Research ArticleAnEco-InefficiencyDominanceProbability Approach forChineseBanking Operations Based on Data Envelopment Analysis

Feng Li 1 Lunwen Wu1 Qingyuan Zhu 23 Yanling Yu1 Gang Kou 1 and Yi Liao 1

1School of Business Administration Southwestern University of Finance and Economics Chengdu 611130 China2College of Economics and Management Nanjing University of Aeronautics and Astronautics Nanjing 211106 China3Research Centre for Soft Energy Science Nanjing University of Aeronautics and Astronautics Nanjing 211106 China

Correspondence should be addressed to Qingyuan Zhu zqynuaanuaaeducn

Received 23 February 2020 Revised 30 March 2020 Accepted 10 April 2020 Published 4 May 2020

Guest Editor Baogui Xin

Copyright copy 2020 Feng Li et alis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Data envelopment analysis (DEA) has proven to be a powerful technique for assessing the relative performance of a set ofhomogeneous decision-making units (DMUs) A critical feature of conventional DEA approaches is that only one or several sets ofoptimal virtual weights (or multipliers) are used to aggregate the ratio performance efficiencies and thus the efficiency scores mightbe too extreme or even unrealistic Alternatively this paper aims at developing a new performance dominance probability approachand applying it to analyze the banking operations in China Towards that purpose we first propose an extended eco-inefficiencymodel based on the DEA methodology to address banking activities and their possible relative performances Since the eco-in-efficiency will be obtained using a set of optimal weights we further build a performance dominance structure by considering all setsof feasible weights from a data-driven perspectiveen we develop two pairwise eco-inefficiency dominance concepts and proposethe inefficiency dominance probability model Finally we illustrate the eco-inefficiency dominance probability approach with 32Chinese listed banks from 2014 to 2018 to demonstrate the usefulness and efficacy of the proposed method

1 Introduction

Forty years have gone by since the great reform and openingpolicy of 1978 and China has made substantial progress ineconomic development with an annual increase of almostnine percent in gross domestic product from 149541 billiondollars in 1978 to 13608 trillion dollars in 2018 It is ratherremarkable that the banking industry of China especiallystate-owned and listed banks has played a great role inChinese economic growth [1 2] roughout the ever-in-creasing national economic development the banking in-dustry in China has also been promoted and developedconsiderably For example the total assets of the Chinesebanking industry reached almost 41 trillion dollars in 2018which is more than three times the gross domestic productin current US dollars in the same year Meanwhile theunprecedented competition within Chinese banks and be-tween Chinese domestic banks and foreign banks has be-come increasingly fierce since the opening of financialmarkets To participate in the ongoing competitive

challenges all over the world it is of vital importance forChinese banks to pay special attention to their operationperformances [1 3 4] In addition it is also an inherentrequirement of guaranteeing and promoting the healthy andsustainable economic development of China to address thebanking performance

Among the family of existing performance evaluationmethods data envelopment analysis (DEA) is one of themajor approaches because of its general applicability [5ndash10]DEA first introduced by Charnes et al [11] and furtherextended by Banker et al [12] is a data analytics approachthat can be used for evaluating the relative performances of agroup of homogeneous decision-making units (DMUs)which in practice consume multiple inputs to gain multipleoutputse basic logic behind the DEAmethodology is thatit compares DMUsrsquo real activity levels relative to the idealstatus by projecting their actual input-output bundle ontothe production frontier To obtain the production frontierall DMUsrsquo observed inputs and outputs are used to constructa production possibility set (PPS) with a set of certain

HindawiComplexityVolume 2020 Article ID 3780232 14 pageshttpsdoiorg10115520203780232

axiomatic hypothesis while the production frontier is anenvelopment of the production possibility set e DEAmethodology has many apparent characteristicsadvantagesand since its inception work in Charnes et al [11] DEA hasbeen applied to many kinds of activities in various differentcontexts [13ndash16] In addition the DEAmethodology has alsoproven to be a powerful and preferable method for per-formance evaluation in the banking industry and has beenfrequently applied to this industry [2ndash4 17]

e conventional DEA methods allow each DMU togenerate a set of relative weights to maximize its ratio ef-ficiency of aggregated weighted outputs to aggregatedweighted inputs while ensuring that the same ratio is nomore than one for all DMUs and the maximum ratio isconsidered the performance index for the evaluated DMU[18 19] e weight determination is critical to performanceevaluation results but there are some significant concernsthat reduce the applicability of DEA-based performanceanalytics and applications [20 21] On the one hand eachDMU selects its most favorable set of weights tomaximize itsefficiency ratio and thus the efficiency score of each DMUmight be too optimistic or even impossible due to the un-realistic set of weights On the other hand each DMU selectsits set of weights separately and as a result their perfor-mance scores are obtained under different standards andthus are not comparable

Many studies have been proposed for more reasonableperformance analytics by focusing on the selection of fea-sible weights Cook et al [22] and Roll et al [23] suggest acommon set of weights method that attempts to find acommon set of weights and the performance assessment isimplemented using that common set of weights Cook andKress [24] also address the common set of weights byminimizing the gap of upper and lower bounds of weightsKao and Hung [25] suggest generating a common set ofweights by minimizing the sum of squared difference be-tween the possible efficiency scores for the common set ofweights and the CCR efficiency score across all DMUsSimilar studies can also be found in Liu amp Peng [26] Kao[27] Zohrehbandian et al [28] Ramazani-Tarkhorani et al[29] Shabani et al [30] Li et al [31] and Li et al [32]Although the common set of weights method can provide acommon evaluation standard for all DMUs its majorproblem is that it still considers only one possibility forweights upon which the performance is estimated More-over the determination of the common set of weights is stilla big problem that will affect the performance analyticsresults relative to different common sets of weights and aconsensus regarding this has not been reached thus far

Another research stream focuses on the cross-efficiencymethod in which each DMU selects a set of commonweights and then the set of common weights is used toevaluate each DMU [33] It is notable that the classic DEAapproach evaluates each DMUrsquos relative efficiency using itsfavorable set of weights [18 34] while the cross-efficiencymethod requires that each DMUrsquos favorable set of weights beused to evaluate itself as well as the other DMUsrsquo relativeefficiency As a result several sets of weights are used tomeasure the relative performance which is a great

improvement relative to classic DEA approaches consid-ering only one set of weights Furthermore each DMU willhave a maximal efficiency score based on self-appraisal andseveral smaller efficiency scores based on peer appraisal eultimate cross-efficiency score can be aggregated with theseself-appraisal and peer appraisal scores for each individualDMU [35ndash41] e DEA cross-efficiency method has somepreferable characteristics such as satisfied discriminationpower between good and poor performances [42] a fullranking of all DMUs [43] and more realistic weights at-tached to various inputs and outputs [44] e literature haswitnessed numerous studies on various cross-efficiencyevaluation approaches for many kinds of real applications[45ndash47] Although the cross-efficiency method considersseveral sets of weights to measure the performance it isinsufficient to involve all performance possibilities More-over the determination of nonunique weights from eachDMUrsquos perspective will also reduce the applicability of cross-efficiency methods as does the aggregation of individualcross-efficiencies [41 48ndash50]

e recent research by Salo and Punkka [21] suggestsdeveloping ratio-based efficiency analysis over all sets offeasible weights Salo and Punkka [21] build ranking in-tervals dominance relations and efficiency bounds to showhow the DMUsrsquo efficiency ratios relate to each other for allsets of feasible weights rather than for some sets of weightsthat are typically used in classic DEA studies More spe-cifically Tang et al [51] propose a novel efficiency proba-bility dominance model and develop the dominanceefficiency probability over all sets of feasible weights buttheir approach is based on a radial model under the constantreturns to scale (CRS) assumption and only traditionaldesirable outputs are considered by ignoring undesirableoutputs Shi [52] extends the Salo and Punkka [21] modelover sets of all feasible weights to a more common andpractical case considering the internal two-stage productionstructure e proposed approach calculates each DMUrsquosefficiency bounds for the overall system as well as the effi-ciency bounds for each subsystem Li et al [53] build theefficiency ranking interval for two-stage production systemsand calculate each DMUrsquos ranking interval for the overallsystem as well as for each substage Li et al [54] propose afixed cost allocation approach based on the efficiencyranking concept which addresses the performance andefficiency ranking interval by considering all relativeweights It is of vital significance to consider all sets offeasible weights because doing so can address all possibilitiesfrom a data analytics perspective and provide evaluationresults that are more logical and fairer

In this paper we will develop a dominance probabilityapproach with undesirable outputs to assess the ratio per-formance based on DEAmodels and the proposed approachis illustrated with Chinese listed banks From a data-drivenanalytics perspective we will consider all sets of feasibleweights that are used for estimating the ratio performanceTowards that purpose we first build a performance eval-uation model to address the banking activities in the classicDEA framework by considering only a set of optimalweights Since banking operations inevitably yield some

2 Complexity

undesirable by-products such as bad debts that are jointlyproduced with incomes an extended eco-inefficiencymodel is developed Furthermore the eco-inefficiencymodel is used to develop the pair dominance structuretaking all sets of weights into account based on Tang et al[51] e performance dominance probability is proposedto calculate the average probability of a certain DMUrsquosperformance dominating all DMUs A larger performancedominance probability indicates that it is easier for thatDMU to dominate all other DMUs implying that its in-efficiency score is more likely to be smaller than that ofother DMUs in the sense of data-driven analytics Finallythe proposed approach is applied to a four-year dataset of32 Chinese listed banks and the empirical results show that(1) the inefficiency dominance probability is largely dif-ferent from inefficiency scores (2) these state-ownedcommercial banks are more likely to have a better domi-nance performance while local rural commercial banksmight have very promising inefficiency performances insome extreme situations but are likely to have poor per-formance in the sense of inefficiency dominance proba-bility and (3) China Construction Bank Industrial andCommercial Bank of China and Industrial Bank are the topthree listed banks based on operations analytics and on thecontrary Suzhou Rural Commercial Bank Rural Com-mercial Bank of Zhangjiagang and Jiangyin Rural Com-mercial Bank are the lowest three banks with inefficiencydominance probability is paper contributes to the lit-erature in at least the following aspects First this paperextends a new approach in the DEA framework by taking allsets of feasible weights into account whereas previousstudies have considered only one or several sets of weightsSecond this paper establishes a pairwise performancedominance concept which can help decision-makers an-alyze the performance relation in any context with priceinformation (ie weights or multipliers) ird this paperanalyzes the performance of Chinese listed banks andprovides some empirical findings which can facilitate thebanking industries in China

e remainder of this paper is organized as followsSection 2 develops the mathematical methodology of anextended eco-inefficiency model with undesirable outputsand performance dominance structure using inefficiencyscores Afterwards the proposed approach is used to studythe empirical performance analytics of 32 listed banks inSection 3 Finally Section 4 concludes and summarizes thispaper

2 Mathematical Modeling

We first propose an extended eco-inefficiency model toaddress banking activities and possible relative perfor-mances in Section 21 Furthermore we build a performancedominance structure considering all sets of feasible weightsin Section 22

21 An Extended Eco-Inefficiency Model Suppose a set of npeer banks with each bank using m inputs for the sake of

producing s traditional desirable outputs as well as q un-desirable outputs such as bad debt in the illustrative ap-plication A bad debt or nonperforming loan is a jointlyproduced and unavoidable by-product in the banking in-dustry Without loss of generality we consider each bank asa homogeneous decision-making unit (DMU) in the DEAframework Furthermore DMUj(j 1 n) consumeinputs Xj (x1j xmj) to produce desirable outputsYj (y1j ysj) and undesirable outputsBj (b1j bqj) respectively Before constructing themodeling a core task is to identify appropriate methods tohandle undesirable outputs It is clear that the strong andweak disposability assumption of undesirable outputs anddesirable outputs are the two most common and naturalmethods in the literature [55ndash57] A significant featurebetween the strong and weak disposability assumption iswhether undesirable outputs can be produced withoutdamage or subsequent cost to desirable outputs [58] Ifundesirable outputs can be freely generated withoutdamage or subsequent cost implying that both inputs andoutputs can change unilaterally without compromisingeach other then undesirable outputs are assumed to bestrongly disposable In contrast if the production of un-desirable outputs indeed has some damage or subsequentcost to inputs or desirable outputs implying that a re-duction in undesirable outputs would result in a reductionof desirable outputs simultaneously [59 60] then unde-sirable outputs are assumed to be weak disposable Here weconsider the weak disposability assumption because it ismore suitable for the real world and more specificallyfor the empirical application of banking operations whereit is very difficult to freely reduce bad debts without af-fecting incomes and changing banking operations To thisend the production possibility set (PPS) under the variablereturns to scale (VRS) assumption can be formulated asfollows

PPS xi yr bp1113872 1113873

1113944n

j1λjxij le xi i 1 m

ρj1113944n

j1λjyrj geyr r 1 s

ρj1113944n

j1λjbpj bp p 1 q

1113944n

j1λj 1

0le ρj le 1 λj ge 0 j 1 n

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(1)

e above formula is nonlinear since both the reductionfactor ρj and intensity variable λj are unknown Further-more we can equivalently change formula (1) into a linearversion which is presented in the following formula

PPS xi yr bp1113872 1113873

1113944n

j1 λj + ηj1113872 1113873xij lexi i 1 m

1113944n

j1λjyrj geyr r 1 s

1113944n

j1λjbpj bp p 1 q

1113944n

j1 λj + ηj1113872 1113873 1 λj ηj ge 0 j 1 n

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2)

Complexity 3

Based on formula (2) both desirable and undesirableoutputs are weighted by nondisposed intensity variables λjwhereas the inputs are weighted by the sum of nondisposedintensity variables λj and disposed intensity variables ηj Inaddition the VRS assumption is ensured by summing thetotal nondisposed intensity variables λj and disposed in-tensity variables ηj to 1 ie 1113936

nj1(λj + ηj) 1

Based on the above PPS in formula (2) we can buildmathematical models to compute the relative performanceof DMUs In this paper we follow the practice of Chen andDelmas [58] to develop an extended eco-inefficiency modelfor performance evaluation since the eco-inefficiency modelhas some advantages in modeling activities with undesirableoutputs compared with four well-established models in theliterature (undesirable outputs as inputs transformation ofundesirable outputs directional distance function andhyperbolic efficiency model Readers can refer Chen andDelmas [58] for details on the comparison) e extendedeco-inefficiency model is formulated as follows

inElowastd Max1113936

mi1g

xi xid + 1113936

sr1g

yr yrd + 1113936

qp1g

bpbpd

m + s + q

st 1113944

n

j1λj + ηj1113872 1113873xij lexid minus g

mi i 1 m

1113944

n

j1λjyrj geyrd + g

yr r 1 s

1113944

n

j1λjbpj bpd minus g

bp p 1 q

1113944

n

j1λj + ηj1113872 1113873 1

λj ηj gmi g

yr gb

p ge 0 j 1 n

r 1 s p 1 q

(3)

In model (3) the variables gxi g

yr and gb

p represent theamount of potential improvements in inputs desirableoutputs and undesirable outputs respectively that theevaluated DMU can make relative to its current input usageand output production to reach its ideal benchmark targeton the efficiency frontiere potential improvements reflectinput reduction potentials and desirable output expansionpotentials (or undesirable output reduction potentials) in-stead of actual input usage and output production [61 62]Model (3) uses a slack-based formula that is similar to thedirectional distance function (DDF) model to maximize theaverage additive inefficiency index across all input andoutput measures where the inefficiency index representspotential improvements divided by observed inputs oroutputs More specifically the optimal direction vector ofmodel (3) can be endogenously obtained in a similar way asthat of Arabi et al [63] For that purpose suppose

gxi β middot gi g

yr β middot 1113957gr and gb

p β middot 1113954gp where the directionvector is g (gi 1113957gr 1113954gp) Solving model (3) determines anoptimal solution (gxlowast

i gylowastr gblowast

p ) then we have β

(gxlowasti gi) (g

ylowastr 1113957gr) (gblowast

p 1113954gp)(i 1 m r 1 s

p 1 q) Hence we have a system of (m + s + q + 1)

unknown variables (β gi 1113957gr 1113954gp) and (m + s + q) linearlyindependent equations gxlowast

1 g2 gxlowast2 g1 gxlowast

m 1113957g1 gylowast1 gm

gylowast1 1113957g2 g

ylowast2 1113957g1 g

ylowasts 1113954g1 gblowast

1 1113957gs gblowast1 1113954g2 gblowast

2 1113954g1 gblowastqminus11113954gq

gblowastq 1113954gqminus1 and gblowast

q β middotgq Together with another equationsuch as 1113936

mi1gi +1113936

sr11113957gr +1113936

qp11113954gp 1 that is used to ensure a

bounded and closed space we then have a system of (m+

s+q+1) unknown variables and (m+s+q+1) linearly in-dependent equations erefore this system has a uniquesolution and we can obtain a unique optimal direction

Model (3) is slightly different from that of Chen andDelmas [58] in several aspects first we also take the inputimprovement into account while Chen and Delmas [58]studied only the output improvement secondly we assumethe weak disposability assumption of desirable outputs andundesirable outputs while the strong disposability as-sumption was modeled in Chen and Delmas [58] In ad-dition the VRS assumption is considered in model (3) suchthat it is more suitable for real applications Since the in-dividual inefficiency index theoretically has a value rangingfrom zero to unity for inputs and undesirable outputs and avalue from zero to infinity for desirable outputs the overallaverage inefficiency of DMUd also takes a value from zero toinfinitye larger the average inefficiency index is the moreinefficient the evaluated DMU is An average inefficiencyindex of zero value means that the considered DMU is on theefficiency frontier and has no slack for improvement andhence the DMU is efficient

Furthermore model (4) is a dual formulation of theabove model (3) in its multiplier formulation

inE lowastd Min 1113944m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0

st 1113944

m

i1vixij minus 1113944

s

r1uryrj + 1113944

q

p1wpbpj + u0 ge 0

j 1 n

1113944

m

i1vixij + u0 ge 0 j 1 n

vi ge 1(m + s + q)xid i 1 m

ur ge 1(m + s + q)yrd r 1 s

wp ge 1(m + s + q)bpd p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(4)

Model (4) computes the inefficient component (ie thedifference between aggregated inputs and aggregated out-puts) for the evaluated DMU yet the inefficient componentis nonnegative for all DMUs and some constraints on

4 Complexity

multipliers are held Solving model (4) for each DMUd(d

1 n) determines a series of inefficiency scores inElowastd usinga series of optimal solutions (udlowast

r vdlowasti wdlowast

p udlowast0 ) e in-

efficiency score can be used for a performance indicatoramong all DMUs and the smaller the inefficiency score isthe better the performance of DMUd is

22 Dominance Probability Based on Inefficiency ScoresIt is notable that the inefficiency score inE lowastd obtained pre-viously can be used to analyze the performance of Chineselisted banks but there are two main concerns On the onehand it is calculated by only considering the optimal weightplan (udlowast

r vdlowasti wdlowast

p udlowast0 ) while ignoring any other feasible

weights and thus the obtained inefficiency score inElowastd mightbe too extreme or even unrealistic On the other hand theinefficiency score inE lowastd is calculated separately for eachDMUd(d 1 n) and different weights will be preferredby different DMUs thus the results are not completelycomparable

