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Operations Research R O Under intense scrutiny for the last few decades, Multiple Objective U ROUGH Decision Making (MODM) has been useful for dealing with the multiple- G criteria decisions and planning problems associated with many important applications in fields including management science, engineering design, D H MULTIPLE OBJECTIVE and transportation. Rough set theory has also proved to be an effective E M mathematical tool to counter the vague description of objects in fields C DECISION MAKING such as artificial intelligence, expert systems, civil engineering, medical I U S data analysis, data mining, pattern recognition, and decision theory. L I O T Rough Multiple Objective Decision Making is perhaps the first book to combine state-of-the-art application of rough set theory, rough N I P approximation techniques, and MODM. It illustrates traditional techniques— M L and some that employ simulation-based intelligent algorithms—to solve E a wide range of realistic problems. Application of rough theory can A remedy two types of uncertainty (randomness and fuzziness) that present K O significant challenges to existing decision-making methods, so the I B N authors describe the use of rough sets to approximate the feasible set, J and they explore use of rough intervals to demonstrate relative coefficients G E C and parameters involved in bi-level MODM. The book reviews relevant T literature and introduces models for both random and fuzzy rough MODM, applying proposed models and algorithms to problem solutions. I V Given the broad range of uses for decision making, the authors offer E background and guidance for rough approximation to real-world problems, with case studies that focus on engineering applications, including construction site layout planning, water resource allocation, and resource- constrained project scheduling. The text presents a general framework of rough MODM, including basic theory, models, and algorithms, as well as a proposed methodological system and discussion of future research. XU JIUPING XU TAO ZHIMIAO TAO K13329 K13329_Cover.indd 1 6/7/11 9:49 AM ROUGH MULTIPLE OBJECTIVE DECISION MAKING TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk ROUGH MULTIPLE OBJECTIVE DECISION MAKING JIUPING XU ZHIMIAO TAO MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MAT- LAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110617 International Standard Book Number-13: 978-1-4398-7236-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Preface Since V. Pareto introduced the concept of optimal solution in 1896, multiple ob- jectivedecisionmaking(alsocalledvectoroptimizationdecisionmaking)hasbeen found many important applications in practical decision-making problems, such as in management science, engineering design, transportation, and so on. To find a solution for multiple objective decision-making problems requires the intervention of a decision maker. The main idea is simple: the system generates reasonable al- ternatives, and the decision maker will make choices. Those choices are used to leadthealgorithmtogeneratemorealternativesuntilthedecisionmakerwillreach the solution that pleases him or her most. Helping decision makers to deal with multiple criteria decision and planning problems has been the subject of intensive studies since the 1970s, but many theoretical concepts were defined much earlier. In the 1970s, research focused on the theory of multiple objective mathematical programming and the development of procedures and algorithms for solving such problems. Many ideas originated from the theory of mathematical programming. Thereareseveralbasicsolutionapproachessuchastheweighted-sumapproach,the utilityfunctionapproach,thecompromiseapproach,andthelexicographicordering approach. Basedonthebasicsolutionapproach, manyresearcherspresentedsome comprehensiveapproachestomultipleobjectivedecision-makingproblems. The rough set theory initialized by Z. Pawlak in 1982 has been proved to be an effectivemathematicaltoolindealingwiththevaguedescriptionofobjects.Afunda- mentalassumptionisthatanyobjectfromauniverseisperceivedthroughavailable