditions baed on the opthnlzation theory of Dubovitskil and
Milytin presmding certain convexity assumptions are presented in Censor
(1977) for Pareto optinmfity in R"atd in Minanfi (1981) for weak Pareto
optimaiity in a llnear topologlcal space. No differentiabillty assumptions are
needed. Necessa[y altd suffiaient conditions for weak, proper and pareto opti-
mality in finite-dlmensionai normed spaces are presented in Staib (1991) under
different assumptlons, h Shhnlzu et al. (1997, pp. 319 322), nondifferentiable
optimaiity conditions assurding constraint qualkficatinns based on directlonal
defivatives are deñvcd.
3.4. Sensitivity Analysis and Duality
The last topics to be mentioned in this chapter are sensitivity analysls,
stabifity ai duallty. Sensitivity analysis studies situations when the input
parameters defining the multiobjectlve optimlzation problem change or cont aih
errors. In sensltivity aalysls, a answer is sought to the question of how much
the paraaneters ca be altered and vaxled wlthout affectitg the solution. More
justification for sensltlvity anaiysis is provided in Rarig and Haimes (1983).
Given a family of parametrized multiobjective optinfization problems, a
set vaiucd perturbation flmctinn is defined in Tanino (1990), such that it as-
socites with each parameter value the set of Pareto optimal points of the
perturbed feasible regon. The behaviour of the perturbation function is anm
lyzed both qualitatively ai quantitatively. In this context stability meas the
study of various continuity properties of the perturbation function of a family
of parmnetrized optimization problems, that is qualitative anaiysis. Sensïtiv-
ity means the study of the derivatlves of the perturbation flanctlon, tht is,
qumatitatlve maalysls.
In general multiobjective optlmlzatinn problems, consider able attention has
been paid to the stabllity of the preference Structure of the decision maker, h
these cases, it is ustally assured that the partial ordering of the objective
space is induced by a ordering cone.
However, mathematicai stability ai sensitivity amlysis are broad areas of
research, ad we do hot intend to go into details here. hstead, we refer, for
example, to Balbg and Guerra (1996), Craven (1988), Ester (1984), Gai and
Wolf (1986), Kltk et al. (1996), Luc (1989), Lucchetti (1985), Papageorgiou
(1985), Tmñno (1988a, b, 1999) and anino and Sawaragi (1980), for further
anaiysis.
Let us still mention that st billty is hot mn unmbiguous notion. As stressed,
for example, in Dauer mtd Liu (1997), the terz and results connected to stablb
ity and sensitlvlty mtaiysis are not miversaily defined in the literature. Different
types of stability tan be defmed and rneasured in many ways. Often stability is
associated with worst case perfornmnce and anaiysing how fasta sdiution de-
grades fo a certain still acceptable leveh Thus, anaiysis of stabillty is important
in Lmplementing solutions in practice. Regardless if its significance, stability has
been widely ignored in the multiple criteria declslon-making context thus far.
A revlew of sensltlvlty aalysis results for both linear ad nonfinear prob-
lems is given in Dauer and Liu (1997). h addition, they study sensitlvity amly-
sis for MOLP problems kr the objective space ad deal with prlorlty structures
in goal programming. Sensitlvlty analysis for MOLP probleus is aiso treted
in Gdi (1995).
Changes that occux in the solution of an MOLP prohlem if the number of
objective flanctions, the nurober of varlables Or the nurober of constralnt flanc-
tions is altered, are exanfined in Eiselt et al. (1987). This is alto a interesting
topic for nonlinear problems, as for example, an objective fimction may have
the solution obtained. For example if a convex objective flanction is added to
tions remain weakly pareto optimal (see Lowe et al. (1984)). The corresponddig
be found in Steuer (1986 p. 179). A result rcgarding the generation of the
timal sets where subsets of the objective flaactinns are used, is proved in Lowe
et al. (1984).
in Nakayama (1985c). Duaiity theory for nonfinear multiobjectivc optimiza-
tion problems is also presented, for example, in Bitran (1981), GSpfert (1986),
Luc (1984b, 1987, 1989), Nakayama (1984, 1985b, 1996), Sdigh et al. (1996)
and Weir (1987); for convex problems in Jaim (1983) ai Martncz-Legaz and
Singer (1987); for more general convex fike problems in Das ai Nanda (1997),
Preda (1992, 1996) and Wang and Li (1992); for nonconvex problems in Luc
and Jahn (1992); md for nonconvex nondifferentiable problems in Preda and
Stancu-Minasian (1997). Some regularity results for mult|objective opthnlza-
tion problems are presented in Martein (I989). On the other halld, duality
theory designed for a declslon mker determiaing preferred solutions in convex
multiobjectlve optimlzatinn pïoblems is derived in Javainen (1996).
Finaily, we stte that an excellent account of stabillty and duality in mul-
tiobjective optimlzatlon can be round in Sawaragl et al. (1985). More than a

Part II
METHODS

1. INTRODUCTION
Generating Pareto optimal solutions plays an important ïole in multiobjec-
ivc optimization, mad matbcmatically the problcm is considered tobe solved
whcn the Parcto optimal set is foud. Thc terre vector optimization is somc
imes used to denote the problem of identifying the Pareto optimal set. How-
ever, this is hot always enough. We want to obtaln only one solution. Tlfis
means that we must find a way to put the Pareto optimal solutions in a com-
plete ol"del". This is why we need a decision maker and ber or his preference
structuïe. Heïe b Part II, we present several methods for solving multiobjective
optimization problems. Usually, this means finding the Pareto optimal solution
that best satisfies the decision maker.
We are not here going to interfere with the formulation of a real-life phe-
nomenon m a mathematlcally wcldefined problcm. We merely stress that a
proper formulation is important. Let us emphasizc that in realife problen
inaccuracy in some form is offen present. Remember that we exclude the ban
dling of stochastic or Mzzy problcms in his contex. Even when the problems
are modelled in a deterministic form, restricting the treatment to Parcto opti
mal solutions only may be misleading. For exoeple, forgetting or misspelling
an objective functioa may affect thc Pareto optimal set. If it is impossible to
model the pra¢tical problem in an explicit and precise mthematical form, we
ceamot automatically lcave non Pareto optimal solutions out of consideraton.
For cxamp]e, imprecision of the data, the measarement or the objective func
tions means that thc Pareto optimal set avallable is only an approximation of
tle real one. Here we bave a gap betwcen theory mad practiee.
Several crucial issues to bear in mind in the formulation of problems are
treated in Halmes (1985) and Nijknp et al. (1988). Among these are risk
ssessment, sufiïcient representativeness of the objective functions and precision
of information. In many complicated, practical cses it may be impossible to
give a correct formulation to the problem belote itis solved. This means that
the modclling and the soltion phases shofld hot be undertakea separately,
which is generally the cas nowadays. In other words, the modelling phase may
requirc interaion with the sohtion phase. A parallel idea of approaching the
modelling phase by including the decision maker in the modelling is suggested
in Brans (1996). The goal is to give more freedom to the decision maker and
hot to limit ber or his way of thinking toa prespecified model and its concepts.

In most methods we are interested in the objective space instead of the de-
cision variable spze. One reaon for this is that the dimension of the objective
space is usually considerably smJ]er than the dimension of the decisinn vari-
able spze. Another reson is that declsion makers are offert more interested
in the objective values. However, calculation still takes ple in the decision
variable space because we do hot usually know the explicit form of the fesi-
ble objective region. In brief, decision makers usually handle objective values
l/ general, multiobj ective optimlzatinn problems are solved by scalarization.
tobe dealt with here, where some simplex-based solution methods can find
set. Another exception, which is presented here, is the multiobjective proximal
As mentioned in Part I, scalarization means converting the problem into a
single or a family of single objective optimizatio problens with a real-valued
objective fonction, termed the scalamzïng function, depending possibly on some
the optimal solutions of multlobjective optiimzatlon problems can be charac
merical difficulties may appeaï if the single objective optimization pïoblem
fesible solutions only with very few parameter values or itis hot solvable with
ail the parameter values. Thus the seemingly promising idea of simplifying the
problem into single objective optimizations hs also its weaknesses. II what
In Sawaragá et al. (1985), three requirements are set for a scalarlzing func-
(1) It cma cover may Pareto optimal solution.
(2) Every solution is Pareto optimal.
If the scalarizing fianction is basd on spiration levels, thei b in addition
(5) Its solution is satisficing ifthe spiration levels used are fesible.
Unfortunately, there is no scalarizing function that cma satisfy ail three require-
An important fact to keep in mind is that standard routines for single objec-
tive optimizatinn problems can only find local optima. This is why only locally
Pareto optimal solutions are usually obtained mad handled when dealin with
scalarizing fianctions. Global Pareto optimality can be guaranteed for exam-
pie, if the objective fonctions mad the fesible rcgion are convex (s stated in
Theorem 2.2.5 of Part I) or qusiconvex mad convex, ïespectively (see Theorem
2.2.4 of Part I). An alternative is to employ global single objective optimiz-
ers. ha the following, however, the solutions are understood to be local, unless
stated otherwise.
Another matter to cotsider is the possibiSty of the scalarizing fonction
having several alternative optimal solutions. [tl this cse, the objective vector
produced depends on the solution chosea. This may affect in an uncontrolled
way the direction in which the solution process proceeds. This fact hs hot been
taken into account in most method developments. Ides fo handliIg alternative
optima in MOLP problems arc prcsented in Sarma mad Mcroumai (1995).
There is a large variety of methods for acomplishing multiobjective opti-
mizatinn. None of them cma be said to be gencrally superinr to ail the others.
When selecting a solution method, the specific features of the problem tobe
soled must be taken into cotsideration. [tl addition, the opininxis of the de
cision maker are importmat. It is hot enough that the maalyst simply prefers
some method. It may happen that the decision maker cmanot or does hot wmat
to use it. The decisinn nmker may be busy or mathematically ignorant. One
can say that selecting ma appropñate multinbj cctive optimization method itself
is a problem with multiple objectives! We shall return to the method selection
problem in Section 1.3 of Part
Methods of multiobjectivc optimizatinn cma be clssified iI many ways ac-
cording to different criterin. [tl Cohon (1985), they are catcgorized into two
relatively distinct subsets: generating methods mad prefercnce bsed methods.
In geneïating methods, the set of Pareto optimal (or efficient) solutions is gen-
erated for the decision maker, who then chooses one of the alternatives. In
preference-bsed methods, the prefcrences of the decision maker are taken into
cotsideratinn s the solution pïocess goes on, mari the solution that best satisfies
the declsion maker's preferences is selected.
Rosenthal (1985) suggests three clsses of solution methods: partial gen-
eration of the Parcto optimal set, explicit value fonction maximizatioa and
iteractive implicit value fianction maximizatinm In Carnfichael (1981), meth-
ods are clssified according to whethe a composite single objective functinn,
a single objective functinn with constraints, or many single objective fianctions
arc the basis foï the approach. One more rough dlvisloI coald be made into
interactive and noninteractie nœethods. These classes can be further divided
in mmay ways.
Here we apply the classification presented in Hwang mad Msud (1979). This
classification is followed, for instmace, in Buchanan (1986), Hwang et al. (1980)
and Lieberma (1991a, b). Hwmag and Msud clssify the methods according
to the participation of the decision maker in the solution process. The clsses
are:
1) metbods where o articulation of preference informatinn is used (no-
preference methods),
2) methods where a posteriori articulation of preference information is used
( a po sterio methods ) ,

3) methods where  prior articulation of preference information is used (a
1. Introduction 65
(1973). A wide collection of methods available (up to the year 1983) is assembled
also in Despontin et al. (1983). Almost 100 methods for both multlobjective
and multiattribute cases are included.
As fat- as chfferent nationalities are concerned, overvlews of multiobjective
optimization methods in the former Soviet Union are presented in Liebermaa
(1991a, b) aad of theoïy and app]catiolts in China in Hu (1990). Nine multi-
objective optimization methods developed in Gernaay are brlefiy introduced
h Ester and Holzmiilleï (1988).
A great number of interactive multiobjectlve optlndzation methods is cog
lected in Shin and Ravindrml (1991) and Vaadeïpooten and Vincke (1989).
Interactive methods are also presented in Narula aad Welstroffer (1989a) aad
Whlte (1983b). Information about applications ofthe mcthods is a/so reported.
Some [iterature ot interactive multiobjective optindzation between the years
1965 aad 1988 is gathered in Aksoy (1990). A set of scalarizlng functions is out-
lined in Wierzblcki (1986b) with special attention to whether weakly, properly
or Pareto optimal solutions arc prodlced.
As to different problem types, aa over,)iew of methods for MOLP problems
can be round in Zionts (1980, 1989). Methods for hieïaïchical multiobjective
optindzation problems are reviewed in Haimes aad Li (1988). Such methods are
needed in large-scale problems. A wide survey on the literature of hlerarchical
multiobjective analysis is also provided.
Methods with applications to large scale systems and industry are presented
in the moiographs Haimes et al. (1990) and Tabucanon (1988), respectlvely.
timization are reported in Eschenaueï (1987), Jendo (1986), Koski aad Silven-
noinen (1987) aad Osyczka aad Koski (1989). The collections of papers edited
by Eschenauer et al. (I990) and Stadleï (1988a) contai mainly applications
we solve problem (2.1.1) defiled in Part 1.

2. NO-PREFERENCE METHODS
In nprefcrence methods, where the opinions of the decision maker axe
hot taken into consideration, tbe miltiobjective opthaization problem is solvcd
using some relativeLv simple method and the solution obtained is presented to
the decision nm2er, The decision maker may either acept or reject the solution.
It seems quite unlikely that the solution best satis fyig the decisiol nmker could
be round with these methods. Tbat is why no prcfcrence mcthods arc suitablc
for situations where the decisio nmker does mt bave mly special expectatlons
of the solution and (s)he is satisfied simply with some optimal solutioL The
working order here is: 1) analyst, 2) none,
As exampl's of this class we present the method of the global criterion and
the multiobjective proximal btmdlc mcthod.
2.1. Method of the Global Criterion
Tbe mcthod of the global criterion is also sometimes caHed iompromise
pro#mming (see Yu (1973) and Zeleny (1973)). In this method, the distance
between somc reference point and the feasible objective region is minimlzed.
The analyst has to select thc reference point and the metric for mesuring the
distances, Ail the objective ftmctions are thought to be equally important.
2.1.1. DifferentMetrics
Herc we exoeline the method where the ideal objective vector is used as a
reference point and Lp-metrics are tsed for measurig. In this case, the Lp-
problem to be solved is
nfinimize
(2,1,1)
subjcct to
From thc definition of the ideal objective vector z* we know that ri(x) _> z*
for ail i - 1,., .,k and ail x ff S. This is why no absolute values are needed if
we know the global ideal objective vector, If the global ideal objective vector

H the ideal objective vector is replaced by some otheï vector, it must be se
lccted cexefully. Pessimlstlc reference points mtst be avoided sin«e the method
The exponent 1/p may be dropped. Problems with or withont the exponent
1/p are equivalent for 1 < p < c, slnce Lv-problem (2.1.1) is  ncreing
funcion of tbe correspondng problem thout tbe exponent.
If p -- oe, tbe metric is Mso cMled a Tchebycheff metric d the L or
tbe Tchebycheff pvblem is of the form
(2.1.2)
Notice that problem (2.1.2) is tondifferentible evet in the bsence of absolute
vMues, h this ce, it c, however, be trmsformed into a diffcrentiable form
if the objectNe d tbe constrMm flmctlons are differentiable. Then, instcad
of problem (2.1.2), he problem
subjectto a>f,(x) z forM1 i-1,...,k,
Tbe solution obtMned depends greatly on tbe vMue chosen for p. Widely
doEerent metrics e shown. The black point is the idem objective vector d
the bold line represents the Peto optimal set. Itis worth noticing that if tbe
origM problem is linear, the choice p - 1 mMntMns tbe lineity. As the vMue
ofp increes, the nonlinear mininfization problem becomes more dicult d
badly conditioned to solve.
For line problenm, the solutions obtalned by tbe Lp-problenm where
1 < p < oe e situated between tbe solutions obtalned by tbe L d L-
problems. It is iflustïated in leny (1973) that ts set of solutions is a pt
of the Pto optimal set, but only a substtiMly smMl part.
hstead of tbe terms ]ri(x) z], denomators nay be added to prob-
len (2.1.1) d (2.1.2) to normMize the conpozmnts, tha is, to use the tern
[ri(x) zî[/]z;[ insead. Some other denonnamrs, like [zï  -z;[, can also
tbe idem objective vector. NaurMly, the denonfinatoïs z cno be used if
Tbe objtive Mnctions may also be normalized by
2.1. Method of the GlobA Criter[on 69
Figure 2.1.1. Different metrics.
before the distmace is minlmized. In thls cse, the ïmage of the new obje
tive functios is [0,1]. Thls normallzing is possible only if the objectives are
bounded. However, it is usually better to employ the ranges of the Pareo opti-
mal set mad replace the max term by the compoxent of tbe approximated nadlr
objective vector z  in (2.1.3).
A variation of tbe Tchebychcff problem is snggested in Osyczka (1989a,
1992), where the problem to bc solved is
n,n,e n [n IlS« )- «, fco) z: 11
(2.1.4) =,..., [ [[ zî ]i(x JJ
subject to x  S.
2.1.2. Theoretical Results
Next, we prescrit some theoretical results concerning the method of the
global crlterion. We ssume that we know the global ideal objective vector mad
can, thus, leave the absobte alues.

70 P Il -- 2. NoPreference Method
Theorem 2.1.1. The solution of Lp-problem (2.1.1) (where 1 _< p < oe) is
Pareto optimal.
Proof. Let x* E S be a sobltlon of problem (9.1.1) with 1 < p < oe. Let us
suppose that x* is hot Pareto optinml. Then there exists a point x E S such
that f(x) < f(x*) for ail i --- 1,..., k and f(x) < f(x*) for at leat one j.
Now (f(x)- z) p _< (ri(x*) zî) P foï alli mad (fj(x) z;) P < (fs(x*) z) .
From tlfis we obtain
k k
When both sldes of the inequality are raised into the power 1/p we have a
contradiction to the sumption that x* is a solution of prohlem (2.1.1). This
completes the proof. []
Yu h polnted out in YI (1973) that if Z is a convex set, then for 1 < p < oe
the solution of problem (9.1.1) is unique.
Theorera 2.1.2. The solutiol of Tchebycheff prob/em (2.1.2) is wcakly Pareto
optimal.
Proof. Let x*  S be a solution of problem (2.1.9). Let us suppose that x* is
hot weakly Pareto optimal. In thi ce, there exists a point x  S such that
f,(x) < ri(x*) for ail i - 1,... ,k. It memas that, f(x) z[ < f(x*) zî
for MI i. Thus, x* cannot be a solution of problem (2.1.2). Heïe we have a
contradiction which completes the proof. [
Theorem 2.1.3. Tchebycheffproblem (9.1.2) has at let one Pareto optimal
solution.
Proof. Let us suppose that note of thc optimal solutions of problem (2.1.2) is
Pareto optimal. Let x*  S be an optimal solution of problem (2.1.2). Since we
sume that it is not Pareto optimal, therc must cist a solution x  S which is
hot optimal for problem (2.1.9) but for whlch ri(x) < ri(x*) for ail i -- 1,..., k
and ](x) < ](x*) for at let one j.
We have now f(x) - zî < ri(x*) - z[ for ail i with the strict inequality
holding for at let one index j, ald fur ther maxi[f(x)
Because x* is an optimal solution of pïoblem (9.1.2), x h to be ma optimal
solution,  we. This contradiction completes the proof. []
Corollary 2.1.4. If Tchebycheff problem (2.1.2) ha a utfique solution, it is
Pareto optimal.
A linear numeïical application exoeple of the method is given in Hwmag mad
Mud (1979, pp. 23 29). Sufficient conditions for the solution of ma Lp-problem
to be stalle with respect to changes in the feible region S are preented in
Jlïkiewicz (1983). Refeïence points more geneïal than the ideal objective vectoï
garanteeing Pareto optimal results are haadled in Skulimowski (1996).
2.1.3. Conaludlng Remarks
The method of the global criterion is a simple method to use if the aim
is simply to obtain a solution wbere no special hopes are set. The properties
of the metrics imply that if the objective functions are hot normalized in any
way, then ma objective function whose ideal objective value is situated nearer
the feaible objective region receives more impotmace.
The solution obtained with the Lp-metric (1 <_ p < oe) is guaranteed to be
Parcto optlnml. If the Tchebycheff metric is used, the solution may be weakly
Parcto optinml. In the latter ce, for instance, problem (2.10.2) of Part I tan
be sed to produce pareto optimal solutions. It is up to the malyst to select
ma appropiate metric.
2.2. Multiobjective Proximal Bundle Method
The multiobjective proximal hundle (MPB) method is an extension of
single-objective hundle-bed methods of nondiffeïentiable optinization into
the multiobjective ce. Itis derived ha Mkel (1993) mad Miettinen mad
Mkel (1995, 1996a) according to thc ide of Kiwlel (1984, 1985a, b) mad
Wang (1989). The underlying pïoximal butdle method, prcsented in Kiwiel
(1990), is ma advmaced version of the bundle foely for convex unconstïahxed
nondifferentiable sitgle objective optirzation. It is generalized for nonconvex
mad constrained problems bi Mkel mad Nelttamamiki (1992, pp. 112 137).
The idea of the MPB method in bïieL is to more in a direction where the
values of ail the objective finctions improve slmditmacously. Here we describe
features of the MPB method fom an implementational viewpolnt. For detal]
see MEkeli (1993).
2.2.1. Introduction
The MPB method is capable of solvin problems with mnlineaï (possi-
bly nondifferentiable) functions. It is sumed that ail the objective and the
contrabat functions are locally Lipschitzima.
The MPB method is not like the other scalaxization methods. Ordinary
scalarization methods trmasform the problem into one with a snle objective
function. This new problem tan thcn be solved with any appropriate method
for no5near proranning. In the MPB method the scalaïizing nction lies
baside a special (lmndifferentiable) optimizeï, which is wby its philosopby is so

by
H(x ,x ) - max[fi(x t) - f,(x), gt(x t) Il - l,...,k, l = l,...,m].
(mad Coto]]ary 3.2.7) in Section 3.2 of Part L
for allj - 1,...,m and f,() < f,(x') for ail i - 1,...,k. ff 9j(2) < 0 for ail
H(,x*) < 0 - H(x*,x'),
Otherwise, that is, if g3() - 0 for some index j, it follows from the Slater
H(,x*) < 0 - H(x*,x*)
Otherwise, we define 10 • {1,...,k) such that f,(x) OE f(x*) > fi() for
(2.2.1) gj(y) < Ag3(x ) + (1 - A)gj(:) < 0
ri(y) < *kf(x) + (1 - ),)k() < .f»(x*) + (1 - ),)ri(x*) =
for MI i e { 1,..., k}  [0. If l0 is nonempty, we choose
objective flmctions, we bave Dr MI i  Io
k(y)  ],= + U ,(=) () ),(*) (k(=) k(=))
- ],(x*) oe(L(x) k()) <
Then, combning the results (2.2.1) and (2.2.2), we obtMn
which s Mn a conradi(tion with the contion that x* mininfized H. Thus,
2.2.2. MPB AIgoritlml
In the foliowing, we take a look at the MPB method. We do hot describe
the method completely but present its idea roughly. The reason is that the
structure of the method is higb]y connected to the underlying nondifferentlable
pïoxinml bundle mcthod.
In the MPB method, the solution is looked for iteratively, tmtli some stop-
ping criterimz is fulfilled. The itcration counter h refers to the inner iterations
ofthe MPB method. Let x h be the current approximation to the solution of
the multiobjective optinfization problem at the iteration h. Then, by Theorem
2.2.1, we seek for the search direction d h as a solution of the unconstrained
optimizatlon problem
minimize H(x h + d, x h)
Since problem (2.2.3) is still nondifferentiable, we must approxlmate it some-
Let us assume for a moment tht çhc problem is convex. We suppose that,
at tbe iteration h in addition to the iteration point x t, we have somc auxillary
subset of {1, • .., h). We linearize the objective and tbe constraint functions at
the point yJ and denote

fi,(x) = f»(y) + (),)T(x --y) for ail i -- 1,...,k, j E Jn and
,(x) - gt(y) + (,)r(x y) for ai!  -- L.",m,  e J.
We can now dem a convex piecewise nem" approximation to the impïov
Rb(x) -- m[f,.(x) - fi(xh), I.j(X) ] i -- l,...,k, I - 1,...,m, j e jh]
d we get an appromation to (2.2.3) hy
nninfize h(x + d) + uh[]d]] 
(2.2.4)
subjecto dR ,
terre u][d[]  is ded to antee that there exists a solution to problem
(2.2.4) and to keep the approxintion locM enouh.
Notice that (2.2.4) is sti/l a nondifferentiable problem, but due o its min-
m-namre itis equivalent to the foowln direntiable quadratic problem
with d  R  and v  R  varibles:
(2.2.5) subject to
v + ½ulldll 
v>_ aî,.+(,)Td, il,...,k, jÇjh
h T
v> ag,.j+(,)
a,.1 = -91d(xh), I = 1,...,m, j e jh
ea-e scailed linearization errors.
and
bI the nonconvex cse, we replace the linearization eïrors by so called sub-
gradient loc«fity mesuïes:
Kiwiel (1985c) is ed m bound he sorage requiremens (i.e., he size of
index set jh) and a modtficatlon of the weight updating aigorithm described in
Kiwiel 0990) is used to update the weight u h. For detai/s, see Miettinen and
Mkel/i (1995, 1998a).
This is roughly the MPB method. Next, some words about optimality ea-e
in order.
2.2.3. Theoretlcal ttesults
According to Theorem 2.2.1 we, on the one hand, kaaow that mininfizing an
improvement frmction produces weakly Pareto optimal solutions. On the other
hand, any weakly Pareto optimal solution of a conve problcm can bc round
rmder minor cond]tions. Wile we do not optimize the improvement frmction
but its approximation, the optimaiity restflts ofthe MPB method are somewht
different. Here we only present some results without proofs, since giving these
would necessitate exp]Jcit expression of the MPB aigorlthm.
Theorem 2.2.2. Let the mu]tiobjectivc optifizatioa problem be convex and
the Slater constraint qualification be satisfied. If the MPB method stops with
a finite number of iterations, then the solution is weakly Pareto optimal. On
the otheï hand, any accumultlon point of the bfinlte sequence of solutions
geneïted by the MPB mcthod is weaidy Pareto optimal.
Proof. See Kiwicl (1985a) or Wang (1989).
If the convexity ssumptlon is hot satisfied, we obtain somewhat weaker
ïesults about substationaïy points (See Definltion 3.2.6 of Part I). This result
involves upper semidiffereatlable fnctions (see Definitioa 2.1.15 of Part
Theorem 2.2.3. Let the objective and the constraint fianctions of the multi
objective optimization pïoblem be upper senfidifferentiable at every x • S. K
the MPB method stops with a finite number of iteïations, then the solution is a
substationea'y point. On the other hand, any a:cumulation point of an infinite
sequence of solutions generated by the MPB method is a substationary point.
Proof. See Wang (1989) and references therein.
Note that only the substationeaïty of the solutions of he MPB routine is
guaxanteed for general maitiobjecive optimization problems.
2.2.4. Concluding Remarks
The MPB method can be used s a method wiere no opinions of the decision
maker ea-e sought. In tbis case, we must select the staxting point so that itis hot
(weakly) Pareto optimal but that every component of the objective vectoï

be improved. The method con also handle or fier thon nonlinear const raints, but
they have hot been included here for the sake of t fie clarity of the present ation.
The MPB routine con also be used s a black-box optimized withLu inter-
active multiobjective optinfizatLun methods. This is the cae witfi tfie vector
version of NIMBUS (see Section 5.12).
The a¢cura¢y ofthe computatLun in the MPB method is an interestlng mat
ter. Accura¢y con De considered in a more extensive meaaiag s a separatLug
factor between ordinary scalarizing fonctions aud inner scalarizLu function, s
Lu the MPB method, lï some ordinary scaiaïlzing fonction is employed, then it
is the a¢curacy of that additional function that cou be followed along with
solution proccss. [t ny happen that when the accuracy of the scalarizing lune-
tion hs reached the desired level, the vahms of the a¢tual objective functlons
could still change considerabLv.
Mauy scalarizing functions have positive fetures whose impor tauce is not to
be tmderest[mated, such s prodlcin only Pea-eto optimal solutions. Itoweve L
employing some scalarizing flmction usually brings alon exra paraueters aud
the dhicdty of specifying their values. This causes additinnal stability concern.
To put it briefiy, scalarizing functions add exra characteristics to thc problem.
ScalarizatLun caunot completely be avoided even in the MPB routine. Itow-
ever, the scalarizatlon is caried out under the surface, invisible to thc user.
Whateveï additional parameters or phses are mcded, they caunot be scen
aud the user does hot have to be finthered wlth them. The weakness of the
MPB routine is that the Pareto opthnallty of the solutLuns ohtained cnnot
be glarmteed. In iheoïy, only the substatlonarlty of tire solutions is certain.
In practlce, it is finwever, very llkely that the solutions are at le,St wekly
Pareto optimal. As a mtteï of fact in the numerical experiments perfoïmed,
the final solutions obtalned bave usually proved to be Pareto optimal at the
final testing.
For problems with nondifferentiable functinns the MPB routLue repïesents
au efficient proximal bundlebsed solution approach. The implemenation of
the MPB routine (called MPBNGC) is descrlbed in M&keli (1993). It cails 
quadrtic solver derived in Kiwlel (1986).
3. A POSTERIORI METHODS
A posteriori methods could also be called method for gene*nting Pareto
optimal solutions. Aer the Pareto optimal set (or a part of it) hs been gen-
erated, itis presented to the declsion nmker, who selects the most pïefeïred
outang the alternatives. The inconvefiences here are that the generation process
is usually computationally expensive and sometlmes in part, at le,st, difficult.
On the other hand, it is hard for the decisinn maker to select from a large set
of alternatives. One more importaut question is how to present or display the
alternatives to the decision maker in an effective way. The working oïder in
these methods is: 1) aualyst, 2) decision maker.
If there are only two objective functions, the Pareto optimal set con be gea-
erated parametrically (see, for exmple, Benson (1979b) aud Gss aud Saaty
(1955)). When there are more objectives, the problem becomes more compll-
cated.
Let us briefiy metion that in solvlng MOLP problems the methods cou be
divided into two subclsses. In the first are the methods that cn find ail the
Pareto optimal solutLuns aud in the second are the methods that cou find only
ail thc Pareto optimal exrem olutions. In the ltter cse, edges connecting
Pareto optimal extrcme points may be Pareto optLual or hot. h nonlinear
proffiems, t fie distictlon lies between convex and nonconvex problems. In otfinï
words, some methods can only generate Pareto optimal solutions of convcx
pïoffiems.
Tfie metfinds presented in detail here are called basic methods, since tfiey
aïe used fequently in practical pïoblems, and they are also uscd s elements
of moïe developed metfinds. Bslc methods arc the weightlng method aud tfin
«-constïaint method. Af¢er them, we give a llnfited overvlew of a metfiod com-
binlng featuïes of botfi the weightlng aud the ¢ constïaint metfind. Tfinn we in-
troduce two more bsic methods. The method of weigfited metïlcs is a weigfit ed
extension of the method of the global crlterlon. It is failowed by tfie handling
of a¢fiievement scalrlzing functlons. FLualiy, some otfier methods i thls clss
are briefiy mentinned.

3.1. Weighting Method
La the weightLag method, presented, for example, in Gass and Saaty (1955)
and Zadeh (1963), the idea is to associnte each objective function with a welght
ing coefficient and nfinimize the weighted suIn ofthe objectives. In thls way, the
multiple objective fuImtioIm are transformed into a single objective function.
We suppose that the welghting coefficients w are real numbers such that wi _> 0
for all i = 1,..., k. Itis also usnally supposed that the weights are normalized,
that is, î_l wï = 1. To be more exoet, the mu]tiobjective optimization prob-
Iem is modified into the following problem tobe called a weighting problem:
k
minimize
(3.1.1) i-I
subject to x  S,
where wi >_ 0 for alli -- 1,...,k and , t w -- L
3.1.1. Theoreticl Results
In the following, several theoïetical results are pïesented concernlng the
welghting method.
Theorem 3.1.1. The solution of weighting problem (3.1.1) is wealdy Pareto
optimal.
Proof. Let x*  S be a sointlon of the weightlng problem. Let us suppose
that it is hot weakly Pareto optimal. In this case, there exlsts a solution x  S
such that f,(x) < f,(x*) for alI i -- 1,...,k. Accordlng to the assumptions
set to the weightlng coefficlents, w5 > 0 for at least one j. Thus we bave
 wlf«(x) < _ wlfi(x*). This is a contradiction to the assnmption that
x* is a solution of the weighting pïoblem. Thus x* is wealdy Pareto optimal.
[]
Theorem 3.1.2. The solution of weighting pïoblem (3.1.1) is Païeto opt[raaI
if the weightlng coefficients m-e positive, that is wl > 0 for all i -- 1,...  k.
Proof. Let x* Ç S be a solution ofthe weightlng problem wlth positive welght-
ing coefficlents. Let us supposc that itis hot Pm'eto optimal. This mem}s that
there exists a solutinn x  S such that f,(x) < fo(x*) for alI i -- 1,...,k and
f(x) < fj(x*) for at least one j. Since w > 0 for all i - 1,...,k, we bave
î-I wifi(x) < î-I wlf*(x*)" This contradlcts the assumptinn tht x* is 
solution of the welghtlng problen and, thus, x* must be Pareto optinml. []
3,1. Weighting Method 79
Theorem 3.1.3. The unique solution of weightlng problcm (3.1.1) is Pareto
optimal.
Proof. Let x*  S be  unique solution of the welghtlng pïoblem. Let us
suppose tht it is not Pareto optLaml. In this case, there exlsts a solution
x Ç S such that f,(x) < f(x*) for ail i -- 1,...,k and f2(x) < f(x*) for
at least one j. Because ail the weighting coefficients wï m-e nonnegative, we
bave _ wïf(x) <_  w,fi(x*). On the other hand, the uniqueness of x*
means that î_ w,f,(x*) < t wfi() for all  Ç S. The two iequalitles
above are contradlctory and, thus, x* must be Pro'oto optinmI. []
As Theoïems 3.1.2 and 3.1.3 state, the solution of the welghtLag method is
alwys pareto optimal if the weightlng coefliclents are ail positive or if the soIu
tion is unique, wlthout oa]y fuït her assumptions. The weakness of the weightlng
method is that not ail of the Pareto optimal solutions can be round unless the
problem is convex. This feature cn be ïelaxed to some extent by convexifying
the nonconvex Pareto optñnal set as suggested in Li (1996). The convexifica-
tion is realized by raising the objective fitnctlons toa hlgh enough power under
certain assumptinns. Howeveï, the nmin resuIt i thc folIowing:
Theorem 3.1.4. Let th nmltiobjective optlmization pïoblcm be convex. If
x* ff S is Pareto optimal, then theïe etsts a weighting vector w (wl > 0,
i = 1,... ,k, _t wï -- 1) such tht x* is a solution of weightlng problem
(3.1.1).
Proof. The proof is presented after Theorem 3.2.6,
According fo Theorem 3.1,4 any Pm'eto optimal sointlon of a convex mul-
tiobjective optimization problem can be found by the weightLag method. Note
that the weighting vector is not necessm-ily unique. The conteats of Theorem
3.1.4 is illustrated in Figure 3.1.1. On the Ief, every Païeto optinal solution
along the bold llne can be obtaLaed by lteïlng the welghting coefficients. On
the right, it is hot possible to obtain the Pareto optñnl solutions in the 'hol-
An equlvalent trigonometric formulation to the welghting problem with two
objective functions is presented in Das and Denins (1997). This formulation can
be used in illustrating geometfically why hot 11 the Pm-eto optimal solutions
of nonconvex problems can be round.
Remark 3.1.5. According to Theorem 3.1.4, all the Pareto optñnal solutions
of MOLP problems can be found by the welghting method.
Let us bave a look at linem- cases for a while. In practice, Remm'k 3.1.5 is
not quite true. The slngle objective optimizatinn ïoutlncs for llnear problems

z 2
Figure 3.1.1. WelgMing method with convex and nonconvex problcms.
usually find only extreme point solutions. Thus, if some lacet of the feasible re-
gion is Pm-eto optimal, then the infinlty of Pm'eto optimal non extreme points
must be described in terres of lhlem- combinatlons of the Pareto optimal ex-
treale solutions. On the other hand, note that if two adjacent Pareto optinml
extreme points for an MOLP problen, are round, the edge connectlng them le
hot necessarily Pareto optimal.
The conditions uader which the wbole Païeto optimal set can be gener-
ated by the welghting method with positive weighting coefficients are presented
in Censor (1977). The solutions that it is possible to reach by the weighting
method with positive weighting coefficients m'e chm'acterlzed in Belkeziz and
Piïlot (1991). Some generallzed ïesults arc also given. More relations between
nonnegative and positive weighting coefficlents, convexlty of S and Z, and
Pareto optimality are studied in Lin (1976b).
If the wcighting coefficients in thc weighting method arc ail positive, we
can say more about the solutions than that they are Païeto optimal. The fol-
lowing ïesults concerning pïopeï Pm'eto optknality weïe originaIly presented in
Geoffrion 0968).
Theorem 3.1.6. The solatlon of welghting problem (3.1.1) is pïoperly pm-eto
optimal if ail the welghthlg coefficlents arc positive (sufficlent condition).
Proof. Let x* E S be a solution of the wclghting problem with positive
welghting coefficients. In Theorem 3.1.2 wc showed that the solution is Païeto
optimal. We shall now show that x* is properly Pm'eto optimal with M -
(k - 1) maxï,(w/ O.
Let us on the contrary suppose that x* is not properly Pareto optimal. Then
for some i (which we fix) and for x  S such that ri(x*) > f,(x) we have
ff(x*) -- ff(x) > M(fj(x) - fj(x*))
for ail j such that f(x*) < )(x). We caxt now write
which means
Heïe we bave a contïadictlon to the assumptlon that x* le a solution of tbe
weighting pïoblem. Thus, x* has tobe pïopeïly Païeto optimal. []
Theorem 3.1.7. K the multiobjcctive optinization problem is convex, then
thc condition in Theorem 3.1.6 is also necessary.
Proof. See Geoffrion (1968) or Chou et al. (1985).
Corollary 3.1.8. A necessary and sufficient condition for a point tobe a
properly Parcto optimal solution of an MOLP problem is that it is a solntinn
of a wcighting problcm with ail the wclghting coefficietts being positive.
The ratio of the weighting coefficients gives an upper bound to global trade
Theorem 3.1.9. Let x* be a solution of weighting problem (3.1.1), when ail
the wcighting coefficicnts wi, i - 1,..., k, are positive. Then
A(x*) < max wa
for every i, j - 1,..., k, i  j.
Proof. See Kallszewski (1994, p. 9g).
Some results concerning weak, proper and Pareto optimality of the solutions
obtained by the weighting method are combined in Wlerzbicki (1986b). Proper
Pareto optimality and the weighting œethod are also discussed in Belkeziz and
Plelot (1991) and Luc (1995). The weighting method is used in Isermann (1974)

