Stochastic Benchmarking: Theory and Applications

154 Pages·2021·2.968 MB·English

by  Alireza Amirteimoori, Biresh K. Sahoo, Vincent Charles, Saber Mehdizadeh

International Series in
Operations Research & Management Science
Alireza Amirteimoori
Biresh K. Sahoo
Vincent Charles
Saber Mehdizadeh
Stochastic
Benchmarking
Theory and Applications
International Series in Operations Research &
Management Science
FoundingEditor
FrederickS.Hillier,StanfordUniversity,Stanford,CA,USA
Volume 317
SeriesEditor
Camille C. Price, Department of Computer Science, Stephen F. Austin State Uni-
versity,Nacogdoches,TX,USA
AssociateEditor
Joe Zhu, Foisie Business School, Worcester Polytechnic Institute, Worcester, MA,
USA
Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/6161
(cid:129) (cid:129)
Alireza Amirteimoori Biresh K. Sahoo
(cid:129)
Vincent Charles Saber Mehdizadeh
Stochastic Benchmarking
Theory and Applications
AlirezaAmirteimoori BireshK.Sahoo
RashtBranch,IslamicAzadUniversity XavierInstituteofManagement
Rasht,Iran XIMUniversity
Bhubaneswar,India
VincentCharles SaberMehdizadeh
CenterforValueChainInnovation, RashtBranch,IslamicAzadUniversity
CENTRUMCatolicaGraduate Rasht,Iran
BusinessSchool
PontificalCatholicUniversityofPeru
Lima,Peru
ISSN0884-8289 ISSN2214-7934 (electronic)
InternationalSeriesinOperationsResearch&ManagementScience
ISBN978-3-030-89868-7 ISBN978-3-030-89869-4 (eBook)
https://doi.org/10.1007/978-3-030-89869-4
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland
AG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof
illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmission or information storage and retrieval, electronic adaptation, computer software, or by
similarordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors, and the editorsare safeto assume that the adviceand informationin this
bookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
We dedicate this book to Professor Rajiv
D. Banker, a pioneer in the field of data
envelopment analysis. His passion for
research has helped to shape our
understanding and thinking around efficiency
and productivity measurement. A Harvard
University graduate Professor Banker has
dedicated his career to advancing the field
and spreading the knowledge across the
Globe while encouraging and supporting
researchers along the way. His extraordinary
contributions to literature and society are
inspirational and enduring and are matched
only by his humble and kind personality.
Preface
In the last three decades, benchmarking as a performance measurement procedure
has become a powerful quantitative and analytical tool for measuring the relative
efficiency of entities. This book introduces benchmarking techniques under a sto-
chastic environment, most notably, the Stochastic Data Envelopment Analysis
(SDEA). Similar to different stochastic theories, the SDEA approach deals with
real-life problems that involve stochastic inputs and outputs. The SDEA models
shown in this book account for the possibility of variations in both inputs and
outputs. To enhance understanding, simple examples along with graphical illustra-
tionsarealsoprovided.Theusesoftheoriesandinterpretationsofthemathematical
programs are emphasized. Moreover, the proposed mathematical programs are
combinedwitheconomicandorganizationalthinking.Realindustry-relevantappli-
cationsarefurtherdiscussed.Theunderlyingtheoriesareintroduced,whichsupport
relevantcalculationsandhelpinthediscussionofapplications.Overall,ourpurpose
is to inform the reader about the advantages of the different methods under deter-
ministic and stochastic environments, as well as about the proper use of these
methodsindifferentcases.Thisbookcanbeusedasareferencetextorasatextbook.
As a textbook, it covers materials that we have taught over the years in several
coursesandmodules.
There are severalexcellentbooks on performance analysisin theliterature. Our
bookisnotdesignedtocompetewiththeexistingbooks,butrathertocomplement
thembycontributingtothetopicofperformancemeasurementusingSDEA.
Wehopethatthereaders findthisbookuseful intheirstudiesandresearch.We
welcomeanycriticalfeedbackandconstructivecomments.
Rasht,Iran AlirezaAmirteimoori
Bhubaneswar,India BireshK.Sahoo
Lima,Peru VincentCharles
Rasht,Iran SaberMehdizadeh
vii
Introduction
In the last three decades, data envelopment analysis (DEA) as the main
benchmarking technique has proven to be an excellent and powerful quantitative
tool for evaluating and measuring the relative efficiency of entities. Classic DEA
modelsassumethattherearenovariationsinthevaluesoftheinputsandoutputs.In
