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Applied Soft Computing 38 (2016) 676–702

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l ho me page: www.elsev ier .com/ locate /asoc

comprehensive fuzzy DEA model for emerging market assessmentnd selection decisions

aveh Khalili-Damghania, Madjid Tavanab,c,∗, Francisco J. Santos-Arteagad

Department of Industrial Engineering, South-Tehran Branch, Islamic Azad University, Tehran, IranBusiness Systems and Analytics Department, Distinguished Chair of Business Analytics, La Salle University, Philadelphia, PA 19141, USABusiness Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, D-33098 Paderborn, GermanyDepartamento de Economía Aplicada II, Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Pozuelo, Spain

r t i c l e i n f o

rticle history:eceived 10 September 2014eceived in revised form 17 August 2015ccepted 23 September 2015vailable online 22 October 2015

eywords:uzzy data envelopment analysismerging marketsreference assessmentndesirable input–outputissing valueimension reduction

a b s t r a c t

The changing economic conditions have challenged many financial institutions to search for more effi-cient and effective ways to assess emerging markets. Data envelopment analysis (DEA) is a widely usedmathematical programming technique that compares the inputs and outputs of a set of homogenousdecision making units (DMUs) by evaluating their relative efficiency. In the conventional DEA model,all the data are known precisely or given as crisp values. However, the observed values of the inputand output data in real-world problems are sometimes imprecise or vague. In addition, performancemeasurement in the conventional DEA method is based on the assumption that inputs should be min-imized and outputs should be maximized. However, there are circumstances in real-world problemswhere some input variables should be maximized and/or some output variables should be minimized.Moreover, real-world problems often involve high-dimensional data with missing values. In this paperwe present a comprehensive fuzzy DEA framework for solving performance evaluation problems withcoexisting desirable input and undesirable output data in the presence of simultaneous input–outputprojection. The proposed framework is designed to handle high-dimensional data and missing values. Adimension-reduction method is used to improve the discrimination power of the DEA model and a pref-erence ratio (PR) method is used to rank the interval efficiency scores in the resulting fuzzy environment.A real-life pilot study is presented to demonstrate the applicability of the proposed model and exhibitthe efficacy of the procedures and algorithms in assessing emerging markets for international banking.

© 2015 Elsevier B.V. All rights reserved.

. Introduction

The intensity of global competition has forced banks to expand their global operations in emerging markets and develop growingetworks of physical branches and subsidiaries in foreign countries. Multinational banks enter emerging markets largely to increase theirrofitability within an acceptable risk profile. Despite a large body of literature on the role of foreign banks in emerging economies, theubject of selecting the most suitable emerging market has received very little attention in the literature and remains a difficult task. Thisifficulty is due to multiple and often conflicting factors that add to the inherent technical complexities and valuation uncertainties involved

n the assessment process. As such, a comprehensive and systematic decision making framework is needed to guide the assessment process,hape the decision outcomes and enable confident choices to be made.

Multinational banks and emerging economies share common incentives regarding the development of the financial market of the hostountry. Banks try to increment their profits by exploiting their competitive advantage in technological and knowledge management termsithin the host financial system. At the same time, emerging markets aim at developing and stabilizing their banking and financial systems

ia competition and spillover effects.

∗ Corresponding author at: Business Systems and Analytics Department, Distinguished Chair of Business Analytics, La Salle University, Philadelphia, PA 19141, USA.el.: +1 215 951 1129; fax: +1 267 295 2854.

E-mail addresses: [email protected] (K. Khalili-Damghani), [email protected] (M. Tavana), [email protected] (F.J. Santos-Arteaga).URLs: http://kaveh-khalili.webs.com (K. Khalili-Damghani), http://tavana.us/ (M. Tavana).

ttp://dx.doi.org/10.1016/j.asoc.2015.09.048568-4946/© 2015 Elsevier B.V. All rights reserved.

dx.doi.org/10.1016/j.asoc.2015.09.048

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dx.doi.org/10.1016/j.asoc.2015.09.048

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K. Khalili-Damghani et al. / Applied Soft Computing 38 (2016) 676–702 677

Clearly, foreign banks have imperfect information regarding the state of the host banking system, which due to its developing qualitys inherently unstable and subject to different types of government and institutional controls. As a result, banks must consider the mainharacteristics of the host country and its banking sector together with the consequences derived from their exposure to the internationalnancial market. Both the characteristic and the consequences comprise desirable and undesirable elements that foreign banks mustimultaneously account for. Moreover, the resulting consequences determine the subsequent policies implemented by the governmentnd the financial structure of the country.

For example, when a country opens its financial system to the international market, the exposure and risk faced by its banks increases1,2].

At the same time, as the financial system of an emerging country develops, so does the scope and range of its financial services, i.e., loansand deposits, among the local population [1].However, the decrease in activity restrictions required from local governments together with the increase in competition and efficiencyof the financial system do not necessarily improve its stability. That is, countries must be macroeconomically stable before opening theirfinancial systems [1].As a result, the relationship between competition and stability would differ among countries depending on multiple hard to quantifyfactors such as their market structure together with their regulatory and institutional environments [3].

These facts add a strategic dimension to the country selection decision of foreign banks. Indeed, foreign banks must consider the coexis-ence of both desirable and undesirable market conditions (inputs) and also expect both desirable and undesirable outputs determined byhe evolution of the host financial and banking systems [4]. Thus, multinational banks must perform a careful analysis of the local bankingystem, its regulatory restrictions and institutional conditions before deciding whether or not to entry a country. Despite the obviousmportance of the emerging market evaluation and selection problem, our knowledge about how this process takes place remains ratherimited [5]. In this regard, most of the research in economics and business has focused on how the firm should select a country based onts financial and institutional environments.

This evaluation process represents a particularly difficult task given the subjectivity displayed in the evaluation of many of the factorshat banks must account for. Such a problem prevails even when evaluators design their surveys so as to avoid it. For example, consider theank Regulation and Supervision Survey performed by the World Bank [6]. It is acknowledged that while quantitative variables comprise aubset of the data available, qualitative ones subject to different value judgments on the side of the respondent constitute also an importantart of the information available. Banks have access to this information together with several other sources subject to the same type ofonstraint. Additional examples are given by the database on bank regulation compiled by Barth et al. [7], or the study of Cihak et al. [8],here fundamental questions such as “What was the impact of moving to Basel II on the overall regulatory capital level of the banking

ystem?” (Fig. 2, p. 27) must be categorically addressed and analyzed.Thus, a formal ranking evaluation model implementable by managers when selecting a country must satisfy the following requirements.

Given the substantial amount of highly diverse information available, it must allow managers to eliminate any redundancy existing inthe data. Econometricians have access to collinearity tests to deal with highly correlated variables. An equivalent mechanism simplifyingthe posterior data envelopment analysis (DEA) implementation is required.The DEA section of the model must account for positive and negative factors comprising the evaluation of the different alternatives. Inthis regard, econometrics allows for both positive and negative factors influencing a given dependent variable.Crisp evaluation factors are generally analyzed alongside fuzzy ones. The coexistence of both types of factors requires models that allowfor the analysis of fuzzy variables based on the subjective evaluations of different information sources.Finally, any evaluation model must deliver a ranking of the available alternatives. That is, the (fuzzy) efficiency scores obtained afterapplying the fuzzy DEA method must be ranked, concluding the corresponding evaluation process.

We propose a comprehensive and structured method for emerging market assessment and selection which is grounded in DEA. A fuzzyersion of DEA is used to capture the ambiguity and vagueness associated with real-world performance measurement problems. At theame time, coexisting desirable input and undesirable output data are considered in the presence of simultaneous input–output projectionn DEA. A dimension-reduction method is used to improve the discrimination power of DEA, while a preference ratio (PR) method ismplemented to rank the interval efficiency scores of different emerging market alternatives in a fuzzy environment. It should be notedhat the non-parametric model introduced in this paper complements and competes with the parametric econometric models commonlysed in the international business literature.

The remainder of this paper is organized as follows. In Section 2 we review the relevant literature on conventional and fuzzy DEA.n Section 3 the comprehensive DEA model proposed in this study is introduced. In Section 4 a real-world pilot study is presented toemonstrate the applicability of the proposed model and exhibit the efficacy of the procedures and algorithms in assessing emergingarkets for international banking. In Section 5 we provide managerial implications and describe how to extend our approach to similar

nvironments within the international business literature. Section 6 presents our conclusions and suggests future research directions.

. Literature review

In this section we review the relevant literature on conventional DEA models with precise input–output data, fuzzy DEA models withmprecise input–output data, DEA models with desirable inputs and/or undesirable outputs, and fuzzy ranking methods.

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78 K. Khalili-Damghani et al. / Applied Soft Computing 38 (2016) 676–702

.1. Conventional DEA models with precise input–output data

DEA is a widely used mathematical programming technique that was originally developed by Charnes et al. [9] and extended by Bankert al. [10] to include variable returns to scale. DEA generalizes the Farrell [11] single-input single-output technical efficiency measureo the multiple-input multiple-output case in order to evaluate the relative efficiency of peer units with respect to multiple performance

easures [12,13]. The units under evaluation in DEA are called decision making units (DMUs). A DMU is considered efficient when no otherMUs can produce more outputs using an equal or lesser amount of inputs. The DEA method generalizes the usual efficiency measurement

rom a single-input single-output ratio to a multiple-input multiple-output ratio by using a ratio of the weighted sum of outputs to theeighted sum of inputs [14]. Unlike parametric methods which require detailed knowledge of the process, DEA is non-parametric and doesot require an explicit functional form relating inputs and outputs (see Cooper et al. [14] and Cook and Seiford [15] for an appraisal of theheoretical foundations and developments in DEA). Numerous applications in recent years have been accompanied by new extensions andevelopments in expanding the concept and methodology of DEA (see Seiford [16] and Emrouznejad et al. [17] for an extensive bibliographyf DEA).

.2. Fuzzy DEA models with imprecise input–output data

The basic DEA methods, known as Charnes, Cooper, and Rhodes (CCR) and Banker, Charnes, and Cooper (BCC), require accurate mea-urement of both the inputs and outputs. However, the observed values of the input and output data in real-world problems are sometimesmprecise or vague. Imprecise evaluations may be the result of unquantifiable, incomplete and non-obtainable information. Fuzzy logicnd fuzzy sets are widely used to represent ambiguous, uncertain or imprecise data in DEA by formalizing the inaccuracies inherent inuman decision-making.

Fuzzy set algebra developed by Zadeh [18] is the formal theory that allows for the treatment of imprecise estimates in uncertain envi-onments. Sengupta [19] proposed a fuzzy mathematical programming approach by incorporating fuzzy input and output data into a DEAodel and defining tolerance levels for the objective function and constraint violations. Triantis and Girod [20] proposed a mathematical

rogramming approach by transforming the fuzziness into a DEA model using membership function values. Kao and Liu [21], León et al.22] and Lertworasirikul et al. [23] proposed three similar fuzzy DEA models that implement uncertainty in fuzzy objectives and fuzzyonstraints using the possibility approach. Lertworasirikul et al. [24] proposed a fuzzy DEA model based on the credibility approach, whereuzzy variables were replaced by expected credits according to credibility measures. Lertworasirikul et al. [25] further extended the fuzzyEA through the possibility and credibility approaches.