3.1.2. Applications and Extensioits
As far as applications m'e concerned, the weighting method is used to gen-
erate Pareto optimal solutions in Sadek et al. (1988- 89) in solving a problem of
the optimal control of a damped beam. The weighting method is also applied in
Weck and F51tsch (1988) to structural systems in the optinñzation of a spindle
bearing system and in the optimization of a table as well as in ReVelle (1988),
where reductlons in strategc mclear weapons for the two superpowers m-e ex-
amlned. Furthcrmore, Parcto optfinal solutions are generated for an anti-lock
brake system control problem by the walghting method in Athan and Papalam-
bros (1997). However, no attention is paid to the possible nonconvexity of the
problem. In addition, the weighting method is an essential compollent in thc
determination of the optimal size of a batch system in Iiechnan and Mehrez
(1992) and a 5oEzy optimal design problem concerning a bridge is solvcd in
Ohkubo et al. (1998).
Linear problems wlth two objective hmctions m'e studied in Gass and Saaty
(1955). Thc systematlc genet ation ofthe Pareto optimal set is possible in thcsc
problems by parametric opt haizatlon. Thc pareto optimal set of multiobjectlve
optimization problems wlth convex quadratic objective functios and linear
equality constraints is characterlzed analyically in Goh and Yang (1996), The
characterization involves the weighting method and active set methods.
Systematic ways of perturbing the weights to obtah different parcto opti-
mal sohltions are suggcsted in Chankong and Haimes (1983a, b, pp, 23436).
/m addition, an algorithm for generating dfferent weighting coefficients auto
matlcally for convex (nonlinear) problems to produce an approximation of the
Pareto optimal set is proposed in Caballero et al. (1997).
A rthod for rcducing the Pareto optimal set (of an MOLP problem) belote
it is presented to the decision maker is suggested in Solovcychik (1983). Pareto
optimal solations are flrst generaicd by the weighting method Then, statistical
analysis (factor analysls) is used to group nd partition the Pareto optimal set
into groups of relatlvely homogermous elements. Fhmlly, typical solutions from
the groups are chosen and presented to the declsion maker.
It is suggcsted in Koski and Silvennoinen (1987) that the weighting method
can be used to reduce tbe number of the objective 5mctions before the actual
solution process. The Original objective functions are dlvided into groups such
that a linear combination of the objective functions in each group forms a new
objective finction and these new objective fimctions forma new multiobjective
optimization problem. Itis stated that every Pareto optimal solution of thc
new problem is also a Pareto optimal solution ofthe original problem, bnt thc
reverse restflt is hot generally truc.
3.1. Weightlng Method 83
As mentloned em'ier, the weighting vector that produces a certain Pareto
optimal solution is hot necessm'ily tmique. Tbis is partlcuJarly truc for
probleras. A method is presented in Steuer (1986, pp 183 187) for determining
ranges for weightlng vectors that produce the saine sohltion. Note that some
weighting vectors may produce unbounded sagle objective optimization prob-
leras. Tins does hot mean that the problem may hot have feasible solutions
wlth some other weightlng vectors.
A property related to producing dfferent Pm'eto optimal solutions by al
tering the welghting coefficients is the woak stability ofthe system. On the orte
hand, a sraall change in the weighting coefficients may cause big changes in
the objective vectors. On the other hand, dramatically different weighting co
efficients nmy produce nearly similar objective vectors. The reason is that the
welghthg problem is hot a Lipschitzian fonction of the weighting coefficients.
In addition, it is emphasized and i[htstrated in Das and Dennis (1997) that
an evenly distributed set of welghting vectors does hot necessm'ily prodce an
evenly distributed representation ofthe Pareto optimal set, even if the problem
is Convex. Furthgn Das artd Dennis demonstrate how an evea spread of Pm-eto
optimal solutions is obtained only for very special shapes of Pm'eto optimal
sers. The treatment concerns two objective functions.
An entrOpy-based formulation of the weighting method is suggested in Sul-
tan and Templeman (1996) The entropy-bsed objective function to be op
timized bas only one pm-ameter no marrer what the mraber of the orlgnal
objective finctions is. A representation of the ParetO optlnml set can be gem
erated by varying the value of the single pm-mmeter. The properties of the
suggested method are the saine as those ofthe welghting noethod, for exmmple,
ail the Pm-eto optimal solations of nonconvex probleras cannot be round.
3.1.3. Weightlng Method as an A Pr]ori Method
The weghting method can be used so that thc dcclsion maker specifies a
weighting vector representing her or his prefercnce information. In this case, the
weighting problem can bc consldered (a negative of) a value function (remember
that value fmlctions are ramximized). Note that according to Remark 2.8.7
of Part I the weighting cocfficients provided by the decision maker are now
notlfing but marginal rates of substitution (mij - wa/w). Whert the welghting
method is used in tbJs foshloa, it can be considered to belong to the class of a
priori methods. Related to thls, a method for assisting in the determination of
thc welghtlng coefficlents is prcscnted in Batishchev et al. (1991). This method
can also be extended into art interactive form by letting the declsion maker
modfy the weighting vectorS after cach iteratinn.
The objective functions shou]d be normalized or scaled so that thelr oh
jcctive values m-e approximately of the same magnitude (see Subsectinn 2.4.3
in Part I). Only in thls way can one control and manoeuvre the method to
produce solutions of a desixable nature in proportion to the ranges of the oh-

jectlve functions. Other¢ise the role of the weighting coefficients may be greatly
3.1.4. Cotcludlag Remarks
The welghting method is a simple way to generate different pareto opti-
nml solutions. Pareto opthnality is gum-anteed if the weighting coefficients
positive or the solution is unique.
Applying Theorem 3.1.2, we know that diffeïent Pareto optimal solutions
can be obtained by the weightlng method by alterhag the positive weightlng
coefficients. However, in practlcal calculations the condition wi _> «, whcre
 > 0, must be used instead of the condition w« > 0 for ail i
This necessitates a correct choice as to the value of e. Ail the Pareto optlnml
solutions in some convex problems ny be round if e is snmll enough. But the
concept of small enough' is problem-dependent and for this reason difficult to
specify in advance, as pohated out in Korhonen and Wallenius (1989a).
As observed before, the wealflmss of the weighting method is that all of the
Pareto optimal points cannot be found if the problem is noaconvex. If this is
the case, a duality gap is sald to occur (according to duality theory). The same
weakness may also occur in problems with discontlnuous objective functions as
demonstrated kn Kitagawa et al. (1982).
Sometimes, the results concerning the weighthag method are presented in a
skmpler form, assuming that zeros are not accepted as weightlng coefficients. It
may seem that the welghting coefficient zero nmkes no sense. It means that we
have included in the problem some objective fm}ction that has no significance
at ail Neveltheless, zero values have here been included to make the presen
turion more general. On the other hand, by also allowing zeros as weighting
coefficients, it is easy to explore how solutions change when some objective
fnctlon is dropped.
Employing the weightlng method as an a priori method presumes that the
decision maker's under lying value fUnCtion is Or con be approximated by a llnem"
function (see Section 2.6 ha Part I). This is in many cases a rather simpltfylng
assumption. In addition, it nmst be noted that altering the weighting vectors
linearly does not have to noean that the values of the objective functions also
change linearly. It is, moreover, difficult to control the direction ofthe solutions
by the weighting coefficients, as illustrated in Nakayama (1995).
3.2. -Constraint Method
Ln the s constralnt method, introduced ha Halmes et al. (1971), one of the
objective functlons is selected tobe optim[zed and alI the other objective fUnC
tions are conver ted hato constralnts by setting an upper bound to each of them.
The problem tobe solved is now of the foïm
minimize f(x)
(3.2.1) subjectto f(x)_< forall j--1,...,k, jTg,
An altcrnative fornlhation is proposed in Lin (1976a, b), where proper
equality constrahats are uscd instead of thc above-nmntioned inequalltics. The
solutions obtalned by thls proper equality method are Pareto optkmal under
certain assmptions. Here, however, we concentrate on formulation (3.2.1).
3.2.1. Theoretical Results on Weak md Pareto Optimallty
Proof. Let x*  S be a solution of the -constralnt problem. Let us assume
that x* is hot weakly pareto optinml. In this case, there exlsts some other
x • S such that f,(x) < ff(x*) for alli = 1,...,k.
This means that f(x) < f(x*) _< e for ail j -- 1,...,k, j 7  g. Thus x
is feasible wlth respect to the -constralnt problem. While ha addition fo(x) <
f(x*), we have a contradiction to the assumption that x* is a solution of the
-constrahat problem. Thus, x* has tobe wealdy Pareto optinml. []
Next, we handle Pareto optinmlity and the «-constraln method.
Theorem 3.2.2. A decision vector x*  S is Pareto optimal if and only if
it is a solution of ¢-constralnt pïoblem (3.2.1) for eveïy g = 1,...,k, wheïe
j-- ](x*)forj=l,...,k, jTg.

solve the e-constraint problem for some  where cj = f/(x*) for j -- 1,.
j ¢ . Then there exists a solution x • S such that ft(x) < f(x*) and
fj(x) < f(x*) when j ¢ L Tbs contradcts the Pareto opthna/ity of x*. In
other words, x* bas to solve the problem for aay objective fimction.
Sufficiency: Since x*  Sis by assumption a solution of the cortstraint
problem for every £ -- l,...,k, there is no x  S such that ri(x)
and f(x) < f(x*) when j ¢ . This is the definition of Pareto optlmality for
ploblem).
of «-constraint problem (3.2.1) for soIae £ with ej - f(x*) for j -- 1, ..., k,
point x ° • S such that f(x ) < f(x*) for 1 i - 1,..., k and for st let one
index jis vid fj(x ) < f (x*). The uniqueness of x* means tht for 1 x Ç S
such that f(x)  f(x*), i ¢ t,  f.(x * ) < f(x). Here we bave a contradiction
me that fe(x * ) < f(x) for 1 x e S when f(x *)  ej for every j -- 1,..., k,
vcctor x °  S such that f(x ) < f(x *) for aH i -- I,..., k and Che Inequity
Ifj = » ts me that f«(x °) < f(x ) d f(x °) < f(x ) < « for 1
On the other hd» ffj ¢ t, then f(x ) < f (x*)  ej, f(x )  f(x*) < e
for l i ¢ j d t, d fe(x °) < f(x*). This is ha contrdlctlon to x* as a
In Figure 3.2.1» different upper bounds for the objective fimction f2 are
given wble the function ri is to be mhahnlzed. The Pareto optimal set is
shown by a bold line. The upper bound level e I is too tight and so the feasible
region is empty. On the other hmd, the level «4 does not restrict the region st
ail If it is used as the upper bound, the point z  is obtained as a solution. It
is Pareto optinml accord[ng to Theorem 3.2.4. Correspond[ngly, for the upper
bound e the point z is obtained as a Pareto optlnml solution. The pohat z 
is the Optimal solution for the upper bound 2. Its pareto optlnmlity can he
proved accordhag to Theorem 3.2.3. Theorem 3.2.2 cm be applied as well.
z 2 
To ensure that a solution produced by the ¢ constraint method is Pareto
optinml, we have to either iolve k different pohlems or obtaln a unique solution.
In general, uniqueness is hot necessarily ea.y to veify. However, if for ezample,
the prohlem is convex md the function f to he nnimized is strictly convex,
we know that the solution is unique without further checking (see ChazoEong
md Haimes (1983b, p. 131)).
According to Theorem 3.2.1 we know that the -constraint method pro
duces wealdy Pareto optinml solutions without my additional assumptions.
We cm show tht any weakly Pareto opthnal solution can be found with the
s-constraint method (for some objective functlon tobe mñfimlzed) if the feasi-
ble egion is convex md ail the objective fimctlons are quasiconvex md Strictly
quasiconvex. Thls result is deïived h Rulz Canales and Rufi&-Lizmm (1995)
md a shortened proof is also given ha Luc and Schaible (1997). The value
of thls result is somewhat questlonable because usually we m-e haterested in
Pareto optinml solutions md we know that my of them can bc found wlth the
sonstrahat method.

3.2.2. Connections with the Weighting Method
The relationshlps between the weightlng method ad the c-constraint
method are presented in the following theorems,
Theorem 3.2.5. Let x* E S be a solution of welghting problem (3.1.1) ad
(1) if wl > 0 x* is  solution of the «-constraht prohlem for ] as the
objective function ad Ci -- ]j(x*) for j -- 1,..., k, j ¢ ; or
(2) if x* is  uique solution of welghtlng problem (3A.1), then x* is a
solution of the ¢ constralnt prohlem when ¢j -- f#(x*) for  -- l,..., k,
j   and for eve£y Je,  -- 1,..., k, as the ohjective function.
Proof. Let x* E S he n solution of the welghtin problem for some welghting
there exlsts a point  e S such tht ]«() < ](x*) and fj() < ]j(x*) when
j-l,,,.,k, jcL Weumedthat w, > 0andw 0wheniCL Nowwe
have
whlch is  contrdictfon with inequity (3.2,2), Thus x* is a solution of the
problem when If is to be nfined, then we c find a solution  Ç S such
that ]() < ]«(x*) d ]() < ]j(x*) when j ¢ L Th means that for y
w > 0 is Z wf«() < Z wï],(x*). Thls contrdlcts inequity
Theorem 3.2.6. t the multlobjectlve optimation probm be convex, H
minimized and ¢ = ] (x*) for j = 1,.. ,, k, j 7  g, then there exists a weighting
k W
vector 0 _< w E R , i=l ï = 1, such that x* is dso a solution of weighting
problem (3.1.1).
Proof. The proof needs a s>called enera[ized Gordan theorem. See Chm}kong
and Himes (1983b, p. 121) and ïefoïenccs theïein,
We have now appropriate toots for proving Theorem 3,1,4 fi'om the pre-ous
Proof. (Proof of Theorem 3A.4) Since x* is Pareto optinml, it is by Theorem
3.2.2 a solution of the «-constïaht pïohlem for every objective functlon ] to
he nfininfized. The pïoof is complet ed with the aid of the convexity assumptlon
and Theorem 3,2,6. []
A diagram representiïig several resutts concerRing the characterization
of Parcto optimal solutions and the optimallty conditions of the weighting
method, the ¢-constïaint method and a so called jth Lgrangian method, thelr
relations and connections is presented in Chkong and Haimes (1982, 1983h,
pp, 119). The jth Lagranglan mcthod, presented in Benson and Moñn (1977),
means solving the problem
mininze fj(x) ÷
(3,2,4) ï
subject to x  S,
where n -- (u,... ,u ,u+,,., ,uk) T and u > 0 for all i  j. The jth
Lagrmgan method is fl-om  computtional vlewpoit almost equal to the
welghting method, Thls is why it i not studled more closely here, Chaong
and Hahnes hve treated the problems separately to emphaslze two ways of
aïrlvlng t the same point.
3.2.3. Theoretical Results on Proper pareto Optimallty
v(y) = êfs{/(x) I ](x) ej _< y for ail j =

Definltion 3.2.7. The «-constraint problen (3.2.1) is said tobe stable whea
v(0) is xfite and there exlsts a scalar/ > 0 sucb that, for ail 0 ¢ y E R  
v(O)llY v(Y) /L
Afer thls, a theorem concerniig the proper Pareto optlmallty of the sifi
tions of the e-constraint problem con be presented.
Theorem 3.2.8. Let the multlobjective optimizatlon problem be COnvex and
let x* E S bc Pareto optimal. Then x* is properly Pareto optimal if and only if
-constraint problem (3.2.1) is stable for eacb g - 1,..., k, where j - ](x*)
for all j - l,...,k,  #L
Proof. Sec Benson and Morin (1977) or Sawaragi et al. (1985, p. 88).
Let Is now suppose that the feasible region is of the form
s - { E R  I g() - (g(),(x), ... ,,,,()) <_ 0}.
The ¢-constraint problem is  constrahmd sigle objective optimization
problem and it can be convoxted ito an unconstrained problem by foïmifitiag
a Lagrange function of the foin
k m
to be minimized. Setting some assumptlons on the Lagrange mult*plïers  Ç
R   and  Ç R TM, we can derlve more conditions for proper Pareto optlmality.
In the following, we need the constralnt qualification of Definltion 3.1.10
of Part I, that is the definition of a rcglar point applied to the e constraiat
problem. In other words, a point x*  Sis regular if the graifients of the active
constraints of thc e constraint problem at x* are llnearly independcnt.
For clarity, we shall now formifiate the classlcal Karush-Kuhn Tucker nec-
essary condition for optinxality (see Kuh and Tuckeï (1951)) applied to the
¢-constÆaint problem. The proof for general nonllnear problcms is presented,
for example, in Kuhn and Tuckcr (1951) and Luenberger (1984, p. 315). The
condition can also be derived from tbe optimality conditions for multlobjective
optimlzatlon problems, presented in Section 3.1 of Part I.
Note 3.2.9. (Karush-Kuhn-Tucker necessary optimality conditio applïed to
the  costraint pvblem) Let the objective and the constraint huctions be
contlnuously differentlable at x*  S whicb is a regular point of the constraints
of the e constraint problem. A necessary condition for x* tobe a solution of the
Note that the (Lagrange) mifitipllers  are in what follows called Karush-
Kuhn Tucker multipliers, when they are associated wlth the Karush Kuhn-
Tucker optimallty condition. The condition stores, for exaple that if the
constraint concernlng ] is hot active, the correspondhg nmltiplier Ai must be
equal to zero.
We can now present the foltowig theorem.
Theorem 3.2.10. Let ail the objective and the constralnt fimctions be con-
tiauously dilïercntlable at x*  S wlùcb is a regular point of the constraints of
the e-constralt problem. Then thc following is valid.
(1) If x* is properly Pareto optimal, then x* solves the e-constraint prob-
lem for some ] being minimlzed and ai - /(x*) (for j --- 1, ...,k,
j ¢ 2) wlth ail the Karush-Kuhn-Tucker multip[lers assoclated with the
constraints ]j(x) <_  for j 1,..., k, j 7  g, belng positive.
(2) If the multlobjective optimizatlon problem is convex, thc x* is prop-
erly Pareto optial if it is a solution of the e-constraint problem with
the Karush-Kuhn-Tucker multipliers associated wlth the constraints
] (x) < ej for j - 1,..., k, j ¢ g, bcig positive.
Proof. Sec Chankong and Haimes (1983b, p. 143).
It can also be proved that if some solution is improperly Pareto optimal
and the problem is convex, hen some of the associated Kaïush-Kuh-Tucker
multlpllers equal zeïo. On the otber hand, if some of the Karush-Kuhn Tucker
multlpliers equal zeïo, then the solution of the e-constrait problem is improp
erly Pareto opthal (sec, e.g., ChaoEong and Haimes (1983a)).
Accorifing to Theorem 3.2.10 we con say that if a multlobjectlve optinfiza-
tion problem is solved by the e-constïaint method, pïoper Pareto optimality
con be checked by employing tbe Lagïange function, ha the previous section in
connection with the walghting method we also presented some conditions for
proper Pareto optimality. Let us meatlon that proper Païeto optlmallty is cbar-
acterized with the help of jth Lagranglan problem (3.2.4) in Benson and Morúa
(1977). There are, however, many methods where proper Pareto opthnality is
difficult to guarantee algorithmlcally.

3.2.4. Connections with Trade-OffRates
The relationshlps betwe+n Karush Kutm:Tucker m]tipllers and tradeoff
rates m'e studed in Chaakong and Haimes (1983b, pp. 159165) and Haimes
and Chaakong (1979). Lndeed, under certain conditions to be presented in the
following, the Karush-Kuhn-Tucker multlpliers of the Lagraage problem are
equivalent to trade-off rates.
For notatlonal convenlence we state the second order sufficient condition
applled to the  constraint problem. See Chaakong and Haimes (1983b, p. 58)
for details.
Note 3.2.11. (Second-ortier suïcinnt condit£on for optimallty applied to the
-constraint pblem) Let the objective d the nstraint 5mctinns be twlce
contuoly fferentiable t x*  S whlch is a regulm" point of the constrnts
of the -oetrnt pblem. A sufficient condition for x* tobe a solution of the
-constrnt problem is that there ex[st vectors 0    R - and 0  p  R m
such tbat the optinmlity condition of Note 3.2.9 is satisfied d the Hessi
amtroe of the corrponding Lage functlon
k m
v/( *) +  v(f( ") - «) + .«v%(')
iv positive de'ire on the set {d  R   Çg(x*)Td -- 0 r  i such that
 > 0},
A connection between Ksh-Km-cker multipliers and tradoff rates
is presented in the followlng theorem. The upper bound vector is denoted by
 e R   and itis sed to be chosen so that feible solutions est.
Theorem 3.2.12. Let x* 6 S be a solution of -constraint problem (3.2.1)
j # , be the correspondlng Karush Kuin-Tucker multipbers associated with
the constraints f(x) <  for j # L If
(1) x* is a reglar point of the constraints of the «-constraint pïoblem,
(2) the second order sufiicie]t condition of Note 3.2.11 is satisfied at x*,
and
(3) there are no degenerate constraints at x* (i.e., the Karush-Kub-Tucker
multlpllers of ail the corrstraints are strictly positive),
then« 1- Off(x*) for all j--1,..,k, j C L
Proof. The proof is based on the implicit function theorem, see Luenberger
(194, p. 313).
3.2. -Coraint Method 93
From the assumption A(f(x*) - ê) -- 0 for all j -- 1, .., k, j #  of the
Karush-Kub-Tuckr necessary opt rmallty condition and the nondegeneracy of
the constraints we know that fj(x*) Q for ail j # . Thus, from Theorem
3.2.12 we rave the trade-off rates
0fo(x*) for all
An important res]t corcerning the relatioashlp between Karush-Kuin-
Tucker multipliers and trade-off rates in a more general situatio: L where zer-
valued multipliers also m'e accepted, is presented in the followng. For hot ational
simplicity we now suppose that the 5rection to be minimized in the z-constraint
are assumed to be chosen so that feas[ble sointlons ex]st. Tb]s does hot lose
any generality. For deta[ls and a more extensive form of the theorem we refer
to Chankong and Haimes (1983b, pp. 161 163).
Let A be the Karush Kutm-Tucker multlpllers associnted wlth the coin
straints fj(x) < ¢, j -- l,.. ,k l Wlthout loss of generallty we can as-
sume that the first p (1 < p < k 1) of the multlpllers m'e strictly positive
(Le., Akj > 0 for j -- 1,...,p) and tbe ïest k - 1 -p multipliers equal zero
(i.e., Aj -- 0 for j = p + 1, .  ,k - 1). We denote the objective vector corre-
spond]ng to x* by z* E Z.
Theorem 3.2.13. Let x* E S be a solution of corrstraint problem (32.1)
(when f is mhimized) such that
(1) x* is a reglar point of the corrstraints of the z-constraint problem,
(2) the sccond-order sufficient conditlo of Note 3.2.11 is satisfied at
and
(3) ail the active constralnts at x* m-e nondegeneratc.
Then we have the following.
1) Ifp -- k - 1, that is, all the m]tlpliers A are strictly positive, then the
Pareto optimal surface h the feasible objective region in the neighbour-
hood of z* caa be represented by a continuously differentiable functinn
f such that fr each (z,z,...,z) T in the neighbourhood of z" is
2) If 1 < p < k - 1  that is, some of the multipliers A, equal zero, then the
Pareto optimal surface in the feaslble objective reginn in the neighbour-
hood of z* cm} be represented by coxtt[nuously ddIeïentiable functions

94
Proof. See Chaakong and Hahnes (1983b, pp. 163 165).
Let us now conslder the contents of Theoreal 3.2.13. Part 1) says that under
the gJven conditions there m'e exactly k - 1 degrees of freedom x specifying
a pohlt on the (locally) pareto optimal surface in the objective space in the
neighbourhood of z*. bi other words, when the values for z, z2,... ,zk- bave
been chosen from the neighbourhood of z*, then the value for z can be calcu-
lated fom the 6iven function and the resulthg point z will llc on the (locally)
Pareto optiaml surface in the objcctive space.
Part 2) of Theorem 3.2.13 extends the result of part 1) by relaxlng the
assumption that ail thc constralnts f(x) _< e, j -- 1,...,k l, should bc
active and nondegenerate, that is, Aij > 0 for ail j - l,...,k - l. When
the number of nondegenerate constraints is p (< k 1), then the degree of
feedom in specifyiaig a point on the (locally) Pareto opthnal surface in the
objective space ax the neighbourhood of z* is the IoEumber of nondegenerate
active constraints (p). The restflts of Theorem 3.2.13 will be needed in Section
5.1 when the «-constralnt aoethod is used as a part of an interactive method.
3.2.5. Applications md Extensioxs
Systematic ways of peturbing the upper bounds to obtain different pareto
optimal solutions m'e suggested in Chankong and Haimes (1983a, b, pp. 283
295). The -constralnt method is used for generating Pareto optlnl solutions
in Osman and Ragab (1986b). Then the solutions are clustered and a global
pareto optimum s located.
Sensitivity analysis of the «-constralnt method is dealt with ñx Rarig and
Haimes (1983). An index is defined approximatla 6 the standard deviation ofthe
optimal solution. The objective and the constraint functfons m'e not supposed
to be known for a certainty.
Now that we have introduced two basic methods it is wothwble to mem
tion a method for nonlinear problenm presented in Osman and Ragab (1986a).
It combhms features fom both the walghtin 6 method and the ¢-constralnt
method. The nonconvex feasible objective region is dlvlded hIto convex and
nonconvex pm'ts. The positive featttre of the weighting method that the feasl
ble reglon is hot dlsturbed in the solution process is util[zed in the convex peurs,
and the capabillty of the e-constraint method to final ail the Pareto optimal
solutions is utilized in the nonconvex peurs. Therefore, merits of both these
baslc aoethods are exploited.
A method related to the e-constraint method is presened in Youness (1995).
It generates the Pareto optimal set for problems with quasiconvex (and lower
seaficonttnuous) objective fmctlons. The method is based on level sets. If we
consideï an objectlvc vector z h and level sets L(z h) -- (x • S ] /(x) _< z h)
for i - 1,..., k, and if we have ç_t Li(zî) - (zh}, then the vector z h is Pareto
optimal.
An entropy based formulation of the e-constraint amthod is suggested in
Sultan and Tctnplcman (1996). The entropy-based objective finctlon tobe oI
timized has only one parameter no amtter what the umber of the origiaal
objective fmctlons fs. A representation of the pareto optimal set can be gener-
ated by varylng the value ofthe stngle parameter. The entropy-based fmctlon
contains logarithms and expoaential functions.
3.2.6. Conchlding Remarks
Theoret ically, every pareto optimal solution of any multlobjective optlmiza
tlon problcm can be found by the -constralat method by alterlng the upper
bounds and thc function to be afinimized. It must be stressed that even duality
gaps in ioEonconvex problems (see, e.6. , Section 2.10 of Part I and ChazoEong and
Halmes (1983b, pp. 135 136)) do hot dlsturb the fmctlonlng ofthe e constraht
method. However, compattlonally, the conditions set by Theorems 3.2.2, 3.2.3
and 3.2.4 are hot always very pïactlcal. For ezample, according to Theorem
3.2.2, the ¢-constraint problem needs tobe salved k tlmes for ail f as objec-
tive functions h} order to generate oae pareto opthnal solution. On the other
hand, the unlqueness of the solution demm}ded in the other theorems is hot
always too easy to check either.
Comput ationally, the  const raint method is more laborious than the weight-
ing method because the number of constrañts increases. It may be difficult to
specify appropriate upper bounds for the objective finctions. Thc components
of the ideal objective vector can be ned to help in the specification. Thet we
cm} set 3 -- z + e for j -- 1,..., k, j ¢ £, where ej is somc relatively smMl
positive real immber that can be altered.
The -constraint method can also be used as an a priori method, where the
declsion maker spccifies fl and the uppeï bounds. Then it can be characterized
as an ad hoc method. It mcans that oae can never be completely sure how
to select the objective finction and the upper bounds to obtaln a desirable
solution. Thls is a conmon wealflless with the a priori weightlng method.

3.3. Hybrld Method
(1980) and Wende[1 and Lee (1977) in slight ly different forms. The imme hybïid
method is introduced in Chankong and Haimes (1983, b).
miimize  wij'(x )
(3.3.1)  
subjectto j')(x)_<ej forall j l,...,k,
Theorem 3.3.1. The solution of hybrid problem (3.3.1) is Pareto optbnal for
any upper bound vector  £ R k , On the other band, if x*  Sis Pareto optimal,
then it is a solution of problem (3,3.1) for «- f(x*).
Proof. See Corley (1980) or Wendell and Lee (1977).
The set of Pareto optimal solutions can be found by solving problem (3.3.1)
with methods for parametrlc constraints (where the parameter is the vector of
upper bounds e), sec, fOr example, Rao (1984, pp. 41821). This means tbat
the weighting coefficlents do hot bave to be altered.
Optlmallty conditions for the solutions of problem (3.3.1) to be properly
Pareto optimal m'e prescnted in Wende[I and Lce (1977).
We can say tbat the positive fetures of the weighting method and the «-
constïalnt method are comblned in the bybrid method. Namely, any Pareto
optimal solution can be found independently of the convexity of the problem
and one does hot bave to solve several problems or th]nk about uniqueness
to guarantee the Pareto optlmality of the solutions. On the other hand, the
specification of the parameter values mŒEy still be difficult. Computationally,
the hybrid method is similar to the e-constraint method (with the increased
nLunber of constraint functions).
3.4. Method of Weighted Metrics 97
3.4. Method of Weighted Metrics
La the method of the global crlterion, introduced in Section 2.1, L»- and
L-metrics were used to generate (weakly) pareto optimal solutions, These
metrics can also bu welghted in order to produce different (weakly) pareto
optimal solutions. The weighted approach is also somctlmes called compromise
programmLag (sec Zeleny (1973)). Here we use the term the method of wcighed
metrics.
3.4.1. Introduction
We assume tbat w > 0 for all i -- 1,...k and Lw -- 1, We obtain
different solutions by alterlng the weighting coefficients w in the weighted L-
and Tcbebycheff metrics. The weghted Lp-problem for nfinlmizing distances is
now of the form
(3.4.1) minimize w]f(x) - z »
subject to x  S
for 1 _< p < c. The weighted Tchebycheff problem is of the form
minhnize max
(3.4.2)
subject to x 6 S.
problem (3.4.2) was originally introduced in Bowman (1976), Again, denonn
tors may be included. Further, the absolute value signs can be dropped because
of the definltion of the ideal objective vecto L if it is known globally. Walghting
vectors can also be used in connectlon wlth problems of form (2.1.4).
If p -- 1, the sum of weighted devlations is minlnzed and the problem tobe
solved is equal to the weighting problem except fo  constant (if z* is known
globally). If p -- 2, we bave a method of least squares. When p gets larger,
case instead of pïoblem (3.4.2), the poblem
(3.4.3) subject to a > wi(f,(x) - z**) for ail i -- 1,...

3.4.2. Theoretical Results
Theorem 3.4.1. The solution of weigbted L problem (3.4.1) (when 1 _ p <
oï Yu (1973).
Theorem 3.4.3. Weighted Tchebycheffproblen (3.4.2) has at least ont Pareto
optimal solution.
Proof. The proof follows directly ffom the proof of Theoren 2.1.3.
Corollary 3.4.4. If weighted Tchebycheff problem (3.4.2) has a uniqne solu-
Convexity of the multiobjective optinñzation problem is needed in order to
garantec that every Pareto optimal solution can be round by the w«ightcd
Lp-problem (see Sawarag et al. (1985, p. 81)). The ïollowing theorem shows
that, on the other hand, every Pareto optinal solution can be round by the
weighted Tchebycheff problem.
Theorem 3.4.5. Let x* E S he Pareto optimal. Then there exists a weighting
vector 0 < w E R  such that x* is a solution of weighted Tchebycheff problem
(3.4.2), where the reference point is the utopian objective vector
problem. We know that ri(x) > zî* for ail i - 1,...,k and for ail x E S. Now
we choose wi -- fl/(f(x*) - zî*) for ail i - 1,...,k, where fl > 0 is some
3.4. Method of Weighted Metrlcs 99
If X* is hot a solution of the weighted Tchehycheff problem, there exists
another point x °  S that is a solution of the weighted Tchebycheff problom,
meanlng that
OEhu wi(f(x*) zî) <  fer ail i - 1,..., . OEhis means thag
£(x ) <
3,4,3. Comments
Theorem 3.4.5 above somds quite promising for the weighted Tchebycheff
problem. Unfortmtely, this is not the whole truth h addition to the fact that
cvery Pareto optimal solution can be round, weakly Pareto optimal solutions
nmy lso be included. Auxáliary calculation is needed in order to identify the
weak ones. Remember that as far as the weighted Lv-problem (1 < p < c) is
conccrncd, it produces Pareto optimal solutions but does not necessarily fld
ail of them.
Selectlng the alue for the exponent pis treated in Ballestero (1997b) fron
the point of view of risk aversion. The conclusion is that for gre&ter risk aversion
we should use grcatcr alues for p. Another gideline is that for a smaller
number of objective fltnctions we should select greater p alucs.
More results conccrning the properties of the Lp-netrics (1 _< p g c) with
and without the weighting coefficients can be round, for exampl% in Bowman
(1976)» Chankong and Halmes (1983b» pp. 144 146), Koski and Silvennoinen
(1987), Nakayanm (1985a) and Yu (1973), the first of thcsc treating especially
the Tchebycheff metric. Some results concernilzg thc proper eflïciency (in the
sense of Henig) of the solutions of the weighted Lp problem are presented briefly
in Wlerzbicki (1986b).

3.4.4. Connections with Trade-Off Rates
Useful snlt s concerning trade-off rates and the weighted Tchebycheff prob-
lem are proved in Yano and Sakawa (1987). The approach is closely related to
what was presented in Subsectlon 3.2.4 in conntlon wih the ¢-co]strMn
problem.
Let us once agaln suppose that the feible region is of ghe form
A the objective oed the constraint functioas e sumed tobe twlce coin
tinuously differentiable, which is why problem (3.4.3) is the oae to be dent
lem with oae objective functioa, the Loege function, of the fom
k m
variable vector being (a,x)  R "+, let us denote the minimal solution of
It is sumed that the sumptlons in Threm 3.2.13 when applled to
constrnts of (3.4.3), the seconorder sufficient condition of Note 3.2.11 ap
plled to problem (3.4.3)  satisfied at y* oed l the active constrMnts at y*
are nondegenerate. (The ]t sumption meoes that the Kaïush-Kuhn-cker
mtipllers of l the active constrats e positive.)
If l the constralnts connected to the objective functions aïe tive, we
have
in Subsecion 3.2.4. For detafls, s Yano and Swa (1987).
3.4.5. Variants of the Weighted Tchebycheff Problem
Thus far, it has been proved that the weighted Tchebycheff problem can find
any Pareto optimal solntion. According to Corollary 3.4.4, the unique solution
of the weigbtcd Tchebycheff pïoblem is Pareto optimal. If the solution is not
unique or the uniqueness is difficult to guarantee, the weakness of the pïoblem
is that it may pïoduce weakly Pareto optimal solutions as we[L This weakness
can be overcome in different ways. 0ne possibklity is to solve some additional
3.4. Method of Weighted Metrics 101
problem, for example problem (2.10.1) or problem (2.10.2), given in Part ].
Problem (2.10.1) can also be modified so that instead ofthe sum of the objective
filnctions, the sum of objective functinns minus utopioe objective values is
minimed. Su an approa is handled in Section 5.4.
ex¢ple in Korhonen (1997). One more way  to use lexicogïaphic ordeïing
(to be troduced  Section 4.2).
the metric. Weoey Peto optim solutions coe be avoided by giving u slit
slope to the contouï of the metric. The price to be pd is that in some ces
it my be impossible to find every Pareto opti solution. For that tenon,
properly Peto optim solutions e of interest heïe. Note that the utopioe
objective vector is used  a refeïence point  in Theorem 3.4.5.
It is snested in Steuer (1986) and Steuer oed Choo (1983) that the
weighted Tchebyeff problem be vied by oe amentation term. In this
ce, the distance between the utopioe objecti vector oed the ible o
jectlve reon is meured by oe augmentoE weghted Tchebycheff metc. The
augment weighted Tchebycheff problem is ofthe rm
k
(3.4.5) -, ,
subject to x  S,
A stigbtly derent rwdified weighted Tchebycheff retmc is used in the
ified weighted Tchebycheff problem
k
(3.4.6) mininfize i- [Wi ( f,(X) zï[ +p
subject to x  S,
w]mre pis a sufficiently sml positive scal. Itis shown in Kaliszcwski (1987)
hat problcm (3.4.6) is equivent (up to scar multiplication) to that proposed
in Choo and Atkins (1983).
The derence between the augmented and the mofied weighted Tcheby-
eff problems is in the way the slope takes ple  the metrics, as iustrated
 Figure 3.4.1. h the augmnted weighted Tchebyeff problem the slope is
fction of the weighting coefficients oed the pmter p. h other words, the
slope may be different r eh objective function, that is, for eh coordate
of the objective space. As f  the modified weighted Tchebycheff problem is
concerned, the slope is a nction of the pmter p
ail the objective fctions, h Fre 3.4.1 we have
fli=arctanl w+pP oed fl=arctan
To ee the comparlsoa, the dotted lines represeat the slope of the weighted
Tebycheff metric. See Kaliszewski (1986, 1987) r

Figure3.4.1. Slopes oftwo metrics.
The augaalented welghted Tchehycheff prohlem is illustrated in Figure 3.4.2,
where the dotted thaes represent the contours ofthe augnmnted metrlc. The con-
tour of the weighted Tchehycheff metric (coutimlous line) bas only heen added
to ease the compaïison. Sensitivity analysis of the augmented and the modified
weighted Tchehycheff metrics is handied in Kaliszewski (1994, pp. 113 119).
z 2
Figure 3.4.2. Augmented weighted Tchehycheff prohlom.
It is valid for hoth the augmented and the modifled weighted Tclehychcff
pïohlem that they generate only properly Pareto optimal solutions and any
pïoperly Pareto optinl solution can he round, ha what follows the symhol M
is the scalar from Definition 2.9.1 of proper pareto optimality in Part I.
3.4. Method of Weighted Metrics 103
Theorem 3.4.6. A decision vector x* E Sis properly pareto optinml if and
only if there exJst a welghtlng vector w E 1  with wi > 0 for ail i -- 1:..., k and
a mmber p > 0 such that x" is a solution of augmented weighted Tchehycheff
pr oblem (3.4.5).
In addition, for each properly Pareto optinl sohation x*  S there exists
a weighting vector 0 < w  1  such tht x* is a unique solution of prohlem
(3.4.5) for every p > 0 satisfying M < afin-l, ,k w/((k 1)p).
Furtheï, the inequality
M< 1- max wi+(k 1)p
is valid for eveïy sohltion x* of prohlem (3.4.5).
Proof. See Kaliszewskl (1994, pp. 5153).
Tte prOOf la Kaliszewski (1994) is hased on acone separatlon technique.
The necessity and the sufficiency components are also proved in Kallszewski
(19S5).
The theorcm for the modified wcighted Tchehycheffprohlem is almost sim-
Theorem 3.4.7. A declsion vector x*  Sis properly Pareto optimul if and
only if there exlst a weighting vector w  R  with wi > 0 for ail i - 1 ...
and a numher p > 0 such that x* is an optinml solution of modified weighted
Tchehycheff prohlem (3.4.6).
In addition, for each properly Pareto optimal solution x*  S there exlsts
a weighting vector 0 < w E R  such that x* is a unique solution of prohlem
(3.4.6) for every p > 0 satisfying M < 1/((k l)p).
Further, the inequality M _< (1 + (k 1)p)/p is valid for eveïy solution
of problem (3.4.6).
Proof. See Kaliszewski (1994, pp. 4850).
This necessity and suflïciency formulation is an extension of the original
theorem in Choo and Atkins (1983). te necessary conditions in Theorems
3.4.6 and 3.4.7 are also proved ha Kaliszewski (1995).
3.4.6. Connections wlth Global Trade-Offs

104 Part II 3. A Posteriori Methods
For the simpliclty of notations we here assume the global ideal objective
vector and, thus, the global utopian objective vector tobe lumwn. This implles
that we can drop the ahsolute value signs.
Ail the properly pareto optimal solutions produced with modiJied weighted
Tchebycheff problem (3.4.6) have bounded global trade-offs. Frther, we have
a conmon bouad for every global trade-off ñvolved.
Theorem 3.4.8. Let x* • S be a solution of modified weighted Tchehycheff
problem (3.4.6) for some weighting vector 0 < w E R k and p > 0. In this case
the global trade-offs ae bounded, that is
A(x*)_< l+p
foï every i, j=l,...,k,i#j.
Proof. See Kaliszewski (1994, pp. 94 95).
Corresponding rcsults can be proved for other types of pïoblems, see Kaliszewski
(1994, pp. 82 113).
Sometimes the decision maloer may wish to set a priori bounds on some
specific global trade-offs. Such a request calls foï a scalaizing functlon of a
speclal form. These toplcs are treated in Kaliszewski and Michalowski (1995,
1997). Thus faL the addltional term multiplied with p was added to gaantee
the proper Pareto optlnmlity of the solutions. If we leave it out, we obtaln
weàghted Tchehycheff problem (3.4.2) and, thus, weakly paeto optimal solu-
tions, ir what follows, wo use metrics wlthout modification oï augmentation
teïms but use other paameters « and o" i . 0 to contïol the bounds of the
global trade-offs involved. Thus, the followbg results deal with weak paeto
The next theoïem handles a case where we wish to set a priori bouads foï
a goup of selected glohal tïade-offs. Let us choose a subset of the objective
functlons lo C 1 = {1,...,k} and define l (i) = {j I J • lo, J # i }.
Theorem 3.4.9. A decision vector x* • Sis weakly Paeto optimal and
(3.4.7) Aji(x )-- for ail i•lo andall j•l(i)
if and only if there exlst a weighting vector 0 < w • R  and a numhcr « > 0
such that x* is a solution of the problem
mlnimize max [max w,((1 ÷ o')(f (x) - zï) + o Z (f,(x) -
subject to x • S.
3.4. Method of Weighted Metrlcs 105
Proof. See Kallszewski and Michalowski (1995).
Result (3.4.7) of Theorem 3.4.9 is utilied so that the decisiolt maker is asked
to specify upper bounds for the selected global trade-offs. These values ae set as
upper bounds to (1 +a)/a. A lower hound for the paameter « is obtalned from
these inequalities. By uslng the calculated  value, we receive different weakly
paeto optimal solutions satising the global tradc off bounds by altering the
weighting coefficients. In other words, we avoid generatlng solutions exceeding
the specified bounds for global trade-off.
iri Theorem 3.4.9 we have a conmmn bound for the selected set of glohal
trade-offs. Thls can further be generalized hy using several diffeïent parameters
Theorem 3.4.10. A declsion vector x*  Sis weakly Pareto optinal and for
each i E Io and each j Ç I0, A(x*) is bounded ¢rom above by a positive finite
numher if and only if there exist a weighting vector 0 < w  R  and numhers
ai > 0 for j Ç I0, such that x* is a solution of the prohlem
A(') <
for i, j
Proof. See KMiszewski and Michalowski (1997).

nfinimize max. w_f,.x.-z__+_,p<fixl-z_*"
(3.4.9)
subject te x E S.
It is proved in Kallszewski and Michalowski (1997) that Theorem 3.4.7
pïoperly pareto optimal ff and only if therc exist a welght ing vector 0 < w 6 B. 
and numbcrs pi > 0 for ail i 1,..., k such that x* is a solution of problem
(3.4.9). Further, for the solution x* of problem (3.4.9) the upper bounds of the
gJobal trade-offs are of the form
A(x*) < 1 +Pi
3.4.7. Applications and Extelasions
A stmpe opt tmlzatlon problcm ofa spillway profile is solved by t fie weighted
L-metrlc (wlth denominators) in Wang and Zhou (1990). Further, an extension
of the method of weig fited metrics called composite progranmiag is presented in
B&dossy et al. (1985). The L»-met tic (p < c) is divided into nestcd parts with
dtfferent exponents. In particular, thls approach can be appficd te problcms
where objective functions consist of several components. The method i app fiod
te problems of multiobjectlve watcrshed management and observation rtetwork
design.
The weighted Tchebycheff metric is used in Kostreva et al. (1995) it de-
riving an iategral approacfi for solving nonlinear problen with discontinuous
objective functions and a discontinuous feasiblc rcgion. Thc close connections
between the weigfited Tclebycfieff metric and the weoEtting method for con-
vex problems are handled in Dauer and Osmma (1985). Karusfi-Kuhn-Tucker
optlmallty conditions for these two methods are treated as well.
One more possibifity of avoiding weakly Pareto optimal solutions is sug-
gested in Hclbig (1991). He/big defines optimality with ordering cones. The
idea is te perturb the orderlng cone se tiret even thougt the met fiod produces
wealdy efficient solutions, they are eflïclmt te the origSnal problem.
One way te apply the weighted Tdmbycheff metric and its augmented ver-
sion successfolly will be introduced in Section 5.4.
3.4.8. Concluding Remaïks
Par ticularly the method of weigfited Tchebycheff metric and its variant s are
popular met fiods for gerterating Pareto optimal solutions. They work for convex
as well as nonconvex problems (unUke tfie weighting metfiod) and alteration of
the parameters is casier ttmn in the e constraint method.
are Pareto optimal but net necessartly ail of them are found (depending on the
degree of nonconvexity of the problem). The weighted Tchebycheff metric can
mentitg or modifying tfie metrlc or by solving another optimization problem
after mialnUzing tfie distance. These alternatives will be furt fier addressed in
Section 5.4. For nonconvcx problems the sccess of the metfiod of weigated
metrics depends on whether tfie global ideal objective vector is known or net.
metric is tfie ability te produce solutions with a priori specilied bounds for
solutions can be generated satlsfybg given fixed bounds.
3.5. Achievement Scalarizing Function Approach
The approach te be presented is related te tiret of weigated metrlcs. Nanmly,
wc handle special types of scalarizing functions, ternmd aclfievement scalarizia8
fun(oEions. They have bcea introduced by Wierzbicki, for example, in Wlerzbicki
(1981, 1982, 1986a, b) (and are aise handied h Wierzbicki (1977, 1980a,
Somewhat similar results for scalarlzïng functions are aise presented in Jahn
(1984) and Luc (1986).
3.5.1. Introduction
in the method of weoEbtcd L»-nmtric or the wcighted Tchebycheff metric,
the distance is mlnimized between the ideal objective vector and the feasible
objective reglon. If the ob ide objective vector is unkaown, we may fMI
in producing (wey) Pareto optimal solutions. In other wor, if the refer
encc point used is oe objective vector inside the ible obctive tenon, Che
solution. We tan overcomc th weakns by replacing metrlcs wlth acevement
FOr example weakly Pareto optimal solutions can be generaed with oey
minimize 2ï[ w(/(x) )]
(3,5.1)
It ders om weigfited Tcfiebycleff problem (3.4.2) only in that the abs
solutions are produced independently of the feibifiy or inbility of the
rcDrence point.

an achievement function.
Because we do hot know the feaible objective regon Z explicit ly, in pr acice
we minimize the flmctinn sz(f(x)) subject to x E S (see, e.g., Figure 2.2.1 in
solved in the feaible objective reg)on.
3.5.2. Theoretlcal/esults
We need to apply som of tbe general properties izttroduced in Part I to
an chlevemcnt functinn ôz, namely the definitinns of strictly increasing (Def-
initinn 2.1.8), strongly increasing (Definitinn 2.I.9) and ¢-strongly increaing
(Defdtion 2.I.10) fltnctinns, lq the lait mentioned concept tlm defition of
the set P is the sanoe a in connection wth «-proper Pareto optimality (see
Definltion 2.9.2 in Part I).
Next we can deflne order-representing mld order-approxlmating achieve-
ncett flmctinns.
Definition 3.5.1. A contlnuous achievcment function sz: Z  tt is orde
representing if it is strictly increaing os  functinn of z E Z fOr any   tt 
and if
(for ail z  ).
(for M1 z  ) with e > e OE 0.
ek 3.5.3. r a continuous order-representng or order-approxinmtlag
achievement function s : Z  R we have
s(z) = 0.
achlevement functinns according to Wierzblçkl (I986a, b). The achievement
problem tobe solved is
mininze s.(z)
(3.5.2) subject to z  Z.
Theorem 3.5.4. If thc achievement fltnctinn s. : Z  Ris strictly increaing,
then the solution of aclievement problem (3.5.2) is weakly Pareto optimal. If
the chievement fltction 0z : Z -- tt is strongly incïeaing, then tbe solution of
problem (3.5.2) is Pareto optimal. Finally, if the achievement functlon sz : Z --
tt is e strongly increaing, tben tbe solution of problem (3.5.2) is «-properly
Pareto optimal.
Proof. Here we only prove the second statcment because of thc slmilarity of
the proofs. We astme that sz is strongly increaing. Let z* Ç Z be a solution
of tbe chievement problem. Let us suppose that itis hot Pareto optimal. In
this cae, there exists an objective vector z • Z such that zi _< z for ail
i -- I,... ,k and zj < z; for some j. Because sz is strongly increaing, we know
that sz(z) < s.(z*), which contradicts the assumptinn that z* minlmizes s..
Tbus z* is Pareto optimal. []
Tbe results of Theorem 3.5.4 can be augmcted by tbe followlng theorem.
Theorem 3.5.5. If tbe achievenent functin s. : Z  tt is incrcaing and the
solution of achievement problem (3.5.2) is unique, then itis Pareto optimal.
Proof. The proof corresponds to tbe proof of Theorem 3.5.4.
Note tht Theorems 3.5.4 and 3.5.5 are alid for any scalarizing function.
Tbus, the Pareto optimality and the weak Pareto optimallty results proved for
the weightlng metbod, the e constralnt metbod and the rnethod of weighted
metrics are explahed by the monotonicity properties of tlm scalarizing çunc
tions in question (see, e.g., Vanderpooten (1990)).
We can now rewrite Theorem 3.5.4 so os tobe able to characterize Pareto
optimal solutions with tbe help oi order representing and order-approxlmatlng
achieiement flmctions. The proof fo[lows from the proof of Tbeorem 3.5.4.
Corollary 3.5.6. Iftbeachievement functinnôz: Z - ttisorder-representing,
then, for any z  R , tbe solutlon of achlevement problem (3.5.2) is weakly
Pareto optinal. If the acbievement fltnctinn s : Z  tt is order pproxiniating
with some  and é a in Definltion 3.5.2, then, for any   tt , tbe solution
of problem (3.5.2) is Pareto optimal. If s in addition is ¢-strongly increaing,
tben the solution of problem (3.5.2) is e-properly Pareto optLmal.

