decision-making for urban planning and regional...

Advances in Operations Research

Decision-Making for Urban Planning and Regional Development

Lead Guest Editor: Marta BotteroGuest Editors: Alessandra Oppio and Chiara D’Alpaos

Decision-Making for Urban Planningand Regional Development

Advances in Operations Research

Decision-Making for Urban Planningand Regional Development

Lead Guest Editor: Marta BotteroGuest Editors: Alessandra Oppio and Chiara D’Alpaos

Copyright © 2019 Hindawi. All rights reserved.

This is a special issue published in “Advances in Operations Research.” All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Juan Aparicio, SpainIgor L. Averbakh, CanadaEduardo Fernandez, MexicoKevin Furman, USAAhmed Ghoniem, USAMhand Hifi, FranceDylan F. Jones, UK

Imed Kacem, FranceIoannis Konstantaras, GreeceDemetrio Laganà, ItalyChing-Jong Liao, TaiwanYi-Kuei Lin, TaiwanViliam Makis, CanadaLars Mönch, Germany

KhosrowMoshirvaziri, USAPanagiotis P. Repoussis, GreeceShey-Huei Sheu, TaiwanKonstantina Skouri, GreeceWolfgang Stadje, GermanyHsien-Chung Wu, Taiwan

Contents

Decision-Making for Urban Planning and Regional DevelopmentMarta Bottero , Chiara D’Alpaos, and Alessandra OppioEditorial (2 pages), Article ID 5178051, Volume 2019 (2019)

A New Robust Dynamic Data Envlopment Analysis Approach for Sustainable Supplier EvaluationHava Nikfarjam , Mohsen Rostamy-Malkhalifeh , and Abbasali NouraResearch Article (20 pages), Article ID 7625025, Volume 2018 (2019)

Multicriteria Evaluation of Urban Regeneration Processes: An Application of PROMETHEEMethod inNorthern ItalyMarta Bottero , Chiara D’Alpaos, and Alessandra OppioResearch Article (12 pages), Article ID 9276075, Volume 2018 (2019)

Measuring Conflicts Using Cardinal Ranking: An Application to Decision Analytic Conflict EvaluationsTobias Fasth , Aron Larsson , Love Ekenberg, and Mats DanielsonResearch Article (14 pages), Article ID 8290434, Volume 2018 (2019)

Minimizing Cost Travel in Multimodal Transport Using Advanced Relation Transitive ClosureRachid Oucheikh , Ismail Berrada, and Lahcen OmariResearch Article (7 pages), Article ID 9579343, Volume 2018 (2019)

Multiobjective Optimization for Multimode Transportation ProblemsLaurent Lemarchand , Damien Massé , Pascal Rebreyend , and Johan HåkanssonResearch Article (13 pages), Article ID 8720643, Volume 2018 (2019)

Integration between Transport Models and Cost-Benefit Analysis to Support Decision-MakingPractices: Two Applications in Northern ItalyPaolo Beria , Alberto Bertolin , and Raffaele GrimaldiResearch Article (16 pages), Article ID 2806062, Volume 2018 (2019)

http://orcid.org/0000-0001-8983-2628http://orcid.org/0000-0001-5072-6721http://orcid.org/0000-0001-6105-7674http://orcid.org/0000-0001-8983-2628http://orcid.org/0000-0002-2324-1021http://orcid.org/0000-0003-0310-0018http://orcid.org/0000-0001-9996-9759http://orcid.org/0000-0002-3235-532Xhttp://orcid.org/0000-0003-0894-4076http://orcid.org/0000-0002-4485-8936http://orcid.org/0000-0003-1015-8015http://orcid.org/0000-0003-4871-833Xhttp://orcid.org/0000-0001-5171-6576http://orcid.org/0000-0003-3804-3551http://orcid.org/0000-0002-7832-2398

EditorialDecision-Making for Urban Planning andRegional Development

Marta Bottero ,1 Chiara D’Alpaos,2 and Alessandra Oppio3

1Department of Regional and Urban Studies and Planning, Politecnico di Torino, Italy2Department of Civil, Environmental and Architectural Engineering, University of Padua, Italy3Department of Architecture and Urban Studies, Politecnico di Milano, Italy

Correspondence should be addressed to Marta Bottero; [email protected]

Received 26 November 2018; Accepted 26 November 2018; Published 10 January 2019

Copyright © 2019 Marta Bottero et al. �is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Urban and regional development can be considered as mul-tidimensional concepts which involve socioeconomic, eco-logical, cultural, technical, and ethical perspectives. Decisionproblems in the domain of urban and regional developmentprocesses represent “weak” or unstructured problems asthey are characterized by multiple actors, many and oenconflicting values and views, a wealth of possible outcomes,and high uncertainty.

Under these circumstances, evaluation of alternativeprojects is therefore a complex decision problem, where dif-ferent aspects need to be considered simultaneously, and bothtechnical elements, based on empirical observations, andnon-technical elements, based on social visions, preferences,and feelings, need to be taken into account. �is complexityrequires multidimensional approaches and specific qualita-tive/quantitative methods to analyse and synthesize the fullvariety of aspects involved in transformation processes, thatrange from the environmental impacts of urban renewal toits impacts on energy consumption/production patterns andmobility; from the social and economic impacts of a specificurban transformation strategy to its effects on landscape andcultural heritage.

�is special issue addresses recent advances on the role ofevaluation in supporting decision-makers in urban planningand regional development. 6 papers are published in thisspecial issue; each paper was reviewed by at least tworeviewers and revised according to review comments. �eaccepted papers show the role of evaluation procedures tosupport decisions in the context of urban management andterritorial transformations.

�e paper “A New Robust Dynamic Data EnvelopmentAnalysis Approach for Sustainable Supplier Evaluation” byNikfarjam et al. presents a new dynamic Data EnvelopmentAnalysis (DEA) approach for suppliers selection which takesinto account social, environmental and economic criteriaand considers differently fromprevious literature, contiguoustime periods. In detail efficient Decision Making Units(DMUs) are identified in each time period and as well asan ideal DMU by implementing a robust scenario-basedoptimization approach.

�e paper “Multicriteria Evaluation of Urban Regener-ation Processes: An Application of PROMETHEE Methodin Northern Italy” by M. Bottero et al. proposes an originalmultimethodological evaluation procedure, which combinesSWOT Analysis, Stakeholders Analysis, and PROMETHEEmethod, to evaluate alternative renewal strategies in an urbanarea in Northern Italy and provide decision-makers withuseful tools in making welfare-maximizing urban planningdecisions.

�e paper “Measuring Conflicts Using Cardinal Ranking:An Application to Decision Analytic Conflict Evaluations”by T. Fasth et al. provides: (a) an application of the cardinalranking method for preference elicitation to inform decision-makers with respect to controversies; (b) and two indexes tomeasure potential conflicts within a group of stakeholders orbetween two groups of stakeholders.

�e paper “Minimizing Cost Travel inMultimodal Trans-port Using Advanced Relation Transitive Closure” by R.Oucheikh et al. proposes a new method for travel cost opti-mization, which can be applied either on path optimization

HindawiAdvances in Operations ResearchVolume 2019, Article ID 5178051, 2 pageshttps://doi.org/10.1155/2019/5178051

http://orcid.org/0000-0001-8983-2628https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/5178051

2 Advances in Operations Research

for graphs or on binary constraint reduction in ConstraintSatisfaction Problem (CSP). In addition, it introduces themathematical background for the transitive closure of binaryrelations.

�e paper “Multiobjective Optimization for MultimodeTransportation Problems” by L. Lemarchand et al. presentsa model to solve service facilities localization problems ina multimode transportation context, by implementing anadapted 𝜀-constraint multiobjective method and exploringthe implementation of heuristic methods based on evolution-ary multiobjective frameworks.

�e paper “Integration between Transport Models andCost-Benefit Analysis to SupportDecision-Making Practices:Two Applications in Northern Italy” by P. Beria et al. con-tributes to the assessment of sustainable mobility transportplans and infrastructure projects, and presents an operativeapplication of Cost Benefit Analysis to the evaluation ofalternative scenarios, complemented by the implementationof transportation models and GIS.

�e papers in this special issue represent a scientificallybased support to address the complexity of decisions makingin urban planning and regional development, improve theeffectiveness and soundness of choices, and increase trans-parency in collective decision-making, by enhancing sharedlearning processes. We hope that this special issue will attractattention for further research into complex urban/territorialtransformation processes, and will prove to be a valuableresource in the improvement of knowledge that the develop-ment of future cities and society requires.

Conflicts of Interest

�is is to confirm that as guest editors of the special issuetitled “Decision-Making for Urban Planning and RegionalDevelopment” we have not any possible conflicts of interestor private agreements with companies.

Marta BotteroChiara D’Alpaos

Alessandra Oppio

Research ArticleA New Robust Dynamic Data Envlopment Analysis Approach forSustainable Supplier Evaluation

Hava Nikfarjam ,1 Mohsen Rostamy-Malkhalifeh ,2 and Abbasali Noura2

1Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran2Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Correspondence should be addressed to Mohsen Rostamy-Malkhalifeh; mohsen [email protected]

Received 8 November 2017; Accepted 14 November 2018; Published 9 December 2018

Academic Editor: Yi-Kuei Lin

Copyright © 2018 HavaNikfarjam et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Supplier selection is one of the intricate decisions of managers in modern business era.There are different methods and techniquesfor supplier selection. Data envelopment analysis (DEA) is a popular decision-making method that can be used for this purpose. Inthis paper, a new dynamic DEA approach is proposed which is capable of evaluating the suppliers in consecutive periods based ontheir inputs, outputs, and the relationships between the periods classified as desirable relationships, undesirable relationships, andfree relationships with positive and negative natures. To this aim various social, economic, and environmental criteria are takeninto account. A new method for constructing an ideal decision-making unit (DMU) is proposed in this paper which differs fromthe existing ones in the literature according to its capability of considering periods with unit efficiencies which do not necessarilybelong to a unique DMU. Furthermore, the new ideal DMU has the required ability to rank the suppliers with the same efficiencyratio. In the concerned problem, the supplier that has unit efficiency in each period is selected to construct an ideal supplier. Sinceit is possible to have more than one supplier with unit efficiency in each period, the ideal supplier can be made with differentscenarios with a given probability. To deal with such uncertain condition, a new robust dynamic DEA model is elaborated basedon a scenario-based robust optimization approach. Computational results indicate that the proposed robust optimization approachcan evaluate and rank the suppliers with unit efficiencies which could not be ranked previously. Furthermore, the proposed idealDMU can be appropriately used as a benchmark for other DMUs to adjust the probable improvement plans.