Kao and Liu [26] transformed fuzzy input and output data into intervals by using �-level sets. Guo and Tanaka [27] used the �-levelets and changed the fuzzy DEA model into a bi-level linear programming model. Kao and Liu [28] presented a method to rank the fuzzyfficiency scores without knowing the exact form of the membership functions. In their procedure, the efficiency rankings were determinedy solving a pair of non-linear programs for each DMU. Guo and Tanaka [29] used the fuzzy DEA model proposed by Guo and Tanaka [27] to

ntroduce a fuzzy aggregation framework for integrating multiple attribute fuzzy values. Furthermore, Guo [30] used the model proposedy Guo and Tanaka [27,29] in a case study for a restaurant location problem in China. The �-level set approach was extended by Saati et al.31], who defined the fuzzy DEA model as a possibilistic-programming problem and transformed it into an interval programming problem.ntani et al. [32] extended the �-level set research by changing fuzzy input and output data into intervals. Dia [33] proposed a fuzzy DEAodel where a fuzzy aspiration level and a safety �-level were used to transform the fuzzy DEA model into a crisp DEA one. Wang et al.

34] also used the �-level set approach to change fuzzy data into intervals.Hougaard [35] proposed a simple approximation procedure for the assessment of productivity scores with respect to fuzzy production

lans. This procedure has a clear economic interpretation and all the necessary calculations can be performed in a spreadsheet making itighly operational. Liu [36] developed a fuzzy DEA model to find the efficiency measures associated with the assurance region concept.e applied the �-level approach and Zadeh’s extension principle to transform the fuzzy DEA model with assurance region into a pairf parametric mathematical programs and work out the lower and upper bounds of the efficiency scores of the DMUs. The membershipunction of efficiency was approximated by using different possibility levels. Jahanshahloo et al. [37] commented on the fuzzy DEA modelroposed by Liu [36] and corrected the proof of his theorem. Liu and Chuang [38] further used the fuzzy DEA model with assurance regionuggested by Liu [36] to evaluate the performance of 24 university libraries in Taiwan. Wang et al. [39] proposed two fuzzy DEA modelsith fuzzy inputs and outputs by means of fuzzy arithmetic. They converted each proposed fuzzy model into three linear programmingodels in order to calculate the efficiencies of the DMUs as fuzzy numbers and rank them.Jahanshahloo et al. [40] defined a methodology for assessing, ranking and imposing weight restrictions in DEA problems with fuzzy

nput–output data. They proposed a slack-based DEA model in a fuzzy environment. Saati and Memariani [41] designed a procedure foreducing weight flexibility in fuzzy DEA. They found a common set of weights across the set of DMUs and assessed upper bounds on theactor weights. They also aggregated the resulting intervals in order to determine a common set of weights. The efficiencies resultingrom their procedure were fuzzy numbers rather than crisp values so it gave more information about the case. Karsak [42] introduced

DEA model with crisp, ordinal and fuzzy data to select advanced manufacturing technologies. Hatami-Marbinia et al. [43] proposedn interactive evaluation process for measuring the relative efficiencies of a set of DMUs in fuzzy DEA taking in consideration the DMs’references. They constructed a linear programming model with fuzzy parameters and calculated the fuzzy efficiency of the DMUs forifferent � levels. Finally, Azadi et al. [44] developed an integrated DEA-enhanced Russell measure model to evaluate and select the bestustainable suppliers within a fuzzy environment.

In general, fuzzy DEA methods can be classified into four primary categories, namely, the tolerance approach [19], the �-level basedpproach [28,31,45], the fuzzy ranking approach [27] and the possibility approach [23]. An exhaustive review and taxonomy of variousuzzy DEA models in the literature can be found in Hatami-Marbini et al. [46].

.2.1. Fuzzy DEA applied to banking evaluationIn this section, we review several papers that employ fuzzy DEA methods to rank different DMUs related to banking systems. This

pplication constitutes a recent extension of the DEA literature and must be developed further to account for the specific requirements of

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ifferent real-world scenarios. For example, Wu et al. [47] were among the first authors to use fuzzy DEA for efficiency analysis in a bankingystem. They introduced fuzzy logic into the DEA formulation to deal with the environmental variables of a banking setting composed byifferent cross-region branches. They also compared their results with the results from traditional DEA analysis.

While describing the models introduced by the recent literature on the topic, we emphasize the specific improvements we implementn our model.

Puri and Prasad Yadav [48] defined a fuzzy DEA model with undesirable outputs to measure the performance of Indian public sectoranks. In their model, undesirable outputs could result from the production process and needed to be minimized. The authors implemented

cross-efficiency technique to rank the efficient DMUs at every �-value in the interval (0, 1]. The computational efforts required to accountor the different �-value choices are generally quite high. Thus, it would be preferable to define a version of the fuzzy DEA model whoseolutions are independent of the �-value considered.

Wang et al. [49] implemented a common scale within a two-stage DEA model to analyze the performance of several bank holdingompanies. They compared their results using a truncated-regression model and found a positive relationship between intellectual capitalnd the performance of these companies. Their paper emphasized the comparability between the parametric regression approach and theon-parametric DEA one, a relationship that will also be emphasized in the current paper.

Tsolas and Charles [50] measured the efficiency of the Greek banking industry using a satisficing DEA model. They approximated thenancial risk taken by the banks through the credit risk provisions and the participation of banks in private sector involvement. These

actors are specific to the risk taking behavior of banks and the authors did not consider several other types of inputs and outputs thatetermine the efficiency of banks. A wider approach, accounting for a larger set of (opposing) factors, is generally required to evaluate theerformance of countries.

The subject of market risk and its specific measurement has also been considered by Chen et al. [51]. These authors applied a modelf fuzzy slack-based measurement to estimate the efficiency scores and value regions, as well as the managerial achievements, of Taiwananking under market risk. Another non-radial fuzzy DEA model was introduced by Puri and Prasad Yadav [52], who defined a noveloncept of fuzzy input mix-efficiency and evaluated the resulting efficiency of the DMUs using an �-cut approach. These authors alsoefined a fuzzy correlation coefficient between the fuzzy inputs and outputs, and a method for ranking the DMUs on the basis of theiruzzy input mix-efficiency. The ranking process designed by these authors highlights the fact that the fuzzy DEA model must be part of aarger evaluation structure. Such a structure should also account for the existing correlations among the different evaluation factors androvide a ranking method.

Finally, several drawbacks associated with the fuzzy DEA method have been emphasized in the literature [53], including the �-cutpproaches, the type of fuzzy numbers considered and the ranking techniques applied. The model defined in this paper accounts for allhese drawbacks. In particular, and similarly to Tavana et al. [54,55], we define a fuzzy linearized DEA model that is independent of the-cut variables considered. This fuzzy DEA model is integrated within a larger evaluation structure that accounts for the existing correlationmong inputs and outputs and provides a fuzzy ranking method.

.3. DEA models with desirable inputs and/or undesirable outputs

In the standard DEA models, the performance of a DMU is improved by decreasing inputs and/or increasing outputs. However, therere circumstances in real-world problems where the performance of a DMU improves as input(s) increase and/or output(s) decrease. Theesirable inputs (also known as “good” inputs) and undesirable outputs (also known as “bad” outputs) have been specifically modeled inhe DEA literature distinctively [56,57]. In order to improve efficiency, the undesirable outputs should be reduced and the desirable inputshould be increased.

Furthermore, various transformation techniques have been proposed in the literature for dealing with desirable inputs and undesirableutputs [58]. Lu and Lo [59] have classified the alternatives for dealing with undesirable outputs in the DEA framework as follows. Therst alternative is to simply ignore the undesirable outputs. The second alternative is either to treat the undesirable outputs in terms of aon-linear DEA model or to treat them as outputs and adjust the distance measurement in order to restrict their expansion [60]. The thirdlternative is either to treat the undesirable outputs as inputs or to apply a monotone decreasing transformation [61]. Seiford and Zhu56] have suggested a different method where they multiply the undesirable outputs by (−1) and then use a translation vector to turn theegative undesirable outputs into positive desirable ones. Färe and Grosskopf [62] proposed an alternative approach to Seiford and Zhu’s56] method by adopting a directional distance function to estimate the DMUs’ efficiencies based on weak disposability of undesirableutputs.

The recent applications of DEA models with desirable inputs and/or undesirable outputs can be found in various industries, includingairy farms [63,64], electric utilities [65,66], agriculture [67], paper mills [56,68], cement manufacturing [69,70], aquaculture [58], airports71,72], petroleum [73], health care [74] and commercial banks [75] among others. There is no unique model in the literature for handlingEA problems with coexisting desirable inputs and undesirable outputs. In this paper we present a comprehensive fuzzy DEA framework

or solving performance evaluation problems with coexisting desirable input and undesirable output data.

.4. Fuzzy ranking methods

Fuzzy set theory has been applied to many ranking problems in fuzzy environments. In order to rank a group of fuzzy numbers, oneumber has to be evaluated and compared with the other ones in the group. However, this is not an easy task since fuzzy numbers areepresented by possibility distributions and they can overlap with each other. As a result of this overlap, it is often difficult to determinelearly whether one number is smaller or larger than another one. A commonly used method for ranking fuzzy numbers is the centroid-basedethod [76–79].
Lee et al. [80] developed a method for ranking sequences of fuzzy values that assigns a preference degree to each ranked sequence.
hen and Chen [81] proposed a method for ranking generalized trapezoidal fuzzy numbers based on their spreads and defuzzified values.hen and Chen [82] introduced a method for ranking generalized trapezoidal fuzzy numbers with different heights and spreads. Chennd Wang [83] proposed a fuzzy ranking approach based on the �-cuts, the belief feature and the signal/noise ratios. They calculated the

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ignal/noise ratio of each �-cut to evaluate the quantity and the quality of a fuzzy number. Abbasbandy and Hajjari [84] introduced aethod for ranking trapezoidal fuzzy numbers based on the left and right spreads at some �-levels. Other ranking methods have been

eveloped based on defuzzification [85]; coefficient of variation index [86]; fuzzy simulation analysis [87]; domain separation [88]; areaethod using the radius of gyration [89]; left and right deviation degree [90] and positive and negative ideal points [91]. Asady and

endehnam [92] developed a fuzzy ranking method using distance minimization. Abbasbandy and Asady [93] proposed a modification ofhe distance-based approach called the sign distance. Asady [94] introduced a revised method based on the left and right deviation degree

ethod proposed by Wang et al. [90].Modarres and Sadi-Nezhad [95] proposed a preference ratio (PR) method that was capable of calculating the overall ranking of the fuzzy

umbers point-by-point and at each point. Modarres and Sadi-Nezhad [96] used the PR method to develop a new fuzzy simple additiveeighting method. Sadi-Nezhad and Ghaleh-Assadi [97] applied the PR method to a fuzzy flow shop scheduling problem. Sadi-Nezhad andhalili-Damghani [98] modified the earlier PR method proposed by Modarres and Sadi-Nezhad [95] and used the modified PR method tossess the performance of traffic police centers. Several other successful recent applications of the PR method have been reported in theEA literature [54,99,100]. In this study we use the PR method proposed by Modarres and Sadi-Nezhad [95] to rank the interval efficiency

cores in the fuzzy environment.

. Proposed framework

The framework proposed in this study is composed of three modules as depicted in Fig. 1.

.1. Module 1: DEA modeling

Module 1 is composed of the following two distinct steps of extended and fuzzy DEA modeling.

