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Computers & Industrial Engineering

journal homepage: www.elsevier.com/locate/caie

Efficiency decomposition and measurement in two-stage fuzzy DEA modelsusing a bargaining game approach

Madjid Tavanaa,b, Kaveh Khalili-Damghanic,⁎, Francisco J. Santos Arteagad, Reza Mahmoudie,Ashkan Hafezalkotobc

a Business Systems and Analytics Department, Distinguished Chair of Business Analytics, La Salle University, Philadelphia, PA 19141, USAb Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, D-33098 Paderborn, Germanyc Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Irand Faculty of Economics and Management, Free University of Bolzano, Bolzano, Italye Department of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan, Iran

A R T I C L E I N F O

Keywords:Fuzzy DEATwo-stageBargaining gameBankingPerformance evaluation

A B S T R A C T

In this paper, a fuzzy two-stage Game-DEA (FTSGDEA) approach is proposed using a bargaining game model.Each decision-making unit (DMU) consists of two serially connected sub-DMUs. The first sub-DMU uses severalinputs to produce several outputs. The outputs of the first sub-DMU, called intermediated measures, are theinputs of the second sub-DMU. Intermediate measures are used by the second sub-DMU to produce the finaloutput of the main DMU. In standard TSGDEA settings, the product of the distance of the efficiency scores ofeach sub-DMU, which defines its breakdown payoff, is maximized. This approach leads to the overall en-hancement of the efficiency scores through a cooperative game environment. We extend this setting to a fuzzyenvironment where the uncertainty of inputs, intermediate measures, and outputs is accounted for using lin-guistic terms parameterized via fuzzy sets. The proposed FTSGDEA associates an interval efficiency score to eachDMU and sub-DMU, which allows the model to handle real-life problems involving uncertainties. We applyFTSGDEA to a case study involving the assessment of sixty branches from the Saman bank in Iran.

1. Introduction

Data envelopment analysis (DEA) is a methodology based on linearprogramming used to measure the relative efficiency of homogenousdecision-making units (DMUs) with multiple inputs and multiple out-puts. The first DEA model was proposed by Charnes, Cooper, andRhodes (1978) under the constant returns to scale assumption and waslater extended by Banker, Charnes, and Cooper (1984) using variablereturns to scale. In DEA models, the efficiency scores of efficient DMUsare equal to one, while they are less than one for inefficient DMUs.Thus, a DMU is called efficient if its relative efficiency score is equal tounity.

Extensive research has been performed in the field of DEA since theseminal work of Charnes et al. (1978). The models proposed haveaimed at enhancing the shortcomings of classic DEA modeling such ascommon weights, multiple objectives, two-stage and network struc-tures, uncertainty effects, different types of data and returns to scale,and productivity analysis. One of the main shortcomings of the classic

DEA models is their incapability to assign efficiency scores to the in-ternal structure of a DMU (Du, Liang, Chen, Cook, & Zhu, 2011). TheDEA literature introduced two-stage models to overcome this short-coming.

In two-stage DEA modeling, the main DMU is composed of twoserially connected sub-DMUs. Inputs are used by the first sub-DMU toproduce outputs, which are also called intermediate measures. Thesemeasures are used as inputs by the second sub-DMU in order to producethe outputs of the main DMU. This type of two-stage structure allows toanalyze the internal relations of a DMU, while the standard DEA ap-proach cannot be used to address potential conflicts between the twostages arising from the intermediate measures (Du et al., 2011). Severalpractical approaches that include combining the efficiency scores of thetwo stages in a multiplicative manner (Kao & Hwang, 2008), using aweighted additive model to aggregate the two stages and decomposingthe efficiency of the overall process (Chen, Cook, Li, & Zhu, 2009), orimplementing a game theoretical structure (Liang, Cook, & Zhu, 2008),have been proposed to cope with such problems.

https://doi.org/10.1016/j.cie.2018.03.010Received 22 August 2017; Received in revised form 22 January 2018; Accepted 5 March 2018

⁎ Corresponding author at: Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran.E-mail addresses: tavana@lasalle.edu (M. Tavana), k_khalili@azad.ac.ir (K. Khalili-Damghani), fsantosarteaga@unibz.it (F.J. Santos Arteaga),

r.mahmoudi@in.iut.ac.ir (R. Mahmoudi), a_hafez@azad.ac.ir (A. Hafezalkotob).URL: http://tavana.us/ (M. Tavana).

Computers & Industrial Engineering 118 (2018) 394–408

Available online 07 March 20180360-8352/ © 2018 Published by Elsevier Ltd.

T

More precisely, Du et al. (2011) applied directly Nash bargaininggame theory to determine the efficiency of DMUs that conform to theclassic two-stage processes. In the approach proposed by these authors,each stage is assumed to act as an independent bargaining playeraiming for a better payoff. In the current paper, the model proposed byDu et al. (2011) is extended to an uncertain environment whose un-certainty is accounted for using linguistic terms parameterized by fuzzysets. In this regard, the α-level based approach (Kao & Liu, 2000; Saati,Memariani, & Jahanshahloo, 2002; Triantis & Girod, 1998; Wang,Greatbanks, & Yang, 2005), the fuzzy ranking approach (Guo & Tanaka,2001; León, Liern, Ruiz, & Sirvent, 2003; Soleimani-damaneh,Jahanshahloo, & Abbasbandy, 2006), and the possibility approach(Lertworasirikul, Fang, Joines, & Nuttle, 2003; Lertworasirikul, Fang,Nuttle, & Joines, 2003) are among the solution categories generallyapplied to analyze fuzzy mathematical programming models.

Although a variety of research studies on fuzzy DEA have alreadybeen developed, there are just a few recent studies on fuzzy two-stageDEA. In particular, several network DEA approaches have been in-troduced in the literature (Avkiran, 2009; Chen et al., 2009; Cook,Liang, & Zhu, 2010; Cook, Zhu, Bi, and Yang, 2010; Färe & Grosskopf,2000; Lewis & Sexton, 2004a, 2004b; Tone & Tsutsui, 2009) amongwhich the relational framework of Kao and Hwang (2008, 2010) andKao (2009a, 2009b) constitutes one of the most fruitful research areas.In this regard, Kao and Liu (2011) applied the relational network DEAapproach to a two-stage system with fuzzy data, while Kao and Lin(2012) considered a system with parallel processes and fuzzy data.

In the current paper, a novel bargaining game-DEA approach isdeveloped to compute process efficiencies in two-stage systems withfuzzy data. An optimistic-pessimistic method based on the α-level ap-proach, and in which the upper and lower bounds of the efficiencyscores are determined, is proposed to handle the associated fuzzymathematical programs. The proposed approach presents several ad-vantages in comparison with other existing methods

a. To the best of our knowledge, a fuzzy bargaining game-DEA modelhas never been developed before.

b. Our model is independent of the α-level variables. Thus, the cal-culation efforts are relatively modest in comparison with classic α-level approaches.

c. Our model is feasible and bounded, independent of the number ofDMUs, the number of inputs and outputs and their values.

The proposed model has been applied to a real case study involvingthe assessment of sixty branches from the SAMAN bank in Iran. Theefficiency scores of each DMU and each sub-DMU defining the twostages of the DEA framework will be calculated within a fuzzy en-vironment.

The paper is organized as follows. In Section 2, we provide somebasic theoretical preliminaries and review the related literature. InSection 3, we discuss the theoretical properties of fuzzy game-DEAmodels. A conceptual two-stage efficiency model for a bank's branch,including a productivity and a profitability stage, is proposed in Section4. The application of the proposed model to the SAMAN Bank case

study is also discussed in Section 4. Finally, Section 5 presents theconclusion and suggests future research directions.

2. Theoretical preliminaries and literature review

In this section, we briefly review the main theoretical preliminariesas well as the related literature. The current study is then placed amongthe existing works.

2.1. Theoretical preliminaries

2.1.1. Conventional DEA modelingFirst, let us review the classic single stage DEA model briefly.

Suppose that there are n DMUs, where each = …DMU j n( 1, , )j consumesm inputs, = …X i m( 1, , )ij to produces s outputs, = …y r s( 1, , )rj .Considering = …u r s( 1, , )r and = …v i m( 1, , )i as the relative importanceof outputs and inputs, respectively, Model (1) calculates the relativeefficiency of a given DMUo (Charnes et al., 1978):

∑ ∑

∑

− ⩽ = …

=

> = …> = …

=

= =

=

sum u ys t

u y v X j n

v X

u r sv i m

Max. .

0, 1, ,

1,

0, 1, , ,0, 1, , .

rs

r ro

r

s

r rji

m

i ij

i

m

i io

r

i

1

1 1

1

(1)

Model (1) is run for each DMU and the efficiency scores as well as therelative importance of inputs and outputs are calculated. One of thedrawbacks of Model (1) is that it is not designed to measure the effi-ciency of internal processes in a DMU. Two-stage DEA models havebeen proposed to handle this issue.

2.1.2. Two-stage DEA modelingSuppose that the internal process of a given DMU can be divided

into two stages, as illustrated in Fig. 1. = …DMU j n, 1, ,j , consumes inputs= …X i m, 1, , ,ij in the first stage to produce intermediate products= …z p q, 1, ,pj . These intermediate products are then used in the second

stage to produce the final outputs = …y r s, 1, ,rj .Several two-stage DEA approaches have been developed in the lit-

erature. Kao and Hwang (2008) proposed the most popular and widelycited. In their model, system efficiency is defined as the product of theefficiencies of the two stages. Such decomposition has been widelydiscussed in the literature and adopted as the basis of many researchworks. The linear form of the two-stage model proposed by Kao andHwang (2008) is briefly revisited as Model (2) for a given

∈ …DMU o n, {1, , }o :

Fig. 1. Two-stage process.