Theorem 3.5.7. Ifthe achievement function sa : Z -- Ris order-representing
and z* E Z is weakly Pareto optimal or Pareto optimal, then itis a solution
of achievement prohlem (3.5.2) with z - z* and the value of the achievement
fonction is zero Ifthe achievement fanction sa  Z  Ris order-approximting
and z* E Z is  properly Pareto optimal then itis a solution of problem (3.5.2)
wïth z - z* and the value of the chlevement fanctlon is zero.
Proof. Here we only prove the statement for Pareto optlmality. The proofs
of the other statements are very similar. (The proof of the necessary condJtlon
for «-proper Pareto optlmality can he round in Wierzbïcki (1986).)
Let z*  Z he Pareto optimal. This means tht there does not exist any
other point •  Z such that z < z for alli = 1,...,k and zj < z for some
j Let us ssume that z* is hot  solution of the achievement problem when
z - z* ir this cse there exists some vector °  Z such tht sa(z °) < 8(z*) -
s(z) -- 0 and z °  z*. Since sa ws ssttrned to he ordcr representing, we hve
°ez htRî_--z* intl.Thsmeansthatz <z: foralli-1, .,k,
which contradlcts the ssumption that z* is Patto optimal. Thus, z* is a
solution of the achievement problem 
Pemark 3.5.8. Aided hy the restflts in Theorem 3.5.7 a certoSn point can he
confirmed hot to be weakly, «-properly or Pareto optimal (if the optimal walue
of the achievement fanction differs from zero)
3.5.3. Comments
the reference point is infesihle, tiret is, z  Z + R, then the minimization of
lution can be obtoSed by moving the reference point only. Itis shown in
Wierzbicki (1986a) that the solution of the achievement function depends
Lipschitz-continuously on the reforence point.
There are many achievement functions satisfying the above-presented condi
for ail i - 1,...,k, then soe(z ) -- max,[wi(z, t - z)] < maxi[w(z 2 - z)] --
s.(z ) and thus the fanction is strictly incresing. If the inequality sa(z) --
max[wi(zi z,)] < 0 holds, then we must have zi < 5i for ail i - 1,..., k, that
(3.5.3)
increasing. Function (3.5 3) is related to augnmnted weigated Tchebycheff prob
() -
approximting with  > 1/# (see Wierzhicki (1980, 1982)) More examples of
pie, in Wierzhfoki (1980h, 1986a h)

112 Part II -- 3. A Posteriori Methods
ff itis hot dominated by any feasible point and it dominates at least one of the
feaslble points.
3.5.4. Coneluding Pemaxks
Achlevement scaiarlzlng fuctions are a set of general scaiarizlng functlons
satlsying certain reqinrements, la generai achievement functions are concep-
tuaily very appealiag for generting weakly, e properly or Pareto optlnml so-
lutioas. They overcome most of the difficulties arising with other methods in
thls class.
The restflts concerinng acbJevemcnt fitnctions will be utillzed, for example,
in Section 56 when derivlag an iateractive method. For interactive methods,
the idea of moving the reference point hstead of the weigbting coefficient seems
more natural and casier for the decision maker. A fact favouring achlevement
scaiaviziag functions against weighted metrlcs is that the global idcai objective
vcctor does not have tobe kmwn. Thus the metbod is more relinble.
3.6. Other A Posterlorl Methods
Finally, we brlefiy mention some other methods ofthe a posteriori type. For
more detailed information, sec the references clted.
The so-cailed hyperplmle method is introduced in Yano and Salwa (1989)
for generatlng Pareto optinml or properly Pareto optimal solutions. It is
shown that the weigbting nmthod, the -constraint method and the method
of weighted metrics can be viewed as special cases of the hyperplanc method.
A theory concerning trade-off rates in the bypcrplane method is provlded in
Salwa and Yano (1990). A generallzcd hyperplane ncthod for generating ail
the efficient solutions (wlth respect to some ordering cone) is presented in
Sakawa and Yano (1992).
Another method for a general characterlzation ofthe Pareto optimai set is
suggested in Solnd (1979). For example, the weigbting method, the method of
weited noetrics and goal programnfing (see Section 4.3) can be seen as speciai
cases of the generai scalar problem of Solad. Further, the weighting method
and the ¢-constraint method are utilized in a so-called envelope approach for
determiinng Pareto optlnml solutions in Li and Haimes (1987). Az application
to dynanfic multlobjective progranming is aiso treated.
The noninferior (meaning here Pareto optimality) set cstinmtion (NISE)
method for MOLP problems can aiso be consldercd to belong to this class
of a posteriori methods. It is a technique for gcneratlng thc Parcto optim;fl
set of two objective functions (sec Cohon (1978)). It can bc gcneralized for
convex problems wlth two objective fitnctions (see, for example, Chankong and
Haimes (1983b, pp. 268 274)). la Baiachandran and Gero (1985), the method
is extended to problems with three objective functions. The weighting method
is the basis of the NISE method.
Multiobjective optinfization problems with polynomiai objective and con-
straint filnctlons are treated in Kostreva et ai. (1992). The method for deter-
mining Pareto optimal solutions is based on problem (2.10.2) of Part I and a
so-cailed homotopy continution. Note that problems with polynonfial fun
tions are highly nonllnear, nonçonvex and nonconcave.
A scalarizatiot method for multiobjective optimiz&tion problems, where
optimallty is defined through ordering cones, is suggested in Pascoletti and
Serafini (1984). By varying the parameters of the scaiar problem itis possible
to find ail the efficient solutions. A fitrthcr investigation is condueted in Sterna-
Karwat (1987).
Itis suggested in Benson and Sayin (1997) that istead of trying to generate
the whole Pareto optinml set one should aim at findlng a truly global repre
sentation of it. This would decrease both the burdcn of the decision maker
computationai costs. Bcnso and Sayin introduce a global shooting procedure
to meet this need. In Armann (1989), a method is presented fr generating
a dispcrscd subset ofthe Pareto optimal set which is then presented to tbe
decislon maker.
One more method for generating an evenly distributed set of Pareto opti-
mal solutions toa differentiable nonlinear multlobjective optimization problem
is suggested in Dos and Dennis (1998). Tbe approach is called the normal
boundary intersection (NBI) method. The idea in broad outline is to intersect
tbe feaslble objective reJon with a nornml to the convex combinations of the
columns of the payoff matrix. Evenly distrlbuted parameters, that is, the coe
ficients in the convex combinations, produce evenly dlstributed solutions. The
weakess of thc approach is the fact that it may produce ton-Pareto optinml
solution to nOnCOnvex problems.
The diffieulty of illustratig the set of Pareto optimal solutions to the deci
sion maker is treated in Bushenkov et ai. (1994, 1995) and Lotov et ai. (1992,
1997). An extension of the Pareto optinml set, a so called Pareto optimai hull
is approximated by polyhedrai sets (see Lotov (1995, 1996)) using convolution
based aigorithnm (sec Bushenkv et al. (1995) and Lotov (1996)). Different
decision maps are generated in this way (see Chapter 3 of Part III). Specific
approaches exist fr linear convex and nonconvex cases but they are ail based
on the saine so-cailed generalized reachable sets method. An implementation
of the generalized reachable sets method is available (sec Section 2.2 in Part
m).

4. A PRIORI METHODS
In the case of a priori methods the decision maker must specify her or
his preferences, hopes and opinions before the solution process. The difficulty
is that the decision makcr does hot necessarily know beforehand what it is
possible to attaln in the problem and how reallstic ber or his expectations are.
The working ortier in these methods is: 1) decision makcr, 2) analyst.
Below, we handle three a priori methods. First, we glvc a short presentatlon
of the vale function method. Then we introduce lexcograplfic orderlng and
goal progamming.
4.1. Value Function Method
The value function optinizatlon approach was already mentioned earller.
Herc we present it again briefly.
4.1.1. Ii.t roductlon
In the value function method, the decision maker must he able to glve an
accurate and expliclt mathematlcal form of the value function U : R   R that
represents }ter or his preferences globally. This function providcs a complete
ordering in the objective space. Then the value unction problera
ma)dmize U(f(x))
(4.1.1)
subject to x E S
is ready tobe solved by some method for single objective optimization as
illustïated ii Figure 4.1.1. The bold line ïepïesents the Preto optimal sel
Remember Theorem 2.6.2 of Part I, whicl says that the solution of problem
(4.1.1) is Pareto optimal if the value functioi is stïongly decreasing.
The walue function method seems to be a very simple method, but the diffi
culty lies in speclfylng the mathematical expression of the value function. The
inabillty to encode the decision Imker's underlying value function reliably is
demonstrated by experiments in de Neufville and McCord (1984). It is shown
that encodiag methods that should theoretically produce identical value lune-
tions ïail: the ïunctions may differ oeom each other by more than 50 %. It is also

Figure 4.1.1. Contours of the value functlon.
pointed out tht there is no actual analysi of the accuracy of the value function
assessment. The consistelcy checks that is, wbetber decision nmkers provide
consistent answers to similar questions, are hot adequate: a hiased instrument
can provide consistent data.
On the other rond, even if it were possihle for the decision nmker to express
ber or his preferences gJobally tbe resulting preference structure nfigbt be too
smple, since walue fanctlons cannot represent intransitivity or incomparbility
(see Rosinger (1985)). More features and wealesses were presented in connec-
tion wlth the defiaition of the value function (Definition 2.6.1) in Section 2.6
of part I.
4.1.2. Comments
The value function metbod coald bc called an 'optimal  way of Solving mul-
tlohjective optinfization problems f the decislon maker could reliahly express
tbe value fanction. The use of the value fuaction method is restricted in prac-
tice to multiattrihute decision analysls problems with a discrete set of feasihle
alterlatives. Tbe tbeory of value and utility functions for multinttribute prob
lems is exanfined broadly in Kcetmy and Ralffa (1976). But, it is believed, for
example, itt Rosenthal (1985), that tbese experiences can also be uti[ized in
Important results concerning value functions and the conditions for theh- ex
istence are collected in Dyer and Surin (1981). Two general classes of value func-
tions, additive and tnultiplicative forms are presented extensively in Keeney
and Ralffa (1976) and bfiefiy in Rosetthal (1985). Tbe existence of value fun
tions and tbe nature of additlve decreaslng value fanctions are handled in Statu
et al. (1985). These topics and tbe construction of value functions are pre-
sented more widely in Yu (1985, pp. 9161), General properties and some
desirable featuïes of certain types of value functions (e.g., additive max min,
nfin sum and expoltential forms) are statcd in Bell (1986), Harrison and Rosen-
thal (I933), Soland (1979) and Sounderpandian (1991). More cxamples of value
functions are given in Tell and Wal]enius (1979). Utility compatible measures
of rlsk are deduced in Bell (1995). Relations betwcen value functions, ordering
cones and (proper) efficiency are studied in Henig (1990).
In some interactlve nethods it is assumed that thc underlying value func-
tion is of some partictflar (e.g. additive or exponential) form, afer which, its
parameters are fitted according to the decision maker's preferences. Such meth
ods are presented, for example, in Rotbermel and Schilling (1986) and Sakawa
and Seo (1980, 1982a, b) (see Section 5.3).
ble to excludc any Pareto optimai or properly Pareto optimal solution from
consideration a priori are identified in Soland (1979).
The convergence properties oï additive vale fanctions (assuming prefer-
ettial indepetdence of tbe objective functions) are investigatcd by simulation
experiments in Stewart (1997). One observation is that piecewise llnear value
Relatiotships hetween tbe method of welghted metrics and the value func
tion metbod are reported in Ballestero and Romcro (1991). It mlgbt he
ined that tbe two methods bave nothing in conmlon, sincc a value function
rlcs does hot take tbe declsion nmker into cotsideration. However, condltiois
can he set on tbe wlue fanction to guarantee tiret its optinmm belongs to the
solution set ohtalnahle hy tbe metbod of weighted metrics. More relatlonships
are presented in Bagestero (1997a). It is demonstrated in Mor6n et al. (1996)
that there are large la.filles of such weg-beImved value fanctions for bl-critcrin
4.1.3. Concluding Remarks
The value function metbod is an excellent method if the declslon ker
happens to know an explicit matbematical formulation for the value functlon
and if that fuaction represents wholly the prefcretces of tbe decision nmker.
These two crucial preconditions arc tbe difficultles of thc approach.
There are certain conditions that tbe decision makers preferences mltst
satis so tiret a value function can be defiaed on them. Tbc decision makeï
nmst, for instance, be able to speciiy consistent (implying transitive) prefer-
ences. Tbus, there my hot itecessarily exist a value function that wIll impose
a total order in the set of feasihle ohjective vectors. Tbe assumption of a total
ortier is ofteit contrary to out intuitive altos and hence is qultc [ikely to lead to
less timon ideal selectioas, as Polak and Payne (1976) refind us. Ttfis fact must
be kept in fiId below, wbeIt several metbods whlch assume the existence of a
value function (at least inplicitly) are introduced.

al. 09ss).
4.2. Lexicographic Ordering
Lexicograp|Uc ordering was meutioned earller as a tool for produclng Pareto
optimal sobztions from wealdy Pareto optimal ones. ]t can also be used as ma
a priori solution met&od.
4.2.1. Introduction
la lexlcographic orderlng the decision maker must arrange the objective
functions according to their absolute importmace. This ordering means that a
more important objective is infinltely more important thma a lcss important
objective. After orderlng, the most importmat objective functlon is mbñmized
subject to the original constraints. H this problem has a unique solutlon itis
the solution of the whole multlobjective optlmization pïoblem. Otherwlse, the
second most importmat objective functlon is minimizcd. Now, in addition to
the original constraints, a new constraint is added. This new constraint is thcre
to garmatee that the most importmat objective functlon preserves its opthnal
value. If thls problem has a unique solution, it is the solution of the original
pïohlem. Otherwise, the process goes on as above. Lexlcographlc ordeïs mad
utilltles are wldely examined in Fishburn (1974).
An example of lexlcographlc ordering is pïesented in Figure 4.2.1. There
are two objective nctlons of whlch the first is the most important. After
mlnimizlng the first objective, there are two points left and after mlnimizlng
the second objective, the point z tis obtalned as the final solution. The bold
llne represents the Pareto optimal set in the figatre. This example is somewhat
too positive since all the objective functions have their effect on the solution
process.
4.2. Le)dcogaphic Ordering 119
z2
Figure 4.2.1. Lexlcographic ordering.
Let the objective functions be arranged accordlng to the lexácogr aphic order
lex minimizc fl(x),f2(x),...f(x)
(4.2.1)
Theorem 4.2.1. The solution of lexicographic problem (4.2.1) is Pareto op
ProoL Let x*  S be a solution of the lexicogaphic problem. Let us assume
that it is not Pareto optbnal. In this case, there exists a point x  S such that
)¢(x) < f(x*) for ail i = 1, ..., k mad for at least one j the inequality is strict,
tht is, f() < f(*).
Let bei -- 1. tom the definitlon of lexlcographlc ordering we know tiret ft
attains its mlnlmmm at x*. Since also ri (x) <_ ft(x*), it is only possihle that
f, () - f, (×').
There are two possibilltles in determlnlng the lexlcogaphlc optlmum. Eit her
a unique solution is found during Che optimization process, or optlmizations are
performed for eveïy i -- l,..., k. In he latter case, where i -- 2, we also have
f2(x) -- f2(x*) and with similar reasonlng we hve that f(x) -- f(x*) for every
i -- 1,..., k. This contradlcts the assumption of at least one strict inequality.
Thus, x* is Pareto optimal.
Ou the otheï lmnd, if lexicogïaphic ordering stops before every objective
fonction has been examined, this memas that a unique solution x* has been

obtalned for ¢. The assumption
which is a contradiction. Thus, x* is Pareto optimal. []
4.2.2. Comments
Numerlcal application exmnples of the method are glven in Hwang and
Masud (1979, pp. 4955). In Ben-Tal (1980), Pareto and lexicographlc optlma
are characterized in convex problems. Duallty theoïy for convex prohlexs with
the help of lexicographic ordering is developed in Martlnez Legaz (1988).
A modification of lexicographic ordering called lfierarchical optlmization,
is applled to a vehicle design prohlem of mechanical engneerlng in Bestlc and
Eherhard 0997). In hierarchical optimizatlon the tpper hounds ohtalned when
minknlzlng more important ohjectlve functlons are relaxed hy so-called wors-
enlng factors. These factors are specified hy the decisinn maker.
Lexicographic oidering corresponds to the weightlng method when the
wei#tting coeflïcients are of very different magultude. The question whether
there exist weightlng vectoïs such that the optimal solution of the weighting
method is identical to the soMtlon ohtalned hy lexlcographic ordeïing is con-
sidered in Sherali (1982) and Shera[i and Soyster (1983). Tloe answeï is positive
for linear prohlems and several discrete problems. In practlce, this means that
the prohlem of lexicographic ordering can he solved as a welghting prohlem
with standard optlmlzeïs.
The notion ahsolute importance of ohjectlve ftmctlons is dlscussed in
and Mousseau (1990). Ry and Mousseau also consldeï under what khd of
conditions one tan say that one ohjectlve functinn is more important than
another.
4.2.3. Concludlng Remarks
The justification for using lexicographic ordeïlng is its simplicity and tho
fact that people usually make decisions successlvely. However, this method bas
several drawhacks. The decision makcr may bave diflïculties in putting the
ohjective functions into an ahsolute order of impoïtance. On the other hand,
the method is usua[ly rohust. It fs very llkely tiret the less important objective
functions are not taken into consideïation at Ml. If the most important objective
function bas a unique solution, the otheï ohjectives do hot bave any influence
on the solution. And even if the most important ohjective had alternative
optima and it was possihle to use the second most important ohjectlve, it is
veïy unlikely that this prohlem would bave alternative optima, and the third
or other less importatt objectives could he used.
Note that lexicographic ordering does hot allow a small increment of an
important ohjectlve function to he traded off with a gïeat decrement of a less
important ohjective functlon. Yet, this ldnd of trading might often he appeuling
to the decision maker.
Lexicographic ordering muy be used as a part of the following solution
method, called goal prograraming.
4.3. Goal Programming
The ideas of goal pïogïamming were origlnally introduced itx Charnes et
al. (1955), but the term goal prograraming was fixed in Charnes and Cooper
(1961). It is one of the first methods expressly created for multlohjective op-
tlmizatlon. Amottg more recent papers, an easy-to-understand presentatlon of
goal prograraming is given in Ignizio (1983a, 1985). Goal prograraming was
origlnally developed for MOLP pïohlems, and this hackground is very evident
in the formulation.
4.3.1. Iutroduction
The hasic idea in goal programmitLg is that the decisinn maker specifies
(optimistic) aspiration levels fur the ohjective functions and any devlatlons
from these aspiration levels are minimized. An ohjectlve function jolntly wlth
an aspiration level forms a goal. Wc can say that, for exmnpl, minirdzing the
price of a prodttct is an ohjective function, httt if we want the price to he less
tlan 500 dollars, it is a goal (and if the price must he less than 500 dollars, it
is a constraint). We denote the aspiration level of the ohjective function , hy
For minimization prohlems, goals are of the form ri(x) _< z (and of the
fOrm ff(x) > zl for maximization problenls). Goals may also he repïesented
as equalities or ranges (for tloe latter, s« Charnes and Coopeï (1977)). The
aspiration leve/s are asmtmcd tobe selected so that they are hot achievahle
simult aneously.
It is worth noticing that the goals are of the s&rne form as the constïaints
of the prohlem. This is why the constralnts may he regaïded as a suhset of the
goals. This way of formulating the pïohlem is culled generahzed goal program-
ming. In this case, the goals can he though of as helng divlded into flexible and
inflexible goals, where the constraints are the inflexihle (oï rigid) ones. More
detailçd presenbations and practical applications of geneïalized goal prograra-
ming are gven, for example, in Igaizio (1983a) and Koïhonen (1991a). See also
Section 5.10.
After the aspiration levels bave heen specifled, the following task is to mha-
imize the under- and overachievements of the ohjective function values with
respect to the aspiration levels. It is su6ïcient to study the devlational vari-
ahles 5 -- zi - ,(x). The devlational variahle 5 may bave positive or negative
values, depending on the prohlem. We can present it as the difference of two
positive variahles, that is, 5i = 5ï 5 +. We can now investigate how we[I each
of the aspiïation levels is attalned hy studying the deviational variahles. We

can write Si(x) + 8- - 8, + - i for ail i = 1,..., k, where 8 is a negative devia-
tion or underachievement and 8i + is a positive deviaion or overachievement in
We now bave the multiobjective optimization problem in a form where
we cm minimize the devintional varinbles. ['or minlmlzation problems itis
sufficient to minimize the k varinbles 8 +. If the ith goal is in the form of m
equahty, we minimize 5 i + 5.
4.3.2. Different Approaches
Thus far, we have only formulated the multiohjective optimlzatinn problem
in an eqtúvalent foïm, where we have deviational variables as the objective
functions. There are several ways to proceed from this point. Here we present a
weighted (also called Archimedim) and a lexlcograpldc (also called preemptlve)
approach. More methods are hmdled in Ignizio (1983a) md some formulations
are expfored in de Kluyvr (1979).
In the weighted approach, see Charnes md Cooper (1977), the weighted
sum of the deviatinnal variables is minlmized. This means that ha addition to
the aspiration levels, the decision makr must specify information about the
importmce of attaining the aspiration levels in the form of weightlng coeffi-
cients The weoEhtlng coefficlents are assumed to be positive md sum up to
one. The bigger the weighting coefficient is, the more importmt is the attain-
ment of that aspiration level. (Sometimes negatlve welghting coefficients are
used to represent a premium instead of a penalty.)
To put the introduction presented above into mathematlcal form md to
reason ahout the usage of the deviation varlables, we can say that the problcm
miniralzc  wi[L(x)
(43A)
subject fo x @ S
is converted into a new foïm by adding the overachlevement variahles
6i = maxi0, Si(x) - zl] or 61 TM [1 - S,(×)I + S,(,q - ]
and underachievement variables
7=max[0,5 Si(x)] or 6;-=[Iz, S,(x)l+z
This mems that the absolute value signs cm he 4ropped from problem (4.3,1)
hy introducing the underachievement md the overachievement varlables. The
resulthag weighted oal pvgramming problem is
Figure 4.3.1. Contours with different weighting vectors.
Even though the constraints 5- " Ç = 0 for ail i = 1, .., k are hot usually
included in the problem formulations, some attention must be pald to guar

124 Px Il -- 4. A Priori Methods
antee that they are alid (sec detal[s in Rosenthal (1983)). An example of the
required conditions is given in Sawar agi et al. (1985 p. 253). The weitted goal
programming problem my be solved by standard sigle objective optimlzatioa
methods, ff the original problem is bnear then the corresponding weighted goal
progamung problem is also linear. The close connection betweea goal pro-
grammlng and MOLP problems explains why the above mentioned constraint
is tmua[ly absent from the problem formulation (it would ma]ce the problem
Note that weighted goal programming is closely related to the method of
weighted metrics or compromise programming. This can be seen particularly
we]l in formulation (4.3.I). Instead of the ideal objective vector, the ïeference
point of the declslon maker is sed in goal programmlng. The distances can be
measured by metrics other than the L-metric. The L1 metric is widely nsed in
connection with goal programmbg because of the origin of the method i linear
programming. (This metric maintains the ]inearlty of the problem.) If some
other L-metric is used thee is another problem in determlning an approprlate
value for p. Note, however, that if we have appropriate sdivers availahle, we can
solve problem (4.3.1) directly withont auy devlatlonal variables and uslng auy
metrlc.
In the lexicographie approach, the decislo maker must speclfy a lexico-
graphie order for the goals in addition to the aspiration leve[s. The goal at the
ldgIœest priority level is suppesed to be infiditely more important than thc goal
at the second priority level, etc. Thls means that no matteï how large  mul-
tiplier is selected  lower priority goal multiplled by it can never be made as
important as a hiter priority goal. After the lexicographlc ordering, the prob-
lem wltb the devlatlotl variables as objective futmtions and the constraints as
in (4.3.2) is solved as explained in Section 4.2. I order tobe able to use the
lexicogr phlc approach, the decision maker's preference order for the objectives
lntt be definite and rigid.
A combination of the weighted mad the lexicogphic approaches, to be
called a combined approach, is quite popular. In this case, seeral objective
functions may belong to the same class of importance in the lexicogïaphic
order. I each prlorlty class, a weighted sure of the deviatlonal variables is
minlmlzed. The same weakesses pïesented in commctioa with lexicographic
orderltg are valid for this and the lexicographlc approach.
It is trot necessary to include the original corstraints (x  S) in the lex-
icographic optidiztlon problem i the normal way. They can be consideïed
to belong to the first prlority level. I this way, they are taken into account
before any objective function is optimlzed and the feaslbility of the solutions
is guaranteed by the nature of the lexicographie ordering.
Next, we prove a result concerning the Pareto optimality of the solutios of
goal programmbg.
Theoren 4.3.1. The solntinn of a weighted or a lexlcogaphic goal progam
ming pïoblem is Pareto optimal if elther the aspiration levels forma Pareto
optimal reference point or ail the devintional varlables (/+ for fnctions to be
minimized and (il for functions to be maximized have positive values at the
optimum.
Proof. For the lexicographic approach the proof corresponds to that of The-
oïem 4.2.1. Heïe we only prescrit a proof for tc welghted approach. For sim-
p[icity of notation, we assume that the problem is of the form (4.3.3). A more
general case is straightforward.
Let x*  S be a solution of the weighted goal progamming problem, wheïe
the deviationdi variables (demted here for clarlty by (fî) are positive. Let us
assume that x* is hot Pareto optimal. In this case, theïe exists a vector x °
such that fi(x °) < f(x*) foï ail i -- 1,..., k md fj(x °) < fj(x*) foï at least
one index j.
We denote fj(x*) - f(x °) --  > 0. Then we set ci ï -- (f* > 0 for i  j and
 -- max [0, 5î ] OE 0, where (f is the devlatlonal variable corresponding to
We have now fi(x °) ci ï < f,(x*) - (f _ zi for ail
then fj(x°) --5ï -- f 3(x°) ; + fj(x*) - f(x°) <_ z3, and lf  Z<_0, then
This mems that x ° satisfies the constralns of pïoblem (4.3.3). We have
(f < (f (this is also valid if (f -- 0 since (f* > 0 for ail i), md ci ï _ (fî for ail
i ¢ j. As the weighting coeflïcients arc positive, we have
which contradicts the fact that x* is a solution of weighted goal pïogrammlng
problem (4.3.3).
For aspiration leve[s formig a Pareto optimal point the pïoof is se]f-evident.
pggramming (suggested in Flave/I (1976)). It is hot as widely used as the
minlmize max (f +
(4.3.4) subject to f(x) - 5 + < 21 for ail i = 1,..., k,

thal (1983), that weighted goal programmlng problem (4.3.2) is equivalent to
OU(f (x)) = [ w ï if f,(x) < 21,
f t. -w if f(x) > ,
substitution in goal programming problems. In lemark 2.8.7 of Part Iit
was mentioned that the marginal rates of substitution may be deflned as
mi(x) = 1 Thus goal programming does hot take into con-
sideratloa the possibillty that it is easler for the declslon maker to let some-
thlng increase a IJttle if (s)he ms got llttle of it than if (s)he has got much of
it. The reason for this is that goal programming impllcltly assumes that the
to the lexicographlc approach (see dctai/s in Rosenthal (1983, 1985)). More
cïitlcal observations about goal progranamlng are presented in Romero (1991)
and Rosenthal (1983).
4.3.4. Applications and Exte]slons
glven in Ignlzlo (1976), mad a stmamary of different var iatlons of goal program-
ming is provided in Chanes mad Cooper (1977). In addltlol L a wide survey of
thc literatuïe around goal programling up to the year 1983 is presented in
Soyibo (1985). Several modifications mad improvements as well as applications
are revlewed. A survey of goal programmlng is also given in Kornbluth (1973)
and the welgated and tbe lexicographlc approaches are applied to problems
wlth fractlonal objective fuactinns, lrthcr, a broad collection of journal pa-
pers and books on goal programming is aSSembled in Schniederjmas (1995a).
Reforences in nine broad areas of applfoatloa are also included.
in terres of practical applications. Welghted goal progranwalng with equal
welghtlng coeflàcients is employed in the plarmlng of public works in Yoshlkawa
et al. (1982). Weighted goal programming wlth sensltivity analysls is atso used
for portfotfo se/ectlon in TanUz et al. (1996).
Lexicographic goal programming is applied in Benlt-Afonso mad Devau
(1981) toa prohlem conceraing the location and slzc of day nurseries, in Sinha
et al. (1988) to storage problems in agriculture and in Mitra and Patankar
(1990) to nid manufacturers in selectlng the prlce mad the warrmaty tlme of
thelr products. Lexlcographic goal progrmnming is also applied in Kumar et
al. (1991) to nonllnear lnu]tstage decision problems in mauufacturing systems,
in Ng (1992) to alr cïaft loadJng and in Brauer and NaadJmut hu (1992) to solve a
mined integor MOLP problem involving inventory mad distñbutlon plmming. In
Hemalda and Kwak (1994)  llnear trmas-shlpment problem and in Current and
Storbeck (1994) a location model are solved by lexicogïaphlc goal programmlng,
mad in Gimmikos et al. (1995) it is applled in ma integer allocation problem, h
Berbel mad Zamora (1996) lexlcographic goal programmlng is appled in wddl] le
management and in Kim et al. (1997) in solvlng a/inear problem of milJtary
budget plmaning. An implementing declsinn support system is also descrlbed.
Hwmag and Masud (1979, pp. 7995). Combined goal programllng is applied

in Levary (1986) to problems of optimal control, in Giokas and Vassiloglou
(1991) to the (lilear) management of the asets and llabilities of a Greek bamk
and in Ghosh et al. (1992) to the ïesource planning of universlty management.
In Sankaïan (1990), the combined approach is also used to solve an integer
MOLP problem in cell formation, and in Schniederjans and Hoffman (1992),
national busiless expansion analysis. Th ideas of comblned goal progïamming
are adapted in Miyaji et al. (1988) in solvilg a transportation problem type
problem of dvidng students into groups. In addition, the combined goal pro-
grmaalng approach is applied in fund and port folio management in Powell and
Premachandra (1998).
The appllcatioIs mentioned here are only a few of the exlsting ones. The
popularlty of goal pïogrammlng is well afiïrmed by the fact that in a bibli-
ography collected in Whlte (1990) on multiobjective optimlzation applications
(coverlng the years from 1955 to 1986) more than a hall involved goal program
Four different goal interpretatlons in multiobjective optimization are pre I
seited in Dinkelbach (1980). Goal progïamming is adapted to multlobjective
generallzed ntworks for integer pïoblems in Ignizio (1983b). In hmlguchi and
Kmme (1991), goal programmiig is extended to llnear problems where the co-
efiïclents and the aspiratio1 levels are glven as intervals. The aspiration level
isfied, but regions where the aspiration levels may vary. A generallzatlon of
Thore et al. (1992).
An extension of goal programming to MOLP problems is gvetl in Martel
and Aoudi (1990). [astead of the deviatlonal variables, some functions descrlb-
ing the wlshes of the decision maker about attalrdng the goals set are used
in the weigbted approach. )al illustrative example is also provlded. Technlcal
presented in Tamlz and Jones (1995). Thls approach is exended in Martel and
Aouni (1998) by alfowlng goals to be intervals instead of exact numbers. Thls
means that indiffeïence thresholds (see Subsectlon 5.9.1) are used in modelllng
the impreclslon of the goals. (Even though we have mentloned some interestlng
MOLP exensions and solution methods, we skip most of them here.)
An adaptation of lexicographlc goal programmlng for cotvex problems is
provlded in Caballeïo et al. (1996). The idea is to produce satisfying solutions
by solving the hybrld problem (in Section 3.3) with the components of the goal
solutions.
Lexácogïaphlc goal programming is modfied significantly in Caballeïo et
al. (1997). No devlational varlables are used and the objective function of each
priorlty level is optimized at each iteratlon. The approach is valid for convex
pïoblems.
A sollltion method for lexicograpllic goal programrlfilg problems where ob-
jective functions are fractions of llnoar or nonlinear functinns is described in
Pal and Basu (1995). More than one objective function can then befong to the
same priority class. The method has clmracteristics of dynardic programming.
A generalized reduced gradient (GRG) method-based solution algorithm for
lexicogrphlc and welghted uonllnear goal programmlng problems is introduced
in Saber and ttavlndran (1996). This partltionlng techráque is demonstrated
to be rellable and robust. Several aspects to take into accourir when aimlng at
the efiïclent hnplementatlon of goal programming approaches are collected in
Tamiz and Jones (1996).
Goal programming can be expanded in an interactive direction in different
ways. One can systematlcally modffy the welghtlng vectors or the lexicographic
order ofthe objective functions or ask for new aspiration levels from the declslon
maker. These topics are coasidered in amiz and Jones (1997a, b).
4.3.5. Concludlng Remarks
Goal progroelming is a very wldely uscd and popular solutioll method for
practlcal multlobjective optimizatlon problems. One of the reasons fs its age.
Another reason is that goal-setting is an understandable and easy way of mak-
ing decisions. The specification oftbe welghting coefiïcients or the lexicographlc
ordaring may be more difiïcult. The weights do hot have so direct an effect on
the solution obtalned as in the a priori welghting method. However, they are
relative to each other. This means that only the relations of the weighting co-
efficient marrer, hot the weights themselves. It may be dfiïcult to specify the
weights because they have no direct pbyslcal meanlng. It is demonstïated in
Nalyama (1995) that desirable solutions are very dfiïcult to obtain by ad
justing the wclghting coefiïcients in the weighted goal programmlng problem.
Aayway, it is as advlsable as in the welghtlng method to nor mallze the objective
functions when weighting coefiïcients are used.
One must be care5al wlth the selectlon of the aspiration levels so that the
Pareto optlmality of the solutions can be guaranteed. The correct selection may
be difiïcult for a decision maker who does hot kuow what the feasible reglon
looks like. Presenting the ranges of the Pareto optimal set, or at least the ideal
objective vector, to the decision maker may help in the selection.
Goal programming is hot an appropïlate method to use ff it is desired to
obtain trade-offs. Another restricting property is the underlying assumption of
a piecewise ]inear value flmction and thus plecewlse constant margálal ïates of
substitution.
Assuming that goal programmlng follows a traditlonal product llfe cycle,
it is inferïed in Schnlederjans (1995b) that the current stage of pïoductivity
is in decllne. It is polnted out that the number of goal programming papers
has been on the decrease for several years. One of the reasons suggested is the
aging of the few active contrlbutors to goal prograraming.