the presence of stochastic variations in the data, the concepts of efficiency and
returns-to-scale(RTS)areinextricablyrelatedtohowentitiesdealwithuncertainty.
Therefore, when there are variations in inputs and outputs due to uncertainty, the
evaluation of efficiency and RTS under a deterministic DEA setting becomes
sensitive to such variations. As a result, one would expect to see whether the
efficiency and RTS characterizations of entities are subject to change under this
stochasticenvironment.
Benchmarking
Benchmarking isanefficiency measurementprocedurethatallows entities tocom-
pare their performance to the top competitors. In Chap. 1, benchmarking (as a
performance measurement tool using a specific indicator) and key performance
indicators (KPI) are first briefly introduced. Then, we introduce and connect the
conceptsofefficiencyandproductivityanddiscussthewaysformeasuringdifferent
typesofefficiencies.
An Introduction to DEA
DEA,asapowerfulestimatorinbenchmarking,hasproventobeanexcellenttoolin
measuringtherelativeefficiencyofhomogeneousentities.Anintroductiontoclassic
DEAmodelsalongwithacomprehensivereviewanddiscussionisgiveninChap.2.
ix
x Introduction
Probability Theory
Probability theory is concerned with the analysis of random phenomena. It is the
branchofmathematicsconcernedwithprobability.Chapter3introduceselementary
terms in probability theory in such a way that readers who are beginners to this
subjectcanquicklybecomefamiliarwithvariousterminologies.
Stochastic DEA
Stochasticity and stochastic variations in inputs and outputs represent a variety of
uncertainty.Inthissense,incorporatingthestochasticvariationsoftheinput/output
data into DEA models is an important subject, which has received considerable
attention over the past two decades. The underlying theories of DEA under a
stochastic environment are presented in Chap. 4. This permits us to replace terms
suchas“efficient” and“inefficient”with “probablyefficient”and“probablyineffi-
cient,”respectively.
Stochastic Network DEA
Two-stagenetworkstructuresinDEAarestudiedincasesinwhichtheinput/output
data set is deterministic. In many real applications, however, we face uncertainty.
Chapter 5 proposes a two-stage network DEA model for input/output data that are
stochastic.
Stochastic Scale Elasticity
Toanalyzetheperformanceofentities,severaleconomicconceptssuchas“econo-
miesofscale”(RTS)and“marginalratesoftechnicalsubstitutions”havebeenused
intheliterature.Inthelastchapterofthisbook,weconcentrateondeterminingand
measuring the RTS in quantitative form (called scale elasticity). As such, Chap. 6
providesadetailedstudyofstochasticscaleelasticityinDEA.
Contents
1 Benchmarking. . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 1
1.1 WhatIsBenchmarking?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 KeyPerformanceIndicator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 EfficiencyandProductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 EfficiencyAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 TechnicalEfficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.2 (Input)AllocativeEfficiency. . . .. . . . . .. . . . . .. . . . . .. 8
1.4.3 (Output)AllocativeEfficiency. . . . . . . . . . . . . . . . . . . . . 10
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 AnIntroductiontoDataEnvelopmentAnalysis. . . . . . . . . . . . . . . . 13
2.1 SymbolsandNotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Technology(ProductionPossibility)Set. . . . . . . . . . . . . . . . . . . . 14
2.3 BasicDEAPrograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 CCRProgram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 BCCProgram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3 AdditiveProgram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.4 AllocativeEfficiencyPrograms. . . . . . . . . . . . . . . . . . . . . 22
2.4 Returns-to-Scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 NetworkDEA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 Non-cooperativeModels(Leader–Follower). . . . . . . . . . . 26
2.5.2 CentralizedModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 ProbabilityTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 ProbabilitySpace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 RandomVariable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 MathematicalExpectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 DiscreteDistributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 ContinuousDistributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 TheNormalDistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
xi