1. Introduction

Supplier selection is an important strategic decision ofmanagers for the economic and industry. In recent decadesscholars and practitioners have paid special attention tothis issue. To name a few relevant samples we can refer toKhan et al. [1] who analyzed the suppliers regarding theirability in transferring the technology. However, they merelyconsidered economic criteria in evaluation of four levels oftechnology transfer among suppliers of auto industries inPakistan. Nowadays, the decision-maker duty (in supplierselection) has become more and more intricate. This meansthat they must care specifically about sustainability criteriawhile supplier selection. Sustainable supplier evaluation andselection concept is resulted from incorporating environ-mental and social responsibility factors into economic factorswhen making decisions regarding supply chain management

(SCM) [2]. In recent years, sustainability factors have playedpivotal role in supplier evaluation and selection process [3].Ratan et al. [4] discussed that sustainability principles forcecompanies to select the suppliers which develop productsand services, preserve environmental resources and lookafter manpower and communities. Beamon [5] introducedethical and social responsibilities criteria as fundamentalrequirements of sustainable SCM for future decades.

A wide range of multiple criteria models and approachessuch as fuzzy, AHP, ANP, TOPSIS and DEA have beenproposed to deal with supplier selection issue over the lasttwo decades. Some of the following researchers applied fuzzy,AHP/ANP-based methods to deal with multi-criteria sup-plier selection problems (e.g. [6–12]). Some other researchersused TOPSIS based methods to evaluate the suppliers (e.g.[9, 13, 14]). For instance, using fuzzy inference system,Amindoust et al. [15] proposed a ranking model based on

HindawiAdvances in Operations ResearchVolume 2018, Article ID 7625025, 20 pageshttps://doi.org/10.1155/2018/7625025

http://orcid.org/0000-0001-5072-6721http://orcid.org/0000-0001-6105-7674https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/7625025


fuzzy inference system for sustainable supplier selection.Wen et al. [3] introduced a model for sustainable supplierevaluation using intuitionistic fuzzy sets’ group decision-making models.

Another applied method is DEA which is capable toevaluate the suppliers based on weighted ratio of outputs toinput. According to Kumar et al. [16], DEA is an applicableand effective tool for supplier selection problem. In the DEAapproach, the importance or weights of inputs and outputsare determined through model itself in a pairwise compari-sonmanner without any human’s interference [17]. Differentmodels of DEA have been developed in the literature toevaluate the potential suppliers. For this purpose, Weber etal. (2000) proposed a hybrid multi-objective programming(MOP)-DEA model. Farzipoor Saen [18] proposed a DEAmodel for ranking suppliers in the presence of imprecisedata, weight restriction, and nondiscretionary factors. Also,Farzipoor Saen [19] suggested a DEA model for supplierselection in the presence of undesirable outputs and impre-cise data. Noorizadeh et al. [20] introduced a model forsupplier selection in the presence of dual-role factors, nondis-cretionary inputs, and weight restrictions. To help managersfor ranking and selecting the best suppliers in the presence ofundesirable outputs and stochastic data, Azadi and FarzipoorSaen [21] developed a new slack- basedmeasuremodel. Azadiet al. [22] developed a chance-constrained DEA (CCDEA)model for supplier selection in the presence of stochastic dataand nondiscretionary factors. Kumar et al. [16] proposed aunified green DEA (GDEA) model for selecting the best sup-pliers using a comprehensive environment friendly approach.

2. Literature Review

Reviewing the relevant literature reveals that traditionalmodels of DEA evaluate efficiency of DMUs merely inone specific past period. Hence, Dynamic DEA (D-DEA)was an appropriate approach which was initially developedby Sengupta [23] and, nowadays, it is used for evaluatingDMUs in different periods [24]. In the literature related todynamic DEA, Färe and Grosspkof [25] proposed a dynamicproduction frontier using an intermediate output whichrelates annual production processes. Tone and Tsutsui [26]introduced a new dynamic slack-based measure (DSBM)model to assess DMUs in different periods, using carry-over variables (links). They introduced four types of carry-overs (links) as desirable, undesirable, discretionary, andnon-discretionary (fixed) links. Nevertheless, one of the defi-ciencies of the existing dynamic DEAmodels is their inabilityin introducing a strictly efficient DMU with efficiency scoreof unity in all periods. In fact, strictly efficient DMUs aredefined as those that have unit efficiency in all periods. Ifefficiency score of a DMU in one of the periods has lessthan unity, it won’t be considered as a strictly efficient unit.This deficiency can be seen in the works by Yousefi et al.[27] and Cook et al. [28]. To overcome this problem, wepropose a new method for constructing the ideal DMU(IDMU) where in each period a different DMU with unitefficiency is selected. In fact, the ideal DMU is constructedby a combination of different strictly efficient DMUs. In

other words, in this paper, we extend the dynamic DEAmodel to evaluate suppliers in different periods based on theirinputs, outputs and the relationships between the periodsclassified as desirable relationships, undesirable relationshipsand free relationships with positive and negative natures.The proposed model is used to evaluate suppliers based onsustainable supplier criteria such as social, economic andenvironmental criteria.

In a methodological point of view, the recent hybrid-approach studies by Tavana et al., 2017; Shabanpour et al.,2017a; Shabanpour et al., 2017b; Yousefi et al., 2016 andYousefiet al., 2015 have played fundamental roles in creating themain idea and the contributions of this study. Tavana etal., [29] developed a hybrid DEA framework for SustainableSupplier Evaluation. They combined the goal programmingand dynamic DEA model to assess efficiency of suppliersover several periods. Their approach enables the decision-maker to provide improved solutions for inefficient suppliersbased on the extent to which the suppliers achieve futuregoals (benchmarks). Merging dynamic DEA with ANNmodels, Shabanpour et al., [30] created a novel frameworkfor assessing and forecasting prospective efficiency of greensuppliers. Likewise, another relevant survey was conductedby Shabanpour et al., [31].They applied robust values to definemanagerial goals (improvement solutions) for evaluatingsuppliers. Given the fact that the management goals areinherently uncertain as well as based on human interference,they created a new robust double-frontier DEA model toassess and rank sustainable suppliers. Yousefi et al., [32]developed a scenario-based robust DEA technique to dealwith sustainable suppliers’ evaluation. Yousefi et al., [33], fur-thermore, combined goal programming and network DEAand proposed a novel DEA framework to evaluate supplychains. Their approach has the potential of predicting theDMUs’ efficiencies in prospective periods. Accordingly, thedecision-maker can not only evaluate suppliers/supply chainsbut also rank them based on their efficiency trend in severaltime periods. Tanskanen et al. [34] consider relationshipstrategies for levels of suppliers with respect to sustainablecriteria. Varoutsa and Scapens [35] evaluate the supply chain’sagents inside the organization. Patala et al. [36] considersustainable criteria such as economic, environmental andsocial criteria in supplier evaluation.

The ideal DMU, as another assessment method, hasbeen used in different performance evaluation problems overthe past decade. Wang et al. [37] created an interval DEAmodel in which efficiency was calculated within the rangeof an interval. The upper bound of the interval was set toone and the lower bound was established by introducing avirtual IDMU, whose performance was superior to any DMU.Jahanshahloo et al. [38] developed two ranking methodsusing positive IDMU.They ranked 20 Iranian bank branchesby two ranking methods. Hatami-Marbini et al. [39] provideda four-phase fuzzy DEA framework based upon the theory ofdisplaced ideal. They made two hypothetical DMUs namelythe ideal and nadir DMUs as reference points to rank theDMUs. Jahanshahloo et al. [40] proposed an interval DEAmodel to attain an efficiency interval, including evaluationsfrom both the optimistic and the pessimistic perspectives. In

Advances in Operations Research 3

their method, the lower bounds of the DMUs are increasedto obtain the maximum value. The derived points from thismethod were called ideal points. Then, the ideal points areemployed to rank DMUs. Wang et al. [41] developed newDEA models for cross-efficiency evaluation by introducing avirtual IDMU and a virtual anti-ideal DMU (ADMU). Thepurpose of their study was to measure the cross-efficienciesin a neutral and more logical way.

This paper aims to evaluate the suppliers of a homeappliance company based on sustainable suppliers’ criteriausing a new robust dynamic DEA model which is capableto evaluate and rank the suppliers with unit efficiencieswhich could not be ranked in the previously developedapproaches. Furthermore, a new method is developed toconstruct an ideal DMU. The proposed ideal DMU is madeup of a combination of DMUs with unit efficiency in eachperiod. It is possible to have more than one DMU with unitefficiency in each period thus resulting in different combi-nations or scenarios. Therefore, a two-step scenario-basedrobust approach is employed to deal with these scenariosall with unit efficiencies. The proposed ideal DMU can beappropriately used as a benchmark for other DMUs to adjustthe probable improvement plans.

The main contributions of this paper are summarized asfollows:

(i) Presenting a new method for constructing an idealDMU in the dynamic DEAmodel (since it is unlikelyto find a DMU with unit efficiency in all periods, ineach period a different DMU with unit efficiency canbe considered)

(ii) Presenting a two-step robust method to deal withdifferent scenarios for ideal DMU (it is possible tohavemore than one DMUwith unit efficiency in eachperiod and therefore different combinations of DMUsresult in different scenarios)

(iii) Presenting robust ranks and improvement plans forall efficient and inefficient units

Correspondingly, the following research questions areexpected to be addressed in this paper:

(i) What is the efficiency of suppliers in different periodswith respect to sustainable criteria?

(ii) What is the rank of suppliers when more than onesupplier with unit efficiency exists?

(iii) How can an ideal DMU be constructed to presentbetter improvement methods when no DMU existwith unit efficiency in all periods?

(iv) What if when multiple DMUs have unit efficiency ina period?

The rest of the paper is organized as follows. In Sec-tion 2 the proposed dynamic DEA model is presented;first the deterministic model (Model D) is introduced andthen the standard form of the model is written (Model S).Section 3 introduces the proposed two-step robust methodwhich are proposed for the dynamic DEA; For step “a,” theproposed robust input/output-oriented dynamic DEAmodel

is defined (Model RIa/ROa) along with the correspondinglinear form (Model LRIa/LROa). Afterwards for step “b,” theproposed robust input/output-oriented dynamic DEAmodelis defined (Model RIb/ROb) along with the correspondinglinear form (Model LRIb/LROb). In Section 4 the proposedrobust dynamic DEAmodels are investigated on a case studyfor supplier selection. Finally, in Section 5 conclusions arebrought along with future research directions.

3. Problem Statement and Formulation

The purpose of this paper is to evaluate suppliers of acompany in consecutive periods based on sustainable criteriawithin three categories: (1) social, (2) economic, and (3)environmental. In each period, there are a number of inputsand outputs for each supplier. Also some materials may betransferred from one period to the next period(s) such asbackorders and uncashed checks. These make relationshipsbetween periods which some of them are desirable and someare undesirable. In the context of DEA, each supplier isconsidered as a DMU. Since the suppliers are evaluated inmultiple periods, dynamic version of DEA is appropriate.Dynamic DEA aims at evaluating 𝑚DMUs during 𝑃 periodswith respect to relationships between periods. For everyDMU, in each period, n inputs and 𝑠 outputs are considered.Also, three types of relationships such as desirable, undesir-able and free are considered which ensure the link betweenperiods. Relationships with desirable and undesirable natureneed to be maximized and minimized, respectively. Freerelationships are those that lack any essence and their naturecannot be recognized some of them have positive nature andsome have negative nature. Figure 1 graphically illustrates theproposed dynamic DEA.