information, and such information may not be sufficient to characterize the object exactly.Aroughsetisthendefinedbyapairofcrispsets,calledthelowerandupper approximations.Sincethedaywhenitwasputforward,roughsettheoryhasbeenap- pliedinmanyfieldssuchasartificialintelligence,expertsystems,civilengineering, medical data analysis, data mining, pattern recognition, and decision theory. This bookmainlyconcentratesontheapplicationofroughsettheoryandroughapproxi- mationtechniquestomultipleobjectivedecisionmakingandsystematicallypresents astate-of-the-artofmultipleobjectivedecisionmakingbasedroughapproximation inbothtechniquesandapplications. Theapplicationofroughsettheoryandrough approximation technique to multiple objective decision making includes rough ap- proximation to feasible regions and the assumption that the parameters are rough intervals. Forexample, whenthequantityofanunregulatedwatersourceishighly imprecise,ithastobeestimatedintermsofadecisionmaker’ssubjectiveexperiences andobjectivehistoricaldata. Anoptimalmanagementstrategycapableofhandling the problem under both normal and special conditions is often preferred. For this type of information, existing methods can neither reflect its dual-layer features nor v vi entirelypassittotheresultingdecisions. Twochallengesthusemerge: oneistofind an effective expression that could reflect dual-layer information (i.e., not only the parameter’smostpossiblevaluebutalsoitsmostreliablevalue),andtheotheristo useanappropriatemethodtogeneratedecisionswithdual-layerinformationdirectly correspondingtothemostpossibleandreliableconditionsofthesystem. Roughin- tervalscanbeasuitableconcepttoexpresssuchinformation. Inthisbook,weapply roughtheorytoclassicmultipleobjectivedecisionmakingintwoways.Oneisusing roughsetstoapproximatethefeasibleset, andtheotherisusingroughintervalsto approximaterelativecoefficients.Forbilevelmultipleobjectivedecisionmaking,we approximatesomeparameterswithroughintervals. In the classical multiple objective decision-making model, all data and informa- tionareassumedtobeabsolutelyaccurate,andtheobjectivesandconstraintsareall assumed to be well expressed by mathematical formation. However, it is difficult toclearlydescribetheobjectivefunctionsandconstraintsbymathematicalequation inmanyrealisticproblems,andthus,themultipleobjectivedecision-makingmodel with certain parameters cannot deal with all real-life problems. The two kinds of uncertainties are randomness and fuzziness. In order to study the randomness and fuzzinessofobjects,twokindsoftheorieswereproducedsuccessivelyinthehistory ofmathematics,thatis,theprobabilitytheoryandthepossibilitytheory. Theprob- abilitytheory,asascience,originatedinthemiddleofthe17thcenturywithPascal, Fermat,andHuygens.Therealhistoryoftheprobabilitytheorybeginswiththework ofJ.Bernoulli, andDeMoivre. Afterthat, S.Poisson, C.Gauss, P.Chebyshev, A. Markov, andA.Lyapunovmadeimportantcontributiontothistheory. Themodern period in the development of the probability theory begins with its axiomatization. ThefirstworkinthisdirectionwasdonebyS.Bernstein,R.Mises,andE.Borel. In 1933,A.Kolmogorovpresentedtheaxiomaticsystemofprobabilitytheorythathas becomegenerallyacceptedandisnotonlyapplicabletoalltheclassicalbranchesof probability theory but also provides a firm foundation for the development of new branchesthathavearisenfromquestionsinthesciencesandinvolveinfinitedimen- sionaldistribution. Sinceitsintroductionin1965byL.Zadeh,thefuzzysettheory has been well developed and applied in a wide variety of real problems. The term fuzzyvariablewasfirst introducedby S.Kaufmann, andthen itwasadoptedby L. ZadehandS.Nahmias. ThepossibilitytheorywasproposedbyL.Zadeh,anddevel- opedbymanyresearcherssuchasD.DuboisandH.Prade. Considering multiple objectives and uncertainties, we can employ the general modeltoformulatemultipleobjectiveuncertainmultipleobjectivedecision-making problemsby  max[(cid:189)f1(x,ξξξ),f2(x,ξξξ),···,fm(x,ξξξ)] g (x,ξξξ)≤0,r=1,2,···,p (1)  s.t. r x∈X wherex=(x ,x ,···,x )T isann-dimensionaldecisionvector,ξξξ=(ξ,ξ,···,ξ) 1 2 n 1 2 n isauncertainvector, f(x,ξξξ)areobjectivefunctions,i=1,2,···,m,g (x,ξξξ)≤0are