5. INTERACTIVE METHODS
The class of inteïactive methods is the most developed of the four classes
of methods presented here. The interest dcvoted to this class cm be explalned
by the fact that assumlng the decision maker has elough time td capabilities
for copeïatlon interactive methods cm be presumed to pïoduce the most
satisfactoïy ïesults. Mauy of the weak points of the methods in the other tin'ee
classes are oveïcomc. Namely only part of the Pareto optimal points has tobe
getcrated md evaluated, and the decisinn makor can specffy nd correct heï
or his preferences md selections as the sointion pïocess continues and (s)he
gets to know the pïoblcm md its potentialitlcs betteï. This a]so means that
the decision tnaker does hot have t(» know my global prefereuce structure. In
additloi, the declslon makor cm be msmed to have more coufldence it the
final solution slnce (s)he is invalved thïoughout the solution process.
In interactlve methods, thc decision maker woïks together with m malyst
or m iatcractive computer progïam. O1m cm say that the analyst tïles to de-
termine the prefeïece structure of the decision maker in m interactive way.
A solutio patteïn is formed and ïepeated scveral tlmes. After cveïy iteïation,
some information is glven to the declsinn maker and (s)he is asked to mswer
some questions or provide some other type of information. The working oï-
dcr in these methods is: 1) malyst, 2) dcclsion maker, 3) malyst, 4) decision
maker, etc. After a reasonable (finite) nunlber ()f iteratios evcry interactive
method should )eld a solution that the decisinn makeï can be satsfied with
and convinced that no considerably betteï solution exists. The l)msic steps in
interactlve algorithms cm be expressed as
a) find m initial feasible solution
b) intcract with the decision maker,
c) obtain a new solution (or a set of ew solutions). If the ew solution
(or ote of them) or one of the pïevious solutions is acceptable to the
decision makeï, stop. Otherwise go to step b).
lztteïactive methods differ from each other by the form in whlch information
is giveI to the decision maker, by the form in which information is provlded by
the decision maker, and how the problem is transformed into a single objective
optimization problem. One problem to be solved when deslgning m interactive
method is what klnd of data one should use to interact wlth the decislon maker.
It should be mcaningul md easy for the decision maker fo comprchend. The

decision maker should understand the meanlng of the parameters for whlch
(s)he is asked to snpp]y values. On the other hmd, the data provided to the
decislou maker should be easily obtainable by the malyst md contain infor-
mation about the system. Too much information should hot be used and the
information obtalned from the decision maker should be uti[zed eflïciently. To
ensure that the greatest possible benefit cm be obtained from the interactlve
method, the decislon maker must find the method worthwhile and acceptable,
md (s)he must be able to use it properly. Thls usually mems that the method
must be understmdable aId sufficiently easy to use. Thls alm calls for research
in understandlng the under[yla 6 decislon processes md how declslons are made.
As stressed in Kok (1986), experlmeats in psychology indlcate that the
amouat of information pïovlded to the declsinn maker has a crucial role. If more
information is glven to tbe decision maker, the percentage of the information
used decreases. In other words, more information is hot necessarily better than
less informatinn. More information may increase the confidence of the decislon
makor in the solution obtained but the quality of the solution may nonetheless
be woïse.
In addition to the fact that the decisloa makor has an essentlal role in
interactlve methods, the analyst should aot be forgotten either. The analyst
can support the declsio maker in many ways md, in the best possible case,
explaln the behavinUr ofthe problem to the decision makor. Thus, the malyst
may play a memfingful foie in Che learnlng pïocess of the decisinn maker.
Interactive methods have been clasificd in many ways, mainly according to
thelï solution appr oaches. Here we do hot follow auy of tlose classhïcatinns. Let
us, howevei, mention two different conceptions re6arding nteractlve appïoaches
according to Vmderpooten (1989a, b, 1992). The approaches are searching
md learnlng. In searching orentcd methods a cohverging sequence of solution
proposais is presented to the decisicn maker. It is asumed that the decisinn
maker pmvides consistent prefereace information, h leaïng-oriened methods
a free expIoratina of alternatives is possible alfowing trial md eïroï. The latter
does aot guide the declslon maker md convergence is hot guaranteed. The best
procedure would be a combination of these two approaches, dïawing on their
pcsitlve features. Such an approach would support the learnlng of preforeaces,
while it would also include guldlng propertles.
Before we preset any methods some critlcal coments are in order. Re-
peatedly, it has been and will be assmned that the decislon makeï makes consls
tent decislons or that (s/he has m underlying (tmpllcltly known) value functio
upon whlch her or his decisinns are made. The purpose is nottc go deeply into
the tbeories of decislon making. However it is worth mentinning that those
asSUmptlons can be called into ques¢ion because they are diflïcult to veriy.
Conslstency of the responses of the declslon maker is one of the most im-
portrait factors guarmteeing the success of mauy interactlve solution methods.
Because of the sabjectivity of the decisioa makers, different startlng points,
different tpes of questions or inteïactlon styles may lead to different final so-
lutions. Some methods axe more sensitive with respect to conslstency than oth-
ers. The hmdling of inconsistency wlth respect to several interactive methods is
treated in Shin md Ravindrm (1991). In general, inconsistency cm be reduced
by consistency tests durlng the solution process or by mlnlmizing the deckslon
maker's cognitlve burden. In other words, interactlve methods assuming con-
sistent maSwers shofld have built-in mechmfisms to deal with inconsistencies.
This is one of the motivations in deveIoplng new metods for multinbjectlve
lenlus (1996, 1997). Korhonen and Wallenius also hmdle several behavlouïal
Coroer (1995) that the methods should become more user-frlendly md descrip-
performed (most prohably) by a speclalized part of the hlman mind." To pro-
Declslon making is apposltely descrlbed i Zeleny (1989) as "searchln 6 for
set). hstead, the dcclslon maker shouId be gulded throu her or hls own

Decision analysis is hOt handled here in more detail. The above-mentinned
aspects are only a fêw examples of the issues involved.
One con underst and several different features as convergence. On the one hmad,
nal solution cma be proved tobe Pareto optimal. One cma also say that the
method converges into a satisficing solution, if the final solution is satisficing.
Finding the best Pareto optimal compromise solution may be understood as
is optimal for ma underlying value function. TlliS kind of mathematfoa conver-
tion. In this case, the observations ment folxed above are valid. If the method is
not based on the assumption on any underlying value fltnctinn, this conception
different methods under consideration. It cma also be claimed that mathemat
lidity ofa method, as stated for example, in Stewart 0997). Tbe saine idea is
also expressed in Gardiner and Vanderpooten (1997) and Zionts (1997a, b). On
interactive procedures should converge we[l inmledintely in the few initial iter-
ations. This is concluded, for example, in Korhonen et al. (1990) after experi-
for progress for a lotlg rime.
Stopping criteria are related to the convergence of interactine methods.
There are three mahl stopping criteria. Either the decisinn maker gets tired of
the solution process, some algorishmàc stopping (convergence) rulc is fulfilled or
the decisinn maker finds a desirable solution mad wants to stop. Itis difcult to
define precisely when a solution is desirable enough to become a final solution.
The convergence of the method has sometimes been considered tobe ma
and Vincke (1989), the solution process should hot be stopped because of any
the decision maker with the solutinn obtalned. This usually means that the
decision maker must feel that (s)he has received enough information about the
problem tobe solved.
The current view is that a solution is a final solution if the decision maker
is convinced that itis preferred to all the other Pareto optimal solutions (see
Korhonen mad Walleinus (1996 1997)). This means that the decision maker
Gardhier and Vanderpooten (1997)). Gardiner mad Vanderpooten have studied
it is due to some other reason. Possibly the decision makers did hot know how
An importmat factor when using interactive solution methods is the selection
of the stalting point. Particularly for nonconvex problems where the objective
desires mad the preforences of the decisfon maker. It is hot desirable that the
final solution is affected by the starting point. In general, tbe starting point
should provide a useful basis for the decision maker in exploring the Pareto
optimal set. The starting point cal, for example, be generated by some of the
points from the point of view of human judgment and decision making is the
sion maker fixes her or iris thinking on some (possible irrelevant) information,
fike the stalting point, mad fails to sufclently adjust and more away from
that anchor. In other words, the decisinn maker is unable to move far from
the stalting point. This kind of behavfoural perspective on interactlve decision
making is handled in Buchanan and Corner (1997). On the basis of a number
Buchanan and Corner conclude that whenever ma maChOring bias is possible, it
is important that the starting poiut refiects the initial preferences of the deci-
sion makcr. The reasoning is that since ally starting point is likely to bias the
decision makeï, it is best to bias her oï him in the right direction.
Evett though interactive methods con be regarded as most promising so-
lution methods for multiobjcctive optimization problems, there are still cases
cision makcr. Such problems include, for instance, mauy eIgineering probles
approximatinns). One must, however, remember that computatfonal faciIities
have devefoped greatly during the last few years. Thus, the number of problems
that cannot be solved by interactive methods has decreased. See Osyczka mad
Zajac (1990) foï a suggestion of handling computationalIy expensive functions.
On the other hmad, the ]arg number of objective functions may make inteïac-
tire methods impractical. In this case, it may be difiïcult for the decision maker
direct the solution process.
worth trade-off method, the Geoffrinn-Dyer-Feinberg method, the sequential

136 Pm II -- 5. lateractive Methods
other metfinds.)
AIl the methods to be presented are based on generatlng mainly weakly,
5.1. Interactive Surrogate Worth Trade-Off Method
The interactlve surrogate worth trade-off (ISWT) method put forward in
Chankong md Halmes (1978, 1983b pp. 371 379/, is an extension of the surro
gare worth trade-off (SWT / method preseated in Halmes md Hall (1974)
Halmes et al. (1975). We do not go into detalls ofthe SWT metfind here, but
preset direct]y the interactive versiom The motivation for cluding the ISWT
method in this book is that it is of theoretical interest.
5.1.1. Introduction
metfind are Pareto optimal (see Section 3.2).
It is assumed that
1. The underlying value fonctioI1 U: R k - R exists and is implicltly
to the declsion maker, la addition, U is contlnuously differentiable
2. The objective md the constraint functions are twlcc contiauously differ
entlable.
3. The feasible reglon Sis compact (so that a finlte solution exlsts for eveïy
fcasible ¢-constralnt problem).
4. The assumptions in Theoem 3.2.13 are satisfied.
5.1.2, ISWT Algorithm
The main features of the ISWT method cm be presented ctrsorily wlth
four steps.
(1) Select the ïcfeïence mctlon f to be mlninized and give upper bounds
to the other objective functlons. Set h - 1.
(2) Solve the curreioEt -constralnt problcm to get a Pareto optimal solution
x h. Trade-off rate [nformatlon is obtalaed fom the connected Karush-
Kuhmcker multip[iers.
(3) Ask the opinions of the decision maker with respect to the trade-off
rates at z a correspondlng tox a.
(4) If some stopping cïlterlon is satisfied, stop wlth x h as the final solu
tion. Otheïwlse update the upper bounds of the objective fonctions
with the help of the answers obtaiaed in step (3) md solve several
constraint problems (to determirie m appïopriate step-size). Let the
decision maker choose the most prefcrred alteroEative. Denote the cor
ïesponding declslon vector by x h+ and sct h -- h + 1. Go to step (3).
First, we exmanne how trade off rate infoïmatiot is obtained from Karush-
Kuhn-cker multipllers. As [oted la Theorem 3.2.13 of Section 3.2, the
Karush-Kuhn-cker multlpliers ïepresent tïade-off rates under the specified
wheïe fiis the fonction to be mi[llmized md the uppeï bounds are Qh for
Theorem 3.2.13. If the Kaïush-Kumckeï multlplieïs Aî assoclated wlth thc
constralnts f:(x) _< gh are strictly positive for ail i

We know now that to move from x h to some other (locly) pareto optimal
solution in the neighbourhood of x h, the value of the function ft decreazes by
Aîi traits for every unit of increment in the value of the function f, (or vice
versa), while the values of ail the other objective functions remaln unaltered.
The opinion of the decision maker with regard to this kind of trade-off rate for
ail i -- 1,... ,k, i  g, is found out by posing the following question.
Let an objective vector f(x h) -- z h be given. If the value of ftis decrcased
by Aî, units, then the value of f, ïs incased by one unit (or vïce versa)
and the other objective values remain unaltered. How deôirable do you find
(é'î,... ,«î_l,é'î+,,... ,e h + ¢,... ,Eî.), WllOre 6  0 is & scalar wlth a S[lall
0() ..((i)__) *.
In qkrvalnen (1984), itis suggested that far fewer choices are given to the
Regardless of the scale selocted, the response of tbe decision makeï is called
a surrogate wo*¢h of the tïde-off rate between/'t and/' at x  and denoteà by
W. At each point x a, a number of k - 1 (or less, if N- ; ) questions of the
pïeviously descïibeà form are presented to the decision makeï and the values
for W]', (i -- 1,..., k, i  g) are obtalned.
a way to update the uppeï bonnds of the objective functio in an appropriate
How to proceed from thls point depcnds ou the seule chosen for the surrogate
It is asumcd to satisfy thc preferences of tbe decislon mak«r indicated by the
[I tbe original version by Chankong mad Haimes, it is snggested that the
npper bounoE are npdated from iteïation h to h + 1 by
for i  N > and
Ci = Ej + t Çfj(xh) T WIfi(x h)
for j  N , where i  N > mad tis a step-size to be determined. For detalls
ste Chmakong eald Haimes (1978) mari ïeferences therein.
Foï simplicity it is assmned in Taïvalnen (1984) that the Kaïusb Kuhn
Tucker multiplieïs are ail strictly positive. The decision mal<er is askcd to spec-
ify small mari meanñgul arounts A/" i for ail i -- 1,..., k, i ¢ g. Tbc scalar
A/' represents the arount of chmage in the vahm of/', that is relevant to the
decisioa mal<er. Tbe uppeï bounds are now updated by
e, +' -- , +
for i - 1..., k, i ; g, where t denotes the step-slze.

5.1.3. Comment s
In pïactice, when the decision maloer is asked to express ber or his prefer-
ences concerning the trade-off rates, (s)he is implicitly asked to compare the
trade off rates with her or his marginal rates of substitution. (Naturally, the
decision maker does not hure to be ble to specify the margiial rates of sub-
stitutlon explicitly.) If me < ),i, then the surrogate worth value is positive
(and the contïaïy ïespectively). If mt - ),tï for ail i -- l,..., k, i  £, meaning
W«{ -- 0, then the stopping cïiterion (2.8.1) introduced in Subsection 2.8.2 of
pat Iis valid. Thus, the condition W -- 0 for ail {  £ is a common stopplng
criterion for the algorlthm. Another possible stopping situation is tht tlle de-
cision maker wants to proceed, but only in an infeaslble direction. The latter
condition is more difficult to check.
The ISWT method tan be classified as non ad hoc in nature If the value
faction is known, then the tïade-off rates are easy to compare with the
marginal rtes of substitution. [hrther, when comparing alternatives, itis easy
to select the one wlth the highest value flmction value.
The convergence rte ofthe ISWT method greatly depends on the accuracy
and the consistency of the answers of the decision maker. It was pointed out in
Section 2.8 of Pat I that it is important to select the ïeference function care-
flflly. Thls coruncnt is also valid when coasiderbg the convergence propeïties.
If theïe i.s a sbarp limit in the values of the refcrence function where there is a
change in stisfaction from vcry satisfactory » to very unsatisfactory,' the so-
lution procedure may stop too early. [hrther references are cited in Chnkong
and Haimes (1978) for convergence ïesults.
A method related to the ISWT method is presented in Chen and Wang
(1984). The method is an interctive versioi of the SWT method, where new
solution alternatives are generated by Lin's proper equality method (see Section
3.2), and the decisinn maker has to specify odiy the sign ofthe surrogate worth
5.2. Geotfrion-Dyer-Feinberg Method 141
5.1.4. Conclud[ng Remarks
The role of the decision maker is quite easy to understand in thc ISWT
method. (S)he is provided with one solution and has to specify the surrogte
woïth values. The complicatedness of glvbg the answeïs depends
perienced the decision maker is in such specification and which varitin of the
method is employed. The set of 21 different alternatives as surrogte worth
values in the origiial version is quite  lot to select fl'om. It may be difficult for
the decision maker to provide consistent aaswers througtout the decision pro-
cess. h addition, if theïe is a large number of objective tlnctions, the decision
maloer has to specffy a lot of surrogte worth values at each iteïation. At least
for some decislon makers it may be easier to mintaln consistency when there
are feweï alteritive values for the surrogate worth avallable (as suggestcd by
Tar vahmn (I984)).
Tzade-off rates play an important ïole in the ISWT method, nd tht is
why the decision maker has to understand the concept of trade off properly.
Attention must diso be pid to the ease of understaadlng and carefll formul
tion of the questions concerning the trade-off rates. Careless formulatioI may,
for exaruple, cause the sign of the surrogte worh value tobe changed.
It is a viïtue tht ail the alternatives during the solution process are Pareto
optimal. Thus, the decision maker is hot botheïed wlth any other kind of soin
5.2. Geoffrion-Dyer-Feinberg Method
The Geoffrion Dyer-Feinbcrg (GDF) method, proposed h Geoffxion et
al. (1972), is an interactive method based i principle on the same idea as
the ISWT method; maximization of the underlying (impllcitly known) value
flmctioa. The realisation is cluite dfferent, though. The GDF method is one of
the most well known interctive methods.
5.2.1. Introduction
The basic idea behind the GDF and the ISWT methods is the sarue. At
ech iteration, a local approxlation of an underlybg value function is gener-
ated and maximized. In the GDF method, the idc is somewhat more clearly
visible. Marginal rates of substitution specified hy the decision maker are used

to approximate the direction of steepest ascent of the value function. Then the
value function is maximized by a gradient-baed method. A g'adlent method
of Frmk md Wolfe (FW) (see Frmk and Wolfe (1956)) has been selected for
optimization because of its simphcity and robust convergence (rapid initial con-
vergence) properties. The GDF method is also sometimes ca[led m interaCtive
Frmk-Wolfe method, because it has been constructed on tbe basis of the FW
method.
The problem to be solved here is
maximize u(x) = U(f(x))
(5.2.1)
subject to x  S.
It is assumed that
1. The underlying value function U : R k  R exists md is implicitly lown
to the decision maker. In addition, u : R'*  Ris a coatinuously differen-
tiable and concave function on S (sufficient conditions for the coacavity
are, for example, that U is a concave decreasing function md the objec-
tive lctions are convex; or U is concave tard the objective functions
are linear), nd U is strongly decreaslng with respect to the ïeference
fuaction (denoted here by ft) so that  < 0.
2. The objective functions are continuously OEerentible.
3. The feasible ïeffion Sis compact and convex.
Let as begin by presenting tbe main principles of the FW method. Let a
point x h  S be given. The idea of the FW method is that when maximizing
some objective function u : R   R subject to constraints x  S, instead of u,
a linear approximation of it at some oint x a  Sis optimlzed. If the solution
obtained is yh, then the direction d = yh x h is a promising direction in
whicb to seek m increased value for the objective functlon u.
At my feasible point x a, a llnear approximation to u(y) is
U(X h )  ÇxU(xh)T(y -- Xh).
The maximization of this hnear approximation, afer excludlng construit terres,
is equivalent to the poblem
maximze VxU(xh)Ty
(5.2.2)
subject to y  S,
where x  is fixed md y  R  is the variable. Let yh  S be the solution•
A well-known condition for x á to be m optimal solution of problem (5.2.1)
is that V.u(x)Td < 0 for ail d  S. Therefoïe, if afteï solving problem 5.2•2)
s yh = xh, then we know that 0 œee ÇxU(xh)T(y h X I')  Çu(xh)T(y -- X h)
for all y  S, and, thus, the optimality condition is fulfilled at x u.
Ifyhxhthenweset dh=yh Xh. Thepointsyhandxh arefeasiblc
and, hecause of the convexity assumpion of S, my new point x TM -- x h + tri h
in the direction d h by maximizing u(x  +td ) subject to 0 ( t < 1.
5.2.2. GDF Algorithm
Below, we shali show that even thogh we do hot know the value functioa
explicit ly, we cm obtain a local llnear approximation foï it or to be more exact,
its gradient, with the help of marginal rates of substitution. This is enough to
permit the FW method tobe applied. Before going into details we present the
basic phases of the GDF algorithm.
(1) Ask the decision maker to specify a reference function f. Choose a
feasible stoztlng point x I . Set h - 1.
(2) Ask the decision maker to specify maïginal rates of sabstitution between
/' and the other objectives at the current solation point x .
(3) Solve problem (5.2.3), where tbe appïoximation of the value fumtion
is maximized. Denote the solution by yh  S. Set the direction d  --
ya-x h.Ifd a-0,gotostep(6)•
(4) Determine wlth the help of the decision mal<er the appropriate step-size
t to be taken in the drection d a. Denote the corïesponding solution
•
(see Remark 2.8.7 of Part I).

The foie of the reference fnction is siafificaat, because marginal rates of
substitution are geilerated with respect to it. The decislon maker must be asld
to speclfy the reference function so that the marginal rates of substitution
are sensible. Note that if the uIlderl3dng value function is linear, then only
one iteratioa is needed to aclfieve the final solution (the marginal rates of
substitution are construit).
It may be difficult for the declslon maker to speclfy the marginal rates
of substitution directly (Or straight away). If this is the case, some auxiliary
pïoceduïes may be bïought it to asslst. One such procedure is presented in
Dyeï (1973a). The idea there is to determiae (at the point f(xh)) small amotmts
of/'« md /' denoted by zl« md zl, respectively, such that m increase in
the value of f by z L is matched for the decisioa maker by a compensatory
decrease by zi in the value of/'«, while the values of ail the other objective
fiuictions renin unaltered, lai other words» the vectors (/1 (xh), ''' , -k(xh)) T
mld(/el(Xh),...,/e(X h ) z],...,fi(xh)+zi,,...,f(xh))Tareindifferentto
the declslon mM<er. We obtaln aow
A '
Figure 5.2.1. An approximation of tlle margfial rate of substitution.
In practice, the z-amouats of change cmmot be made arbitrarily small near
0, as emphaslzed in Sawaragi et al. (1985, pp. 259 260). The reason is that
human belngs cmnot recognlze small chmges beyond a certaixi point. Thls
threshold of human recognition is called a just noticeable differnce. Tbat is
why the naxglnal rates of substltutloIl are always approxlmations ofthe correct
5.2. Geoffxion-Dyer-Felnberg Method 145
values. An example of the effects of the just notlceable dffi'erence is given in
Nakayama (1985.) by i]]ustrating bow the solution process may termñaate at
a wrong solution. For çhis reason one may have doubts about the adequacy of
marghal rates of substitution m a meais of provldig prefeence information.
They seem to be difllcult for the decislon maker to specify and theh- accuracy
is questlonable.
However, we must now assume that the naxginal rates of substitution are
provided accurately enough. According to the FW method, the maxlnñzation
of the linear approximation of U is equlvalent to the problem
k T
subject to y  N
with y  R' beng the variable. The solution is denoted by yh. The existence
of the optimal solution is esured by the compactness of S md the continuity
of ail the fimctlons.
The search direction is now dt' - yh X h. Provided that the naxgiaal rates
of substltutiox are ïeasonably accurate, the search direction should be usable.
Let us mention that a scaling idea presented in Clinton and Toutt (I988) can
be included in the method. Heterogeneous objective ftmctions can be scaled to
have equal effect in problem (5.2.3) by adjusting the norms of the gradierts of
the objective functions with scalar coefllcients.
The following problem is to find an appropïlate step-size for gofitg in the
search direction. The oaly variable is the stelsize. The decisiox maker ca
be offered objective vectors, wheïe zi - /,(x a +td ) for i - 1,... ,k, aad t
varles stepwise betwecn 0 and 1 (e.g., t -  where j - 1,..., P, aad Pis the
number of the alternative objective vectoïs tobe prescnted). Another possibillty
is to draw the objective values as a function of t, provlded no serious scaling
problen exlst. Ax example of the ga'aphical presentation is given in Hwang
aad Masud (1979, p. 109). Graphlcal illustratio of the alternative objective
vectors is handled in Chapteï 3 of part III. Note that the alternatives are hot
neccssarily pareto optimal. From the information given to the decision maker
(s)he selects the most pïefeïred objective vector and the corresponding value of
tis selectcd as t h. It is obvious that tlm task of selectloI bccomes more difficult
for the decision mM<er as the number of objectlve functions increses.
The opinions ofthe decisiot maker and the situation y : x h are used heïe
as stoppbg crlteria. 0ther possible crlteria are pïesentcd in Hwaag aad Masud
(1979, pp. 108 110) aad Yu (1985, p. 327).

The GDF method cm be characterlaed tobe a non ad hoc method, if one
knows the value function, itis easy to specify the marginal rates of substitution
the GDF method are closely related to the convergence properties of ttoe FW
method. The convergence ofthe FW algorithm under the assumptlans prodded
at the beglaning of this section, is proved in Zmgvill (1969), However, it must
be kept in mind that the correctness of the marginal rates of substitution
thc step-sizes affects the convergence conslderably, If it is assumed that the
continues, it is asserted in Geoffvion et al. (1972) that infirfite Convergence
holds.
be round in a reasonable number of iteratlans. Itis claimed in Geoffrion et
al. (1972) that the error in the objective function values is at least halved at
each of the first H itcratlans (H is unknown). The convergence becomes slower
The effects of errors in estlmatln 8 tbe gradient of the value function are
investigated in Dyer (1974). The result is that even if the answers of the decislon
maker are hot strlctly consistent md the just noticeable difference affecs
marginal rates of substitution, ttœe method is stable md converges (only slower)
5.2.4. Applications and Extenslons
Tbe GDF method is applied in Geoffrlon et al. (1972) to the operatlon
of an academlc depaïtment. Numerical examples are also given, for example,
in Hwmg and Masud (1979, pp. 111 121) and Steuer (1986, pp. 37779). A
time-sharing computer program implementlng the GDF algorithm is suggested
in Dyer (1973a). The GDF method is implemented for convex problems by a
so-called projection-relaxation proceduïe bi the objective space in Ferreiïa and
Machado (1996). An application in water resources allocation is also given. The
GDF method is adapted for continuous eciuflibrium network dcsigx prohlems
in Friesz (1981).
In Dyer (1972), a method called interactive goal programming is pïesented.
It is a combination of tbe GDF method and goal programmlng. The vector yh
is obtalned by toe means of welghted goal programmhlg with the marginal rates
of substitution as weights. Some convergence results are also givem The GDF
method md the hteractlve goal programming mcthod are applied bt Jedrze-
jowicz md Rosicka (1983) to multiobjective rchability optimization problems
appeaïing irt multiple classes of system failures.
Tœe GDF method has beea a subject of many modifications in toe llt-
erature. New versions have heen malnly developed to overcoIlm some of the
weaknesses ofthe GDF method. In Hemmin 8 (1981), a simplex-based direction-
finding problem is proposed for MOLP problems to avoid the specification of
optimal solutions. In addition the GDF method is modfied for MOLP prol>
lcms in Winkels md Meika (1984) so that when determining tbe step size at
projection onto the Pareto optimal set. This is done by solving a parametric
linear pro'ammin 8 pïoblem.
Tbe GDF method is altered in Rosinger (1981) by constructlng a wide fam-
The so-called proxy approach is intïoduced in Oppenheimer (1978). TOe
value fnctlan is no longer appïoxlmated llnearly. The idea is to glve a local
proxy to the value functlon at each iteratlan. A sum-of-powers or a sum oç
the pïoblem. Now, direction finding md step-size determlnation problems are
rate thm tbc origfial GDF metbod. Even this umthod does not 8uarantee
Raindrm (1986). First, the FW method is ïeplaced by a generalized ïeduced
gradient method. Then tbe role of the declslon makcr is facilitated by asklng
for intervals for the marginal ïates of substitution bstead of exact values. The
step-slze is computed wlth the help of upper md lower bounds for the objective
functlons without the declslon mal(er.
In Musselman md Talavage (1980), tœe idea of the adaptation is to reduce
the fiasible region according to tbe marginal rates of substitution specified by
current solution are àropped. The method peïmits sensitivity malysis of the
Ideas ofthe GDF met bod are applied in the inteïactive inte-ated approach
for quasiconcave value fmctlons in Al-alvani et al. (1992). A large set of Pareto
optimal solutions is first generated md the foïm of the underlyïn8 implïcit
by the weighted Tchebycheff metïlc. Tbe stopping crlterion is based on tïade-off

Finally we mention a modification of the GDF method known as the sub-
gradlent GDF method, for nondifferentinble mditinbjective optimizatinn prob-
lems presented in Miettinen (1994) md Miettinen md Mikel (1991 1993,
1994). The twice continuous differentiability of the objective 5mctions is re-
laxed md they are assmned to be locally Lipschitzim, but the value fnction
has to still be continuously differentiable. The FW method is replaced by the
subgradient method (see Shot (1985)) in optimiziag the approximated value
functlon.
In addition to being able to hmdle nondifferentiable functions the modifi-
cation has mother advmt age. It produces only Pareto optimal solutions, tmllke
the original GDF method. Each calculated solution is set as a reference point
optimal. Naturally additional optimlzations increase the computational bur-
den but this is the price tobe paid for the certainty that the decision maker
does not have to hmdle nomPareto optimal solutions. (A strongiy decreasing
value fnction implles that less is preferred to more in the mlnd of the decisinn
maker.)
Some applications solved wlth the subgradient GDF method are presented
in Miet tinen (1994). The subgradient GDF method is used in solvlng ma optimal
oontrol problem concernlng ma elastlc string in Miettinen md Mi&el (1993)
and continnous casting of steel in Miettlnen and Mikel (1994).
5.2.5. ConcIuding Remarks
ha the GDF method the decision mal<er is first givea one solution where
(s)he has to specify the marginal rates of substittlom After that thc decislon
mal<er must select the most prefered solution from a set of alternativcs. Thus,
the ways of interaction are versatile.
ha splte of the plausible theoretical fomdation of the GDF method, it is hot
so convlncing md powerfl in practice. The most ñztportmt difllculty for the
declslon mal<er is the determlnlng of the k 1 marginal rates of substitution
at each iteration. Even moe difiqcult is to give consistent md correct marginal
rates of substitution at every iteration. The difi$culties of the decisinn mal<er in
determining the marginal rates of substitution are demonstrated, for example,
in Wallenius (1975) by comparative tests. The same point can be iilustrated
by ma example from Hemming (I981) w]lere a polJtician is asked to specify the
exact marginal rate of substitution between unemployment and a decrease of
1% in the inflation rate.
A drawback ofthe GDF method is that the final solution obt ained is hot nec-
essarily Pareto optimal. Naturally, it cm always be projected onto the Pareto
optimal set with ma auxàliary problem. A more serlous objection is that when
several alternatives are gioen to the decision mal<er from which to select the
step-size, itis liliely that mauy of them are hot Pareto optimal. They can also
be projected onto the Pareto optimal set befre presentation to the decision
mal<er, but this neoessitates extra effort. The projection may be done, for ha
stmace, by lexJcographic ordering or by the memas preseated ha Section 2.10 of
part I. The use o acbàevement fimctinns is demonstrated in the subgadient
GDF method. The weakness in the projection is that the computatlonal burde
increes. It le for the maalyst mari the decision mal<er to declde which of the
two shortcomings is less inconvenlent.
Theoretically, the Pareto optimallty of the final sdittion is guarmateed if the
value ftmctioa is strongiy decreasing (by Theorem 2.6.2 of part I). In may case,
marginal rates of substitution are crucial in approxñzating the value fiction,
mad for mmay decision makers they are difi$cult mad troublesome to specify.
For mauy people it is easier to thlnk of desired changes in the objective
function values than to specffy indifference relations. This may, especially, be
the ce if the objective vector at wbJch the marginal rates of suhstltution are
to be specified i not particularly desirable. Then it may bc frustrating to think
of indifferent solutions instead of the improvements sought.
The Frank-Wolfe gradient ïaethod has been selected as the maxJmization
algorlthm for its fast initial convergence. In some cases, other gradient-based
methods may be more appropriate. For example, the subgradient method is
employed in the subgradient GDF method.
There are a lot of assumptions that the problem to be solved must satisfy
in ortier the method to work md converge. Several sufiqcient conditions on the
decislon makr's preferences are presented in Sawaragi et al. (1985, pp. 258-
259) to guarantee the d[fl'erentiability and the concaity of the value function.
Even these conditions are not very easy to check. For more critical diacussion
concerning the GDF method, see Sawaragi et al. (1985, pp.25761).
5.3. Sequential Proxy Optimization Technique
Like the two pïevlous methods, thc sequential proxy optimzation technique
(SPOT), presented in Sakawa (I982), is based on the idea of maxlmizlng the
decision maker's underlying value ftmctian, which is once agaln assumed tobe
known implicitly. SPOT includes some properties of the ISWT md the GDF
methods, md that is why we describe it here briefiy.
5.3.1. Introduction
As in the two interactivc methods presel]ted thus far, the search oeection
in SPOT is obtained by approxlmatlng locally the gradient of the mderlying
vaine fnction, md the step-size is determlned according to the preferences
of the decision maker. Here, both marginal rates of substitution md trade-off
rates are used in approxàmating the value function.

to the decision maker. In addition U is a continuously differentiable,
strongly decreaslng md coi*cave function on the subset of Z where the
2. The objective and the constraint functions are co*vex md twice contin-
uously dilïerentïable.
3. The feasible region Sis compact and convex (md there exist some upper
botmd for the -oenstraint problem so that the solution is finite)
4. The assumptions in Theorem 32.13 are satisfied.
The ¢-constraint problem is used to generate Pareto optimal solutions The
solution of e-constraint problem (3.21) is denoted by x h. It is assumed to be
unique so that Pareto optlmality is &armteed. Throughout this section itis
this is hot the case, then the upper botmds must be slightly modified.) Then
f (x ) = « for all j -- 1,..., k, j ; g. The optimal value of fl, that is, f« (xh), is
denoted by zî. Itis also assumed that ail the Karush-Kuhn-Tucker multipliers
Theorem 3.2.13 are assumed to be satisfied so that trade-off rate information
cm be obtained from the Kaush-Kuhn-'lcker multipliers.
Here, the value ftmction is ot maximized in form (4.1.1) as before Instead,
the set of asible alternatives is restricted to the Pareto optimal set. According
to the assumption above stating that f(x ) =  for ail j = l, ..,k, j ; g,
(5.3.1) maximize U(, , .. ,1_1, Zl ,g-f+l, , ,).
It is proved in Sakawa (1982) that tbe new function s concave with respect
also claires that the partial dcrivative of (5.3.1) wïth respect to «h, j = 1,.. , k,
j ; g, is equivalent to  (mîj Aîj), wbere mî is tbe marginal rate of substi-
tution between ri and f at x h (obtained from the declsion maker, see Section
5.2 md Ah • "
) O ls the poztal trade off rate between f and f at x  (obtained from
the Karush-Kuhn-Tucker multipliers, see Sect[ons 3.2 and 51).
know that  <
for j = 1,... ,k j ; g and it repreoents the drection of steepest ascent of the
value function (5.3.l) at the currcnt point x h for j ; g. According to Sakawa,
5.3.2. SPOT Algorithm
We cm now present the basic ideas of the SpOT algorithm.
(1) Choose a reference functlon le and upper bounds e I  P  for whicb
ail the constraints of the e-constraint problem are active. Set h = L

(2) Solve the current (active)  constraint problem for œe to obtain a solu-
(3) Denote the pareto optimal objective vector corresponding to x a by z h
and the corresponding Karush-Kuhn-Tucker multlpliers by 2î, j --
1,...,k,j£.
(4) Ask the decision maker for the marginal rates of substitution mî for
j -- 1,..., k, j  , at x h. Test the consistency of the marginal rates of
substitution and ask the decision maker to respecify them if necessary.
(5) If ]mî ),îj] < 8, where  is  prespeeified positive tolerance, thon stop
with x  as the final solution. Otherwise, determàne the components
A«, j  t, of the seaïch direction vector.
(6) Select an appropriate form of the pïoxy ftmction and calculate
rameters. If the obtained proxy flanction is hot strongiy decreasing and
coIcave, then ask the decision maker to specify new marginal rates of
substitution.
(7) Determine the stel>size by solving the v-constralnt problem wth the
upper botmds h + tact, j -- 1,..., k, j  , fo diffèrent values of t.
Denote the optimal value of the objective fonction by zî (t). A step-sze
t h maximizing the proxy fonction is selected, ff the new objective vector
(eî + thAŒEî,..., Zî(th),..., e + hAe)T is prefeïred to z a, denote the
(3). If the decision maker prefers z h to the new solutlon reduce t h to
be 21t h, 4£th, . . . until an impïovement is achieved.
The maximum of the proxy function is determined by altering the step size
t, calculating the corresponding Pareto optimal solution and oearching for three
t values, 1, t h and t2 so that t < t h < t2 and p(t ) < p(t h) > P($2), wheïe pis
the proxy fonction. Wben the condition above is satisfied, the local maximum
of the proxy function p(t) is in the neighbouïhood of t h.
Under assmnptions 1-4 (in Subsection 5.3.1), the optimality condition for
prohlem (5.3.1) at œe is that the gradient equals zero at that point. This means
that mî -- ),îj for j -- 1,... k, j ¢ g. This is the backgïound of the absolute
value cbecking at stop (4) (see also (2.8.1) in Part I).
5.3.5. Comments
The consistency of the marginal ïates of substitution is checked because it
is krportant for the successful convergence of the algorithm. The consistency
at a single point is tested by the chain rule and by limiting the discrepancy (the
formula is given in Sakawa (1982)) by a given tolerance level. The consistency
at successive points is tested by checking the concavity and monotonicity of
the proxy flanction (the prOXy ftmction must flalfill the same assumptions as
the value function). A theorem giving conditions for different types of proxy
fonctions is presented in Sakawa (1982).
checked that a sufficient impr ovement is obtalned. If the decision maker prefers
be estimated.
It is remarked in Sakawa (1982) tht the SPOT algoñthm is nothing but
a feasible direction method as for the conveïgence ïate. The convergence can
be demonstrated by the convergence ofthe modifled feasible direction method.
Foï thls statemcnt tobe true, an ideal (i.e., consistent with coïrect answers)
decisinn makeï must be assumed.
SPOT can be classified among met hods of a non ad hoc nature. H the value
5.3.4. Applications ad Extensions
The finictioning of the SPOT algorithm is demonstrated in Sakawa (1982)
by an academic example. Itis shown that even though the margial rates of
substitution are only approximations, this does not neoessarily worsen the re-
sults remarkably. A problem concerning industrial pollution in Osaka City in
Japan is solved by SPOT h Sakawa and Seo (1980, 1982a, b). The pïoblem is
defined as a largc-scale problem in Sakawa and Seo (1980) and a dual decom
position metbod is used to solve the  constraint problems.
A flazzy SPOT is presented in Sakawa and Yano (19&5). The decision maker
is assmned to asess the marginal rates of substitution in a fozzy form. In
Sakawa and Moïl (1983), a new method for nonconvex problems is proposed,
where the welghted Tchebycheff problem is used to generate Pareto optimal
solutions instead of the e-constraint method, and trade-off rates are hot uoed.
A method reinted to tffè preceding one is pïesented in Sakawa and Mort (1984).
Tbe différence is a penalty scalarizing function uscd in generating Pareto op-
timal solutions (sec Section 3.5). This method is also applicable to onconvex
problems.
5.5.5. Concludlng Remarks
Ideas from several methods are combined in SPOT and several concepts
are utilized. As far as the ïole of the decislon makeï is concened, (s)he is only
requiïed to determine the marginal ïates of substitution. Difficulties related
to this determination were mentioncd in Section 5.2 and they are still valid.
However, the consistency of the marginal rates of substitution in SPOT is
even more important than in the GDF method. This is a very demanding
requirement.
A positive featuïe of SPOT when compared to the GDF method is that only
Pareto optimal solutions are hand]ed. Because the multiobjective optkrdzation

probletn was asstmed tobe convex, globlly Pareto optimal solutions are ob-
talned. The bttrden on tbe decision maker is decreased by employ[ilg a prox
fnction when selecting tbe step-slze.
Many assumptions are set to guarantee tbe proper functioning of the al-
goïithm. Some of these are quite diflicult to check in practice (see concluding
remarks concerning toe GDF method in Subsection 5.9.5).
5.4. TchebycheffMethod
The Tchebycbeff method, proposed in Steuer (1986, pp. 419150) and Steuor
and Choo (1983) and refined in Steuer (1989a), is an interactive weighting vec
tor space reduction metbod. Originally, it was called the interactive weighted
Tchebycheffprocedure. A notable difference when compared to the methods de-
scribed thus far is that a value functio is aot used in the Tchebycheff method.
In addition, the role of the decision maker is differeat and somewhat simpler.
Here, we introduce thc Tchebycheff algorithm according to the refmed version
but modified for minlmlzation problems.
5.4.1. Introduction
Tbe Tchebycheff metbod has been designed tobe user-friendly for the deci-
sion mal<er, and, thus, complicated information is hot required. To start with,
a utopian objective veCtOr below tbe idem objective vector is estabfished. Then
the distance from the utoplm objective vector to the feasible objective region,
measured by a welghted Tchebycheff metric, is miaimized. Different solutions
are obtained with dhïerent weighting vectors in the metïic, as introduced in
Section 3.4. The solution space is reduced by working wlth sequences of smaller
and smaller subsets ofthe weighting vector space. Thus, the idea is to develop a
sequence of progressively smaller subsets of tbe Pareto optimal set untll a final
solution is located. At each iteration, differeat alternative objective vectors are
presented to the decision maker and (s)he is asked to select the most prerred
of them. Tbe feasible region is tben reduced and alternatives from the reduced
space are presented to the decision maker for selection.
Contrary to the previous illteactive metbods for multiobjectlve optimlza
tion, the Tchebycbeff method does hot presmne many assulIlptions regardblg
the pïoblem tobe solved. It is assumed that
1. Less is pïeforrcd to more by the declslon maker.
2. The objective functions are boundcd (from below) over thc feasible region
S.
In what follows we assume that tbe global ideal objective vector and, tbus,
the global utopian ohjective vector are known, and we can leave tbe absolute
value signs from the metrics. The metrlc to be used for measuring the distances
5.4. Tchebycheff Method 155
to a utopian objective vector is the weighted Tchebycheff metric (see Section
3.4). That is, the fnctinn tobe minimized is
(5.4.1) i _n,a.ï,  [ w, (f, (x) zï)],
wtœerew é W -- (w  R  ] 0 < wl < 1, lw -- 1). We havea family
of nletrics since w  W ean vary widely. This nondffi'erentLuble problem can
be solved as a fferentiable weighted Tcbebyctœeff problem (3.4.3) (where the
ideal objective vector is repl¢ed by the utopian objective vector).
According to Tbeorem 3.4.5, we lmow that every pareto optimal solution of
any multiobjective optimization problem can be found by solving the weigbted
Tchebycheff problem witb z**. The negative aspect with is problem is that
some of tbe solutions may be weakly Pareto optimal. This weakness was handled
1 Subscctlon 3.4.5. Producing weakly Pareto optimal solutions is overcome in
the Tchebycheff method by formalating the distance minimization problem as
a lexicogphic weighted Tchebycheff pr»blem:
k
(5.4.2) lex minlmize
subject to x
The foactioning of problem (5.4.2) is described in Figure 5.4.1 by a problem
with two objective functions. The bold line illustrates the Pareto optimal set.
The weighted Tcbebycheff problem has L-sbaped contours (the rhin continu-
ous fine) whose vertices fie along thc fiae emanatblg from z** in the direction
(1/w, l/w2,..., 1/wk). When minimizing the dist mace, a contour is determined
which is closest to z** and intersects Z. If this problem does hot have a urfique
solution, that is, there are several feasible points on the optimal contour in-
tersectlng Z, then some of them may not be Pareto optimal. In praCtlce, the
uniqueness is usually difli('ult to check, and, tobe on tbe safe side, the fol
lowing step musç be taken, la this case, the sum term is minimized subject to
the obtained points to determine whlch of them is closest to z** accotding to
the Lt-metric (the dotted line). Thus a Pareto optimal solution (see Theorem
3.4.1) is obtalned.
]le following theorems formulate tbe commction between the lexicographic
weighted Tchebycheff problem and Pareto optimal solutions.
Theorem 5.4.1. The solution of lexicographic weighted Tchebycheff problem
(5.4.2) is Pareto optimal.
ProoL Let x*  S be a solution of problem (5.4.2). Let us assume that is is not
Pareto optimal. In thls case tbeïe exists some x °  S such that f(x °) _< f(x*)
for ail i - 1,..., k and ] (x °) < ri (x*) for at least one j. This and the positivity
ofthe weights impfies that wi(f,(x °) z *) < wi(f,(x') - zî*) for eveïy i and
tbus moe If; (x °) -- z, «] _ maxï[f, (x*) -- zî*].