In the dynamic DEA, usually a strictly efficient unit calledideal DMU is specified to be considered as a benchmark sothat inefficient DMUs try to reach it. In fact, the ideal DMU isthe one that in all periods has unit efficiency and it is a strictlyefficient unit. In most cases such a DMU which is efficientin all periods does not exist and according to the existingdefinition no ideal DMUcan be recognized. To overcome thisproblem, we introduce a new method for constructing theideal DMU where in each period a different DMU with unitefficiency in that period is selected. Obviously, the proposedideal DMU is virtual and does not exist in reality. To clarify,consider 3 DMUs in 5 periods (Figure 2). According to theexisting method for constructing the ideal DMU, no DMU isselected as an ideal DMUwhereas according to the proposedmethod in this paper, the ideal DMU is composed of DMU 1,DMU 2, DMU 3, DMU 2, and DMU 1, respectively.

In the dynamic DEA, the efficient frontier is constructedin a pairwise comparison between units where units withmaximum ratio of outputs to inputs are selected to constructthe efficient frontier as presented by dotted line in Figure 3.The improvement method is presented for inefficient units topush them towards this efficient frontier. This issue can bementioned as another shortcoming of the existing dynamicDEAmodels where the benchmark(s) as well as improvementplans are merely introduced for inefficient units. Actually,those models do not present improvement methods for


zijpd,zijpu ,zijpfPeriod p

Yijp

Period p+1

Yijp+1

Xijp xijp+1

Figure 1: Representation of the proposed dynamic DEA.

DMU1

Period1 Period2 Period5Period4Period3

DMU2

Period1 Period5Period4Period3Period2

DMU3

Period1 Period4Period3Period2 Period5

ideal DMU4

Period1 Period5Period4Period3Period2

1 0.91 0.79 0.55 1

0.87 1 0.81 1 0.75

0.67 0.78 1 0.85 0.94

DMU1 DMU2 DMU3 DMU4 DMU5

Figure 2: Demonstrating the proposed ideal DMU.

I DUM

02468

101214

OU

T PU

T

4 62 8 10 12 140IN PUT

DUM

Figure 3: Efficient frontiers by the ideal DMU and efficient units.

benchmarks themselves. The proposed ideal DMU, shown byan asterisk in Figure 3, can be introduced as a benchmarkfor both inefficient units and efficient units. The boundarywhich is presented by the solid line in Figure 3 is the frontierwhich has been constructed by the ideal DMU. Asmentionedearlier, the proposed ideal DMU consists of periods with unitefficiencies which do not necessarily belong to a DMU. It ispossible to have more than one DMU with unit efficiencyin a specific period. Therefore, different scenarios may existfor the ideal DMU. To deal with these scenarios, we employa scenario-based robust optimization approach and proposea robust dynamic DEA model to present improvement plans

for all DMUs. Even when all DMUs obtain similar efficiency,the proposed model can rank those units by considering anabsolutely efficient unit called ideal DMU. Actually, the idealDMU can be used as a unique benchmark based on whichimprovement methods can be presented for other DMUs.As a matter of fact, initially DMUs are evaluated based ona dynamic DEA approach and then different scenarios areconstructed for the ideal DMU (ideal supplier) through acombination of efficient DMUs in each period.

Altogether, the proposed ideal DMU addresses the fol-lowing concerns about the existing DEA models:

(i) Presenting the improvement methods for efficientunits (in existing models the improvement methodscan only be presented for inefficient units)

(ii) Considering requirements and opinions of experts inpresenting the improvement methods (these opinionsare considered in inputs and outputs values of theideal DMU).

(iii) Modifying the benchmarks in case these units are notacceptable from DM’s perspective.

Since in this paper different models are presented, to preventmisunderstanding, we number the models using the follow-ing acronyms.

Acronyms for ModelsD: Deterministic modelS: Standard modelRIa: Robust Input-oriented model step aRIb: Robust Input-oriented model step bROa: Robust Output-oriented model step aROb: Robust Output-oriented model step bLRIa: Linear Robust Input-oriented model step aLRIb: Linear Robust Input-oriented model step bLROa: Linear Robust Output-oriented model step aLROb: Linear Robust Output-oriented model step bBefore describing the models, the used notations can be

described as follows.

Notationsm: Index of DMUsn: Index of inputss: Index of outputsp: Index of time periods


𝑛𝑑, 𝑛𝑢,𝑛𝑓: Total number of desirable, undesirable, and freerelationships, respectively.𝑥𝑖𝑗𝑝: 𝑖th input of the jth DMU in period p (i= 1, 2, 3, . . . . . ..,n) 𝑦𝑖𝑗𝑝: 𝑖th output of the jthDMU inperiod p (i= 1, 2, 3, . . . . . .,s) 𝑧𝑑𝑖𝑗𝑝: Desired relationship for the jth DMU in period p,(i=1,..,𝑛𝑑), (j=1,. . .,m), (p=1,. . ., P).𝑧𝑢𝑖𝑗𝑝: Undesirable relationship for the jth DMU in periodp. (i=1,..,𝑛𝑢), (j=1,. . .,m), (p=1,. . ., P).𝑧𝑓𝑖𝑗𝑝: Free relationship for the jth DMU in period p.(i=1,..,𝑛𝑓), (j=1,. . .,m), (p=1,. . ., P).𝜆𝑝𝑗 : Benchmark for the jth inefficient DMU in period p.

o: Index for the under investigation DUM𝑤−𝑖 : Weight of the 𝑖th input𝑤+𝑖 : Weight of the 𝑖th output𝑤𝑝: Weight of the pth period𝜙∗𝑜𝑝: Efficiency of the under investigated DMU in periodp (input-oriented model)Φ∗0 : Total efficiency of the under investigated DMU(input-oriented model)𝜓∗𝑜𝑝: Efficiency of the under investigated DMU in periodp (output-oriented model)Ψ∗𝑜 : Total efficiency of the under investigated DMU(output-oriented model)

3.1. �e Proposed Mathematical Model for the DeterministicDynamic DEA (Model D)

maxΦ∗0 = 𝑃∑𝑝=1

𝜙∗𝑜𝑝𝑃 (D-1-IN)maxΨ∗𝑜 = 𝑃∑

𝑝=1

𝜓∗𝑜𝑝𝑃 (D-1-OUT)𝑥𝑖𝑜𝑝 ≥ 𝑚∑

𝑗=1

𝑥𝑖𝑗𝑝𝜆𝑝𝑗(𝑖 = 1, . . . , 𝑛; 𝑝 = 1, . . . , 𝑃)

(D-2)

𝑦𝑖𝑜𝑝 ≤ 𝑚∑𝑗=1

𝑦𝑖𝑗𝑝𝜆𝑝𝑗(𝑖 = 1, . . . , 𝑠; 𝑝 = 1, . . . , 𝑃)

(D-3)

𝑧𝑑𝑖𝑜𝑝 ≤ 𝑚∑𝑗=1

𝑧𝑑𝑖𝑗𝑝𝜆𝑝𝑗(𝑖 = 1, . . . , 𝑛𝑑; 𝑝 = 1, . . . , 𝑃)

(D-4)

zuiop ≥ m∑j=1zuijp𝜆pj

(𝑖 = 1, . . . , 𝑛𝑢; 𝑝 = 1, . . . , 𝑃)(D-5)

zfiop: free of sign

(𝑖 = 1, . . . , 𝑛𝑓; 𝑝 = 1, . . . , 𝑃) (D-6)𝜆pj ≥ 0 (j = 1, . . . ,m : p = 1, . . . , P) (D-7)

Objective function (D-1-IN) and (D-1-OUT) maximizesthe efficiency of the under investigation DMU. (D-1-IN)is the objective function of the input-oriented model and(D-1-OUT) is the objective function of the output-orientedmodel. At each time either objective function (D-1-IN) or(D-1-OUT) is considered. Constraint set (D-2) ensures thatthe 𝑖th input of the under investigation DMU in period 𝑝 begreater than or equal to weighted sum of input 𝑖 in period𝑝 for all DMUs. Constraint set (D-3) ensures that the 𝑖thoutput of the under investigation DMU in period 𝑝 be lessthan or equal to weighted sum of output 𝑖 in period 𝑝 forall DMUs. Constraint set (D-4) indicates that the value ofthe 𝑖th desirable relationship of the under investigation DMUin period 𝑝 be less than or equal to weighted sum of thedesirable relationship 𝑖 in period 𝑝 for all DMUs. Constraintset (D-5) indicates that the value of the 𝑖th undesirablerelationship of the under investigation DMU in period 𝑝 begreater than or equal to weighted sum of the undesirablerelationship 𝑖 in period 𝑝 for all DMUs. Constraint set(D-6) defines the free relationship variables which are freeof sign. Constraint set (D-7) defines the weight variables ofimprovement plans.

Note that the right hand sides of the above constraints,i.e., 𝑥𝑖𝑗𝑝, 𝑦𝑖𝑗𝑝, and 𝑧𝑑𝑖𝑗𝑝 𝑧u𝑖𝑗𝑝, are positive values. The left handside of the mentioned constraints, i.e., 𝑥𝑖𝑜𝑝, 𝑦𝑖𝑜𝑝, 𝑧𝑑𝑖𝑜𝑝, 𝑧𝑢𝑖𝑜𝑝, and𝑧𝑓𝑖𝑜𝑝, is connected together through 𝜆pj . The continuity of theflow representing the relationship between pth and (p+1)thperiods is ensured through (1), where ∝ is a general indexwhich can be d, u, or 𝑓 representing desirable, undesirable,and free relationships. These constraints are important in theproposed dynamic DEA model, since they connect period 𝑝to period p+1 and ensure having a series of time periods.

m∑j=1x∝ijp𝜆pj = m∑

j=1z∝ijp𝜆p+1j (𝑖 = 1, . . . , 𝑛 ∝; 𝑝 = 1, . . . , 𝑃) (1)

3.2. �e Standard Mathematical Model for the DeterministicDynamic DEA (Model S). After writing the standard formfor the constraints of model (D), the final standard model (S)whose constraints have equal sign is obtained as follows:

minΦ∗0 = 1𝑃P∑

p=1w𝑝 [[1 −

1𝑚 + 𝑛𝑢 + 𝑛𝑓 (

n∑i=1

w−i s−ip

xiop+ n𝑢∑

i=1

suipzuiop

+ n𝑓∑i=1

s 𝑓−ipzfiop

)]] (S-1-IN)


max 1Ψ∗𝑜 =1𝑃𝑃∑𝑝=1

𝑤𝑝 [[1 −1

𝑠 + 𝑛𝑑 + 𝑛𝑓 (𝑠∑𝑖=1

𝑤+𝑖 𝑠+𝑖𝑝𝑦𝑖𝑜𝑝 +𝑛𝑑∑𝑖=1

𝑠𝑑𝑖𝑝𝑧𝑑𝑖𝑜𝑝 +n𝑓∑i=1

s 𝑓+ipzfiop

)]] (S-1-OUT)

xiop = m∑j=1xijp𝜆pj + s−ip (𝑖 = 1, . . . , 𝑛; 𝑝 = 1, . . . , 𝑃) (S-2)

𝑦𝑖𝑜𝑝 = 𝑚∑𝑗=1

𝑦𝑖𝑗𝑝𝜆𝑝𝑗 − 𝑠+𝑖𝑝 (𝑖 = 1, . . . , 𝑠; 𝑝 = 1, . . . , 𝑃) (S-3)

𝑧𝑑𝑖𝑜𝑝 = 𝑚∑𝑗=1

𝑧𝑑𝑖𝑗𝑝𝜆𝑝𝑗 − 𝑠𝑑𝑖𝑝 (𝑖 = 1, . . . , 𝑛𝑑; 𝑝 = 1, . . . , 𝑃) (S-4)

zuiop = m∑j=1zuijp𝜆pj + suip (𝑖 = 1, . . . , 𝑛𝑢; 𝑝 = 1, . . . , 𝑃) (S-5)

zfiop = m∑j=1zfijp𝜆pj + sfip (𝑖 = 1, . . . , 𝑛𝑓; 𝑝 = 1, . . . , 𝑃) (S-6)

sfip : free of sign (∀i,p) ,suip ≥ 0,spip ≥ 0,s+ip ≥ 0,s−ip ≥ 0,𝜆pj ≥ 0

(S-7)

In model (S) like model (D), two alternative cases are con-sidered to calculate the efficiency of the under investigatedDMU.One is input-oriented (S-1-IN) and the other is output-oriented (S-1-OUT) which are described in the followings.As a matter of fact, the choice of these objective functionsdepends on the DEA approach, i.e., whether it is input-oriented or output-oriented, which is used for presentingimprovement plans.