.1.1. DEA modeling with both desirable inputs and undesirable outputsConsider n DMUs. Each DMU uses m, i = 1, 2, . . ., m, different positive inputs to produce s, r = 1, 2, . . ., s, different positive outputs.

et Xgj

= (xg1j

, xg2j

, . . ., xgm1j

) and Xbj

= (xbm1+1j

, xbm1+2j

, . . ., xbmj

) be vectors of desirable and undesirable inputs, respectively. We have j such

ectors, where each one is associated with a particular DMU and xgij

(which is a component of Xgj

) represents the amount of desirable

nput i consumed by DMUj. The same type of description applies to the undesirable input case. Similarly, let Ygj

= (yg1j

, yg2j

, . . ., ygs1j

) andbj

= (ybs1+1j

, ybs1+2j

, . . ., ybsj

) be vectors of desirable and undesirable outputs, respectively. The decision variables �j represent the weights of

he DMUs under assessment. The decision variable �o represents the relative technical efficiency of DMUo, where the subscript o refers tohe DMU under assessment. The standard input-oriented DEA model with variable returns to scale [10] is extended to account for desirablend undesirable inputs and outputs as follows:

Min �o = �o − ϕo − ε

(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �oxgio

=n∑

j=1

�jxgij

+ s−i

, i = 1, . . ., m1

1�o

× xbio =

n∑j=1

�jxbij + s−

i, i = m1 + 1, . . ., m

ϕoygro =

n∑j=1

�jygrj

− s+r , r = 1, . . ., s1

1ϕo

× ybro =

n∑j=1

�jybrj − s+

r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

�j, s−i

, s+r ≥ 0, ∀j, i, r

(1)

Model (1) expands desirable outputs and contracts undesirable outputs. It also contracts desirable inputs and expands undesirablenputs. Unfortunately, Model (1) is nonlinear, so its global optimum is hard to find. Using the translation invariance in the BCC model [101],erived from Seiford and Zhu’s [56] and Lewis and Sexton’s [57] approaches, we multiply each undesirable input/output by “−1” and thennd proper translation values wi, i = m1 + 1, . . ., m and pr, r = s1 + 1, . . ., s that allow to transform all the negative undesirable input/outputalues into positive ones as follows:

x′b = −xb + wi > 0, i = m1 + 1, . . ., m
ij ij
y′brj = −yb

rj+ pr > 0, r = s1 + 1, . . ., s

(2)

here wi = max{xbij

+ 1}, i = m1 + 1, . . ., m and pr = max{ybrj

+ 1}, r = s1 + 1, . . ., s.

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By [2], Model (1) becomes the following linear programming problem:


(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �oxgio

=n∑

j=1

�jxgij

+ s−i

, i = 1, . . ., m1

�ox′bio =

n∑j=1

�jx′bij + s−

i, i = m1 + 1, . . ., m

ϕoygro =

n∑j=1

�jygrj

− s+r , r = 1, . . ., s1

ϕoy′bro =

n∑j=1

�jy′brj − s+

r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

�j, s−i

, s+r ≥ 0, ∀j, i, r

(3)

Model (3) can handle both desirable and undesirable inputs/outputs coexisting in a unique formulation. Model (3) expands desirableutputs and undesirable inputs and contracts desirable inputs and undesirable outputs, concurrently. Letting the optimal solution of Model3) be represented by (�∗

o, �∗o, ϕ∗

o, s∗−i

, s∗+r , �∗

j), we have:

efinition 3.1. A DMUo is said to be extreme-efficient using Model (3) if and only if:

a) �∗o = 0

b) �∗o = ϕ∗

o = 1.c) s∗−

i= 0, i = 1, . . ., m; s∗+

r = 0, r = 1, . . ., s.

efinition 3.2. The projection (efficiency-image) of an inefficient DMUo on the efficient frontier is calculated as follows:

xgio

= �oxgio

− s−i

=n∑

j=1

�jxgij, i = 1, . . ., m1

x′bio = �ox′b

io − s−i

=n∑

j=1

�jx′bij, i = m1 + 1, . . ., m

ygro = ϕoyg

ro + s+r =

n∑j=1

�jygrj

, r = 1, . . ., s1

y′bro = ϕoy′b

ro + s+r =

n∑j=1

�jy′brj, r = s1 + 1, . . ., s

(4)

efinition 3.3. The inverse of the reverse scoring transformation for computing the target value of the undesirable inputs-outputs isbtained as follows:

xbio

= −x′bio + wi > 0, i = m1 + 1, . . ., m

ybro = −y′b

ro + pr > 0, r = s1 + 1, . . ., s

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a


.1.2. Fuzzy DEA modeling with both desirable inputs and undesirable outputsThe basic fuzzy form of Model (3) is given by Model (5). Note that the parameters of Model (5) have the same definitions as the

arameters in Model (3), where the symbol ∼ indicates the fuzziness of the associated parameter.


(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �oxgio

=n∑

j=1

�jxgij

+ s−i

, i = 1, . . ., m1

�ox′bio =

n∑j=1

�jx′bij + s−

i, i = m1 + 1, . . ., m

�oygro =

n∑j=1

�jygrj

− s+r , r = 1, . . ., s1

�oy′bro =

n∑j=1

�jy′brj − s+

r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

�j, s−i

, s+r ≥ 0, ∀j, i, r

(5)

Without loss of generality, we consider Trapezoidal Fuzzy Numbers (TrFNs) as inputs and outputs. Let xgij

= (x1gij

, x2gij

, x3gij

, x4gij

) and xbij

=x1b

ij, x2b

ij, x3b

ij, x4b

ij) be TrFNs in the left and right spread format representing the ith good and bad inputs of the jth DMU, respectively. Also let

˜grj

= (y1grj

, y2grj

, y3grj

, y4grj

) and ybrj

= (y1brj

, y2brj

, y3brj

, y4brj

) be TrFNs in the left and right spread format representing the rth good and bad outputsf the jth DMU, respectively.

The lower and upper bounds of the membership functions of these TrFNs can be calculated for an arbitrary �-cut level for each inputnd output. Thus, the following pair of models can be used to calculate the upper and lower bounds of the efficiency values:

Min �Lo = �o − ϕo − ε

(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �oxUgio

=n∑

j = 1

j /= o

�jxLgij

+ �oxUgio

+ s−i

, i = 1, . . ., m1

�ox′Lbio =

n∑j = 1

j /= o

�jx′Ubij + �ox′Lb

io + s−i

, i = m1 + 1, . . ., m

ϕoyLgro =

n∑j = 1

j /= o

�jyUgrj

+ �oyLgro − s+

r , r = 1, . . ., s1

ϕoy′Ubro =

n∑j = 1

j /= o

�jy′Lbrj + �oy′Ub

ro − s+r , r = s1 + 1, . . ., s

(6)

n∑j=1

�j = 1

�j, s−i

, s+r ≥ 0, ∀j, i, r

Udp

3

da


Min �Uo = �o − ϕo − ε

(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �oxLgio

=n∑

j = 1

j /= o

�jxUgij

+ �oxLgio

+ s−i

i = 1, . . ., m1

�ox′Ubio =

n∑j = 1

j /= o

�jx′Lbij + �ox′Ub

io + s−i

, i = m1 + 1, . . ., m

ϕoyUgro =

n∑j = 1

j /= o

�jyLgrj

+ �oyUgro − s+

r , r = 1, . . ., s1

ϕoy′Lbro =

n∑j = 1

j /= o

�jy′Ubrj + �oy′Lb

ro − s+r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

�j, s−i

, s+r ≥ 0, ∀j, i, r

(7)

Models (6) and (7) should be solved for several �-cut levels in order to achieve the lower and upper bound of the efficiency scores.nfortunately, this standard procedure for �-cut approaches is time consuming and may result in conflicting ranks for a given DMU forifferent �-cut levels. Moreover, Models (6) and (7) are non-linear and it is hard to find their global optimum solutions. Therefore, weropose a procedure to achieve the crisp-linear equivalent of these models.

.1.3. Crisp linear equivalent of the proposed DEA modelsLet us consider ˛i = ˛, i = 1, . . ., m for all inputs and define � j = ˛�j, where 0 ≤ � j ≤ �j. Assume also that ˛r = ˇ, r = 1, . . ., s for all outputs and

efine j = ˇ�j, where 0 ≤ j ≤ �j. Then, the upper and lower bounds of the inputs and outputs for an arbitrary �-cut level can be denoteds follows:

n∑j=1

�j(xLgij

)˛

=n∑

j=1

�j[x1gij

+ ˛(x2gij

− x1gij

)] =n∑

j=1

[�jx1gij

+ �j(x2gij

− x1gij

)], i = 1, . . ., m1

n∑j=1

�j(xLbij )

˛=

n∑j=1

�j[x1bij + ˛(x2b

ij − x1bij )] =

n∑j=1

[�jx1bij + �j(x

2bij − x1b

ij )], i = m1 + 1, . . ., m

n∑j=1

�j(xUgij

)˛

=n∑

j=1

�j[x4gij

− ˛(x4gij

− x3gij

)] =n∑

j=1

[�jx4gij

− �j(x4gij

− x3gij

)], i = 1, . . ., m1

n∑j=1

�j(xUbij )

˛=

n∑j=1

�j[x4bij − ˛(x4b

ij − x3bij )] =

n∑j=1

[�jx4bij − �j(x

4bij − x3b

ij )], i = m1 + 1, . . ., m

n∑j=1

�j(yLgrj

)ˇ

=n∑

j=1

�j[y1grj

+ ˇ(y2grj

− y1grj

)] =n∑

j=1

[�jy1grj

+ j(y2grj

− y1grj

)], r = 1, . . ., s1

n∑j=1

�j(yLbrj )

ˇ=

n∑j=1

�j[y1brj + ˇ(y2b

rj − y1brj )] =

n∑j=1

[�jy1brj + j(y

2brj − y1b

rj )], r = s1 + 1, . . ., s

n∑�j(y

Ugrj

)ˇ

=n∑

�j[y4grj

− ˇ(y4grj

− y3grj

)] =n∑

[�jy4grj

− j(y4grj

− y3grj

)], r = 1, . . ., s1

(8)

j=1 j=1 j=1n∑

j=1

�j(yUbrj )

ˇ=

n∑j=1

�j[y4brj − ˇ(y4b

rj − y3brj )] =

n∑j=1

[�jy4brj − j(y

4brj − y3b

rj )], r = s1 + 1, . . ., s


Fig. 1. The proposed framework.

