Uncertain Data Envelopment Analysis

157 Pages·2015·1.297 MB·English

by  Meilin Wen (auth.)

Uncertainty and Operations Research
Meilin Wen
Uncertain
Data
Envelopment
Analysis
Uncertainty and Operations Research
Forfurthervolumes:
http://www.springer.com/series/11709

Meilin Wen
Uncertain Data Envelopment
Analysis
123
MeilinWen
ScienceandTechnologyonReliability
andEnvironmentalEngineeringLaboratory
SchoolofReliabilityandSystemsEngineering
BeihangUniversity
Beijing,Beijing,China
ISSN2195-996X ISSN2195-9978(electronic)
ISBN978-3-662-43801-5 ISBN978-3-662-43802-2(eBook)
DOI10.1007/978-3-662-43802-2
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014944128
©Springer-VerlagBerlinHeidelberg2015
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
In the past 30 years, data envelopment analysis (DEA) has grown into a powerful
quantitative, analytical tool for measuring and evaluating relative efficiency of
decision-making units (DMUs). DEA has been successfully applied to many
different types of entities engaged in a wide variety of activities in many contexts
worldwide. Although DEA offers more advantages than many other statistical
approaches, some limitations have to be considered. One important problem is its
sensitivity to data. Therefore, a key to the success of the DEA approach is the
accurate measure of all factors, including that of inputs and outputs. However, in
manysituations,suchasinamanufacturingsystem,inaproductionprocess,orina
servicesystem,inputsandoutputsaresovolatileandcomplexthattheyaredifficult
to measure in an accurate way. Thus, various uncertain DEA methods have been
proposedtohandlethedatavariationinDEA.Thisbookisintendedtopresentthe
milestonesintheprogressionofuncertainDEA.
UncertainTheories
Based on different uncertain theories, uncertain DEA methods deal with the
problems involving uncertain inputs and uncertain outputs. Chapter 1 will present
the basic introduction to uncertain theories which are crucial for the following
chapters, including probability theory, credibility theory, uncertainty theory, and
chancetheory.
Introduction toDEA
Chapter2isintendedtorepresentamilestoneintheprogressionofDEA,including
acomprehensivereviewanddiscussionofbasicDEAmodelsandextensionstothe
basicDEAmethods.
v
vi Preface
StochasticDEA
Incorporation of random variations into DEA analysis has received considerable
attention in recent years. Chapter 3 will present some stochastic DEA model
extensionsoftheusualdeterministicDEAformulations.Thiskindofmodelsmakes
itpossibletoreplace“efficient”and“notefficient”with“probabilityefficient”and
“probabilitynotefficient,”respectively.
FuzzyDEA
Fuzzinessisabasictypeofsubjectiveuncertainty,whichhasbeenappliedinDEA
recently. Chapter 4 will provide some fuzzy DEA methods based on credibility
measure, including fuzzy DEA models, fuzzy sensitivity analysis, fuzzy fully
rankingmethods,andfuzzycongestion.
UncertainDEA
Uncertainty theory is a branch of axiomatic mathematics for modeling human
uncertainty. Chapter 5 will apply uncertainty theory to DEA so as to produce a
new method of dealing with the empirical data. Besides uncertain DEA models,
uncertain sensitivity analysis, uncertain fully ranking methods, and uncertain
congestionwillalsobetakenintoaccount.
Hybrid DEA
In order to evaluate the DMUs in which uncertainty and randomness appear
simultaneously, Chap.6 will introduce a hybrid DEA method based on chance
measure.HybridDEAmodels,hybridfullyrankingmethods,andhybridcongestion
willbeincludedinChap.6.
Purpose
Thepurposeofthisbookistoprovidesomemethodsindealingwithuncertaintyin
DEA.ThebookpresentsfouruncertainDEAmethodsbasedondifferentmeasures
includingprobabilitymeasure,credibilitymeasure,uncertaintymeasure,andchance
measure.Thebookissuitableforresearchers,engineers,andstudentsinthefields
of management science, economics, operations research, industrial engineering,
informationscience,andsoon.
Preface vii
Acknowledgment
This work was supported by the National Natural Science Foundation of China
(No.71201005).
Beijing,China MeilinWen
October1,2013

Contents
1 UncertainTheories........................................................... 1
1.1 ProbabilityTheory...................................................... 1
1.1.1 ProbabilityMeasure ............................................ 1
1.1.2 RandomVariable................................................ 2
1.1.3 OperationalLaw................................................ 4
1.1.4 ExpectedValue.................................................. 6
1.2 CredibilityTheory ...................................................... 11
1.2.1 CredibilityMeasure............................................. 11
1.2.2 FuzzyVariable.................................................. 17
1.2.3 ExpectedValue.................................................. 22
1.3 UncertaintyTheory ..................................................... 23
1.3.1 UncertainMeasure.............................................. 24
1.3.2 UncertainVariable.............................................. 24
1.3.3 OperationalLaw................................................ 27
1.3.4 ExpectedValue.................................................. 31
1.4 ChanceTheory.......................................................... 35
1.4.1 UncertainRandomVariable.................................... 35
1.4.2 UncertainRandomMeasure.................................... 36
1.4.3 OperationalLaw................................................ 38
1.4.4 ExpectedValue.................................................. 40
References..................................................................... 44
2 IntroductiontoDEA......................................................... 45
2.1 SymbolsandNotations ................................................. 46
2.2 CCRModel.............................................................. 46
2.3 BCCModel.............................................................. 50
2.4 AdditiveModel.......................................................... 53
2.5 SBMModel ............................................................. 56
2.6 RussellMeasureModel................................................. 57
References..................................................................... 58
ix