Objective function (S-1-IN) represents the total efficiencyof the input-oriented model. This objective function is basedon the nonredial input-oriented model which considersundesirable relationships, i.e., 𝑠𝑢𝑖𝑝 and s 𝑓−ip , along with surplusof inputs, i.e., 𝑠−𝑖𝑝, which should be simultaneouslyminimized.If all these variables become zero, the efficiency of theconsidered DMU in period p is one. Obviously, a DMU withoverall efficiency equal to one is the one which has unitefficiency in all periods.This objective function calculates the

weighted mean of the efficiencies in all periods whose valueis between 0 and 1, i.e., (0 ≤ Φ∗𝑜 ≤ 1), (0 ≤ 𝜙∗𝑜𝑝 ≤ 1). Theoptimal value for the efficiency of period p in input-orientedmodel is according to

𝜙∗𝑜𝑝= 1− 1𝑚 + 𝑛𝑢 + 𝑛𝑓 (

n∑i=1

w−i s−ip

xiop+ n𝑢∑

i=1

suipzuiop

+ n𝑓∑i=1

s 𝑓−ipzfiop

) ,(𝑝 = 1, . . . , 𝑃)

(2)

Objective function (S-1-OUT) represents the total efficiencyof the output-oriented model. The optimal value for theefficiency of period 𝑝 in output-oriented model is accordingto

𝜓∗𝑜𝑝 = 11 − (1/ (𝑠 + 𝑛𝑑 + 𝑛𝑓)) (∑𝑠𝑖=1 (𝑤+𝑖 𝑠+𝑖𝑝/𝑦𝑖𝑜𝑝) + ∑𝑛𝑑𝑖=1 (𝑠𝑑𝑖𝑝/𝑧𝑑𝑖𝑜𝑝) + ∑n𝑓i=1 (s 𝑓+ip /zfiop)) , (𝑝 = 1, . . . , 𝑃) (3)


The denominator of the objective function deals with theslack of outputs, 𝑠+𝑖𝑝, free relationships with positive nature,s 𝑓+ip , and desirable relationships, 𝑠𝑑𝑖𝑝. If these values becomezero, the denominator becomes one and therefore the effi-ciency of the considered DMU in period 𝑝 is one. If theseslacks get values more than one, the denominator becomesmore than one. Therefore, the efficiency of the consideredDMU in period 𝑝 is less than one. Consequently, the totalefficiency in the objective function gets a value between zeroand one, i.e., (0 ≤ Ψ∗𝑜 ≤ 1), (0 ≤ 𝜓∗𝑜𝑝 ≤ 1).

In Constraint (S-2), s−ip represents the slack for the 𝑖thinput in period 𝑝. The left hand side of this constraint isthe inputs of the underinvestigated DMU in period 𝑝. Ifthe value of s−ip be zero, it means that the supplier doesnot have excess consumption for that input in period 𝑝.In constraint (S-3), s+ip represents the surplus for the 𝑖thoutput in period 𝑝. In the rest of the constraints, 𝑠𝑑𝑖𝑝, 𝑠𝑢𝑖𝑝, 𝑠𝑓𝑖𝑝,respectively, represent the slack of the desirable relationship,surplus of the undesirable relationship, and the deviation ofthe free relationship. Note that the auxiliary variables usedto standardize constraints have negative natures. For inputsit means excess consumption, for outputs it means shortagein production, for desirable relationships it means shortagein this relationship, for undesirable relationships it meansexcess in this relationship, and for free relationships it meansdeviation in this relationship. Constraint (S-6) contains a freeof sign variable. The deviation of the free relationship caneither be stated as slack or surplus.Therefore, to deal with thefree of sign variable sfip, two positive variables, s

f−ip and s

f+ip ,

are defined and the following constraints are considered:

sfip = s f−ip − s f+ips f+ip ∗ s f−ip = 0,

s f−ip ≥ 0, s f+ip ≥ 0(4)

Consequently, the following constraints are substituted forConstraint (S-6):

zfiop = m∑j=1zfijp𝜆pj + s 𝑓−ip

(𝑖 = 1, . . . , 𝑛𝑓; 𝑝 = 1, . . . , 𝑃)(S-6-1)

zfiop = m∑j=1zfijp𝜆pj − s 𝑓+ip

(𝑖 = 1, . . . , 𝑛𝑓; 𝑝 = 1, . . . , 𝑃)(S-6-2)

4. Robust Dynamic DEA Model

As mentioned previously, one of the deficiencies of theexisting dynamic DEA is the lack of a strictly efficient unit asa unit that can be introduced as a benchmark. To overcomethis deficiency, we introduce a new method for constructingthe ideal DMU which is constructed by making use of the

results obtained from the dynamic DEA. The proposed idealDMU is made up of different periods each of which containsDMUs with unit efficiency. As a matter of fact, in eachperiod, DMUs are evaluated andDMU(s) with unit efficiencyare selected to construct the ideal DMU. Since more thanone DMU with unit efficiency may exist in each period,different combinations of DMUs may be generated for theideal DMU. Each of these combinations is called a “scenario”.More than one DMUwith unit efficiency in each period leadsto different combinations or scenarios for the ideal DMUwhose probabilities of occurrence are considered the samein this paper. Different scenarios for the ideal DMU resultin different improvement plans. By taking the advantages ofthe scenario-based robust optimization method and applyingit for the studying dynamic DEA, we evaluate and rank thesuppliers with respect to these scenarios for the ideal DMU.This process is done in two steps. The first step (step a)formulates the robust optimization model where one of thescenarios is under investigation. The second step (step b)formulates the robust optimizationmodelwhere otherDMUsalong with the selected scenario unit are under investigation.

The procedure for the proposed supplier evaluation andrank model is summarized in the following procedure.

Procedure: Supplier Evaluation and Rank through the ProposedRobust Dynamic DEA

Begin

(1) Determine inputs, outputs, desirable, and undesirablerelationships for suppliers in each period.

(2) Consider each supplier as a DMU and employthe dynamic (input/ouput-oriented) DEA model todetermine the efficiency values of each supplier ineach period.

(3) If there is a DMU with unit efficiency values in allperiods consider it as a strictly efficient unit.

(4) Otherwise, build a virtual ideal DMU whose periodsbelong to DMUs with unit efficiency thus leading todifferent scenarios for the ideal DMU.

(5) In step “a,” evaluate and rank scenarios (with equalprobability and based on the punishment and encour-agement values considered for each scenario) usingthe proposed linear (input/output-oriented) robustdynamic DEAmodel (LRIa/LROa).The best scenariois considered as a unique benchmark for presentingimprovement plans.

(6) In step “b,” consider the selected scenario from step“a” along with other suppliers (resulting in m+1number of DMUs) and evaluate the suppliers throughmodels (LRIb/LROb).

End.

Notations Used in the Proposed Robust Method^𝑠: The probability of occurrence for scenario s𝑘𝑠𝑖 : The unit cost for 𝑖th input of sth scenario in period p.𝑔𝑠𝑖 : The unit cost for 𝑖th undesirable relationship of sth

scenario in period p.


𝑣𝑠𝑖 :The unit cost for 𝑖th free relationship of sth scenario inperiod p.ℎ𝑠𝑖 : The unit revenue for 𝑖th output of sth scenario in

period p.𝑏𝑠𝑖 : The unit revenue for 𝑖th desirable relationship of sth

scenario in period p.𝑒𝑠𝑖 : The unit revenue for ith free relationship with positive

nature of sth scenario in period p.

4.1. Step a: A Scenario Unit Is under Investigation. In thissection the efficiency of scenarios are investigated throughmodels RIa or ROa, depending on the decision-maker’sapproach which could be input-oriented or output-oriented.

At each time one of the scenarios is under investigation andthe efficiencies of scenarios for the ideal DMU are calculatedand they are ranked. The high ranked scenario is selectedbased on which the improvement plan is presented. As amatter of fact, the best scenario has more distance from otherDMUs (see Figure 3) and the improvement plan for otherDMUs is presented in the worst case.Therefore, the proposedimprovement plan is robust against different scenarios whichcould be considered for the ideal DMU.

