Rough multiple objective decision making

445 Pages·2011·4.361 MB·English

by  Jiuping Xu, (Professor of management science), Zhimiao Tao

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
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K13329_Cover.indd 1 6/7/11 9:49 AM
ROUGH
MULTIPLE OBJECTIVE
DECISION MAKING
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ROUGH
MULTIPLE OBJECTIVE
DECISION MAKING
JIUPING XU
ZHIMIAO TAO
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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-
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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