M. Tavana et al. Computers & Industrial Engineering 118 (2018) 394–408

395

∑

∑

∑ ∑

∑ ∑

∑ ∑

=

=

− ⩽ = …

− ⩽ = …

− ⩽ = …

⩾ = …⩾ = …⩾ = …

=

=

= =

= =

= =

E u y

v X

u y v X j n

w z v X j n

u y w z j n

v ε i mw ε p qu ε r s

max

s.t.

1,

0, 1, , ,

0, 1, , ,

0, 1, , ,

, 1, , ,, 1, , ,, 1, , .

or

s

r ro

i

m

i io

r

s

r rji

m

i ij

p

q

p pji

m

i ij

r

s

r rjp

q

p pj

i

p

r

1

1

1 1

1 1

1 1

(2)

The main characteristic of Model (2) is that the multipliers wp as-signed to the intermediate products Zpj are jointly determined, re-gardless of whether these products are treated as outputs of the firststage or inputs of the second stage. The first inequality constraint de-fines the upper bound of the efficiency scores for the whole system.Similarly, the second and third inequality constraints determine theupper bound for stages 1 and 2, respectively. Given the optimal solu-tions of Model (2) for a DMUj, ∗ ∗u v,r i , and ∗wp, the efficiency of thesystem, Ej, and that of stage 1, Ej

1, and stage 2, Ej2, can be calculated

using (3)

= = …

= = …

= × = = …

∑

∑

∑

∑

∑

∑

=∗

=∗

=∗

=∗

=∗

=∗

E j n

E j n

E E E j n

, 1, , ,

, 1, , ,

, 1, , .

jw z

v X

ju y

w z

j j ju y

v X

1

2

1 2

pq

p pj

im

i ij

rs

r rj

pq

p pj

rs

r rj

im

i ij

1

1

1

1

1

1 (3)

A DMU is considered efficient if and only if the associated stages areefficient.

2.1.3. Fuzzy DEA modelingBuilding on the assumptions and notations of Model (1), the fol-

lowing model can be used to calculate the fuzzy efficiency scores of thedifferent DMUs

∑

∑ ∑

∑

− ⩽ = …

=

> = …> = …

∼

∼ ∼

∼

=

= =

=

u y

s t

u y v X j n

v X

u r sv i m

max

. .

0, 1, , ,

1,

0, 1, , ,0, 1, , .

r

s

r ro

r

s

r rji

m

i ij

i

m

i io

r

i

1

1 1

1

where the “∼” symbol denotes the fuzzy quality of inputs and outputs.Four main solution categories have been introduced in the literature

to handle the above fuzzy DEA model (Emrouznejad, Tavana, &Hatami-Marbini, 2014), namely, the tolerance (Kahraman & Tolga,1998; Sengupta, 1992), α-level (Hatami-Marbini, Saati, & Tavana,2010; Kao & Liu, 2003; Triantis & Girod, 1998), fuzzy ranking (Guo,2009; Guo & Tanaka, 2001), and possibility (Lertworasirikul, Fang,Joines, et al., 2003; Lertworasirikul, Fang, Nuttle, et al., 2003) ap-proaches. In the current paper, we use the α-cut based approach, sinceit is the most efficient and straightforward one (Khalili-Damghani &

Taghavifard, 2012; Khalili-Damghani & Tavana, 2013; Tavana &Khalili-Damghani, 2014).

Khalili-Damghani and Abtahi (2011) proposed a fuzzy DEA modelto measure the efficiency of just in time implementation.

2.2. Literature review

In this section, we briefly review the literature on fuzzy and two-stage DEA modeling.

2.2.1. Fuzzy DEASeveral methods based on the formal developments introduced by

Zadeh (1965) have been defined to deal with fuzzy data in DEA. Kaoand Liu (2000) and Entani, Maeda, and Tanaka (2002) built on α-levelsets to convert uncertain data into intervals. Similarly, Liu (2008) andLiu and Chuang (2009) introduced the assurance region framework byextending the α-level set one within a fuzzy DEA environment, whileWang, Luo, and Liang (2009) relied on fuzzy arithmetic to deal withfuzzy inputs and outputs. Dia (2004) developed a crisp DEA model froma fuzzy one using a fuzzy aspiration and a safety α-level.

In this regard, Soleimani-damaneh et al. (2006) designed a fuzzyDEA model that provided crisp efficiencies while accounting for severaldrawbacks of the models introduced by Kao and Liu (2000), León et al.(2003) and Lertworasirikul, Fang, Joines, et al. (2003). Soleimani-da-maneh (2008) defined an additive DEA model using the concepts offuzzy upper bound and signed distance. Moreover, Soleimani-damaneh(2009) illustrated that a distance-based upper bound existed for theobjective function introduced by Soleimani-damaneh (2008).

Khodabakhshi, Gholami, and Kheirollahi (2010) proposed two sto-chastic and fuzzy additive DEA models that could be used to determinethe type of returns to scale. In the same line of research, Tavana,Khanjani Shiraz, Hatami-Marbini, Agrell, and Paryab (2012) for-mulated three fuzzy DEA models accounting for different possibility,necessity, credibility as well as probability constraints. Hatami-Marbini,Emrouznejad, and Tavana (2011) reviewed and provided a taxonomy ofthe main fuzzy DEA models introduced in the literature.

2.2.2. Two-stage DEAStandard DEA models consider the DMUs being analyzed as black

boxes where inputs are consumed to produce outputs (Avkiran, 2009).As a result, single-stage DEA models could deliver inaccurate evalua-tions of efficiency (Rho & An, 2007). On the other hand, two-stage DEAmodels were designed to account for the internal processes and struc-ture of the DMUs, allowing the decision makers to identify the sourcesof inefficiency across the sub-DMUs composing a given DMU (Färe &Grosskopf, 2000; Kao & Hwang, 2008; Li, Chen, Liang, & Xie, 2012).Cook, Liang, et al. (2010) and Cook, Zhu, et al. (2010) provided a four-categories classification of two-stage DEA environments: standard,network, efficiency decomposition, and game-theoretic. Kao and Liu(2011) extended the two-stage DEA framework of Kao and Hwang(2008) to a fuzzy environment through a two-level optimization pro-cess.

Seiford and Zhu (1999) designed a two-stage DEA method based onprofitability and marketability to evaluate different branches of com-mercial banks. Profitability was measured in the first stage by con-sidering labor and assets as inputs, while profits and revenue con-stituted the resulting outputs. Marketability was measured in thesecond stage using the profits and revenue obtained from the first stageas inputs, while market value, earnings per share and returns wereconsidered as outputs. Their two-stage DEA model was applied by Zhu(2000) to analyze the efficiency of the companies composing the For-tune Global 500 list. Casu and Molyneux (2003) studied the efficiency

M. Tavana et al. Computers & Industrial Engineering 118 (2018) 394–408

396

of the European banking system considering total costs and deposits asinputs and total loans and earning assets as outputs.

Khalili-Damghani, Sadi-Nezhad, and Aryanezhad (2011) extendedthe fuzzy formal setting introduced by Abtahi and Khalili-Damghani(2011) to measure the agility exhibited by different supply chains andrank the resulting interval efficiencies. This scenario was further de-veloped and extended to a fuzzy two-stage DEA framework by Khalili-Damghani, Taghavifard, and Abtahi (2012), to a fuzzy network en-vironment by Khalili-Damghani and Tavana (2013), and to a settingwith ordinal Likert-based data by Khalili-Damghani et al. (2012).Khalili-Damghani and Taghavifard (2013) designed several models tocompute the stability radius in two-stage DEA environments facingsubstantial variation and uncertainty in their inputs and outputs.

Khalili-Damghani, Taghavifard, Olfat, and Feizi (2012) measuredthe performance of a fresh food supply chain using a two-stage DEAmodel.

3. Proposed fuzzy bargaining game DEA model

In this section, we describe our approach to measure the efficiencyof uncertain two-stage processes modeled using fuzzy sets through abargaining game setting.

3.1. Bargaining game model

We adopt the notation of Du et al. (2011) to describe the payoffstructure of a Nash bargaining game, which consists of dividing thebenefits obtained between the two players according to their competi-tion in the market place. The Nash bargaining model requires a compactand convex feasible set defining the potential payoff vectors of theplayers (Nash, 1950), so that each individual attainable payoff isgreater than the corresponding breakdown points.

Breakdown points determine the minimum payoff pairs attainablewhen one player decides not to bargain with the other player. As statedin Binmore, Rubinstein, and Wolinsky (1986), the choice of breakdownpoint is an issue of modeling judgment. Thus, if → =u u u( , )1 2 and→

=b b b( , )1 2 are the respective payment and breakdown payoff vectors,the players must maximize ∏ −= u b( )i i i1

2 by solving the following pro-blem

− −

⩾⩾

u b u bs tu bu b

max( )( ). .

,.

1 1 2 2

1 1

2 2 (4)

Du et al. (2011) identified the two stages with the two players of thebargaining game, the efficiency ratios with the payoffs, and the effi-ciency weights with the strategies.

Let =x xmax { },i j ijmax =y ymin { },r j rj

min =z zmin { }p j pjmin and

=z zmax { }p j pjmax . In the first stage, consider a situation in which a DMU

consumes the maximum amount of inputs and produces the minimumamount of intermediate measures, i.e., =x z i( , ),i d

max min

… = …m d D1, , , 1, , . Clearly, this vector represents the anti-ideal DMU ofthe first stage. Similarly, consider = … = …z y d D r s( , ), 1, , , 1, ,d r

max min , as theanti-ideal DMU of the second stage. Note that, in this case, the DMUconsumes the maximum amount of intermediate measures while pro-ducing the minimum amount of outputs.