4.1.1. Robust Optimization for Input Oriented DEA ModelStep a (RIa)

min𝛽 × Average + (1 − 𝛽) × 𝑆∑𝑠=1

^𝑠

×1𝑃

P∑p=1

w𝑝 [[1

1 − (1/ (𝑚 + 𝑛𝑢 + 𝑛𝑓)) (∑ni=1 (w−i s−ip/xiop) + ∑n𝑢i=1 (suip/zuiop) + ∑n𝑓i=1 (s 𝑓−ip /zfiop))]] − Average

+ 𝑆∑𝑠=1

^𝑠( 𝑛∑𝑖=1

𝑘𝑠𝑖xsisp𝜆Sps + 𝑛𝑢∑𝑖=1

𝑔𝑠𝑖 zsuisp𝜆sps +𝑛𝑓∑𝑖=1

V𝑠𝑖𝑧𝑠𝑓−𝑖𝑠𝑝𝜆𝑠𝑝𝑠 − 𝑠∑𝑖=1

ℎ𝑠𝑖𝑦𝑠𝑖𝑠𝑝𝜆𝑠𝑝𝑠 − 𝑛𝑑∑𝑖=1

𝑏𝑠𝑖 zsdisp𝜆sps −𝑛𝑓∑𝑖=1

𝑒𝑠𝑖𝑧𝑠𝑓+𝑖𝑠𝑝𝜆𝑠𝑝𝑠)

(RIa-1)

Akerage = 𝑆∑𝑠=1

^𝑠 × 1𝑃P∑

p=1w𝑝 [[1 −

1

𝑚 + 𝑛𝑢 + 𝑛𝑓 (n∑i=1

w−i s−isp

xiop+ n𝑢∑

i=1

sui𝑠pzuiop

+ n𝑓∑i=1

s 𝑓−ispzfiop

)]] (RIa-2)

xisp = m∑j=1xijp𝜆

pj + xsisp𝜆Sps + s−isp (𝑖 = 1, . . . , 𝑛; 𝑝 = 1, . . . ,𝑃; 𝑠 = 1, . . . , 𝑆) (RIa-3)

𝑦𝑠𝑖𝑠𝑝 = 𝑚∑𝑗=1𝑦𝑖𝑗𝑝𝜆𝑝

𝑗 + 𝑦𝑠𝑖𝑠𝑝𝜆𝑠𝑝𝑠 − 𝑠+𝑖𝑠𝑝 (𝑖 = 1, . . . , 𝑠; 𝑝 = 1, . . . ,𝑃; 𝑠 = 1, . . . , 𝑆) (RIa-4)z𝑠fiop = m∑

j=1zfijp𝜆

pj + zsfisp𝜆sps + s 𝑓−isp (𝑖 = 1, . . . , 𝑛𝑓; 𝑝 = 1, . . . ,𝑃) (RIa-5)

zsfiop = m∑j=1zfijp𝜆

pj + zsfisp𝜆sps − s 𝑓+isp (𝑖 = 1, . . . , 𝑛𝑓; 𝑝 = 1, . . . ,𝑃) (RIa-6)

𝑧𝑠𝑑𝑖𝑜𝑝 = 𝑚∑𝑗=1𝑧𝑑𝑖𝑗𝑝𝜆𝑝

𝑗 + zsdisp𝜆sps − 𝑠𝑑𝑖𝑠𝑝 (𝑖 = 1, . . . ,𝑛𝑑; 𝑝 = 1, . . . ,𝑃) (RIa-7)zsuisp = m∑

j=1zuijp𝜆

pj + zsuisp𝜆sps + suisp (𝑖 = 1, . . . ,𝑛𝑢; 𝑝 = 1, . . . ,𝑃; 𝑠 = 1, . . . , 𝑆) (RIa-8)

suisp ≥ 0,s+ips ≥ 0,s−isp ≥ 0,𝜆pj ≥ 0,

sps ≥ 0,


s 𝑓+isp ≥ 0,s 𝑓−isp ≥ 0,𝑠𝑑𝑖𝑠𝑝 ≥ 0 (RIa-9)

The objective function (RIa-1) consists of three terms. Thefirst term calculates the average efficiency of scenarios. Thesecond term calculates the deviation of efficiency of scenariosfrom the average value. The third term calculates the totalprofit or loss resulted from scenarios. In fact, in each scenario,outputs, desirable relationships and free relationships withpositive natures, yield return. Whilst inputs, undesirablerelationships, and free relationships with negative naturesresult in cost. Note that in this term, the values of returns aresubtracted from the values of costs. Therefore, if this term isnegative it means that the considering scenario is profitable;otherwise it makes losses.

It is worth noting that since each scenario is a combi-nation of DMUs with unit efficiency selected from differentperiods, the efficiency of each scenario is one. Therefore, theaverage efficiency of all scenarios is one and consequently thestandard deviation is zero.

(1) Linear Robust Input-Oriented Model Step a (LRIa). To dealwith the absolute function and make the model linear, twopositive variables 𝑄+𝑠 and 𝑄−𝑠 are defined.

minΦ∗0

= 𝛽 × Akerage + (1 − 𝛽) × 𝑆∑𝑠=1

^𝑠 × (𝑄+𝑠 +𝑄−𝑠 )+ 𝑆∑𝑠=1

^𝑠 ( 𝑛∑𝑖=1

𝑘𝑠𝑖xsisp𝜆Sps + 𝑛𝑢∑𝑖=1

𝑔𝑠𝑖 zsuisp𝜆sps +𝑛𝑓∑𝑖=1

V𝑠𝑖𝑧𝑠𝑓−𝑖𝑠𝑝𝜆𝑠𝑝𝑠 − 𝑠∑𝑖=1

ℎ𝑠𝑖𝑦𝑠𝑖𝑠𝑝𝜆𝑠𝑝𝑠 − 𝑛𝑑∑𝑖=1

𝑏𝑠𝑖 zsdisp𝜆sps −𝑛𝑓∑𝑖=1

𝑒𝑠𝑖𝑧𝑠𝑓+𝑖𝑠𝑝𝜆𝑠𝑝𝑠)(LRIa-1)

1

𝑃

P∑p=1

w𝑝 [[1 −1

𝑚 + 𝑛𝑢 + 𝑛𝑓 (n∑i=1

w−i s−isp

xiop+ n𝑢∑

i=1

suispzuiop

+ n𝑓∑i=1

s 𝑓−ispzfiop

)]] − Akerage = 𝑄+𝑠 -𝑄−𝑠 (LRIa-2)

Other constraints of model (RIa) hold.

4.1.2. Robust Optimization for Output-Oriented DEA ModelStep a (ROa). The robust optimization for the output-oriented model differs with the input oriented model in the

objective function and also in the average and the linearizedconstraints. Other constraints are the same for both.

min𝛽 × Akerage + (1 − 𝛽) × 𝑆∑𝑠=1

^𝑠

×1𝑃

𝑃∑𝑝=1𝑤𝑝 [[1 −

1𝑠 + 𝑛𝑑 + 𝑛𝑓 (

𝑠∑𝑖=1

𝑤+𝑖 𝑠+𝑖𝑝

𝑦𝑖𝑜𝑝+ 𝑛𝑑∑𝑖=1

𝑠𝑑𝑖𝑝

𝑧𝑑𝑖𝑜𝑝+ n𝑓∑

i=1

s𝑓+ipzfiop

)]] − Akerage

+ 𝑆∑𝑠=1

^𝑠( 𝑛∑𝑖=1𝑘𝑠𝑖xsisp𝜆S

ps + 𝑛𝑢∑𝑖=1𝑔𝑠𝑖zs

uisp𝜆s

ps +𝑛𝑓∑𝑖=1𝑣𝑠𝑖𝑧𝑠𝑓−

𝑖𝑠𝑝𝜆𝑠𝑝𝑠 − 𝑠∑𝑖=1ℎ𝑠𝑖𝑦𝑠𝑖𝑠𝑝𝜆𝑠

𝑝𝑠 − 𝑛𝑑∑𝑖=1𝑏𝑠𝑖zs

disp𝜆s

ps −𝑛𝑓∑𝑖=1𝑒𝑠𝑖𝑧𝑠𝑓+

𝑖𝑠𝑝𝜆𝑠𝑝𝑠)

(ROa-1)

Akerage = 𝑆∑𝑠=1

^𝑠 × 1𝑃𝑃∑𝑝=1𝑤𝑝 [[1 −

1𝑠 + 𝑛𝑑 + 𝑛𝑓 (

𝑠∑𝑖=1



𝑠𝑑𝑖𝑝


i=1

s 𝑓+ipzfiop

)]] (ROa-2)



Like the input-oriented model, the objective function(ROa-1) consists of three terms. The first term minimizesthe average efficiency of scenarios with weight importanceof 𝛽. The second term minimizes the standard deviation of

efficiencies with weight importance of 1- 𝛽. Finally, the thirdterm calculates the total profit or loss resulted from scenarios.

(1) Linear Robust Output-Oriented Model Step a (LROa). Byconsidering two positive variables 𝑄+𝑠 and 𝑄−𝑠 and substitut-ing it with the absolute function, the model is linearized.

min𝛽 × Akerage + (1 − 𝛽) × 𝑆∑𝑠=1

^𝑠 × (𝑄+𝑠 + 𝑄−𝑠 )+ 𝑆∑𝑠=1

^𝑠( 𝑛∑𝑖=1𝑘𝑠𝑖xsisp𝜆S

ps + 𝑛𝑢∑𝑖=1𝑔𝑠𝑖zs

uisp𝜆s

ps +𝑛𝑓∑𝑖=1𝑣𝑠𝑖𝑧𝑠𝑓−𝑖𝑠𝑝𝜆𝑠𝑝𝑠 − 𝑠∑𝑖=1ℎ𝑠𝑖𝑦𝑠𝑖𝑠𝑝𝜆𝑠

𝑝𝑠 − 𝑛𝑑∑𝑖=1𝑏𝑠𝑖zs

disp𝜆s

ps −𝑛𝑓∑𝑖=1𝑒𝑠𝑖𝑧𝑠𝑓+𝑖𝑠𝑝𝜆𝑠𝑝𝑠)

(LROa-1)

[[1𝑃

𝑃∑𝑝=1𝑤𝑝 [[1 −

1𝑠 + 𝑛𝑑 + 𝑛𝑓 (

𝑠∑𝑖=1



𝑠𝑑𝑖𝑝


i=1

s 𝑓+ipzfiop

)]] − Akerage]] = 𝑄

+𝑠 −𝑄−𝑠 (LROa-2)


4.2. Step b: A DMU from Other DMUs Is under Investigation.In the previous section the under investigation DMUwas oneof the scenarios and we evaluated scenarios and selected thesuitable one. In fact, model (RIa/ROa) calculates benefit orloss resulted from each scenario and we can select the bestone accordingly. In this section, other DMUs are evaluatedalong with the selected scenario and therefore the number ofDMUs increases by one (i.e.,m+1). Actually the best scenariois considered in model (RIb/ROb).4.2.1. Robust Optimization for Input-OrientedDEAModel Stepb (RIb). The following model (RIb) evaluates other DMUs

along with the strictly efficient DMU (i.e., the selected sce-nario as an ideal DMU) as a benchmark. Then the improve-ment methods can be presented for other DMUs based ontheir distance from the efficient frontier and image inefficientDMUs to the efficient frontier. The main difference of model(RIb) with model (RIa) is that in model (RIb) the numberof DMUs is m+1. Another difference is that the objectivefunction consists of two terms, i.e., the average efficiency andthe standard deviation of efficiencies of the considered DMUin different periods respectively with importance weights 𝛽and 1 − 𝛽. Note that the objective function of model (RIb)does not consider the cost since by taking the ideal DMU intoconsideration; we can rank DMUs and present improvementmethods.

min Φ∗0= 𝛽 × Average + (1 − 𝛽) × 𝑆∑

𝑠=1

^𝑠 × ( 1𝑃P∑

p=1w𝑝 [[1 −

1𝑚 + 𝑛𝑢 + 𝑛𝑓 (

n∑i=1

w−i s−ip

xiop+ n𝑢∑

i=1

suipzuiop

+ n𝑓∑i=1

s 𝑓−ipzfiop

)]] − Average)(RIb-1)

s.t. Average = 𝑆∑𝑠=1

^𝑠 × 1𝑃P∑

p=1w𝑝 [[1 −

1𝑚 + 𝑛𝑢 + 𝑛𝑓 (

n∑i=1

w−i s−ip

xiop+ n𝑢∑

i=1

suipzuiop

+ n𝑓∑i=1

s 𝑓−ipzfiop

)]] (RIb-2)

xiop = m+1∑j=1

xijp𝜆pj + s−ip (𝑖 = 1, . . . , 𝑛; 𝑝 = 1, . . . , 𝑃; 𝑗 = 1, . . . , 𝑚 + 1) (RIb-3)𝑦𝑖𝑜𝑝 = 𝑚+1∑

