Sequential approximate multiobjective optimization using computational intelligence

206 Pages·2009·3.9 MB·English

by  Hirotaka Nakayama

Sequential Approximate Multiobjective
Optimization Using Computational Intelligence
SeriesEditor:
JohannesJahn
UniversityofErlangen-Nürnberg
DepartmentofMathematics
Martensstr.3
81058Erlangen
Germany
[email protected]
Vector Optimization
The series in Vector Optimization contains publications in various fields of opti-
mization with vector-valued objective functions, such as multiobjective optimiza-
tion, multi criteria decision making, set optimization, vector-valued game theory
and border areas to financial mathematics, biosystems, semidefinite programming
andmultiobjectivecontroltheory.Studiesofcontinuous,discrete,combinatorialand
stochasticmultiobjectivemodelsininterestingfieldsofoperationsresearcharealso
included.Theseriescoversmathematicaltheory,methodsandapplicationsineco-
nomicsandengineering.ThesepublicationsbeingwritteninEnglishareprimarily
monographsandmultipleauthorworkscontainingcurrentadvancesinthesefields.
· ·
Hirotaka Nakayama Yeboon Yun Min Yoon
Sequential Approximate
Multiobjective Optimization
Using Computational
Intelligence
123
ProfessorHirotakaNakayama AssistantProfessorMinYoon
Dept.ofIntelligenceandInformatics DivisionofMathematicalScience
KonanUniversity PukyongNationalUniversity
8-9-1Okamoto,Higashinada 599Daeyeon3-dong,Nam-gu
Kobe658-8501 Busan608-737
Japan Korea
AssociateProfessorYeboonYun
Dept.ofReliability-basedInformation
SystemEngineering
FacultyofEngineering
KagawaUniversity
2217-20Hayashi-cho
Takamatsu
Kagawa761-0396
Japan
ISBN978-3-540-88909-0 e-ISBN978-3-540-88910-6
DOI10.1007/978-3-540-88910-6
SpringerSeriesinVectorOptimizationISSN1867-8971
LibraryofCongressControlNumber:2008943999
(cid:2)c Springer-VerlagBerlinHeidelberg2009
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations
areliabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneral descriptive names,registered names, trademarks, etc. inthis publication does not
imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective
lawsandregulationsandthereforefreeforgeneraluse.
Coverdesign:WMXDesignGmbH,Heidelberg
Printedonacid-freepaper
springer.com
Preface
Manykindsofpracticalproblemssuchasengineeringdesign,industrialman-
agement and financial investment have multiple objectives conflicting with
eachother.Thoseproblemscanbeformulatedasmultiobjectiveoptimization.
In multiobjective optimization, there does not necessarily a unique solution
which minimizes (or maximizes) all objective functions. We usually face to
the situation in which if we want to improve some of objectives, we have
to give up other objectives. Finally, we pay much attention on how much to
improve some of objectives and instead how much to give up others. This
is called “trade-off.” Note that making trade-off is a problem of value judg-
ment of decision makers. One of main themes of multiobjective optimization
is how to incorporate value judgment of decision makers into decision sup-
port systems. There are two major issues in value judgment (1) multiplicity
of value judgment and (2) dynamics of value judgment. The multiplicity of
value judgment is treated as trade-off analysis in multiobjective optimiza-
tion. On the other hand, dynamics of value judgment is difficult to treat.
However,itisnaturalthatdecisionmakerschangetheirvaluejudgmenteven
in decision making process, because they obtain new information during the
process. Therefore, decision support systems are to be robust against the
change of value judgment of decision makers. To this aim, interactive pro-
grammingmethodswhichsearchasolutionwhileelicitingpartialinformation
on value judgment of decision makers have been developed. Those methods
arerequiredtoperformflexiblyfordecisionmakers’attitude.Atearly1980s,
many interactive programming methods for solving multiobjective optimiza-
tion have been developed. Above all, the aspiration level approach to multi-
objective programming problems has been widely recognized to be effective
in many practical fields.