The efficiency scores of the above anti-ideal DMUs are the worstamong the existing DMUs. The efficiency scores of the anti-ideal DMUsin the first and second stage are denoted by θmin

1 and θmin2 , respectively.

In this paper, these values (i.e., θmin1 and θmin

2 ) are taken as the break-down points of the model.

3.2. Two-stage process with fuzzy data

Fuzzy numbers can be used to represent the uncertainties inherentto inputs, outputs, and intermediate measures. A fuzzy interval ∼A is aset of real numbers with a continuous, compactly supported, uni-modal,and normalized membership function ∼μ A of the following form (Dubois& Prade, 1980):

=

⎧

⎨

⎪⎪

⎩⎪⎪

− ⩽ ⩽ >

⩽ ⩽ + >

−

−∼

( )( )

μ xL a c x a c

R b x b d d

otherwise

( )

1,

, , 0

, , 0

0,

A

a xc

x bd

(5)

where L and R are left- and right-shape functions, respectively. Thefunctions � →L R, : [0,1] are non-increasing and continuous. To gen-eralize the discussion and simplify the notation, it will be assumed thatall the observations retrieved consist of interval fuzzy numbers, since acrisp number can be considered as a degenerated interval fuzzy numberwith only one value in the domain.

Denote by ∼Xij,∼Yrj and

∼Zpj the fuzzy counterparts of Xij, Yrj, and Zpj,respectively. Since the observations retrieved are fuzzy numbers, theresulting efficiency scores ∼EK should also be fuzzy numbers. The cor-responding −α cuts of ∼Xij,

∼Yrj, and ∼Zpj are given by the intervals=X X X( ) [( ) ,( ) ],ij α ij α

Lij α

U =Y Y Y( ) [( ) ,( ) ],rj α rj αL

rj αU and =Z Z Z( ) [( ) ,( ) ],pj α pj α

Lpj α

U

respectively (Kao & Liu, 2000).Let E x y z( , , )O represent the solution of Model (2), emphasizing the

fact that given a set of crisp values for the inputs, outputs, and inter-mediate products the system efficiency can be uniquely determined.

The membership function can be obtained using Zadeh’s extensionprinciple (Zadeh, 1972, 1975, 1978) as follows:

= =∼∼ ∼∼μ e μ x μ y μ z e E x y z( ) supmin{ ( ), ( ), ( )| ( , , )}Ex y z i r p j x ij Y rj z pj o

, , , , ,o ij rj pj (6)

This definition allows for the determination of the lower and upperlimits of the −α cuts of the system efficiency, =E E E( ) [( ) ,( ) ]o α o α

Lo α

U , bysolving the following two-level models

=⩽ ⩽ ∀ ∀⩽ ⩽ ∀ ∀⩽ ⩽ ∀ ∀

EX x X i jY y Y r j

Z z Z p j

E x y z( ) max( ) ( )( ) ( )

( ) ( )

{max ( , , )}o αU

ij αL

ij ij αU

rj αL

rj rj αU

pj αL

pj pj αU

o

(7)

The two-level Model (7) determines the maximum achievable of themaximum efficiency score.

=⩽ ⩽ ∀ ∀⩽ ⩽ ∀ ∀⩽ ⩽ ∀ ∀

EX x X i jY y Y r j

Z z Z p j

E x y z( ) min( ) ( )( ) ( )

( ) ( )

{max ( , , )}o αL

ij αL

ij ij αU

rj αL

rj rj αU

pj αL

pj pj αU

o

(8)

Similarly, the two-level Model (8) determines the minimumachievable of the maximum efficiency score.

3.3. Mathematics of the fuzzy bargaining game DEA model

Given the bargaining game framework, the fuzzy inputs and out-puts, and two-stage structure described in the previous sections, themaximum and minimum efficiency scores achievable by each DMU canbe computed using Models (9) and (10) below, respectively.

3.3.1. The optimistic viewpointModel (9) is defined to compute the upper bound of the efficiency

score of a two-stage process with fuzzy data based on an optimistic

M. Tavana et al. Computers & Industrial Engineering 118 (2018) 394–408

397

viewpoint. That is, the DMU under assessment is in its best situationwhile the other DMUs are in their worst situation

⎜ ⎟⎜ ⎟

=

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎛⎝

− ⎞⎠

⎛⎝

− ⎞⎠

⩾

⩾

⩽ ∀ ≠

⩽

⩽ ∀ ≠

⩽

⩾ = …⩾ = …⩾ = …

⩽ ⩽ ∀ ∀

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

E

θ θ

s t θ

θ

j o

j o

v i mw d Du r s

( ) max

max

. .

1 ,

1

1 ,

1,

0, 1, , ,0, 1, ,0, 1, , .

o αU

Z Z Z j d

w z

v X

u y

w z

w z

v X

u y

w z

w z

v X

w z

v X

u y

w z

u y

w z

i

d

r

( ) ( )

·( ) min1 ·( )

min2

·( ) min1

·( )min2

·( )

·( )

·( )

·( )

dj αL dj dj α

U

dD

d do

im

i io αL

rs

r ro αU

dD

d do

dD

d do

im

i io αL

rs

r ro αU

dD

d do

dD

d dj

im

i ij αU

dD

d do

im

i io αL

rs

r rj αL

dD

d dj

rs

r ro αU

dD

d do

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

(9)

Model (9) is a two-level mathematical optimization problem. Its inneroptimization section is an input-oriented DEA bargaining model for aspecific DMUo. Note that the inner optimization model cannot uniquelyidentify the parameter Zdj since it is associated to both stages. At thesame time, the inner optimization parameter Zdj constitutes a decisionvariable for the outer optimization section. Therefore, the optimumvalue of Zdj must be determined using the outer optimization problemin Model (9).

Denote all the constraints of the inner optimization model by S,which represents the feasible set for this bargaining problem. Du et al.(2011) proved that the feasible set S is both compact and convex. Sincethe directions of optimization for the inner and outer programs inModel (9) are the same, the model can be easily transformed into thesingle stage optimization problem described in Model (10).

⎜ ⎟⎜ ⎟= ⎛⎝

− ⎞⎠

⎛⎝

− ⎞⎠

⩾

⩾

⩽ ∀ ≠

⩽

⩽ ∀ ≠

⩽

⩽ ⩽ ∀ ∀⩾ = …⩾ = …⩾ = …

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

E θ θ

s t

θ

θ

j o

j o

Z z Z j dv i mw d Du r s

( ) max

. .

,

,

1 ,

1,

1 , ,

1,

( ) ( ) , ,0, 1, ,0, 1, ,0, 1, , .

o αU w z

v X

u y

w z

w z

v X

u y

w z

w z

v X

w z

v X

u y

w z

u y

w z

dj αL

dj dj αU

i

d

r

·( ) min1 ·( )

min2

·( ) min1

·( )min2

·( )

·( )

·( )

·( )

dD

d do

im

i io αL

rs

r ro αU

dD

d do

dD

d do

im

i io αL

rs

r ro αU

dD

d do

dD

d dj

im

i ij αU

dD

d do

im

i io αL

rs

r rj αL

dD

d dj

rs

r ro αU

dD

d do

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

(10)

3.3.1.1. Linearization of the optimistic model. The previous model is anon-linear fractional mathematical programming problem. Thus, itsglobal optimum solution is hard to find. The following procedure is

proposed in order to linearize Model (10).First, multiply each term in the constraints ⩽ ⩽Z z Z( ) ( )dj α

Ldj dj α

U by

wd. Then, let = ∑ = ∑ ==−

=−t v X t w z γ( ·( ) ) , ( ) ,i

mi io α

LdD

d do i1 11

2 11

= = =t v z t w z μ t u μ t u, · , ,i dj d dj r r r r1 1 1 1 2 2 . Note that =μ t ur r1 1 and =μ t ur r2 2

imply the linear relationship =μ μrtt r1 212

between μr1 and μr2. Therefore,

replacing tt12with ⩾β 0, we have =μ βμr r1 2, r= 1,… , s. As a result,

Model (10) can be rewritten as Model (11).

∑ ∑ ∑

∑

∑

∑

∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

= × − −

+

⩾

⩾

=

=

− ⩽ ∀ ≠

− ⩽

× − ⩽ ∀ ≠

× − ⩽

⩽ ⩽ ∀ ∀>> = …

> = …

= = =

=

=

=

=

= =

= =

= =

= =

E β μ y θ μ y θ z

θ θs t

z θ

μ y θ

γ X

z β

z γ X j o

z γ X

β μ y z j o

β μ y z

t w Z z t w Z j dβγ i mμ r s

max ( ) ·( ) ·( )

·. .

,

·( ) ,

·( ) 1,

,

·( ) 0 ,

·( ) 0,

·( ) 0 ,

·( ) 0,

· ·( ) · ·( ) ,0,0, 1, , ,

0, 1, , .

o αU

r

s

r ro αU

r

s

r ro αU

d

D

do

d

D

do

r

s

r ro αU

i

m

i io αL

d

D

do

d

D

dji

m

i ij αU

d

D

doi

m

i io αL

r

s

r rj αL

d

D

dj

r

s

r ro αU

d

D

do

d dj αL

dj d dj αU

i

r

12 min

1

12 min

2

1

min1

min2

1min1

12 min

2

1

1

1 1

1 1

12

1

12

1

1 1

2 (11)

Clearly, ∑ = = ∑ ⩾= =γ X β z θ·( ) 1,im

i io αL

dD

do1 1 min1 and ∑ =d

D1

− ∑ ⩽=z γ X·( ) 0do im

i io αL

1 . Thus, we have ⩽ = ∑ =θ β dD

min1

1 ⩽ ∑ ==z γ X·( ) 1do i

mi io α

L1 .