Since the resulting inefficiency indexes can change rel-ative to different sets of weights it is important to explore theperformance associated with all sets of feasible inputoutputweights To this end we focus on the efficiency dominanceconcept of Salo and Punkka [21] As well defined and dis-cussed in Salo and Punkka [21] and Tang et al [51] thedominance relation is determined through a pairwise effi-ciency comparison among DMUs Furthermore a certainDMU dominates another DMU if and only if its efficiencyscore is as large as that of the other for all sets of feasibleinputoutput weights and is larger for at least some sets offeasible inputoutput weights Here we follow the same ideaof Salo and Punkka [21] and Tang et al [51] to determine thedominance relation of Chinese listed banks For comparisonwe take inevitable undesirable outputs in banking opera-tions such as bad debts into account Furthermore wefollow Chen and Delmas [58] in focusing on eco-inefficiencyscores rather than efficiency scores through a nonradialdirectional distance function model To this end we firstbuild the inefficiency dominance concept as follows

Definition 1 DMUd dominates DMUk (denoted asDMUd ≻DMUk) if and only if DMUd always has a smallerinefficiency score relative to DMUk for all sets of feasibleinput and output weights

e dominance relation based on inefficiency scores isdetermined if the inefficiency score for a certain DMUd is assmall as that of DMUk for all sets of feasible inputoutputweights and is smaller for at least some sets of feasible inputoutput weightse dominance relation between DMUd andDMUk is determined through their inefficiency comparisonBy rethinking the idea of model (3) and model (4) whichcalculate the maximal inefficiency score we can formulatemodel (5) to calculate the inefficiency range of DMUd whenDMUk is fixed with a prespecified inefficiency level of zeroby requiring the constraint that inEk 1113936

mi1vixikminus

1113936sr1uryrk + 1113936

qp1wpbpk + u0 0 In fact the inefficiency

level of DMUk can be set to any nonnegative valueM and bysubstituting u0 with u0 minus M we can get the same dominance

probability as given in Definition 2 and Definition 3 hencewe immediately set the inefficiency level of DMUk to zero forsimplification in the following model

inEmaxd1113957k

inEmind1113957k

MinMax

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xid

i 1 m

ur ge1

(m + s + q)yrd

r 1 s

wp ge1

(m + s + q)bpd

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(5)

An additional constraint that 1113936mi1vi + 1113936

sr1ur+

1113936qp1wp 1 is inserted into model (5) to make the feasible

weight space closed and bounded e optimal objectivefunction formodel (5) shows the lower and upper bounds onhow much different DMUdrsquos inefficiency score can be rel-ative to DMUk across all sets of feasible inputoutputweights Using model (5) the dominance structure can bedetermined for any pairwise DMUs More specifically if theinefficiency level of DMUk is fixed to zero and if inEmax

d1113957klt 0

which means that DMUd will always have a smaller inef-ficiency score compared with DMUk then DMUd dominatesDMUk In contrast if inEmin

d1113957kgt 0 which means that DMUd

will always have a larger inefficiency score than DMUk

(which has an inefficiency level of zero) then DMUd isdominated by DMUk For more general cases however wecannot obtain the complete dominance relation which isusually true in practice erefore we propose determiningthe performance dominance probability To this end wefollow the work of Tang et al [51] in building the inefficiencydominance probability concept as given in Definition 2

Definition 2 When DMUk has an inefficiency score of inEkthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byPd1113957≻k (inEk minus inEmin

d1113957kinEmax

d1113957kminus inEmin

d1113957k)

It is clear that if inEk le inEmind1113957k

which means that DMUd

will always have a larger inefficiency by remaining theinefficiency inEk for DMUk the inefficiency dominance

Complexity 5

probability of DMUd relative to DMUk would take anonpositive value at is it is impossible for DMUd todominate DMUk For the sake of a bounded range weassume that Pd1113957≻k 0 if inEk le inEmin

d1113957k For a more general

case in which inEmind1113957kle inEk le inEmax

d1113957k the probability that

DMUd dominates DMUk takes a value from zero to unity Alarger value of Pd1113957≻k means that it is more likely for DMUd tohave a smaller inefficiency score compared with DMUk overall sets of feasible inputoutput weights

Note in addition that Definition 2 fixes the inefficiencylevel of DMUk to calculate the inefficiency range of DMUd

and further computes the inefficiency dominance proba-bility of DMUd relative to DMUk In contrast we can alsocompute the inefficiency dominance probability of DMUd

relative to DMUk by fixing the inefficiency level of DMUd

and calculating the inefficiency range of DMUk e aboveidea is formulated in model (6) which is very similar tomodel (5) but substitutes DMUk for DMUd

inEmaxk1113957d

inEmink1113957d

MinMax

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xik

i 1 m

ur ge1

(m + s + q)yrk

r 1 s

wp ge1

(m + s + q)bpk

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(6)

An alternative inefficiency dominance probability ofDMUd relative to DMUk by fixing the inefficiency level ofDMUd is given in Definition 3

Definition 3 When DMUd has an inefficiency score of inEdthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byP

d≻1113957k (inEmax

k1113957dminus inEdinEmax

k1113957dminus inEmin

k1113957d)

Furthermore the overall probability that the inefficiencyscore of DMUd dominates that of DMUk is the average of thetwo probabilities by fixing the inefficiency level of DMUd

and DMUk More specifically Definition 4 gives the pairwise

performance dominance probability with regard to ineffi-ciency scores

Definition 4 e pairwise performance dominance prob-ability of DMUd relative to DMUk over all sets of feasibleinput and output weights is Pd≻k ((Pd1113957≻k + Pd≻k1113957)2)

e classic DEA methods allow each DMU to generate aset of relative weights to maximize its ratio of aggregatedweighted outputs to aggregated weighted inputs while en-suring that the same ratio is no more than one for all DMUsand the maximum ratio is considered the efficiency score forthe evaluated DMU By taking all sets of feasible weights andthe inefficiency dominance structure into account theoverall inefficiency dominance probability for a certainDMUd among all DMUs can be calculated by the average ofpair inefficiency dominance probabilities across all DMUse above idea is given in Definition 5

Definition 5 e performance dominance probability ofDMUd across all DMUs over all sets of feasible input andoutput weights is Pd 1113936

nk1Pd≻kn

e classic DEA approaches use deterministic (in)effi-ciency scores to measure the relative performance while theperformance dominance probability is an alternative per-formance indicator from a data analytics perspective thatconsiders all sets of feasible weights that are stochastic todetermine the relative performance It is rather remarkablethat different sets of weights will cause different performancemeasures and the performance dominance probability in-volves all possibilities over all sets of feasible input andoutput weights e performance dominance probability ofDMUd calculates the probability of its performance domi-nating the set of all DMUs A larger performance dominanceprobability indicates that it is much easier for that DMU todominate other DMUs implying that its inefficiency score ismore likely to be smaller than that of other DMUs

3 Illustrative Application of ChineseListed Banks

In this section we illustrate the proposed approach usingempirical performance analytics for 32 Chinese listed banksSince the proposed approach considers all sets of feasibleweights the performance relations and ratio index resultsare more comprehensive and reasonable

31 Data Description is section addresses the perfor-mance of listed banks in China For simplification and re-search purposes we consider only those banks that havebeen registered in China and that are listed in the mainlandof China More specifically only banks that are owned byChinese organizations and are listed on the Shenzhen StockExchange and Shanghai Stock Exchange are collected Incontrast neither Chinese banks listed on other stock ex-changes nor foreign banks listed on the Shenzhen StockExchange and Shanghai Stock Exchange are involved in thisstudy As a result we have 32 listed banks For the research

6 Complexity

purpose we give these 32 banks and their correspondingcodes in Table 1

In practice each bank will consume multiple inputs togenerate multiple outputs and more specifically mainly forprofits In this study we follow similar studies such as Wanget al [4] Zha et al [1] Fukuyama andMatousek [64] Li et al[65] and Zhu et al [2] in taking employment referring tohuman resource investment and manpower fixed assetreferring to the asset value of physical capital that can beused for business activities and operation cost as threeinputs Note in addition that the operation cost in this studyexcludes the expense of labor input that occurs in bankingoperations because the employment has already taken thelabor input into account Furthermore we consider threedifferent outputs generated in banking operations withinterest income and noninterest income being two desirableoutputs and nonperforming bad loan percentage in thecurrent year as an undesirable outpute interest income isderived directly from the gap between the interest paid ondeposits and the interest earned from loans while thenoninterest income is primarily derived from commissionssecurities investments fees and other business activity in-comes However a bad loan is a jointly produced andunavoidable by-product that will greatly harm the bank einputs and outputs to be used in this study are summarizedin Table 2

Our empirical study contains operation data for 32Chinese listed banks over the 2014ndash2018 period accountingfor 160 observations All data for these listed banks werecollected from official sources of bank annual reports and thefinancial reports of banks in China StockMarket AccountingResearch (CSMAR) during 2014ndash2018 Table 3 shows thedescriptive statistics of the inputs desirable outputs andundesirable outputs of these 160 observations It can befound that both the average fixed asset and operation costamong the 32 listed banks are increasing year by year but theaverage employment and bad debt percentage increased to apeak in 2016 and then decreased continuously Furthermoreboth interest income and noninterest income show an in-creasing trend but decreased in a year

32 Result Analysis To provide data analytics of bankperformance we first calculate the inefficiency scores usingmodel (3) or model (4) in the nonradial DDF-based for-mulation under the VRS property e inefficiencies of these32 banks from 2014 to 2018 are given in Table 4 Since theinefficiency score represents potential improvements di-vided by the observed inputs or outputs an inefficiency scoreof zero indicates that there will be no improvement po-tential Table 4 shows that by selecting the optimal set ofweights to aggregate inputs and outputs many listed bankswill be extremely efficient without improvement potentialsere are always ten banks that have an inefficiency scorelarger than zero but the average inefficiency score acrossthese 32 listed banks fluctuates according to the year Sincethe VRS analysis is adopted in this study we may not drawany significant conclusion as to inefficiency changes year byyear Furthermore some banks (DMU6 Bank of Guiyang

DMU8 Huaxia Bank DMU15 Bank of Ningbo and DMU32Zijin Rural Commercial Bank) will always have a nonzeroinefficiency meaning that these banks always show veryterrible performance compared with banks that have a zero-value inefficiency index

Since the previous inefficiency scores are separatelyderived by considering only one optimal set of weights theresulting performance information might be unrealistic andunreasonable erefore we can use the proposed eco-in-efficiency dominance probability approach in this paper toanalyze the banking performance considering all sets offeasible weights To this end we first use model (5) to

Table 1 Codes for 32 Chinese listed banks

DMUs Banks AbbreviationDMU1 Bank of Beijing BOBDMU2 Changshu Rural Commercial Bank CRCBDMU3 Bank of Chengdu BCD

DMU4Industrial and Commercial Bank of

China ICBC

DMU5 China Everbright Bank CEBDMU6 Bank of Guiyang BOGDMU7 Bank of Hangzhou BOHDMU8 Huaxia Bank HBDMU9 China Construction Bank CCBDMU10 Bank of Jiangsu BOJDMU11 Jiangyin Rural Commercial Bank JRCBDMU12 Bank of Communications BCDMU13 China Minsheng Bank CMSBDMU14 Bank of Nanjing BONDMU15 Bank of Ningbo BNDMU16 Agricultural Bank of China ABCDMU17 Ping An Bank PBDMU18 Shanghai Pudong Development Bank SPDBDMU19 Bank of Qingdao BOQDMU20 Qingdao Rural Commercial Bank QRCBDMU21 Bank of Shanghai BOSDMU22 Suzhou Rural Commercial Bank SRCBDMU23 Wuxi Rural Commercial Bank WRCBDMU24 Bank of Xian BOXDMU25 Industrial Bank IBDMU26 Rural Commercial Bank of Zhangjiagang RCBZDMU27 Bank of Changsha BCSDMU28 China Merchants Bank CMBDMU29 Bank of Zhengzhou BOZDMU30 Bank of China BOCDMU31 China CITIC Bank CCBDMU32 Zijin Rural Commercial Bank ZRCB

Table 2 Input and output variables

Inputoutput Variable Notation Unit

InputsEmployment x1

Personcount

Fixed assets x2 Million yuanOperation costs x3 Million yuan

Desirable outputsInterest income y1 Million yuanNoninterestincome y2 Million yuan

Undesirableoutputs

Bad debtpercentage z1 Percentage

Complexity 7

Table 3 Descriptive statistics for input-output measures

Year Statistics x1 x2 x3 y1 y2 z1

2014

Max 493583 196238 271132 849879 165370 27100Min 1089 41201 115141 353390 10948 07500Mean 67582 2849915 4823162 15411269 2568599 12531SD 135316 5589804 7815135 23578569 4416322 04237

2015

Max 503082 221502 306395 871779 189780 23900Min 1180 41804 124526 348611 13069 08300Mean 69772 3133597 5738723 16210941 3034405 15308SD 136444 6026226 8745917 24329478 4941992 04226

2016

Max 496698 243619 282712 791480 204045 24100Min 1352 41399 123588 326889 16262 08700Mean 70266 3477886 5828596 15179021 3556052 16013SD 134489 6511810 8516872 22069357 5718980 03678

2017

Max 487307 245687 331518 861594 204424 23900Min 1426 46293 126089 378914 16605 08200Mean 69782 3638326 6007996 16692792 3516824 15431SD 132071 6662176 9168104 23946371 5304531 03240

2018

Max 473691 253525 368966 948094 201271 24700Min 1454 41808 173355 456700 20317 07800Mean 69354 3838034 6744402 18305026 3918308 15034SD 129697 7006174 10096472 26175120 5500040 03382

Table 4 Inefficiency scores of 32 listed banks from 2014 to 2018

Banks 2014 2015 2016 2017 2018DMU1 00000 00000 00000 00000 00000DMU2 00000 03483 06442 05430 03483DMU3 00000 00979 02730 00000 00979DMU4 00000 00000 00000 00000 00000DMU5 00000 00590 00000 01087 00590DMU6 02345 03469 03273 05898 03469DMU7 01798 00000 00000 00000 00000DMU8 02880 03059 02288 02768 03059DMU9 00000 00000 00000 00000 00000DMU10 00000 00000 00000 00000 00000DMU11 00000 00000 00000 00000 00000DMU12 01449 00000 00000 00000 00000DMU13 00000 00000 00000 00000 00000DMU14 02088 01072 01415 00000 01072DMU15 02793 02415 02018 02225 02415DMU16 01926 00000 00000 00000 00000DMU17 00000 00000 00000 00000 00000DMU18 00567 00000 00000 00000 00000DMU19 00000 00000 00827 00000 00000DMU20 00000 00000 00000 00000 00000DMU21 00000 00000 00000 00000 00000DMU22 00000 00000 00000 02032 00000DMU23 00000 00000 00000 00000 00000DMU24 00000 00000 00000 00000 00000DMU25 00000 00000 00000 00000 00000DMU26 00000 00000 00000 00000 00000DMU27 00000 02287 02670 06379 02287DMU28 00000 00000 00000 00000 00000DMU29 00000 00000 01373 00967 00000DMU30 00000 00000 00000 00000 00000DMU31 00634 00799 00000 00000 00799DMU32 05982 02792 03526 05700 02792Mean 00702 00655 00830 01015 00655

8 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

undesirable by-products such as bad debts that are jointlyproduced with incomes an extended eco-inefficiencymodel is developed Furthermore the eco-inefficiencymodel is used to develop the pair dominance structuretaking all sets of weights into account based on Tang et al[51] e performance dominance probability is proposedto calculate the average probability of a certain DMUrsquosperformance dominating all DMUs A larger performancedominance probability indicates that it is easier for thatDMU to dominate all other DMUs implying that its in-efficiency score is more likely to be smaller than that ofother DMUs in the sense of data-driven analytics Finallythe proposed approach is applied to a four-year dataset of32 Chinese listed banks and the empirical results show that(1) the inefficiency dominance probability is largely dif-ferent from inefficiency scores (2) these state-ownedcommercial banks are more likely to have a better domi-nance performance while local rural commercial banksmight have very promising inefficiency performances insome extreme situations but are likely to have poor per-formance in the sense of inefficiency dominance proba-bility and (3) China Construction Bank Industrial andCommercial Bank of China and Industrial Bank are the topthree listed banks based on operations analytics and on thecontrary Suzhou Rural Commercial Bank Rural Com-mercial Bank of Zhangjiagang and Jiangyin Rural Com-mercial Bank are the lowest three banks with inefficiencydominance probability is paper contributes to the lit-erature in at least the following aspects First this paperextends a new approach in the DEA framework by taking allsets of feasible weights into account whereas previousstudies have considered only one or several sets of weightsSecond this paper establishes a pairwise performancedominance concept which can help decision-makers an-alyze the performance relation in any context with priceinformation (ie weights or multipliers) ird this paperanalyzes the performance of Chinese listed banks andprovides some empirical findings which can facilitate thebanking industries in China

e remainder of this paper is organized as followsSection 2 develops the mathematical methodology of anextended eco-inefficiency model with undesirable outputsand performance dominance structure using inefficiencyscores Afterwards the proposed approach is used to studythe empirical performance analytics of 32 listed banks inSection 3 Finally Section 4 concludes and summarizes thispaper

2 Mathematical Modeling

We first propose an extended eco-inefficiency model toaddress banking activities and possible relative perfor-mances in Section 21 Furthermore we build a performancedominance structure considering all sets of feasible weightsin Section 22

21 An Extended Eco-Inefficiency Model Suppose a set of npeer banks with each bank using m inputs for the sake of

producing s traditional desirable outputs as well as q un-desirable outputs such as bad debt in the illustrative ap-plication A bad debt or nonperforming loan is a jointlyproduced and unavoidable by-product in the banking in-dustry Without loss of generality we consider each bank asa homogeneous decision-making unit (DMU) in the DEAframework Furthermore DMUj(j 1 n) consumeinputs Xj (x1j xmj) to produce desirable outputsYj (y1j ysj) and undesirable outputsBj (b1j bqj) respectively Before constructing themodeling a core task is to identify appropriate methods tohandle undesirable outputs It is clear that the strong andweak disposability assumption of undesirable outputs anddesirable outputs are the two most common and naturalmethods in the literature [55ndash57] A significant featurebetween the strong and weak disposability assumption iswhether undesirable outputs can be produced withoutdamage or subsequent cost to desirable outputs [58] Ifundesirable outputs can be freely generated withoutdamage or subsequent cost implying that both inputs andoutputs can change unilaterally without compromisingeach other then undesirable outputs are assumed to bestrongly disposable In contrast if the production of un-desirable outputs indeed has some damage or subsequentcost to inputs or desirable outputs implying that a re-duction in undesirable outputs would result in a reductionof desirable outputs simultaneously [59 60] then unde-sirable outputs are assumed to be weak disposable Here weconsider the weak disposability assumption because it ismore suitable for the real world and more specificallyfor the empirical application of banking operations whereit is very difficult to freely reduce bad debts without af-fecting incomes and changing banking operations To thisend the production possibility set (PPS) under the variablereturns to scale (VRS) assumption can be formulated asfollows