M
Note that, these conversions are essential for linearizing Models (6) and (7). Using the equation set [8], Models (6) and (7) become
odels (9) and (10), respectively:


(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �o[x4gio

− ˛(x4gio

− x3gio

)] =n∑

j = 1

j /= o

[�jx1gij

+ �j(x2gij

− x1gij

)] + [�ox4gio

− �o(x4gio

− x3gio

)] + s−i

, i = 1, . . ., m1

�o[x′1bio + ˛(x′2b

io − x′1bio )] =

n∑j = 1

j /= o

[�jx′4bij − �j(x′4b

ij − x′3bij )] + [�ox′1b

io + �o(x′2bio − x′1b

io )] + s−i

, i = m1 + 1, . . ., m

�o[y1gro + ˇ(y2g

ro − y1gro )] =

n∑j = 1

j /= o

[�jy4grj

− j(y4grj

− y3grj

)] + [�oy1gro + o(y2g

ro − y1gro )] − s+

r , r = 1, . . ., s1

�o[y′4bro − ˇ(y′4b

ro − y′3bro )] =

n∑j = 1

j /= o

[�jy′1brj + j(y′2b

rj − y′1brj )] + [�oy′4b

ro − o(y′4bro − y′3b

ro )] − s+r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

s−i

≥ 0, i = 1, . . ., m

s+r ≥ 0, r = 1, . . ., s

�j ≥ 0, j = 1, . . ., n

�j ≥ 0 j = 1, . . ., n

j ≥ 0, j = 1, . . ., n

�j − �j ≥ 0, j = 1, . . ., n

�j − j ≥ 0, j = 1, . . ., n

(9)



(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �o[x1gio

+ ˛(x2gio

− x1gio

)] =n∑

j = 1

j /= o

[�jx4gij

− �j(x4gij

− x3gij

)] + [�ox1gio

+ �o(x2gio

− x1gio

)] + s−i

, i = 1, . . ., m1

�o[x′4bio − ˛(x′4b

io − x′3bio )] =

n∑j = 1

j /= o

[�jx′1bij + �j(x′2b

ij − x′1bij )] + [�ox′4b

io − �o(x′4bio − x′3b

io )] + s−i

, i = m1 + 1, . . ., m

�o[y4gro − ˇ(y4g

ro − y3gro )] =

n∑j = 1

j /= o

[�jy1grj

+ j(y2grj

− y1grj

)] + [�oy4gro − o(y4g

ro − y3gro )] − s+

r , r = 1, . . ., s1

�o[y′1bro + ˇ(y′2b

ro − y′1bro )] =

n∑j = 1

j /= o

[�jy′4brj − j(y′4b

rj − y′3brj )] + [�oy′1b

ro + o(y′2bro − y′1b

ro )] − s+r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

s−i

≥ 0, i = 1, . . ., m

s+r ≥ 0, r = 1, . . ., s

�j ≥ 0, j = 1, . . ., n

�j ≥ 0, j = 1, . . ., n

j ≥ 0, j = 1, . . ., n

�j − �j ≥ 0, j = 1, . . ., n

�j − j ≥ 0, j = 1, . . ., n

(10)

6

m(


Models (9) and (10) are nonlinear due to the terms ˛�o and ˇϕo. Consequently, some variable conversions are required to obtain linearodels. We define ıo = ˛�o, where 0 ≤ ıo ≤ �o and o = ˇϕo, where 0 ≤ o ≤ ϕo. The associated final linear models are represented by Models

11) and (12).


(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �ox4gio

− ıo(x4gio

− x3gio

) =n∑

j = 1

j /= o

[�jx1gij

+ �j(x2gij

− x1gij

)] + [�ox4gio

− �o(x4gio

− x3gio

)] + s−i

, i = 1, . . ., m1

�ox′1bio + ıo(x′2b

io − x′1bio ) =

n∑j = 1

j /= o

[�jx′4bij − �j(x′4b

ij − x′3bij )] + [�ox′1b

io + �o(x′2bio − x′1b

io )] + s−i

, i = m1 + 1, . . ., m

�oy1gro + o(y2g

ro − y1gro ) =

n∑j = 1

j /= o

[�jy4grj

− j(y4grj

− y3grj

)] + [�oy1gro + o(y2g

ro − y1gro )] − s+

r , r = 1, . . ., s1

�oy′4bro − o(y′4b

ro − y′3bro ) =

n∑j = 1

j /= o

[�jy′1brj + j(y′2b

rj − y′1brj )] + [�oy′4b

ro − o(y′4bro − y′3b

ro )] − s+r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

s−i

≥ 0, i = 1, . . ., m

s+r ≥ 0, r = 1, . . ., s

�j ≥ 0, j = 1, . . ., n

�j ≥ 0, j = 1, . . ., n

j ≥ 0, j = 1, . . ., n

�j − �j ≥ 0, j = 1, . . ., n

�j − j ≥ 0, j = 1, . . ., n

ıj ≥ 0, j = 1, . . ., n

j ≥ 0, j = 1, . . ., n

(11)

�j − ıj ≥ 0, j = 1, . . ., n

�j − j ≥ 0, j = 1, . . ., n

3

aip



(m∑

i=1

s−i

+s∑

r=1

s+r

)

s.t. �ox1gio

+ ıo(x2gio

− x1gio

) =n∑

j = 1

j /= o

[�jx4gij

− �j(x4gij

− x3gij

)] + [�ox1gio

+ �o(x2gio

− x1gio

)] + s−i

, i = 1, . . ., m1

�ox′4bio − ıo(x′4b

io − x′3bio ) =

n∑j = 1

j /= o

[�jx′1bij + �j(x′2b

ij − x′1bij )] + [�ox′4b

io − �o(x′4bio − x′3b

io )] + s−i

, i = m1 + 1, . . ., m

ϕoy4gro − o(y4g

ro − y3gro ) =

n∑j = 1

j /= o

[�jy1grj

+ j(y2grj

− y1grj

)] + [�oy4gro − o(y4g

ro − y3gro )] − s+

r , r = 1, . . ., s1

ϕoy′1bro + o(y′2b

ro − y′1bro ) =

n∑j = 1

j /= o

[�jy′4brj − j(y′4b

rj − y′3brj )] + [�oy′1b

ro + o(y′2bro − y′1b

ro )] − s+r , r = s1 + 1, . . ., s

n∑j=1

�j = 1

s−i

≥ 0, i = 1, . . ., m

s+r ≥ 0, r = 1, . . ., s

�j ≥ 0, j = 1, . . ., n

�j ≥ 0, j = 1, . . ., n

j ≥ 0, j = 1, . . ., n

�j − �j ≥ 0, j = 1, . . ., n

�j − j ≥ 0, j = 1, . . ., n

ıj ≥ 0, j = 1, . . ., n

j ≥ 0, j = 1, . . ., n

�j − ıj ≥ 0, j = 1, . . ., n

ϕj − j ≥ 0, j = 1, . . ., n

(12)

.1.4. Theoretical properties of the proposed DEA modelsThe formal contribution of the model requires guaranteeing that it delivers a full ranking evaluation procedure that extends the ones

vailable in the literature and provides a setting directly applicable by managers. This implies that the DEA model must be feasible andts objective function bounded so an efficiency score can be obtained for each DMU before proceeding to the next stage of the evaluationrocess. The current subsection verifies both these facts, i.e., it describes the theoretical properties of Models (11) and (12).

6

T

P

McvT

T

P

3

•

•

•

•

3

ao

m

nn

wa

wo


heorem 1. Model (11) is always feasible and its objective function value is bounded.

roof. Consider an arbitrary solution for Model (11) given by:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

�Lo = 0

�o = ϕo = 1

s−i

= 0, i = 1, . . ., m

s+r = 0, r = 1, . . ., s

�j = 0, j = 1, . . ., n, j /= o

�o = 1

�j = 0, j = 1, . . ., n

j = 0, j = 1, . . ., n

ıj = 0, j = 1, . . ., n

j = 0, j = 1, . . ., n

(13)

It follows that for independent values of the inputs and outputs, solution [13] is always a feasible solution for Model (11). Therefore,odel (11) always has at least one feasible solution such as [13]. Since the objective function value for [13] is equal to zero (i.e., �L

o = 0),onsidering the minimization form of the objective function of the linear programming Model (11), it can be concluded that the optimumalue of the objective function of Model (11) is always less than or equal to zero (i.e., �∗L

o ≤ �Lo = 0). Hence, Model (11) is always bounded.

his completes the proof.�

heorem 2. Model (12) is always feasible and its objective function value is bounded.

roof. Given the proof of Theorem 1, the result is obvious.�

.1.5. Managerial, computational, and practical advantages of the proposed DEA modelsModels (11) and (12) have several attractive features described below:

Models (11) and (12) can handle both desirable and undesirable inputs and outputs in a unique formulation. This unique formulation iswell-posed and suitable for analyzing complicated real-life problems with coexisting desirable and undesirable inputs and outputs.Models (11) and (12) can account for the uncertainties inherent to real-life data using fuzzy sets. This provides more flexibility forhandling real-life problems.Models (11) and (12) are linear and independent from the �-cut concept. Hence, there is no need to solve the problem for different �-cutlevels. The model will calculate the best value for the �-cut variables. This will significantly decrease the computational effort. That is,given n DMUs, this treatment of the �-cut concept decreases the number of linear programs needed to solve the fuzzy DEA problem to2 × n. Moreover, there is no need to determine the step-size of the �-cut levels heuristically, which is usually considered to be a practicalproblem in the fuzzy DEA modeling based on the �-cut approach.Solving these models leads to a unique interval efficiency score (i.e., optimistic and pessimistic efficiency values) for each DMU. Thisprevents the occurrence of conflicting efficiency scores, which is a common problem in fuzzy DEA modeling based on the �-cut approach.In addition, the post-screening analysis of interval efficiency scores will be simplified using the proposed modeling approach.

.2. Module 2: Preference ratio modeling

In Module 2, we consider the efficiency intervals of the DMUs determined by the fuzzy DEA model presented in the previous section as special case of TrFN and use an efficient fuzzy ranking method to rank them. The fuzzy ranking method used in this module is developedn the basis of the PR concept.

Consider the problem of ranking I fuzzy numbers. Let Ni be the ith fuzzy number defined over a real domain Si ⊂ R and identified by aembership function (Ni

(x), x ∈ Si), with Ni(x) ∈ [0, 1].

Let Si be the support of the ith fuzzy number Ni (i.e., Si = {x, Ni(x) > 0}) and be defined as the union of the support of all fuzzy

umbers (i.e., = ∪Ii=1Si). In other words, fuzzy numbers are ranked over ˝. In order to rank fuzzy numbers, we assume their spans are

ot disjoint, because the ranking of fuzzy numbers with disjoint spans is evident.A fuzzy number is evaluated by a function called the preference function. At each point ∈ ˝, this function is defined as follows:

G(˛)

∫ U

˛(x)dx∫ U

L(x)dx

(14)

here (x) is the membership function of the fuzzy number, L = min {x : x ∈ ˝} and U = max {x : x ∈ ˝}. This function has the same definitions 1 − F(˛) in probability theory, where F(˛) = P[X ≤ ˛] is the distribution function.

For ∈ ˝, let p(˛) = i denote the ith fuzzy number that is the most preferred one. Therefore:

p(˛) = i, if Gi(˛) = max{Gj(˛), j ∈ I}.here Gj(˛) is the preference function of the jth fuzzy number. Let, ˝i be the set of points for which the ith fuzzy number is ranked number

ne. That is, ˝i = ( ∈ ˝, p(˛) = i}.

Dp

wd

D

(

3

dfdbtwmc

3

cTdt

3

oA(

t

(


efinition 3.2. The PR of the ith fuzzy number p(˛) = i, denoted by R(i) is the percentage of for which the ith fuzzy number is the mostreferred one. That is:

R(i) =∣∣˝i

∣∣˝

(15)

here |˝i| and |˝| are the lengths of the real sets ˝i and ˝, respectively. Modarres and Sadi-Nezhad [96] developed an algorithm foretermining the PR in the TrFN case. In order to make the PR of two fuzzy numbers A and B equivalent, they defined the following concept:

efinition 3.3.

(a) Two fuzzy numbers A and B are said to be “PR equivalent” if R(A) = R(B) = 0.5, where R(A) and R(B) are the PR of A and B, respectively.The PR equivalence is denoted as follows:

APR≡B (16)

b) If k × APR≡B, then k is said to be the “equivalence multiplier” of A with respect to B.

The PR method used in Module 2 is coded using Visual Basic 6.0 software.

.3. Dimension-reduction procedure for DMUs with fuzzy data

If the number of inputs/outputs is high with respect to the number of DMUs, the discrimination power of a DEA model significantlyecreases. That is, many DMUs may be recognized as efficient. In order to overcome this problem, we develop a variable reduction procedureor DMUs with fuzzy data based on the one introduced by Amirteimoori et al. [102]. These authors proposed a procedure to reduce theimension of crisp data in DEA models based on mathematical programming. Amirteimoori et al. [102] used linear correlation coefficientsetween criteria to determine the highly correlated criteria. Afterwards, they developed a pair of mathematical models to determinehe weighted sum of highly correlated inputs/outputs. They suggested an algorithmic procedure in which the dimension of the dataas decreased iteratively. They replaced highly correlated inputs/outputs with the weighted sum of correlated criteria in a way that theinimum perturbations occurred for the relative preference of DMUs. Unfortunately, the original procedure by Amirteimoori et al. [102]

annot be utilized with fuzzy data. Therefore, we need to reformulate their procedure in order to handle fuzzy data.