3.3.1.2. Fuzzification of the optimistic model. In this sub-section, weintroduce the fuzzy form of Model (11) using trapezoidal fuzzy numbers(TrFNs).

Definition 3.1. A TrFN in left and right spread format is a vector of theform =∼x x x x x( , , , )1 2 3 4 , where the membership function ∼μx of ∼x is givenby:

=

⎧

⎨⎪

⎩⎪

⩽ ⩽

⩽ ⩽

⩽ ⩽

−−

−−

μ x

x x x

x x x

x x x

( )

( )

1 ( )

( )

x xx x

x xx x

1 2

2 3

3 4

12 1

44 3 (12)

The interval x x[ , ]2 3 is the mode of ∼x , and the values x1 and x4 define thelower and upper limits of ∼x , respectively.

We use TrFNs in left and right spread format to define the differentinputs, intermediate measures, and outputs of the n DMUs consideredwithin a two-stage process. In the first stage, each = …DMU j n( 1,2, , )j

utilizes m fuzzy inputs =∼x x x x x( , , , )ij ij ij ij ij1 2 3 4 , i=1, 2,… , m, to produce D

M. Tavana et al. Computers & Industrial Engineering 118 (2018) 394–408

398

fuzzy intermediate measures =∼z z z z z( , , , )dj dj dj dj dj1 2 3 4 , d= 1, 2,… , D. These

fuzzy intermediate measures constitute the inputs that will be used inthe second stage to produce s fuzzy outputs =∼y y y y y( , , , )rj rj rj rj rj

1 2 3 4 , r=1,2,… , s.

Given an arbitrary α-cut value defined for each fuzzy input, inter-mediate measure, and output, we can compute the lower and upperbound of the membership functions using Eqs. (13)–(18) (Khalili-Damghani et al., 2012).

= + − ∈ = … = …X x α x x α i m j n( ) ( ), [0,1], 1, ; 1, ,ij αL

ij ij ij1 2 1

(13)

= − − ∈ = … = …X x α x x α i m j n( ) ( ), [0,1], 1, ; 1, ,ij αU

ij ij ij4 4 3

(14)

= + − ∈ = … = …Z z α z z α d D j n( ) ( ), [0,1], 1, ; 1, ,dj αL

dj dj dj1 2 1

(15)

= − − ∈ = … = …Z z α z z α d D j n( ) ( ), [0,1], 1, ; 1, ,dj αU

dj dj dj4 4 3

(16)

= + − ∈ = … = …Y y α y y α r s j n( ) ( ), [0,1], 1, ; 1, ,rj αL

rj rj rj1 2 1

(17)

= − − ∈ = … = …Y y α y y α r s j n( ) ( ), [0,1], 1, ; 1, ,rj αU

rj rj rj4 4 3

(18)

The upper bound of the overall efficiency value for DMUo at theα-level (i.e., E( )O α

U ) can be calculated using Model (19), where the ex-pressions described in Eqs. (13)–(18) have been introduced in Model(11).

∑ ∑

∑

∑

∑

∑

∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

= × − − − − −

− +

⩾

− − ⩾

+ − =

=

− − − ⩽ ∀ ≠

− + − ⩽

× + − − ⩽ ∀ ≠

× − − − ⩽

+ − ⩽ ⩽ − − ∀ ∀

⩽ ⩽> = …

> = …

= =

=

=

=

=

=

= =

= =

= =

= =

Max E β μ y α y y θ μ y α y y

θ z θ θ

s t

z θ

μ y α y y θ

γ x α x x

z β

z γ x α x x j o

z γ x α x x

β μ y α y y z j o

β μ y α y y z

t w z α z z z t w z α z z j d

θ βγ i mμ r s

( ) ·( ( )) ·( ( ))

·

. .

,

·( ( )) ,

·( ( )) 1,

,

·( ( )) 0 , ,

·( ( )) 0,

·( ( )) 0 ,

·( ( )) 0,

( · ·( ( )) ( · ·( ( )) , , ,

1,0, 1, , ,

0, 1, , .

O αU

r

s

r ro ro ror

s

r ro ro ro

d

D

do

d

D

do

r

s

r ro ro ro

i

m

i io io io

d

D

do

d

D

dji

m

i ij ij ij

d

D

doi

m

i io io io

r

s

r rj rj rjd

D

dj

r

s

r ro ro rod

D

do

d dj dj dj dj d dj dj dj

i

r

12

4 4 3min1

12

4 4 3

min2

1min1

min2

1min1

12

4 4 3min2

1

1 2 1

1

1 1

4 4 3

1 1

1 2 1

12

1 2 1

1

12

4 4 3

1

11 2 1

14 4 3

min1

2

(19)

3.3.1.3. Linearization of the fuzzy model. Model (19) is a non-linearmathematical programming problem, whose global optimum solution ishard to find. Moreover, it is also dependent on an α-cut variable, whichreduces the computational efficiency. The following variable exchangesare introduced in order to resolve these issues.

Let =η α γ·i i, =λ α μ·r r2, and =θ α t w· ·d d1 , where ⩽ ⩽η γ0 i i,⩽ ⩽λ μ0 r r2, ⩽ ⩽θ t w0 ·d d1 and =ω t w·d d1 . Then, Eqs. (20)–(25) illus-

trate how the upper and lower bounds of the inputs, intermediatemeasures and outputs can be rewritten so as to account for any given α-cut level

∑ ∑+ − = + −= =

γ x α x x γ x η x x·( ( )) · ( )i

m

i ij ij iji

m

i ij i ij ij1

1 2 1

1

1 2 1

(20)

∑ ∑− − = − −= =

γ x α x x γ x η x x·( ( )) · ( )i

m

i ij ij iji

m

i ij i ij ij1

4 4 3

1

4 4 3

(21)

∑ ∑− − = − −= =

μ y α y y μ y λ y y·( ( )) · ( )r

s

r rj rj rjr

s

r rj r rj rj1

24 4 3

12

4 4 3

(22)

∑ ∑+ − = + −= =

μ y α y y μ y λ y y·( ( )) · ( )r

s

r rj rj rjr

s

r rj r rj rj1

21 2 1

12

1 2 1

(23)

+ − = + −t w z α z z t w z θ z z( · ·( ( )) ( · · ( ))d dj dj dj d dj d dj dj11 2 1

11 2 1

(24)

− − = − −t w z α z z t w z θ z z( · ·( ( )) ( · · ( ))d dj dj dj d dj d dj dj14 4 3

14 4 3

(25)

Replacing Eqs. (20)–(25) within Model (19) results in Model (26).

∑ ∑

∑

∑

∑

∑

∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

= × − − − + −

− +

⩾

− − ⩾

+ − =

=

− − − ⩽ ∀ ≠

− + − ⩽

× + − − ⩽ ∀ ≠

× − − − ⩽

+ − ⩽ ⩽ − − ∀ ∀⩽ ⩽⩽ ⩽⩽ ⩽

⩽ ⩽> = …

> = …

= =

=

=

=

=

=

= =

= =

= =

= =

Max E β μ y λ y y θ μ y λ y y

θ z θ θ

s t

z θ

μ y λ y y θ

γ x η x x

z β

z γ x η x x j o

z γ x η x x

β μ y λ y y z j o

β μ y λ y y z

ω z θ z z z ω z θ z z j dλ μη γθ ω

θ βγ i mμ r s

( ) ( · ( )) ( · ( ))

·

. .

( · ( ))

( · ( )) 1

( · ( )) 0 ,

( · ( )) 0

( · ( )) 0 ,

( · ( )) 0,

( · ( )) ( · ( )) ,0 ,0 ,0 ,

1,0, 1, ,

0, 1, , .

O αU

r

s

r ro r ro ror

s

r ro r ro ro

d

D

do

d

D

do

r

s

r ro r ro ro

i

m

i io i io io

d

D

do

d

D

dji

m

i ij i ij ij

d

D

doi

m

i io i io io

r

s

r rj r rj rjd

D

dj

r

s

r ro r ro rod

D

do

d dj d dj dj dj d dj d dj dj

r r

i i

d d

i

r

12

4 4 3min1

12

1 2 1

min2

1min1

min2

1min1

12

4 4 3min2

1

1 2 1

1

1 1

4 4 3

1 1

1 2 1

12

1 2 1

1

12

4 4 3

11 2 1 4 4 3

2

min1

2

(26)

Solving Model (26) we obtain the upper bound of the system effi-ciency. Model (26) is a linear programming problem whose global op-timum can be easily found using standard operations research software.Moreover, its independence of the α-cut values increases the compu-tational efficiency.

3.3.2. The pessimistic viewpointModel (27) has been designed to calculate the lower bound of the

M. Tavana et al. Computers & Industrial Engineering 118 (2018) 394–408

399

efficiency score of a two-stage process with fuzzy data from a pessi-mistic viewpoint. In this scenario, the DMU under assessment is in itsworst situation while the other DMUs are in their best situation.

⎜ ⎟⎜ ⎟

= ⩽ ⩽∀ ∀

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎛⎝

− ⎞⎠

⎛⎝

− ⎞⎠

⩾

⩾

⩽ ≠

⩽

⩽ ≠

⩽

⩾ = …⩾ = …⩾ = …

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

E z z zd j

max θ θ

s t

θ

θ

j o

j o

v i mw d Du r s

( ) min( ) ( ),

. .