𝑗=1

𝑦𝑖𝑗𝑝𝜆𝑝𝑗 − 𝑠+𝑖𝑝 (𝑖 = 1, . . . , 𝑠; 𝑝 = 1, . . . , 𝑃; 𝑗 = 1, . . . , 𝑚 + 1) (RIb-4)zuiop = m+1∑

j=1zuijp𝜆pj + suip (𝑖 = 1, . . . , 𝑛𝑢; 𝑝 = 1, . . . , 𝑃; 𝑗 = 1, . . . , 𝑚 + 1) (RIb-5)

z𝑠fiop = m+1∑j=1

zfijp𝜆pj + s 𝑓−ip (𝑖 = 1, . . . , 𝑛𝑓; 𝑝 = 1, . . . , 𝑃; 𝑗 = 1, . . . , 𝑚 + 1) (RIb-6)


zsfiop = m∑j=1zfijp𝜆pj − s 𝑓+ip (𝑖 = 1, . . . , 𝑛f ; 𝑝 = 1, . . . , 𝑃; 𝑗 = 1, . . . , 𝑚 + 1) (RIb-7)

𝑧𝑠𝑑𝑖𝑜𝑝 = 𝑚∑𝑗=1

𝑧𝑑𝑖𝑗𝑝𝜆𝑝𝑗 − 𝑠𝑑𝑖𝑝 (𝑖 = 1, . . . , 𝑛𝑑; 𝑝 = 1, . . . , 𝑃; 𝑗 = 1, . . . , 𝑚 + 1) (RIb-8)suip ≥ 0,s+ip ≥ 0,s−ip ≥ 0,𝜆pj ≥ 0,s 𝑓+ip ≥ 0,s 𝑓−ip ≥ 0,𝑠𝑑𝑖𝑝 ≥ 0

(RIb-9)

(1) Linear Robust Input-Oriented Model Step b (LRIb). Byconsidering two positive variables𝑄+𝑠 and𝑄−𝑠 and substitutingit with the absolute function, the model is linearized.

min Φ∗0 = 𝛽 × Average + (1 − 𝛽) × 𝑆∑𝑠=1

^𝑠 × (𝑄+𝑠 + 𝑄−𝑠 ) (LRIb-1)

s.t. Average = 𝑆∑𝑠=1

^𝑠 × 1𝑃P∑

p=1w𝑝 [[1 −

1𝑚 + 𝑛𝑢 + 𝑛𝑓 (

n∑i=1

w−i s−ip

xiop+ n𝑢∑

i=1

suipzuiop

+ n𝑓∑i=1

s 𝑓−ipzfiop

)]] (LRIb-2)1𝑃

P∑p=1

w𝑝 [[1 −1

𝑚 + 𝑛𝑢 + 𝑛𝑓 (n∑i=1

w−i s−ip

xiop+ n𝑢∑

i=1

suipzuiop

+ n𝑓∑i=1

s 𝑓−ipzfiop

)]] − Average = 𝑄+𝑠 − 𝑄−𝑠 (LRIb-3)

Other constraints of model (RIb) hold.

4.2.2. Robust Optimization for Output-Oriented DEA ModelStep b (ROb)

min Φ∗0 = 𝛽 × Average + (1 − 𝛽)× 𝑆∑𝑠=1

^𝑠 × ( 1𝑃𝑃∑𝑝=1𝑤𝑝 [[1 −

1𝑠 + 𝑛𝑑 + 𝑛𝑓 (

𝑠∑𝑖=1



𝑠𝑑𝑖𝑝


i=1

s 𝑓+ipzfiop

)]] − Average)(ROb-1)

s.t. Akerage = 𝑆∑𝑠=1

^𝑠 × 1𝑃𝑃∑𝑝=1𝑤𝑝 [[1 −

1𝑠 + 𝑛𝑑 + 𝑛𝑓 (

𝑠∑𝑖=1



𝑠𝑑𝑖𝑝


i=1

s 𝑓+ipzfiop

)]] (ROb-2)

Other constraints of model (RIb) hold. (1) Linear Robust Output-Oriented Model Step b (LROb). Byconsidering two positive variables 𝑄+𝑠 and 𝑄−𝑠 and substitut-


ing them with the absolute function, the model is linearized.

min Φ∗0 = 𝛽 × Average + (1 − 𝛽) × 𝑆∑𝑠=1

^𝑠 × (𝑄+𝑠 + 𝑄−𝑠 ) (LROb-1)

s.t. Akerage = 𝑆∑𝑠=1

^𝑠 × 1𝑃𝑃∑𝑝=1𝑤𝑝 [[1 −

1𝑠 + 𝑛𝑑 + 𝑛𝑓 (

𝑠∑𝑖=1



𝑠𝑑𝑖𝑝


i=1

s 𝑓+ipzfiop

)]] (LROb-2)1𝑃

𝑃∑𝑝=1𝑤𝑝 [[1 −

1𝑠 + 𝑛𝑑 + 𝑛𝑓 (

𝑠∑𝑖=1



𝑠𝑑𝑖𝑝


i=1

s 𝑓+ipzfiop

)]] − Average = 𝑄+𝑠 -𝑄−𝑠 (LROb-3)

Other constraints of model (RIb) hold.

5. Case Study Implementation andPerformance Evaluation

In this section, the performance of the proposed robustdynamic DEA approach is investigated via a case study takenfrom NANIWA (http://www.naniwa.ir) appliances produc-tion plant. The mentioned firm aims at evaluating its 35suppliers in 4 time periods based on environmental, social,and economic criteria. For the purpose of evaluation, for eachsupplier as a DMU, 4 inputs, 3 desirable and undesirablerelationships, and 4 outputs are considered.

Inputs(1) Price offered by suppliers (as an economic criterion):

It is the money (in $1000) that is paid to suppliers foreach unit of products.

(2) Cost of recoverable packages (as an environmentalcriterion): This input is the cost (in $100) that isimposed to the company by the supplier for usingrecoverable packages. It is worth noting that it ismandatory for Naniwa Company to ship their prod-ucts in suitable pallets with recoverable packages toprevent damaging the environment.

(3) Final transportation cost (as an economic criterion):This is the final cost (in $100) which is imposedby the supplier to the firm for transportation ofshipped pallets. The farthest the supplier is and theless accessible the paths are, the more cost imposed.This cost is the main concern of the decision-makersin the company.

(4) Work safety and labor health (as a social criterion):This is the cost that each supplier pays for dangers thatexist and accidents which happen in the workplace.The less damage and casualties are, the less cost is paidby each supplier.

Relationships(1) Used technology in the production line of suppliers

(as a desirable relationship and an economic cri-terion): The used technology is scored by expert’s

opinion using 9-point Likert spectrum according toTable 1. Likert scale is used to convert qualitativefactors into quantitative values [42].There are varietyof scales which can be rated as 1 to 5, 1 to 7, and 1to 9. Valuation of factors in this scale is performedaccording to concept of each factor [43].

(2) Green research and development (as a free relation-ship and an environmental criterion): The corre-sponding budget per year (in $100). Green researchand development is a dual-role factor which plays therole of both undesirable and desirable factors. Greenresearch and development can be considered as anundesirable criterion since it is cost of performinggreen researches. On the other hand, green researchand development is a desirable criterion, because itimplies innovations inmanufacturing green productsand services and environmental efficiency enhance-ment.

(3) Shortages (as an undesirable relationship and aneconomic criterion): The amount of shipments (interms of pallets) that have not delivered in the pastperiod and should be met in the next period by thesupplier.

Outputs

(1) Obtaining ISO certificates and observance of stan-dards: this kind of output is scored by expert’s opinionusing 9-point spectrum with respect to qualitativeand environmental criteria and standards and alsoworkplace standards. Table 2 shows the 9-point spec-trum.

(2) Quality (as an economic criterion): Quality of prod-ucts is evaluated by Likert scale. In Table 3, usinga 9-point Likert scale, valuation of quality of partssupplied by suppliers is presented.

(3) Supply capacity (g 2): maximum amount of materialsthat supplier can send to Naniwa Co.

(4) Efficiency of energy consumption (as an environmen-tal criterion): Efficiency of the energy consumption isthe third output which is an environmental criterion.To determine an appropriate scale for evaluating

http://www.naniwa.ir


Table 1: 9-point Likert spectrum for Technologic power.

Value 9 7 5 3 1 2-4-6-8

Technology HighTechnology Good TechnologyMedium

Technology Weak TechnologyVery weakTechnology

Intermediatevalues forTechnology

Table 2: 9-point Likert spectrum for Standards.

Value 9 7 5 3 1 2-4-6-8

ISO standards HighStandards Good Standards Medium Standards Weak StandardsVeryWeakStandards

Intermediate valuesfor Standards

efficiency of energy consumption we apply a scoringmethodwhich is shown byA+++ toG scale.The letterA+++ shows the lowest energy consumption. Theletter G indicates the highest energy consumption.Using 100-point scale. Table 4 shows the energyconsumption of the suppliers.

Figure 4 illustrates the proposed dynamic DEA approach onthe case study to evaluate suppliers in periods 2011 to 2014.The inputs, outputs, and relationships between periods isillustrated in this figure.

The inputs, relationships, and outputs data for35 suppliers in 4 time periods are presented inTable 5 in the Appendix.

In Table 6, the DMUs (i.e., 35 suppliers) are evaluatedby making use of the existing input-oriented dynamic DEAmodel on data shown in the Table 5 presented in theAppendix. The DEA model is selected according to the DM’sapproach for which he/she want to present their proposedimprovement plans. Table 6 shows the efficiency values for35 suppliers in 4 periods from 2011 to 2014.

The results of Table 6 show that none of the suppliershave unit efficiency values in all 4 periods. Therefore, noneof them are strictly efficient. To construct a strictly efficientunit as a benchmark for other units, in each period the DMUwith unit efficiency is selected as a candidate. As a result, 8scenarios are generated for ideal DMU which are differentcombinations of DMUs (suppliers) with unit efficiency ineach period.These resulting scenarios are presented inTable 7with probability of 0.125 for each one. If these scenarios aretaken into consideration and evaluated along with other 35suppliers, the efficiency of scenarios becomes one becausethey consist of DMUs with unit efficiencies in all periods.Therefore, we can claim that the existing dynamicDEAmodelcannot evaluate and rank the scenarios. In the proposedmodel (RIa), since all scenarios have unit efficiency, theaverage efficiency is also one and the deviation of efficienciesis zero. Therefore, we set 𝛽 = 1, 1 − 𝛽 = 0. To deal with thisdifficulty, a punishment (Cost) and encouragement (Income)value is considered for the inputs, outputs, and relationshipsof each scenario.