PPS xi yr bp1113872 1113873

1113944n

j1λjxij le xi i 1 m

ρj1113944n

j1λjyrj geyr r 1 s

ρj1113944n

j1λjbpj bp p 1 q

1113944n

j1λj 1

0le ρj le 1 λj ge 0 j 1 n

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(1)

e above formula is nonlinear since both the reductionfactor ρj and intensity variable λj are unknown Further-more we can equivalently change formula (1) into a linearversion which is presented in the following formula

PPS xi yr bp1113872 1113873

1113944n

j1 λj + ηj1113872 1113873xij lexi i 1 m

1113944n

j1λjyrj geyr r 1 s

1113944n

j1λjbpj bp p 1 q

1113944n

j1 λj + ηj1113872 1113873 1 λj ηj ge 0 j 1 n

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2)

Complexity 3

Based on formula (2) both desirable and undesirableoutputs are weighted by nondisposed intensity variables λjwhereas the inputs are weighted by the sum of nondisposedintensity variables λj and disposed intensity variables ηj Inaddition the VRS assumption is ensured by summing thetotal nondisposed intensity variables λj and disposed in-tensity variables ηj to 1 ie 1113936

nj1(λj + ηj) 1

Based on the above PPS in formula (2) we can buildmathematical models to compute the relative performanceof DMUs In this paper we follow the practice of Chen andDelmas [58] to develop an extended eco-inefficiency modelfor performance evaluation since the eco-inefficiency modelhas some advantages in modeling activities with undesirableoutputs compared with four well-established models in theliterature (undesirable outputs as inputs transformation ofundesirable outputs directional distance function andhyperbolic efficiency model Readers can refer Chen andDelmas [58] for details on the comparison) e extendedeco-inefficiency model is formulated as follows

inElowastd Max1113936

mi1g

xi xid + 1113936

sr1g

yr yrd + 1113936

qp1g

bpbpd

m + s + q

st 1113944

n

j1λj + ηj1113872 1113873xij lexid minus g

mi i 1 m

1113944

n

j1λjyrj geyrd + g

yr r 1 s

1113944

n

j1λjbpj bpd minus g

bp p 1 q

1113944

n

j1λj + ηj1113872 1113873 1

λj ηj gmi g

yr gb

p ge 0 j 1 n

r 1 s p 1 q

(3)

In model (3) the variables gxi g

yr and gb

p represent theamount of potential improvements in inputs desirableoutputs and undesirable outputs respectively that theevaluated DMU can make relative to its current input usageand output production to reach its ideal benchmark targeton the efficiency frontiere potential improvements reflectinput reduction potentials and desirable output expansionpotentials (or undesirable output reduction potentials) in-stead of actual input usage and output production [61 62]Model (3) uses a slack-based formula that is similar to thedirectional distance function (DDF) model to maximize theaverage additive inefficiency index across all input andoutput measures where the inefficiency index representspotential improvements divided by observed inputs oroutputs More specifically the optimal direction vector ofmodel (3) can be endogenously obtained in a similar way asthat of Arabi et al [63] For that purpose suppose

gxi β middot gi g

yr β middot 1113957gr and gb

p β middot 1113954gp where the directionvector is g (gi 1113957gr 1113954gp) Solving model (3) determines anoptimal solution (gxlowast

i gylowastr gblowast

p ) then we have β

(gxlowasti gi) (g

ylowastr 1113957gr) (gblowast

p 1113954gp)(i 1 m r 1 s

p 1 q) Hence we have a system of (m + s + q + 1)

unknown variables (β gi 1113957gr 1113954gp) and (m + s + q) linearlyindependent equations gxlowast

1 g2 gxlowast2 g1 gxlowast

m 1113957g1 gylowast1 gm

gylowast1 1113957g2 g

ylowast2 1113957g1 g

ylowasts 1113954g1 gblowast

1 1113957gs gblowast1 1113954g2 gblowast

2 1113954g1 gblowastqminus11113954gq

gblowastq 1113954gqminus1 and gblowast

q β middotgq Together with another equationsuch as 1113936

mi1gi +1113936

sr11113957gr +1113936

qp11113954gp 1 that is used to ensure a

bounded and closed space we then have a system of (m+

s+q+1) unknown variables and (m+s+q+1) linearly in-dependent equations erefore this system has a uniquesolution and we can obtain a unique optimal direction

Model (3) is slightly different from that of Chen andDelmas [58] in several aspects first we also take the inputimprovement into account while Chen and Delmas [58]studied only the output improvement secondly we assumethe weak disposability assumption of desirable outputs andundesirable outputs while the strong disposability as-sumption was modeled in Chen and Delmas [58] In ad-dition the VRS assumption is considered in model (3) suchthat it is more suitable for real applications Since the in-dividual inefficiency index theoretically has a value rangingfrom zero to unity for inputs and undesirable outputs and avalue from zero to infinity for desirable outputs the overallaverage inefficiency of DMUd also takes a value from zero toinfinitye larger the average inefficiency index is the moreinefficient the evaluated DMU is An average inefficiencyindex of zero value means that the considered DMU is on theefficiency frontier and has no slack for improvement andhence the DMU is efficient

Furthermore model (4) is a dual formulation of theabove model (3) in its multiplier formulation

inE lowastd Min 1113944m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0

st 1113944

m

i1vixij minus 1113944

s

r1uryrj + 1113944

q

p1wpbpj + u0 ge 0

j 1 n

1113944

m

i1vixij + u0 ge 0 j 1 n

vi ge 1(m + s + q)xid i 1 m

ur ge 1(m + s + q)yrd r 1 s

wp ge 1(m + s + q)bpd p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(4)

Model (4) computes the inefficient component (ie thedifference between aggregated inputs and aggregated out-puts) for the evaluated DMU yet the inefficient componentis nonnegative for all DMUs and some constraints on

4 Complexity

multipliers are held Solving model (4) for each DMUd(d

1 n) determines a series of inefficiency scores inElowastd usinga series of optimal solutions (udlowast

r vdlowasti wdlowast

p udlowast0 ) e in-

efficiency score can be used for a performance indicatoramong all DMUs and the smaller the inefficiency score isthe better the performance of DMUd is

22 Dominance Probability Based on Inefficiency ScoresIt is notable that the inefficiency score inE lowastd obtained pre-viously can be used to analyze the performance of Chineselisted banks but there are two main concerns On the onehand it is calculated by only considering the optimal weightplan (udlowast

r vdlowasti wdlowast

p udlowast0 ) while ignoring any other feasible

weights and thus the obtained inefficiency score inElowastd mightbe too extreme or even unrealistic On the other hand theinefficiency score inE lowastd is calculated separately for eachDMUd(d 1 n) and different weights will be preferredby different DMUs thus the results are not completelycomparable

Since the resulting inefficiency indexes can change rel-ative to different sets of weights it is important to explore theperformance associated with all sets of feasible inputoutputweights To this end we focus on the efficiency dominanceconcept of Salo and Punkka [21] As well defined and dis-cussed in Salo and Punkka [21] and Tang et al [51] thedominance relation is determined through a pairwise effi-ciency comparison among DMUs Furthermore a certainDMU dominates another DMU if and only if its efficiencyscore is as large as that of the other for all sets of feasibleinputoutput weights and is larger for at least some sets offeasible inputoutput weights Here we follow the same ideaof Salo and Punkka [21] and Tang et al [51] to determine thedominance relation of Chinese listed banks For comparisonwe take inevitable undesirable outputs in banking opera-tions such as bad debts into account Furthermore wefollow Chen and Delmas [58] in focusing on eco-inefficiencyscores rather than efficiency scores through a nonradialdirectional distance function model To this end we firstbuild the inefficiency dominance concept as follows

Definition 1 DMUd dominates DMUk (denoted asDMUd ≻DMUk) if and only if DMUd always has a smallerinefficiency score relative to DMUk for all sets of feasibleinput and output weights

e dominance relation based on inefficiency scores isdetermined if the inefficiency score for a certain DMUd is assmall as that of DMUk for all sets of feasible inputoutputweights and is smaller for at least some sets of feasible inputoutput weightse dominance relation between DMUd andDMUk is determined through their inefficiency comparisonBy rethinking the idea of model (3) and model (4) whichcalculate the maximal inefficiency score we can formulatemodel (5) to calculate the inefficiency range of DMUd whenDMUk is fixed with a prespecified inefficiency level of zeroby requiring the constraint that inEk 1113936

mi1vixikminus

1113936sr1uryrk + 1113936

qp1wpbpk + u0 0 In fact the inefficiency

level of DMUk can be set to any nonnegative valueM and bysubstituting u0 with u0 minus M we can get the same dominance

probability as given in Definition 2 and Definition 3 hencewe immediately set the inefficiency level of DMUk to zero forsimplification in the following model

inEmaxd1113957k

inEmind1113957k

MinMax

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xid

i 1 m

ur ge1

(m + s + q)yrd

r 1 s

wp ge1

(m + s + q)bpd

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(5)

An additional constraint that 1113936mi1vi + 1113936

sr1ur+

1113936qp1wp 1 is inserted into model (5) to make the feasible

weight space closed and bounded e optimal objectivefunction formodel (5) shows the lower and upper bounds onhow much different DMUdrsquos inefficiency score can be rel-ative to DMUk across all sets of feasible inputoutputweights Using model (5) the dominance structure can bedetermined for any pairwise DMUs More specifically if theinefficiency level of DMUk is fixed to zero and if inEmax

d1113957klt 0

which means that DMUd will always have a smaller inef-ficiency score compared with DMUk then DMUd dominatesDMUk In contrast if inEmin

d1113957kgt 0 which means that DMUd

will always have a larger inefficiency score than DMUk

(which has an inefficiency level of zero) then DMUd isdominated by DMUk For more general cases however wecannot obtain the complete dominance relation which isusually true in practice erefore we propose determiningthe performance dominance probability To this end wefollow the work of Tang et al [51] in building the inefficiencydominance probability concept as given in Definition 2

Definition 2 When DMUk has an inefficiency score of inEkthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byPd1113957≻k (inEk minus inEmin

d1113957kinEmax

d1113957kminus inEmin

d1113957k)

It is clear that if inEk le inEmind1113957k

which means that DMUd

will always have a larger inefficiency by remaining theinefficiency inEk for DMUk the inefficiency dominance

Complexity 5

probability of DMUd relative to DMUk would take anonpositive value at is it is impossible for DMUd todominate DMUk For the sake of a bounded range weassume that Pd1113957≻k 0 if inEk le inEmin

d1113957k For a more general

case in which inEmind1113957kle inEk le inEmax

d1113957k the probability that

DMUd dominates DMUk takes a value from zero to unity Alarger value of Pd1113957≻k means that it is more likely for DMUd tohave a smaller inefficiency score compared with DMUk overall sets of feasible inputoutput weights

Note in addition that Definition 2 fixes the inefficiencylevel of DMUk to calculate the inefficiency range of DMUd

and further computes the inefficiency dominance proba-bility of DMUd relative to DMUk In contrast we can alsocompute the inefficiency dominance probability of DMUd

relative to DMUk by fixing the inefficiency level of DMUd

and calculating the inefficiency range of DMUk e aboveidea is formulated in model (6) which is very similar tomodel (5) but substitutes DMUk for DMUd

inEmaxk1113957d

inEmink1113957d

MinMax

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xik

i 1 m

ur ge1

(m + s + q)yrk

r 1 s

wp ge1

(m + s + q)bpk

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(6)

An alternative inefficiency dominance probability ofDMUd relative to DMUk by fixing the inefficiency level ofDMUd is given in Definition 3

Definition 3 When DMUd has an inefficiency score of inEdthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byP

d≻1113957k (inEmax

k1113957dminus inEdinEmax

k1113957dminus inEmin

k1113957d)

Furthermore the overall probability that the inefficiencyscore of DMUd dominates that of DMUk is the average of thetwo probabilities by fixing the inefficiency level of DMUd

and DMUk More specifically Definition 4 gives the pairwise

performance dominance probability with regard to ineffi-ciency scores

Definition 4 e pairwise performance dominance prob-ability of DMUd relative to DMUk over all sets of feasibleinput and output weights is Pd≻k ((Pd1113957≻k + Pd≻k1113957)2)

e classic DEA methods allow each DMU to generate aset of relative weights to maximize its ratio of aggregatedweighted outputs to aggregated weighted inputs while en-suring that the same ratio is no more than one for all DMUsand the maximum ratio is considered the efficiency score forthe evaluated DMU By taking all sets of feasible weights andthe inefficiency dominance structure into account theoverall inefficiency dominance probability for a certainDMUd among all DMUs can be calculated by the average ofpair inefficiency dominance probabilities across all DMUse above idea is given in Definition 5

Definition 5 e performance dominance probability ofDMUd across all DMUs over all sets of feasible input andoutput weights is Pd 1113936

nk1Pd≻kn

e classic DEA approaches use deterministic (in)effi-ciency scores to measure the relative performance while theperformance dominance probability is an alternative per-formance indicator from a data analytics perspective thatconsiders all sets of feasible weights that are stochastic todetermine the relative performance It is rather remarkablethat different sets of weights will cause different performancemeasures and the performance dominance probability in-volves all possibilities over all sets of feasible input andoutput weights e performance dominance probability ofDMUd calculates the probability of its performance domi-nating the set of all DMUs A larger performance dominanceprobability indicates that it is much easier for that DMU todominate other DMUs implying that its inefficiency score ismore likely to be smaller than that of other DMUs

3 Illustrative Application of ChineseListed Banks

In this section we illustrate the proposed approach usingempirical performance analytics for 32 Chinese listed banksSince the proposed approach considers all sets of feasibleweights the performance relations and ratio index resultsare more comprehensive and reasonable

31 Data Description is section addresses the perfor-mance of listed banks in China For simplification and re-search purposes we consider only those banks that havebeen registered in China and that are listed in the mainlandof China More specifically only banks that are owned byChinese organizations and are listed on the Shenzhen StockExchange and Shanghai Stock Exchange are collected Incontrast neither Chinese banks listed on other stock ex-changes nor foreign banks listed on the Shenzhen StockExchange and Shanghai Stock Exchange are involved in thisstudy As a result we have 32 listed banks For the research

6 Complexity

purpose we give these 32 banks and their correspondingcodes in Table 1

In practice each bank will consume multiple inputs togenerate multiple outputs and more specifically mainly forprofits In this study we follow similar studies such as Wanget al [4] Zha et al [1] Fukuyama andMatousek [64] Li et al[65] and Zhu et al [2] in taking employment referring tohuman resource investment and manpower fixed assetreferring to the asset value of physical capital that can beused for business activities and operation cost as threeinputs Note in addition that the operation cost in this studyexcludes the expense of labor input that occurs in bankingoperations because the employment has already taken thelabor input into account Furthermore we consider threedifferent outputs generated in banking operations withinterest income and noninterest income being two desirableoutputs and nonperforming bad loan percentage in thecurrent year as an undesirable outpute interest income isderived directly from the gap between the interest paid ondeposits and the interest earned from loans while thenoninterest income is primarily derived from commissionssecurities investments fees and other business activity in-comes However a bad loan is a jointly produced andunavoidable by-product that will greatly harm the bank einputs and outputs to be used in this study are summarizedin Table 2

Our empirical study contains operation data for 32Chinese listed banks over the 2014ndash2018 period accountingfor 160 observations All data for these listed banks werecollected from official sources of bank annual reports and thefinancial reports of banks in China StockMarket AccountingResearch (CSMAR) during 2014ndash2018 Table 3 shows thedescriptive statistics of the inputs desirable outputs andundesirable outputs of these 160 observations It can befound that both the average fixed asset and operation costamong the 32 listed banks are increasing year by year but theaverage employment and bad debt percentage increased to apeak in 2016 and then decreased continuously Furthermoreboth interest income and noninterest income show an in-creasing trend but decreased in a year

32 Result Analysis To provide data analytics of bankperformance we first calculate the inefficiency scores usingmodel (3) or model (4) in the nonradial DDF-based for-mulation under the VRS property e inefficiencies of these32 banks from 2014 to 2018 are given in Table 4 Since theinefficiency score represents potential improvements di-vided by the observed inputs or outputs an inefficiency scoreof zero indicates that there will be no improvement po-tential Table 4 shows that by selecting the optimal set ofweights to aggregate inputs and outputs many listed bankswill be extremely efficient without improvement potentialsere are always ten banks that have an inefficiency scorelarger than zero but the average inefficiency score acrossthese 32 listed banks fluctuates according to the year Sincethe VRS analysis is adopted in this study we may not drawany significant conclusion as to inefficiency changes year byyear Furthermore some banks (DMU6 Bank of Guiyang

DMU8 Huaxia Bank DMU15 Bank of Ningbo and DMU32Zijin Rural Commercial Bank) will always have a nonzeroinefficiency meaning that these banks always show veryterrible performance compared with banks that have a zero-value inefficiency index

Since the previous inefficiency scores are separatelyderived by considering only one optimal set of weights theresulting performance information might be unrealistic andunreasonable erefore we can use the proposed eco-in-efficiency dominance probability approach in this paper toanalyze the banking performance considering all sets offeasible weights To this end we first use model (5) to

Table 1 Codes for 32 Chinese listed banks

DMUs Banks AbbreviationDMU1 Bank of Beijing BOBDMU2 Changshu Rural Commercial Bank CRCBDMU3 Bank of Chengdu BCD

DMU4Industrial and Commercial Bank of

China ICBC

DMU5 China Everbright Bank CEBDMU6 Bank of Guiyang BOGDMU7 Bank of Hangzhou BOHDMU8 Huaxia Bank HBDMU9 China Construction Bank CCBDMU10 Bank of Jiangsu BOJDMU11 Jiangyin Rural Commercial Bank JRCBDMU12 Bank of Communications BCDMU13 China Minsheng Bank CMSBDMU14 Bank of Nanjing BONDMU15 Bank of Ningbo BNDMU16 Agricultural Bank of China ABCDMU17 Ping An Bank PBDMU18 Shanghai Pudong Development Bank SPDBDMU19 Bank of Qingdao BOQDMU20 Qingdao Rural Commercial Bank QRCBDMU21 Bank of Shanghai BOSDMU22 Suzhou Rural Commercial Bank SRCBDMU23 Wuxi Rural Commercial Bank WRCBDMU24 Bank of Xian BOXDMU25 Industrial Bank IBDMU26 Rural Commercial Bank of Zhangjiagang RCBZDMU27 Bank of Changsha BCSDMU28 China Merchants Bank CMBDMU29 Bank of Zhengzhou BOZDMU30 Bank of China BOCDMU31 China CITIC Bank CCBDMU32 Zijin Rural Commercial Bank ZRCB

Table 2 Input and output variables

Inputoutput Variable Notation Unit

InputsEmployment x1

Personcount

Fixed assets x2 Million yuanOperation costs x3 Million yuan

Desirable outputsInterest income y1 Million yuanNoninterestincome y2 Million yuan