1,

1

1,

1

0, 1, , ,0, 1, , ,0, 1, , .

o αL

dj αL

dj dj αU

w z

v X

u Y

w z

w z

v X

u Y

w z

w z

v X

w z

v X

u Y

w z

u Y

w z

i

d

r

·( ) min1 ·( )

min2

·( ) min1

·( )min2

·( )

·( )

·( )

·( )

dD

d do

im

i io αU

rs

r ro αL

dD

d do

dD

d do

im

i io αU

rs

r ro αL

dD

d do

dD

d dj

im

i ij αL

dD

d do

im

i io αU

rs

r rj αU

dD

d dj

rs

r ro αL

dD

d do

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

(27)

The optimization objectives of the inner and outer programs inModel (27) differ, implying that the conversion of Model (27) into asingle-level optimization program is not so straightforward as in theoptimistic setting. We make use of the duality conditions in linearprogramming to resolve this situation (Dantzig, 1963).

Note first that the inner optimization program described in Model(27) is not linear. Therefore, the inner optimization program should belinearized. We do so by considering the following decision variablechanges = ∑ =

−t v X( ·( ) ) ,im

i io αU

1 11 = ∑ =

−t w z( ) ,dD

d do2 11 =z t w zdj d dj1 , =γ t v ,i i1

=μ t u ,r r1 1 =μ t ur r2 2 when defining Model (28).

∑ ∑ ∑

∑

∑

∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

= − − +

⩾

⩾

=

− ⩽ ≠

− ⩽

− ⩽ ≠

− ⩽

> = …> = …> = …

= = =

=

=

=

= =

= =

= =

= =

μ Y θ μ Y θ z θ θ

set

z θ

μ Y θ

γ X

z γ X j o

z γ X

μ Y z j o

μ Y z

γ i mμ r sμ r s

Max Z ·( ) ·( ) ·

·( )

·( ) 1

·( ) 0,

·( ) 0,

·( ) 0,

·( ) 0,

0, 1, , ,0, 1, , ,0, 1, , .

r

s

r ro αL

r

s

r ro αL

d

D

do

d

D

do

r

s

r ro αL

i

m

i io αU

d

D

dji

m

i ij αL

d

D

doi

m

i io αU

r

s

r rj αU

d

D

dj

r

s

r ro αL

d

D

do

i

r

r

11 min

1

12 min

2

1min1

min2

1min1

12 min

2

1

1 1

1 1

11

1

11

1

1

2

(28)

After defining proper dual variables, we define the dual form ofModel (28) as Model (29).

∑

∑

= + + +

+ − ⩾ −− ⩾ ∀ ≠

+ ⩾ = …

⩾ − = …

− − ⩾ = …

⩽⩾ ∀

≠

≠

W θ Q θ Q Q θ θsetQ Q Q θQ Q j o

Y Q Y Q Y r s

Y Q θ Y r s

X Q X Q X Q i m

Q QQ Q Q Q jQ free in sign

Min ·

,0 ,

( ) ( ) ( ) , 1, ,

( ) ( ) , 1, ,

( ) ( ) ( ) 0, 1, ,

, 0, , , 0 , ,

j j

j orj α

U jro α

Lro α

L

ro αL

ro αL

io αU

j oij α

L jio α

U

j j

min1

1 min2

2 3 min1

min2

1 5 7 min2

4 6

6 7

2 min1

3 4 5

1 2

4 5 6 7

3 (29)

Theorem #1. Model (29) is always feasible independently of the valuestaken by the inputs and the outputs. Moreover, the upper bound of itsobjective function equals one.

Proof. Consider the following arbitrary solution:

⎧

⎨

⎪⎪

⎩⎪⎪

= = = == =

== = ∀=

Q Q Q Qθ θQQ Q jW

01

10 ,

1

j j

1 2 3 5

min1

min2

7

4 6

The above solution is feasible and independent of inputs, outputs, andintermediate measures. Thus, a feasible solution independent of inputs,outputs, and intermediate measures always exists for Model (29).Consequently, Model (29) is always feasible and independent ofinputs, outputs, and intermediate measures.

The minimization form of Model (29) implies that the optimumvalue of the objective function, denoted by ∗W , is not greater than thevalue of the above feasible solution. That is, ⩽∗W W . Also, since =W 1for the above arbitrary feasible solution, we have ⩽∗W 1. This com-pletes the proof. □

Corollary. By virtue of the duality theorem in linear programming, it can beconcluded from Theorem #1 that the optimal objective function of Model(29) is equal to the objective function of Model (28), i.e., =∗ ∗W Z . Hence,Model (29) can be used in place of the inner optimization program defined inModel (27).

Introducing Model (29) within the inner optimization program ofModel (27) results in the single-level optimization Model (30). Thislevel reduction is due to the fact that after the replacement both theinner and outer optimization programs in Model (27) have minimiza-tion objective functions.

The procedures required to introduce and linearize the α-cut basedvariables in Model (30) are like those described in the optimistic sce-nario. Thus, these procedures are briefly described as follows:

∑

∑

= = + + +

+ − ⩾ −− ⩾ ∀ ≠

+ ⩾ = …

⩾ − = …

− − ⩾ = …

⩽ ⩽ ∀ ∀⩽

⩾ ∀⩾ ∀

⩾

≠

≠

E θ Q θ Q Q θ θsetQ Q Q θQ Q j o

Y Q Y Q Y r s

Y Q θ Y r s

X Q X Q X Q i m

t w z z t w z j dQ Qw dQ Q jQ Q tQ free in sign

( ) Min W ·

0 , ,

( ) ( ) ( ) , 1, , ,

( ) ( ) , 1, , ,

( ) ( ) ( ) 0, 1, , ,

· ·( ) · ·( ) , , ,, 0,

0 , ,, 0 , ,, , 0,

o αL

j j

j orj α

U jro α

Lro α

L

ro αL

ro αL

io αU

j oij α

L jio α

U

d dj αL

dj d dj αU

dj j

min1

1 min2

2 3 min1

min2

1 5 7 min2

4 6

6 7

2 min1

3 4 5

1 1

1 2

4 6

5 7 1

3 (30)

M. Tavana et al. Computers & Industrial Engineering 118 (2018) 394–408

400

The lower bound of the overall efficiency values (i.e., E( )O αL) can be

calculated using Model (31), which is obtained by replacing Eqs.(13)–(18) in Model (30).

∑

∑

= = + + +

+ − ⩾ −− ⩾ ∀ ≠

− − + + − − ⩾ = …

+ − + + − ⩾ = …

− − − + − − − − ⩾

= …+ − ⩽ ⩽ − − ∀ ∀

⩽⩾ ∀

⩾ ∀⩾

≠

≠

E Min W θ Q θ Q Q θ θsetQ Q Q θQ Q j o

y α y y Q y α y y Q r s

y α y y Q θ y α y y r s

x α x x Q x α x x Q x α x x Q

i mt w z α z z z t w z α z z j dQ Qw dQ Q jQ Q tQ free in sign

( ) ·

,0 , ,

( ( )) ( ( ))( 1) 0, 1, , ,

( ( )) ( ( )) 0, 1, , ,

( ( )) ( ( )) ( ( ))

0, 1, , ,· ·( ( )) · ·( ( )) , , ,

, 0,0 , ,

, 0 , ,, , 0,

o αL

j j

j orj rj rj

jro ro ro

ro ro ro ro ro ro

io io ioj o

ij ij ijj

io io io

d dj dj dj dj d dj dj dj

dj j

min1

1 min2

2 3 min1

min2

1 5 7 min2

4 64 4 3

61 2 1

7

1 2 12 min

1 1 2 1

4 4 33

1 2 14

4 4 35

11 2 1

14 4 3

1 2

4 6

5 7 1

3

(31)

For the sake of linearization, we define =ε αQj j1 6 , = −ε α Q( 1)2 7 ,

=ε αQ3 2, =ε αθ4 min1 , =ε αQ5 3, =ε αQj j

6 4 , =ε αQ7 5, =ε αt wdd8 1 , and

=ω t w·d d1 , where ⩽ ⩽ε Q0 j j1 6 , ⩽ ⩽ −ε Q0 12 7 , ⩽ ⩽Q ε 02 3 , ⩽ ⩽ε θ0 4 min

1 ,⩽ ⩽ε Q0 5 3, ⩽ ⩽ε Q0 j j

6 4 , ⩽ ⩽ε Q0 7 5, and ⩽ ⩽ε ω0 dd8 . As a result,

Model (31) is transformed into Model (32).

∑

∑

= = + + +

+ − ⩾ −− ⩾ ∀ ≠

− − + − + − ⩾ = …

+ + − + ⩾ = …

− + − − − + − ⩾

= …+ − ⩽ ⩽ − − ∀ ∀

⩽ ⩽ − ⩽ ⩽ ⩽⩽ ⩽ ⩽

⩽⩾ ∀

⩾ ∀⩾

≠

≠

E W θ Q θ Q Q θ θsetQ Q Q θQ Q j o

Q y ε y y Q y ε y y r s

Q θ y y y ε ε r s

x Q Q x x ε ε Q x ε x x

i mω z ε z z z ω z ε z z j d

ε Q ε Q Q ε ε θ ε Qε Q ε Q ε ωQ Q εω ε dQ Q ε ε jQ Q ε ε ε εQ free in sign

( ) Min ·

,0 , ,

( ( )) ( 1) ( ) 0, 1, , ,

( ) ( )( ) 0, 1, , ,

( ) ( )( ) ( ( ))

0, 1, , ,( ) ( ) , , ,

, 1, , , ,, , ,

, , 0,, 0 , ,, , , 0 , ,, , , , , 0,

.

o αL

j j

j o

jrj

jrj rj ro ro ro

ro ro ro

io io ioj o

jij

jij ij

d djd

dj dj dj d djd

dj djj j

j j dd

dd

j j j j

min1

1 min2

2 3 min1

min2

1 5 7 min2

4 6

64

14 3

71

22 1

2 min1 1 2 1

3 4

43 5

4 37 5 4

16

2 1

18

2 1 48

4 3

1 6 2 7 2 3 4 min1

5 3

6 4 7 5 8

1 2 3

8

4 6 1 6

5 7 2 4 5 7

3 (32)

Solving Model (32) we obtain the lower bound of the system effi-ciency for the two-stage structure with fuzzy inputs, intermediatemeasures, and outputs. Model (32) is a linear programming problemwhose global optimum can be easily found using standard operationsresearch software. Moreover, its independence of the α-cut values in-creases the computational efficiency.

