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