i r uncertainconstraints,r=1,2,···,p,andX isafixedsetthatisusuallydetermined byafinitenumberofinequalitiesandequalitiesinvolvingfunctionsofx. Thereexist vii severalmethodstodealwiththeuncertaintiesof(1). However,theexistingmethods have significant drawbacks from the viewpoint of information. For random multi- pleobjectivedecisionmaking,theexistingmethodsmayleadtolossofinformation due to statistical deviation. For fuzzy multiple objective decision making, the ex- isting methods may lead to a loss of information due to subjective deviation. The applicationofroughtheoryto(1)canremedythesedrawbacks. This book takes real-life problems as the background and guidance and devel- ops the general framework of rough multiple objective decision making, including the basic theory, model, and algorithm. In addition, the application of the multiple objective decision making model based on rough approximation to the real world are presented. The issues selected are for engineering applications, including con- structionsitelayoutplanningproblem,waterresourceallocationproblem,resource- constrainedprojectschedulingproblem,andearth-rockworkallocationproblem. We shall now give a brief indication of the contents and organization of Rough MultipleObjectiveDecisionmaking.Chapter1,onelementsoftheroughsettheory, including materials that will be familiar to most readers: basic concepts and prop- erties of rough sets, rough membership, and rough intervals. These materials are thetheoreticalbasisofourbook,whichwillhelpreadersunderstandthesubsequent chaptersbetter. Chapter2dealswithmultipleobjectiveroughdecisionmakinganditsapplication toconstructionsitelayoutplanningproblems. Themultipleobjectiveroughdecision makingreferredtohereincludestwotypesofmodels: oneinwhichthefeasibleset isapproximatedbytheroughset,andtheotherinwhichtheparametersareassumed as rough intervals. For the latter, expected value rough model using the expected value operator and the chance-constrained rough model, and the dependent-chance rough model using the chance operator are presented. For each model, we deduce the equivalent model of those models in special cases and propose a technique for theroughsimulation-basedgeneticalgorithmsforgeneralcases. Arealapplication fortheLongtanprojectisillustratedinthelastsectionofChapter2. ThemainobjectiveinChapter3isthebilevelmultipleobjectiveroughdecision- making problem and its application to water resource allocation problems. The roughnessofthisclassofmodelmanifestsintheroughintervalsofparameters. We present three methods similar to the methods in Chapter 2. In some special cases, we obtain the equivalent models and solve them by traditional methods. For gen- eralcases,roughsimulation-basedtabusearchalgorithmsareproposed.TheGan-Fu Plainwaterresourceallocationisdiscussedinthelastpartofthischapter. In order to apply the rough set theory to random multiple objective decision- making problems, we propose random multiple objective rough decision-making modelsinChapter4.Thischapterfirstreviewstheliteraturesonresource-constrained projectschedulingproblems. Thenextthreesectionsintroducetherandomexpected value rough model by the expected value operator, the random chance-constrained roughmodel,andtherandomdependent-chanceroughmodelbythechanceopera- tor. Ineachsection,wededucetheequivalentmodelforthoseproblemsandpropose a technique of rough simulation-based multiobjective particle swarm optimization algorithm to deal with those problems. In the last section, the propose models and viii algorithmshavebeenappliedtosolvetheproblemsintroducedinthefirstsection. Chapter 5 gives an introduction to random multiple objective rough decision- making. Inthischapter,weconsidertheapplicationoftheroughsettheorytofuzzy multipleobjectivedecision-makingproblemscalledfuzzymultipleobjectiverough decision-making. The allocation problem literature is reviewed in the first section, andthenthefuzzymultipleobjectiveroughdecision-makingmodelisproposed.The nextthreesectionsintroducethefuzzyexpectedvalueroughmodelbytheexpected valueoperator,thefuzzychance-constrainedroughmodel,andthefuzzydependent- chanceroughmodelbythechanceoperator. Ineachsection,wededucetheequiva- lentmodelforthoseproblemsandproposeasimulation-basedsimulatedannealing