Undesirableoutputs

Bad debtpercentage z1 Percentage

Complexity 7

Table 3 Descriptive statistics for input-output measures

Year Statistics x1 x2 x3 y1 y2 z1

2014

Max 493583 196238 271132 849879 165370 27100Min 1089 41201 115141 353390 10948 07500Mean 67582 2849915 4823162 15411269 2568599 12531SD 135316 5589804 7815135 23578569 4416322 04237

2015

Max 503082 221502 306395 871779 189780 23900Min 1180 41804 124526 348611 13069 08300Mean 69772 3133597 5738723 16210941 3034405 15308SD 136444 6026226 8745917 24329478 4941992 04226

2016

Max 496698 243619 282712 791480 204045 24100Min 1352 41399 123588 326889 16262 08700Mean 70266 3477886 5828596 15179021 3556052 16013SD 134489 6511810 8516872 22069357 5718980 03678

2017

Max 487307 245687 331518 861594 204424 23900Min 1426 46293 126089 378914 16605 08200Mean 69782 3638326 6007996 16692792 3516824 15431SD 132071 6662176 9168104 23946371 5304531 03240

2018

Max 473691 253525 368966 948094 201271 24700Min 1454 41808 173355 456700 20317 07800Mean 69354 3838034 6744402 18305026 3918308 15034SD 129697 7006174 10096472 26175120 5500040 03382

Table 4 Inefficiency scores of 32 listed banks from 2014 to 2018

Banks 2014 2015 2016 2017 2018DMU1 00000 00000 00000 00000 00000DMU2 00000 03483 06442 05430 03483DMU3 00000 00979 02730 00000 00979DMU4 00000 00000 00000 00000 00000DMU5 00000 00590 00000 01087 00590DMU6 02345 03469 03273 05898 03469DMU7 01798 00000 00000 00000 00000DMU8 02880 03059 02288 02768 03059DMU9 00000 00000 00000 00000 00000DMU10 00000 00000 00000 00000 00000DMU11 00000 00000 00000 00000 00000DMU12 01449 00000 00000 00000 00000DMU13 00000 00000 00000 00000 00000DMU14 02088 01072 01415 00000 01072DMU15 02793 02415 02018 02225 02415DMU16 01926 00000 00000 00000 00000DMU17 00000 00000 00000 00000 00000DMU18 00567 00000 00000 00000 00000DMU19 00000 00000 00827 00000 00000DMU20 00000 00000 00000 00000 00000DMU21 00000 00000 00000 00000 00000DMU22 00000 00000 00000 02032 00000DMU23 00000 00000 00000 00000 00000DMU24 00000 00000 00000 00000 00000DMU25 00000 00000 00000 00000 00000DMU26 00000 00000 00000 00000 00000DMU27 00000 02287 02670 06379 02287DMU28 00000 00000 00000 00000 00000DMU29 00000 00000 01373 00967 00000DMU30 00000 00000 00000 00000 00000DMU31 00634 00799 00000 00000 00799DMU32 05982 02792 03526 05700 02792Mean 00702 00655 00830 01015 00655

8 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

Based on formula (2) both desirable and undesirableoutputs are weighted by nondisposed intensity variables λjwhereas the inputs are weighted by the sum of nondisposedintensity variables λj and disposed intensity variables ηj Inaddition the VRS assumption is ensured by summing thetotal nondisposed intensity variables λj and disposed in-tensity variables ηj to 1 ie 1113936

nj1(λj + ηj) 1

Based on the above PPS in formula (2) we can buildmathematical models to compute the relative performanceof DMUs In this paper we follow the practice of Chen andDelmas [58] to develop an extended eco-inefficiency modelfor performance evaluation since the eco-inefficiency modelhas some advantages in modeling activities with undesirableoutputs compared with four well-established models in theliterature (undesirable outputs as inputs transformation ofundesirable outputs directional distance function andhyperbolic efficiency model Readers can refer Chen andDelmas [58] for details on the comparison) e extendedeco-inefficiency model is formulated as follows

inElowastd Max1113936

mi1g

xi xid + 1113936

sr1g

yr yrd + 1113936

qp1g

bpbpd

m + s + q

st 1113944

n

j1λj + ηj1113872 1113873xij lexid minus g

mi i 1 m

1113944

n

j1λjyrj geyrd + g

yr r 1 s

1113944

n

j1λjbpj bpd minus g

bp p 1 q

1113944

n

j1λj + ηj1113872 1113873 1

λj ηj gmi g

yr gb

p ge 0 j 1 n

r 1 s p 1 q

(3)

In model (3) the variables gxi g

yr and gb

p represent theamount of potential improvements in inputs desirableoutputs and undesirable outputs respectively that theevaluated DMU can make relative to its current input usageand output production to reach its ideal benchmark targeton the efficiency frontiere potential improvements reflectinput reduction potentials and desirable output expansionpotentials (or undesirable output reduction potentials) in-stead of actual input usage and output production [61 62]Model (3) uses a slack-based formula that is similar to thedirectional distance function (DDF) model to maximize theaverage additive inefficiency index across all input andoutput measures where the inefficiency index representspotential improvements divided by observed inputs oroutputs More specifically the optimal direction vector ofmodel (3) can be endogenously obtained in a similar way asthat of Arabi et al [63] For that purpose suppose

gxi β middot gi g

yr β middot 1113957gr and gb

p β middot 1113954gp where the directionvector is g (gi 1113957gr 1113954gp) Solving model (3) determines anoptimal solution (gxlowast

i gylowastr gblowast

p ) then we have β

(gxlowasti gi) (g

ylowastr 1113957gr) (gblowast

p 1113954gp)(i 1 m r 1 s

p 1 q) Hence we have a system of (m + s + q + 1)

unknown variables (β gi 1113957gr 1113954gp) and (m + s + q) linearlyindependent equations gxlowast

1 g2 gxlowast2 g1 gxlowast

m 1113957g1 gylowast1 gm

gylowast1 1113957g2 g

ylowast2 1113957g1 g

ylowasts 1113954g1 gblowast

1 1113957gs gblowast1 1113954g2 gblowast

2 1113954g1 gblowastqminus11113954gq

gblowastq 1113954gqminus1 and gblowast

q β middotgq Together with another equationsuch as 1113936

mi1gi +1113936

sr11113957gr +1113936

qp11113954gp 1 that is used to ensure a

bounded and closed space we then have a system of (m+

s+q+1) unknown variables and (m+s+q+1) linearly in-dependent equations erefore this system has a uniquesolution and we can obtain a unique optimal direction

Model (3) is slightly different from that of Chen andDelmas [58] in several aspects first we also take the inputimprovement into account while Chen and Delmas [58]studied only the output improvement secondly we assumethe weak disposability assumption of desirable outputs andundesirable outputs while the strong disposability as-sumption was modeled in Chen and Delmas [58] In ad-dition the VRS assumption is considered in model (3) suchthat it is more suitable for real applications Since the in-dividual inefficiency index theoretically has a value rangingfrom zero to unity for inputs and undesirable outputs and avalue from zero to infinity for desirable outputs the overallaverage inefficiency of DMUd also takes a value from zero toinfinitye larger the average inefficiency index is the moreinefficient the evaluated DMU is An average inefficiencyindex of zero value means that the considered DMU is on theefficiency frontier and has no slack for improvement andhence the DMU is efficient

Furthermore model (4) is a dual formulation of theabove model (3) in its multiplier formulation

inE lowastd Min 1113944m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0

st 1113944

m

i1vixij minus 1113944

s

r1uryrj + 1113944

q

p1wpbpj + u0 ge 0

j 1 n

1113944

m

i1vixij + u0 ge 0 j 1 n

vi ge 1(m + s + q)xid i 1 m

ur ge 1(m + s + q)yrd r 1 s

wp ge 1(m + s + q)bpd p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(4)

Model (4) computes the inefficient component (ie thedifference between aggregated inputs and aggregated out-puts) for the evaluated DMU yet the inefficient componentis nonnegative for all DMUs and some constraints on

4 Complexity

multipliers are held Solving model (4) for each DMUd(d

1 n) determines a series of inefficiency scores inElowastd usinga series of optimal solutions (udlowast

r vdlowasti wdlowast

p udlowast0 ) e in-

efficiency score can be used for a performance indicatoramong all DMUs and the smaller the inefficiency score isthe better the performance of DMUd is

22 Dominance Probability Based on Inefficiency ScoresIt is notable that the inefficiency score inE lowastd obtained pre-viously can be used to analyze the performance of Chineselisted banks but there are two main concerns On the onehand it is calculated by only considering the optimal weightplan (udlowast

r vdlowasti wdlowast

p udlowast0 ) while ignoring any other feasible

weights and thus the obtained inefficiency score inElowastd mightbe too extreme or even unrealistic On the other hand theinefficiency score inE lowastd is calculated separately for eachDMUd(d 1 n) and different weights will be preferredby different DMUs thus the results are not completelycomparable

Since the resulting inefficiency indexes can change rel-ative to different sets of weights it is important to explore theperformance associated with all sets of feasible inputoutputweights To this end we focus on the efficiency dominanceconcept of Salo and Punkka [21] As well defined and dis-cussed in Salo and Punkka [21] and Tang et al [51] thedominance relation is determined through a pairwise effi-ciency comparison among DMUs Furthermore a certainDMU dominates another DMU if and only if its efficiencyscore is as large as that of the other for all sets of feasibleinputoutput weights and is larger for at least some sets offeasible inputoutput weights Here we follow the same ideaof Salo and Punkka [21] and Tang et al [51] to determine thedominance relation of Chinese listed banks For comparisonwe take inevitable undesirable outputs in banking opera-tions such as bad debts into account Furthermore wefollow Chen and Delmas [58] in focusing on eco-inefficiencyscores rather than efficiency scores through a nonradialdirectional distance function model To this end we firstbuild the inefficiency dominance concept as follows

Definition 1 DMUd dominates DMUk (denoted asDMUd ≻DMUk) if and only if DMUd always has a smallerinefficiency score relative to DMUk for all sets of feasibleinput and output weights

e dominance relation based on inefficiency scores isdetermined if the inefficiency score for a certain DMUd is assmall as that of DMUk for all sets of feasible inputoutputweights and is smaller for at least some sets of feasible inputoutput weightse dominance relation between DMUd andDMUk is determined through their inefficiency comparisonBy rethinking the idea of model (3) and model (4) whichcalculate the maximal inefficiency score we can formulatemodel (5) to calculate the inefficiency range of DMUd whenDMUk is fixed with a prespecified inefficiency level of zeroby requiring the constraint that inEk 1113936

mi1vixikminus

1113936sr1uryrk + 1113936

qp1wpbpk + u0 0 In fact the inefficiency

level of DMUk can be set to any nonnegative valueM and bysubstituting u0 with u0 minus M we can get the same dominance

probability as given in Definition 2 and Definition 3 hencewe immediately set the inefficiency level of DMUk to zero forsimplification in the following model

inEmaxd1113957k

inEmind1113957k

MinMax

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xid

i 1 m

ur ge1

(m + s + q)yrd

r 1 s

wp ge1

(m + s + q)bpd

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(5)

An additional constraint that 1113936mi1vi + 1113936

sr1ur+

1113936qp1wp 1 is inserted into model (5) to make the feasible

weight space closed and bounded e optimal objectivefunction formodel (5) shows the lower and upper bounds onhow much different DMUdrsquos inefficiency score can be rel-ative to DMUk across all sets of feasible inputoutputweights Using model (5) the dominance structure can bedetermined for any pairwise DMUs More specifically if theinefficiency level of DMUk is fixed to zero and if inEmax

d1113957klt 0

which means that DMUd will always have a smaller inef-ficiency score compared with DMUk then DMUd dominatesDMUk In contrast if inEmin

d1113957kgt 0 which means that DMUd

will always have a larger inefficiency score than DMUk

(which has an inefficiency level of zero) then DMUd isdominated by DMUk For more general cases however wecannot obtain the complete dominance relation which isusually true in practice erefore we propose determiningthe performance dominance probability To this end wefollow the work of Tang et al [51] in building the inefficiencydominance probability concept as given in Definition 2

Definition 2 When DMUk has an inefficiency score of inEkthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byPd1113957≻k (inEk minus inEmin

d1113957kinEmax

d1113957kminus inEmin

d1113957k)

It is clear that if inEk le inEmind1113957k

which means that DMUd

will always have a larger inefficiency by remaining theinefficiency inEk for DMUk the inefficiency dominance

Complexity 5

probability of DMUd relative to DMUk would take anonpositive value at is it is impossible for DMUd todominate DMUk For the sake of a bounded range weassume that Pd1113957≻k 0 if inEk le inEmin

d1113957k For a more general

case in which inEmind1113957kle inEk le inEmax

d1113957k the probability that

DMUd dominates DMUk takes a value from zero to unity Alarger value of Pd1113957≻k means that it is more likely for DMUd tohave a smaller inefficiency score compared with DMUk overall sets of feasible inputoutput weights

Note in addition that Definition 2 fixes the inefficiencylevel of DMUk to calculate the inefficiency range of DMUd

and further computes the inefficiency dominance proba-bility of DMUd relative to DMUk In contrast we can alsocompute the inefficiency dominance probability of DMUd

relative to DMUk by fixing the inefficiency level of DMUd

and calculating the inefficiency range of DMUk e aboveidea is formulated in model (6) which is very similar tomodel (5) but substitutes DMUk for DMUd

inEmaxk1113957d

inEmink1113957d

MinMax

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xik

i 1 m

ur ge1

(m + s + q)yrk

r 1 s

wp ge1

(m + s + q)bpk

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(6)

An alternative inefficiency dominance probability ofDMUd relative to DMUk by fixing the inefficiency level ofDMUd is given in Definition 3

Definition 3 When DMUd has an inefficiency score of inEdthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byP

d≻1113957k (inEmax

k1113957dminus inEdinEmax

k1113957dminus inEmin

k1113957d)

Furthermore the overall probability that the inefficiencyscore of DMUd dominates that of DMUk is the average of thetwo probabilities by fixing the inefficiency level of DMUd

and DMUk More specifically Definition 4 gives the pairwise

performance dominance probability with regard to ineffi-ciency scores

Definition 4 e pairwise performance dominance prob-ability of DMUd relative to DMUk over all sets of feasibleinput and output weights is Pd≻k ((Pd1113957≻k + Pd≻k1113957)2)

e classic DEA methods allow each DMU to generate aset of relative weights to maximize its ratio of aggregatedweighted outputs to aggregated weighted inputs while en-suring that the same ratio is no more than one for all DMUsand the maximum ratio is considered the efficiency score forthe evaluated DMU By taking all sets of feasible weights andthe inefficiency dominance structure into account theoverall inefficiency dominance probability for a certainDMUd among all DMUs can be calculated by the average ofpair inefficiency dominance probabilities across all DMUse above idea is given in Definition 5

Definition 5 e performance dominance probability ofDMUd across all DMUs over all sets of feasible input andoutput weights is Pd 1113936

nk1Pd≻kn

e classic DEA approaches use deterministic (in)effi-ciency scores to measure the relative performance while theperformance dominance probability is an alternative per-formance indicator from a data analytics perspective thatconsiders all sets of feasible weights that are stochastic todetermine the relative performance It is rather remarkablethat different sets of weights will cause different performancemeasures and the performance dominance probability in-volves all possibilities over all sets of feasible input andoutput weights e performance dominance probability ofDMUd calculates the probability of its performance domi-nating the set of all DMUs A larger performance dominanceprobability indicates that it is much easier for that DMU todominate other DMUs implying that its inefficiency score ismore likely to be smaller than that of other DMUs

3 Illustrative Application of ChineseListed Banks

In this section we illustrate the proposed approach usingempirical performance analytics for 32 Chinese listed banksSince the proposed approach considers all sets of feasibleweights the performance relations and ratio index resultsare more comprehensive and reasonable

31 Data Description is section addresses the perfor-mance of listed banks in China For simplification and re-search purposes we consider only those banks that havebeen registered in China and that are listed in the mainlandof China More specifically only banks that are owned byChinese organizations and are listed on the Shenzhen StockExchange and Shanghai Stock Exchange are collected Incontrast neither Chinese banks listed on other stock ex-changes nor foreign banks listed on the Shenzhen StockExchange and Shanghai Stock Exchange are involved in thisstudy As a result we have 32 listed banks For the research

6 Complexity

purpose we give these 32 banks and their correspondingcodes in Table 1

In practice each bank will consume multiple inputs togenerate multiple outputs and more specifically mainly forprofits In this study we follow similar studies such as Wanget al [4] Zha et al [1] Fukuyama andMatousek [64] Li et al[65] and Zhu et al [2] in taking employment referring tohuman resource investment and manpower fixed assetreferring to the asset value of physical capital that can beused for business activities and operation cost as threeinputs Note in addition that the operation cost in this studyexcludes the expense of labor input that occurs in bankingoperations because the employment has already taken thelabor input into account Furthermore we consider threedifferent outputs generated in banking operations withinterest income and noninterest income being two desirableoutputs and nonperforming bad loan percentage in thecurrent year as an undesirable outpute interest income isderived directly from the gap between the interest paid ondeposits and the interest earned from loans while thenoninterest income is primarily derived from commissionssecurities investments fees and other business activity in-comes However a bad loan is a jointly produced andunavoidable by-product that will greatly harm the bank einputs and outputs to be used in this study are summarizedin Table 2

Our empirical study contains operation data for 32Chinese listed banks over the 2014ndash2018 period accountingfor 160 observations All data for these listed banks werecollected from official sources of bank annual reports and thefinancial reports of banks in China StockMarket AccountingResearch (CSMAR) during 2014ndash2018 Table 3 shows thedescriptive statistics of the inputs desirable outputs andundesirable outputs of these 160 observations It can befound that both the average fixed asset and operation costamong the 32 listed banks are increasing year by year but theaverage employment and bad debt percentage increased to apeak in 2016 and then decreased continuously Furthermoreboth interest income and noninterest income show an in-creasing trend but decreased in a year

32 Result Analysis To provide data analytics of bankperformance we first calculate the inefficiency scores usingmodel (3) or model (4) in the nonradial DDF-based for-mulation under the VRS property e inefficiencies of these32 banks from 2014 to 2018 are given in Table 4 Since theinefficiency score represents potential improvements di-vided by the observed inputs or outputs an inefficiency scoreof zero indicates that there will be no improvement po-tential Table 4 shows that by selecting the optimal set ofweights to aggregate inputs and outputs many listed bankswill be extremely efficient without improvement potentialsere are always ten banks that have an inefficiency scorelarger than zero but the average inefficiency score acrossthese 32 listed banks fluctuates according to the year Sincethe VRS analysis is adopted in this study we may not drawany significant conclusion as to inefficiency changes year byyear Furthermore some banks (DMU6 Bank of Guiyang

DMU8 Huaxia Bank DMU15 Bank of Ningbo and DMU32Zijin Rural Commercial Bank) will always have a nonzeroinefficiency meaning that these banks always show veryterrible performance compared with banks that have a zero-value inefficiency index

Since the previous inefficiency scores are separatelyderived by considering only one optimal set of weights theresulting performance information might be unrealistic andunreasonable erefore we can use the proposed eco-in-efficiency dominance probability approach in this paper toanalyze the banking performance considering all sets offeasible weights To this end we first use model (5) to