Anothermajorissueisthatinmanypracticalproblems,inparticularinen-
gineeringdesign,thefunctionformofcriteriaisnotgivenexplicitlyintermsof
designvariables.Giventhevalueofdesignvariables,underthiscircumstance,
the value of objective functions is obtained by real/computational experi-
ments such as structural analysis, fluid mechanic analysis, thermodynamic
v
vi Preface
analysis, and so on. Usually, these experiments are time consuming and ex-
pensive. One of recent trends in optimization is how to treat these expensive
criteria.Inordertomakethenumberoftheseexperimentsasfewaspossible,
optimization is performed in parallel with predicting the form of objective
functions.Thisiscalledsequentialapproximateoptimizationwithmetamod-
eling. It has been observed that techniques of computational intelligence can
be effectively applied for this purpose. Moreover, techniques of multiobjec-
tive optimization themselves can also be applied to develop effective meth-
odsincomputationalintelligence.Forexample,theauthorsdevelopedseveral
kinds of support vector machines using multiobjective optimization and goal
programming.
Recently, researches of generating Pareto frontier are actively made. It is
useful to visualize Pareto frontier, because decision makers can make trade-
off analysis very easily on the shown figures of Pareto frontier. However, it
is difficult to generate Pareto frontier in cases with more than two or three
objectives. At this event, a method combining aspiration level approach and
sequential approximate optimization using computational intelligence was
proposed and recognized to be effective in many practical problems.
This book describes those sophisticated methods for multiobjective opti-
mization using computational intelligence along with real applications. This
topicseemsquitenew.Nobookonthistopichasbeenseentoourknowledge
inspiteofitsimportance.Thebookisself-containedandcomprehensive.The
potential readers are researchers, practitioners in industries and students of
graduate course and high grade of undergraduate course.
Wehopethatthereadersofboththeoreticalresearchersandpractitioners
canlearnthroughthisbookseveralmethodologiesonanewtrendofmultiob-
jectiveoptimizationwhichisveryimportantandapplicableinmanypractical
fields.
Japan and Korea Hirotaka Nakayama
November 2008 Yeboon Yun
Min Yoon
Contents
List of Tables ................................................. xi
List of Figures................................................ xiii
1 Basic Concepts of Multiobjective Optimization ........... 1
1.1 Mathematical Foundations............................... 1
1.2 Preference Order and Domination Set..................... 4
1.3 Scalarization........................................... 5
1.3.1 Linearly Weighted Sum ........................... 7
1.3.2 Tchebyshev Scalarization Function ................. 8
1.3.3 Augmented Tchebyshev Scalarization Function....... 8
1.3.4 Constraint Transformation Method ................. 10
1.4 Scalarization and Trade-off Analysis ...................... 11
2 Interactive Programming Methods for Multiobjective
Optimization ............................................. 17
2.1 Goal Programming ..................................... 17
2.2 Why is the Weighting Method Ineffective? ................. 20
2.3 Satisficing Trade-off Method ............................. 22
2.3.1 On the Operation P .............................. 22
2.3.2 On the Operation T .............................. 24
2.3.3 Automatic Trade-off .............................. 25
2.3.4 Exact Trade-off .................................. 29
2.3.5 Interchange Between Objectives and Constraints ..... 29
2.3.6 Remarks on Trade-off for Objective Functions
with 0-Sensitivity ................................ 30
2.3.7 Relationship to Fuzzy Mathematical Programming ... 32
2.4 Applications ........................................... 34
2.4.1 Feed Formulation for Live Stock.................... 34
2.4.2 Erection Management of Cable-Stayed Bridge........ 35
2.4.3 An Interactive Support System for Bond Trading..... 37
2.5 Some Remarks on Applications........................... 42
vii
viii Contents
3 Generation of Pareto Frontier by Genetic Algorithms..... 45
3.1 Evolutionary Multiobjective Optimization ................. 45
3.1.1 Vector Evaluated Genetic Algorithm................ 46
3.1.2 Multiobjective Genetic Algorithm .................. 47
3.1.3 Elitist Nondominated Sorting Genetic Algorithm
(NSGA-II) ...................................... 51
3.1.4 Strength Pareto Evolutionary Algorithm (SPEA2).... 53
3.2 Fitness Evaluation Using DEA ........................... 56