Figure 5.4.1. Lexicographic weighted Tchebycheff problen.
Ontheotherhmad, f(x °) zî<f(x*)-zîfralli 1,..,kmadat
least one of the inequalities is strict. That is why we havc  (f (x ° ) -zî) <
î_(f(x*) zï). Here we hve a contraction with x  helng a solution of
vector 0 < w  R  such tha¢ x   a unique solution of lexicographic welghted
Tchebycheff problem (5.4.2).
weighting ctor w > 0 such Cht x* is a uique solution of problem (5A.2).
If x* is hot n unique solution of (5.4.2), there exists unother point x °  S
that is a solution of this lecogïuphlc weighted Tchebycheff pïoblem. This
implles that x ° must he a solution of the weighted Tchebycheff problem. This
5.4. Tchebycheff Method 157
(r,«x.,) (,.(xo,
After slmplifyg the expression we bave
A(x °) 
r every i - 1...,k. Because x* is Peto optimal, we must hve fi(x °) -
f(x*) for ail i. ha other words the weighted Tchebycheff problem and tbus
also the lexicographic weoEhted Tchebycheff problem, h  ique solution.
An alternative proof of Th«rcms 5.4.1 and 5.4.2 is gaven in Steuer (1986
p. z145) mad Steuer and Choo (1983). Now we know that the lexicogïaphic
weighted Tchebycheff problem produces Pareto optimal solutions mad any
Pareto optimal solution can be round.
In thc Tchebycheff method diffeïent Pareto optimal solutions are obtalned
hy alterlng the weighting vector. At each iteration h, the welghting vector space
W t - {w h e R  I 1 h < w/h < u h, 1 wh - 1} is reduced to W h+l, wbere
W n+ C W . With a sequence of progïessively smaller subsets of the weighting
vector space, a sequence of smaller subsets of the pareto optimal set is sampled.
At the first iteration, a sample of the whole Pareto optimal set is generated
by solving the lexicographlc welght ed Tclebycheff problem with well dlsper8ed
weighting vectors from W - W  (with I - 0 and ul - 1). The reduction of W/
is done by tlghtening the upper and the lower hounds for the welghtlng vectors.
Let z h be thc objective vector that the decision maker chooses from the smaxple
at the iteration h mad let w h be the corresponding weigbting vector in problem
(5.4.2). Now a concentrated group of weighting vectors centred around w h is
formed. In this way a smaxple of pareto optimal solutions centred about z h is
ohtalned. It is advised to use norlnalized objective Binctloas in the calculations.
The number of the alternative objective vectors to be presented to the de
cision maker is denoted by P. The numher is usually specified by the decision
maker. It my be fixed oï different at eacb itertion. The algorlthm becomes
more reliahle, if as mmay alternatives as possible cma he evaluated effectlvely
at each iteration. Human capabilitles are yet limited and some khd of a con]-
promise is desirable.
Wben reduchg the weighHng vector space at each iteration, a reduction
factor ris needed. The larger the reduction factor is, the fster the weightlng
vector space is reduced mad the smaller are the declslon maker's posslbilities

for maldng errors and ehanging her or his mind conceruing her or his deslres
in Steuer (1986) and Steuer and Choo (1983) that
where v is the final interval length of the weighting vectors with g < v < 3
H is the numher of iterations tobe caxried out and < stands for %pproximately
5.4.2. Tchebycheff Algorithm
We can now preseut the mab features of the Tchebycheff algorithm.
(1) Specify alues for the set size P( h), a reduction factor ï < 1 and
an approximation for the numher of iterations H( k). Set Il  - 0 and
u,  - 1 for ail i - 1,...k. Construct the utopian objective vector. Set
(2) Form the weighting vectoï space W h -- {w   R k [ l,  < wl a <
/3) Geuerate 2P dispersed weightlng vectors w h ff W a.
(4) $olve lexlcographic weighted Tchebycheff problem f5.4.2) for each of
the 2P weighthg vectors.
(5) Present the P most different of the resulting ohjective vectors to the
decision nker and let her or him choose the most preferred among
them, denoting it hy z a.
(6) If h -- H go to step (8). Otherwise, modJfy, if necessaïy the weght
ing vector correspondng to a such that if problem /5.4.2) was solved
again, z  would be a ufiquely generated solution at the vertex of the
(1) Speclfy lî + and uî+ for the reduced weightlng vector æpace W TM,
(8) The final solution is x a coïresponding to z h.
Dispersed weightig vectors are generated from W ñ in step f3). I pïactlce,
this can he realized by generating randomly a large set (e.g., 50k) of welghting
vectors. Then the vectoïs are filtered (see Steuer f1986, pp. 3112)} or clus-
tered. The clustering is pïactical shce subroutines for it are avaflable h many
subïoutiue lihraries (such as IMSL). Whde we want to obt aih 2P well dlæpersed
weigbtlng ectors, we form 2P clusters and ehoose one candidate from each of
them elther aïbltrarily or near the centre.
Computatlona[ly, the following algorithm can he used to obtain random
weightiug vectors in W n. We omlt the index h for clarity.
wei -- li + ra(ui
where r n neans raising r to the power h. fi Steuer (1989a) an auxiliaxy scalar
w is determined so that the ratio of the volumes of W ñ+ and W h is r. Then
w is used in the reduction hmtead of the term r h.
The predetermlned number of iteratlons is not necessarily conclusive. The
declslon maker can stop iteratlng when (s)he ohtalns a satisfactory solution or
continue the solution process longer if necessary.
5.4.3. Comments
Itis suggested in Steuer (1986, 1989a) that the sampling of the Pareto
optimal set works in the most unhiased way if the ïanges of the ohjective
hmction values over the Pareto optimal set are approximately the saine. This
can be accompfished by re-scaling the objective functions in a way sbnilar to
that presented in Suhsection 2.4.3 of Part I, when necessary. It is advisahle

decision maker in the original form. More suggestions for modifications of the
algorithm are presented in Steuer (1989a).
The convergence rate of the Tchehycheff method is very difficult to estah-
lish. Itis stïessed in Steueï (1989a) that the Tchehycheff method is ahle to
that is satisfactory enough to he a final solution (see Steuer and Choo (1983)).
The Tchehycheff method can be characterized as a non ad hoc method. If
We do not here go into detalls of the alternative version of the TchehychoE
method. We only mention that the possihillty of getting wealdy Pareto optinal
solutions may be overcome hy using angment ed weighted Tchehycheff prohlem
(3.4.5) (see Figure 3.4.2). This means that properly Pareto optimal solutions
are handled instead of Pareto optimal ones (see Theorem 3.4.6). In this way, the
lexicogaphic optimization is avoided, but the Tchebycheff algerithm is more
for the augmentation parameter p hrings additional prohlcms. Itis proved in
Steuer (1986, pp. 440-4A) and Steuer and Choo (1983) that the augnented
welghted Tchehycheff prohlem can he used to char acterize Pareto optimal solu-
tions if the feasible region is fiinte or ail the coDstralnts are linear. A numerical
illustration of the algorltbm is presented in Steuer (1986 pp. 468472).
hnplementing the Tchebycheff method in a spreadsheet (Excel) environ-
ment is suggested in Steueï (1997). The TchehychoE method in its augmented
form is applied in Wood et al. (1982) to water allocation prohlems of a river
basin and in Silverman et al. (1988) to manpoweï supply forecasting. The au
mented method form is also used in Agreil et al. (1998) when solving an MOLP
prohlem of reservolr management. In Oison (1993), the Tchehycheff method is
apphed toa sausage blending pïohlem and in Kallszewski (1987) itis proposed
that modified weighted Tchehycheff problem (3.4.6) is used to mlnbnize the
distances in the Tchehycheff method.
5,4,4. Concluding Remarks
A positive feature of the TchebychoE method is that the ïole of the decision
maker is qulte easy to understand. (S)he does hot need to realize new concepts
or speciy numerical answers as, for example, in the ISWT and the GDF meth-
ods. Ail (s)he has to dois to compare several alternative objective vectors and
select the most preferred one. The ease of the comparison depends on tbe mag-
nitude of P and on the number of ohjective factions. The peïsonal capahiSties
of the decision makers also play an ilnpoant role. [t is also positive that
the alternatives are Pareto optimal
5.5. Step Method 161
The fl exibiilty of the method is reduced by the fact that the discar ded parts
of the weihting vector space cannot he restored if the decislon maker changes
ber or his mind. Thus some consistency is reqtfired.
The weakness of the Tchehycheff method is that a great deal of calcu/ation
is needed at each iteratinn and many of the results are discarded. For large
and complex problems, whee the evaluation of the values of the ohjective
5mctions may he lahorious, the TchebychoE method is hOt a realistic choice.
On the other hand, it is possihle to utilize parailel computing slnce ail the
lexicographic prohlems can he solved independently.
Although no absolute supeiority can he attrihuted, itis worth mentioning
that the TcbebychoE metbed pefformed best in the comparative evaluation of
four methods (the ZW the SWT, the Tcbehycheff and the GUESS mcthods) in
Buchanan and Daeilenhacb (1987) (see Subsection 1.2.3 of Part III). However, a
difficulty was encountered in comprehendlng the information provided. The test
example had only three objective 5tnctions and six alternatives were presented
at each iteration. And the cognitive burden only becomes larger when the
numher of the ohjectlve functions is increased.
5.5. StepMethod
Thc step method (STEM), presented in Benayotm et al. (1971), contains e
ements somewhat simi/ar to the Tchehycheff method, hut is hed on a diffœerent
idea. STEM is one of the first interactive methods developed for multinbjec-
tive optimization prob/ems. It was origlnaily designed for the maximization of
MOLP prohlems hut can he extended for nonilnear prohlems as descrihed,
for example, in ]schenauer et al. (1990b) and Sawaragi et al. (1985, pp. 268-
269). It can he considered to aspire at finding satisfactory solutions instead
of optimizing an underlying value functiom We describe the method for the
minimization of non[inear prohlcms.
Introduction
[tis assumed in ST]M that at a certain Pareto optimal ohjectlve vector
the declsion maker can indicate hoth finctions that have acceptahle values and
those whose alues are too high. The latter can be sald tobe unacceptable. The
decision makeï is now assumed to allow the values of some acceptahle objective
functioDs to increae so that the unacceptahle finctions can have loweï values.
In other words, (s)he must give up a little in the value(s) of some objective
functinn(s) f (i  I>) in odeï to improve the values of some other objective
functions ri (i  1<) such that I > I < -- (1,...,k}.
ST]M uses the weighted Tchehycheff prohlem (3.4.2) to generate new se.
lutions. The ideal ohjective vector z * is used as a reference point in the calcu-
lations. According to Theorem 3.4.2 tbe solutions are weakly Pareto optimal.

Itis assumed that
1. Less is preferred to more hy the decision nmkeï.
2. The ohjective functions are hotmded over the feaslble region S.
Information conceïning the ranges of the pareto optimal set is neeffèd in
deteïmluing the woighting vector for the inetric. The idea is to make the scales
of ail the objective f/mctions similar with the help of the weighting coefficients.
The nadiï ohjective vector z «d is approximated from the payoff table as
explaimd in Suhsection 2.4.2 of Part I. Thus» the maximal elemcnt of the
column i is called zï ad . The weighting vector is calculatcd hy the formula
where for every i -- L... k
as suggested il Eschenauer et al. (1990h) or
as suggested in Vanderpooten and Vincke (1989). (Thc denominatoïs aïe not
allowed to he zero.) The weight is laïgeï for thosc objcctive functions that are
5.5.2. STEM Algorlthm
The hasic phases of the STEM algoñthm are the fogowing:
(1) Calcu]ate the ideal and the nadir ohjective vectors and the weighting
coefficients. Set h - l. Solve welghted Tchehychcffprohlem (3.4.2) with
the calculated weights. Denote the solution by x h E S and the corre-
spoIding ohjective vector hy z h E Z.
(2) Ask the decislon mak«r to clasify the ohjective functions at z h into
satisfactory I > and unsatisfactory nes I <. If the latter class is empty,
go to step (4). Otherwise, ask the decisiol maker to specify relaxed
uppcï hounds i for the satisfactory ohjective functlons.
(3) Solve prohlem (5.5.l), where the upper hounds are taken into account.
Detmte the solution hy x h+  S and the corresponding ohjectlue vector
hy z h+  Z. Set h- h + l. Go to step (2).
(4) Stop. The final solution is x h.
In the first step the distance hetween the ideal ohjective vector and the
feasihle ohjective region is minimlzed by thc welghted Tchehycheff metric (the
weightlng coefficients specified as ahove). Tac solutiol ohtained is presented to
5.5. Step Method 163
the declsion maker. Then the decision maker is asked to specify those objective
function(s) whose value(s) (s)he is willing to relax (i.e., weaken) to decrease
the values of some other ohjective fuactions. (S)he must also determilie the
amount(s) of acceptable relaxation. Ways of helpiag the ffècision makeï in this
phase are preseated in BenayOlln et al. (197l).
The feasihle region is restricted according to the infoïmatlon of the decision
maker and the welghts of the relaxed objective functions are set equal to zero,
that is wi - 0 for i  I >. Tben a new distance minimization pïohlem
mirdmize max [ wilfi (x)
(5.5.1) suhject to ri(x)  ci for ail i  I >,
f,(x) _< fi(x t') for ail i • /'<,
is solved. The first new constraint set allows tlœe relaxed (acceptable) ohjectlve
fimction values to increase up till the spccified level and the second new con-
stralnt set nmkes sure that the unsatisfactory ohjective fmmtion VallleS dO hot
increase, that is, get worsc. The procedure continues untll the decision makor
docs aot want to change any comportant of the current ohjective vector. If thc
decision maker is hot satisficd with any of the components, then the procedurc
must also be stopped. In this case, STEM falls to find a satisfactory solution.
Diffèrent versions of the method vary in the formulation of the constralnt
set. Itt some versions, a new constraint set is generatcd ai every iteration and
in some other versions new constraints are included to açcompany the old ones.
In the latter model the decision maker must he somewhat consistent in ber or
hls actions hecause it is hot possible to wlthdraw the restrlctions set on the
feasihle regiom
5.5.3. Comments
STEM does hot assume the existence of an underlying Valll(! function. Even
ifone were avallahle, it would hot help in answerlng tlae questions. Thus STEM
ca bc eharacterlzed as an ad hoc method. Naturally, nothing con he said
abolit the convergence of STEM wih respect toa value functlun, tIowever, the
ffèvclopers of tffè method meltion that the algorithm produces a final solution
fast if the new constraints constructed during the solution proccss hecome
inebglhle for furtffèr relaxations.
A [inear numerical application example of STEM is givox in tIwang and
Masud (1979, pp. 174 182). Tlle pïopertles of the solution set of STEM are
studled in Crama (1983). A so cal[ed exterior hranching algorithm is presented
in Auhin and Niislund (1972). It is another kind of extension of STEM into
nonlinca prohlems. There are several dlfferenees whet compared with tlc orig-
inal method. For example, the decision maker does hot nced to speclfy any
amounts of change and an implicit Vaille function is assumed to exist. Some

164 Part II 5. Iaterotive Methods
extensioDs and modificatios of ST]M are also mentloned in Chankong and
Halmes (1983b, p. 329).
5.5.4. Coneluding Remarks
Becaase we are moving arotmd the (weakly) Pareto optbnal set, a decrement
in some objective function values can be achieved only by paylng the prlce of
an tncrement in some other objective function values. The idea of specffying
objective functions whose values should be decreaed or can be increaed seems
quite simple and appealing. Howeveï, it may be difficult to estimate appropïlate
amounts of increment that would allow the desired amotmt of tmprovement in
those functions whose values should be decreased. In other words, the contïol
of the solution is somewhat indirect. On the other hand, a positive feature is
that the iffonnatlon handled is eay to understand. No complicated concepts
are introduced to the decision nmker.
According to the results presented in Section 3.4 the solutions of STEM
are hot necessarily Pareto optimal but weakly Pamto optimal solutions may
be obtalned. It must also be kept in mind that the global ideal objective vector
ha to be knovai.
STEM w&s the first interactlve mcthod tobe b&sed on the claificatioa
idea. Numerons other methods adapting this idea in one way or the other have
appeared slnce. In what follows, we pïeseat several methods wheïe the decision
maker can specify both the amounts of relaxation and desirable apiration
levels. In this way the decision nmker caa control the solution process in a
more direct way than in STEM.
5.6. Reference Point Method
As its naine suggests the refcrence point method, presented in Wierzbicki
(1980, b, 1981, 1982), is ba.sed on a reference point of apiration lcvels. The
reference point is a feaalble or infeasible point in the objective space which is
reaonable or desirable to the declslon maker. The reference point is used to
derive achlevement scalarizin 6 functions having nnnimal solutions at weakly, 
properly oï Pareto optimal points a introdced in Section 3.5. In this method,
generating Pareto optimal solutions is baed on reference points, hot on value
functions or weightin 6 vectors. No specific aumptions are set ou the pïoblem
tobe solved. The refercnce point idea ha been utilized in several methods
in different ways. Wierzbicki's refcrence point method wa among the first of
them.
5.6.1. Introduction
The baic idea behind the reference point method is to reconsider how
declslon makers make decisions. Itis doubted in Wlerzblcki (1980a, b) that
individuals nmke everyday decisions by maxinfizing a certain value function.
Instead, Wierzbicki claires that decision makers want to attaln certain apL
ration levels (e.g. when making purcha.ses accord}ng toa shopping list). He
sug6ests that while thousands of consumers may behave on the average a
if they were maxàmizing a value function, no indlvldual behaves in that way.
The baslc idea is satisficlng (introduced in Section 2.6 of Part I) rather than
optimizing. In addition refeence points aJe intuitive and eay for the decisioa
maker to specify and thei consistency is hot an essentlal requirement.
Classifying the objective functions into acceptable and unacceptable ones
(at a current objective vector) wa mentloned in connectlon with ST]M. Spec-
ifying a reference point can be considered a way of claSifying the objective
functlons. I tbe apiration level is lower than the current objective value, that
objective function is currently unacceptable, and if the apiration level is equal
to or lñgher than the curïent objective value that function is acceptable. The
difference here is that the referencc point can be infeaible in every component.
In other words, where the set of acceptable objective functions is empty, the
reference point bsed approach can still be utihzed. Natuxally this does hot
mean that all the objective values could be decrea3ed but a diffeïent solution
can be generated.
Further information concerning the matters addressed in this section can be
found in Wierzblcki (1977, 1980b 1981, 1982, 1986a, b). By a refereace point
method we herc mean that of Wierzbickïs. The reference point method relies
heavily on the propertles of achievcment functlons, which were dealt with in
Section 3.5. Of partlculaJ interest are Coïollary 3.5.6 and Theoïem 3.5.7. As
far a thc preference structure of the declslon maker is concened, it is asumed
that
1. Less is preferred to more by the decislon maker.
5.6.2. Reference Point Algorlthm
The inteïactive multiobjective optimization technique of Wierzbicki is very
simple and practical. Before the solution process starts, ome information is
give to tbe decision nmker about the problem. If possible, the ideal objective
vector and the (approximated) nadir objective vector are presented to i[lus
trate the ranges of the PaJeto optimal set. Another possiblhty is to minimize
and maxinfize the objective functions iidlvidually in the feaiblc region (if it
is bounded). Both declsion variable and objective values are presented. An
appropïlate form for the achievement function must also be selected.
The baic steps of the reference point method are the following:
(1) Present information about the problem to the declsion maker. Set h - 1.

(2) Ask the decision maker to specify a reference point z h  R  (an aspi-
ration level for every objective function).
(3) Minimize the achievement flmctinn and obtaln a (weakly, e-properly or)
decisinn maker.
(4) Calculate a number of k other (wealdy, -properly or) Pareto optimal
solutions by mininizing the achievement function with perturbed ref-
z(i) -- z  + dhe i,
where d h -- IIz h zhll and e iis the ith unit vector for i -- 1,...,k.
(5) Present the alternatives to the decision maker. If (s)he finds one of the
Otherwise, ask the decision maker to specify a new reference point +t.
Set h- h + 1 and go to step (3).
The reasou for writing the words weoey or e-properly in parentheses in the
algorithm is that it depends on the achievement function selccted whether the
The advantage of perturbing the reference point in step (4) is that the de
cision maker gets a hetter conception of the possible solutions. If the reference
optimal set, then a finer description of the Pareto optimal set is gven. The
in Figure 5.6.1.
z 1
Figure 5.6.1. Altering the reference points.
5.6.3. Comnents
As to the infinite COnvergence of the algoritinn, the following result is stated
in Wierzbicki (1980b).
Theorem 5.6.1. If the solutions of the achievelnent function in the algorithm
are unique and if the miinmal value of I[z z[[ subject to Pareto optimal
objective vectors is equal to the minimal value of the achievement function sz
subject to Z for   Z+R, then for any metric in Ra the solution procedure
is convergent. In other words, 5moe [[z h - z h+ [[ -- 0.
Proof. See references in Wierzbicki (1980b).
A modification of the algorithm guaranteeing the convergence is presented in
Wierzbicki (1980b).
The reference point nmthod can be characterized as an ad hoc nethod or a
nethod having both non ad hoc and ad hoc featres. Alternatives are easy to
compare if the value function is known. On the other hatd, a reference point
cannot be directly defined with the help of the value function. However, it is
possihle to test whether a new reference point has a highcr value function value
than the enfiler solutions.
A different way of genet ating new reference points is suggested in Wierzbicki
(1997b). It is a way of reaIizing the idea of a reference ball where a set of
additinnal reference points in a hall of a fixed radius centered on the outrent
solution is produced.
An appendix to the reference point method is suggested in Wierzbicki
(1997b). After the dccision nmker has found a al solution (s)he can check
whether more satisfactory solutions exist hy a so-called outranking trials
method. In the spirit of outranking methods of (discrete) multiattribute de-
cision analysis, the decision maker is asked to specify preference, indifference
and veto thresholds (sec Subscctlon 5.9.1) for each objective function. Differ-
ent states of outranking relations are estahllshed and a sequential questioning
procedure is gone through with the decision maker to check whether there
exist objective vectors whose components outrank the current final solution.
This procedure may involve a lot of questions. The convergence of many other
interactive nethods can be investigated by the outrafldng trials method, as
5.6.4. Implementation
A software fanfily called DIDAS (Dynamic Interactive Decision Analysis
and Support) has been developed on the basis of the reference point ideas of
Wierzhicki. Tne nonlinear version of DIDAS has been created and developed
in several phases. For example, the International hstitute for Applied Systems

Analysis (IIASA) in Austria and the Warsaw Technical Ulliversity bave been
[nvolved. The latest version is called ]AC-DIDASN++. There is a lot of lit-
eratttre describing the variotm phases in the development work (see Granat et
al. (1994a, b), Grauer (1983a, b), Grauer et al. (1984), Kreglewski (1989), Kre-
glewski et al. (1987, 1991), Lewandowski and Grmmï (1982), Lewandowski et
al. (1987) and logowski et al. (1987)),
DIDAS is a dynamlc decision support system which abris at helping to
achieve better decisions. Thc ideology bas been extended from the ïeference
point method with ïeservation levels. Reservation levels 5i are objective ftmc-
tion values the user wants to avoid. For the objective functions to be mininfizcd
they must be al)ove the apixation levels forming the reference point z. Iii DI-
DAS, the user is aked to speci bath apiration and reservation levels for each
objective function. The achievement function bas to he reforlaulated to take
the reservatioti levels into account. Several achievement 5mctions bave been
suggested in different versions of the system.
The user can eaily obtMn different Pareto optimal solutions by cbanging
the apiYation levels and the reseïvation levels. ne objective functions are
scaled and the user is asumed to specLy apiration levels hetween thc ideal
objective vector and the nadlr objective vector. In this setting, the user can
ilaplicitly attach more importance to attMning a partlcular apiration level
by placing it near the ideal objective value. In tbat case, the corresponding
objective fmction is weighted stronger in the achievement fraction.
We gvc an example of achievement functions, inchding both apiration
and reservation levels. I5 ail the objective functions are to he miinmized, an
order-approximating achievement function tobe m2xilaized caz be of the form
mln [mini ' f,(x)
al, (1991)).
derivatives is a tii:ae-consui:alng and laboïlous tank, and secondly, errors and
i:alstakes are likely to occur. (Mistakes have been found to be a main ïeason for
the fa[htre of non]inear optinlization methods in convergence.) The difllculties
can be oveïcome wlth symbolic differentiation. This fs in'iefly handled in Kre-
glewski (1989). One more alteruative is to use automatic differentintion (see,
e.g., Rail (1981)). However, no results of doing this have been ïeported.
DIDAS is general and can thus bandle objective functions needlng tobe
lainimized, maxlmlzed or stabilized (Le., the objective funcinn should bave a
value a close to the gven level a possible). Different objective function types
imply changes in the achievement function used (see, for example, Granat et
al. (1994a)).
5.6.5. Applications and Extensions
The reference point method is applled to econometric modeLs in Olbrisch
(1986). Some experiences in applying DIDAS to macroeconomlcs planning are
:eported in Grauer et al. (1984). DIDAS is used in empirical tests in Bischoff
(1985) to experiment with different scalarizing functlons. A problem of de-
termining the optilaal temperature in a g2eenhonse is solved by DIDAS in
Udink ten Cate (1985). In Statu et al. (1992), DIDAS is used in analysing
the acld tain problem in Europe. A trajectory-orlented extension of DIDAS is
descrlbed and applied in Lewandowski et al. (1985a, b). Thïee applications of
IACDIDASN+ + for englneering desiga are epored in Wierzbicki and Granat
(1997). They handle the desiga of a spur gear transmission unit, ship navigation
support and automatlc controL
The refcrence point idea is modified in Mocci and Prtmlceïlo (1997) to
better hand/e nonconvex problems and avoid local optlma. At each iteration,
the achievement function is lainimized in a reduced feaslble reglon determined
by the declsion maker. This modified method is applied to a problem of :ing
network design.
The reference point method is generalized for several decision makes or
several reference points in Song and Çheng (1988) and Vetschera (1991a). The
eference point i:aethod is also essentlal iii a group decision support system,
descrlbed in Vetschera (1991b) where each group meI:aber uses the reference
point method.
A so-called colabined procedure cornbining he Tcbabycheff method and the
refe:ence point method is ktroduced in Steuer et al. (1993). Tbere the deci-
sion maker is asked to specffy both reference points and the most satisfactory
solntion alaong the alternatives produced by the means of the two methods.
Scalled reference sets, extersions of reference polnts are the bais of an in-
teractivc procedure described in Skulimowski (1996).
An extension ofthe reference point method, called he preenptive reference
point method, is introduccd bi Ogryczak (1997a). The approach formulates rcf-
crence poit prohlelas in the form of goal prog2amlning. Instead of considering

5,6,6. Concludlng Remarks
Wierzhicki's refereace point method is quite easy for the decislon maker to
understand. The decision maker only has to speciy appropriate aspiration lev-
els and compare ohjective vectors. What has heen sald ahout the comparison of
alternatives in connectloa with the prevlous methods is also valld here. The so-
lutions are weakly, -properly or Pareto opthnal depending on the achievement
fmction employed.
The freedom of the decision makeï has hoth positive and negat]ve aspects.
The decision maker can direct the solution process and is ïTee to change her or
5,7. GUESS Method
The GU]SS method is a simple interactive method related to the reference
prohlems).
5.7. GUESS Method 171
5.7.1. Introduction
The GUESS method does not involve any special assumptioIs. The only
requirement is that the ideal ohjtive vector z * d the nadir ohjective vecr
1. Less is preferred to more hy the decision mer.
2. The ohjective funct]ons c bounded over the feihle region S.
The method procds  follows. ae decision mer specifies a reference
point (or a guess) z  d a solution with equal proportional achievements is
geerat. Then the decision ner specifies a new reference point and the iter-
at[on continues until the decision nmker is satisfied with the solution produced.
The sech procedure is hot isted in any otlmr way.
The scales of the ohjective functions e normMized with denomators
weoEhtcd deviation from thc n ohjective vector. Thus, the idea is opp
site to, for example, the weited Tchehycheff problem where the mmum
weighted deviation om the ideal ohjecti vector is minimized.
çVe c put the sm rcofing k other words. We coe say that the ohjective
functions  rescMed so tha they MI have the roege [0,1]. This meoes that
each ohjective function ri(x) is repld hy a nornmlized functior
z d - f(x) for ail i = 1,...,k.
zï d - zî
Let us once again emphize that the global idem ohjectlve vector and the
ohjtive vector are sumed to he known.
The weiyhted max min problem to he solved is
(5.7.1) *=,...,* [w zï d zî ]
suhject to x  S,
where the weighing coefficients wi i -- 1,..., k, e positive and the denomi-
We bave the folwing result.
Theorem .7.1. The solution of weighted mmx-miIt prohlem (5.7.1) is weakly
Proof. Let x*  S he a solution of he weighd n rein prohlem. Let us
suppose that x* is hot weakly Peto optimal. In this c, there exts
point x  S such that f,(x) < f,(x*) for every i -- l,...,k. Tlfis meoes that,
zï-f,(x)>zï-f,(x*)fori.çilewehawi>Oandzï  z>0,

1 zï a*l ri(x) > 1 zïa*l-f(x*) for every it,...,k.
nfin
nornmllzed piration lcvels. In other wor, we
With the speced weighting coefficients we can write the prohlem fo he
solved in the form
nn
(5.7.2) nmize i=L .k L zï " zï J
Notice that the piratlon lcvels specified hy the decion mer h to he
ohjective functions e doEerentlahle prohlem (5.7.2) can be written in a f-
entiah form with Che help of an additional ile, where the nondiffer
eatiahle formulation can he solved with appropriate sgleohjcctive optimizers.
(5.7.2).
(5.7.2) with  -- f(x*).
of (5.7.2) with z -- f(x'). In ts ce there cs another x ° e S such Chat
This means that f(x °) < fi(x') for every i -- 1, ..., k which is a contradiction
(5.7.2).
5.7. GUESS Method 173
5.7.2. GUESS Algorithm
The GUESS method has rive baoic steps.
(1) Calculate the ideal objective vector and the nadir ohjective vector and
present them to the decisinn maker. Set h = 1.
(2) Let the deciion maker specify llpper or lower hounds to the ohjective
flmctions if (s)he so desires. Update prohlem (5.7.2), if necessary.
(3) Ask the decisinn nmker to specify • refere**ce point z h 6 R  hetween
the ideal and the nadk ohjectlve vectors.
(4) Solve prohlem (5.7.2) and ohtain a weakly Pareto optimal solution x h.
Prese**t the corresponding ohjective vector z h to the decisinn maker.
(5) If the decision maker {s satisfied with z h, set x h as a final solution and
stop. Otherwise, set h -- h + 1 and go to step (2).
In step (2) the specification of upper or lower hounds means adding con-
stralnts to prohlem (5.7.2). Nevertheless, the components of the ideal Or the
nadir ohjective vector needed in the calculation elsewhere are hot changed Or
affected.
The only stopplng rule is the satisfaction of the decision nmker. No guidance
is given to the decsion maker in setting new aspiration levels. This {s typical
of nmny reference point-hased mctho&.
5.7.3. Commeats
The GUESS method is hased on trial and error. The decision maker can
examine what kind of an effect her or his input has on the solution oht ained and
then modify the input, if necessary. The system fines rot provide any additional
Or supporting information ahout the prohlem to he solvd.
As long as no additional constiaints are included in the prohlem, the com
ponents of the solution ohtalned are in equal proportion with tfin components
of the reference point specified. In other words, when the sointlon ohtained
and the corresponding reference point are ormalized, the quotiettts of tfinir
component are the sme for each componet. The reason for this hehaviour i
that the reference point is contalned in the weighting vector.
The GUESS method is an ad finc method. The existence of a value functinn
would hot help in determlning ŒEew reference points or upper or lower hounds
for the ohjective ftuctions.
An interesting practical ohservation is mentioned in Buchaaaa (1997).
Namely, decision makers are easily satisfied if there is a small difference hetween
the reference point and the solution ohtalned. Somehow they feel a need to he
satisfied when they have almost achieved vhat they wnted. In this case they
may stop iterting 'too early.' The decision maker is nattrally finwed to stop
the solution process if the solution really is satisfactory. But, the coincidence
of settig the reference point near an attalnhle solution may mnecessarily
increase the decision maker's satist«ctlon.

5.7.4. Coneludiag Remarks
The GUESS method is simple to use and does hot set any specific assump-
tions on the hehaviou or the preference structure of the decisioa maker. The
decision amker can change her or his mlnd since no consistency is requred.
The only information ïeqinred ïTom the decision maker is a ïeforence point and
possible upper and lower botmds.
The method lins been compared to several otheï interactlve methods in dif-
ferent comparative evaluations (tobe described in Subsectioa 1.2.3 of Part III).
It has been received relatively well in the experiments ïeported. The reasons
may he its simplicity and fiexibi[lty.
The optioaal upper or inwer bounds specified by the declsion maker are
not checked in any way in the method. Inappropriate lower botmds may lead
into solutions that are not weakly Pareto optinml. In other words, additional
constraints may invalidate the result of Theorem 5.7.1. This can be avoided,
for example, by allowiag only upper bounds.
The weakness of the GUESS method is its heavy rellance on the availabli-
ity of the nadir objective vcctor. As mentloned in Subsectlon 2.4.2 of Par I,
the nadir objective vectoï is not easy to deteïmine and it is usua[ly only an
approximation.
5.8. Satisficing Trade-Off Method
The satisficing trade-off method (STOM), presented in Nakayma (1989,
1995), Nakayama and Frukawa (1985), Nakayama and Sawaragi (1984) and
Nakayama et al. (1986), is based on ideas slmilar to the reference point method
of Wierzblckl and the GUESS mcthod. The dfferentlating factor is thc trade.
off info.mation uti[ized. The method is here presented according to Nakayama
(1995).
5.8.1. Introduction
STOM origlnates ffom classification and aspiration levels. It is based on
satisficing decision maldng, as can be deduccd ïT0m its name.
The ftmctioinng of STOM is the following. Affer a weakly or a properly
Pareto optimal solution has been obtained by optiafizing a scalarlzing flmction,
itis presented to the decisioa maker. On the basis of this information (s)he is
asked to c]asslfy the objective Mactinns into three classes. The classes are the
unacceptable objective functions whose values (s)he wants to improve (I<),
the acceptable objective functinns whose values (s)he agïees to relax (impair)
(I>) and the acceptable objective functions whosc values (s)he accepts as they
are (I-) (such that I < L I > U I- -- {1,..., k}). Trade-off rate information is
utiSzed so that the decision maker only has to specify aspiration levels for the
decision maker is asked to classlfy tbe ohjective fllnctions at the new solutioll.
weighted Tchebycheff problem that can find any Pareto optimal solution. Hence
it and its augaented variant are used in STOM.
DoEercnt forms of scainrlzlng functlons have been suggested for use in
STOM. In the original formulation the weightlng coefficients are set as
(5.8.1) w/h _ 5, h 1 Z*** for every i = 1,...,k,
fl > z**, and the scalarizing fanction tobe minimized is once again (5.4.1).
If weakly Pareto optlaml solutions are tobe avolded, the scalarlzlng function
used is
(5.8.2)
10 -. Both these scalarlzlng functlons presume that the ideal objective vector
and, thus, the utoplm objective vector are known globally. However, if some
objective functlon fj is not bounded from befow in S, then some small scalar
value can be selected as zf ,
If the problem is bounded, then the solutions obtalned by functfon (5.4.1)
are guaranteed to be wealdy Pareto optimal (sec Theorem 3.4.2) and every
Pareto optimal solution can be found (sec Theorem 3.4.5). Further, it is proved
in Nakayama (1985a) and Sawaragi et al. (1985, pp. 271-272) that the solution
obtained is satisficig (i.e., f,(x" ) < z, h foï ail i - 1,..., k) if t he reference palnt
is feasible and weighting coefficients (5.8.1) are empfoyed. For function (5.8.2)
ail the solutions are properly Pareto optimal and any properly Pareto optimal
solution can be found. Even though the formulation s5ghtly differs ïTom (3.4.5),
the results of Theorem 3.4.6 are still valid. Unforunately, function (5.8.2) docs
not satisfy the thiïd ïequirement concerning satisficing decision maldng (sec
Nakayama (1985a)).
Other fo.ms of weighting coefficlcats can also be used. The selection affects
the results obtalned. Thls is demonstrated in Nakayama (1995). The refereace
the utopian objective vector is repinced by tbe refeïence point.
Both the scalarizing functlons mentloned are aondfferentlable but they can
be written in a differentinble form assuming the differentlability of the functlons
involved. This is carïled out by intïoducing a scalar variable a as in (3.4.3).

In what foilows, we refor to che cfiIferentinble form where all Che ohjective
As mentloned la Subsection 3.4.4 trade off rate information can be oht ained
with the help of differentiable foïmulation (3.4.3). Both weighting coefficients
and Karush-Kuhn-Tuckeï multipileïs are then utilized. That is why it must he
assumed that
1. Less is preferred to more hy the decLsion makeï.
2. The ohjectine and the coastraint fuactinns are twice continuously differ-
eatiahle.
The availahiilty of trade-oIf rates also necessitates Che fulfillment of other
aumptioas mentioned in Suhsection 3.4.4. They are parallel to Chose in The-
orem 3.2.13; s« also Yano and Sakawa (1987). This fact has not earlier heea
sufficicntly emphasized when introducing the method.
5.8.2. STOM Algorithm
Let us now write down che steps of the algorithm.
(1) Calculate the utopian objective vector z *. Set h - 1.
(2) Ask Che decision maker to specify a refereace poknt £a E R  such that
 > z* for everyi -- l,...,k.
(3) MinLmize che scalarizing function used. Denote che solution hy x h. Let
che corresponding ohjective veCtOr he gh. Present it to Che decision
(4) Ask the decislon maker to classify the objective functions into the classes
I <, I > and I =. If I < = O, go to step (6). Otherwise, ask the decisinn
maker to specify acw a.piration levels 2, a+ for the functions in I < . Se
2i h+ = fi(x h) for i  I =.
(5) Let ).h  R  he the Karush-Kuhn-Tacker multipllers connectcd tox .
Use automatic trade-off to obtala aw levels (upper bounds) 5î +1 foï
Che hmctions in/>. Set h - h ÷ 1 and go to step (3).
(6) Stop with the final solutinn x .
[t is aaturally possihle that che decision maker wiil also specify new levels
for those objective functions whose values (s)he agrces to rclax (i.e., increase).
But particularly when the aumher of ohjectve functioas is great, the declsina
maker may appreciate the autonmtic trade-off feature when (s)he does hot have
to specify new asplr atin leve/s for ail the fuactinns at each iteration. Naturally,
che decLsinn maker can modify che calculated aspiration levcls if they are
As long as trade-off rates are ohtalnable from Che Karush Kuhn-Tucker
mu]tipilers and the weightlng coefficients, Che huïden set on Che declsion maker
tan be decreased hy employing automatic tde off in specifying che aspir
tinn lcvels (upper houads) for Che functinas to be relaxed. They are derved
5.8.3. Comments
Ifthe prohlem is linear or quadratic, we tan go even further than Che au-
tomatic trade-off. In thLs case parametric optimlzatina is used in generatlng
so-ca]led exact trade-off Thls means chat we tan calctlate exactly how much
the ohjective functinn values must he relaxed in ordeï to stay in che Pareto
optinml set. Thus, we get a new Pareto optimal sohttinn wlthout having to
re-optinfize Che scalarizing functina (see Nakayama (1991h, 1992a, h)).
Trade-off informatina tan also he used to check che foaslhillty of Che refer-
ence point specified by che decisinn maker. If itis Rot feaslhle, the numher of
mininfizatinns of the scalarizing function tan he reduced hy directly specify-
ing higher aspiration levels (reemher that satisficing solutinas are ohtalaed
when Che reference point is fea.slhle in scalarizing functinn (5.4.1)). Sec detalls
in Nakayama (1985a, 1989), Nakayanm and Furukawa (1985) and Nakayama
and Sawarag (1984).
Trade-off information is valuahle even if sonne Karush-Kulm-Tucker aulti-
pllers are equal to zero. For example, if all Che Kar ush- Kuhn-Tucker multipliers
of Che functios to be relaxed cqual zero, we know Chat it is not possible fo
improve the desircd ohjective functinn vahes with this classification. In other
words, Che fonctions fo he relaxed cannot compenste foï Che improvement de
sired. The reason is that Che ohjective functlons to he relaxed are positlvely
affected by other ohjectlve function(s) to be improved (see Nakayama (1995)).

and increased or the amounts of change. Thus the method is characterlzed as
ma ad hoc method. On the other hmad one must ïememher that the aim of the
It is poiated out in Nakayama (1995) and Nakayama et al. (1995) that the
foies of the ohjective and the constraint functions can easily he interchanged
when the noadiffeïentiahle scalarizing fimction is solved in differentiable form
(3.4.3). This is carried out hy adding constant mfltipliers Bi to the artificial
variahle ( in each additional constraint. The ohjective function ri hecomes a
constïaint fnctinn hy changing the value of ï from one to zeïo. For frther
detalls, see Nakayama et al. (1995). The idea of intcrchanglng the roles of the
finctions is also handled in Suhsections 5.10.3 and 5.19.7.
5.8.zL Implementation
A STOM implementation hz heen carried ot in Bulgaria. The software is
ca[[ed MONP-19; see Vzsilev et al. (1990). This pïoram has heen developed
for nonlinear mdtiohjectlve optimizatinn prohlems. The system suggests new
zpiration levels for ail the ohjective fimctions, and the user can fTcely change
them. The fezihility of the zpiration levels is checked in the scnse of linear
pproximations. If it is impossihlc to satisfy ail the specified aspiration levels,
the user can either modify the levels or go ahead and optimize anyway. In
the latter czc the solution will he wealdy pareto optimal but hot satisficing.
MONP 19 can handle ohjective fnctions hoth to he minimized and to be
maximized.
5.8.5. Applications and Extenslons
Somc theoretical specifications concerning the STOM aIgorithm are pre
sented in Nakayama and Furukawa (1985). The method is also applied to the
zeismic design of a tower-pler system for  long span sspension hridge. Soft-
ware implementing STOM for interactive construction accuïacy control sys-
tems of cahle-stayed bridges is introduced and applications are descrihed in
Nakayama et al. (1995, 1997). A linear diet prohlem and the eïection of cahle-
stayed hridges are mentioned z STOM applications in Nakayama (1994). The
latter prohlem is also haadled in Nakayama (1995). An application toa water
cuallty control prohlem of a river hzin is presented in Nakayama (1985a) and
Nakayama and S,waragi (1984). In Olson (1993), the method is employed to
solve a smsage hlending prohlem and, in Nakaymxa et al. (1989), to solve 
hleding problem of indstrial plztic materials. STOM is adapted for linear
fractional ohjective functions in Nakayama (1991a) with an application con-
cerning material hlending in cernent production. A hlending prohlcm in feed
formflation for ]ive stock is descrihed in Nakayama (1995) z well z an inter-
active sppor system for hond trading. In addition, STOM is applied in a diet
planning prohlem in Mitani and Nakayama (1997).
5.8.6. Concluding Remarks
STOM contains identical elements with STEM, the reference point method
and the GUESS method. Therefoïe, the comments gven there are not repeated
here. The role of the decislon makeï is easy to understand. STOM requlres even
less input ffom the decisinn maker than the ahove-mentioned methods hecause
on]y a part of the aspiration levais need to he gJven. The so]utinns ohtalncd are
properly Pareto optimal or wealdy Parcto optimal depending on the scalarizing
function used.
As sald hefoïe, in practice, classifying the ohjective functions into three
clzses and speckfyiag the amourits of increment and decrement for their valtles
is a suhset of specifying a new ïeference point. A new refeïence point is implic
it]y formed. Either the new zpiration ]evels are larger, smal]er, or the samc z
in the crrent solution. Thus the same otcome can he ohtained with differcnt
rezoning. A positive diffcrentiating feature in STOM when compared to other
clzsification-hzed methods is the automatic or exact tradeoff. This decïezes
the mamunt of information inquired fTom the decision maker. STOM is in a
sense opposlte to STEM. In STOM, only desired impïovements are specified,
wherez only amounts of relaxation are used in STEM.
Becmue the method is hzed on satisficiag decision maldng, the decisinn
maker can freely search for a satisficing solution and change her or his mind,
if necessary. No convergence hzed on value functinns hz even heen iatended.
5.9. Light Beam Seoech
Light beam search, descrihcd in Jzzkiewicz and Slowifiski (1994, 1995),
combines the ïefeïence point idea and tools of mu]tiattïlhte decision analysis.
That is why itis an intercstlng methad for inc]usion here to reprcsent how the
henefits of different pïohlem solving arez can he put to use. Here we modify
the original method for mlnimization prohlems.