The costs and incomes associated with inputs, outputs,and relationships are presented in Table 8 with the followingnotations:

𝑘𝐴𝐿𝐿𝑗 : Unit cost for input j for all scenarios (j: price,packaging, transportation cost, and workforce health cost):this is a penalty that is imposed to a scenario by the decision-maker for each unit of inputs.ℎ𝐴𝐿𝐿𝑗 : Unit income for output j for all scenarios (j:

quality, production capacity, the number of acquired ISO andstandards, and efficiency of the energy consumption): this isthe encouragement that is considered for a scenario for eachunit of outputs.𝑔𝐴𝐿𝐿1 : Unit cost of shortages in shipped pallets for all

scenarios (undesired relationships): this is the penalty that isdecided by the decision-maker for each unit of this undesiredrelationship. In this paper for backlogs or goods shipped witha delay a penalty is considered for the supplier in that period.𝑣𝐴𝐿𝐿1 : Unit cost of green research and development for all

scenarios (free relationships): if the free relationship be anundesired relationship a penalty value will be assigned foreach unit of this relationship. In our case study, if the greenR&D is considered as an undesired relationship for the underinvestigation DMU, a penalty is considered for that supplierfor each unit of this relationship.𝑒𝐴𝐿𝐿1 : Unit income of green research and development for

all scenarios (free relationships): if the free relationship is adesired relationship an encouragement value will be assignedfor each unit of this relationship. In our case study, if the greenR&D is considered as a desired relationship for the underinvestigationDMU, an encouragement value is considered forthat supplier for each unit of this relationship.𝑏𝐴𝐿𝐿1 : Unit incomeof technological power for all scenarios

(desirable relationships).As mentioned earlier, the existing dynamic DEA model

cannot rank different scenarios for the ideal supplier. In thispaper, first we evaluate and rank these scenarios throughthe proposed input-oriented robust dynamic DEA model(RIa). In Table 9, the punishment and encouragement foreach scenario resulting from model (RIa) is presented. Aspresented in Table 9, the second scenario is the best scenariosince it has the maximum value of benefit. Generally, if theobjective function of model (RIa) is positive, the scenariowill generate loss while it is negative the scenario will makebenefits. If all scenarios generate loss, the scenario withminimum loss will be selected. The values of landa obtainedfrommodel (RIa) admit that the scenarios can be considered


Table 3: 9-point Likert spectrum for Quality.

Value 9 7 5 3 1 2-4-6-8

Quality High Quality Good Quality Medium Quality Weak Quality VeryWeak Quality Intermediate valuesfor Quality

Table 4: Spectrum for efficiency of the energy consumption.

Score 100 90 80 70 60 50 40 30 20 10Efficiency of the energy consumption A+++ A++ A+ A B C D E F G

as a benchmark. In the next step, the second scenario whichis the best scenario is considered as the ideal DMU and it isevaluated along with other DMUs (suppliers) through model(RIb). The ideal DMU which is actually the ideal scenariois considered as a unique benchmark which owns both theproperty of a real supplier in that it consists of some periodsand the property of a virtual supplier in that it is strictlyefficient and consists of periods with unit efficiency. There-fore, the proposed ideal DMU can present improvementplans for all suppliers and can rank the suppliers with sameefficiency.The results obtained frommodel (RIb) are reportedin Table 10. As presented in Table 10, the ideal DMU givesthe ability of ranking the suppliers that had the same rankin Table 6. From Table 6 we can see that suppliers 2 and 13,19 and 3, 8 and 31, 6 and 4, two by two have the same rankbecause the total efficiency of these DMUs are equal. Usingthe proposed model (RIb) for evaluating the suppliers andconstructing an ideal DMU help us to rank the suppliers.This ranking considers the standard deviation of efficienciesin calculating the efficiency of each supplier. Furthermore, theproposed ideal DMU gives us the opportunity of presentingimprovement methods for all DMUs. As a matter of fact, theimprovement methods are presented based on their distancefrom the ideal DMU (supplier). The high ranked scenarioresulting frommodel (RIa), which is the second scenario, hasmaximum distance from other suppliers and it is the worstcase that can be happen for an ideal scenario based on whichimprovement plans can be presented. Through model (RIb)other suppliers can be ranked based on their distance fromthis worst case ideal scenario. Therefore, we could claim thatthe resulting ranks and improvement plans cannot get worsewhen each of the other scenarios, which could be happenwitha given probability, is considered thus resulting in a robustsolution.

6. Conclusions and Future Research Directions

This paper proposed a new DEA approach for evaluatingsuppliers based on sustainable supplier criteria such as social,economic, and environmental criteria. The proposed modelconsiders suppliers in different periods thus leading to adynamic DEA model. Existing dynamic DEA models justpresent improvement plans and do not have the ability ofranking DMUs. The proposed model apart from having theability of ranking DMUs can present improvement plans.In most DEA models, when different time periods areconsidered in evaluating DMUs, no DMUs can be introduced

as a strictly efficient unit which is efficient in all periods. Ourcontribution is introducing a new method for constructingthe ideal DMU such that apart from the previous definitionfor ideal DMU which considers one of the DMUs which hasunit efficiency in all periods as an ideal DMU, a virtual DMUis considered as an ideal DMU.Thenew proposed ideal DMUhas the ability of presenting an improvement method for allsuppliers and also ranking the suppliers with the same effi-ciency.Theproposed idealDMU introduces a strictly efficientunit by building a combination of DMUs with unit efficiencyin each period. It is possible that multiple DMUs have unitefficiency in a period. Therefore, different combinations ofthese units lead to different scenarios for the ideal DMU eachof which can happen with a specific probability. The existingdynamic DEA model cannot evaluate and rank the scenarioswhich all have unit efficiency. To deal with these scenarios,a scenario-based robust optimization model for the dynamicDEA is developed which is capable of ranking the scenariosbased on a punishment and encouragement value assigned toeach scenario.

The proposed robust method is implemented in twosteps and it is is able to obtain a solution which is immu-nized against different scenarios exist in constructing theideal DMU. In the first step, at each time one of thescenarios is under investigation. The high ranked scenariowhich has more distance from other DMUs is selected as aunique benchmark based on which the improvement planis presented in the worst case. In fact, in the second stepother DMUs along with the worst case scenario are underinvestigation and improvement plans are proposed based ontheir distance from the ideal DMU. Therefore, the proposedmethod can rank the suppliers with the same efficiencyand propose an improvement plan which is robust againstdifferent scenarios which could be considered for the idealDMU.

For future study on this work, apart from the uncertaintyin different scenarios for ideal DMU, one can consider theinput, output, and relationship values as uncertain param-eters which can vary in a convex uncertainty set. Then ahybrid robust optimization approach, i.e., a hybridization ofthe scenario-based and robust counterpart methods, can bedeveloped to deal with these uncertainties.

Appendix

See Table 5.


Table5:Th

einp

uts,relatio

nships

andou

tputsd

atafor

35supp

liersin

4tim

eperiods.

DMU

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

2930

3132

3334

35

2011

IN1

2631

2930

2634

3216

2930

3328

2431

2926

2529

2831

3515

3321

2624

2823

3027

2631

2529

33IN

215

1823

2220

1914

917

1819

2522

2018

1716

2319

2219

1019

1315

1617

1920

1812

1616

1821

IN3

1312

1110

96

72

810

69

911

145

86

95

73

98

125

67

89

105

812

9IN

410

97

89

64

810

98

118

99

106

97

810

29

98

66

78

910

68

89

OUT1

44

65

84

96

57

65

44

56

78

63

49

58

74

57

65

46

59

8OUT2

86

59

56

97

87

57

86

79

68

77

99

77

68

78

79

97

68

7OUT3

7853

5964

6971

8868

6359

5549

6972

6667

8171

6365

7395

6256

8479

8378

6669

5974

9576

85OUT4

6090

8080

9090

60100

8090

6070

9080

8060

7050

6050

40100

7070

8060

6050

7060

4070

6050

80LINK1

87

86

79

69

55

79

78

68

95

56

79

98

65

88

67

86

67

8LINK2

2831

3032

2928

3518

2626

2729

3033

2725

2930

2729

2715

2526

3732

3426

2929

2826

3334

25LINK3

2016

1718

1315

152

1518

1618

1513

68

97

68

113

1219

1513

146

169

1112

1415

10

2012

IN1

1933

3227

2929

2830

3233

3629

2832

3129

2832

3328

3015

2933

2830

2929

2732

3033

2925

38IN

212

1625

2422

2016

1816

1720

2619

2322

1918

2016

1820

1016

2119

1420

1615

2016

2018

1618

IN3

39

1312

105

67

79

57

814

197

95

79

105

79

149

57

97

97

109

10IN

44

118

98

910

69

79

109

1312

98

108

69

511

109

87

56

79

86

710

OUT1

96

78

66

57

68

67

86

58

67

58

69

78

56

68

77

65

86

7OUT2

96

78

65

58

99

69

87

76

86

98

79

66

78

58

67

89

78

8OUT3

9559

6070

7766

6852

7059

6352

7378

7269

7369

7472

6989

7366

7866

9165

7372

6085

8066

80OUT4

100

8080

8090

8050

6080

7050

7070

8080

7070

5060

6040

100

7070

7060

6060

8060

5070

7050

80LINK1

96

78

88

77

66

78

67

77

86

67

89

87

77

77

76

87

78

7LINK2

1230

2831

3329

2925

1928

3436

3328

3429

2532

3330

2620

3031

2826

2930

3225

2729

3226

28LINK3

512

1011

1716

1513

1716

1411

1011

1512

1217

1412

165

1010

1710

1211

1814

1315

1812

10


Table5:Con

tinued.

DMU

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

2930

3132

3334

35

2013

IN1

3132

3426

3026

2932

3532

3026

3030

2922

2931

3025

3226

2832

2931

3028

2633

2830

2624

15IN

218

1827

2524

2219

1919

1822

2817

2525

1719

2419

1922

1918

2317

1821

1717

2218

2119

159

IN3

1410

1413

118

98

810

89

911

158

107

88

1210

810

128

89

106

78

84

2IN

415

1412

1110

1012

87

910

1210

1514

1110

149

910

1015

1410

1112

105

88

119

104

OUT1

79

79

89

98

77

98

96

97

88

99

88

69

87

69

97

88

67

9OUT2

68

98

69

78

68

79

98

67

99

87

98

67

89

98

89

87

98

9OUT3

6370

7264

8670

6465

6560

7060

6585

8075

8070

6583

8270

6580

8175

7655

6580

5080

7765

95OUT4

7070

8060

9070

5060

7050

4070

6080

7070

7070

6060

5060

7070

7080

6050

8060

6070

7060

100

LINK1

87

78

98

79

65

79

67

85

85

87

86

87

86

87

86

88

78

9LINK2

3332

2729

3034

3139

2830

3036

3329

3130

2627

2930

3136

3433

2930

3229

3729

3433

2628

10LINK3

1519

2120

1918

1716

1415

1918

1617

1512

1817

1315

1614

1012

1411

1315

1614

1215

1612

2

2014

IN1

2034

3729

3433

3236

3734

3435

3432

3328

3234

3331

3030

3234

3033

3230

3035

3332

2828

15IN

214

2230

2827

2625

2023

2420

2620

2729

2022

2620

2125

2119

2619

2123

2120

2421

2220

1712

IN3

414

1515

1312

1210

1012

1010

1114

1716

1218

1011

1512

914

159

710

1214

1016

910

5IN

48

1514

1312

129

1012

1010

1110

1813

1211

1310

1013

1214

1112

1114

129

1110

1411

97

OUT1

97

79

98

88

97

78

99

78

69

79

88

98

77

89

98

87

98

9OUT2

99

99

79

79

79

89

98

87

99

88

98

88

99

99

99

89

98

9OUT3

8959

6372

7372

6868

6068

6773

7266

6469

6543

7072

7174

7370

7266

6759

6875

6378

6959

95OUT4

100

9080

8090

7060

6090

7050

6060

9070

8070

8070

6050

5060

8070

8060

70100

6060

8070

70100

LINK1

97

78

99

89

77

89

78

86

86

97

97

86

97

88

88

88

98

9LINK2

2030

2930

3232

3335

2932

3235

3430

3332

2829

3029

3035

3634

3132

3432

3032

3630

2531

22LINK3

59

1110

1416

1311

1210

1416

1618

517

1618

1510

1013

1214

167

1021

1814

1113

1510

3


Table 6: Efficiency values for 35 suppliers in 4 periods (Input-oriented dynamic DEA model).