Table 1 Codes for 32 Chinese listed banks

DMUs Banks AbbreviationDMU1 Bank of Beijing BOBDMU2 Changshu Rural Commercial Bank CRCBDMU3 Bank of Chengdu BCD

DMU4Industrial and Commercial Bank of

China ICBC

DMU5 China Everbright Bank CEBDMU6 Bank of Guiyang BOGDMU7 Bank of Hangzhou BOHDMU8 Huaxia Bank HBDMU9 China Construction Bank CCBDMU10 Bank of Jiangsu BOJDMU11 Jiangyin Rural Commercial Bank JRCBDMU12 Bank of Communications BCDMU13 China Minsheng Bank CMSBDMU14 Bank of Nanjing BONDMU15 Bank of Ningbo BNDMU16 Agricultural Bank of China ABCDMU17 Ping An Bank PBDMU18 Shanghai Pudong Development Bank SPDBDMU19 Bank of Qingdao BOQDMU20 Qingdao Rural Commercial Bank QRCBDMU21 Bank of Shanghai BOSDMU22 Suzhou Rural Commercial Bank SRCBDMU23 Wuxi Rural Commercial Bank WRCBDMU24 Bank of Xian BOXDMU25 Industrial Bank IBDMU26 Rural Commercial Bank of Zhangjiagang RCBZDMU27 Bank of Changsha BCSDMU28 China Merchants Bank CMBDMU29 Bank of Zhengzhou BOZDMU30 Bank of China BOCDMU31 China CITIC Bank CCBDMU32 Zijin Rural Commercial Bank ZRCB

Table 2 Input and output variables

Inputoutput Variable Notation Unit

InputsEmployment x1

Personcount

Fixed assets x2 Million yuanOperation costs x3 Million yuan

Desirable outputsInterest income y1 Million yuanNoninterestincome y2 Million yuan

Undesirableoutputs

Bad debtpercentage z1 Percentage

Complexity 7

Table 3 Descriptive statistics for input-output measures

Year Statistics x1 x2 x3 y1 y2 z1

2014

Max 493583 196238 271132 849879 165370 27100Min 1089 41201 115141 353390 10948 07500Mean 67582 2849915 4823162 15411269 2568599 12531SD 135316 5589804 7815135 23578569 4416322 04237

2015

Max 503082 221502 306395 871779 189780 23900Min 1180 41804 124526 348611 13069 08300Mean 69772 3133597 5738723 16210941 3034405 15308SD 136444 6026226 8745917 24329478 4941992 04226

2016

Max 496698 243619 282712 791480 204045 24100Min 1352 41399 123588 326889 16262 08700Mean 70266 3477886 5828596 15179021 3556052 16013SD 134489 6511810 8516872 22069357 5718980 03678

2017

Max 487307 245687 331518 861594 204424 23900Min 1426 46293 126089 378914 16605 08200Mean 69782 3638326 6007996 16692792 3516824 15431SD 132071 6662176 9168104 23946371 5304531 03240

2018

Max 473691 253525 368966 948094 201271 24700Min 1454 41808 173355 456700 20317 07800Mean 69354 3838034 6744402 18305026 3918308 15034SD 129697 7006174 10096472 26175120 5500040 03382

Table 4 Inefficiency scores of 32 listed banks from 2014 to 2018

Banks 2014 2015 2016 2017 2018DMU1 00000 00000 00000 00000 00000DMU2 00000 03483 06442 05430 03483DMU3 00000 00979 02730 00000 00979DMU4 00000 00000 00000 00000 00000DMU5 00000 00590 00000 01087 00590DMU6 02345 03469 03273 05898 03469DMU7 01798 00000 00000 00000 00000DMU8 02880 03059 02288 02768 03059DMU9 00000 00000 00000 00000 00000DMU10 00000 00000 00000 00000 00000DMU11 00000 00000 00000 00000 00000DMU12 01449 00000 00000 00000 00000DMU13 00000 00000 00000 00000 00000DMU14 02088 01072 01415 00000 01072DMU15 02793 02415 02018 02225 02415DMU16 01926 00000 00000 00000 00000DMU17 00000 00000 00000 00000 00000DMU18 00567 00000 00000 00000 00000DMU19 00000 00000 00827 00000 00000DMU20 00000 00000 00000 00000 00000DMU21 00000 00000 00000 00000 00000DMU22 00000 00000 00000 02032 00000DMU23 00000 00000 00000 00000 00000DMU24 00000 00000 00000 00000 00000DMU25 00000 00000 00000 00000 00000DMU26 00000 00000 00000 00000 00000DMU27 00000 02287 02670 06379 02287DMU28 00000 00000 00000 00000 00000DMU29 00000 00000 01373 00967 00000DMU30 00000 00000 00000 00000 00000DMU31 00634 00799 00000 00000 00799DMU32 05982 02792 03526 05700 02792Mean 00702 00655 00830 01015 00655

8 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

multipliers are held Solving model (4) for each DMUd(d

1 n) determines a series of inefficiency scores inElowastd usinga series of optimal solutions (udlowast

r vdlowasti wdlowast

p udlowast0 ) e in-

efficiency score can be used for a performance indicatoramong all DMUs and the smaller the inefficiency score isthe better the performance of DMUd is

22 Dominance Probability Based on Inefficiency ScoresIt is notable that the inefficiency score inE lowastd obtained pre-viously can be used to analyze the performance of Chineselisted banks but there are two main concerns On the onehand it is calculated by only considering the optimal weightplan (udlowast

r vdlowasti wdlowast

p udlowast0 ) while ignoring any other feasible

weights and thus the obtained inefficiency score inElowastd mightbe too extreme or even unrealistic On the other hand theinefficiency score inE lowastd is calculated separately for eachDMUd(d 1 n) and different weights will be preferredby different DMUs thus the results are not completelycomparable

Since the resulting inefficiency indexes can change rel-ative to different sets of weights it is important to explore theperformance associated with all sets of feasible inputoutputweights To this end we focus on the efficiency dominanceconcept of Salo and Punkka [21] As well defined and dis-cussed in Salo and Punkka [21] and Tang et al [51] thedominance relation is determined through a pairwise effi-ciency comparison among DMUs Furthermore a certainDMU dominates another DMU if and only if its efficiencyscore is as large as that of the other for all sets of feasibleinputoutput weights and is larger for at least some sets offeasible inputoutput weights Here we follow the same ideaof Salo and Punkka [21] and Tang et al [51] to determine thedominance relation of Chinese listed banks For comparisonwe take inevitable undesirable outputs in banking opera-tions such as bad debts into account Furthermore wefollow Chen and Delmas [58] in focusing on eco-inefficiencyscores rather than efficiency scores through a nonradialdirectional distance function model To this end we firstbuild the inefficiency dominance concept as follows

Definition 1 DMUd dominates DMUk (denoted asDMUd ≻DMUk) if and only if DMUd always has a smallerinefficiency score relative to DMUk for all sets of feasibleinput and output weights

e dominance relation based on inefficiency scores isdetermined if the inefficiency score for a certain DMUd is assmall as that of DMUk for all sets of feasible inputoutputweights and is smaller for at least some sets of feasible inputoutput weightse dominance relation between DMUd andDMUk is determined through their inefficiency comparisonBy rethinking the idea of model (3) and model (4) whichcalculate the maximal inefficiency score we can formulatemodel (5) to calculate the inefficiency range of DMUd whenDMUk is fixed with a prespecified inefficiency level of zeroby requiring the constraint that inEk 1113936

mi1vixikminus

1113936sr1uryrk + 1113936

qp1wpbpk + u0 0 In fact the inefficiency

level of DMUk can be set to any nonnegative valueM and bysubstituting u0 with u0 minus M we can get the same dominance

probability as given in Definition 2 and Definition 3 hencewe immediately set the inefficiency level of DMUk to zero forsimplification in the following model

inEmaxd1113957k

inEmind1113957k

MinMax

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xid

i 1 m

ur ge1

(m + s + q)yrd

r 1 s

wp ge1

(m + s + q)bpd

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(5)

An additional constraint that 1113936mi1vi + 1113936

sr1ur+

1113936qp1wp 1 is inserted into model (5) to make the feasible

weight space closed and bounded e optimal objectivefunction formodel (5) shows the lower and upper bounds onhow much different DMUdrsquos inefficiency score can be rel-ative to DMUk across all sets of feasible inputoutputweights Using model (5) the dominance structure can bedetermined for any pairwise DMUs More specifically if theinefficiency level of DMUk is fixed to zero and if inEmax

d1113957klt 0

which means that DMUd will always have a smaller inef-ficiency score compared with DMUk then DMUd dominatesDMUk In contrast if inEmin

d1113957kgt 0 which means that DMUd

will always have a larger inefficiency score than DMUk

(which has an inefficiency level of zero) then DMUd isdominated by DMUk For more general cases however wecannot obtain the complete dominance relation which isusually true in practice erefore we propose determiningthe performance dominance probability To this end wefollow the work of Tang et al [51] in building the inefficiencydominance probability concept as given in Definition 2

Definition 2 When DMUk has an inefficiency score of inEkthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byPd1113957≻k (inEk minus inEmin

d1113957kinEmax

d1113957kminus inEmin

d1113957k)

It is clear that if inEk le inEmind1113957k

which means that DMUd

will always have a larger inefficiency by remaining theinefficiency inEk for DMUk the inefficiency dominance

Complexity 5

probability of DMUd relative to DMUk would take anonpositive value at is it is impossible for DMUd todominate DMUk For the sake of a bounded range weassume that Pd1113957≻k 0 if inEk le inEmin

d1113957k For a more general

case in which inEmind1113957kle inEk le inEmax

d1113957k the probability that

DMUd dominates DMUk takes a value from zero to unity Alarger value of Pd1113957≻k means that it is more likely for DMUd tohave a smaller inefficiency score compared with DMUk overall sets of feasible inputoutput weights

Note in addition that Definition 2 fixes the inefficiencylevel of DMUk to calculate the inefficiency range of DMUd

and further computes the inefficiency dominance proba-bility of DMUd relative to DMUk In contrast we can alsocompute the inefficiency dominance probability of DMUd

relative to DMUk by fixing the inefficiency level of DMUd

and calculating the inefficiency range of DMUk e aboveidea is formulated in model (6) which is very similar tomodel (5) but substitutes DMUk for DMUd

inEmaxk1113957d

inEmink1113957d

MinMax

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xik

i 1 m

ur ge1

(m + s + q)yrk

r 1 s

wp ge1

(m + s + q)bpk

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(6)

An alternative inefficiency dominance probability ofDMUd relative to DMUk by fixing the inefficiency level ofDMUd is given in Definition 3

Definition 3 When DMUd has an inefficiency score of inEdthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byP

d≻1113957k (inEmax

k1113957dminus inEdinEmax

k1113957dminus inEmin

k1113957d)

Furthermore the overall probability that the inefficiencyscore of DMUd dominates that of DMUk is the average of thetwo probabilities by fixing the inefficiency level of DMUd

and DMUk More specifically Definition 4 gives the pairwise

performance dominance probability with regard to ineffi-ciency scores

Definition 4 e pairwise performance dominance prob-ability of DMUd relative to DMUk over all sets of feasibleinput and output weights is Pd≻k ((Pd1113957≻k + Pd≻k1113957)2)

e classic DEA methods allow each DMU to generate aset of relative weights to maximize its ratio of aggregatedweighted outputs to aggregated weighted inputs while en-suring that the same ratio is no more than one for all DMUsand the maximum ratio is considered the efficiency score forthe evaluated DMU By taking all sets of feasible weights andthe inefficiency dominance structure into account theoverall inefficiency dominance probability for a certainDMUd among all DMUs can be calculated by the average ofpair inefficiency dominance probabilities across all DMUse above idea is given in Definition 5

Definition 5 e performance dominance probability ofDMUd across all DMUs over all sets of feasible input andoutput weights is Pd 1113936

nk1Pd≻kn

e classic DEA approaches use deterministic (in)effi-ciency scores to measure the relative performance while theperformance dominance probability is an alternative per-formance indicator from a data analytics perspective thatconsiders all sets of feasible weights that are stochastic todetermine the relative performance It is rather remarkablethat different sets of weights will cause different performancemeasures and the performance dominance probability in-volves all possibilities over all sets of feasible input andoutput weights e performance dominance probability ofDMUd calculates the probability of its performance domi-nating the set of all DMUs A larger performance dominanceprobability indicates that it is much easier for that DMU todominate other DMUs implying that its inefficiency score ismore likely to be smaller than that of other DMUs

3 Illustrative Application of ChineseListed Banks

In this section we illustrate the proposed approach usingempirical performance analytics for 32 Chinese listed banksSince the proposed approach considers all sets of feasibleweights the performance relations and ratio index resultsare more comprehensive and reasonable

31 Data Description is section addresses the perfor-mance of listed banks in China For simplification and re-search purposes we consider only those banks that havebeen registered in China and that are listed in the mainlandof China More specifically only banks that are owned byChinese organizations and are listed on the Shenzhen StockExchange and Shanghai Stock Exchange are collected Incontrast neither Chinese banks listed on other stock ex-changes nor foreign banks listed on the Shenzhen StockExchange and Shanghai Stock Exchange are involved in thisstudy As a result we have 32 listed banks For the research

6 Complexity

purpose we give these 32 banks and their correspondingcodes in Table 1

In practice each bank will consume multiple inputs togenerate multiple outputs and more specifically mainly forprofits In this study we follow similar studies such as Wanget al [4] Zha et al [1] Fukuyama andMatousek [64] Li et al[65] and Zhu et al [2] in taking employment referring tohuman resource investment and manpower fixed assetreferring to the asset value of physical capital that can beused for business activities and operation cost as threeinputs Note in addition that the operation cost in this studyexcludes the expense of labor input that occurs in bankingoperations because the employment has already taken thelabor input into account Furthermore we consider threedifferent outputs generated in banking operations withinterest income and noninterest income being two desirableoutputs and nonperforming bad loan percentage in thecurrent year as an undesirable outpute interest income isderived directly from the gap between the interest paid ondeposits and the interest earned from loans while thenoninterest income is primarily derived from commissionssecurities investments fees and other business activity in-comes However a bad loan is a jointly produced andunavoidable by-product that will greatly harm the bank einputs and outputs to be used in this study are summarizedin Table 2

Our empirical study contains operation data for 32Chinese listed banks over the 2014ndash2018 period accountingfor 160 observations All data for these listed banks werecollected from official sources of bank annual reports and thefinancial reports of banks in China StockMarket AccountingResearch (CSMAR) during 2014ndash2018 Table 3 shows thedescriptive statistics of the inputs desirable outputs andundesirable outputs of these 160 observations It can befound that both the average fixed asset and operation costamong the 32 listed banks are increasing year by year but theaverage employment and bad debt percentage increased to apeak in 2016 and then decreased continuously Furthermoreboth interest income and noninterest income show an in-creasing trend but decreased in a year

32 Result Analysis To provide data analytics of bankperformance we first calculate the inefficiency scores usingmodel (3) or model (4) in the nonradial DDF-based for-mulation under the VRS property e inefficiencies of these32 banks from 2014 to 2018 are given in Table 4 Since theinefficiency score represents potential improvements di-vided by the observed inputs or outputs an inefficiency scoreof zero indicates that there will be no improvement po-tential Table 4 shows that by selecting the optimal set ofweights to aggregate inputs and outputs many listed bankswill be extremely efficient without improvement potentialsere are always ten banks that have an inefficiency scorelarger than zero but the average inefficiency score acrossthese 32 listed banks fluctuates according to the year Sincethe VRS analysis is adopted in this study we may not drawany significant conclusion as to inefficiency changes year byyear Furthermore some banks (DMU6 Bank of Guiyang

DMU8 Huaxia Bank DMU15 Bank of Ningbo and DMU32Zijin Rural Commercial Bank) will always have a nonzeroinefficiency meaning that these banks always show veryterrible performance compared with banks that have a zero-value inefficiency index

Since the previous inefficiency scores are separatelyderived by considering only one optimal set of weights theresulting performance information might be unrealistic andunreasonable erefore we can use the proposed eco-in-efficiency dominance probability approach in this paper toanalyze the banking performance considering all sets offeasible weights To this end we first use model (5) to

Table 1 Codes for 32 Chinese listed banks

DMUs Banks AbbreviationDMU1 Bank of Beijing BOBDMU2 Changshu Rural Commercial Bank CRCBDMU3 Bank of Chengdu BCD

DMU4Industrial and Commercial Bank of

China ICBC

DMU5 China Everbright Bank CEBDMU6 Bank of Guiyang BOGDMU7 Bank of Hangzhou BOHDMU8 Huaxia Bank HBDMU9 China Construction Bank CCBDMU10 Bank of Jiangsu BOJDMU11 Jiangyin Rural Commercial Bank JRCBDMU12 Bank of Communications BCDMU13 China Minsheng Bank CMSBDMU14 Bank of Nanjing BONDMU15 Bank of Ningbo BNDMU16 Agricultural Bank of China ABCDMU17 Ping An Bank PBDMU18 Shanghai Pudong Development Bank SPDBDMU19 Bank of Qingdao BOQDMU20 Qingdao Rural Commercial Bank QRCBDMU21 Bank of Shanghai BOSDMU22 Suzhou Rural Commercial Bank SRCBDMU23 Wuxi Rural Commercial Bank WRCBDMU24 Bank of Xian BOXDMU25 Industrial Bank IBDMU26 Rural Commercial Bank of Zhangjiagang RCBZDMU27 Bank of Changsha BCSDMU28 China Merchants Bank CMBDMU29 Bank of Zhengzhou BOZDMU30 Bank of China BOCDMU31 China CITIC Bank CCBDMU32 Zijin Rural Commercial Bank ZRCB

Table 2 Input and output variables

Inputoutput Variable Notation Unit

InputsEmployment x1

Personcount

Fixed assets x2 Million yuanOperation costs x3 Million yuan

Desirable outputsInterest income y1 Million yuanNoninterestincome y2 Million yuan

Undesirableoutputs

Bad debtpercentage z1 Percentage

Complexity 7

Table 3 Descriptive statistics for input-output measures

Year Statistics x1 x2 x3 y1 y2 z1

2014

Max 493583 196238 271132 849879 165370 27100Min 1089 41201 115141 353390 10948 07500Mean 67582 2849915 4823162 15411269 2568599 12531SD 135316 5589804 7815135 23578569 4416322 04237

2015

Max 503082 221502 306395 871779 189780 23900Min 1180 41804 124526 348611 13069 08300Mean 69772 3133597 5738723 16210941 3034405 15308SD 136444 6026226 8745917 24329478 4941992 04226

2016

Max 496698 243619 282712 791480 204045 24100Min 1352 41399 123588 326889 16262 08700Mean 70266 3477886 5828596 15179021 3556052 16013SD 134489 6511810 8516872 22069357 5718980 03678

2017

Max 487307 245687 331518 861594 204424 23900Min 1426 46293 126089 378914 16605 08200Mean 69782 3638326 6007996 16692792 3516824 15431SD 132071 6662176 9168104 23946371 5304531 03240