5.9.1. Introduction
The basic set ting in the light beam search is identical to the reference point
method of Wierzbicki in the spiñt of satificing decisinn making. The achieve-
ment function tobe minimized is ftmction (3.5.3) where weighting coeItlcients
are used only in the maximum part. They take into account the ideal mad the
nadir objective values. This achievement ftmction memas that c-properly parer o
optimal solutions are generated. The reference point is here assumed tobe an
infeaalble objective vector.
Itis assumed that
1. Less is preferred to more by the decision maker.
2. The objective and the constraint functions are continuously differen-
3. The objective functions are bounded over the feasible region S.
4. None of the objective functions is more importmat than ail the others
together.
Assumption 3 is needed in order to have the ideal mad the nadir objec-
tive vectors vailable. The otber sumptions are related to the generatlon of
alternative solutions.
bi the llght beam search it is acknowledged that reference points provlde a
practical and an easy way for the declslon maker to direct the solution process.
Howeveï, the learnlng process of the decisinn mer is suppoled better if the
decisioa maker receives additloaal information about the Pareto optimal set
at each iteratlon. This mens that otber solutions in tbe neighbourhood of
the current solution (bed on the reference point) are displayed. Thus far,
the motivation is the same as in the reference point method. But wht if the
comparison of even a small number of alterimtive solutions is diflictflt for the
decisioI1 maker? Or what if ail the alternatives provided are indifferent to the
decision maker? In such cases tbe decision maker may even stop the solution
process and never get as far as the satlsfactory solutions.
An attempt is made to avoid frustration on the part of the decision maker
in the light beam search by the help of concepts used in multiattribute declslon
maalysis and particularly in ELECTRE methods (see, for example, Roy (1990)
mad Vincke (1992, pp. 5ç69)). Tbe idea is to este.blish autranking relations
between alternatives. It is sald tbat the alternative z z outrmaks the alternative
additional alternatives near the curreat solution are generated so that they
outrank the crrent one. hicompar ble or indifferent alternatives are not shown
to the declslon maker.
To be able to compare alternatives and to define outrmaking relations, we
need several thresholds from the decision maker. Assumptlon 4 is related to
this. Because of the just notfoeable differeace or for some other reasons itis
hot always possible for the declslon maker to distingulsh between different
alternatives. Thls means that there is an inteïval where indifference prevai]s.
For this reason the decision maker is asked to provide indiffence threshoIds
q, for each objective ftmction (i - 1,...,k). In fact the thresholds should be
functions of the objective values, that is qi(zi), but in the llght beam search
The line between indifference mad preference does hot have to be sharp
preferece thresholds Pi for i - 1,..., k. Applying the same reasoning as above,
but constmat, in addition, we must have pi >_ qi >_ 0 for i - 1,...,k.
Given these thresholds we crue distinguish three preference relations between
pairs of alternative objective vectors (z r mad z 2) for each component, t]at is,
each objective function. We can say that as far as the ith components (i -
],..., k) of thc two objective vectors are concerned,
z I and z  are indifferent if [z z] <_ q
z llsweaklypreferredtoz if qi<z-z<P
z  is preferred to z  if z - z > Pi.
One more type of threshold, namely a veto threshold v for i - 1,..., k can
threshold to be constmat and have the relation v >_ p for i - 1,..., k. in this
case z  cannot be prefcrred to z
We crue now define outranking relatioils on the basis of for how mmay com-
ponents indiffereace, weak preference or preference Ís vahd or preference cmanot
be validi Let us compare the objective vector of the current iteration z h and
that is, objective functions, for which the condition mentioned holds. We defom
m»(z,z ) as #i where z is indifferent, weakly preferred or preferred to z ,
mq(z h,z) as #i where z h is weakly preferred to z,
mv(z a,z) as
rn (zt
the declsion maker has specified ail the thresholds, that is the indifference, the
preference and the veto thresholds, it is proposed in Jaszkiewicz and Slowifiski
(1994, 1995) that
zSz  if m(zh,z) -- 0, rn(z,z) < l mad mq(z,z) +m»(zh,z) < m(Z,Z h)
be defined. Thls definition must be modified if no veto thresholds are available.
] thls case
zSz h if mp(zh, z)--0 and mq(zh,z)<m(z,Z).

If no preference thresholds have heen spechïed, the denition is
zSz h if (zh,z)=O and m«z^,z)_<l.
Finally, if on]y indiffeence thresholds are availahle, tbe outranking relation is
defined hy
zSz ^ if mq(zh,z) = 0-
5.9.2. Light Beam Algorlthm
Let us now outline the light heam algorithm.
(1) If the decisioa maker wants to or can specify the hest and the worst
»lues for each objective function, denote the corïespondlng vectors hy
z* and znd, respective]y. Alternatively calculate z * and znd. Set h = 1
and the reference point z h = z*. lnitialize the set of saved solutioas
z B = {. Ask the decision maker to specify an indifference threshold
for every objective function. If desired, (s)he can also spech*y preference
and veto tkresho]ds for them.
(2) Calculate a current solution x h and the corresponding zh hy minimizing
the achievement function vAth z h.
(3) Present z h to the decision maker. Calculate k Pareto optimal character-
istic neighhours of z  and pïesent them z well to the decision maker.
If the decision maker wants to see alternatives hetween any two of the
k + 1 alterrmtlves displayed, set their difference z a search direction,
take different steps in this direction and project them onto the pareto
optimal set hefore showing them to the decision makeï. If the decision
maker wants to save the current solution z , set B = B U {zh}.
(4) If desired, the decision maker can revise the thtesholds. If this is the
cze, set z h = z +, h --- h + I and go then to step (3). Othewise,
if the decislon maker wants to gve another reference point, denote it
by n+, set h --- h + 1 and go to step (2). If, on the other bmad, the
decision maker wants to select one of the alternatives displayed or one
solution in B z a current solution, set it z z h+ , set h = h + 1 and
go to step (3). FinaIly if one of the alternatives is satisfactory, set the
corresponding decisinn vector to he x h+ , set h = h + 1 and go to step
(5).
(5) Stop with x  z the final sohtion.
The option of saving desirahle solutions in set B increzes the fiexJhility
of the method. The decisior maker can explore dflïerent diïectios and select
the best among different trin]s. The possihility of having  look at solutions
hetween any two alternatives is elated to the GDF method. The same idea
will also he handled in Section 5.12. The alternative solutions in step (3) can
he projected hy minimizing the achievement function.
Let us corsider how the Pareto optimal characteristic neighhours of z h
are generated. The thresho]ds specified hy the decisior maker are needed in
defining outankg relations as descfibed in Subsection 5.9.1. Characteristic
solution. The number of characteristic neighbours z(i) is equal to the numher
of objective functions. For each i --- l,..., k, the neighbour z(i) is the point
in the outranking neighhourhood of z h with maximal distance fl'om z h in the
The nalghbours are determined by projecting the gradient of one ohjective
active in z h with gradient projection methods (this necessitates differentiahility
and, thus, zsumption 2) ;see Jzzkiewicz and Slowifiski (1994, 1995) for detalls.
The fezihle direction in the ohjective space offering the greatest improvement
for the ith comportent of z  is denoted by d . The outranking characterlstic
neighbour in that direction is ohtalned with the prohlem
5.9.3. Comments
The idea of the ]ight beam search is analogous to projecting a focused beam
of light from the reïerence point onto the Pareto optimal set. The lighted part
of the Pareto optimal set changes if thc location of the spotlight, that is, the
reference point or the poirt of interest in the Pareto optimal set are changed.
This conncction explalns the name of the method. An implementation of the
light beam search is availahle Ifom its developers (see Section 2.2 in par [1[).
The light beam search can be characterized z an ad hoc method. If a value
function were avallable, it could hOt directly determine new reference points. It
cou]d, however, be used in comparing the set of alternatives. Yet, the thresho]ds
are important in the method and they must corne from the decision maker.
This method comhines elements of multiohjective optimization and multi-
attrlhute decision analysis in an interesting way. A extension is suggested in
Wierzhicki (1997b), where hoth zpiration levels forming a reference point and
reservtion levels (tohe avoided) are used. In ttis cze the reference point still
determlnes the sourcc of light hut the reservatlon ]eve]s are used to generate a
cone of light. Some convergence idez are put forward in Wierzhicki (1997h) z

5.9.4. ConcIudlng Remarks
The light heam search is a rather versatile solution method where the decb
sion m aller can specify reference points compare a set of alternatives and affect
the set of alternatives in different ways. Thresho]ds are used to try to make
sure that the alternatives generated are hot worse than the current solution.
La addition, they are di/ïereat enough to he compared and comparahle on the
whole. This should decrease the htrden on the decislon maker.
Specifyiag different thresholds is a new aspect when compaïed to the meth-
ods presented earlier. This may he demandlng for the decision tnaker. Auyway
it is positive that the thresholds are hot assumed to he glohal hut can he al-
teïed at any time. In other woïds, outranking relations hased o the threshold
values are on]y used as local preference mode]s in the neighhourhood of the
cuïrent solution.
The idea of combining strengths fTom different areas ceïtalrdy deserves fur-
ther study. Neverhe]ess thls approach also has its weaknesses. As noted in
Jmzkiewicz and Slowifiski (1994, 1995), it may he computationally ïather de
tnanding to find the exact characteïistic nalghhouïs in a general case, para[lel
computing is one solution. If this is hot possih]e one can at least present differ-
ent nelghhouïs as soon as they are ca]ctflated instead of walting till ail of them
bave heen generated. The visua5zatlon of alternatives is handled in Chapteï 3
of part III.
5o10. Reference Direction Approach
The refereuce oeection approach was iutroduced in Korhonen and Loekso
(1984, 1985, 1986a) by the name visual interactive approach. It contains
ideas ffom, for example, the GDF nethod and the reïerence point method of
Wieïzhicki. However, more information is provided to the decision naker. The
algofitun works hest for MOLP prohlems if itis desred to check the optimal-
ity of the final solution• Otherwise, the algorithm can he applied to nonlinear
prohlems as welh The algorithm was oïigitally designed for the maxSmization
of proh]ems hut here it is presented in the form of mlnimization. The reference
direction approach and its extensions are also hriefiy descrihed in Korhonen
(1997).
5.10.1. Introduction
In reference point hased methods, a reference point consisting of aspiration
levels for each ohjective function is pïojected onto the Pareto optilval set hy
an achievement functlon. Thls idea is exLended here so that a whole scal]ed
ïefeïence direction is pïojected onto the Pareto optimal set. The reference di-
rection is a vector from the current iteration point to the refeïence point. After
the projection the decision maker can examine this Paeto optimal curve or a
representation of it by the means of computer graphics.
An interesting feature in the refeïence direction approach is that no ex-
pliclt knowledge is assumed ahout che properties of the value fmction during
the solution process. However, suflïcient conditions for optlmality can he es-
tahlished for the terminatinn point of the algorlthtn, if the decislon maker's
underlylng value function is assumed to he psendoconcave (and differentiahle)
at that point (and several other assumptions to ho hsted Inter are fulfilled).
The optimafity conditions are nccessary only for MOLP prohlems.
5.10.2. Reference Direction Approach Algorithm
The algoritln is as follows. Once agala, in the notation we employ ohjectlve
ectors for simplicity. Naturally, the actual calculations are performed in the
decislon varlahle space,
(1) Find an arhitrary starting ohjectlve vectoï z   R . Set h = 1.
(2) Ask the decislon maker to spectfy a ïefeïence point h  R  and set
d h+l  Z h --Z h ç a [lew ïeference direction,
(3) Find the set Z h+ of (weakly) Pareto optimal solutions z that solve the
prohlem
minimize s,(z)
suhjectto z=z h+td ,
z  Z is Pareto optinal,
where s., is an achlevement functlon, w is a weightlng vector and t
has diffeïent discrete nonnegative values.
(4) Ask the decision maker to select the most preferred solution z h+ in
(5) ]ff z h  z h+l, set h = h + 1 and go to step (2), Otherwise, check the
optimality conditions. ]ff the corditions are satisfied, stop with x h+l
corresponding to z h+ as the faal solution• Otherwise, set h = h + 1
and set d h+ tobe a new search direction identified hy the optinality
checking procedure. Go to step (3),
The settin of the algorithm nakes it possihle for the starLing point to he
any point in the objective space. [t does hot have to he feasihle, much less
Païeto optimal, since it is projected onto the (weakly) Pareto optimal set in
step (3). As a weighting vector one can use, for e×ample, the reference point
specified by the declslon maker.
The straight line from the current iteratio point z h (or its Pareto optimal
projection at the fLrst iteration) to the houndary of the Pareto optimal set is
discïetized and pïojected onto the set of Païeto optimal points. The discretiz
tion means using seveïal different values for t. For linear prohlems parametïlc
llnear programming can he used to obt ain a Pareto optinal curve when the pro
rameter t bas values fTom zer0 to infitity. The idea is to plot the ohtalned values

of the objective ftmctinns oI a computer screen as value paths (see Subsectinn
3.3.1 in Part III) with different colours l]ustrating each of the objectives. The
decision maker can move the cursor hack and forh and see the corresponding
The achievement functinn s,.. (z) is ofthe same form as presented in Section
3.5, namely
(5.10.1) sï,w(z) = max --,
where I = {i I wï > 0} C {1,...,k},wis aweighting vector, z Ç Zis an
if t is desired to avoid weakly Pareto opthadi solutions, then an augmentathan
3.4.5 or Section 3.5 (see also Steuer (1980, pp. 422431)). An alternative fs
suggested in Korhonen (1997) aad Korhonen and Haknc (1990), wheïe iexico-
graphic ordering fs used to g-aarantee the Païeto opttmahty of the sohathans.
However, orlginMly such kind of actions wele hot considered to he necessary be-
cause the purpose was simply to produce differelt solutions effectlvely. Distance
measure (5.10.1) bas bcen chosen to faclitate parametlic linear programming
(even though the solutions are only guaranteed to be wealdy pareto opthnal).
formed into an eqdivalent, differentiable form aasuming the different]ahi]ity of
the functioas havolved. Let us, for clarity, formulate the problem in the decisioxx
variable space wheïe it is solved as
mirñmize 
suhjectto ],(x)--awi<_z+tdh, +l forall iI,
If we have z h = z h+ in step (5), we know that the projection of d a in
the weakly Pareto optimal set is hot a directio of improvement. Then we can
apply the fo[lowing suflïcient condition for optimdiity.
Theorem 5.10.1. Let assumptions 2 be satisfied. Let z +  Z and let C
be acone contalning ail the feasible dtrectlons at z h+ (as in (5.10.2)). Let us
U(z h+l) _> U(z h+ +/3d(j)) for ail /3 >_ 0 and j  1,... ,p.
Then z + is a glohally optimal solution (w]th respect to U).
Proof. See Koïhonen and Laakso (1986a).
For MOLP problems we know that if the ctrrent sdiution is not opthnal,
tlen ont of the feasihle directioas of com C must be a direction of improvement.
This direction is then used as a new reference direction in step (3). In other
woïds, to be ahle to apply Theorem 5.10A at a certain point, the decision
maker must first check every feasible direction at that point for impïovement.
maker. If is demonstïated in Halme and Korhonen (1989) and Korhonen and
Laakso (1986a, b) how the number of search drections can be ïeduced. For
checked ha practice (an iafiùte number of checks would ho nceded).
5.10.3. Comments
Note that the termination condition of Theoiem 5.10.1 is analogous to the
Kaïush-Kuhn-Tucker optinality conditions. This is pïoved in Halme and Kor-
honen (1989). If the value function is ktown, it  easy to compare alternative
ohjective vectors. However, what was sdid concerning the difficulty in deter-
mlning new ïeference points in coimecthan vdth the ïeference pohat method in
Section 5.6 is also valid here. Thus the rcference direction approach can he
characterized as an ad hoc method.
The graphical illustration of the alternatives has been an important as-
pect in the developmeat of multiobjective optimLzation methods that seek to
impïove and faciitate the co-operatlon between the declsion makeï and the
analyst (computer). That is why graphical illustration of the alternatives is
here emphasized.
The computathan ttmc for large problems can he reduced by presenting one
piece ata tlme of the wealdy Pareto optimal curve or its ïepïesentathan to the

decision maker, if (s)he finals the end point to he the most satisfactory one
then the nex piece can he presented, if the nuraher of objective functions is
large, the quality of graphical illustration suffers. For tlds reazn, it is advisable
hot to bave more than ten objective fnctions ata rime.
If it is hot desired to check the optimality of the final result, the prohlem
to he solved does hot have to satisfy any special sumptions. This means that
the reference direction approach cœ bc app]Jed to more general problems. The
reverse is va[id z well. If the zsumptions set are not satisfied, the optimality
cannot be checked, hut the method can, of course, be used in any other way.
A similar interactive line search algoïithm for MOLP prob]ems is presented
in Benson and Aksoy (1991). The procedure generates only Pareto optimal
points and is able to automatically correct possible errors in the decision
maker's judgement.
The idez of the ïeference direction approach are adapted to the goal pr
gramming enviroament in Korhoncn and Laakso (1986h). The intention is to
relax the predetermined foies of the objective fnctions and the constralnts,
that is, to enable the foies tobe interchanged. For that reazn the prohlem to
he solved is now zsumed to he in the generalzed goal programming form (see
Section 4.3). The ohjective functions are considered to he flexible goals and the
constralnt fnctions infie)=ihle goals. At each iteration, the decision maker can
ezily convert flexihle geais into inflexible ones and vice versa. This increzes
the Ifeedom ofthe decisioï maker. Comhialng achievement functions into goal
programming also eliminates the problems caused by fezib]e zpration levels
(see Section 4.3),
The idea of changing the roles of the fonctions is refaed in Korhonen and
Narula (1993). A systematic way of changlng the foies of the objective functions
vd the constralnts is descrihed therein. The presentation examines where and
finw the changes can be carried out. This systematic handling conce.ns MOLP
probleras, hut the idea can in principle be generalized to other prohlems.
A dynamic user interface to the reference direction appïoach and its adapta-
tion to generabzed goal programming is introduced in Korhonen and Wallenius
(1988). This method hz heen designed for MOLP prohlems and is called the
Pareto race. The software system implementing the pareto race is cal[ed VIG
(Visual Iteractive Goal programming) and it is descrihed in Korhonen (1987,
1990, 1991a) and Korhone]l and Waenius (1989c, 1990). VIG is a dynmJc,
visual and hteractive solution system for MOLP prohlems with the emphazis
on graphical illustration.
The pareto race develops reference directions in a dynamic way. In VIG,
the reference directions and the step-szes are updated accoïding to the actions
of the decision maker who can thus feel that (s)he is in control. The decision
maker can travel around the (weakly) Pareto optimal set z if driving a car.
The pioneering idez of rea5zig user interaces in VIG are supported by a
compaïison of rive MOLP programs in Korhonen and Wallenius (1989b). VIG
wz round to he superior. The main reason wz that tle decision makers round
the zpiration levels tobe a comfortable way of expressing preference relations.
The Pareto race is extended into a computer graphics-bzed decision sup-
por system in Korhonen et al. (1992b). The new method is especially useful
for large-scale MOLP prohlems.
5.10.4. Concluding emarks
I the reference direction approach the role ofthe decision maker is reminis-
cent of the reference point method. (S)he hz to hoth specify reference points
and select the most preferred alternatives. I the reforence point methoŒEs, how-
ever, there are fewer choices to select Ifom. If the prohlem is set in a genet alized
goal progcamming form, the decision maker can also interchange the ïoles of
the ohjective and the constraint lhnctions. By the ïeference direction approach,
the decision maker can explore a wider part of the weakly Pareto optimal set
than hy the refeïence point method, even by providing slmilar reference point
information. This possihility brings the tzk of compaing the alternatives and
selecting the most preferred of them.
The reference direction approach works hest for MOLP prohlems, z it hz
hzical[y heen desiffaed for them. It is interesting that the method requires no
additional zsumptions about the prohlem and the underlying value lhction
u]til the optimaSty of the final solution is to be examined. The optimality can
be gaaranteed tmder certain zsumptions and with some effort.
The performance of the method depends greatly on how well the decision
maker manages to specify the reference directions that lead to improved solu-
tions. Korhonen and Laakso (1986a) mention that particularly when the num-
ber of ohjective fnctions is large, the specification of reference points may
be quite laborious for the decision makeï. I this cse, they suggest that ran-
dom directions in conjunctlon with decision maker-defaed reference directions
should be used. See Korhonen and Laakso (1986a) for a discussion concerning
otheï ways of specifying the reference directions. Natral[y, the choice of the
weighting coeflïcients affects the direction of the projection even though the
selectlo of their values hz hot been stressed here.
The consitency of the decision maker's answers is hot important and itis
not checked in the algorithm. Thus the algorithm may cycle. This can also he
seen z a positive featuïe, slnce the decision maker is ahle to return to such
parts that (s)he already hz examined, if (s)he changes her or his mind.

5.11. Peference Direction Method
The reference direction (RD) method, introduced in Naruin et al. (1994, h)
is closely related to the reference directio11 approach. As its name suggests it is
also hased on refereace directioas. To avoid confusion hetween these two meth-
ods with very similar names we use the name RD method when referring to the
reference direction method in what follows. The RD method has heen designed
for aonlinear maximization pïohlems hut heïe we revise it for midimization and
gener alize it. In addition, we ïelax the original convexity assumptinn and settle
for local optima.
5.11.1. Introduction
In the ID method, ohjectivc fmctinn values z h calcdiated at a point x  are
pïesented to the decisinn maker nd (s)he is asked to specify a reference point
h consisting of desired levels for the ohjectlve fuactions. Once again we move
around the weakly pareto optimal set, which is why some ohjective functions
must he allowed to incïease in order to attaln loweï values for some other
ohjective functinns. La other words, somc components of the reference point
have to he lower and some others higher or equal when compared to the current
solution. Allowing the set of higher values to bc empty is a genet alization of the
original form of the method. (Weakly Pareto optimal solutions crue he made
Pareto optimal.)
As mentloned earlie L specifying a reference point is equivalent to an impliclt
classification using three classes and indicating those ohjective functions whose
values should he decreased till they reach some acceptahle aspiration level,
those whose values are satisfactoïy at the momen L mad those whose values are
alIowed to increase to some upper hound. Let us denote the sers of funcions hy
I< I and I >, respectively. We deaote the components of the reference point
coïresponding to the set I > hy h hecause we have upper hounds i ques-
tioa. To put it hriefly, a reference point is heïe sensihle and the corresponding
classification feasihle if I <  { and I >
It is once again assumed that
L Less is preferred to more h$ the decisioa makeï.
The reference diectlon z h - z h is a fundamental element i the RD method.
The decision mer specifies a priori the numher of steps to he taken in the
reference direction. The idea is to move step hy step as loag as the decisio
maker wants to. La other wods, extra computatinn is avoided hy calculating
oaly those alternatives the decision maker wants to see.
5.11. leference Direction Method 191
The alternatives are produced hy solving the RD problem
minimize max f (x) - z
(5.11.1) suhjectto f(x)+(z-eî) for l le I >,
f(x) z for 1 ici ,
where z h is the current solutlon 0   < 1 is the stesize h the reference
direction, z < z for i Ç I < and  > z for i  [>. The prohlem is nondiffcr
entiable hut it coe be trsformed ito a fferetiahle form hy introduci 
ditina vahle  descrihed earlier (see, e.g., pmhlem (3.4.3)). If some of
the ohjective or the constraint functions e nondifferentinhle, a single ohjective
solver appllcahle to nonfferentinhle problems is needed.
The D prohlem produces weoey Pto optimal solutions.
Theorem 5.11.1. The solution of RD prohlem (5.11.1) is wely Pareto op-
tlm for every 0 <  < 1.
Proof. Let x*  S he a solution of the ID prohlem for some 0 < a < 1. Let
us assume that itis not weakly pareto optimal. La this case there exists some
point x ° ff S such that /,(x °) < /,(x*) for every i - I,..., k.
Because x* is feasihle in prohlem (5.11.1), x °, heing weakly pareto optimal,
must also he feasihle. In addition, zî 2î > 0 for every i Ç I < ald that is why
f(°)  * 
< f( )  foreveïy iI <.
z, z
This implies that
and, thus, x* cammt he a solution of pïohlem (5.11.1). This contradiction
completes the proof and x* is weakly pareto optimal. []
A result conceïning the opposlte direction mad Pareto optlma5ty can also
he estahlished.
Theorem 5.11.2. Let x*  S he Pareto optimal. Then there exists a reference
poLat and a real numheï 0 _< a < 1 such that x* is a solution of RD prohlem
(5.11.1).
Proof. Let x*  S he Pareto optimal. Let us assume that there does not exlst
and a such that x* is a solutio of the RD prohlcm. Let us suppose that

Let us choose f(x*) z a reference point. This means chat we set 2 = /(x*)
for those indices where z h > il(x*) and we denote this index set by I <. Flr ther
we set i = /i(x*) foç indice i E I > satisfylng z h < /i(x*). Finally, the set
of indices wheïe z = /;(x*) is valid is dellotad hy I . This setting is possihle
aCCOrding to Theorem 5.11.1. That is why I <  { and I- k91 >  {. In addition
x ° E S that is a solution of the RD problem, memalug that
max fi(:°) z' f(x*) - z h •
Thus f(x °)-z h < (z -f;(x*)),thatis, f(x °) < fi(x*) for aHi  I <.
Because x ° is a solution of prohlem (5.11.1) it muet he fezihle. In other
words, we bave f;(x °) < ff(x*) ÷0 for   I > and fi(x °) < f(x*) for i  I-.
Accoïding to Theorem 5.11.2 we know that any pareto optimal solution can
he found with an appropriate clzsifiction.
An augmented formulatioz of the RD prohlem is presented in Nartfla et
al. (1994a, h) in order to produce only Pareto optimal solutions.
5,11.2. RI) Algorithm
The steps of the RD algerithrn are the following:
(1) Calcuinte a starting solution x I hy solving auxiliary prohlem (5.11.2).
Show the corresponding ohjectivc vector z z to the decision maker. If
(s)he wants to stop, go to step (5). Otherwise, set h - 1.
(2) If the decisin maker does hot want to decrese any component of z a,
go to step (5). Otherwise, zk the decision maker to specify a reference
point z where some of the components are lower and some higher or
equal when compared to those of z . If there are no higher va]ues set
P - r = 1 and go to step (3). Otherwise ask the decisinn maker also to
specify the mKximum numher of alternatives p (s)he wants to see. Set
(3) Set  = 1 - r/P. Solve RD pïohlem (5.1L1) to obtalll a solutinn xU(r)
and the correspoading z(r). Set r - r + 1.
(4) Show zh(r) to the decisinn maker. If (s)he is satisfied, go to step (5).
step (3). Otherwise, ifr > P or the decision maker wants to change the
reference point, set z h+ - za(r), h - h + 1 and go to step (2).
(5) Stop with x a corresponding to z h z the final solution.
The starinfi solution is calcuinted hy solving the prohlem
(5.11.2) i-..k
suhject to x  S.
Naturly, this prohlem can he formulated   diffe:entiable prohlem with, if
necessy and possihle, the help of oe dition vable ( elier).
5.11.3. Comments
The RD method is an ad hoc method. The existence of a value fnction
wou]d hot help in specifying reference points Or the numhers of steps to he
taken. It could imt even help in salecting the most preferred alternative. The
reoson is that one must declde for one point at a rime whether to calcu]ate new
alternatives or hot. If the new alternative turned out to ho less preferred than
its pïedecessor, one could not go hack anyway.
In Miettinen and M£keli4 (1998a), a water quality nlanagemet problem
is solved hy thc RD method. A modification of the RD method for covex,
mnlinear integer prohlems is introduced in Gouljoshki et al. (1997).
5.11.4. Concluding Remrks
The RD method can he considered an interactive clzsificatlon-h sed meth
od. It does hot require artificial or comp]icated ifformation from the decision
maker; on]y ïeference points and the numheï of intermediate solutions are used.
The decisfoa maker is hot zked to compare several diffeïet alternatives but
on]y to declde whether another alternative is to be generated Or hot.
The decislon maker must a priori deteïmine the numher of steps to he taken
in the refeïeace direction and then intermediate solutions are calcu]ated one
hy one z long z the decision maker wants to. This can he seen z a henefit z
wel] z a weakaess. O the one hand, itis computatinal]y efficient since it may
he unnecessary to ca]cu]ate ail the intermediate solutions. On the other hand,
the decision maker is unah]e to return toa solution oce it hz heen discarded,
which may he a disadvantage. Further, the numher of steps to he taken cannot
he changed.

Tbus fr, we have described 8everal dtfferen methods for multinbjective
optLmizatlon. Tbe question of differentib[lity bas hot been empbaslzed. How
ever nondifferentibilty and many kLads of irregularities and disconginuitles
are cbaracteristic of real-world optimlzation problems, for example, in ecoe
been solved (e.g.,  Hllnger and Neittaanmiki (i988)) by firSt scalarlzing the
is avaiLable.
5.12. NIMBUS Method 195
the weaknesses detected in the older methods. Most of tbe methods previoly
descrlbed lmve had an effect on the development of NIMBUS, Either they hve
offered usefl ideas to adopt or unsatisfactory properties to avoid.
Trafle-off rate information cannot be exploited in nondifferentiable prol>
lems in tbe way it is used La the ISWT method and in SPOT and STOM,
The natural reason is that obtainhlg trade-off information from the Karusb-
Kuhn-Tucker multipflers necessitates that the functions are twice contLauos
diffeïentiable. How to obtain trafle-off information in nondifferentiable cases
needs and deserves more research.
Tbe ideas of reference points mad satisficing decision makLag seem tobe
general[zable to nondifferentiable problems. We can adopt the ideas of classify.
ing the objective functions and reference points and m]x them with some ideas
from nondifferentiable analysis, The otcome is described La tbe next sectloi.
5.12. NIMBUS Method
. NIMBUS (Nondifferentiable Interactive Multinbjective BUndle-based opti-
mzatlon System) presented in Miettinen (1994) mad Miettinen and Mikel/i
(1995, 1997), is an interactive multLabjective optimation method designed
especinlly to be able to handle nondifferentinble functions efficlently. For thi
reason, it is capable of solving comp]cated real-world problems. We introduce
two versions of NIMBUS. They have diffences La both their tbeoretlcal and
computational aspects. Theoretically, the versions differ in handlLag tbe infor-
matlon requested from the user. Numerical experiments idicate differences in
the computational eflïciency and cotrollability of the solution processes.
5.12.1. Introduction
The starting point La developLag tbe NIMBUS metbod has been somewhat
the opposite to theoretical sounchess. EmphasizLag theoretical aspects may
lead to difficulties on the declsioa makeï's side and more or less instable results,
not to mention higher computational costs. IJ the NIMBUS method, tbe idca
bas been to oveïcome the difficultles encountered wlth many other interactive
met bods. The most important aspects bave appeared to be the effectlveness and
tbe comfoïtableness of tle decision makeï. Tlms, the interactlon phase has been
aimed at being comparatively simple and easy to undeïstand for the declsion
maker. NIMBUS offers flexible ways of performing Lateractive evaluation of
the problem and determinig the prefereaces of the de¢ision maker duïing
the solution process. At each iteration of tbe interactive soLatlon process tbe
decisLan maker can direct the search aCCOrding to her or his wishes.
Aspiration tevels and classification have becn selected as tbe meaas of inter
action between the decision maker and the algoïithm. It has been emphasized
on several occasions (e.g., in Nakayama (I995)) that an aspiration level-oEed

approach is effective in praCtical fields. Among the vMidating facts for this
statement are the followlng. Aspiration levels do hot require consitency from
the decision maker and they refiect her or his wishes we]]. In ddition, they are
easy to implement. Using aspiration levels as a way of receivii*g iaformation
Tom the decision maker means avoiding diflïcult and artificial concepts.
It is assumed that
1. Less is preferred to more by the decislon maker.
2. The objective and the constraint fnctions are locally Lipschitzian.
3. The objective functions are bounded (ïrom below) over the fezible reglon
The second assumption cornes fvom nondiffeïentiable analysis, and the third
assumption from the requirement of having the ideal objective vector avallable.
h the clsification of the objective fnctions, the decision maker can ezily
indicate what kind of improvemelts are desirable and what kbld of impalrmcnt s
are tolerable. The idea is that the decision maker examines at every iteration h
the values of the objective functions calculated at the current solution x h and
fl whose values
o should be decrezed (i • I<),
o should be decremed to a certabl zpiration level z h (i • I-<),
o are satisfactory at the moment (i E I-),
o are a[lowed to increze to a certala upper bound c h (i • I>), and
o are a[lowed to cblge freely (i • I°),
h ddition to the clzsification, the decision maker is zked to specLfy the
for i E I > suCh that  > f,(x). Notice that the two somewhat parallel clzses
I < and I < are available. The difference between them is that the functions bi
I < are to be minimized z far z possible but the fnctions iIx I < only z far z
The clzsification is the cote of NIMBUS. However, the decision makeï can
weighting coeflïcients w summing up to ont (for numerical stbility). ]ff the
equal to ont. Note that the weighting coeflïcients do hot change the primary
NIMBUS hz more clzses than STEM, STOM or the RD method. In this
way the decision maker hz more fveedom in speciïying the desired changes in
the objective values and (s)he can select a clzs refiecting hcr or his desircs
h practicc, it means that hot ail the objective fuctions have to be clzsified
at ail. Naturally ail the clzses do hot have tobe employed.
5.12. NIMBUS Method 197
Aer the decision maker has classhïed the objective nctlons, one of che two
alternative subproblems, called vector and scalar subproblems, is formed. Thus,
the original mtfltiobjective optimization problem is transformed into either a
Imw multiobjective or a sin#e objective optimization problem, accordingly.
The subproblems lead to two different versions of NIMBUS, to be ca[led vecor
version and scalar version. We first introduce the older, that is, the vector
version.
5.12.2. Vector Subproblem
According to the classification and the connected inïormation, a vector sub
problem
{«x i «<, «ma ma [-,
(5.12.1) subject to /,(x) _< /,(x h) for ail i  I-,
](x) < for ail i•I >,
is formed (set Miettinen (1994) and Miettinen and Mkel& (1995)).
The vector subproblem seems to be even more complicat ed than the origánal
problem. Nonetheless, the advantage of this fmrmulation is the fact that the
opinions and the hopes of the decision maker are taken careful]y into account.
Notice that if I < # , we have  nondifferentiable problem regardless of the
differexxtiability of the original problem. Fnis fact does hot bring may dditional
diflïculties since we are in any cze prepared for handling nodiffcrentiabilities.
The vector version is quite general. The clzsification of the objective franc-
tions can be performed z if the  constralnt method, the weighting method,
lexicographir ordering or goal programming were used to produce new solu
h order to be able to solve the vector subproblem, we need the MPB method
(introduced in Section 2.2). If the constraints are inequalities, that is S -
{x • R" I g(x) - (g(x),g2(x),... ,g,(x)) T < 0}, the improvement ïunction
H: R  × R' -* R applied to problem (5.12.1) is of the form
H(x',x ) =max {fï(x)/w .h, f;(x)/w h, (i • I<),
f,(x 1) £(x), (i •
( = 1,...,m)}.

As explained ha Section 2.2, the mbaLralzation ofthe irapr ovement functlon takes
place iteratively. For details, see also Mikel£ (1993) and Miettinen and M/kel
(1995, 1998a).
bi the scalar version of NIMBUS afer the classification, a scaIar subproblem
minitize mmx w/h(fi(x)--zî),whmax[fj(x)--5, 0]]
(5.12.2) subject to /(x) 5/i(xh) rl iI <UI gUI =,
A(x) e forl iI>,
. (1996b)), where zî for i  I < e components ofthe ide objective vector
(sumed to be kno ob).
Noice that problem (5.12.2)  nondiffercntiable bu h one objective func-
tion. It can be solved by y method for aonffereniable singlc objective
minimize mx
(5.12.3) subjectto f(x) f(x u) rall iI <I 5I =,
/(x) < e fo 1 i e I >,
where the piration level is constant 5 = zî for i  I < . This formution seems
somewhat slmpler but the idea is the saine. Subproble (5.12.2) d (5.12.3)
following, we refer to problcm (5.12.2)  the scM subproblem but problem
(5.12.3) could equally be used stead.
We prove in Subsection 5.12.5 tt the solutions of the vector d the scMar
5.12.4. NIMBUS Algorlthm
The solution of vector subproblcm (5.12A) or scalar subproblem (5.12.2) is
denoted by h. If the decisfo maker does hot like the objective vector f(h) for
some reason, (s)he cern explore other solutlons between x h zmd h. This means
that we calculate a search direction d h = h x h and provide more solutions
5.12. NIMBUS Method 199
by taking steps of different sizes in tbàs direction. The step-size is determined
by the decision maker as in the GDF method. Objective vectors f(xh œee td h) are
calculated wth ditferent values of t (0 < t _< 1). Their weakly Pareto optimal
counterpars are presented to the decision maker, who then selects the most
satisf36ng solution arnong the alternatives.
A detailed algorithm of the NIMBUS method is given below. The sarne
algoritfon i valid for both of the NIMBUS versions. Note that the decision
maker must be reody to give up something in order to attain improvement
for some other objective fonctions. The search procedure stops if the decision
maker does not want to improve any objective function value.
(1) Select subproblem (5.12.1) or (5.12.2) tobe used in the contlnuation.
Choose a starting point x E R  and project it onto the feasible region
by solving auxiliary problem (5.12.4). Denote the new point by x °.
Calculate its wealdy Pareto optimal counterpart x 1 by etting I < -
{1,... ,k} and by solving the selected subproblem. Set the iteration
counter h - 1.
(2) Ask the decision maker to divide the objective function into the
classes I <, I <-, I-, I >, and I ° at the point z h - f(x h) such that
I U I >  I °#  and I <  I < # . If either of the unions is empty,
go to step (9). Ask the decision maker for the aspiration levels zi  for
i  I < and the upper bounds Qh for i  I >. Ask also for the optional
weighting coefficients w h > 0 for i  I <  I <, summing up to one.
(3) Calculate ±h by solvàng the subproblem. If h -- X h, ask the decisfon
maker whether (s)he wants to try another classification. If yes, set
x h+t = x h, h = h + 1, and go to step (2); if no, go to step 9 .
(4) Now  s a new solution. Let us denote h = f(h). Present z h and
h to the decision maker. If the decision maker wants to see different
alternatives between z h and h, set d h -- n _ x h and go to step (6).
If the decision maker prefers z h, set x h+ -- x  emd h -- h + 1, and go
to step (2).
(5) Tho decision maker wants now to continue from h. If i < # , set
xt*+l = h, h - h + 1, and go to step (2). Otherwise (I< = ), the
weak Pareto opt bnality must bc guaranteed by setttng I < = { 1,..., k}
and solving the subproblem. Let the solution be h. Set x h+ - Rb
emd h - h + 1, and go to step (2).
(6) Ask the decision maker to specify the desired number of alternatives P
and calculate vectors f(x h + tdh), j _ 2,..., P - 1, where ri - Pt'
(7) Produce weakly Pareto optimal objective vectors from the vectors
above by solving auxiliary problem (5.12.5).
(8) Present the P alternatives to the decision maker and let her or him
choosc the most preferred one aong them. Denote the corresponding
decision vector by x h+r and set h - h+ 1. Ifthe declslon maker wants
to continue, go to step (2).
(9) Check the Pareto optimality of x h by solving auxiIiary problom
(2.10.2) of Part I with x h as x*. Let the solution be (,).