DMU 𝜙2011 𝜙2012 𝜙2013 𝜙2014 Φ𝑗 RANK1 0.856 1 0.986 1 0.9605 12 0.763 0.943 0.915 0.966 0.8967 43 0.88 0.849 0.965 0.824 0.8795 114 0.867 0.825 0.814 0.894 0.85 265 0.728 0.831 0.824 0.872 0.8137 336 0.809 0.941 0.769 0.881 0.85 267 0.911 0.817 0.964 0.764 0.864 198 1 0.866 0.813 0.821 0.875 139 0.766 0.835 0.849 0.938 0.847 2910 0.849 0.961 0.827 0.933 0.846 3111 0.867 0.985 0.863 0.74 0.8637 2012 0.857 0.938 0.917 0.768 0.87 1613 0.892 0.846 0.965 0.884 0.8967 414 0.893 0.851 0.824 0.825 0.8482 2815 0.846 0.837 0.764 0.864 0.8277 3216 0.937 0.84 0.819 0.897 0.8732 1517 0.967 0.869 0.893 0.831 0.89 718 0.955 0.815 0.84 0.843 0.8632 2119 0.934 0.834 0.933 0.817 0.8795 1120 0.871 0.769 0.968 0.934 0.8855 921 0.768 0.941 0.824 0.92 0.8632 2222 1 1 0.866 0.819 0.9212 323 0.869 0.911 0.814 0.8349 0.8572 2424 0.942 0.771 0.867 0.822 0.8505 2525 0.852 0.764 0.789 0.796 0.8002 3426 0.789 0.859 0.984 0.753 0.8462 3027 0.844 0.915 0.943 0.749 0.8627 2328 0.809 0.864 0.972 0.921 0.8915 629 0.78 0.91 0.928 0.966 0.8842 1030 0.922 0.752 0.846 0.946 0.8665 1831 0.818 0.92 0.837 0.925 0.875 1332 0.846 0.881 0.819 0.933 0.8697 1733 0.856 0.825 0.935 0.928 0.886 834 0.736 0.714 0.966 0.784 0.8 3535 0.855 0.891 1 1 0.9365 2

Table 7: Different scenarios for Ideal DMU.

Ideal DMU Period 2011 Period 2012 Period 2013 Period 2014Scenario 1 Supplier 22 Supplier 22 Supplier 35 Supplier 1Scenario 2 Supplier 22 Supplier 22 Supplier 35 Supplier 35Scenario 3 Supplier 22 Supplier 1 Supplier 35 Supplier 1Scenario 4 Supplier 22 Supplier 1 Supplier 35 Supplier 8Scenario 5 Supplier 8 Supplier 22 Supplier 35 Supplier 1Scenario 6 Supplier 8 Supplier 22 Supplier 35 Supplier 35Scenario 7 Supplier 8 Supplier 1 Supplier 35 Supplier 1Scenario 8 Supplier 8 Supplier 1 Supplier 35 Supplier 35


Workforce health

Cost of recoverable packages

Final transportation cost

price

Quality

Efficiency of energy

consumption

ISO standards

Production capacity Technological power Workforce health Cost of recoverable packages

priceFinal transportation cost

Green research anddevelopment

Deduction or shortages

Figure 4: Demonstration of the proposed dynamic DEA on the considered case study.

Table 8: Costs and incomes associated with inputs, outputs and relationships.

Parameters 𝑘𝐴𝐿𝐿1 𝑘𝐴𝐿𝐿2 𝑘𝐴𝐿𝐿3 𝑘𝐴𝐿𝐿4 𝑔𝐴𝐿𝐿1 V𝐴𝐿𝐿1Costs 10 5 7 3 11 9Parameters ℎ𝐴𝐿𝐿1 ℎ𝐴𝐿𝐿2 ℎ𝐴𝐿𝐿3 ℎ𝐴𝐿𝐿4 𝑒𝐴𝐿𝐿1 𝑏𝐴𝐿𝐿1Incomes 20 15 18 12 15 18

Table 9: Evaluating different scenarios for the Ideal DMU (Model RIa).

Scenario 1 2 3 4 5 6 7 8 Worst caseEfficiency 1 1 1 1 1 1 1 1 1Benefit/loss 10027 10988 10103 9655 10320 10768 9435 9883 10147.375

Table 10: Evaluating suppliers along with the ideal supplier (Model RIb).

DMU 1 2 3 4 5 6 7 8 9Efficiency 0.489 0.478 0.445 0.354 0.317 0.359 0.412 0.437 0.341RANK 2 5 13 28 34 27 20 15 30DMU 10 11 12 13 14 15 16 17 18Efficiency 0.325 0.406 0.428 0.475 0.348 0.321 0.432 0.469 0.401RANK 32 21 17 6 29 33 16 8 22DMU 19 20 21 22 23 24 25 26 27Efficiency 0.449 0.462 0.392 0.481 0.377 0.362 0.309 0.332 0.384RANK 12 10 23 4 25 26 35 31 24DMU 28 29 30 31 32 33 34 35 Ideal DMUEfficiency 0.472 0.458 0.416 0.439 0.422 0.466 0.305 0.484 0.5RANK 7 11 19 14 18 9 36 3 1


Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

[1] Z. Khan and J. D. Nicholson, “Technological catch-up bycomponent suppliers in the Pakistani automotive industry: Afour-dimensional analysis,” Industrial Marketing Management,vol. 50, pp. 40–50, 2015.

[2] T. Dyllick and K. Hockerts, “Beyond the business case forcorporate sustainability,” Business Strategy and the Environment,vol. 11, no. 2, pp. 130–141, 2002.

[3] L. Wen, L. Xu, and R. Wang, “Sustainable supplier evaluationbased on intuitionistic fuzzy sets group decision methods,”Journal of Information and Computational Science, vol. 10, no.10, pp. 3209–3220, 2013.

[4] S. R. A. Ratan, A. Sekhari, andM. Rahman, “Sustainable supplychain management: State-of-the-art,” in Proceedings of theInternational Conference on So�ware, Knowledge. InformationManagement and Applications, Paro, Bhutan, 2010.

[5] B. M. Beamon, “Environmental and sustainability ethics insupply chain management,” Science and Engineering Ethics, vol.11, no. 2, pp. 221–234, 2005.

[6] S. Yahya and B. Kingsman, “Vendor rating for an entrepreneurdevelopment programme: a case study using the analytichierarchy process method,” Journal of the Operational ResearchSociety, vol. 50, no. 9, pp. 916–930, 1999.

[7] C. Muralidharan, N. Anantharaman, and S. G. Deshmukh,“A Multi-Criteria Group Decisionmaking Model for SupplierRating,” Journal of Supply Chain Management, vol. 38, no. 3, pp.22–33, 2002.

[8] C.Kahraman,U.Cebeci, andZ.Ulukan, “Multi-criteria supplierselection using fuzzy AHP,” Logistics Information Management,vol. 16, no. 6, pp. 382–394, 2003.

[9] S. Önüt, S. S. Kara, and E. Işik, “Long term supplier selectionusing a combined fuzzy MCDM approach: A case study fora telecommunication company,” Expert Systems with Applica-tions, vol. 36, no. 2, pp. 3887–3895, 2009.

[10] A. Ishizaka, C. Pearman, and P. Nemery, “AHPSort: An AHP-based method for sorting problems,” International Journal ofProduction Research, vol. 50, no. 17, pp. 4767–4784, 2012.

[11] A. Ishizaka, “Clusters and pivots for evaluating a large numberofalternatives in AHP,” Pesquisa Operacional, vol. 32, no. 1, pp. 87–101, 2012.

[12] A. Ishizaka, “Comparison of fuzzy logic, AHP, FAHP andhybridfuzzy AHP for new supplier selection and its performance anal-ysis,” International Journal of Integrated Supply Management,vol. 9, no. 1-2, pp. 1–22, 2014.

[13] A. Zouggari and L. Benyoucef, “Simulation based fuzzy TOPSISapproach for group multi-criteria supplier selection problem,”Engineering Applications of Artificial Intelligence, vol. 25, no. 3,pp. 507–519, 2012.

[14] I. Chamodrakas, I. Leftheriotis, and D. Martakos, “In-depthanalysis and simulation study of an innovative fuzzy approachfor ranking alternatives in multiple attribute decision makingproblems based on TOPSIS,” Applied So� Computing, vol. 11, no.1, pp. 900–907, 2011.

[15] A. Amindoust, S. Ahmed, A. Saghafinia, and A. Bahreininejad,“Sustainable supplier selection: a ranking model based on fuzzyinference system,” Applied So� Computing, vol. 12, no. 6, pp.1668–1677, 2012.

[16] A. Kumar, V. Jain, and S. Kumar, “A comprehensive environ-ment friendly approach for supplier selection,” OMEGA - �eInternational Journal of Management Science, vol. 42, no. 1, pp.109–123, 2014.

[17] A. Charnes, W. W. Cooper, and E. Rhodes, “Measuring theefficiency of decision making units,” European Journal of Oper-ational Research, vol. 2, no. 6, pp. 429–444, 1978.

[18] R. Farzipoor Saen, “A decision model for ranking suppliers inthe presence of cardinal and ordinal data, weight restrictions,and nondiscretionary factors,” Annals of Operations Research,vol. 172, no. 1, pp. 177–192, 2009.

[19] R. F. Saen, “Restricting weights in supplier selection decisionsin the presence of dual-role factors,” Applied MathematicalModelling: Simulation and Computation for Engineering andEnvironmental Systems, vol. 34, no. 10, pp. 2820–2830, 2010.

[20] A.Noorizadeh,M.Mahdiloo, andR. F. Saen, “Supplier selectionin the presence of dual-role factors, nondiscretionary inputs,and weight restrictions,” International Journal of Productivityand Quality Management, vol. 8, no. 2, pp. 134–152, 2011.

[21] M.Azadi andR. F. Saen, “Developing a new chance-constrainedDEA model for suppliers selection in the presence of undesir-able outputs,” International Journal of Operational Research, vol.13,

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