2018

Max 473691 253525 368966 948094 201271 24700Min 1454 41808 173355 456700 20317 07800Mean 69354 3838034 6744402 18305026 3918308 15034SD 129697 7006174 10096472 26175120 5500040 03382

Table 4 Inefficiency scores of 32 listed banks from 2014 to 2018

Banks 2014 2015 2016 2017 2018DMU1 00000 00000 00000 00000 00000DMU2 00000 03483 06442 05430 03483DMU3 00000 00979 02730 00000 00979DMU4 00000 00000 00000 00000 00000DMU5 00000 00590 00000 01087 00590DMU6 02345 03469 03273 05898 03469DMU7 01798 00000 00000 00000 00000DMU8 02880 03059 02288 02768 03059DMU9 00000 00000 00000 00000 00000DMU10 00000 00000 00000 00000 00000DMU11 00000 00000 00000 00000 00000DMU12 01449 00000 00000 00000 00000DMU13 00000 00000 00000 00000 00000DMU14 02088 01072 01415 00000 01072DMU15 02793 02415 02018 02225 02415DMU16 01926 00000 00000 00000 00000DMU17 00000 00000 00000 00000 00000DMU18 00567 00000 00000 00000 00000DMU19 00000 00000 00827 00000 00000DMU20 00000 00000 00000 00000 00000DMU21 00000 00000 00000 00000 00000DMU22 00000 00000 00000 02032 00000DMU23 00000 00000 00000 00000 00000DMU24 00000 00000 00000 00000 00000DMU25 00000 00000 00000 00000 00000DMU26 00000 00000 00000 00000 00000DMU27 00000 02287 02670 06379 02287DMU28 00000 00000 00000 00000 00000DMU29 00000 00000 01373 00967 00000DMU30 00000 00000 00000 00000 00000DMU31 00634 00799 00000 00000 00799DMU32 05982 02792 03526 05700 02792Mean 00702 00655 00830 01015 00655

8 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

probability of DMUd relative to DMUk would take anonpositive value at is it is impossible for DMUd todominate DMUk For the sake of a bounded range weassume that Pd1113957≻k 0 if inEk le inEmin

d1113957k For a more general

case in which inEmind1113957kle inEk le inEmax

d1113957k the probability that

DMUd dominates DMUk takes a value from zero to unity Alarger value of Pd1113957≻k means that it is more likely for DMUd tohave a smaller inefficiency score compared with DMUk overall sets of feasible inputoutput weights

Note in addition that Definition 2 fixes the inefficiencylevel of DMUk to calculate the inefficiency range of DMUd

and further computes the inefficiency dominance proba-bility of DMUd relative to DMUk In contrast we can alsocompute the inefficiency dominance probability of DMUd

relative to DMUk by fixing the inefficiency level of DMUd

and calculating the inefficiency range of DMUk e aboveidea is formulated in model (6) which is very similar tomodel (5) but substitutes DMUk for DMUd

inEmaxk1113957d

inEmink1113957d

MinMax

1113944

m

i1vixik minus 1113944

s

r1uryrk + 1113944

q

p1wpbpk + u0

st 1113944

m

i1vi + 1113944

s

r1ur + 1113944

q

p1wp 1

1113944

m

i1vixid minus 1113944

s

r1uryrd + 1113944

q

p1wpbpd + u0 0

1113944

m

i1vixij + u0 ge 0 j d k

vi ge1

(m + s + q)xik

i 1 m

ur ge1

(m + s + q)yrk

r 1 s

wp ge1

(m + s + q)bpk

p 1 q

vi ur ge 0 i 1 m r 1 s

wp u0 are free p 1 q

(6)

An alternative inefficiency dominance probability ofDMUd relative to DMUk by fixing the inefficiency level ofDMUd is given in Definition 3

Definition 3 When DMUd has an inefficiency score of inEdthe probability that DMUd dominates DMUk over all sets offeasible input and output weights is calculated byP

d≻1113957k (inEmax

k1113957dminus inEdinEmax

k1113957dminus inEmin

k1113957d)

Furthermore the overall probability that the inefficiencyscore of DMUd dominates that of DMUk is the average of thetwo probabilities by fixing the inefficiency level of DMUd

and DMUk More specifically Definition 4 gives the pairwise

performance dominance probability with regard to ineffi-ciency scores

Definition 4 e pairwise performance dominance prob-ability of DMUd relative to DMUk over all sets of feasibleinput and output weights is Pd≻k ((Pd1113957≻k + Pd≻k1113957)2)

e classic DEA methods allow each DMU to generate aset of relative weights to maximize its ratio of aggregatedweighted outputs to aggregated weighted inputs while en-suring that the same ratio is no more than one for all DMUsand the maximum ratio is considered the efficiency score forthe evaluated DMU By taking all sets of feasible weights andthe inefficiency dominance structure into account theoverall inefficiency dominance probability for a certainDMUd among all DMUs can be calculated by the average ofpair inefficiency dominance probabilities across all DMUse above idea is given in Definition 5

Definition 5 e performance dominance probability ofDMUd across all DMUs over all sets of feasible input andoutput weights is Pd 1113936

nk1Pd≻kn

e classic DEA approaches use deterministic (in)effi-ciency scores to measure the relative performance while theperformance dominance probability is an alternative per-formance indicator from a data analytics perspective thatconsiders all sets of feasible weights that are stochastic todetermine the relative performance It is rather remarkablethat different sets of weights will cause different performancemeasures and the performance dominance probability in-volves all possibilities over all sets of feasible input andoutput weights e performance dominance probability ofDMUd calculates the probability of its performance domi-nating the set of all DMUs A larger performance dominanceprobability indicates that it is much easier for that DMU todominate other DMUs implying that its inefficiency score ismore likely to be smaller than that of other DMUs

3 Illustrative Application of ChineseListed Banks

In this section we illustrate the proposed approach usingempirical performance analytics for 32 Chinese listed banksSince the proposed approach considers all sets of feasibleweights the performance relations and ratio index resultsare more comprehensive and reasonable

31 Data Description is section addresses the perfor-mance of listed banks in China For simplification and re-search purposes we consider only those banks that havebeen registered in China and that are listed in the mainlandof China More specifically only banks that are owned byChinese organizations and are listed on the Shenzhen StockExchange and Shanghai Stock Exchange are collected Incontrast neither Chinese banks listed on other stock ex-changes nor foreign banks listed on the Shenzhen StockExchange and Shanghai Stock Exchange are involved in thisstudy As a result we have 32 listed banks For the research

6 Complexity

purpose we give these 32 banks and their correspondingcodes in Table 1

In practice each bank will consume multiple inputs togenerate multiple outputs and more specifically mainly forprofits In this study we follow similar studies such as Wanget al [4] Zha et al [1] Fukuyama andMatousek [64] Li et al[65] and Zhu et al [2] in taking employment referring tohuman resource investment and manpower fixed assetreferring to the asset value of physical capital that can beused for business activities and operation cost as threeinputs Note in addition that the operation cost in this studyexcludes the expense of labor input that occurs in bankingoperations because the employment has already taken thelabor input into account Furthermore we consider threedifferent outputs generated in banking operations withinterest income and noninterest income being two desirableoutputs and nonperforming bad loan percentage in thecurrent year as an undesirable outpute interest income isderived directly from the gap between the interest paid ondeposits and the interest earned from loans while thenoninterest income is primarily derived from commissionssecurities investments fees and other business activity in-comes However a bad loan is a jointly produced andunavoidable by-product that will greatly harm the bank einputs and outputs to be used in this study are summarizedin Table 2

Our empirical study contains operation data for 32Chinese listed banks over the 2014ndash2018 period accountingfor 160 observations All data for these listed banks werecollected from official sources of bank annual reports and thefinancial reports of banks in China StockMarket AccountingResearch (CSMAR) during 2014ndash2018 Table 3 shows thedescriptive statistics of the inputs desirable outputs andundesirable outputs of these 160 observations It can befound that both the average fixed asset and operation costamong the 32 listed banks are increasing year by year but theaverage employment and bad debt percentage increased to apeak in 2016 and then decreased continuously Furthermoreboth interest income and noninterest income show an in-creasing trend but decreased in a year

32 Result Analysis To provide data analytics of bankperformance we first calculate the inefficiency scores usingmodel (3) or model (4) in the nonradial DDF-based for-mulation under the VRS property e inefficiencies of these32 banks from 2014 to 2018 are given in Table 4 Since theinefficiency score represents potential improvements di-vided by the observed inputs or outputs an inefficiency scoreof zero indicates that there will be no improvement po-tential Table 4 shows that by selecting the optimal set ofweights to aggregate inputs and outputs many listed bankswill be extremely efficient without improvement potentialsere are always ten banks that have an inefficiency scorelarger than zero but the average inefficiency score acrossthese 32 listed banks fluctuates according to the year Sincethe VRS analysis is adopted in this study we may not drawany significant conclusion as to inefficiency changes year byyear Furthermore some banks (DMU6 Bank of Guiyang

DMU8 Huaxia Bank DMU15 Bank of Ningbo and DMU32Zijin Rural Commercial Bank) will always have a nonzeroinefficiency meaning that these banks always show veryterrible performance compared with banks that have a zero-value inefficiency index

Since the previous inefficiency scores are separatelyderived by considering only one optimal set of weights theresulting performance information might be unrealistic andunreasonable erefore we can use the proposed eco-in-efficiency dominance probability approach in this paper toanalyze the banking performance considering all sets offeasible weights To this end we first use model (5) to

Table 1 Codes for 32 Chinese listed banks

DMUs Banks AbbreviationDMU1 Bank of Beijing BOBDMU2 Changshu Rural Commercial Bank CRCBDMU3 Bank of Chengdu BCD

DMU4Industrial and Commercial Bank of

China ICBC

DMU5 China Everbright Bank CEBDMU6 Bank of Guiyang BOGDMU7 Bank of Hangzhou BOHDMU8 Huaxia Bank HBDMU9 China Construction Bank CCBDMU10 Bank of Jiangsu BOJDMU11 Jiangyin Rural Commercial Bank JRCBDMU12 Bank of Communications BCDMU13 China Minsheng Bank CMSBDMU14 Bank of Nanjing BONDMU15 Bank of Ningbo BNDMU16 Agricultural Bank of China ABCDMU17 Ping An Bank PBDMU18 Shanghai Pudong Development Bank SPDBDMU19 Bank of Qingdao BOQDMU20 Qingdao Rural Commercial Bank QRCBDMU21 Bank of Shanghai BOSDMU22 Suzhou Rural Commercial Bank SRCBDMU23 Wuxi Rural Commercial Bank WRCBDMU24 Bank of Xian BOXDMU25 Industrial Bank IBDMU26 Rural Commercial Bank of Zhangjiagang RCBZDMU27 Bank of Changsha BCSDMU28 China Merchants Bank CMBDMU29 Bank of Zhengzhou BOZDMU30 Bank of China BOCDMU31 China CITIC Bank CCBDMU32 Zijin Rural Commercial Bank ZRCB

Table 2 Input and output variables

Inputoutput Variable Notation Unit

InputsEmployment x1

Personcount

Fixed assets x2 Million yuanOperation costs x3 Million yuan

Desirable outputsInterest income y1 Million yuanNoninterestincome y2 Million yuan

Undesirableoutputs

Bad debtpercentage z1 Percentage

Complexity 7

Table 3 Descriptive statistics for input-output measures

Year Statistics x1 x2 x3 y1 y2 z1

2014

Max 493583 196238 271132 849879 165370 27100Min 1089 41201 115141 353390 10948 07500Mean 67582 2849915 4823162 15411269 2568599 12531SD 135316 5589804 7815135 23578569 4416322 04237

2015

Max 503082 221502 306395 871779 189780 23900Min 1180 41804 124526 348611 13069 08300Mean 69772 3133597 5738723 16210941 3034405 15308SD 136444 6026226 8745917 24329478 4941992 04226

2016

Max 496698 243619 282712 791480 204045 24100Min 1352 41399 123588 326889 16262 08700Mean 70266 3477886 5828596 15179021 3556052 16013SD 134489 6511810 8516872 22069357 5718980 03678

2017

Max 487307 245687 331518 861594 204424 23900Min 1426 46293 126089 378914 16605 08200Mean 69782 3638326 6007996 16692792 3516824 15431SD 132071 6662176 9168104 23946371 5304531 03240

2018

Max 473691 253525 368966 948094 201271 24700Min 1454 41808 173355 456700 20317 07800Mean 69354 3838034 6744402 18305026 3918308 15034SD 129697 7006174 10096472 26175120 5500040 03382

Table 4 Inefficiency scores of 32 listed banks from 2014 to 2018

Banks 2014 2015 2016 2017 2018DMU1 00000 00000 00000 00000 00000DMU2 00000 03483 06442 05430 03483DMU3 00000 00979 02730 00000 00979DMU4 00000 00000 00000 00000 00000DMU5 00000 00590 00000 01087 00590DMU6 02345 03469 03273 05898 03469DMU7 01798 00000 00000 00000 00000DMU8 02880 03059 02288 02768 03059DMU9 00000 00000 00000 00000 00000DMU10 00000 00000 00000 00000 00000DMU11 00000 00000 00000 00000 00000DMU12 01449 00000 00000 00000 00000DMU13 00000 00000 00000 00000 00000DMU14 02088 01072 01415 00000 01072DMU15 02793 02415 02018 02225 02415DMU16 01926 00000 00000 00000 00000DMU17 00000 00000 00000 00000 00000DMU18 00567 00000 00000 00000 00000DMU19 00000 00000 00827 00000 00000DMU20 00000 00000 00000 00000 00000DMU21 00000 00000 00000 00000 00000DMU22 00000 00000 00000 02032 00000DMU23 00000 00000 00000 00000 00000DMU24 00000 00000 00000 00000 00000DMU25 00000 00000 00000 00000 00000DMU26 00000 00000 00000 00000 00000DMU27 00000 02287 02670 06379 02287DMU28 00000 00000 00000 00000 00000DMU29 00000 00000 01373 00967 00000DMU30 00000 00000 00000 00000 00000DMU31 00634 00799 00000 00000 00799DMU32 05982 02792 03526 05700 02792Mean 00702 00655 00830 01015 00655

8 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

purpose we give these 32 banks and their correspondingcodes in Table 1

In practice each bank will consume multiple inputs togenerate multiple outputs and more specifically mainly forprofits In this study we follow similar studies such as Wanget al [4] Zha et al [1] Fukuyama andMatousek [64] Li et al[65] and Zhu et al [2] in taking employment referring tohuman resource investment and manpower fixed assetreferring to the asset value of physical capital that can beused for business activities and operation cost as threeinputs Note in addition that the operation cost in this studyexcludes the expense of labor input that occurs in bankingoperations because the employment has already taken thelabor input into account Furthermore we consider threedifferent outputs generated in banking operations withinterest income and noninterest income being two desirableoutputs and nonperforming bad loan percentage in thecurrent year as an undesirable outpute interest income isderived directly from the gap between the interest paid ondeposits and the interest earned from loans while thenoninterest income is primarily derived from commissionssecurities investments fees and other business activity in-comes However a bad loan is a jointly produced andunavoidable by-product that will greatly harm the bank einputs and outputs to be used in this study are summarizedin Table 2

Our empirical study contains operation data for 32Chinese listed banks over the 2014ndash2018 period accountingfor 160 observations All data for these listed banks werecollected from official sources of bank annual reports and thefinancial reports of banks in China StockMarket AccountingResearch (CSMAR) during 2014ndash2018 Table 3 shows thedescriptive statistics of the inputs desirable outputs andundesirable outputs of these 160 observations It can befound that both the average fixed asset and operation costamong the 32 listed banks are increasing year by year but theaverage employment and bad debt percentage increased to apeak in 2016 and then decreased continuously Furthermoreboth interest income and noninterest income show an in-creasing trend but decreased in a year

32 Result Analysis To provide data analytics of bankperformance we first calculate the inefficiency scores usingmodel (3) or model (4) in the nonradial DDF-based for-mulation under the VRS property e inefficiencies of these32 banks from 2014 to 2018 are given in Table 4 Since theinefficiency score represents potential improvements di-vided by the observed inputs or outputs an inefficiency scoreof zero indicates that there will be no improvement po-tential Table 4 shows that by selecting the optimal set ofweights to aggregate inputs and outputs many listed bankswill be extremely efficient without improvement potentialsere are always ten banks that have an inefficiency scorelarger than zero but the average inefficiency score acrossthese 32 listed banks fluctuates according to the year Sincethe VRS analysis is adopted in this study we may not drawany significant conclusion as to inefficiency changes year byyear Furthermore some banks (DMU6 Bank of Guiyang

DMU8 Huaxia Bank DMU15 Bank of Ningbo and DMU32Zijin Rural Commercial Bank) will always have a nonzeroinefficiency meaning that these banks always show veryterrible performance compared with banks that have a zero-value inefficiency index

Since the previous inefficiency scores are separatelyderived by considering only one optimal set of weights theresulting performance information might be unrealistic andunreasonable erefore we can use the proposed eco-in-efficiency dominance probability approach in this paper toanalyze the banking performance considering all sets offeasible weights To this end we first use model (5) to

Table 1 Codes for 32 Chinese listed banks

DMUs Banks AbbreviationDMU1 Bank of Beijing BOBDMU2 Changshu Rural Commercial Bank CRCBDMU3 Bank of Chengdu BCD

DMU4Industrial and Commercial Bank of

China ICBC

DMU5 China Everbright Bank CEBDMU6 Bank of Guiyang BOGDMU7 Bank of Hangzhou BOHDMU8 Huaxia Bank HBDMU9 China Construction Bank CCBDMU10 Bank of Jiangsu BOJDMU11 Jiangyin Rural Commercial Bank JRCBDMU12 Bank of Communications BCDMU13 China Minsheng Bank CMSBDMU14 Bank of Nanjing BONDMU15 Bank of Ningbo BNDMU16 Agricultural Bank of China ABCDMU17 Ping An Bank PBDMU18 Shanghai Pudong Development Bank SPDBDMU19 Bank of Qingdao BOQDMU20 Qingdao Rural Commercial Bank QRCBDMU21 Bank of Shanghai BOSDMU22 Suzhou Rural Commercial Bank SRCBDMU23 Wuxi Rural Commercial Bank WRCBDMU24 Bank of Xian BOXDMU25 Industrial Bank IBDMU26 Rural Commercial Bank of Zhangjiagang RCBZDMU27 Bank of Changsha BCSDMU28 China Merchants Bank CMBDMU29 Bank of Zhengzhou BOZDMU30 Bank of China BOCDMU31 China CITIC Bank CCBDMU32 Zijin Rural Commercial Bank ZRCB

Table 2 Input and output variables

Inputoutput Variable Notation Unit

InputsEmployment x1

Personcount

Fixed assets x2 Million yuanOperation costs x3 Million yuan

Desirable outputsInterest income y1 Million yuanNoninterestincome y2 Million yuan