Table 1Comparison of efficiencies: Kao and Liu (2011) vs. current model.

DMU Main DMU Stage 1 Stage 2

DMU class Kao and Liu (2011) Proposed Model DMU class Kao and Liu (2011) Proposed Model DMU class Kao and Liu (2011) Proposed Model

EL EU EL EU EL EU EL EU EL EU EL EU

DMU1 E− 0.49 0.91 0.679 0.9 E+ 0.89 1 0.797 1 E− 0.56 0.91 0.297 0.904DMU2 E− 0.44 0.8 0.604 0.8 E+ 0.89 1 0.801 1 E− 0.5 0.8 0.27 0.795DMU3 E− 0.49 0.76 0.669 0.86 E− 0.55 0.76 0.549 0.861 E+ 0.89 1 0.448 1DMU4 E− 0.21 0.43 0.284 0.43 E− 0.64 0.8 0.572 0.903 E− 0.33 0.53 0.181 0.548DMU5 E− 0.56 0.96 0.771 1 E−, E+ 0.76 0.96 0.324 1 E+, E+

+0.74 1 1 1

DMU6 E− 0.28 0.51 0.369 0.51 E+ 0.87 1 0.773 1 E− 0.32 0.51 0.244 0.51DMU7 E− 0.2 0.38 0.26 0.38 E− 0.61 0.74 0.613 0.918 E− 0.33 0.51 0.387 0.66DMU8 E− 0.2 0.37 0.258 0.37 E− 0.6 0.73 0.593 0.885 E− 0.34 0.51 0.381 0.625DMU9 E− 0.16 0.3 0.206 0.29 E+ 0.88 1 0.917 1 E− 0.18 0.3 0.231 0.359DMU10 E− 0.34 0.64 0.449 0.64 E− 0.77 0.95 0.697 1 E− 0.44 0.67 0.466 0.831DMU11 E− 0.12 0.22 0.147 0.22 E− 0.59 0.71 0.606 0.904 E− 0.21 0.31 0.261 0.395DMU12 E− 0.55 0.95 0.773 0.94 E+ 0.91 1 0.869 1 E− 0.61 0.95 0.32 0.942DMU13 E− 0.15 0.28 0.192 0.28 E− 0.61 0.74 0.664 0.989 E− 0.25 0.38 0.441 0.66DMU14 E− 0.21 0.39 0.273 0.39 E− 0.61 0.74 0.591 0.883 E− 0.35 0.53 0.356 0.636DMU15 E− 0.45 0.8 0.598 0.79 E+ 0.91 1 0.925 1 E− 0.5 0.8 0.517 0.861DMU16 E− 0.23 0.44 0.304 0.43 E−, E+ 0.81 0.98 0.743 1 E− 0.29 0.45 0.262 0.472DMU17 E− 0.26 0.49 0.344 0.48 E− 0.58 0.69 0.59 0.883 E− 0.46 0.7 0.693 0.875DMU18 E− 0.19 0.35 0.243 0.35 E− 0.71 0.88 0.651 0.969 E− 0.27 0.4 0.189 0.458DMU19 E− 0.3 0.51 0.395 0.5 E+, E+

+0.91 1 1 1 E− 0.33 0.51 0.284 0.497

DMU20 E− 0.41 0.74 0.532 0.72 E+ 0.84 1 0.761 1 E− 0.48 0.74 0.659 0.934DMU21 E− 0.15 0.27 0.189 0.25 E− 0.66 0.81 0.621 0.903 E− 0.22 0.34 0.178 0.313DMU22 E− 0.44 0.65 0.483 0.78 E− 0.48 0.65 0.51 0.783 E+, E+

+0.91 1 1 1

DMU23 E− 0.3 0.58 0.409 0.56 E−, E+ 0.76 0.93 0.711 1 E− 0.4 0.62 0.242 0.667DMU24 E− 0.1 0.18 0.12 0.18 E− 0.41 0.47 0.393 0.493 E− 0.24 0.39 0.203 0.395

Mean 0.3015 0.54 0.398 0.54 – 0.72 0.86 0.678 0.932 – 0.42 0.62 0.396 0.681Standard Deviation 0.1426 0.24 0.199 0.25 – 0.15 0.15 0.163 0.112 – 0.2 0.23 0.232 0.226

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Table 2Comparison of results: ANOVA and Confidence Interval analysis.

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3.3.3. Proper breakdown pointsThe main objective of the proposed fuzzy two-stage DEA bar-

gaining-game models is to find the best achievable efficiency throughnegotiation, not the best overall efficiency score or the best solution.Therefore, choosing (0, 0) as a breakdown point would lead to the bestoverall efficiency score, but not necessarily to the best achievable ef-ficiency for either stage 1 or 2. Also, (0, 0) constitutes a breakdownpoint when the two stages do not negotiate with each other and obtainan efficiency score of zero. Thus, (0, 0) is not a good candidate for abreakdown point in the proposed procedure.

3.3.4. Ranking of DMUs based on interval efficiency scoresThe uncertainty inherent to inputs and outputs has been modeled

using fuzzy numbers. As a result, each DMU has been assigned a giveninterval efficiency score. In this regard, the equations

=E E E( ) ( ) /( )oL

jL

oU1 2 and =E E E( ) ( ) /( )o

Uj

Uo

L1 2 represent the lower andupper bound of the minimum achievable efficiency for the first sub-DMU, respectively. The lower and upper bound of the minimumachievable efficiency for the second sub-DMU can therefore be calcu-lated as =E E E( ) ( ) /( )o

Lj

Lo

U2 1 and =E E E( ) ( ) /( )oU

jU

oL2 1 , respectively. The

corresponding interval efficiency scores for each DMU and sub-DMUcan be categorized through Eqs. (33) and (34) as follows

For each DMU:

= ∈ == ∈ < == ∈ <

++

+ ∗

− ∗

E j J EE j J E EE j J E

{ |( ) 1},{ |( ) 1 and ( ) 1},{ |( ) 1}.

jL

jL

j

j (33)

For each Sub-DMU:

= ∈ = == ∈ < = == ∈ < =

++

+

−

E j J E kE j J E E kE j J E k

{ |( ) 1}, 1,2,{ |( ) 1 and 1}, 1,2,{ | 1}, 1,2.

ok L

ok L

ok

ok (34)

where J represents the set of DMUs endowed with a cardinality of n(i.e., =j n| | ), such that =∗E Max E E( ) (( ) ,( ) )j j

Lj

U and=E Max E E([ ] ,[ ] )o

kok L

ok U with =k 1,2.

3.3.5. Theoretical and managerial significance of the proposed modelIn this sub-section, the theoretical and managerial significance of

the model defined in the current paper is discussed by comparing itsperformance to that of other approaches proposed in the two-stage DEAliterature. A fundamental feature of two-stage DEA models is efficiencydecomposition, that is, how the models decompose the total efficiencyof the main DMU into the efficiency scores of the first and secondstages. Indeed, efficiency decomposition plays an essential role inpractical two-stage DEA models. Managers aim at disaggregating totalefficiency into the sub-efficiency score of each stage so as to identifystrengths and weaknesses within the two-stage process. However, effi-ciency decomposition is subject to a major practical constraint.

The seminal two-stage DEA model defined by Kao and Hwang(2008) tackled the uniqueness problem of efficiency decomposition asfollows. If the linear problem used to calculate the total efficiency scoreof the main DMU has alternative optimal solutions, then there existseveral combinations of the scores obtained for the different stagesdelivering the same total efficiency. Although Kao and Hwang (2008)identified this problem and proposed a test to check whether or notthere existed alternative optimal solutions, their model could notidentify them. Chen et al. (2009) introduced additive efficiency de-composition into two-stage DEA models. They showed that their ad-ditive decomposition, but not the multiplicative one proposed by Kaoand Hwang (2008), could be extended to variable returns to scale set-tings. However, they were unable to guarantee the uniqueness of effi-ciency decomposition. Wang and Chin (2010) compared both previousdecomposition models and the weighted harmonic one. Once again, nosolution for the uniqueness problem of efficiency decomposition wasproposed.

More in general, cooperative two-stage DEA models that focus onthe total efficiency score of the main DMU, including the three just

Fig. 2. Network efficiency model for the bank’sbranch network.

Fig. 3. Linguistic variables and their associated TrFNs.

Table 3Linguistic variables and their associated TrFNs.