(10) Stop wlth the final solution .
The projection in step (1) is conneeted to the fact that most solvers ne
cessitate feasible starting points. Thus, itis a more implementatlonal thon
algofithmlc marrer. Let us mention that, for exarnple, ff the feasible region
consists of inequality constraints gï (x) < 0 for i -- 1,..., m, ny staring point
can be ptojected onto the feaslble region by solving the auxiliary problem
minmize max[O, gl(x),g2(x),...,gm(x)]
(5.12.4) snbject to x E R u.
The justification of step (5) is given in the optimality results of Subsection
5.12.5. If we only employ the class I < to nize lhnctions (nd the class I <
is empty) we do hot necessarily stay within he weakly Pareto optimal set. In
this case we project the obtalned result onto the wealdy Pareto optimal set.
This is acceptable according to assumption 1.
The intermediate solutions between z h and h are hot necessarily wekly
Pareto optbnal. That is why they have tobe projected onto the weakly Pareto
optimal set. A practical wy of dong sois to employ the results of Coro[lary
3.5.6 nd solve the auxiliry problem
mirdmize max [ri(x) f(x h + td)]
(5.12.5)
subject to x E S
for every j -- 2,...  P- 1. Tbs tretment works for convex as well as nonconve
p[oblems. An alternative method cn be applied if the vectoï subptoblem is
used and the problem is convex (see Miettinen nd Mkel (1995)). In this case
weak Païeto optLmality cn be guaïanteed by solvlng the vector subproblem
wlth I < - { 1, ..., k} starng from each inteïmediate solution.
Since the Pareto optimality ofthe solutions produced cammt be guarnteed
(see Subsection 5.12.5) we check the final solution ih the end by solving n
additional problem introduced in Theorem 2.10.3 of Paït I. As the decision
makeï was assumed to ptefer less to more, we cn presume that (s)hc is satisfied
with the Pareto optimal final solution even where it was hot heï or his choice.
For clarity of notation, it is hot stted in the algoit hm that the decision maker
may check Pareto optimality at ny tLme durng the solution process. Then
problem (2.10.2) of Part Iis solved with the coErent solution as x*.
Note that, if scalar subproblem (5.12.2) is employed in the algorithn, we
have to calculate the components of the ideal objective vector z* in the first
step. Iowever, presentig * to the decision makeï ves valuable information
bout the pïoblem in both NIMBUS versions.
We must ïemember tht we cnnot guarantee global optimality. If the solu-
tion obtalned is hot completely stisfactoïy one cn always solve the problem
agaln from  différent starig point. Tbs action is also advised if the decision
maker has to stop the solution process with )h = x  aller step (3).
5.12. NIMBUS Method 201
If is also possible fo improve the algorithm in step (3) fo avoid thc case
h -- X h. If the upper bounds specified by tffé decision inaker are too tight,
one cn use them as a refeence point nd project them wlth (5.12.5) onto
the (weakly) Pareto optLmal set. Show[ng the new solution fo the decision
maker proddes ber oï him with information concerning the possibiSties and
the ILmitations of the ptoblem, nd some dead ends can bc avoided as well.
Unlike some other methods based on classification, the success of the solu-
tion process does not depend entirely on how well the decision mker manages
in specifyig the classffication nd the appopïiate paraneter valms. If is im-
portant that the classification is not irreversible. Thus, no iïrevocable damage
is caused in NIMBUS if the solution f() is not what was expected. The
decision maker is IYee to go back or exploïe intermediate points. (S)he cn cas
ily get fo loow the problem and ifs possibilities by specifying, foï exmnple,
loose upper bounds mtd examining internediate solutions. NIMBUS is indeed
lear ning-oriented.
5.12.5. Optimality Results
First, we stae two theoretical results concerning the opthnality of the so-
lutions of vector subproblem (5.12.1) and scalar subpïoblem (5.12.2).
Theorem 5.12.1. The Pareto optinal solution of vectoï subproblem (5.12.1)
is wealdy Païeto optimal (to the original multiobjective optimization pïoblem)
if the set I < is nonempty.
Proof. Let us denote the feible ïegion of vector subproblem (5.12.1) by
Let x*   be a Pareto optimM solution of the vector subproblem with some
sets 1% I<  I > mld I , w'e I <  . In other woïds thre does hot
eist other decision vector x  S such that f(x) g f,(x*) for Ml i  I < and
'jel < [,n[fj(x)- z, 0]] < ,X3el< [mx Ifs(x*)- z, 0]] d at let
one of the inequaliçies is strict.
Let ts sume that x* is hot weiy Peto optimM ï the originM pïobiem.
Ts es that here ests a decision vectoï x
foï alli -- l,...,k.
Bccause x  is a feible solution of problem (5.12.1), we have f,(x ) <
fi(x*). f(x h) for i  I and f(x °) < f,(x*)  eî r i  I >. Thus, also
For ail i  I
z', 0]  x [f,(x*) -Sî, 0] for all i  I <, and, further,
[mx

for all i C I < # 0, the point x* camaot be a Pareto optimal solution of the
opt[ral. The proof is also valld if smne of the classes I <, /-, I > or I  are
empty as long as I > U I ° # 0. []
Theorem 5.12.2. The solutlon of scalar subproblem (5.12.2) is weakly Pareto
optlmal if the set I < i nonempty.
Proof. Let us denote the objective functlon of scalar subproblem (5.12.2) by
'(x) to be miMmlzed and che feasible reglon by . Let x* E  be a solution of
the scalar subproblem with sme sets I<t I <, I , I >, eaid I °, where I < # 0.
In other words, f(x*) < f(x) for MI x C S.
exists a vector x °  S such that /(x ° < /i(x*) for ail i -- 1, ..., k.
Note in the following that all the weighting coefficients are strictly positive.
Because x* Ç , we have fi(x °) < f,(x*) < flux h) for i  I < U I < D I ad
f,(x °) < /i(x*) _< ci for i  I>. Thus, also x ° E S.
Since z** _< f(x °) < f,(x*) for ail i  I < # 0, we have f(x*) - z** > 0 for
Let us conslder
f(x °) = max wi(.f,(x ° -zî ,wjmax[fj x °)-2, 0]1.
The maximum cm be attained either in thc class I < or in I < (or, naturally, in
both of them), lai the first case wc have
/(x °) - w(/i(x °) î) < (/,(*) - ) <
optM. T proof  Mso valid if some of the cluses 15 , I-, I > or I  arc
subproblem (even the proofs tan be combined).
5.12. N1MBUS Method 203
Proof. Let x*  S be Pareto optbnal. Let us assmne that there does not
exist a classification such that x* is a solution of the vector or the scalar
subproblem. Let us suppose that we hve the current NIMBUS solution x h
raid thc corresponding z h availz*ble.
Let us choose f(x*) as a referencc point. This mcns that we choose z, --
f(x*) for those indices where z, h > f(x*) and set i • I <. Further, we set
- f(x*) for indices i  I > satisfying z, a < f(x*). Finally, the set I
Consists of indices where z, h  ri(x* . Thls settig is possible becallsc x* is
assmned to be Pa'eto optimal and x ' is weakly Pareto optimal according to
Theorems 5.12.1 eaid 5.12.2 eatd the structure of the NIMBUS algorithm. That
whyI <#$andI PI >#o.Inaddition,wcsetwi-1 fori6I <.
Bccause x* is nOt  solution of the vector or the scalar subproblem, thcre
exlsts anothcr point x ° C S that is a solution, meeafing that
Thus, mx [fj(x °) f(x*), 0] < 0 for every j  I <. In other words, we have
f;(x °) < ]j(x*) for every j  I <. Because x ° is a solution of problems 5.12.1
eaid 5.12.2, it must be feasible. In other words, we have ri(x °) _< ri(x*) for
5,12,6, Comparison ofthe Two Versions
The vector eaid the scalar versions of NIMBUS differ in the form of the
subproblem uscd. The origln of thc development of the scalar version lies in
the drawbacks discovcred in the vector version.
Theorctlcally, the solution of the vector subproblem has to be Pareto opti-
mal in order to guarantee weakly Parcto optimal solutions to the original mub
tiobjective optimizatlon problem. This assumption is quite demandlng. With
the scalar subproblem we do hot bave probloms of this kind. Further, the vector
version nccds a special soiution tool MPB. In additioa to thls limitation, the

role of the weiating coeflficients is hot commensurable between the classes I <
nd i -<. This imples that the controllabillty of the method suffers.
The dvautoge of hving a single objective Mnction in the scalar veïslon
optim[zation. This gJves more generallty nd applcabillty fo the method. Fur-
consistent way nd, thus the roles of the weiting coefficients are also iden-
Notice tht in ddition fo the difference in the objective nctlons of the
subproblems, theïe is also devation in the constïalnt part. Due to the goal of
the classes I < nd I-< we have to make sure Chat the values of these broc-
tions do hot bcïease. This is the reaSOn foï modfyh}g the constïalnts of scar
In the vectoï subpïoblem the MPB method does ot allow incïement ht
l<. However, there is no guarntee that the values ofthe functions in I< could
hot increase. If is clear that [ncluding oAditlonal constïalnts in n optlmizatlon
occuïs very rarely in Che vectoï version, no oAditlonal constralnts have been
used ii oïdeï fo emphasze computational efficiency. Thus either we pay the
price of additlonal computational costs oï tke the risk of blcremeRt Idepending
On Che one hnd, thc calculton of the ideal objective vectoï used n the
scalar version also needs computatonal effoït. On the otheï hnd the ideal
objective vector cn providc suppoïtbtg infoïmatlon for the decision maker in
any kitd of multobjective solution process.
A numercl comparon of the two versions of NIMBUS is ïeported in Miet
tben nd Mikel 11996b, 1998a) wlth versatile multiobjectve optimlzatlon
problems. The standards of comparison chosen are computatonal efficlency
nd the opinion ofthe decsion maker concernng the contïoflablty of the dif-
subïoutne contalnlng the objcctive functons is ced. The coatïollabillty sde
must be elicited from the decision makcï. It is measuïed in the form of a ïatng
Ibetween I nd ).
5.12. NIMBUS Method 205
5,12.. Comment s
The NIMBUS method has hot been developed to converge in the t ïaditional
sense. While the method does mt assume the existence of ny underlying value
function, no explicit convergence ïesults cn be put foïward on the basis ofthe
assumptions about the properties of the function. It pariculaï, the intention
has been to release the decision makeï from the assumption of an underlying
value function. What is hnportant is that the method satlsfies two desiïable
properies of interactive methods: not fo place too demmding assumptions
on the decision maker or the information exchnged, nd to bc able fo find
(weakly) Pareto optimal solutions quickly and efficiently.
The alto has been to foïmulate a method where the decision makeï cn
easily explore the (weakly) Pareto optbnal set. When the decision maker no
longer wnts fo change any objective fuactlon value nd the solution pïocess
stops the solution is then optñnal.
An impoïtnt factoï is that the fial solution is always Pareto optbnal
because of the structure of the algorlthm. In addition,  the intermediate
points arc ai least substatlonary points nd they cn be pïojected onto the
Pareto optimal set, if so desh-ed.
The method is oA hoc in natuïe, since the ex,tente of a value function
would hot directly advise the decislon makcï how fo act to attaln he or hls
desires. A value function could only be used to compare different alteïnatives.
The posslb[lity of inter chnging the roles of the objective and the constïalnt
functions has been mentloned tbus faï ii connection with some methods. This
is easy to carry out also in NIMBUS because the class I > is nothng but
constïaints with uppeï bounds. One cat even go that far as fo foïmulate ail
the constralnt fUnctions as objective functions nd modify thalï uppe bounds
oï roles duriag the solution pïocess from one iteïtion fo thc otheï.
5.12.8. Imple ment atiorLs
The NIMBUS algorithm was origJnally implemented in the malnframe en-
vlronment at thc Unlversity of JyvskylR F[aland. This appïoach is suitable
ior even large-scale problems, but the lack of a flexible user inter face decïeases
its usabi5ty. It is evldent that the user interface plays a crucial foie n realizlng
interaztive algorithms.
An alteïnative is to use microcomputers to develop a functional user inter-
face by paying the price of reduced computational capaclty. However, both the
malnfrzane and the microcomputer envirormmnt share weaknesses in common
from the viewpobit of both the user and the developer of the hnplementation
As far as the user is concerned, the system and some spec[fic compilers have to
be installed. Itis 5mlting that the programs are approprlate only for certain
computer environments and operating systems. For the developer the defivery
of the software updates is laborious. Implementing and keeping up separate
vcïsions foï different environments requires also extra effort.

Miettlnen and Mkel (1998b)).
and avallable to any Interner user (http:]]nimhus.math.jy.fi]). No special
5.12.9. Applications
In Miettnen (1994), two academic problens and a state-constraned optinal
control problem concerning an elastic string are solved hy the vector version of
NIMBUS, The vector version is applied to solve an academic nondifferentiahle
test prob/em and a river pollntion prohlem in Miettincn and M/ikelh (1995). A
noadifferentiable version of the pollution prob/em is dealt with in Miettinen and
M/kelti (1997), A structural design problem is solved by the vector version of
NIMBUS in Miettinen et al. (1996a), In Miettinen and M£kel/ (1996b, 1998a),
a water quality management problem is solved by both the vector and the
scalar versions,
An optimal control problem related to the continuons castitg of steel is
solved by the vector version in Miettinen (1994) zmd by the scalar version in
Miettffien et al. (1996h), This problem is an example of the case where the
modetlng phase ends up with an empty feasihle ïeglom The so-called techno-
iogical constralnts are so tight that there does noL exist any feasihle solution.
When this happeas, the constraints can be treated as objective functions with
the oïiginal objective function(s) thus forming a multiobjective optimzation
5.12. NIMBUS Method 207
problem. One of the goals is thon to find a solution as close to the feaslble
region as possible.
5.12.10. Concluding ttemarks
NIMBUS is one of the few efficient, inteïactive methods especially devel-
oped for solvng nondifferentiable nultiobjective optinization problens. Nat-
urally, differentiable problems can be solved as we[1. h two different versions
of NIMBUS, the decision maker moves around the weakly Pareto optimal set
and expresses iteratively her Or his desires by spccifying thoee objectives whose
values should improve and those whose values are allowed to deteriorate with
the help of rive available classes. The selection of the most preferred alternative
from a glven set is also possible, The questions poeed to the decision maker are
hot demanding. The method altos at being flexible and the decision maker can
select to what exent (s)he exploits the versatile possibilities of the method,
The calculations are hot too massive, either. The use of efficient buadle meth-
ods as the underlying nondifferentiable optimizers is recommended (see
and Neittammikl (1992, pp. 138 143)).
In NIMBUS, the decision maker is free to explore the (weakly) Pareto Op-
timal set and also to change her or his mind if necessary, Previous acts do not
limit the movements. The decision maker tan also extract undesirahle solu-
tiots from further consideratiom Naturally, the decision maker does hot have
to employ ail of the rive classes if (s)he feels uncomforable with somc of them.
However, itis important to provide the decision maker with alternative courses
of action.
The classification of the ffinctions and the specification of the appropriate

ciency. Thus, the user has to choose between these aspects when selecting 
solution method.
Eventually itis up to the user ixiteïface to make the most ofthe possibilities
of the method and provide them to the user« When the first WWW version of
the NLMB US algorithm was hnplement ed in 1995 it was a pioneering it eïactive
optimiztion system in the I}ternet. The realizatioit is hased on the ideas of
centralized computing and a distïibuted interface.
Naturally, there are many challengs in the fur ther development of the NIM-
BUS method and its bnplementations« One of the challenges, applicable to
software development in general, is to create illustïatlve and easy touse user
iter faces. If the inteïface is ah/e to adapt to the decision maker's style of mak-
ing decisions and is of help in analyzilg the alternatives and restflts, and can
perhps give suggestions or dvice, then the interface may even oveïcome some
of the deflciencies of the method itselL
5.13. Other Interactive Methods
The number of existing interactive methods is large« That  why it is neit her
the purpose nor practical noï possible to discuss dil of them heïe. Nevehe
less, in addition to those presetted in the preous sections, some methods are
5sted below. Only the baslc concepts and ideas of the methods are mentioned
together wlth referetces. The methods are roughly divlded accorditg to thelï
basis ilt goal programming in weghted metrics in reference points and in
miscel/aneOuS ideas. Some methods for linear prohlems are included because of
their interesting basic ideas or bec.use they are referred to in connection with
method compañsons in Section 1.g of Part III.
5.13.1. Methods Based on Goal Programmiag
A rather straightforward extetsion of goal programming into an inteïactive
form is presented in Masud and Hwang (1981). The method is called the i-
teractive sequential goal progranmling (ISGP). The itteractive mtfltiple goal
programmig (IMGP) method, described in Nijkaznp and Spronk (1980) and
Spronk (1990), has also been created to combbe the fiexibihty of goal pro-
grammiltg and the robustness o interactive approaches. The decisiot maker
indicates whlch objective value(s) should be improved and either revises the
aspiration levels of the correspondlng goals or the problem is automatically
modffied with additional constralnts.
The sequential multiobjective problem solving (SEMOPS) technique is
briefiy outlned in Monarchi et al. (1973). Five types of goal specffications in
the form of points and intervals are allowed, A different measure of deviation is
utilized for each type. (For example, if the goal is of the form f(x) < 51, then
the correspoi}dig measure of deviation is 5 -- fi(x)/z,.) At each iteration, a
5.13. Other Inteactive Methods 209
subset of deviations is sununed up and then minimized. The decision maker
may change that suhset and specify new aspiration levels. Unfortunately the
solutions are hOt guarant eed to be Pareto opt hnal« A related met hod, called the
sequential information generator for multiple objective problems (SIGMOP), is
ntroduced in Monarchl et al. (1976). SIGMOP is a flexible method where the
declslon maker can altcr aspiration levels d weighting coefficents as (s)he
separatcs attalnable solutions £rom anmng the desired ones. As an app[ic
tion,  pollution problem in water rcsources is solvcd by the SEMOPS and the
SIGMOP methods n the refeïences mentloned.
The ideas of goal progranm]ing the oe constraltt method and trdc offs are
combned n the dizectiomsearching method proposed in Masud and Zheng
(lgg9). The method aims at reducitg the cognitive burden on the decsion
makeï while not increasng computatiomxl complexity. The algorithm is illus-
trated by a immeical example« The propeies of the method are also compared
wlth those of several other interactive methods.
The general purpose interactive goal progïanming algorlthm is suggested
in Tamlz and Jones (199gg).  interactive goal programraing algorithm for
nonlitear problems based ot different noïms and updtitg thc aspiration/evels
is presented in Weistroffer (1983).
5.13.2. Methods Based on Weighted Metrics
The idea of the method in Moldavskiy (1981) is to form a gwid in the space
of the weigatlng vectors and to map thls grid onto the Pareto optimal set.
Weighted L» metrics are used as scalañzing functions to produce a represen
tatlon of the Pareto optimal set. The decisiot maker can colttract the space of
the weightng vectors untll the most satisfactory solution is obtalned.
A method based on sesitivity analysls and the weighted Tchebycheff metrlc
is prescnts it Diaz (1987), where the effects of chagig apiratiot levels are
studied by sesltlvity analysis« The method in Sungga et Ll. (1988) utflizes
also thc weighted Tchebycheff metric. It trforms the constrained mmmax
problem into a series of (differentlable) unconstrained problems by penalty
functions.
The interactlve cutting-plane algoritkm is presented b Loganathan and
Sherali (1987) with applicatiots. The idea is to madraizc the underlying value
function. The weighted Tchebycheff etric is utdized with marginal rates of
substitution as weightitg coefficiets. The convergence of the algorithm is also
treated.
The method proposed ii} Msilti and To[Ia (1993) combines features from
the ¢-costralnt method and the augmented weighted Tchebyclmff metric. The
global Pareto optimality of the sohtons obtalned is checked.
The method of the displaced ideal foï MOLP problems, descïibed in Zeleny
(1973, 1974, 197g), can be characterlzed as an iteïactive extension of the
method of weightcd metrics. A subset of the Pareto optimal set is obtalned
by mlnimizig the distance between the ideal objective vector and the feaslble

objective region by the weighted L-metrics with altered exponents p. The
subset is reduced by moving the reference point towards the feasible objective
reion until the subset of tbe Pareto optbnal solutions is small enough for tbe
decision maker to select the most preferred solution. The method is bsed on
empirlcal studles of the decision maker's behaviour.
The distame to the utopian objective vector is mlnimized by weighted Lp-
metrics in KSksalal and Moskowitz (1994). Tbs interactlve method is bsed
on determining the welghtig coefficients accoïdlng to the preferences of the
decision maker. The preference information is obtalaed from pairwise compar
Ways of approaching dlscrete multinttributc decision analysis problems have
been included in the method introduced in Kok and Lootsma (1985). The ideal
objective vector is used as a reference point. Pairwise comparlson methods are
applled between the reference point ad thc (possibly approximated) nadir ob-
jective vector. The distances re measured by solg the augmented wclghted
TchebychoE problem.
An interesting method is suggested in Kaliszewski et al. (1997). h this hy-
brld interactive decision making tecinique, the decision maker ca select what
kind of information to specify. (S)lie caz cithe classlfy the objective functions
or specif$ upper bounds o global trade-offs. New properly Pareto optimal
solutions are geerated by modied weighted Tchebycheff pïoblem (3.4.9) ac-
cordlng to the restflts derived in Subsection 3.4.6. This is the way of t aking the
bolnds on the global tradc offs into account.
5.13.3. Methods Based on Reference Points
Multiple ïeference points and a gradient projected method are bases of
the mcthod of Costa and Clbnaco (1994) for MOLP problems. The method is
related to pareto race (in Subsection 5.10.3) but it utilizes parallel processing
when haoeing several referece points simultaeously.
The method of Wierzblcki forms the basis of the interacLive reference point
methods introduced in Bogetof et al. (1988). The multiobjectlve optimization
probiem is assumed to be convex. Karush Kuhn-Tucker multiplier information
is presented to the decision maker to guide the specffication of new reference
points. Several différent modifications are also presented ad their couvergence
properties are studied.
In the method presented in Tapia ad Murtagh (1989), the decision maker
is asked to express preferential desires to attaln her or his rcfereace point. So-
called preference criteria are formed from this information. These preference
cri¢eria are then used as a reference point in ¢he achlevement function. The
authors also report some encourang nmnerical experiments.
A method combinig the ideas of reference points and measuring distances
is suggested in Ha[lefjord ad JSrnsten (1986). Afer the declsion maker hs
specified the reference point, the distalce between it ad the feasible objective
5.13. Other Interactive Methods 211
regiorl is ninbized by an entropy fnction. The mathematical bckground of
the method is widely handled in the reference.
The method presented in Weistroffér (1982) assumes that the decislon maker
specifies requied values or ma×immn achlevement lcvels. The strplus is then
maxinfized to the Pareto optimal set. Farther, the methods in Nardia ad
Weistroffer (1989b) and Weistroffér (1984, 1987) expect the decision maker to
provide both the required ad desired values for every objective functlon. Thea
al acbievement functlon is optlmlzed. The required and the desired values are
modified until the most preferred solution is obtalned. Some convergence rcsults
are also dealt with.
The bl-reference procedure presented in Michalowski md Szaplro (1992)
has been developed for MOLP problems. The declsion maker is asked to specify
the worst acceptable objective vector, md a search direction is obtained as the
difference between the worst md the ideal objective vectors. As long a step as
possible is taken in that direction and the decision maker is asked to divlde the
objective functlons into three classes (to be improved, to be kept unchanged
and to be reiaxed). Then thc worst and the ideal objective vectors are replaced
5.13.4. Methods Based on Miscellaneous Ideas
An interactlve exenslon of the weighting method is presented in Steuer
(1986, pp. 39499). Many of its ideas are related to those of the Tchebycheff
method. The set of the walghting vectors is reduced according to the choices of
the decisinn maker. Weigating vectors are generated ïandomly from the reduced
space and filtered to obtain a well dispersed set. Below, this approacb will be
called the method of Steuer.
Another interactive method based on the wcigating method is introduccd
in Hussaln and EI-Ghaffar (1996). It can handle convcx problems and is bsed
on solvlng systems of equatlons formed according to the Karush Kuhn-Tucker
type optimallty conditions.
Two différent interactive relaxation methods are put forward in Nakayama
et al. (1980) and Lazimy (1986b). The latter is applicable to both continu-
ous ad integer problems. The methods are based on the maximization of
underlying value function in a new but equivalent form with addltlonal con-
stralnts. Marginal rates of substitution and other estbnates of the value lune-
tion are required from the decision maker. Similar ingredients are utilized in

the decomposition method presented in Lazimy (1986a). Itis based oll the du-
ality theory for nordinear pïograrnnfi-ng. The original problem is decomposed
a relaxatiompïojection technique, especially for bi-objective problems, is pro-
posed in Ferreira and Geromel (1990). An application to sctedulillg is also
handled.
A iteractive algoritkra with several alteïnative subproblens is proposed
i Mukal (1980). The subproblens generate feasible directions in which the val-
ues of ail the objective functions improve. The decision naker can then indicate
what objective functions to bnprove at the expense of others, and a new direc-
tion is generated. Tools for extending the applicability of Mukaïs algorithns
to nondifiérentlable objective functions are presented in Kiwiel (1984 1985a,
b) and Wang (1989). The ideas were applied in thc MPB method in Section
and valte functions and convex feasible regions in Roy and Wallenius (1992).
A nore general case of nonlinear objective 5ancrions, noneonvex feasible re-
gions and concave value 5ancrions is also dlscussed. This approach uses the
generalized reduced grdient method instead of the original simplcx.
The method in Kbn and Gai (1993) is intended for MOLP problems. It is
based on the concept of a maxinally changeable dominance cone and margilal
rates of substitutiom The effectiveness of the method is illustrated by a numer-
ical exarnple.
Ideas for reducig the burden on the decislon naker in interactive methods
are introduced in Korhonen et al. (1984) and frther devcloped in Ramesh
et al. (1988). An underlying quaslconcave valte function is assumed to exist.
Convex cones are formed according to the preference relatios of the decision
naker. The cones are formed so that the solutions in them can be dropped from
further cosideration, becanse they are domlnated by some other solutions.
Thus, fewer qmstions ]ave tobe put to tbe decision maker i chartilg the
preferences. These ideas concernlng convex coes can be applied equally to
multiobjective optinization as to multiattribute declslon analysis. The ideas
are utilized, for example, in Rarnesh et al. (1989a, b).
A method for complex pïoblems with high dimensionality is proposed in
Baba et al. (1988). The method uses a random optimizatio method and is
also applicable to nondifferentiable objective fonctions.
The pararneter space investigation (PSI) method is described briefly in
Lieberman (1991b) and in morc detall in Statnikov and Matusov (1996) and
Steuer and Sun (1995). It bas been developed for complicated noflbear prob-
le,ris involving possible differentlal equations. Suct probiems occur, for exam-
pie in engieering. The method is very simple and intended to be applicable
to problens where more sophisticated methods are useless. The PSI method is
a nà/ve sampling tectniqm ratheï than an optimization method. Both the col-
straint functions and variables are assumed to have upper and iower bounds.
Thus, the feasible region is a parallelepiped. The Pareto optimal set is approx-
imated by generating randoly uniforly distributed points between the varl-
able bounds. Infeasible solutions are dropped as well as soiutions not satisfying
the zpper bounds specified by the decision maker. Pareto optimal solutions are
selected from this set. The sample size can be altered and the decision maker
can adjust the upper bomlds. The method does hot assnme differentiab[lity.
It works for zonconvex problems since its structure enables global scarch. The
(1997) and Statfikov and Matusov (1996). Thc PSI method has its origins in
(1992).

Part III
RELATED ISSUES

1. COMPARING METHODS
As has been stressed many times tbus far a large variety of methods e×ists
for multiobjectlve optimization problems and aorte of them can be claimed tobe
superlor to the otbers in every pect. Selecting a multiobjective optimizatlon
metbod is a problem wlth multiple objectives itself. Tbus some matters of
comparison and selection between the methods are worth consideritg.
The theoretical properties of tbe metbods can ratber easily be compared. We
summarize some of tbe fcatures of the interactlve methods treated in this book
in a comparative table at the beginnlng of this chapter. However, in addition
to tbeoretical properties, practical applicability also plays an important role
in tbe selection of ah approprlate method for tbe problem to be solved. The
difficulty is that practical app[icability is hard to determine wlthout experience
and experimentatlon.
More iYuit fui information relating to t be question of method selection would
likely emerge if computational applications were more extensively repor ted. Un-
fortumtely, hot too many actual computational app[icatioas of multiobjective
optiñzatioa techaiques have been publisbed, htstead, methods have mainly
been presented without computatioaal experieaces or with simple academic
test problems. As itis aptly remarked in Biscboff (1986), most of tl. applica-
tions preseated are merely proposais for applications or they deal with hlghly
idealized problems. For most intcractive methods a natural reason is the dlffi
culty (in findiag and) ii testing wlth real decisloa makers. A complicatixg fact
is also tbe eormous dlversity oi declslon make-s.
Oae more thing to keep in mid is that for tbe most part only successful
applications are publlsbed. This means that we cannot draw a complete picture
of the applicabibty of a method on the basis of the experiences reported.
The evldeat lack of benctmark type test problems for nonllnear multiobjec
tire optimization complicates tbe comparison of différent methods. Naturally,
some metbods are useful for some problens and otber metbods for other types
of problems. However, beactmark problems could be used to point out such
behaviotr.
In this section wc outline some comparisos of metbods reported in the
literature. We also consider selected issues in deciding upon a met bod, including
a decision tree.

1.1. Comparative Table of Interactive Methods
Presented
In Figure 1.1.1, we preseit a comparative table of the twelve interactive
mtfltlobjective optimization methods described in Chapter 5 of Part II. T[fis
can be regarded as a brief summoey of these methods tIowever, a ceptical
attitnde should always be taken towards such attempts to compress matters
to an extreme. The table is subjective d there is no point in even trying to
deny it.
Different problems arise when one tries to pt together a table of this kind.
Among them are, for example, declding what property is important enough
to be included, bow it should be formulated, and whether it is a positive or a
mgative one.
Figure 1.1.1 preserts some of the properties described when the methods
were i11troduced. They are related to the general features of the method8 and
Figure 1.1.1. Properties of interactive methods.
1.2. Comparisons Available in the Literature
tIere we briefiy mention some of the comprisons available together with
a few results and some conclusions. For more detafled information, see the
references cited.

220 Pt III -- 1. Compaing the Methods
1.2.1. Introduction
by emplo3dng several different vaine functions. If, for example, moeginal rates
numbers. These mearm are employed in Shh and Ravindran (1992). On the
One crucial factor that can affect the performance of the methods in the
compoeisons is the user interace. NothitLg is usually mentloned concerning the
support a 'poor' method vth a good interface. In aldition to the illustration
of the (intermedlate) results, a good user interface also means a clear and
intelligible input phase.
Itis interesting to observe that most of the multiobjective optimization
problems solved when testing the metbods (and reported in the literature)
have been llneoe. It is true that complex notflineoe functlons cause difllculties of
their own and the cboe acterlstlcs oftbe solution methods may be csturbed. On
with MOLP problems. On the wbole, the comparlsons avMlable are hot of too
much help ifone is looking for a metbod for a noniinear problem, and more
comparisons publisbe d.
1.2.2. Noninteractive Tests
An MOLP problem for determining the most economical comblnatlon of
grape growing and wine production in HmLgoey is solved by the weigbting
method, tbe -constralnt method, lexicographic ordefing and tbe weigbted L-
and L2-metrlcs with normalized objective rctions in Szidarovszky and Szen-
tele!d (1987). It is observed that different solutions oee obtalned with eacb
metbod. It is also stated tbat tbe welgbtcd L and L2-metrics with normal-
ized objective nctions produce the most urñform distribution of objective
vectors. Finally, tbe weigbted L1-metrlc is seen as the most convenient way for
geneïating Pareto optimal solutions in loege-scale MOLP problems.
A liner problem in tlm mitling industry is solved in Peterson (1984) by the
welghting method, tbe -constralnt method, tbe metbod of welgbted metrlcs
with and witbout denominators, and by lexicographic ordering. Tbe solutions
obt alned from tbe otber metbods oee utilized ha tbe method of weigbted metrics
and ail tbe solu¢ions effe aalysed. Tbe conclusion is tbat solution methods
should be appiied so tbat tbey complement eacb otber.
1.2.3. Interactive Tests with Human Decision Makers
No inteïactive metbods were included in the compaxsons mentloned tbus
foe. The following comprisons involve interactive methods.
Itis described in Dyer (1973b) bow nine (student) decision makers were
presented with an MOLP problem involving cboosing an engine for a cœ. Tbey
were first asked to suggest an approach and tben compoee it wlth tbe GDF
metbod and a trlal-and-error procedure. In the trial and error procedure the
decision maker was simply asked to enter an objective vector and the procedure

maker.
The results of Dyer and Wallenius concerning the GDF metbad differ re-
The capabilities of the ZW, the SWT, the Tchebycheff and the GUESS
Dællelbach (1987) from the point of view of tba user in solving a linear three-
(students) and two three-objectlve MOLp problems are repoïted in Buchanan
(1994). The methods involved were the Tchebycheff metbad, the GUESS
(SIMOLP) method (by Reeves and Franz) based on fitting a hyperplane
basically the saine as in Buchanan and DaeIlenbch (1987).
fitted accord]ng to the ratings and the declsion malter was asked for foedback
The motivation for using two phases was tbe following. It bs been suggested
at times tbat the opporunity of familiarizlng the decision maker with the
problem tobe solved should improve the actual solution process. However, the
structure tobe fitted was incorrect or the ratings were too dilllcult to provlde.
Tbe methods tested had remoekble philosophical differences. SIMOLP was
the most structured and GUESS the most unstructured method. Iii general, the
GUESS method was the most favoured of the three. Added to thls, the fact that
SIMOLp was the least popu]ar method, there is clerly a preference for less
structure in the solution method. The SIMOLp method requlred the decision
be difficult. It seemed tbat it is easler to select the best rather than the worst
alternative. (Thls observation must be related to the nature of the problem to
be solved. The opposite may be true in some other cases.)
It is interesting tbat the Tchebycheff method was rated better than the
GUESS method in Bucbanan and Dællenbach (1987), whereas GUESS was
better than the Tcbebycheff method in Buchanan (1994). In the latter test,
the GUESS method had been supplemented by allowing the decislon maker
to specify upper and lower hounds for the objective functlons. This may bave
improved the flmctionality of the method. Once again, the decision malers
liked being in control of tbe solution pïocess. However, they would have liled
proved to be çamiliarity witb the method.
The method of Steuer and the ZW method (in Subsection 5.13.4 ofprt II)
oee compared in Michalows!d (1987). Five decision rouliers from the planning
depaïtment ofa factory were employed. A liner production planning problem
with three objective unctions was solved and the evaluation criterla were hot
fixed in advance, although the main interest was in the decision phase. The
declsion makers bad cïitical comments concerning both the metbads, and each
of them obtained a different final solution. One can say that the decision pro-
cesses by the ZW method termbaated sligbtly faster than those by the method

The method of Steuer and STEM are tested in Brockhoff (1985). A total of
cars. The results and progress are malysed accordig to several criteria, with
the method of Steuer emerging with the best outcomes on the average.
(1995, 1997). In Corner md Buchanan (1997) the continuous GUESS method,
a modlfied ZW method and a discrete SMART method (based on construct-
ing a alue fnction) were used to solve a pïodactlon plminng problem wlth
tbree objective functions by 84 dergaduate students a decision makers. The
pïoblem was nonlinear md tmd continuons variables. The main interest wa to
maker. In other words, how weli the methods were able to find desirable solu-
tions md how mach the declslon makers liked the methods. The tlme spent on
fauter thm tbe dlscrete method. The GUESS method wa rated eaiest to use
md to understad. Ail tbe methods produced different solutions of w[sch the
one generated by the GUESS method wa rmked the best. The order of the
the cae when SMART wa ased first. Then the solutions obtalned with the
other methods were statistically the saura. In addition, it wa observed ttmt a
weighted addltive value fimctlon explalnilg their rmking behaviour COuld be
found for most decislon makers.
Anotber experimental test involving the ZW method md the GUESS
method is reported in Bucimnan md Corner (1997). The empbasis wa in test-
ing whether any mchorlng effect cm be explalned by the structure of the
with the free search method GUESS. Thus it cm be dedltced that the selection
with less structured methods.
Some comparisons of contmous md dlscrete methods are also presented
in Korhonen and Wallellius (1989b). A continuons MOLP problem with rive
objective functinns concerding the alioctlon of a stadent's time between study,
woïk md leisure wa solved by 65 student declsion makers. The rive methods
ficatlon of the reference direction varied. The original way of uslng aspiration
A more detalled review of the above descïibed md some other empirical
studies fivolvlng real decision makers is given in Oison (1992). tIoweveï, no
final conclusions can be drawn from the experiments. The reason is that the
1.2.4. Interactive Tests with Value Functions
Next we review some comparisons utillzlng value functions to replace deci-
An MOLP problem wlth three objective functions conceïning operatios
planning in the natural ga business is solved by several methods in Mote et
al. (1988). A nonlinear value fimction was employed instead of htman decislon
makers. The problem wa solved by the GDF, the SWT md the ZW methods,
STEM goal programaing, and the method of Steuer. Only standard LP codes
were utilized in the calculations. No slngle technique wa shown to be supeïior.
The methods had differences cocernlng the burden upon the decislon malter
md ad hoc and non ad hoc propertles.
The method of Steuer md the method of Frmz (m interactive adapta-
tion of welghted and lexicograp]c goal programming) are compared in Gibson
et al. (1987) in solving several rmdomly generated MOLP problems. Different
ison wa to investigate the applicability of the methods to differeat situations
wlth the belp of statistlcal tests. The number of iteratlons wa also recorded.
The conclusion is that, for example, the number of iteratlons md whether ail
in the selection of a method. These crlteia lead to different recommendations
The Tchebycheff method md the SIMOLP method (by Reeves and Franz)
are compared in 15 MOLP test problems with fouï objective fanctlons using
both linear md nonlinear value functlons to replace the decislon maker in
Reeves md Gonzalez (1989). The comparison criteria were the quallty of the
solution (how far the best solution round wa from the best extreme point),
user friendliness, computational reqdirements, whether nonextreme solutions
could be fod uumber of iteratlons needed md fiexibility. The Tchebycheff
the test reported in Buchnm md Dællenbach (1987). The main difference
between the two methods is that the SIMOLP method moves away fiom the
leat preferred alternative wherea the Tchebycheff method moves toward the
most preferred one. Thus, the SIMOLP method is more flexible md it is easier
çor the decision maker to change her or ]s mind. Far ther, the SIMOLP method
needs mach less calculatlon.
The SIMOLP method wa able to find sllghtly better solutions at less com-
putatlonal cost in most problems even with the nolinear value function. The
fact that the SIMOLP method is Imited to Pareto optimal extreme points
did hot seem importmt in the tests, tIowever, Reeves and Gonzalez (1989)
suggest comblnlng the advantages of both methods. Either the declsio maker
the fle×ibility of the SIMOLP method can be used first md thc ability ofthe
iterations.

226 Part III 1. Compaxing the Methods
Another summmy of comparisons pubgshed is given h Aksoy et al. (1996).
The presentation suxnmm'izes six comparisons with human decisinn makers ald
fotLrteen studies utffizhg value fnctions. The apects treated re the compati
son criteria, types of decisinn makers, nature of the test problems tobe solved,
form of value fimctlons, strting solutions, stopping crlteria, md ordering of
methods md test problems. The obviotts conchmion ffom this is that there is
at urgent need for fuher comparative evaluations of nonliner multiobjective
optlmlzatinn methods.
1.2.5. Çomparlsous Bsed ou Intuition
A chractcristic shared by the evahmtlons tobe described next is that they
are based on intuition and inslght rRther thall practica] experiences and tests.
The comparative table in Section 1.1 cotdd equally have been included here.
A collection of the features of rive nonllner interactlve methods is presented
in Masud md Zheng (1989). The methods are compred wlth regard to eleven
items, for example, the certalnty of obtainlng a Preto optimal solutio L the
optlmization techniqlte used, the type of information requred from the deci-
sion maker, computational complexity compared to the GDF method, and the
number of iterations needed with input from the declslo} maker compred to
the GDF method. A similar table compring the declsinn maker's burden, ease
in actual use effectiveness md hmdiing of inconsistency s collected in Shin
md Pvlndran (1991) for ten methods. A further classification and evaluation
of method according to 21 criteria is give1 in Rietveld (1980).
The nmnber of items a decision maker has to assess sknultmeolmly md
per iteration for elght different methods in a medum size linear problem re
tabulated in Kok (1985). It is concluded that the method of displaced ideal,
the interactive multiple goal programmlng method and STEM are p-omiing
because thelr premmnptions are realistic. In Kok (1986), the lernlng effects,
information load, effort of teclmical support and grop declsion capabilitles are
evaluated for rive methods: ZW, interactive multiple goal progl'anlming, ISWT
STOM and palrwie-compaïlso}s. No strict preference can be expressed.
A total of 19 interactive method for MOLP pïoblems aïe listed accoïding
fo three chracteristics in Laricbev et al. (1987). The chaïacteristics re the
rel]ability of the way information is elicited from the decision makcr, insignif
i«ant sensitivity fo rmdom errors on the part of the decisinn malr and good
speed of covergence. The basic prlnciples ofthc methods re also introduced.
In addition, the lectures of STEM, the GDF and the Tchebycheff method, the
reference point method and the ïeference direction approach, among others
re tabtdated in Vanderpooten and Vincke (1989) and Vincke (1992, p. 105).
The criteri are, for instance, assumptions of the existence of a value function,
applicability, trial and erïor support, mathematical convergence, the mmnber
of questions posed and the computtional burden.
The bi-reference procedtre is compared to STEM, the GDF method, the
ZW method md the reference direction approach in Minhalowski and SzaplrO
1.3. Selecting a Method 227
(1992). The idea is to compre the performance of the breference procedlre
to the pllblished results of the otheï methods.
Finally, we mention some other comparative studles. Characterlstic values
in optimlziLg the multiobjective layout of a conical shell by tbe GDF method,
STEM and three othet methods effe reported in Eschenauer et al. (1990b).
As far as the relative performance of STOM and the Tchebycheff metbod is
concerned in finding a solutiolt toa llnear sausage blending problem in Olson
(1993), the main intentinn is to emphasize the power of the weighted Tcheb
cheff metïic in multlobjective optlmlzation.
1.3. Selecting a Method
Chooslng an appropriate solution method for a certain multiobjective op-
timizatio} problem is hot easy, as has been made abundantly cler. None of
the existhg methods can be labellcd . the best for every situation, since there
is a multiplicity of aspects to consider and many of the comprison criterin
effe of a somewhat fzzy character. The fcatures of the problem tobe solved
mad the capabillties and the type of the decision maker have to bc charted be
fore a solution method can be chosen. Some methods may suit some problcms
and some decision makers better than others. Let us sum up by offering some
general guidelines and a declsion trce.
1.3.1. General Guldelines
Several dLfïerent comparlson criteria were already mentioned in Section 1.2
in connectlon with the tests ïepoïted. Some of thc criteria to consider when
evaluatlng methods effe also collected in Hobbs (1986). These selection crltezia
are appropïinteness, ease of use, validity and the sensitinity of the results to the
choice of method. Approprlateness means that the method is pproprlate to the
problem to be solved, to the people who effe to use it and to thc instltutional
setting where it is tobe implemented. Ease of use refers to the effort and the
knowledge reqltlred from the analyst and the declsinn maker. Validity means
that the method measures what it is supposed to and the assumptlons set are
consistent with reality. The sensitivity of the restlts to the choice of method
expresses the desire that solutions obtalned by the method do hot slgnificantly
differ from those of other methods. If the method chosen has a significant
effect on decisiox}s, then the relative validity of different methods should be
consideïed. Ifthe form ofthe method does hot marrer, then the most hnportant
criterla effe ease of use and ppropriateness.
The nuxnber of crucial criteria in selecting a solution method is reduced to
three in Stewart (1992). The inptt required from the declsinn maker must be
meaningful and unequlvocal, the method must be as trasparent as possible
and it mst be simple and efficient.