.3.1. Fuzzy correlation coefficientFirst, the correlation coefficient for fuzzy data must be calculated. In order to do so, we customize the fuzzy measures for the correlation

oefficient of fuzzy numbers proposed by Liu and Kao [103]. This allows us to calculate the fuzzy coefficient values of the inputs and outputs.hen, we reformulate the procedure introduced by Amirteimoori et al. [102] for fuzzy data. The highly correlated inputs and outputs areetermined based on the calculated fuzzy correlation coefficients. These highly correlated dimensions are then aggregated in a way thathe minimum perturbation occurs. This procedure is implemented so as to increase the discrimination power of the fuzzy DEA approach.

.3.2. Inputs/outputs dimension reduction in fuzzy DEA modelsLet us recall the Amirteimoori et al. [102] model:

Min x(i,t)o = vixio + vtxto

s.t. vixij + vtxtj ≥ 1, j = 1, . . ., n,

vi, vt ≥ ε.

(17)

Model (17) determines the weighted sum of two highly correlated inputs such that the minimum potential perturbation is imposedn the efficiency scores of the DMUs. Then, the highly correlated variables can be substituted into the objective function of Model (17).mirteimoori et al. [102] proposed Model (18) to calculate the weighted sum of two highly correlated outputs. The performance of Model

18) for outputs is the same as that of Model (17) for inputs.

Max y(i,t)o = uiyio + utyto

s.t. uiyij + utytj ≤ 1, j = 1, . . ., n,

ui, ut ≥ ε.

(18)

Models (17) and (18) can be used for highly correlated input/outputs variables or even for non-correlated variables. However, they have
he following pitfalls that pertain to our particular setting:
(a) They cannot consider undesirable inputs and outputs.b) They cannot handle fuzzy inputs and outputs.

6

Wv

D

vT

c

c


We develop a set of models based on those proposed by Amirteimoori et al. [102] to address and resolve the aforementioned pitfalls.hen undesirable inputs and outputs coexist in a problem, the transformation defined in [2] can be used to account for the undesirable

ariables in Models (17) and (18). For fuzzy inputs/outputs we have:

Min x(i,t)o = vixio + vt xto

s.t. vixij + vt xtj ≥ 1, j = 1, . . ., n,

vi, vt ≥ ε.

(19)

Max y(i,t)o = uiyio + utyto

s.t. uiyij + utytj ≤ 1, j = 1, . . ., n,

ui, ut ≥ ε.

(20)

The �-cut approach is used to model fuzzy mathematical programming involving TrFNs. Considering the pessimistic situation for aMU. The lower bound of the weighted sum of the correlated inputs i and t can be calculated using Model (21) below:

Min [x(i,t)o ]

L = vi(x1io

+ ˛(x2io

− x1io

)) + vt(x1to + ˛(x2

to − x1to))

s.t. vi(x1io

+ ˛(x2io

− x1io

)) + vt(x1to + ˛(x2

to − x1to)) ≥ 1,

vi(x4ij

− ˛(x4ij

− x3ij)) + vt(x4

tj− ˛(x4

tj− x3

tj)) ≥ 1, j = 1, . . ., n, j /= o

vi, vt ≥ ε.

(21)

Model (21) is non-linear and it is hard to find its global optimum solution. Moreover, Model (21) depends on the values of the �-cutariables and should be solved for several values of �-cuts. This would impose a substantial volume of computational effort on the model.he following variable exchange will help resolve the non-linearity problem of the model:

v′i= vi˛, 0 ≤ ≤ 1

v′t = vt˛, 0 ≤ ≤ 1

(22)

Replacing [22] in Model (21) will result in the following model, which is linear and independent from the �-cut variables:

Min [x(i,t)o ]

L = vix1io

+ v′i(x2

io− x1

io) + vtx1

to + v′t(x

2to − x1

to)

s.t. vix1io

+ v′i(x2

io− x1

io) + vtx1

to + v′t(x

2to − x1

to) ≥ 1

vix4ij

− v′i(x4

ij− x3

ij) + vtx4

tj− v′

t(x4tj

− x3tj

) ≥ 1, j = 1, . . ., n, j /= o

v′i≤ vi

v′t ≤ vt

vi, vt ≥ ε

(23)

The procedure is similar for the optimistic situation where the upper bound of the weighted sum of highly correlated inputs can bealculated. For the sake of brevity, only the final Model (24) is presented here:

Min [x(i,t)o ]

U = vix4io

− v′i(x4

io− x3

io) + vtx4

to − v′t(x

4to − x3

to)

s.t. vix4io

− v′i(x4

io− x3

io) + vtx4

to − v′t(x

4to − x3

to) ≥ 1

vix1ij

+ v′i(x2

ij− x1

ij) + vtx1

tj+ v′

t(x2tj

− x1tj

) ≥ 1, j = 1, . . ., n, j /= o

v′i≤ vi

v′t ≤ vt

vi, vt ≥ ε

(24)

The procedure for highly correlated output variables is similar. Again, for the sake of brevity, we present the final Models (25) and (26)orresponding to the pessimistic and optimistic situations, respectively:

Max [y(i,t)o ]

L = uiy1io

+ u′i(y2

io− y1

io) + uty1

to + u′t(y

2to − y1

to)

s.t. uiy1io

+ u′i(y2

io− y1

io) + uty1

to + u′t(y

2to − y1

to) ≤ 1

uiy4ij

− u′i(y4

ij− y3

ij) + uty4

tj− u′

t(y4tj

− y3tj

) ≤ 1, j = 1, . . ., n, j /= o

′(25)

ui≤ ui

u′t ≤ ut

ui, ut ≥ ε

tt

3

••••••

4

ttRTfBL

tsmsittwm

4

peW

sgswii


Max [y(i,t)o ]

U = uiy4io

− u′i(y4

io− y3

io) + uty4

to − u′t(y

4to − y3

to)

s.t. uiy4io

− u′i(y4

io− y3

io) + uty4

to − u′t(y

4to − y3

to) ≤ 1

uiy1ij

+ u′i(y2

ij− y1

ij) + uty1

tj+ u′

t(y2tj

− y1tj

) ≤ 1, j = 1, . . ., n, j /= o

u′i≤ ui

u′t ≤ ut

ui, ut ≥ ε

(26)

The following variable exchange has been implemented to obtain Models (25) and (26):

u′i= ui˛, 0 ≤ ≤ 1

u′t = ut 0 ≤ ≤ 1

(27)

Following the above procedure, the optimistic or pessimistic weighted sum of highly correlated input variables can be used to reducehe dimension of the correlated data. In this way, minimum data will be lost while the discrimination power of DEA will be increased. Inhis paper, we have used the mean of the optimistic and pessimistic situations for the inputs and outputs.

.3.3. Main properties of the proposed modelsModels (23)–(26) possess several positive properties. The proposed models:

Can handle dimension-reduction for both highly correlated inputs and outputs.Can implement the dimension-reduction process in the presence of uncertainty (which has been modeled using fuzzy sets).Can calculate lower and upper bounds for the weighted sum of highly correlated inputs/outputs.Are independent from �-cut variables (therefore, minimum computational efforts are needed).Are linear, which implies that the global optimum solution can be found in a reasonable CPU time.Are always feasible and bounded.

. Pilot study: the Bank of North America

Given the fact that financial markets have become increasingly globalized, many countries have liberalized banking activities that tradi-ionally had been heavily regulated and protected from competing foreign banks. As a result, the Bank of North America (BONA)1 is studyinghe feasibility of establishing a substantial presence in 24 emerging markets including: Algeria, Argentina, Brazil, Bulgaria, China, Czechepublic, Estonia, Georgia, Hungary, India, Latvia, Lithuania, Mexico, Morocco, Poland, Romania, Russia, Slovakia, Slovenia, South Africa,unisia, Turkey, Ukraine, and Vietnam. BONA’s management has identified 19 factors relevant to the evaluation of emerging markets. Theseactors include: Bank Entry Requirements, Bank Run, Banking Sector Regulations, Corporate Tax Rate, Country Size (Population), Customerase, Delinquency Rate, Deposit Level, Deposit Rate, Economic Risks, GDP Per Capita, Income Per Capita, Interest Income, Investment Risks,ending Rate, Personal Income Tax Rate, Political Risks, Quality of Lending, and State of Profitability.

The authors were invited by BONA to join a group of six researchers specialized in international banking and form an assessmenteam. The team was assembled to work on a pilot study and develop an analytical model for evaluating potential emerging markets. Aftertudying the available data and the bank’s overall goals and objectives, the assessment team agreed to use the comprehensive fuzzy DEAodel proposed in this paper for evaluating the 24 emerging markets under consideration at BONA. Initially, the assessment team carefully

tudied the 19 factors proposed by management and classified them into desirable and undesirable input and output variables as shownn Table 1 The team then collected the data for the 19 factors and the 24 countries under consideration. The data were then normalizedo eliminate the scale effect. The scores presented in Table 2 are the normalized data used in the assessment process. It should be notedhat the data designated as “n/a” in this table were not readily available or reliable. However, the assessment team decided to continueith the pilot study without any further attempt to find the missing values since we were also interested in analyzing the behavior of theodel in the presence of missing data.

.1. On the fuzziness of factors and alternative parametric approaches

The international business literature generally deals with linguistic variables that are converted into crisp real values to be used in aarametric econometric regression in order to obtain some endogenous dependence among them. At the same time, it is widely acknowl-dged that many of these variables are determined by subjective interpretations of the state of the world and the data at different levels.e will provide two explicit examples of fuzzy variables that are subjectively converted (though not explicitly) into crisp ones.Consider, for example, the deposit insurance variable that is extensively employed by economists to measure the fragility of the banking

ystem within a country [104,105]. These authors create a Safety Net Index that aims at describing the coverage of deposit insurance andovernment guarantees on the balance sheets of banks. The index is based on a set of standardized variables designed to increase with theupport of the government and, therefore, with moral hazard. The index includes the coverage limit relative to GDP per capita ratio, together
ith a set of dummy (integer) variables that account for a series of institutional factors relating to deposit insurance and protection. The
ndex merges a highly heterogeneous data set into a unique real value, which allows for comparability among countries. However, thentrinsic imprecision caused by the use of categorical variables of diverse types should be explicitly acknowledged.

1 The name of the bank and their proprietary data are changed to protect their anonymity.


Table 1The input and output factors.

Input or output Desirable or undesirable Factor

Inputs

Desirable (−)

Political risksEconomic risksInvestment risksBank entry requirementsCorporate tax ratePersonal income tax rateDeposit rateBanking sector regulations

Undesirable (+)

Income per capitaGDP per capitaLending rateCountry size (population)

OutputsDesirable (+)

Interest incomeState of profitabilityDeposit levelQuality of lendingCustomer base

httarc

b

wcdfp

iaoaoHmg

obtDnoa

aasFo

afwp

Undesirable (−)Delinquency rateBank run

Consider now the indicator of the World Bank for the control of corruption within a given country, which also comprises a set of highlyeterogeneous variables (open to subjective interpretation) converted into real integer values. The description of the variables composinghe indicator is provided in http://info.worldbank.org/governance/wgi/index.aspx#doc, and includes highly diverse concepts such as publicrust in politicians and perceptions regarding how widespread is the level of corruption. While some answers consist of binary terms, suchs yes or no, others are presented in a natural scale, with the subjective interpretation of those answering the questions playing an essentialole in the answer. The same intuition applies when considering the regulatory environment to which banks are subject within a givenountry [106].