Linguistic variable TrFN scale

Extreme low (EL) (0.0, 0.1, 0.2, 0.3)Very low (VL) (0.1, 0.2, 0.3, 0.4)Low (L) (0.2, 0.3, 0.4, 0.5)Medium low (ML) (0.3, 0.4, 0.5, 0.6)Medium (M) (0.4, 0.5, 0.6, 0.7)Medium high (MH) (0.5, 0.6, 0.7, 0.8)High (H) (0.6, 0.7, 0.8, 0.9)Very high (VH) (0.7, 0.8, 0.9, 1)Extreme high (EH) (0.8, 0.9, 1, 1)

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described, are subject to the uniqueness of efficiency decompositionproblem. On the other hand, non-cooperative (i.e., leader-follower andStackelberg) two-stage DEA settings rarely face this problem. In addi-tion, the uniqueness problem worsens when uncertainties in inputs andoutputs are introduced into cooperative two-stage DEA environments.For instance, Kao and Liu (2011) extended the model proposed by Kaoand Hwang (2008) to allow for fuzzy inputs and outputs. The inclusionof fuzzy sets and α-cut levels complicated considerably the uniquenessof efficiency decomposition problem, which remained unsolved by theauthors. Moreover, the model proposed by Kao and Liu (2011) depends

on the α-cut parameter and must therefore be solved for several α-cutvalues, which requires a considerable computational effort.

The fuzzy two-stage DEA model introduced in this paper has beendesigned using a bargaining game approach, which allows us to tacklethe problem of uniqueness of efficiency decomposition. Our modeldecomposes the total efficiency score of the main DMU into those ofeach sub-DMU, guaranteeing that there is no alternative combination ofthe efficiency scores through the different stages. This uniquenessproperty allows managers and decision makers to identify the mainsources of inefficiency in each stage, which helps them to design sui-table improvement plans for the inefficient DMUs and sub-DMUs.Finally, our model determines the optimal value of the α-cut level forthe main DMUs and their corresponding first and second stages, de-creasing considerably the computational efforts required.

3.3.6. Comparative numerical exampleWe compare now the current model with that of Kao and Liu (2011)

using a numerical example adapted from the latter authors so as toaccount for the dependence of their model on the value assigned to theα-cut level. Table 1 presents the lower and upper bounds of the effi-ciency scores for the current model and that of Kao and Liu (2011).Several relevant conclusions can be inferred when comparing both setsof results:

• First, the optimal α-cut value for the upper bound of the main DMUsas well as their corresponding first and second stages is equal tozero, while the optimal lower bound equals one. Thus, these α-cutlevels have been used when implementing the model of Kao and Liu(2011) so as to favor a fair comparison between both approaches.

• Second, Eqs. (33) and (34) have been applied to determine the ef-ficiency class of the main DMUs as well as their corresponding firstand second stages. As illustrated in Table 1, the classes assigned tothe DMUs and sub-DMUs by both models are identical in most of thecases. The existing differences have been highlighted in bold redcharacters.

• Third, the main descriptive statistics of the efficiency scores for allDMUs and sub-DMUs, including their means and standard devia-tions, display a very similar behavior across models.

A formal comparison between our results and those of Kao and Liu(2011) requires evaluating whether or not there exists a meaningfulstatistical difference between the mean efficiency scores assigned to theDMUs and sub-DMUs by both models. To this end, we have performedan analysis of variance (ANOVA) test and a confidence interval (CI)analysis. Given the fact that we are considering the lower and upperefficiency scores of the main DMU and its corresponding first andsecond stages, six ANOVA tests and CI analyses have been performed.Table 2 displays the results obtained using a 0.95 confidence level.

As illustrated in Table 2, all p-values are bigger than 0.05, meaningthat there is no statistical evidence to reject the null hypothesis. That is,this result implies that there is no meaningful statistical difference be-tween the performance of the current model and that of Kao and Liu(2011). The results derived from the CI analysis also validate the sta-tistical equivalence of both methods. However, the model proposed inthis study should be preferred at the managerial level given the lowercomputational effort required, its independence from the α-cut valuesand its capacity to uniquely decompose the total efficiency score of themain DMUs into those of the sub-DMUs.

4. Case study: efficiency measurement of the Saman Bank

A considerable amount of attention has been focused on the effi-ciency analysis of bank branch performance due to the complexity andrelative importance of this industry. Sherman and Gold (1985) pub-lished the first paper considering the application of DEA to a given setof (US) bank branches. Since then, many other papers have applied DEA

Table 4DMUs' aggregated opinions.

Branch DMU X1 X2 X3 Z1 Y1 Y2 Y3

Shiraz 1 VH ML ML EH ML H VHMarvdasht 2 H MH M H M ML MLShariati-Shiraz 3 M ML L MH H ML HKazeroun 4 M ML M ML ML L LFasa 5 MH M ML VH MH M MModarres-Shiraz 6 H M L H ML ML MHJahrom 7 M M M M ML L LAbadeh 8 ML ML ML ML ML ML VLFirouzabad 9 ML M ML M MH ML MLTabriz 10 M ML L L L L HEqlid 11 M M ML M ML ML LLaar 12 L ML L L ML L MSadi-Shiraz 13 ML L ML ML L L HDarab 14 M L ML M M ML MPalestine-Shiraz 15 ML L ML MH MH M MDaneshjoo-Shiraz 16 ML H L ML L L MFarhangshahr-Shiraz 17 M ML ML ML VL ML MHBahonar-shiraz 18 M ML VH MH ML L MHEstahban 19 ML ML VH L MH MH LNourabad Mamasani 20 M M VH ML M ML MMoalem-Mrvdasht 21 M VH M L M M MLRahmatabad-Shiraz 22 ML VH L VH H M HSibuye Blvd. 23 M VL ML ML L ML MHEram Blvd. Shiraz 24 L VH VL M ML MH MMotahari-Shiraz 25 M VH L VH ML M MHParseh 26 L MH H ML M ML MLPezeshkan-Shiraz 27 L L VH ML ML ML LNasr Blvd. Shiraz 28 ML ML ML M MH ML MLEdalat Blvd. Shiraz 29 ML ML ML ML ML M MPasargad 30 L VH H VL M MH MQaem-Shiraz 31 ML ML L H M M MAmirkabir-Shiraz 32 ML L ML ML MH L MLQuar 33 ML ML ML ML M M HKazeroon Gate-Shiraz 34 L L L ML MH L MEnghelab-Marvdasht 35 MH M M H ML L LQaani No-Shiraz 36 L L ML L ML L MKarim Khan Zand-Shiraz 37 MH ML ML VH ML MH HTeimouri-Shiraz 38 M M MH M VL L MHMirzaye Shirazi 39 M M ML VH MH H VLHedayat-Shiraz 40 L H ML L L VL MHHazarati-Kazeroun 41 L ML ML MH MH M MLLamerd 42 ML ML L ML M M LHang-Shiraz 43 L L L H H VH MPasdaran-Shiraz 44 M ML L MH ML ML HSourian-Bavant 45 L M H VL ML VL VLArsanjan 46 L ML ML L VH M MSepidan 47 L L ML L MH M MLTakhti-Shiraz 48 L L ML L ML M MImam Khomeini Bazaar-Shiraz 49 ML ML VL M M M VHImam Khomeini-Firouzabad 50 L H MH L L ML VLImam Khomeini-Abadeh 51 L MH ML L ML L LKhoram Bid 52 L M H VL M L MLKhonj 53 L M L ML ML MH MHakim-Shiraz 54 ML ML L H ML M HAretesh Sevom-Shiraz 55 M L L M L ML MH30 Meters Sadi Cinema 56 ML ML L MH MH ML MLSatarkhan-Shiraz 57 ML ML L H MH MH HQirokarzin 58 L ML L VL L ML LGolestan Town 59 L H VL ML M ML M22 Bahman-Jahrom 60 L H L VL L L VL

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to measure the efficiency of the banking industry in different countries:Vassiloglou and Giokas (1990) in Greece, Paradi, Rouatt, and Zhu(2011) in Canada, Zhou, Sun, Yang, Liu, and Ma (2013) in China, Oraland Yolalan (1990) in Turkey, Yang and Liu (2012) in Taiwan, Lovelland Pastor (1997) in Spain, Golany and Storbeck (1999) in the US, and

Ebrahimi, Garami, and Mozafari (2014) in Iran, among many others.Also, several papers have studied the efficiency of bank branches

using fuzzy DEA. Wu, Yang, and Liang (2006) applied fuzzy DEA toanalyze the efficiency of 808 Canadian cross-region bank branches.They proposed an alternative methodology to build a BCC model

Table 5Interval efficiency scores and classification of the DMUs and sub-DMUs.