The role of the decision maker is important and should be taken seriously.
Many experin]ents bave sbown that decisinn makers prefer simpler methods
because they can more eaily understand such method and they feel more in
control. The valuation placed on some metlod may increae if trie decision
makers tan practice using triem or obtain advice. An important fact to keep
in irfind is that trieoretically ixrelevant apects, sucb a question pbraiag, may
affect the confidence that trie doclslon maker feels in trie method. The concept
of trie decision maker's confidence is analysed further in Biscboff (1986).
Otber important crlteria for trie decision maker in selectlng the solution
metbod are, for exanple, the simplicity of trie concepts involved posslbi[ities of
interaction the eae with whicb the results tan be interpreted and the chances
of cboosing the most preferred solution from a vade enongh set of alterimtives.
The metbod must also fit trie decision maker's way of rhin!ring. Trie languge of
communication between the decision maker and the method (solution system)
must be understandable to trie decision mker. (S)he wants also to see that the
information (s)be provides bas a (desirable) effect on the solutions obtained.
One more element, hot mentioned trms far, in trie selection of a method is
how well the decision maker knows trie problem to be solved. If (s)he does hot
know its limitations, posslbillties and potentiahties well, (s)le need a metbod
that can provide support in getting acqnainted with the problem. In trie oppo-
site cae a method that makes it possible to focus dlrectly on some interesting
sector is advisable. Ways of identifying approprinte metbod for d]fferent types
of declslon makers are needed.
1.3.2. Method Selection Tools
Few universally applicable guidelines have been given for trie method se-
lection problem in the literature. Let us mention some of triem incIucSng even
approaches for cSscrete problems.
An attempt to aslst in the selection of a solution method is presented in
Gershon and Duckstein (1983). Trie selectlon problem is modefied a a mlfitiob-
jective optlmization problem. A set of 28 criterla for the selection are suggested
and they are dlvided into four groups. Only trie criteria in the lait group bave to
be considered every tlme trie selection algoritlma is applied. The criteria take
into account the characterlstlcs of trie problem, trie decision maker and the
metbods. Many types of problems are taken into consideration in the crlteria
(e.g., discrete and continuous variables). The modeI conteàns 13 solution metb-
ods from wlfich to select. The set of metbads can naturally be modified. Trie
number of selectlon criteria tan also be varied to include only those relevant
to trie problem to be solved. Finally, after the methods have beea evaluated
according to the selection crlteria the resulting multiobjectlve optlmization
problem is solved by trie metbad of trie global criterinn (e.g. Ll-metric ).
A related procedure is snggested in Tecle and Duckstein (1992). There, a
set of 15 metbad is evaluated with respect to 24 criteria in four clases. Trie
weigbted Lv metrlc is used in each clas and another welghted Lp-metric is used
to combine the clases and obtain the best method. For the example problem
proded, the weigbted L»-metrlc turns out to be tle best method. One may
wonder whether the weighted L» metric favours itself or whether this is a mere
coincidence. Some critique of the approach is also expressed in Pmmero (1997).
Different decision trees and rules for providing asistance in selecting a
metbod for multiattribute decision analysis problems are descrlbed in Hwang
and Yoon (1981) and Teglem et al. (1989). However, a criticoEed in Ozernoy
(1992a), to design a comprebensive and versatile decision tree usually restdts
in an explosion in trie ilumber of nodes. Another problem with decision tree
dlag7am is what to do when the user answers I do hot know.'
An expert system for advising in the selection of sobtion methods for prob-
lems with discrete alternatives in proposed in 3elasi and Ozernoy (1989). Steps
are method for discrcte problems are described in Ozernoy (1992a, b). Trie
questions posed by trie system are based on if/then rules. Tbey lead to recom-
mending a method or statlng that no metbod tan be recommended. The user
of trie system can also ahvays ak wby a paricular question is posed.
1.3.3. Decislon Tree
tions are satisfied (a stated, for exanple, in Zionts (1997a, b)). Or it may even

The nodes containing only capital letters are used in two different cases.
The first is to avoid repetition, tn the second, no method can be round along
allow as many previous answors as possible to be exploited. Thus, some dead
ends may be avoided.
The same method may be reached by followlng differeat paths. In this case
Figure 1.3.1. Tree diagam.

2. SOFTWARE
The development of computers and the improvement in the speed, stor-
age capacities and fiexibility of computing facilitles have mado it possible to
produce more sophlstlcated and demaading software for solving multiobje
tlve optimizatlon problems. Efficieit computers e11able, for exmnple, the im-
plementation of interactive alorlthms since they can produce sufficiently fast
responses for the dcision maker wlthout the user geting frustrated waitiag.
Nevertheless, takig into account the multipllclty of methods developed for
solvlng nofllnear multiobjective optlmlzation problems, the number of widely
tested and user friendly computer programs that are generally available is
smaIl. At least they are difficult to find. Most implementations are done for
acadomic testing puïposes and their existence is ot advertised. In oçher words,
there is a real need for functional and reliable software foï solvlng nonlinear
multiobjec¢ive optlmization problems.
2.1. Introduction
Most of the software packages developed for multiobjective optlmization
problems crut be termed multiobjectivc decision support systems, and they
form one c[ass of decision suppoï* systems. Decision uppo2 systems (DSSs)
can be defined as inteïacive computer-based systems designed foï helping and
asslsting in the decision-maklng pïocess. Their main objects are to help decislon
makeïs in solving problems more efllciently and making better decisions.
The main components of a dccisio suppor system are a modeI, an op-
timizer (solver) and an interface between the model, the optlmizeï and the

The foie and the requirement s of the model, the opti1izer and the intedace
in the multiobjective opti1ization envlronment are outllned, for exanple, in
JeIsl et al. (1985). Itis usehI to bave capabi[ities of self learning and model
updating h  decision suppor system. Tbo interface is an important factor in
relation to the user-fi-iendllness of tbe system.
One can state that developlng software for multlobjective optlmizatlon
problems is once agaln a mtfltinbjective optlmlzation problem in itseff, and
proper plarminŒE is essential. Several (cofllctlng) objectives tobe taken into
considcratlon in multinbjective software desig] and realization are collected
in Olkucu (1989). Among them are a short deveIopment tirae, long product
llfe easy and cheap maintenance, rellable implementation of tbe algorittma,
an efficient user interace and a large number of potentlal users. Of no mi-
nor importance in tbis regard are tbe selection of tbe realization envlroarnent
(including tbe operating systcm) and development tooIs.
Features tobe taken into cosideratlon wben deslgnlng dcclslon suppor
systems are also handled in Lewandowski (1986). Differert definitlons of user-
friendlhaess and rules for dialogue design are glven.
It sbould be polntcd out that wtdlc a great deal of effort has gone into
developing tbe methodological and computatinml aspects of the systcms, tbe
interface between tbe system and its user is ofen of poor quality. Thls is a
serious wcakness, since no marrer how brilliant tbe metbodology ad its imple-
mentatlon are it will be dscarded if the interface does hot suit tbe user. In any
case tbe algorltlms must be implemented in such a manner that computer-
technical requirements do hot overshadow tbe real problem and non-sldlied
persons can also use tbe programs. One way to try to improve the situation
is to provide different interace possibilities for the saine system for computer
specialists, tralned users and average users.
An effort of measuring thc cffectivemss of decisinn support systems is de-
scribtl in Salnfort et al. (1990). Even though itis widely assumed that decision
suppor systems really do belp in declslon maldng d problem solving, research
results in ttds important area are few. Added to ttfis is tbe fact that tbere is
currently no geneïal tbeory about problem solvhg because of its complexity.
Group decision suppor systems are mostly handled in Salnfort et al. (1990),
but tbe conchslons favouring decision support systems are general. [tis demon-
strated that decision support systems increase tbe understanding of the prob-
lem, reduce frustration in tbe problem SOlvlng and contribute to progress in
the solution process.
To put if briefiy, a decisioa support system sbould be easy to use, it should
capture tbe ttfinkig procadure ofthe decision maker, it should support differ-
eut decislon styles and it should heIp the decislon mal(er to structure diff'rent
situations. Other desirable cbaracteristics of dccision support systems are 5sted
in Welstroffer and Narula (1997).
2.2. Review
Existing software packages up to the year 1980 are listed in Hwang et
al. (1980). Tbese progranls were malnly developed for linear and goal pro-
grammffig problerns. Tbey are rather primitive when compared with modern
computer facilities. A somewhat more up t(>date llst of decision support sys-
tems developed to ald in multlobjective optlralzation and multiattribute deci-
sion analysis problerns up to the year 1988 is collected in Eom (1989). However,
tbe presentation is only cursory. In addition, a classhïcation of tbe system ap-
plications is provided. Some software implcmeatations are also mentloned in
Welstroffer and Narula (1991), wbereas tbe overview in Buede (1996) handles
software for discrete problems only.
The latest state of decision support systeas for multiple criteria dccision
making problcms up to the year 1997 is presented in Weistroffer and Narula
(1997). Systems for both continuous and dlscrete problems are listed wlth in-
formatloa abolit w}ere they cm be obtahed. Unfortunately, from anong tbe
seventeen continuous products mentioned only six are applicable to general
noflinear mltiobjective optimizatlon problems. Of these GRS generates and
illustrates tge Pareto optimal set implcmenting the gcneralized reactmble sets
metbod (sec Section 3.6 in part II). The actual solvers are CAMOS, DIDAS,
LBS, MONP 16 and NIMBUS, where LpS is an implementation of tbe ligbt
bean search (sec Section 5.9 of par II). Tbe otbers bave already been in-
troduced in par Il ezccpt for CAMOS (see tbe eIoEd of tbis section). DIDAS,
t}e implementation of tbe reference point metbod, was described in Subsectlon
5.6.4, MONP 16 implemnting STOM was mentioned in Subsectlon 5.8.4 and
implementations of NIMBUS were handled ioE Subsection 5.12.8.
Tbe Curreat state of tbe software dcvelopment can be inquired from A. L(>
tov, Russia (GRS), A. Osyczka, Poland/Japan (CAMOS), J. Granat, Poland
(DIDAS), A. Jaszkiewicz, Poland (LBS), L. Kirilov, Bulgarla (MONP 16) and
K. Miettinen, Finland (NIMBUS). Tbe WWW NIMBUS system is avallable at
bttp://nlmbus.mat h.jyu.fi/.
Software for discrete problerns is more easily avallable than software for con-
tinuous problems. For exanple, scveral dlscrete systems havc been developad

to commercial products. Yet a emphaized in Buede (1996) evea those soft-
ware developers concentrate too closely on features of analysls at the expense
of user-friendllness a meatloned ear[ier.
Among software products for solving MOLP problems is VIG by P. Kor-
honen, Finland (see Subsectlon 5.10.3 in part II). Let s also mention a pack-
age of subroutines, callcd ADBASE by R.E. Steuer USA (see Steuer (1986
pp. 254267)). ADBASE contalns for example tools for generating Pareto op-
tlmal extreme points. Tbese are examples of the generally available products
for linear pïoblems.
Tbe situation is worse for continuous on[inear problems. Most of the soft-
ware [mplementig the extensive amount of existing multiobjectlve opt[miza-
tion methods is neither commonly avallable nOr wldely loown.
One explanatloa sometimes mentloned is the lack of a free and reliable
nonlhear solver tiret could be integrated and distributed with the software.
Most software products bave been implemeated for academic testing purposes
and have hot been updted along with the development of computer facilities.
Consequently, their existence is hot alvertised. Simply desoEning and reaSzing
a fanctional USer intcrîce is demandig. Oae must asume tht tbe need ha
and Teicb (1989). One of them is VIG and tbe other seve are for discrete
(see Osyczk (1989b, 1992) and for an earlier version Osyczka (1984)). CAMOS
Tbe method for identlï'ing (weakly) Preto optimal solutions arc the
weighted Tcbebycheff metric. Problem (2.1.4) of pacL II is also used. For more
detalls see for exaniple Osyczka (1984, 1992). DLfïerent underlyJng singe ob-
jective optimiztinn algorithms my be used.
The fanctioning of CAMOS is i[lustrated by two practical pïoblems in Osy-
czk (1992 pp. 9325). They are the optimal desigz of multiple clutch brakcs
and the optimal counterweight balancing of robot
NOA a collection of subroutines for minimlzing nondifferentible fanctions
subject to linear and nonSnear (nondifferentlable) constralnts is described in
Kiwlel and St achurs!d (1989). NOA s applicable to multlobjective optimlzation
problems since the single objective fanction to be mlnimized is assumed to be a
mxJmum of several fanctlons. Tbus, for example some acbevement fanctlons
can be optimized.
Let us finally mention the optimlzation toolbox of tbc MATLAB system
including the weighting method, the «-constralnt method and a modification
of goal proamming. Nturally other multiobjectlve optimlzatinn algorithms
may be coded within the MATLAB envlronment, t aking alvant age of the pow
erful singIe objective solvers and grphlcs avallable.

3. GRAPHICAL ILLUSTRATION
3.1. Introduction
Graphical illustration plays an essential foie when designfng modern soft-
ware user interfaces. Graphics may be LLsed t0 describe the problem, to assist
the decision maker in specifying values for problem parameters or to fllustrate
thc contents and the meaning of questions posed by the algorithms. In such
reafizations, the upper limit lies in one's imaginatiom
In spire of the more gcneral possibi]ities, we restrict out treatment in this
chapter. By graphical illustration we here mean the woE»'s of presenting several
alternative objective vectors to the decision makcr. To be convinced of the need
for such illustration one has only to examine the interactive methods described
in Part II. Good graphlcal illustration helps the decision maker to gain a better
isight into the problem and the different alternatives generated.
As computers have developed, more attention has been pald towards the
role and the possibilities of computer graphics in building human-computer in-
retraces. Nevertheless, utibzing graphical illustration does not mean that the
limits on human information processig capaelty are transcended. This means
that there is no sense in trying to offer too many objective vectors for evalua-
tion, no matter how clear the illustrations are.
Several psychologcal tests are summarized in Miller (1956) to prove that
the span of absolute judgment and the span of immediate memory in human
beings is rather fimited. We cannot receive, proccss or remcmber large mmounts
of information. The magical number seven plus or mhaus two appears in several
tests and in several ways. However, no numbcr (:an be regarded as an absolute
limit. Everything depeIds on the circumstances. Still, the findings of Millet are
tobe kept in mind when deciding the number of alternatives to be presented to
the decision maker or the mmaber of objective fitnctions tobe treated (if these
can be affected). Mi[ler's fidings must also be remembered when expecting
exact information from the decision maker. Let us mention tha L for exmmple,
seven ways of decreaslng the number of alternatives are presented in Graves et
al. (1992).
Naturally, may different ways for illustrating objective vectors can be
thought of. However, elegant graphics must hot be an end in itseff. The graph
ics must be easy to comprehend by the decision maker. On the one hand, not

240 Part III -- 3. Graphical Illustration
too much irfformatlon should be allowed tobe lost and, on the other hand, no
extra unintentional irfformation should be included in the presentation.
3.2. Illustrating the Pareto Optimal Set
In the case of two objective flmctions, graphical illustratiot of the objective
Space is effective. The feasible objective regfion and, especlally, its Pareto opti-
mal subspace can be sketched on a plane. If thls is hot possible, the available
objective vectors can be pfotted in the objective space. As far as tbree objective
fltnctions are concerned, the Pareto optimal set can be cxpressed by three pro
jections on a plane, as suggested, for examplo, in Meisel (1973). However, tbe
interpretation of such information is far more difficult for the decision maker.
Another way of illustrating the Pareto optinal set of tbree objective flmc-
tions is to draw a tw(>dimensional plot with fixed values assigaled to the third
objective function. There is a resemblance here with topographic maps. Such
an approach is handfod in Bushenkov et al. (1995) and Lotov et al. (1997),
where so-called decision maps are used. Sevoral level sets of the third objective
fltnction are drawn in the picture of the Pareto optimal hull of the first two
objective functions. These sets are called efficiency frontiers. If there are more
than three objective fonctions, several different pictures can be drawn each
having fixed values for the other objective fltnctions. For example, in the case
of rive objective fltnctions a matrix of declsion maps may be displayed. There,
the fourth objective fonction has tbe same fixed value i¢ every picture in each
row and the fifLh objective 5mction has the same fixed value in every picturc
in each colmnn. In addition, scroll-bars and antmations can be used. According
toits devefopers, this approach works for up till seven objective flmctinns.
3.3. Illustrating a Set of Alternatlves
Below, we p:esent some ways of ifiustrating a set of alternative objective
vectors graphically Some of the ways are clarified by applyiag thcm to an
x ample of three alternative objective vectors of a problcm with three objective
5mctions.
3.3.1. Value Path
A widely used way of representing sets of objective vectors is to use value
paths, as suggested, for example, in Geoffion et al. (1972) and Schilling et
al. (1983). This means that horizontal lines of different colours or of different
llne styles represent the values of the objective fonctios at different alter
natives. In otber words, one fine displays oe alternative. Tbls i. depicted in
Figure 3.3.1. The bars in the figure show the ranges of the objective ftmctions
Zl z 2
z 3
Figure 3.3.1. Value paths.
In value path illustrations the foies of the lines and the bars can also be
interchanged. Tben bars denote alternatives and lines represent objective froc
tions. In this case, possible difforent scales of the objective functlons have to
be interpreted differently (see, e.g., Hwang and Masud (1979, p. 109)). This
reversal of roles has been utilized, for instance, in the first implementations of
the reference direction approat'h (described in Section 5.10 of Part II), and its
counterpart for discrete problems, called VIMDA, see Korhonen (1986, 1991a).
The idea ia VIMDA is that when the user horizontally moves the cursor to a
bar representing an alternative, the corresponding aumeïical objeç%ive values
arc presented.

Value paths are an effective means of presenting information without over
ladhag the decision maker. Another general mode of illustration i to se bar
«harts. This means that a group of bars represents the alternative values of a
single objective function, as ha Figure 3.3.2. The bars of the same colour indi-
cate one alternative. Separate ranges for objective functions are possible as we[I.
Par allel ideas have been realized for example, in DIDAS and WVW-NIMBUS,
treated in Subsections 5.6.4 and 5.12.8 of Part II, respeetively.
30t
2O t
lot
Figure 3.3.2. Bar chart.
Naturally the foies of the alternatives and the objective functions can be
interchanged in bar charts as well as in value paths. This, of course, means
tbat the order ofthe bars is alteïed. This is possible, for example, in WWW
NIMBUS.
An alternative to using separate ranges for the objective finctions is to
provide bar charts and value paths using both absolute and relative scales.
This is advisable ñ partlcular if the ranges of the objective finctions vary
widely. Thi option is also awilable in WWW-NIMBUS.
3.3.3. Star Coordinate System
It is suggested in Mafias (1982) that objective vectors can be represented
in  star coordinate system. For exmmple, an alternative of three objective
fnctions is represented as an irregular triangle. Thts requlres the ideal objective
vector and the (possibly approximated) nadir objective vector to be known.
An example is given in Figure 3.3.3. Eexh circle ïepïesents one alternative
objective vector. The ideal objective value is at the centre and the component
of the nadiï objective vector is at the «ircumference. Each ray repreents one
objective fimction. The area of the star depicts each alternative. See detalls in
Mmïas (1982).
z I
Figure 3.3.3. Star coordinate system.
as stated in Tan and Pïaser (1998). Tan and Pïaser also suggest a modified star
3.3.4. Spider-Web Chvxt
(possibly approximated) nadr objective vector, the imer triangle (the darkest
Kasnen et al. (1991).

Figure 3.3.4. Spider web chart.
3.3.5. Petal Diagram
Somewhat parallel ideas to the two previous representations are utifized in
Angehrn (1990a, b) when illustrating discrete alternatives in a program called
Triple C (Circular Criteria Comparison). A circle is divided iato k (the altmber
of objective functions) eqaal sectors. The size (radius) of each slice indicates the
magaitude of the objective value. Here we have one circle for each alternative
objective vector. The same idea is suggested in Tan and Fïaser (1998) and it
is called a petal diagram. Each segnent of the diaam, that is, each objective
flmction cma he aociated with a diiïerent colour, as in Figure 3.3.5. Notice
that the order of the objective flctions has o effect on the shape of the
diaam. The relations of the different segments are clearly shown. A way of
connecting weighting information in the petal figures is suggested in Tan and
Fraser (1998). In tbls case the segments are not of equal size but refiect the
weightiag coefficients.
z2 z 2
Figure 3.3.5. Petal diagram.
It is mainly a matter of taste in the star coordinate system, the spider web
chaït and the petal diagram how the ideal objective vector is situated. One
may think that wben minimizing the objective functions it is logical to bave
the ideal area as small as possible. However, the foies cma be interchmaged so
that the ideal objective value is located on the circumference and the nadir
objective value at the centre. In this case the larger the area the better. If tbis
is the settlng, the ideal objective values cma be replaced with, for exampIe,
average objective values. Tbis memas that the figures can extend beyond the
circumferencc, stresslng values better thma the average.
3.3.6. Scatterplot Matrix
The scatte'plot matrix described in Clevelmad (1994) cma be adapted for
visualizing differ ent alternatives. The scatterplot matrix consist s of pmaels each
representing one objective function pair. The dimension of the square matrix
is the number of objective functions. Different alternatives cma be denoted
by different symho[s or colours. As cma be seen in Figuïe 3.3.6, each pair is
graphed twice with the scales inter chmaged. This memas that either the lower or
the upper triangle could be dropper wlthout losing may informatlom Hovever,
displaying the whole matrix makes it easier to compare the objective function
values. One cma measure the peïfoïmance of one objective function against the
otheï objectives by having a look atone column or one ïow. Each objective
flmction can naturally bave a range of its own in the pmaels, as in Figure 3.3.6.
X X

3.3.7. Other Illustrative Meaus
A graphical display system called GPADS is introdimed in Khmbexg (1992).
GPADS is dynamic and can be app[ied to problems with abont rive to twelve
objective fmactions. The decision maker is first asked to indicate two objective
flmctions whose values in the diffeïent alternatives are drawn as points in a
plmae. In thls space, the adjacent alternatives are connected with fines. In
other words, we have one value path The decision maker obtalns informatinn
about the other objective values for one alternative ata rime by indlcating
that point with a mouse. Then, the other objective valucs are depicted as lines
originating from the point consldercd. The lengths ofthe lines are proportfinal
to thc objective values. The end points of the lines are connected tfins forming
triangles of a diffeïent cofinlr. The percentage achlevements of the alternative
in question axe also displayedi They are calcu/ated as the difference between the
nadir objective valtle and tfin Ctlrrent objective value divided by tfin range. The
decision maker can ctmnge the alternative consideïed and the two objectives
whose value paths form the base of the dlsplay.
Dflïerent ideas of graphical ilblstratinn are also handled in Korhonen
(1991b). One of the ideas is Chernoff's faces, orlginally developed to illustrate
numerical information. The idea is to represent the values of up to 18 objective
functions as the characteristics of a face. In other words, the values of each
objective function are parametrized to represent some feature of an icon. The
icon used mtlst be suçh tht the user can see the icon becoming 'better' as tfin
value of the objective fimction improves. This is why concepts like symmetry
and harmony are hnportant. An icon that people have been used to seeing
in a harmoniot and symmetrical form is a house. Tfas, Korhonen suggests
s(>called harrnonlaus bouses to be tŒEed as icons. Objective functions are
clated with the corner points of the finuse, the door, the winfinws or thc roof.
The aim is that wfinn the values of the objective flnctions are close to thc idcal
objective vector, the bouse is qulte harmonious and synmetïical. This type of
i]instration bas especlally been intended for palrwise colnparison.
Literature descrihlng the graphic presentation of data is summarized in Le-
wandowski and Granat (1991). It can be concluded that the reseaïch done does
hot provide clear answers regarding what types of data presentation to favour
in the decision-making context. Lewandowski and Gïanat suggest a technique
called BIPLOT for the gïaptfical presentation of matriccs of tank 2. The set of
3.3. Illustratlng a Set of Alternatives 247
curves with the aid of Founer seies. In this way ail the vectors can be plotted
on tbe same coordinate system for compm'ison.
Other proposMs for the graphlcal illustration of alternatives are given, for
example, in Vetschera (1992). They are based on indifference regons and finear
underlying value functions.
Let us finally mention a projection idea called GAIA (Geometrical Anal-
ysis for Interactive Aid). It is a part of the discrete multiattribute decision
analysis method PROMETHEE and it is described, for example, in Brans and
Mareschal (1990) and Mareschal and Brans (1988). The objective fanctions are
first modified to incbtde some preference inïonnation of the decision maker and
then normalized. These objective fimctions have sonne benefits when compared
to the original ones. Namely, they are in the same scales, big differences in the
objective values are emphasized and snall differences are lessened.
Principal comportent analysis is used in order to find a plane (two dimen-
sions) in whioh the new objective fimctions can be projected. The idea is to lose
as ]ittle infornation and variatiol as possible. In other words, the two largest
principal components are selected to form the projection plane. Thc weakness
here is that if the objective fitnctions have nonfinear relations, principal coin-
portent analysis cannot find it.
If selecting the plane is managed well enough, the relations betwcen the new
objective functlons and the alternative sobltions can be seen in their projec-
tions. Objective functlans are deplcted as vectors and alternatives as points on
the plane. For examplo, if two objective fiinctions are highly confiicting, their
vectors go in opposite direc%ions, whereas independent objective functions are
orthogonal and similar objective fimctions are oriented approximately in the
same direction. From the location of the alternatives one can see how well they
perform with respect to each objective fonction, that is, how near or far they
are fro eaoh other
It seens that this GAIA plan ideology is a rather clear method of illustra-
tion. Howcver, it has two naln limitation. Ffintly, the plane contalns only a
part of the diformatlon aval/able. Secondly, the conflfat characteristics of the
objective functions arc not absolute but depend on the alternatives considered.
3.3.8. General Rernarks
The problem ofhow we can deternine a priori whether the graphical formats
tlsed will ald rather than hinder decision naking is xamlned in Jarvenpaa
(1989) by comparative studies. The conclusion is that knowledge concerning
the relationslfip between the presentation format and the decision strategy can
facibtate the selection of the presentation format. Specinl attention is given to
the benefits of bar charts and grouped bar oharts.
Sfinfinr matters are handled in connectlon with visual interactlve simulation
in Bell and O'Keefe (1995) For example, itis conchded that the use of visual
displays generates solutions that are demonstrably better than those that nake

limited use of suçh ¢fisplays. This means that dffi'erent levels of usage of spe-
experiments, bar çharts were the most favoured visual displays.
Several existing stu¢fies on the applleabi]ity of graphs versus tables are anal-
ysed in Vessey (1991). According to the theory developed, it is concluded that
tables perform better in information acquisition taks in both rime and accu-
racy of performance. Thus tables are in order when specific data values must
be extracted, since they represent discrete data values. If formation must be
viewed ata glance, evaluated or relationships in the data are of interest, graphs
are recommendable. Thus, graphs and tables emphaize ¢fifferent çharacteristics
Uslng colours h ilfostratios ha advantages and disadvantages. Above ail,
colours may have specific connotations to the user. Such colours should be
avoided a far a possible.
An experimental evaluation of graphical and colour-enhanced information
prcsentation is glveI* in Benbaat and Dexter (1985). Colours improve the read-
ability and understandabifity of both symbolic ad graphical displays. Colours
make it eaier for the decision maker to asociate visually bafoïmation belong-
ing to the saine context or unit since such data are coded i the sêzne colour.
Encouaging results with multi-colour repos are mentioned by Benbaat and
Dexter (1985), who also stress that tabular representatioi is the best when
a simple retrieval of data is important and a graphical representation is the
best when relationships among the data have tobe shown. Graphs are visu-
A recommended way of presenting haformation to the decision maker is to
supplement each other. The declsion maker can change her or his attention
from one figure to another and possibly skip undesirable alternatives belote
making the final selection. A simple tabular format may be one of the figures.
Corïesponding idea are suggested, for instace, in Silverman et al. (1985) and
Steuer (1986, pp. 52522).
in Matos and Borges (1997). The idea is to illustrate alternatives in natural
for every objective fonction defining firzzy bounds, for exêznple, vry
wahlng machine selection pblem could be 'most cheap, medium power saver
ald little fat.' The declsion maker is aked for some descriptive information
a the bais of the membership fonctiots belote the solution process. This is a
3.. Illustrating a Set of Alternative 249

4. FUTURE DIRECTIONS
In this chapter, we out]ine some chalienging topics for the future deve
opmelt of multiabjective optimizatinn, malnly from a mathcmatical point of
view. In addition, we gave examples of promising ideas for research where Che
first steps have been taken but fitrther work is needed. Ail the issues mentioned
and many othcrs merit furher research and examination.
Multiobjective optimization is important, and knproved solution methods
can brbg about change in many areas and aspects of lire. Even though mul.
tiobjective optimization methods have been app]ed to solving a variety of
problems in many areas of life such as design problems in engineering produc
tion problems in economics, and environmental control problems in ecology,
there continue to exist many new problem types which could benefit highly
from multiobjective optknization. Particularly challenging h this respect are
real-life problcms. There fs clearly a need for more contribu¢ions reporting on
practical applications (making good use of more developed methods).
One ffiterestln i type of problems is so-called multidisciplinary re-engineer-
ing. It means that old engineering problems for example, in optimal desiga,
whose sohtions have been revised one feature at a time over the course of
years, are solved again rom the very beganing, taking various aspirations and
aspects into consideration at the same tkne. Obously this reqdires too/s of
multiobjcctive optimization.
An important challenge for the developers of interactive methods and ap
proaches is how to approach the decision maker. For example, a real experb
ment with problem-relaCed decision makers in Hobbs et al. (1992) shows that
the decision makers were sceptical of the value of multiobjective optimization
methods and they in some cases preferred unalded dcctsion making. This means
that the methods should hot only be user-friendly but also of real help to deci
sion nakers. Combining knowledge from the behavioural sciences with method
development could usefdily serve in thts direction.
The methodology of multiobjective optimization znust be improvedi This
means, for example, creatag computationally efficient ways of generating trade-
off information for more general problem types under less restrictlng assump
tions than those employed thus far. Another aspect is the structure of the
methods. On the one hand, providing the dectsion maker with the opportudity
for ree search is important. On the other hand, gidance and support must

be avallable, if desired. This necessitates developing mechalisms for dealilag
with inconsistncies. In additlon ways of identLfying appropriate methods for
different problems and differnt types of declslon makers are certalnly neededi
An exaïnple is the moaograph by Janssen (I992) where methods and decisinn
An alternative to creating new methods is to use different methods in differ
ent phases of the solution process. In this way, the positive features of various
methods can be expldited to theft best advantage in approprinte phases of
the solution process. In addition, it may be possible to overcome some of the
weakaesses of the existing methods.
An example of the combination of several method. is a meta algorithm e-
deavouring at COnsolidating different methods of multlobjectivc optimizatinm
This is proposed in Steuer and Whisman (1986). The idea is that the saine
parameters. The GDF the Tchebycheff and the reference point methods with
the reference direction appreach, STEM, the e-constralnt method and two in-
terextive versions of the weighting method are avaiinble. This idea is further
developed in Steuer and Gardiner (1990). An bnportant fact to consider, when
switching from one metbod to another in the mlddle of the solution process,
that is, how to malntaln the convergence properties, needs fltrther investiga-
tion. In Gardiner and Steuer (1994a, b), the meta algorithm is extended into a
unified algorithm contalning thirteen different interactive methods. A vtal ele-
ment of the algorithm is a matrix describing what kinds of switches are allowed
between the methods.
Similar ideas of combining several methods are proposed in Clnaco and
Antunes (1991). The system (only for MOLP problems) contalns, for exaïn
p/e, the ZW method, STEM and VIG. Odiy problems with three objective
flmctions can be handiedi The system bas also been implementedi A fltrther
developed bnplementation of the above-mentloned ideas is described in Antunes
et al. (1992a) and C]imaco and Antunes (1994). The method base package has
explicatedi TOMMIX is fftrther extended into SOMMIX for more than three
(llnear) objective functions in Clfi0aaco et al. (1997).
Another approexh to be elaborated is combining methods for contiauous and
discrete problems. It may, for examp/e, be that a set of solutions is generated
for the continuous problem and then ranked by means of discrete methods.
Examples of this are presented in Bard (1986), Kok and Lootsm (1985) and
Slowbmki (1991). One example of these methods, the light bem search, was
described in Section 5.9 of Part II.
One can also combine methods of global optimization with multiobjective
opt bnization methods. In this way, one can aire at being able to handle globally
,!
P-eto optimal sointlots, instead of locally Pareto optb0aal ones, in nonconvex
problems as well. ldeas of global multiobjective optlmlzation on the basis of
clustering are proposecl in TSrn (1983).
Stochastic global optbnizatinn method, lik genetic algoritinns, can also be
app[ied in multiobjective optimizatlon. An example of this approexh is given
in Osyczka and Kundu (1995). Another possibiSty for avdiding jmnaing into
locally Pareto optimal solutions is to use simdiated annealing or tabu search
as an underlying solver.
In Arbel and Korhonen (1996a, 1997a) a iew aspiration level-bascd method
is developed in the spirit of iaterior point methods (of linear programmlng) for
MOLP problems. The idca is to wander in the interior of the feasible objec-
tive region and only at the end to asccnd to the Pareto optimal surface. Here,
the generally adopted idea that decision makers should handie only (wealdy)
Pareto optimal solutions is called into question. One can justify such an ap
procl by the fext that the decision maker can see some improement in each
obj ectlve function instead of having to trade off ail the tbne. The interior point
method used is an affinscaling prbnal algoritllm (also treated in Arbel (1993,
1994b, c)). The saine idea is implemeated by using an interior point method
called th pñmal dual algorithm in Arbel and Korhonen (1996b, 1997b) (also
treated in Arbel (1994a, 1995)). Aimther modification of interior point meth-
ods for MOLP problems is described in Arbel and Oren (1994, 1996). In this
mehod, the gradient of an imp[icitly known value fitnctina is approximated
and a method of multiattrlbute decision analysis (naïnely AHP) is employed
in comparing alternatives. The gradients of an implicitly known value function
are also approximated and prinaal-dual linear methods used in Arbel (1997).
Results ftom other fields of research, for xaïnp/e, 8aïne theory, can also
be used in the solution processes. Among others, Rao studies the relatlonship
between Pareto optimal solutions and gaine theory in Rao (1987). He also
Another bnportant area of development is software desigaaed to implement
diflërent methods and, especially, the user inter face. As has been demonstrated,
few we[I-known software products exist for nonlinear multiobjectlve optbniz
tion prob/ems. As more and more adwnced computers and graphical devices
joyment of use. This in turn involves new ideas for representiag information,
of symbols. If the interface is able to adapt to the decisinn maker's style of
making decisions and is of help in analyzing the alternatives and results, nd
some of the deficiencies of the method itself.
As far as large-scale problems are concerned, the possibilities of parallel
computing are worh examining in making the solution processes more efficient.
in Grauer and Meren (1995).

properties of the problem and the preferences of the decision maker, and in the
solution process itself througa supporting the decision maker. As ma exmnple,
interactive MOLP methods and expert system techniques are integrated in
Antunes et al. (1992b). The system described includes rive methods, among
them, STEM and the ZW method. When the user ofthe system expresses her or
his hopes for further extions (sueh as a wish to get to know the neighbottrhood
of the current solution), the system suggests one of the available interactive
methods. Computer graphics are also available. There are many features that
deserve further researeh and development, but this is certainly an interesting
path to fullow.
A way of utilizing ar tificiai neurai networks in developing interactive multi-
objective optimization methods in proposed in Sun et al. (1996). The decision
solutions. The preference infurmation can be spccified by a prefcrence value for
ties of artificiai intelligence and neural networks in multiobjective optimization
are also eharted in Gai and Hanne (1997).
viewed in Antunes and Tsoukiàs (1997). The topics handled are fuzzy sets
multimedia, distributed computing, expert systems, object-oriented program-
ming, neural networks, and the World-Wide Web. For example, the possibilities
of the World-Wide Web in implementing interextive methods and making them
easily available were deait with in Subsection 5.12.8 of Part II. Thc example
given was WWW-NUVlBUS.
One more tlfing to mention are spreadsheets. They are widely used and thus
provide a familiar environment fur implementing interextivity in the methods.
This idea is realized in Steuer (1997) but it deserves further examination.
Flexibility in the matbematicai modelling of the problem is often desirable.
Flexibility includes the possibility of interehanglng the foies of objective and
constralnt functions and updating the model if necessary. The decision maker
certain aspirations. This means that int egrating the modelling and the solution
Itis hot to be forgotten that deaiing with incomplete information or un-
though it has hot been included in this book.
5. EPILOGUE
We have presented a seff-cont alned survey of the state of the art of nonlinear
multiobjective optimization together with a great number of further references.
After treating severai important concepts and their relations, we have consid-
ered some theoreticai results and connections.
We have demonstrated the methodology of multiobjectlve optimization by
describing severai methods and by gvJng references in respect of a large number
of other methods. Methods have been classified into four groups according to
the contribution of the decision maker in the solution process. Because the
group of interextive methods has been developed most, it has received the
In general, one can say that the theory and the methods of multiobjective
optimization have been extensively developed during the past couple of decades.
Software implementation are considerably less in evidence. There is aiso a
lexk of documentation in solvJng reai-life multiohjectivc optimization pïoblems
(using more developed methods). The reasons fur this may be ignorance of
the full range of possibilities contained in existing methods as well as the lack
of suitable methods. For our part, we have filled a gap in the literature by
co[lecting severai nonlinear multiobjective optimization nethods between the
In the development of methods the obvJous conclusion is that it is important
to continue in the dixection of user-ftiendliness. Methods must be even betteï
able to correspond to tbe clar exteristics of the declsion maker. If the aspirations
of the dectsion maker change durlng the solution process, the aigorithra must
be able to cope with this situation. Computationai tests bave cotffirmed thc
idea that decision makers want to feel in control of the sobttion process, and

REFERENCES
ebleau (1990b).
(1993), 149 164.

References 261
(1994), 283 290

(1967), 103 124.

References 267

6 (1987), I101 1121.

No. 2 (I991), 128 138,
References 275

(1988), 160 170.

INDEX
A posteriori methods 63,77 Conditional proper Preto optimality
A priori methods 64,115 30
Achievement fanction 108, 164, 168, Cone 6
180, 184, 186 ConnoEtcdness of (weakly) Peto
order-approximating 108 optimal set 20

Index 295
Intertive welghted TchebychoE L-metric 67
procedure ee Tchebycheff welghted 97
method Lp problem 67
Interchann 8 roles of func$ions 178, weighted 97
188, 205

296 Index
Iference direction 184,190 Strictly convex fnction 8
(SIGMOP) 209 Threshold

Twicd[fferentiable fction 9
Underhievement 122
Unified algoritkm 252
Utihty functioa 21
Utopi objective vector 16,98,101,
154,175
Value function 21,27, 64,115,132,138,
141,149,186, 220, 225
Value function method 115 118
Value fu=ction problem I15
Value paOl 240
Vector optimization 61
Vector subpïoblem 197
Vectoï version (of NIMBUS) 197
Veto tllreshold 181
VIG 188, 236
Reference dirtion approach
Weak efiïciency 25
Weak Peo op«imality 19
Weighted gond programing 122
Weighted goal progring problem
Weighted L-metric 97
Weighted L-problem 97
Weighted max-min problem 171
Weighted Tchebycheffmetrïc 97, I55
augmented 101
Weighted Tchebycheff problem 97,155,
augmented 101,160
lexicographïc 155
modified 101
Walghting method 78 85
Wighting problem 78
WWW-NIMBUS 206, 235
ZiontWallenJ (ZW) method 212

NONLINEAR MULTIOBJECTIVE OPTIMIZATION
Kaisa M. Miettinen
Problems with multiple objectives and criteria are generally known as
multiple criteria optimization or multiple criteria decision-making
(MCDM) problems. These types of problems typically focus on linear
programming methods. However, many phenomena are of a nonlinear
nature, which is why we need tools for nonlinear programming capable
of handling several conflicting or incommensurable objectives. In this
case, methods of traditional single objective optimization are not
enough; we need new ways of thinking, new concepts, and new
methods--like nonlinear multiobjective optimization. NONLINEAR
MULTIOBJECTIVE OPTIMIZATION provides an extensive, up-to-date,
self-contained and consistent survey, review of the literature and of the
state of the art on nonlinear (deterministic) multiobjective optimization,
its methods, its theory and its background.
The amount of literature on multiobjective optimization is immense. The
treatment in this book is based on approximately 1500 publications in
English printed mainly after the year 1980. Problems related to real-life
applications often contain irregularities and nonsmoothnesses. The
treatment of nondifferentiable multiobjective optimization in the
literature is rather rare. For this reason, we also include in this book
material about the possibilities, background, theory and methods of
nondifferentiable multiobjective optimization. Even though this book is
restricted fo deterministic nonlinear multiobjective methods, we would
like fo emphasize that promising new developments in multiobjective
methods can be undertaken in researching problems that are stochastic
and fuzzy in nature.
This book is intended for both researchers and students in the areas of
(applied) mathematics, engineering, economics, operations research and
management science; it is meant for both professionals and practitioners
in many different fields of application. The intention has been to provide
a consistent summary that may help in selecting an appropriate method
for the problem to be solved. The extensive bibliography will be of value
to researchers.
ISOR12
0-7923-8278-1
ISBN 0-7923-8278-I