An example of the parametric approach to banking and financial development is provided by Beck and Feyen ([2], p. 21), whoseenchmarking model estimates the following relation

FDi,t = ˇXi,t + εi,t (28)

here FD is an indicator of financial development, X is composed of several structural country-specific factors, the subscripts i and t refer toountries and years, respectively, and ε is a standard normally distributed error term. The authors regress several characteristics assumed toetermine the level of financial development of a country, such as: GDP per capita; population and population density proxies that accountor market size and geographic barriers, respectively; the age dependency ratio, which is used to approximate the savings behavior of theopulation, and several country-specific dummies affecting the behavior of the country’s financial systems.

The general findings of this literature are quite diverse. Demirguc-Kunt and Detragiache [107] point out that instruments such as depositnsurance may force banks to take excessive risks, particularly in economies characterized by weak supervisory frameworks, which arelso prone to corruption. Anginer et al. [108] find that banking systems with weak supervision, low competition and more governmentwnership are more fragile: banks taking on less diversified risks and making the banking system are more vulnerable to shocks. Amidund Wolfe [109] reach a similar conclusion, illustrating how competition increases the diversification of the income generating activitiesf banks and, therefore, the stability of the banking system for a sample of 55 emerging and developing countries. At the same time,ryckiewicz [110] describes the lack of empirical consensus regarding the optimal bailout programs and intervention mechanisms thatust be defined by a government, while finding an increment in the risk and a decrease in the stability of the banking sector caused by

overnment intervention.The combination of positive and negative characteristics determining the behavior of banks in emerging countries leads to our definition

f desirable and undesirable input and output factors. Generally, DEA models aim at minimizing inputs and maximizing outputs. The cost-enefit input–output intuition behind this structure is clear, as firms will generally try to produce the maximum amount of output withhe minimum amount of inputs. The efficiency definition in the basic DEA model is based on such intuition. This implies that standardEA models are constrained in their capacity to account for inputs constituting a positive factor in the production function of firms or foregative outputs arising from their production processes. As illustrated above, when dealing with financial intermediaries, the spectrumf possible inputs and outputs broadens. However, he intuition remains similar to the cost-benefit analysis on which standard DEA modelsre built.

We consider all those regulatory and institutional factors (inputs) that should be minimized when establishing the foreign branch of bank in a host country and, in accordance with the DEA nomenclature, define them as desirable. Thus, banks should aim at decreasingll potential risks and regulatory constraints before entering an emerging market. At the same time, banks require developed economicystems and large customer bases in order to provide their services efficiently, leading to our DEA-based definition of undesirable inputs.inally, banks expect a solid customer base and a consistent flow of profits to result from their activity in the host country (desirableutputs), while being subject to instabilities such as runs and corruption that they should try to avoid (undesirable outputs).

We acknowledge the fuzzy nature of the variables used to define the set of input and output factors considered by the researchers
t BONA. In this regard, the uncertainty arising from these variables is emphasized through the use of trapezoidal (instead of triangular)uzzy numbers in our analysis. That is, we consider the interval of potential and equally viable values that may be taken by these factorsithin their respective membership functions as proxies for the uncertainty faced by the banks. As a result, the non-parametric perspectiverovided by DEA allows us to acknowledge explicitly the fuzzy nature of the variables defining the set of inputs and outputs and account
http://info.worldbank.org/governance/wgi/index.aspx








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693Table 2The crisp scores for the input and output factors.

Country Inputs

Desirable Undesirable

Politicalrisks

Economicrisks

Investmentrisks

Bank entry requirements Corporate tax rate Personal income tax rate Deposit rate Banking sectorregulations

Income percapita

GDP per capita Lendingrate

Country size(population)

Algeria 0.13 0.04 0.08 0.23 n/aa 0.66 0.56 0.03 0.27 0.04 0.09 0.003

Argentina 0.08 0.09 0.49 0.23 0.96 0.19 0.83 0.01 0.35 0.04 0.16 0.004Brazil 0.89 0.68 0.96 n/aa 0.74 0.21 n/aa 0.01 0.43 0.06 0.88 0.020

Bulgaria 0.06 0.12 0.14 0.14 0.15 0.13 0.20 0.03 0.26 n/aa 0.03 0.001China 0.04 0.06 0.16 0.08 0.30 0.22 0.97 0.02 0.31 0.05 0.98 0.134

Czech Republic 0.05 0.06 0.27 n/aa 0.24 0.20 0.96 0.04 0.22 0.59 0.05 0.001Estonia 0.06 0.10 0.05 0.05 0.23 0.14 0.09 0.03 0.28 0.57 0.01 0.000

Georgia 0.14 0.10 0.06 0.31 0.20 0.02 0.05 0.02 0.27 0.05 0.03 0.000Hungary 0.08 0.06 0.68 0.09 0.16 0.19 0.73 0.04 0.24 0.41 0.04 0.001India 0.09 0.08 0.56 0.25 0.88 0.35 0.88 0.04 0.31 0.06 0.96 0.117

Latvia 0.08 0.15 0.20 n/aa 0.15 0.12 0.13 0.03 0.27 0.43 0.11 0.000Lithuania 0.06 0.11 0.16 0.04 0.15 0.23 0.16 0.03 0.27 0.54 0.16 0.000

Mexico 0.04 0.05 0.36 0.14 0.86 0.34 0.97 0.02 0.38 0.06 0.45 0.011Morocco 0.10 0.04 0.56 0.15 0.46 0.32 0.27 0.03 0.33 0.05 0.11 0.0030

Poland 0.10 0.04 0.45 0.17 0.19 0.12 0.98 0.03 0.27 0.05 0.17 0.004Romania 0.04 0.08 0.16 0.25 0.16 0.09 0.75 0.01 0.21 0.05 0.09 0.002

Russia 0.06 0.14 0.07 0.16 0.24 0.06 0.98 0.02 0.30 0.04 0.75 0.014Slovakia 0.08 0.05 0.29 0.21 0.19 0.09 0.33 0.03 0.21 0.06 0.02 0.001

Slovenia 0.07 0.06 0.23 0.13 0.25 0.15 0.22 0.03 0.25 0.05 0.09 0.000South Africa 0.23 0.11 0.32 0.50 0.88 0.50 0.79 0.01 0.45 0.06 0.17 0.0050

Tunisia 0.14 0.05 0.48 0.07 n/aa 0.26 0.11 0.02 0.32 0.05 3.66 0.0010Turkey 0.11 0.10 0.40 0.20 0.30 0.35 0.95 0.02 0.33 0.05 0.24 0.008

Ukraine 0.03 0.25 0.10 0.38 0.25 0.15 0.59 0.03 0.26 0.04 0.21 0.005Vietnam 0.05 0.23 0.49 0.15 0.28 n/aa 0.28 0.03 0.30 0.04 0.47 0.009

Country Outputs


Interest income State of profitability Deposit level Quality of lending Customer base Delinquency rate Bank run

Algeria 0.070 0.034 0.005 0.001 0.008 0.04 0.11

Argentina 0.087 0.055 0.026 0.001 0.007 0.05 0.16Brazil n/aa 0.057 0.053 0.001 0.020 0.17 0.15

Bulgaria 0.022 0.041 0.009 0.001 0.002 0.04 0.13China 0.848 0.060 0.798 0.093 0.144 0.98 0.11

Czech Republic 0.093 0.139 0.024 0.001 0.015 0.14 0.11Estonia 0.009 0.122 0.003 −0.500 0.001 0.02 0.12

Georgia 0.005 0.047 0.002 −0.010 0.000 0.01 0.13Hungary 0.068 0.112 0.018 0.001 0.011 0.11 0.13India 0.127 0.008 0.048 0.001 0.019 0.32 0.12

Latvia 0.012 0.088 0.004 −0.001 0.001 0.02 0.12Lithuania 0.015 0.088 0.006 0.001 0.002 0.03 0.10

Mexico 0.257 0.081 0.055 0.001 0.029 0.31 0.13Morocco 0.026 0.019 0.003 0.001 0.002 0.04 0.11

Poland 0.106 0.089 0.066 0.001 0.018 0.20 0.12Romania 0.065 0.056 0.029 0.001 0.005 0.08 0.23

Russia 0.365 0.161 0.222 0.035 0.047 0.29 0.20Slovakia 0.031 0.220 0.012 −0.037 0.007 0.07 0.13

Slovenia 0.023 0.296 0.005 0.017 0.003 0.03 0.13South Africa 0.077 0.054 0.012 0.001 0.009 0.09 0.13

Tunisia 0.010 0.030 0.002 0.001 0.002 0.02 0.11Turkey 0.161 0.119 0.102 −0.006 0.014 0.19 0.33

Ukraine 0.057 0.074 0.033 −0.050 0.007 0.08 0.15Vietnam 0.024 0.028 0.249 0.063 0.006 0.08 0.12a The term not available (n/a) implies a missing value for the associated criteria.


Table 3The linguistic variables for the alternative ratings.

Linguistic variable TrFNs

Very low (VL) (0, 0.1, 0.2, 0.3)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)

fe

(TnFia

4

ntat

To

iorp

4

a

wvss

tmri

tp

4

Tr

5

f

High (H) (0.6, 0.7, 0.8, 0.9)Very high (VH) (0.7, 0.8, 0.9, 1.0)

or their effect on the efficient behavior of any DMU. This constitutes an important advantage over the parametric analysis performed byconomists and business scholars who do not consider such uncertainties when performing their respective analysis of the market.

That is, together with quantifiable macroeconomic and financial data, such as the size and efficiency of banks, the design of indicatorsas well as the decisions made by banks) is based on subjective evaluations of the political, social and regulatory environments [7,111].hus, while these indexes allow for comparability among countries and provide a natural guiding scheme, their crisp real values shouldot be considered as definitive, since despite the (yes/no) binary answers to some indicators others synthesize more complex variables.or example, while most studies agree on the fact that regulatory restrictions undermine the performance of the banking system, thenteraction of these and other variables in the decision process of the banks has never been quantified, as we have done here, whilecknowledging the fuzzy nature of the indicators.

.2. Pre-screening of data

Throughout the data collection process, the team recognized that the data gathered for the emerging markets under consideration wereot highly reliable and involved a considerable amount of vagueness and impreciseness. Due to the vague and imprecise nature of the data,he team decided to use the linguistic terms and fuzzy numbers presented in Table 3 to fuzzify the crisp values in Table 2 collected by thessessment team. The result of the fuzzification process is represented in Table 4. The missing values in Table 4 have been estimated usinghe geometric mean of the values obtained by the other alternatives for the same criteria.

The highly dimensional data presented in Table 4 reduce the discriminating power of DEA and cannot be used directly in the model.herefore, we calculate the fuzzy correlation coefficient proposed by Liu and Kao [103] among the desirable and undesirable inputs andutputs. The resulting fuzzy correlation coefficients are presented in Table 5.

Models (23)–(26) are solved for highly correlated input and output variables. The mean of the objective function for Models (23) and (24)s substituted for highly correlated inputs. The mean of the objective function for Models (25) and (26) is substituted for highly correlatedutputs. The number of factors before dimension-reduction was 19 (12 inputs and 7 outputs). After the implementation of the dimension-eduction procedure we were able to reduce the number of factors from 19 to 13. This dimension-reduction improves the discriminationower of the fuzzy DEA model.