algorithmtodealwiththoseproblems. Inthelastsection,theproposedmodelsand algorithmsareappliedtosolvetherealproblemintroducedinthefirstsection. Asconclusion, Chapter6providesthemainproblems, models, methods, andal- gorithmstoobtaintheorganicsystemsandproposesthemethodologicalsystemfor theroughmultipleobjectivedecisionmaking. Additional material including MATLAB(cid:176)R code for some numerical examples willbepostedintheAppendix. ThismonographhasbeensupportedbytheNationalNaturalScienceFoundation ofChina(GrantNo.79760060,70171021),theNationalScienceFoundationforDis- tinguishedYoungScholars,P.R.China(GrantNo.70425005),andtheKeyProgram of National Natural Science Foundation of China (Grant No. 70831005). We are greatlyindebtedtoanumberprofessors,suchasS.WangandV.Kachitvichyanukul, fromwhomwehavereceivedmuchhelpinoptimizationtheoryandparticleswarm optimization. For discussions and advice, the authors also thank researchers from theUncertaintyDecision-MakingLaboratoryofSichuanUniversity,particularly,X. Zhou, L. Yao, Z. Zhang, Y. Tu, Z. Li, Y. Ma, Z. Zeng, and C. Ding, who have done valuable work in this field and have made a number of corrections to this book. Finally,theauthorsexpresstheirdeepgratitudetoCRCPress,Taylor&Fran- cis Group professional editorial staffs, especially Leong Li-Ming, Amber Donley, SharmaAastha,AndrewShih,JimMcGovernandtheproofreader. SichuanUniversity, JiupingXu December,2010 ZhimiaoTao MATLAB(cid:176)R isaregisteredtrademarkofTheMathWorks,Inc. Forproductinfor- mation,pleasecontact: TheMathWorks,Inc. 3AppleHillDrive Natick,MA01760-2098USA Tel: 5086477000 Fax: 5086477001 E-Mail: [email protected] Web: www.mathworks.com List of Tables 1.1 Asetofpersons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Efficientsolutionswithdifferentweights . . . . . . . . . . . . . . . 43 √ 2.2 Interactiveprocessas 3/2<α≤1 . . . . . . . . . . . . . . . . . 61 √ √ 2.3 Interactiveprocessas1− 3/6<α≤ 3/2 . . . . . . . . . . . . 61 √ √ 2.4 Interactiveprocessas32 3/3−16<α≤1− 3/6 . . . . . . . . 62 √ 2.5 Interactiveprocessas0≤α<32 3/3−16 . . . . . . . . . . . . . 62 2.6 Efficientsolutionswithdifferentweights . . . . . . . . . . . . . . . 76 2.7 Facilitiestobedistributedandtheirrequiredareaduringeachphase 120 2.8 Thedistancesbetweenlocations(m) . . . . . . . . . . . . . . . . . 121 2.9 Thestartupcostsoffacilitiesduringthefirstperiod(1000CNY) . . 122 2.10 Theroughintervalsoftheinteractivecosts(1000CNY/km) . . . . . 123 2.11 Theoperationalcostsoffacilitiesatdifferentlocations(1000CNY) 123 2.12 ThesafestdynamicsitelayoutplanintheLongtancase . . . . . . . 124 2.13 TheminimumcostofdynamicsitelayoutplanintheLongtancase . 124 3.1 ListingoftheECTSparameters . . . . . . . . . . . . . . . . . . . 144 3.2 Theparametersofroughintervalsd˜ (i=1,2,3,4;k=1,2,3) . . . 181 ik 3.3 Theotherparametersofwaterusersinthewaterallocationsystem . 181 3.4 Theotherparametersofwaterdistrictsinthewaterallocationsystem 182 3.5 Theexpectedvalueofroughintervalsd˜ (i=1,2,3,4;k=1,2,3) . 182 ik 3.6 Theresultsofsolvingfivesingleobjectiveproblems(millionCNY) 182 3.7 Sensitivityanalysis(i=1,2,3,4) . . . . . . . . . . . . . . . . . . . 183 4.1 Theoptimalsolutionbyroughsimulation-basedPSO . . . . . . . . 218 4.2 TheoptimalsolutionbyAPSO . . . . . . . . . . . . . . . . . . . . 232 4.3 Computationalresults . . . . . . . . . . . . . . . . . . . . . . . . . 240 4.4 Executedmodes,successors,andexpectedprocessingtime . . . . . 263 4.5 Project-relateddatasetofdrillinggroutingproject . . . . . . . . . . 264 4.6 Optimalresultoftheproject . . . . . . . . . . . . . . . . . . . . . 266 5.1 Payofftable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 5.2 PayofftableofExample5.8 . . . . . . . . . . . . . . . . . . . . . 318 5.3 Transportationschemefortheexampleofroadsections . . . . . . . 338 5.4 QuantitiesofEPandFP(×104m3) . . . . . . . . . . . . . . . . . . 343 5.5 CapacitiesofBA,TS,andDS(×104m3) . . . . . . . . . . . . . . . 344 5.6 Matchrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 ix

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