Undesirableoutputs

Bad debtpercentage z1 Percentage

Complexity 7

Table 3 Descriptive statistics for input-output measures

Year Statistics x1 x2 x3 y1 y2 z1

2014

Max 493583 196238 271132 849879 165370 27100Min 1089 41201 115141 353390 10948 07500Mean 67582 2849915 4823162 15411269 2568599 12531SD 135316 5589804 7815135 23578569 4416322 04237

2015

Max 503082 221502 306395 871779 189780 23900Min 1180 41804 124526 348611 13069 08300Mean 69772 3133597 5738723 16210941 3034405 15308SD 136444 6026226 8745917 24329478 4941992 04226

2016

Max 496698 243619 282712 791480 204045 24100Min 1352 41399 123588 326889 16262 08700Mean 70266 3477886 5828596 15179021 3556052 16013SD 134489 6511810 8516872 22069357 5718980 03678

2017

Max 487307 245687 331518 861594 204424 23900Min 1426 46293 126089 378914 16605 08200Mean 69782 3638326 6007996 16692792 3516824 15431SD 132071 6662176 9168104 23946371 5304531 03240

2018

Max 473691 253525 368966 948094 201271 24700Min 1454 41808 173355 456700 20317 07800Mean 69354 3838034 6744402 18305026 3918308 15034SD 129697 7006174 10096472 26175120 5500040 03382

Table 4 Inefficiency scores of 32 listed banks from 2014 to 2018

Banks 2014 2015 2016 2017 2018DMU1 00000 00000 00000 00000 00000DMU2 00000 03483 06442 05430 03483DMU3 00000 00979 02730 00000 00979DMU4 00000 00000 00000 00000 00000DMU5 00000 00590 00000 01087 00590DMU6 02345 03469 03273 05898 03469DMU7 01798 00000 00000 00000 00000DMU8 02880 03059 02288 02768 03059DMU9 00000 00000 00000 00000 00000DMU10 00000 00000 00000 00000 00000DMU11 00000 00000 00000 00000 00000DMU12 01449 00000 00000 00000 00000DMU13 00000 00000 00000 00000 00000DMU14 02088 01072 01415 00000 01072DMU15 02793 02415 02018 02225 02415DMU16 01926 00000 00000 00000 00000DMU17 00000 00000 00000 00000 00000DMU18 00567 00000 00000 00000 00000DMU19 00000 00000 00827 00000 00000DMU20 00000 00000 00000 00000 00000DMU21 00000 00000 00000 00000 00000DMU22 00000 00000 00000 02032 00000DMU23 00000 00000 00000 00000 00000DMU24 00000 00000 00000 00000 00000DMU25 00000 00000 00000 00000 00000DMU26 00000 00000 00000 00000 00000DMU27 00000 02287 02670 06379 02287DMU28 00000 00000 00000 00000 00000DMU29 00000 00000 01373 00967 00000DMU30 00000 00000 00000 00000 00000DMU31 00634 00799 00000 00000 00799DMU32 05982 02792 03526 05700 02792Mean 00702 00655 00830 01015 00655

8 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

Table 3 Descriptive statistics for input-output measures

Year Statistics x1 x2 x3 y1 y2 z1

2014

Max 493583 196238 271132 849879 165370 27100Min 1089 41201 115141 353390 10948 07500Mean 67582 2849915 4823162 15411269 2568599 12531SD 135316 5589804 7815135 23578569 4416322 04237

2015

Max 503082 221502 306395 871779 189780 23900Min 1180 41804 124526 348611 13069 08300Mean 69772 3133597 5738723 16210941 3034405 15308SD 136444 6026226 8745917 24329478 4941992 04226

2016

Max 496698 243619 282712 791480 204045 24100Min 1352 41399 123588 326889 16262 08700Mean 70266 3477886 5828596 15179021 3556052 16013SD 134489 6511810 8516872 22069357 5718980 03678

2017

Max 487307 245687 331518 861594 204424 23900Min 1426 46293 126089 378914 16605 08200Mean 69782 3638326 6007996 16692792 3516824 15431SD 132071 6662176 9168104 23946371 5304531 03240

2018

Max 473691 253525 368966 948094 201271 24700Min 1454 41808 173355 456700 20317 07800Mean 69354 3838034 6744402 18305026 3918308 15034SD 129697 7006174 10096472 26175120 5500040 03382

Table 4 Inefficiency scores of 32 listed banks from 2014 to 2018

Banks 2014 2015 2016 2017 2018DMU1 00000 00000 00000 00000 00000DMU2 00000 03483 06442 05430 03483DMU3 00000 00979 02730 00000 00979DMU4 00000 00000 00000 00000 00000DMU5 00000 00590 00000 01087 00590DMU6 02345 03469 03273 05898 03469DMU7 01798 00000 00000 00000 00000DMU8 02880 03059 02288 02768 03059DMU9 00000 00000 00000 00000 00000DMU10 00000 00000 00000 00000 00000DMU11 00000 00000 00000 00000 00000DMU12 01449 00000 00000 00000 00000DMU13 00000 00000 00000 00000 00000DMU14 02088 01072 01415 00000 01072DMU15 02793 02415 02018 02225 02415DMU16 01926 00000 00000 00000 00000DMU17 00000 00000 00000 00000 00000DMU18 00567 00000 00000 00000 00000DMU19 00000 00000 00827 00000 00000DMU20 00000 00000 00000 00000 00000DMU21 00000 00000 00000 00000 00000DMU22 00000 00000 00000 02032 00000DMU23 00000 00000 00000 00000 00000DMU24 00000 00000 00000 00000 00000DMU25 00000 00000 00000 00000 00000DMU26 00000 00000 00000 00000 00000DMU27 00000 02287 02670 06379 02287DMU28 00000 00000 00000 00000 00000DMU29 00000 00000 01373 00967 00000DMU30 00000 00000 00000 00000 00000DMU31 00634 00799 00000 00000 00799DMU32 05982 02792 03526 05700 02792Mean 00702 00655 00830 01015 00655

8 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

calculate the inefficiency score intervals and then use Def-inition 2 to calculate the first-level pairwise dominatingprobability in 2018 Since the pairwise dominating proba-bility involves an n times n matrix that is hard to present in thispaper we arbitrarily take the Bank of Jiangsu (BOJ) Ag-ricultural Bank of China (ABC) and Rural CommercialBank of Zhangjiagang (RCBZ) for instance and the first-level inefficiency dominating probability for these threebanks across all 32 banks is listed in the second third andfourth columns of Table 5 namely P1113958BOJ≻k

P 1113958ABC≻k and

P1113959RCBZ≻k(it is also possible to consider any other banks as

examples to show the calculation results) e results rep-resent the probability of the considered bank having asmaller inefficiency index across any other bank with dif-ferent sets of weights being attached to inputs and outputs toensure an efficient status for other banks For example thevalue of 02699 implies that by fixing the inefficiency scoreof the Bank of Beijing (DMU1) to zero the inefficiency scoreof the Bank of Jiangsu will be smaller with a probability of02699 and larger with a probability of 07301 (1ndash02699) Atthe same time we can use model (6) and Definition 3 tocalculate another dominating probability and the second-level results of Bank of Jiangsu (BOJ) Agricultural Bank ofChina (ABC) and Rural Commercial Bank of Zhangjiagang(RCBZ) in 2018 are given in the last three columns of Table 5

Without loss of generality each DMU will alwaysdominate itself as these three banks will have a dominatingprobability of 1 to itself regardless of whether the first-leveldominance probability or the second-level dominanceprobability is considered From Table 5 we can find that allthree banks have a larger second-level dominance proba-bility than the first-level dominance probability to otherbanks and it is indeed also held for all banks is differenceis due to the definition style of pairwise dominances and wecannot arbitrarily use one dominating probability to rep-resent the performance assessment with the other beingignored By averaging the two pairwise dominating prob-abilities we can calculate the pairwise dominance proba-bility as well as the dominated probability Reconsidering thethree banks in Table 5 we show the pairwise dominanceprobability as well as dominated probability for Bank ofJiangsu Agricultural Bank of China and Rural CommercialBank of Zhangjiagang in Table 6 Taking the inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing(DMU1) for example the arithmetic mean of 02699 and03515 by Definition 4 is exactly the pairwise inefficiencydominance probability of Bank of Jiangsu to Bank of Beijing03107 Since the pairwise dominance probability shows theprobability of the inefficiency score of Bank of Jiangsurelative to any other banks by considering all sets of feasibleinput and output weights it means that the inefficiency scoreof Bank of Jiangsu is smaller than that of Bank of Beijing witha probability of 03107 In contrast the inefficiency of Bankof Jiangsu will be dominated by Bank of Beijing with aprobability of 06893

Furthermore by aggregating the pairwise inefficiencydominance probability across all banks we can obtain theaverage performance dominance probability in terms ofinefficiency scores for these three banks as given in the last

column of Table 6 Furthermore proceeding in the samemanner as above we can determine the inefficiency dom-inance probability for all 32 Chinese listed banks from 2014to 2018 as shown in Table 7

As Table 7 shows these banks have very different inef-ficiency dominance probabilities over all sets of feasibleweights relative to the inefficiency scores derived from only anoptimal set of weights For example Huaxia Bank (DMU8)always has a positive inefficiency in each year and that bankwill be ranked after most banks However Huaxia Bank willhave an inefficiency dominance probability of 06105 0588505749 05640 and 05857 in the 2014ndash2018 period re-spectively is phenomenon implies that Huaxia Bank ismore likely to have an inefficiency index smaller than that ofhalf of all banks In contrast by considering only the optimalset of weights the Rural Commercial Bank of Zhangjiagang(DMU26) is extremely efficient with an inefficiency score ofzero for all years but it has a relatively small inefficiencydominance probability in each year (02405 02264 0228902039 and 01864 respectively) implying that the perfor-mance of Rural Commercial Bank of Zhangjiagang in thesense of inefficiency scores is at a disadvantage to other banksby addressing all weight possibilities

Table 5 Two kinds of dominating probabilities for BOJ ABC andRCBZ in 2018

BanksDefinition 2 Definition 3

BOJ ABC RCBZ BOJ ABC RCBZDMU1 02699 05964 00093 03515 09452 02111DMU2 08578 06238 02563 09850 09959 04689DMU3 08520 06207 00496 09662 09910 01966DMU4 00288 03164 00016 03339 03822 03216DMU5 01671 05897 00074 03446 09119 02663DMU6 08592 06197 00386 09599 09887 01905DMU7 08518 06159 00253 09337 09821 01961DMU8 03056 06073 00106 04337 09419 02603DMU9 00315 01127 00017 03121 01311 02989DMU10 10000 06052 00088 10000 09589 01523DMU11 08514 06230 03883 09915 09977 04059DMU12 00835 05315 00042 03563 07917 03073DMU13 01096 05695 00054 03154 08709 02684DMU14 08678 06123 00146 09135 09736 01657DMU15 09666 06173 00286 09704 09791 02599DMU16 00411 10000 00023 03948 10000 03768DMU17 02062 05856 00089 04109 09140 03081DMU18 00921 05531 00046 03014 08476 02634DMU19 08593 06219 00995 09780 09943 02510DMU20 08454 06226 01170 09772 09942 02986DMU21 05626 06048 00133 05695 09618 02139DMU22 08492 06230 02009 09916 09978 02256DMU23 08463 06225 01454 09886 09971 01876DMU24 08473 06222 00825 09808 09951 01796DMU25 00724 05560 00038 02395 08469 02161DMU26 08477 06232 10000 09912 09977 10000DMU27 08790 06204 00512 09669 09892 02431DMU28 01159 05628 00058 03840 08491 03310DMU29 08588 06202 00662 09644 09908 02669DMU30 00389 03718 00021 03274 04493 03107DMU31 01246 05690 00060 03564 08716 02997DMU32 08494 06222 01199 09848 09960 02093

Complexity 9

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

Furthermore it can be found that all banks have arelatively stable inefficiency dominance probability in theperiod of 2014ndash2018 with the largest variation being 01343(02956ndash01613) for Zijin Rural Commercial Bank (DMU32)and the smallest variation being 00087 (07331ndash07244) forShanghai Pudong Development Bank (DMU18) All banksalways have an eco-inefficiency dominance probability thatis either larger than 050 or less than 050 in the five-yearsample (05 is a threshold where the dominating probabilityis equal to the dominated probability) implying that allbanks can be divided into two groups one for superior banksand another for inferior banks e categories are shown inTable 8 From the average sense we find that China Con-struction Bank (DMU9) Industrial and Commercial Bank ofChina (DMU4) Industrial Bank (DMU25) Bank of China(DMU32) and Shanghai Pudong Development Bank(DMU18) are the top five listed banks for operation per-formance In contrast the lowest five banks are SuzhouRural Commercial Bank (DMU22) Rural Commercial Bankof Zhangjiagang (DMU26) Jiangyin Rural Commercial Bank(DMU11) Zijin Rural Commercial Bank (DMU32) andChangshu Rural Commercial Bank (DMU2) all of which

have an average inefficiency dominance probability of lessthan 02500

e eco-inefficiency and eco-inefficiency dominanceprobabilities of these 32 listed banks are given in Tables 4 and7 respectively It is clear that the proposed approach will giveperformance indexes that are different from those of previousapproaches Furthermore we give the ranking comparison ofeco-inefficiency and eco-inefficiency dominance probabilitiesin Table 9 It can be found from Table 9 that on the one handthe proposed approach will give performance rankings thatare largely different from those of previous approaches Onthe other hand the traditional DEA model cannot dis-criminate all banks and more seriously more than twentybanks are ranked as the first based on inefficiency scoreswhile the eco-inefficiency dominance probability approachcan indeed give a full ranking of all banks From this per-spective the proposed approach can give a more reasonableand discriminating performance assessment

All 32 listed banks can be mainly categorized into fourgroups according to the ownership namely state-ownedbanks joint-stock banks city commercial banks andrural commercial banks Table 10 shows the divisions andTable 11 gives the average inefficiency dominance proba-bilities for different kinds of banks

Table 6 Pairwise dominance probability for BOJ ABC and RCBZin 2018

BanksDominating Dominated

BOJ ABC RCBZ BOJ ABC RCBZDMU1 03107 07708 01102 06893 02292 08898DMU2 09214 08098 03626 00786 01902 06374DMU3 09091 08058 01231 00909 01942 08769DMU4 01813 03493 01616 08187 06507 08384DMU5 02559 07508 01369 07441 02492 08631DMU6 09096 08042 01145 00904 01958 08855DMU7 08928 07990 01107 01072 02010 08893DMU8 03696 07746 01354 06304 02254 08646DMU9 01718 01219 01503 08282 08781 08497DMU10 10000 07820 00805 10000 02180 09195DMU11 09214 08104 03971 00786 01896 06029DMU12 02199 06616 01557 07801 03384 08443DMU13 02125 07202 01369 07875 02798 08631DMU14 08906 07929 00902 01094 02071 09098DMU15 09685 07982 01442 00315 02018 08558DMU16 02180 10000 01895 07820 10000 08105DMU17 03086 07498 01585 06914 02502 08415DMU18 01968 07004 01340 08032 02996 08660DMU19 09187 08081 01752 00813 01919 08248DMU20 09113 08084 02078 00887 01916 07922DMU21 05661 07833 01136 04339 02167 08864DMU22 09204 08104 02132 00796 01896 07868DMU23 09174 08098 01665 00826 01902 08335DMU24 09141 08086 01311 00859 01914 08689DMU25 01559 07014 01099 08441 02986 08901DMU26 09195 08105 10000 00805 01895 10000DMU27 09229 08048 01472 00771 01952 08528DMU28 02499 07060 01684 07501 02940 08316DMU29 09116 08055 01666 00884 01945 08334DMU30 01832 04106 01564 08168 05894 08436DMU31 02405 07203 01529 07595 02797 08471DMU32 09171 08091 01646 00829 01909 08354Mean 06096 07375 01864 04217 02938 08448

Table 7 Inefficiency dominance probability of 32 listed banks from2014 to 2018

Banks 2014 2015 2016 2017 2018 MeanDMU1 06175 06149 06142 06173 06235 06175DMU2 02872 02633 02243 02184 02293 02445DMU3 04257 04131 03831 03685 03971 03975DMU4 07964 07936 07786 07920 08056 07932DMU5 06770 06678 06790 06815 06411 06693DMU6 03007 03382 03529 03928 04268 03623DMU7 04582 04853 04794 04922 04908 04812DMU8 06105 05885 05749 05640 05857 05847DMU9 08039 08091 07945 08038 08132 08049DMU10 05439 05812 05893 06105 06096 05869DMU11 02742 02571 02469 02246 02099 02425DMU12 07623 07480 07216 07106 07032 07291DMU13 07042 06892 06916 07231 06982 07013DMU14 05130 05143 05269 05392 05413 05269DMU15 04937 04849 04806 04678 04710 04796DMU16 07135 07209 07130 07232 07375 07216DMU17 06272 06105 05922 06169 06155 06125DMU18 07315 07255 07244 07324 07331 07294DMU19 03614 03654 03772 03510 03212 03552DMU20 02796 02947 03033 02929 03048 02951DMU21 05859 05885 06007 05769 05712 05846DMU22 01967 01937 01941 01786 02062 01939DMU23 02875 03021 02897 02705 02559 02812DMU24 02823 02992 03333 03240 03304 03138DMU25 07851 07710 07642 07895 07747 07769DMU26 02405 02264 02289 02039 01864 02172DMU27 03743 03810 04116 03980 04038 03938DMU28 07060 06765 06573 06634 06801 06766DMU29 04041 04260 04448 04405 03855 04202DMU30 07721 07665 07577 07673 07795 07686DMU31 07224 06996 07011 06738 06721 06938DMU32 01613 02040 02686 02909 02956 02441

10 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

It can be learned from Table 11 that the four kinds oflisted banks exhibit considerably different performancedominance probabilities More specifically those five state-owned banks have the highest average inefficiency domi-nance probability which is almost three times that of thelowest rural commercial banks is result shows that thesestate-owned banks are more likely to have better perfor-mance compared with other banks over all sets of weights Incontrast those rural commercial banks are more likely tohave worse performance relative to other banks Further-more joint-stock banks are inferior to state-owned banksand superior to city commercial banks which are furthersuperior to rural commercial banks

By using the proposed inefficiency dominance proba-bility approach we can provide a performance analysis of 32Chinese listed banks that is derived from the real worldSince the proposed approach considers all sets of feasibleweights which is different from classic DEA approaches thatfocus on only one or several optimal sets of weights theresulting performance analytics are more reasonable due totaking full weights and all possibilities into account Fur-thermore by considering all sets of feasible weights theperformance dominance probability is largely different fromtraditional performance indexes that are obtained with someextreme weights and the corresponding ranking orders arealso changed considerably erefore it makes sense for the

Table 8 Superior and inferior listed banks

Category Banks

Superior(gt05000)

Bank of Beijing Industrial and Commercial Bank of China China Everbright Bank Huaxia Bank China ConstructionBank Bank of Jiangsu Bank of Communications China Minsheng Bank Bank of Nanjing Agricultural Bank of ChinaPing An Bank Shanghai Pudong Development Bank Bank of Shanghai Industrial Bank China Merchants Bank Bank

of China China CITIC Bank

Inferior(lt05000)