.3. Results of the first module

In the first module (the new fuzzy DEA model with coexisting desirable inputs and undesirable outputs) each country is representeds a DMU. The efficiency scores of the DMUs (countries under consideration) are summarized in Tables 6 and 7.

In Table 6, the PEV column refers to the pessimistic efficiency values in which the evaluated state is considered under its worst scenariohile the remaining states are considered under their best scenario. Similarly in Table 7, the OEV column refers to the optimistic efficiency

alues in which the evaluated state is considered under its best scenario while the remaining states are considered under their worstcenario. In both Tables 6 and 7 the reference sets and the associated Lambda values for each DMU are presented in the column �j. Theseets provide the guidelines required for the improvement of inefficient DMUs.

A virtual DMU is constructed through a linear combination of its reference set members. The ratios used in this combination are equalo the �j values obtained for the corresponding DMU. Note that taking the best rank or a good rank among the available DMUs does not

ean that there is no need/possibility for improvement. Even in this case, the dummy DMUs constructed on the basis of the �j columnatio can dominate the DMU under consideration. Various strategies could be adopted to improve the efficiency of each DMU according tots associated reference sets. Fig. 2 summarizes the pessimistic and optimistic efficiency scores for different DMUs.

The construction of virtual efficient DMUs based on their corresponding � values relates the approach followed in the current paper tohe setting introduced by Hatami-Marbini et al. [45]. In particular, the potential efficiency improvements defined by these virtual DMUsrovide a reference for managers when evaluating the implementation of different policies at a country level.

.4. Results of the second module

The results of the first module are the interval efficiency values for the DMUs. These interval values are extracted from Table 6 andable 7. Table 8 presents the associated TrFN for each interval value as well as the final output of the PR. Table 8 and Fig. 3 present the finalanking of the DMUs (emerging markets under consideration) based on the PR.

. Beyond banks: toward the international business literature

The current paper has concentrated on the choice of countries by multinational banks that want to expand their global operations viaoreign branches. However, our model can be directly extended and applied to the literature on foreign direct investment. That is, the main

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Table 4The fuzzified scores for the input and output factors.

Country Inputs


Politicalrisks

Economicrisks

Investment risks Bank entryrequirements

Corporate tax rate Personal incometax rate

Deposit rate Banking sectorregulations

Income per capita GDP per capita Lending rate Country size(population)

Algeria VL VL VL ML VLa VH VL M L VL VL VLArgentina VL VL VL ML VH L VL VL MH L VL VL

Brazil VH VH VH La ML ML VLa VL VH VH VL LBulgaria VL VL VL L VL L VL MH L Ma VL VLChina VL VL VL VL VL ML VH ML M MH VH VHCzech Republic VL VL VL La VL ML VL VH VL H VL VLEstonia VL VL VL VL VL L VL M ML H VL VL

Georgia VL VL VL MH VL VL VL ML L M VL VLHungary VL VL VL VL VL L VL H VL M VL VLIndia VL VL VL M VH M L H M H MH VH

Latvia VL VL VL La VL L VL M L ML VL VLLithuania VL VL VL VL VL ML VL M L MH VL VL

Mexico VL VL VL L H M ML L H H VL VLMorocco VL VL VL L VH M VL M M MH VL VL

Poland VL VL VL L VL L VL MH L M VL VLRomania VL VL VL M VL VL VL VL VL MH VL VL

Russia VL VL VL L VL VL ML L ML L VL VLSlovakia VL VL VL ML VL VL VL MH VL VH VL VLSlovenia VL VL VL L VL L VL H L MH VL VL

South Africa VL VL VL VH H H VL VL VH VH VL VLTurkey VL VL VL ML VL M L L M M VL VL

Tunisia VL VL VL VL VLa ML VL ML M MH VL VLUkraine VL VL VL H VL L VL H L VL VL VL

Vietnam VL VL VL L VL La VL MH ML ML VL VL

Country Outputs


Interest income State of profitability Deposit level Quality of lending Customer base Delinquency rate Bank run

Algeria VL VL VL ML VL VL VLArgentina VL L VL ML VL VL L

Brazil VLa L VL ML VL L LBulgaria VL VL VL ML VL VL VLChina VH L VH VH VH VH VLCzech Republic VL M VL ML VL VL VLEstonia VL ML VL ML VL VL VL

Georgia VL VL VL L VL VL VLHungary VL ML VL ML VL VL VLIndia L VL VL ML VL ML VL

Latvia VL L VL ML VL VL VLLithuania VL L VL ML VL VL VL

Mexico ML L VL ML L L VLMorocco VL VL VL ML VL VL VL

Poland VL L VL ML VL L VLRomania VL L VL ML VL VL M

Russia ML M L MH ML L MLSlovakia VL H VL VL VL VL VLSlovenia VL VH VL M VL VL VL

South Africa VL L VL ML VL VL VLTurkey L ML VL ML VL L VH

Tunisia VL VL VL ML VL VL VLUkraine VL L VL VL VL VL L

Vietnam VL VL ML H VL VL VLa The missing values are estimated through the geometric mean of the available values for each criterion.


Table 5The fuzzy correlation coefficients between the input and output factors.

Desirable inputs

xg1 xg

2 xg3 xg

4 xg5 xg

6 xg7 xg

8

xg1 – (0.9, 0.92, 0.93) (0.9, 0.95, 0.94) (0.9, 0.91, 0.93) (−0.6, −0.4, −0.2) (−0.6, −0.4, −0.2) (0.1, 0.12, 0.15) (−0.9, −0.8, −0.7)

xg2 – (0.8, 0.9, 0.96) (0.1, 0.11, 0.13) (0.1, 0.13, 0.17) (0.8, 0.91, 0.93) (0.91, 0.94, 0.96) (−0.6, −0.4, −0.2)

xg3 – (0.7, 0.8, 0.85) (0.1, 0.11, 0.13) (0.11, 0.12, 0.15) (−0.8, −0.75, −0.7) (0.09, 0.1, 0.12)

xg4 – (−0.2, −0.17, −0.14) (−0.8, −0.75, −0.7) (−0.3, −0.2, −0.1) (−0.4, −0.25, −0.16)

xg5 – (0.9, 0.95, 0.94) (0.7, 0.8, 0.85) (−0.2, −0.17, −0.14)

xg6 – (0.1, 0.11, 0.13) (−0.6, −0.4, −0.2)

xg7 – (−0.2, −0.17, −0.14)

xg8 –

Undesirable inputs

xb1 xb

2 xb3 xb

4

xb1 – (0.1, 0.11, 0.13) (0.7, 0.8, 0.85) (0.9, 0.95, 0.94)

xb2 – (0.09, 0.1, 0.12) (0.1, 0.11, 0.13)

xb3 – (0.75, 0.81, 0.86)

xb4 –

Desirable outputs

yg1 yg

2 yg3 yg

4 yg5

yg1 – (0.9, 0.95, 0.94) (0.75, 0.81, 0.86) (0.09, 0.1, 0.12) (0.7, 0.8, 0.85)

yg2 – (0.9, 0.95, 0.94) (0.09, 0.1, 0.12) (0.75, 0.81, 0.86)

yg3 – (0.7, 0.8, 0.85) (0.75, 0.81, 0.86)

yg4 – (0.7, 0.8, 0.85)

yg5 –

Undesirable outputs

yb1 yb

2

yb1 – (0.4, 0.5, 0.6)

yb2 –

Table 6The pessimistic efficiency values (PEVs), rankings and reference sets.

Country PEV Rank �j ˇj

China 0.922 1 �1 = 0.8208623, �22 = 0.8025577E–01, �30 = 0.9888188E–01 1Estonia 0.914 2 �1 = 0.8958704E–01, �14 = 0.4238491, �22 = 0.4865639 0.4693157Czech Republic 0.912 3 �1 = 0.1480467, �18 = 0.7208100, �22 = 0.1311433 1Lithuania 0.900 4 �1 = 0.5393124E–01, �23 = 0.5866644, �24 = 0.3594044 1Slovakia 0.896 5 �17 = 0.1011603, �18 = 0.5498137, �22 = 0.3490260 0.6676800Slovenia 0.847 6 �17 = 1.000000 0Latvia 0.820 7 �1 = 0.3388634, �22 = 0.1821371, �30 = 0.4789994 0Bulgaria 0.795 8 �22 = 0.2894399, �30 = 0.7105601 1Georgia 0.784 9 �21 = 0.9504665E–01, �24 = 0.9049534 1Russia 0.771 10 �22 = 0.6162137, �30 = 0.3837863 0Hungary 0.766 11 �1 = 0.8725700E–01, �22 = 0.2301893, �30 = 0.6825537 0Vietnam 0.765 12 �22 = 0.6048418E–02, �30 = 0.9939516 1Mexico 0.733 13 �1 = 0.6787855, �11 = 0.8790555E–02, �17 = 0.3124239 0.7135698E−01Romania 0.714 14 �1 = 0.1910711, �3 = 0.7346200, �22 = 0.7430883E–01 1Poland 0.702 15 �22 = 0.5107330, �30 = 0.4892670 0South Africa 0.644 16 �1 = 0.2329878, �22 = 0.7670122 0Tunisia 0.635 17 �22 = 0.3959285, �30 = 0.6040715 1Argentina 0.626 18 �22 = 0.4245334, �30 = 0.5754666 1Brazil 0.622 19 �22 = 0.2742965, �30 = 0.7257035 1Turkey 0.617 20 �22 = 0.5257981, �30 = 0.4742019 0India 0.613 21 �1 = 0.2793413, �22 = 0.2171254, �30 = 0.5035333 1Algeria 0.576 22 �1 = 0.5197800, �22 = 0.4802200 1Ukraine 0.565 23 �1 = 0.2988909, �22 = 0.6973280, �30 = 0.3781128E–02 0

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Morocco 0.497 24 �1 = 0.2625553, �22 = 0.2322442, �30 = 0.5052005 1

mplications of the current model at the managerial level extend beyond financial intermediaries and encompass any multinational firmhat must enter a foreign emerging market.

a) Our model divides the inputs and outputs into desirable and undesirable ones. Moreover, it has been developed based on the envel-
opment form of DEA modeling, which allows us to calculate efficiency scores and find the projection of inefficient DMUs toward theefficient frontier.


Table 7The optimistic efficiency values (OEVs), rankings and reference sets.

Country OEV Rank �j ˇj

China 0.948 1 �1 = 0.9013925, �22 = 0.9860752E–01 0Czech Republic 0.932 2 �1 = 0.1213521, �11 = 0.7686049, �22 = 0.1100430 0Estonia 0.925 3 �1 = 0.2306050E–01, �22 = 0.9769395 1Lithuania 0.924 4 �1 = 0.2339868, �23 = 0.7660132 0Slovakia 0.897 5 �1 = 0.1703854E–01, �11 = 0.8343735, �22 = 0.1485879 0Slovenia 0.869 6 �1 = 0.6840801, �9 = 0.7433126E–01, �11 = 0.2415886 0Latvia 0.826 7 �1 = 0.5504064E–01, �23 = 0.7959747E–01, �24 = 0.8653619 1Bulgaria 0.803 8 �2 = 0.9917009, �23 = 0.8299122E–02 1Georgia 0.789 9 �1 = 0.1712055, �23 = 0.8287945 1Vietnam 0.788 10 �1 = 0.5994879E–01, �4 = 0.9400512 1Russia 0.783 11 �1 = 0.1522864, �14 = 0.8477136 1Hungary 0.775 12 �1 = 0.9458030, �23 = 0.5419698E–01 1Mexico 0.751 13 �1 = 0.6099043, �22 = 0.1352463, �24 = 0.2548494 0Romania 0.720 14 �1 = 0.5777022, �11 = 0.1362435E–01, �22 = 0.4086735 0Poland 0.717 15 �1 = 0.4735759, �22 = 0.5264241 1Tunisia 0.646 16 �1 = 0.6743305, �22 = 0.3256695 1South Africa 0.645 17 �9 = 0.6416898, �22 = 0.2035458, �24 = 0.1547643 0Argentina 0.638 18 �1 = 0.6822475, �22 = 0.3177525 1Turkey 0.628 19 �1 = 0.4132512, �22 = 0.5867488 1Brazil 0.625 20 �1 = 0.8672588, �22 = 0.1327412 1India 0.623 21 �1 = 0.9622953, �23 = 0.3770474E–01 1Algeria 0.591 22 �1 = 0.4514032, �22 = 0.5485968 1Ukraine 0.581 23 �1 = 0.1559770, �9 = 0.7439890E–01, �11 = 0.1510448E–01, �22 = 0.7545196 0Morocco 0.498 24 �1 = 0.9940281, �22 = 0.5971897E–02 1

Fig. 2. The optimistic and pessimistic efficiency scores.