DMU Class eU eL −Sub DMU1 class eU eL −Sub DMU2 class eU eL

DMU1 −E 0.9837 0.8911 +E 1 0.6719 −E 0.8834 0.5591DMU2 −E 0.9850 0.8801 +E 1 0.5795 +E 1 0.3406DMU3 −E 0.9850 0.8911 +E 1 0.5649 −E 0.8666 0.3806DMU4 −E 0.9861 0.8487 +E 1 0.4054 +E 1 0.2533DMU5 −E 0.9843 0.8664 +E 1 0.6436 −E 0.8929 0.4638DMU6 −E 0.9847 0.8801 +E 1 0.5841 −E 0.8446 0.4369DMU7 −E 0.9861 0.8487 +E 1 0.4762 +E 1 0.2755DMU8 −E 0.9874 0.8251 +E 1 0.4167 +E 1 0.1643DMU9 −E 0.9851 0.8487 +E 1 0.4416 +E 1 0.3277DMU10 −E 0.9861 0.8487 +E 1 0.2970 +E 1 0.2187DMU11 −E 0.9861 0.8487 +E 1 0.5172 +E 1 0.2363DMU12 −E 0.9861 0.8251 +E 1 0.3333 +E 1 0.1920DMU13 −E 0.9861 0.8251 +E 1 0.4292 +E 1 0.2000DMU14 −E 0.9851 0.8487 +E 1 0.4344 +E 1 0.4842DMU15 −E 0.9844 0.8251 +E 1 0.4522 −E 0.9896 0.2883DMU16 −E 0.9861 0.8801 +E 1 0.2857 +E 1 0.2624DMU17 −E 0.9866 0.8487 +E 1 0.4054 +E 1 0.2218DMU18 −E 0.9843 0.8911 +E 1 0.6772 −E 0.8456 0.4916DMU19 −E 0.9840 0.8911 +E 1 0.6452 −E 0.8893 0.4874DMU20 −E 0.9844 0.8911 +E 1 0.6665 −E 0.9163 0.4441DMU21 −E 0.9851 0.8911 +E 1 0.2325 +E 1 0.4990DMU22 −E 0.9839 0.8911 +E 1 0.6378 +E 1 0.8451DMU23 −E 0.9861 0.8487 +E 1 0.5667 −E 0.6586 0.3255DMU24 −E 0.9851 0.8911 +E 1 0.6004 −E 0.7690 0.3821DMU25 −E 0.9843 0.8911 +E 1 0.6585 −E 0.8009 0.5245DMU26 −E 0.9851 0.8801 +E 1 0.4252 −E 0.9688 0.2581DMU27 −E 0.9861 0.8911 +E 1 0.4540 −E 0.8994 0.2278DMU28 −E 0.9851 0.8251 +E 1 0.5306 −E 0.9582 0.2999DMU29 −E 0.9851 0.8251 +E 1 0.4167 +E 1 0.3000DMU30 −E 0.9844 0.8911 −E 0.8257 0.2342 +E 1 0.6363DMU31 −E 0.9844 0.8251 +E 1 0.5897 −E 0.8598 0.4522DMU32 −E 0.9861 0.8251 +E 1 0.4578 −E 0.9614 0.2100DMU33 −E 0.9844 0.8251 +E 1 0.4167 +E 1 0.3429DMU34 −E 0.9861 0.7921 +E 1 0.5302 −E 0.9233 0.2207DMU35 −E 0.8751 0.8663 +E 1 0.5714 −E 0.8889 0.2593DMU36 −E 0.9861 0.8251 +E 1 0.3333 +E 1 0.1920DMU37 −E 0.9841 0.8663 +E 1 0.6658 −E 0.8796 0.4836DMU38 −E 0.9857 0.8663 +E 1 0.6020 −E 0.9620 0.2900DMU39 −E 0.9841 0.8487 +E 1 0.6589 −E 0.8803 0.4890DMU40 −E 0.9876 0.8801 +E 1 0.3010 +E 1 0.1522DMU41 −E 0.9851 0.8251 +E 1 0.6034 −E 0.8166 0.3980DMU42 −E 0.9861 0.8251 V 1 0.4379 −E 0.9952 0.2012DMU43 −E 0.9843 0.7921 +E 1 0.4788 −E 0.8963 0.4341DMU44 −E 0.9851 0.8487 +E 1 0.6696 −E 0.8490 0.3806DMU45 −E 0.9876 0.8801 −E 0.6466 0.1048 +E 1 0.2209DMU46 −E 0.9844 0.8251 +E 1 0.3438 +E 1 0.5745DMU47 −E 0.9851 0.8251 +E 1 0.3333 +E 1 0.2400DMU48 −E 0.9851 0.8251 −E 0.7471 0.4225 +E 1 0.6203DMU49 −E 0.9844 0.8251 +E 1 0.5809 −E 0.7898 0.4134DMU50 −E 0.9876 0.8801 +E 1 0.3333 +E 1 0.1200DMU51 −E 0.9861 0.8663 +E 1 0.3333 +E 1 0.1920DMU52 −E 0.9861 0.8801 +E 1 0.1265 +E 1 0.3656DMU53 −E 0.9851 0.8487 +E 1 0.5573 −E 0.8268 0.3024DMU54 −E 0.9846 0.8251 +E 1 0.5969 −E 0.8474 0.4434DMU55 −E 0.9859 0.8487 +E 1 0.5649 −E 0.8303 0.3055DMU56 −E 0.9851 0.8251 +E 1 0.6114 −E 0.8205 0.3959DMU57 −E 0.9838 0.8251 +E 1 0.5964 −E 0.8591 0.4890DMU58 −E 0.9861 0.8251 −E 0.8458 0.2129 +E 1 0.5193DMU59 −E 0.9851 0.8801 +E 1 0.5293 −E 0.8674 0.2882DMU60 −E 0.9876 0.8801 +E 1 0.2506 +E 1 0.3449

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designed to deal with evaluations across different systems. Yang and Liu(2012) surveyed managerial efficiency in the Taiwanese banking in-dustry using two-stage fuzzy DEA. To do so, they integrated a two-stageseries performance model and a fuzzy multi objective one. Similarly,Wang, Lu, and Liu (2014) constructed a fuzzy multi-objective two-stageDEA model to evaluate the performance of US bank holding companies.Finally, Ebrahimi et al. (2014) evaluated and ranked the branches of theSaman Bank in Iran using fuzzy DEA.

The efficiency of a bank branch can be assumed to be a measure ofthe level of success in the conversion of costs from the productivitystage into incomes at the profitability stage. In practice, production andintermediation activities should be integrated rather than considered asindividual activities when evaluating banking efficiency. Based on theexisting complementarities between production and intermediationactivities, this study uses the two-stage process proposed by Denizer,

Dinc, and Tarimcilar (2007) to estimate the overall performance of abank branch, as illustrated in the conceptual framework of Fig. 2.

The existing relations between the productivity and profitabilitystages in the provision of efficiency motivated us to build the proposedfuzzy two-stage game-DEA (FTSGDEA) framework.

4.1. Description of the case study, measurement scales, and data gathering

Saman Bank is one of the largest privately-owned Iranian bankswith over 150 branches. In this section, the proposed FTSGDEA modelwill be applied to assess the efficiency of sixty Saman Bank branches. Itshould be emphasized that each branch has been an independent DMUconsisting of two sequential stages.

(a) Upper bound of the efficiency scores for the main DMUs

(b) Lower bound of the efficiency scores for the main DMUs

Fig. 4. The efficiency scores for the main DMUs.

Fig. 5. Dispersion of the lower and upper bounds of the efficiency scores.

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4.1.1. Measurement scalesLinguistic terms parameterized using TrFNs have been used as

measurement indices for the inputs, intermediate measures and outputsof the branches composing the bank’s two-stage network. Fig. 3 illus-trates the membership functions associated to the respective TrFNs,while Table 3 describes the linguistic terms considered together withtheir corresponding TrFNs.

4.1.2. Data collectionA comprehensive study was conducted to gather the information

required from the set of 60 branches of the Saman bank being analyzed.The raw data consisted of documents retrieved from each of the bran-ches together with the data used by Ebrahimi et al. (2014) in theirstudy. In addition to these, supplementary information was acquiredthrough the personal experiences of the managers from the selectedbranches. Managers rated the different factors determining efficiencyusing the linguistic terms described in Table 1. The opinions of themanagers together with the additional data retrieved were used toobtain the values of the indices describing the two stages of eachbranch, which have been summarized in Table 4.

4.2. Experimental results

We start by setting θmin1 and θmin

2 equal to 0.01. After running theproposed FTSGDEA models, i.e., Models (26) and (32), we obtain aunique interval efficiency score for each DMU. The efficiencies dis-played by each DMU and its associated sub-DMUs are presented inTable 5. This information allows us to identify the sub-processes andDMUs exhibiting a weak efficiency, and can be used to improve theoverall performance of the different bank branches.

Fig. 4 shows the upper and lower bounds of the efficiency scoresobtained for the different DMUs. The robustness of these efficiencyscores is validated by plotting their standard deviations and ranges,both of which are illustrated in Fig. 5.

As can be concluded from Fig. 5, the ranges and standard deviationsof the efficiency scores obtained for each DMU are relatively small,which emphasizes the robustness of the DMUs analyzed.

5. Conclusions

We have designed a FTSGDEA model to assess the relative intervalefficiency score of DMUs both overall, as well as for the segmentscomposing a two-stage process. A bargaining game framework has beenbuilt for a two-stage process DMU with uncertain inputs, intermediatemeasures, and outputs. Linguistic terms parameterized through fuzzysets have been used to account for the uncertainty inherent to inputs,intermediate measures and outputs. Finally, optimistic and pessimisticscenarios have been considered within the corresponding multi-leveloptimization problems.

The main contributions of the approach proposed in this study re-lative to those existing in the literature can be summarized as follows:1) Our model accounts for fuzzy inputs, intermediate measures andoutputs within a bargaining game approach to a two-stage DEA process;2) The fuzzy DEA models defined are independent of the α-cut vari-ables, which reduces considerably the computation efforts; 3) Theproposed models have been shown to have feasible solutions that areindependent of the values of inputs, outputs, and intermediate mea-sures.

The proposed FTSGDEA model has been applied to assess the effi-ciency levels of 60 Saman Bank branches by considering their pro-ductivity and profitability stages. The required computations werestraightforward and the results quite promising. Thus, our formal ap-proach can be properly applied in different areas of management andengineering, wherein a bargaining process can be assumed to take placebetween two players and the evaluation criteria are uncertain. An im-mediate extension of the FTSGDEA model should focus on analyzing the

sensitivity and stability of the efficiencies obtained.

Acknowledgement

The authors would like to thank the anonymous reviewers and theeditor for their insightful comments and suggestions.

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