Changshu Rural Commercial Bank Bank of Chengdu Bank of Guiyang Bank of Hangzhou Jiangyin RuralCommercial Bank Bank of Ningbo Bank of Qingdao Qingdao Rural Commercial Bank Suzhou Rural CommercialBank Wuxi Rural Commercial Bank Bank of Xian Rural Commercial Bank of Zhangjiagang Bank of Changsha Bank

of Zhengzhou Zijin Rural Commercial Bank

Table 9 Ranking of inefficiency and inefficiency dominance probabilities

Banks 2014 2015 2016 2017 2018DMU1 1 13 1 12 1 12 1 12 1 12DMU2 1 26 32 28 32 31 29 30 32 29DMU3 1 20 25 21 29 22 1 23 25 22DMU4 1 2 1 2 1 2 1 2 1 2DMU5 1 11 23 11 1 10 25 9 23 11DMU6 29 24 31 24 30 24 31 22 31 20DMU7 26 19 1 18 1 19 1 18 1 18DMU8 31 14 30 15 27 16 28 16 30 15DMU9 1 1 1 1 1 1 1 1 1 1DMU10 1 16 1 16 1 15 1 14 1 14DMU11 1 29 1 29 1 29 1 29 1 30DMU12 25 5 1 5 1 6 1 8 1 7DMU13 1 10 1 9 1 9 1 7 1 8DMU14 28 17 26 17 25 17 1 17 26 17DMU15 30 18 28 19 26 18 27 19 28 19DMU16 27 8 1 7 1 7 1 6 1 5DMU17 1 12 1 13 1 14 1 13 1 13DMU18 23 6 1 6 1 5 1 5 1 6DMU19 1 23 1 23 23 23 1 24 1 25DMU20 1 28 1 27 1 26 1 26 1 26DMU21 1 15 1 14 1 13 1 15 1 16DMU22 1 31 1 32 1 32 26 32 1 31DMU23 1 25 1 25 1 27 1 28 1 28DMU24 1 27 1 26 1 25 1 25 1 24DMU25 1 3 1 3 1 3 1 3 1 4DMU26 1 30 1 30 1 30 1 31 1 32DMU27 1 22 27 22 28 21 32 21 27 21DMU28 1 9 1 10 1 11 1 11 1 9DMU29 1 21 1 20 24 20 24 20 1 23DMU30 1 4 1 4 1 4 1 4 1 3DMU31 24 7 24 8 1 8 1 10 24 10DMU32 32 32 29 31 31 28 30 27 29 27

Complexity 11

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

proposed approach because it can provide a comprehensiveperformance assessment instead of only some extremeperformances from a data analytics perspective

4 Conclusion

is paper proposes a new DEA-based approach forassessing the operation performance of Chinese listed banksSince the conventional DEA approaches consider only a setof optimal and extreme weights to measure the relativeperformance the resulting performance indexes might beunreasonable and even unrealistic in practice From a data-driven decision-making perspective this paper proceeds totake all sets of feasible inputoutput weights into accountrather than only some special weights For that purpose we

first propose an extended eco-inefficiency model to addressbanking activities and build a pairwise performance dom-inance structure in terms of inefficiency scores Further-more we calculate the overall inefficiency dominanceprobability based on all sets of feasible weights and theinefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed bankse proposed approach can provide data analytics on rel-ative performances instead of only some extreme possibil-ities and it is further used for the empirical analytics ofoperation performances for 32 listed banks in China

is paper can be extended with regard to several as-pects First this paper considers the possible inefficiencyscore range but ignores its associated possibility at is weconsider each possible performance score coequally but it iscommon that some performance scores are more likely thanothers erefore future research can be developed to takethe possibilities of various performances based on differentsets of weights into account Second an important researchavenue in the DEA field is how to address the internalproduction structure of DMUs and thus similar studies canbe designed for situations with complex internal structuresand linking connections ird similar approaches based onall sets of weights can also be developed for other purposessuch as fixed cost and resource allocation and target settingin real applications

Data Availability

e illustration data used to support the findings of thisstudy are collected from annual financial reports that arepublicly produced by these listed banks in Shenzhen StockExchange and Shanghai Stock Exchange in China In ad-dition the data are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (nos 71901178 7190408471910107002 and 71725001) the Natural Science Founda-tion of Jiangsu Province (no BK20190427) the Social Sci-ence Foundation of Jiangsu Province (no 19GLC017) theFundamental Research Funds for the Central Universities atSouthwestern University of Finance and Economics (nosJBK2003021 JBK2001020 and JBK190504) and at NanjingUniversity of Aeronautics and Astronautics (noNR2019003) and the Innovation and EntrepreneurshipFoundation for Doctor of Jiangsu Province China

References

[1] Y Zha N Liang M Wu and Y Bian ldquoEfficiency evaluationof banks in China a dynamic two-stage slacks-based measureapproachrdquo Omega vol 60 pp 60ndash72 2016

Table 10 Categories of different listed banks

Category Banks

State-owned banks (5)

Industrial and Commercial Bank ofChina

China Construction BankBank of CommunicationsAgricultural Bank of China

Bank of China

Joint-stock banks (8)

China Everbright Bank Huaxia BankChina Minsheng Bank Ping An BankShanghai Pudong Development BankIndustrial Bank China Merchants

BankChina CITIC Bank

City commercial banks(12)

Bank of Beijing Bank of ChengduBank of Guiyang Bank of HangzhouBank of Jiangsu Bank of NanjingBank of Ningbo Bank of QingdaoBank of Shanghai Bank of XianBank of Changsha Bank of

Zhengzhou

Rural commercial banks(7)

Changshu Rural Commercial BankJiangyin Rural Commercial BankQingdao Rural Commercial BankSuzhou Rural Commercial BankWuxi Rural Commercial BankRural Commercial Bank of

ZhangjiagangZijin Rural Commercial Bank

Table 11 Average inefficiency dominance probabilities for fourkinds of listed banks

Banks 2014 2015 2016 2017 2018 MeanState-ownedbanks 07696 07676 07531 07594 07678 07635

Joint-stockbanks 06955 06786 06731 06806 06751 06806

Citycommercialbanks

04467 04577 04662 04649 04644 04600

Ruralcommercialbanks

02467 02488 02508 02400 02412 02455

12 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

[2] N Zhu J L Hougarrd Z Yu and BWang ldquoRanking Chinesecommercial banks based on their expected impact onstructural efficiencyrdquoOmega vol 94 Article ID 102049 2019

[3] N K Avkiran ldquoOpening the black box of efficiency analysisan illustration with UAE banksrdquo Omega vol 37 no 4pp 930ndash941 2009

[4] K Wang W Huang J Wu and Y-N Liu ldquoEfficiencymeasures of the Chinese commercial banking system using anadditive two-stage DEArdquo Omega vol 44 pp 5ndash20 2014

[5] Q An X Tao B Dai and J Li ldquoModified distance frictionminimization model with undesirable output an applicationto the environmental efficiency of Chinarsquos regional industryrdquoComputational Economics 2019

[6] Q An X Tao and B Xiong ldquoBenchmarking with data en-velopment analysis an agency perspectiverdquoOmega Article ID102235 2020

[7] J Chu J Wu C Chu and T Zhang ldquoDEA-based fixed costallocation in two-stage systems leader-follower and satis-faction degree bargaining game approachesrdquo Omega vol 94Article ID 102054 2019

[8] Y Li L Wang and F Li ldquoA data-driven prediction approachfor sports team performance and its application to NationalBasketball Associationrdquo Omega Article ID 102123 2019

[9] P Yin J Chu J Wu J Ding M Yang and YWang ldquoADEA-based two-stage network approach for hotel performanceanalysis an internal cooperation perspectiverdquo Omega vol 93Article ID 102035 2020

[10] Q Zhu M Song and J Wu ldquoExtended secondary goal ap-proach for common equilibrium efficient frontier selection inDEA with fixed-sum outputsrdquo Computers amp Industrial En-gineering vol 144 Article ID 106483 2020

[11] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal ofOperational Research vol 2 no 6 pp 429ndash444 1978

[12] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data en-velopment analysisrdquo Management Science vol 30 no 9pp 1078ndash1092 1984

[13] W Chen R He and Q Wu ldquoA novel efficiency measuremodel for industrial land use based on subvector data en-velope analysis and spatial analysis methodrdquo Complexityvol 2017 Article ID 9516267 11 pages 2017

[14] F Li Q Zhu and L Liang ldquoA new data envelopment analysisbased approach for fixed cost allocationrdquo Annals of Opera-tions Research vol 274 no 1-2 pp 347ndash372 2019

[15] Q Zhu X Li F Li and A Amirteimoori ldquoData-driven approachto find the best partner for merger and acquisitions in bankingindustryrdquo Industrial Management amp Data Systems 2020

[16] Q Zhu X Li F Li and D Zhou ldquoe potential for energysaving and carbon emission reduction in Chinarsquos regionalindustrial sectorsrdquo Science of the Total Environment vol 716Article ID 135009 2020

[17] J C Paradi and H Zhu ldquoA survey on bank branch efficiencyand performance research with data envelopment analysisrdquoOmega vol 41 no 1 pp 61ndash79 2013

[18] H-H Liu Y-Y Song and G-L Yang ldquoCross-efficiencyevaluation in data envelopment analysis based on prospecttheoryrdquo European Journal of Operational Research vol 273no 1 pp 364ndash375 2019

[19] J Wu J Chu J Sun and Q Zhu ldquoDEA cross-efficiencyevaluation based on Pareto improvementrdquo European Journalof Operational Research vol 248 no 2 pp 571ndash579 2016

[20] R G Dyson R Allen A S Camanho V V PodinovskiC S Sarrico and E A Shale ldquoPitfalls and protocols in DEArdquo

European Journal of Operational Research vol 132 no 2pp 245ndash259 2011

[21] A Salo and A Punkka ldquoRanking intervals and dominancerelations for ratio-based efficiency analysisrdquo ManagementScience vol 57 no 1 pp 200ndash214 2011

[22] W D Cook Y Roll and A Kazakov ldquoA dea model formeasuring the relative eeficiency of highway maintenancepatrolsrdquo INFOR Information Systems and Operational Re-search vol 28 no 2 pp 113ndash124 1990

[23] Y Roll W D Cook and B Golany ldquoControlling factorweights in data envelopment analysisrdquo IIE Transactionsvol 23 no 1 pp 2ndash9 1991

[24] W D Cook and M Kress ldquoA data envelopment model foraggregating preference rankingsrdquo Management Sciencevol 36 no 11 pp 1302ndash1310 1990

[25] C Kao and H-T Hung ldquoData envelopment analysis withcommon weights the compromise solution approachrdquoJournal of the Operational Research Society vol 56 no 10pp 1196ndash1203 2005

[26] F-H F Liu and H Hsuan Peng ldquoRanking of units on theDEA frontier with common weightsrdquo Computers amp Opera-tions Research vol 35 no 5 pp 1624ndash1637 2008

[27] C Kao ldquoMalmquist productivity index based on common-weights DEA the case of Taiwan forests after reorganizationrdquoOmega vol 38 no 6 pp 484ndash491 2010

[28] M Zohrehbandian A Makui and A Alinezhad ldquoA com-promise solution approach for finding common weights inDEA an improvement to Kao and Hungrsquos approachrdquo Journalof the Operational Research Society vol 61 no 4 pp 604ndash6102010

[29] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabianand F Nuri-Bahmani ldquoRanking decision-making units usingcommon weights in DEArdquo Applied Mathematical Modellingvol 38 no 15-16 pp 3890ndash3896 2014

[30] A Shabani F Visani P Barbieri W Dullaert and D VigoldquoReliable estimation of suppliersrsquo total cost of ownership animprecise data envelopment analysis model with commonweightsrdquo Omega vol 87 pp 57ndash70 2019

[31] Y Li F Li A Emrouznejad L Liang and Q Xie ldquoAllocatingthe fixed cost an approach based on data envelopmentanalysis and cooperative gamerdquo Annals of Operations Re-search vol 274 no 1-2 pp 373ndash394 2019

[32] F Li Q Zhu and Z Chen ldquoAllocating a fixed cost across thedecision making units with two-stage network structuresrdquoOmega vol 83 pp 139ndash154 2019

[33] T R Sexton R H Silkman and A J Hogan ldquoData envel-opment analysis critique and extensionsrdquo New Directions forProgram Evaluation vol 1986 no 32 pp 73ndash105 1986

[34] F Li Q Zhu and L Liang ldquoAllocating a fixed cost based on aDEA-game cross efficiency approachrdquo Expert Systems withApplications vol 96 pp 196ndash207 2018

[35] M Davtalab-Olyaie ldquoA secondary goal in DEA cross-effi-ciency evaluation a ldquoone home run is much better than twodoublesrdquo criterionrdquo Journal of the Operational Research So-ciety vol 70 no 5 pp 807ndash816 2019

[36] F Li Q Zhu Z Chen and H Xue ldquoA balanced data en-velopment analysis cross-efficiency evaluation approachrdquoExpert Systems with Applications vol 106 pp 154ndash168 2018

[37] L Liang J Wu W D Cook and J Zhu ldquoAlternative sec-ondary goals in DEA cross-efficiency evaluationrdquo Interna-tional Journal of Production Economics vol 113 no 2pp 1025ndash1030 2008

[38] J L Ruiz and I Sirvent ldquoOn the DEA total weight flexibilityand the aggregation in cross-efficiency evaluationsrdquo European

Complexity 13

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity

Journal of Operational Research vol 223 no 3 pp 732ndash7382012

[39] Y-M Wang and K-S Chin ldquoe use of OWA operatorweights for cross-efficiency aggregationrdquo Omega vol 39no 5 pp 493ndash503 2011

[40] Y M Wang and S Wang ldquoApproaches to determining therelative importance weights for cross-efficiency aggregation indata envelopment analysisrdquo Journal of the Operational Re-search Society vol 64 no 1 pp 60ndash69 2013

[41] G-L Yang J-B Yang W-B Liu and X-X Li ldquoCross-effi-ciency aggregation in DEA models using the evidential-rea-soning approachrdquo European Journal of Operational Researchvol 231 no 2 pp 393ndash404 2013

[42] A Boussofiane R G Dyson and E anassoulis ldquoApplieddata envelopment analysisrdquo European Journal of OperationalResearch vol 52 no 1 pp 1ndash15 1991

[43] J R Doyle and R H Green ldquoCross-evaluation in DEAimproving discrimination among DMUsrdquo INFOR Informa-tion Systems and Operational Research vol 33 no 3pp 205ndash222 1995

[44] T R Anderson K Hollingsworth and L Inman ldquoe fixedweighting nature of a cross-evaluation modelrdquo Journal ofProductivity Analysis vol 17 no 3 pp 249ndash255 2002

[45] M Falagario F Sciancalepore N Costantino andR Pietroforte ldquoUsing a DEA-cross efficiency approach inpublic procurement tendersrdquo European Journal of Opera-tional Research vol 218 no 2 pp 523ndash529 2012

[46] S Lim K W Oh and J Zhu ldquoUse of DEA cross-efficiencyevaluation in portfolio selection an application to Koreanstock marketrdquo European Journal of Operational Researchvol 236 no 1 pp 361ndash368 2014

[47] S Lim and J Zhu ldquoDEA cross-efficiency evaluation undervariable returns to scalerdquo Journal of the Operational ResearchSociety vol 66 no 3 pp 476ndash487 2015

[48] M Carrillo and J M Jorge ldquoAn alternative neutral approachfor cross-efficiency evaluationrdquo Computers amp Industrial En-gineering vol 120 pp 137ndash145 2018

[49] L Liang J Wu W D Cook and J Zhu ldquoe DEA gamecross-efficiency model and its Nash equilibriumrdquo OperationsResearch vol 56 no 5 pp 1278ndash1288 2008

[50] Y-M Wang and K-S Chin ldquoSome alternative models forDEA cross-efficiency evaluationrdquo International Journal ofProduction Economics vol 128 no 1 pp 332ndash338 2010

[51] X Tang Y Li M Wang and L Liang ldquoA DEA efficiencyprobability dominance methodrdquo Systems Engineering-lteoryamp Practice vol 36 no 10 pp 2641ndash2647 2016

[52] X Shi ldquoEfficiency bounds for two-stage production systemsrdquoMathematical Problems in Engineering vol 2018 Article ID2917537 9 pages 2018

[53] Y Li X Shi A Emrouznejad and L Liang ldquoRanking intervalsfor two-stage production systemsrdquo Journal of the OperationalResearch Society vol 71 no 2 pp 209ndash224 2020

[54] F Li Z Yan Q Zhu M Yin and G Kou ldquoAllocating a fixedcost across decision making units with explicitly consideringefficiency rankingsrdquo Journal of the Operational ResearchSociety pp 1ndash15 2020

[55] F Li A Emrouznejad G-l Yang and Y Li ldquoCarbon emissionabatement quota allocation in Chinese manufacturing indus-tries an integrated cooperative game data envelopment analysisapproachrdquo Journal of the Operational Research Societypp 1ndash30 2019

[56] X Li F Li N Zhao and Q Zhu ldquoMeasuring environmentalsustainability performance of freight transportation seaports

in China a data envelopment analysis approach based on theclosest targetsrdquo Expert Systems Article ID e12334 2018

[57] M Song Q An W Zhang Z Wang and J Wu ldquoEnvi-ronmental efficiency evaluation based on data envelopmentanalysis a reviewrdquo Renewable and Sustainable Energy Re-views vol 16 no 7 pp 4465ndash4469 2012

[58] C-M Chen andM A Delmas ldquoMeasuring eco-inefficiency anew frontier approachrdquo Operations Research vol 60 no 5pp 1064ndash1079 2012

[59] T Kuosmanen ldquoWeak disposability in nonparametric pro-duction analysis with undesirable outputsrdquo American Journalof Agricultural Economics vol 87 no 4 pp 1077ndash1082 2005

[60] F Li Q Zhu and J Zhuang ldquoAnalysis of fire protectionefficiency in the United States a two-stage DEA-based ap-proachrdquo OR Spectrum vol 40 no 1 pp 23ndash68 2018

[61] M Asmild J L Hougaard D Kronborg and H K KvistldquoMeasuring inefficiency via potential improvementsrdquo Journalof Productivity Analysis vol 19 no 1 pp 59ndash76 2003

[62] P Bogetoft and J L Hougaard ldquoEfficiency evaluations basedon potential (non-proportional) improvementsrdquo Journal ofProductivity Analysis vol 12 no 3 pp 233ndash247 1999

[63] B Arabi S Munisamy and A Emrouznejad ldquoA new slacks-based measure of Malmquist-Luenberger index in the pres-ence of undesirable outputsrdquo Omega vol 51 pp 29ndash37 2015

[64] H Fukuyama and R Matousek ldquoModelling bank perfor-mance a network DEA approachrdquo European Journal ofOperational Research vol 259 no 2 pp 721ndash732 2017

[65] F Li L Liang Y Li and A Emrouznejad ldquoAn alternativeapproach to decompose the potential gains from mergersrdquoJournal of the Operational Research Society vol 69 no 11pp 1793ndash1802 2018

14 Complexity