Fig. 3. The preference ratio (PR) scores.


Table 8The second module outputs.

Efficiency score Preference ratio procedure

Country Interval Associated TrFN k 1/k Rank

China [0.922, 0.948] [0.922, 0.922, 0.948, 0.948] 1.421203 0.704 1Estonia [0.795, 0.803] [0.795, 0.795, 0.803, 0.803] 1.758727 0.569 2Czech Republic [0.714, 0.720] [0.714, 0.714, 0.720, 0.720] 1.868964 0.535 3Lithuania [0.765, 0.788] [0.765, 0.765, 0.788, 0.788] 1.919741 0.521 4Slovakia [0.497, 0.498] [0.497, 0.497, 0.498, 0.498] 1.924254 0.520 5Slovenia [0.617, 0.628] [0.617, 0.617, 0.628, 0.628] 1.992938 0.502 6Latvia [0.626, 0.638] [0.626, 0.626, 0.638, 0.638] 1.996243 0.501 7Bulgaria [0.733, 0.751] [0.733, 0.733, 0.751, 0.751] 2.002433 0.499 8Georgia [0.644, 0.645] [0.644, 0.644, 0.645, 0.645] 2.122771 0.471 9Russia [0.702, 0.717] [0.702, 0.702, 0.717, 0.717] 2.316989 0.432 10Hungary [0.896, 0.897] [0.896, 0.896, 0.897, 0.897] 2.364912 0.423 11Vietnam [0.613, 0.623] [0.613, 0.613, 0.623, 0.623] 2.379539 0.420 12Mexico [0.635, 0.646] [0.635, 0.635, 0.646, 0.646] 2.383544 0.420 13Romania [0.771, 0.783] [0.771, 0.771, 0.783, 0.783] 2.394997 0.418 14Poland [0.622, 0.625] [0.622, 0.622, 0.625, 0.625] 2.434543 0.411 15South Africa [0.766, 0.775] [0.766, 0.766, 0.775, 0.775] 2.647787 0.378 16Tunisia [0.847, 0.869] [0.847, 0.847, 0.869, 0.869] 2.802267 0.357 17Argentina [0.912, 0.932] [0.912, 0.912, 0.932, 0.932] 2.848156 0.351 18Brazil [0.576, 0.591] [0.576, 0.576, 0.591, 0.591] 3.002272 0.333 19Turkey [0.565, 0.581] [0.565, 0.565, 0.581, 0.581] 3.057519 0.327 20India [0.784, 0.789] [0.784, 0.784, 0.789, 0.789] 3.118356 0.321 21Algeria [0.914, 0.925] [0.914, 0.914, 0.925, 0.925] 4.018687 0.249 22

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Ukraine [0.900, 0.924] [0.900, 0.900, 0.924, 0.924] 4.20868 0.238 23Morocco [0.820, 0.826] [0.820, 0.820, 0.826, 0.826] 4.644964 0.215 24

There exists a clear strategic side to the entry decision of foreign firms in host emerging countries [112]. This strategic quality extendsmmediately to banks when selecting the country where to interact with the branches of the local financial intermediaries [4]. That is, theapacity of multinational corporations to develop networks in different host economies leads to the creation and transmission of knowl-dge at the local level. This may result in knowledge spillovers that can be absorbed by the local firms as a mechanism of technologicalonvergence [113], with the quality of the institutional environment determining the differences in innovation and technological devel-pment across countries [114]. The internationalization strategies of multinationals into developing host economies should account forhis fact and other potentially negative outcomes.

That is, multinational banks (and corporations) require a host economy with a sufficiently developed consumer base so as to increasehe size of its deposits, together with a safe regulatory and institutional framework. At the same time, multinationals aim at minimizingheir knowledge exposure and the potential information spillovers that may lead to an increase in competition from the local firms. Bothypes of elements must be considered when selecting and entering an emerging market. We extend this point below and relate it to ourecond implication.

b) We have developed and extended a formal procedure to reduce the dimension of high-dimensional data while considering fuzzy inputsand outputs.

The economics and international business literatures have consistently analyzed the substantial amount of factors determining thenternationalization decisions of firms and the resulting behavior of foreign direct investment. These factors consider the two-way flowf information and knowledge that takes place between the foreign subsidiaries or branches and the local ones, both from the foreignubsidiaries to the local firms and the other way around if the subsidiaries are trying to acquire knowledge from the host location [115].he choice of foreign location is highly conditioned by the fact that subsidiaries and branches will not only concentrate on productionnd distribution but also on technology and research activities. These latter activities lead to information and knowledge flows that cane exploited by the local firms depending on their level of development and its potential evolution [116–118]. That is, emerging marketsave started to increase their importance both in the reception and creation of foreign direct investment [119], leading to the creation ofmerging multinationals, which are conditioned by their capacity to access knowledge abroad and assimilate it [115].

The (parametric) econometric models employed by economics and business scholars have the capacity to analyze large panels of datahen determining the relations existing between several (desirable and undesirable) variables. In this regard, DEA models may be subject

o a dimensionality problem when considering the substantial amount of different input and output variables determining the relativefficiency of DMUs.

c) Our model addresses the existing uncertainty in inputs and outputs using fuzzy variables. We have also customized and applied awell-known and powerful fuzzy number ranking method, the preference ratio, to fully rank the DMUs based on their efficiency scores.

Measuring quantitatively the performance of a government and its main market institutions is a complex issue. As a result, a substantialubset of the country indicators that must be considered by multinationals are approximations calculated through statistical discrimi-
ation methods, particularly so the indexes constructed and provided by international organism such as the World Bank or the IMF. Forxample, among the studies dedicated to construct institutional quality indicators, we have the Worldwide Governance Indicator createdy Kaufmann et al. [120], who employed 352 indicators collected from different international organizations and rating agencies to definen index accounting for the institutional quality of countries over different time periods.


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Fig. 4. Our DEA non-parametric environment versus parametric econometric regressions.

Assuming fuzzy inputs and outputs allows to account for the uncertainty in the interactions of firms derived from the approximationsonditioning the available data. This uncertainty will be essential when determining both the country and the mode of entry based on theotential complementarities with the local firms [121]. In this case, DEA models as the one constructed in this paper have a considerabledvantage over the econometric methods employed in the international business literature, which have not yet been suitably defined toccommodate fuzzy variables in their statistical regressions.

It should be noted that several models based on the fuzzy-DEA method have started to consider explicitly the parametric approach tohich they relate or complement. Indeed, research comparing and integrating both approaches has already been undertaken. For example,zadeh et al. [122] integrated fuzzy regression and DEA within an algorithm designed to estimate and optimize oil consumption withncertain and ambiguous data. They used an output-oriented DEA model without inputs to examine the efficiency of 15 fuzzy regressionodels.In summary, the complex relationship defined by the internationalization decision of multinational firms together with the main

haracteristics of the host economy, such as the level of technological development and the quality of the institutional framework, determineot only the choice of country but also the activities carried by the subsidiaries and the multinationals afterwards [123,124]. In thisegard, the economics and international business literatures are able to identify the main variables that condition the selection decision ofultinationals but lack an analysis of the decision process of firms in terms of the comparative efficiency of countries (and firms) that DEA

an provide. Fig. 4 summarizes and compares the main characteristics of our DEA non-parametric approach with those of the parametricconometric regressions employed by economists and business scholars.

. Conclusions and future research directions

The field of DEA has grown immensely since the pioneering papers of Farrell [11] and Charnes et al. [9]. In this regard, the modeleveloped in this paper arises from the necessity of bank managers to rank different countries or DMUs given a substantial amount of

nformation on both desirable and undesirable evaluation factors. Moreover, the data acquired to evaluate the different alternatives are
enerally subject to personal value judgments and other imprecisions. That is, the data are highly heterogeneous and include fuzzy andubjective opinions typical of any research assessment group. Though other ranking evaluation models accounting partially for these factsxist in the literature, the comprehensive one developed herein provides a real-world-based structure that can be extended and appliedo several scenarios beyond the banking one described in the current paper.

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The ranking evaluation structure defined throughout the three modules composing our model illustrates how its main contributionoes beyond the application of the fuzzy DEA method to banking selection models.

The initial fuzzy correlation-based reduction in the number of evaluation factors mitigates the potential problem of obtaining too manyefficiency scores equal to one, which would substantially decrease the discriminating power of the resulting ranking.Moreover, the �-cut independency of our model together with its capacity to account for desirable and undesirable inputs and outputsprovide a real-world based ranking structure directly implementable by firm managers.

We have presented a real-world pilot study assessing 24 emerging markets for international banking to demonstrate the applicabilityf the proposed model and exhibit the efficacy of the procedures and algorithms. It should be highlighted that the existing proceduresenerally rank the alternatives without any practical benchmark for improvement. The procedure proposed in this study uses a practicalenchmarking system for the inefficient DMUs. The outputs of the first module determined a reference set for each inefficient DMU in bothhe optimistic and pessimistic situations. A virtual efficient DMU was constructed through a linear combination of each reference set. The

embers of each reference set were treated as a real benchmark instance of the DMU under consideration.Our model is intended to help decision makers think systematically about complex performance evaluation problems. In order to

o so, we have decomposed the emerging market assessment process into manageable steps and integrated the results to arrive at aeasonably good solution. This decomposition encouraged the assessment team at BONA to carefully consider the elements of uncertainty.he proposed structured framework does not imply a deterministic approach to emerging market assessment. That is, while our approachnables decision makers to assimilate the information from different sources and use fuzzy numbers to partially overcome the uncertaintiesn input and output data, it should be used in conjunction with experience and expertise since subjective judgments could bias the finalesults.

A stream of future research can extend our framework to other variations of the DEA method. It would also be interesting to developybrid approaches for an integrated use of our framework, not only hybrids of different DEA methods but also hybrids of numericalptimization.

cknowledgements

The authors would like to thank the anonymous reviewers and the editor-in-chief for their constructive comments and suggestions.his research is supported in part under a research plan entitled “A Comprehensive Fuzzy DEA Model for Emerging Market Assessmentnd Selection Decisions” by the South-Tehran Branch, Islamic Azad University, Tehran, Iran.

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dx.doi.org/10.1080/02331934.2012.684354

dx.doi.org/10.1080/02331934.2012.684354

dx.doi.org/10.1080/02331934.2012.684354

dx.doi.org/10.1080/02331934.2012.684354

dx.doi.org/10.1080/02331934.2012.684354

dx.doi.org/10.1080/02331934.2012.684354

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