3.2.1 Quantitative Evaluation of Fitness.................. 56
3.2.2 Data Envelopment Analysis........................ 57
3.2.3 DEA Models..................................... 59
3.2.4 DEA Method .................................... 63
3.3 Fitness Evaluation Using GDEA ......................... 64
3.3.1 GDEA Model.................................... 64
3.3.2 GDEA Method .................................. 66
3.4 Comparisons of Several Fitness Evaluations................ 67
4 Multiobjective Optimization and Computational
Intelligence ............................................... 73
4.1 Machine Learning ...................................... 73
4.1.1 Learning and Generalization ....................... 75
4.1.2 The Least Square Method ......................... 77
4.2 Radial Basis Function Networks .......................... 79
4.3 Support Vector Machines for Pattern Classification ......... 83
4.3.1 Hard Margin SVM ............................... 84
4.3.2 MOP/GP Approaches to Pattern Classification ...... 86
4.3.3 Soft Margin SVM ................................ 88
4.3.4 ν-SVM.......................................... 90
4.3.5 Extensions of SVM by MOP/GP ................... 90
4.3.6 Comparison of Experimental Results................ 96
4.4 Support Vector Machines for Regression................... 98
4.5 Combining Predetermined Model and SVR/RBFN.......... 110
5 Sequential Approximate Optimization .................... 113
5.1 Metamodels ........................................... 113
5.2 Optimal Design of Experiments .......................... 115
5.3 Distance-Based Criteria for Optimal Design................ 120
5.4 Incremental Design of Experiments ....................... 122
5.5 Kriging and Efficient Global Optimization ................. 126
5.5.1 Kriging and Prediction............................ 127
5.5.2 Efficient Global Optimization ...................... 133
5.6 Distance-Based Local and Global Information.............. 141
Contents ix
6 Combining Aspiration Level Approach and SAMO........ 151
6.1 Sequential Approximate Multiobjective Optimization
Using Satisficing Trade-off Method ....................... 152
6.2 MCDM with Aspiration Level Method and GDEA.......... 159
6.3 Discussions ............................................ 167
7 Engineering Applications ................................. 169
7.1 Reinforcement of Cable-Stayed Bridges.................... 169
7.1.1 Dynamic Characteristics of Cable-Stayed Bridge...... 170
7.1.2 Discussions ...................................... 174
7.2 Multiobjective Startup Scheduling of Power Plants ......... 176
7.2.1 Startup Scheduling for Combined Cycle Power Plant.. 176
7.2.2 Sequential Approximate Multiobjective Optimization
for Optimal Startup Scheduling .................... 178
7.2.3 Application Results............................... 179
7.2.4 Discussions ...................................... 183
References.................................................... 185
Index......................................................... 193
List of Tables
3.1 Fitness assignment of MOGA ................................ 50
3.2 Case of single input and single output......................... 58
3.3 DEA efficiency of Example 3.1 ............................... 59
4.1 Classification rate by GP .................................... 97
4.2 Classification rate by SVM ............................... 97
hard
4.3 Classification rate by SVM ............................... 98
soft
4.4 Classification rate by ν-SVM................................. 99
4.5 Classification rate by SVM ............................... 100
total
4.6 Classification rate by μ-SVM................................. 101
4.7 Classification rate by μ-ν-SVM............................... 102
4.8 Comparison of the results (unit: %)........................... 110
5.1 Results by distance-based local and global information
in Example 5.5............................................. 144
5.2 Simulation results of example (5.29) with C0 =50
x
and C0 =30 ............................................... 148
f
5.3 Comparison among several methods .......................... 149
6.1 Result by SQP using a quasi-Newton method for real objective
functions .................................................. 157
6.2 Result by SAMO with 100 function evaluations................. 158
7.1 Result for Case 1 ........................................... 174
7.2 Result for Case 2 ........................................... 175
7.3 Approximation error of the RBFN models (unit: %) ............ 181
7.4 Results of the objective functions............................. 183
xi