# An Eco-Inefficiency Dominance Probability Approach for Chinese Banking Operations Based on Data Envelopment Analysis

Authors: Feng Li; Lunwen Wu; Qingyuan Zhu; Yanling Yu; Gang Kou; Yi Liao
Journal: Complexity
 (2020)

---

## Abstract

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


---

## Body

## 1. Introduction

Forty years have gone by since the great reform and opening policy of 1978, and China has made substantial progress in economic development with an annual increase of almost nine percent in gross domestic product from 149.541 billion dollars in 1978 to 13.608 trillion dollars in 2018. It is rather remarkable that the banking industry of China, especially state-owned and listed banks, has played a great role in Chinese economic growth [1, 2]. Throughout the ever-increasing national economic development, the banking industry in China has also been promoted and developed considerably. For example, the total assets of the Chinese banking industry reached almost 41 trillion dollars in 2018, which is more than three times the gross domestic product in current US dollars in the same year. Meanwhile, the unprecedented competition within Chinese banks and between Chinese domestic banks and foreign banks has become increasingly fierce since the opening of financial markets. To participate in the ongoing competitive challenges all over the world, it is of vital importance for Chinese banks to pay special attention to their operation performances [1, 3, 4]. In addition, it is also an inherent requirement of guaranteeing and promoting the healthy and sustainable economic development of China to address the banking performance.Among the family of existing performance evaluation methods, data envelopment analysis (DEA) is one of the major approaches because of its general applicability [5–10]. DEA, first introduced by Charnes et al. [11] and further extended by Banker et al. [12], is a data analytics approach that can be used for evaluating the relative performances of a group of homogeneous decision-making units (DMUs), which in practice consume multiple inputs to gain multiple outputs. The basic logic behind the DEA methodology is that it compares DMUs’ real activity levels relative to the ideal status by projecting their actual input-output bundle onto the production frontier. To obtain the production frontier, all DMUs’ observed inputs and outputs are used to construct a production possibility set (PPS) with a set of certain axiomatic hypothesis, while the production frontier is an envelopment of the production possibility set. The DEA methodology has many apparent characteristics/advantages, and since its inception work in Charnes et al. [11], DEA has been applied to many kinds of activities in various different contexts [13–16]. In addition, the DEA methodology has also proven to be a powerful and preferable method for performance evaluation in the banking industry and has been frequently applied to this industry [2–4, 17].The conventional DEA methods allow each DMU to generate a set of relative weights to maximize its ratio efficiency of aggregated weighted outputs to aggregated weighted inputs while ensuring that the same ratio is no more than one for all DMUs, and the maximum ratio is considered the performance index for the evaluated DMU [18, 19]. The weight determination is critical to performance evaluation results, but there are some significant concerns that reduce the applicability of DEA-based performance analytics and applications [20, 21]. On the one hand, each DMU selects its most favorable set of weights to maximize its efficiency ratio, and thus, the efficiency score of each DMU might be too optimistic or even impossible due to the unrealistic set of weights. On the other hand, each DMU selects its set of weights separately, and as a result, their performance scores are obtained under different standards and thus are not comparable.Many studies have been proposed for more reasonable performance analytics by focusing on the selection of feasible weights. Cook et al. [22] and Roll et al. [23] suggest a common set of weights method that attempts to find a common set of weights, and the performance assessment is implemented using that common set of weights. Cook and Kress [24] also address the common set of weights by minimizing the gap of upper and lower bounds of weights. Kao and Hung [25] suggest generating a common set of weights by minimizing the sum of squared difference between the possible efficiency scores for the common set of weights and the CCR efficiency score across all DMUs. Similar studies can also be found in Liu & 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 a common evaluation standard for all DMUs, its major problem is that it still considers only one possibility for weights upon which the performance is estimated. Moreover, the determination of the common set of weights is still a big problem that will affect the performance analytics results relative to different common sets of weights, and a consensus regarding this has not been reached thus far.Another research stream focuses on the cross-efficiency method, in which each DMU selects a set of common weights and then the set of common weights is used to evaluate each DMU [33]. It is notable that the classic DEA approach evaluates each DMU’s relative efficiency using its favorable set of weights [18, 34], while the cross-efficiency method requires that each DMU’s favorable set of weights be used to evaluate itself as well as the other DMUs’ relative efficiency. As a result, several sets of weights are used to measure the relative performance, which is a great improvement relative to classic DEA approaches considering only one set of weights. Furthermore, each DMU will have a maximal efficiency score based on self-appraisal and several smaller efficiency scores based on peer appraisal. The ultimate cross-efficiency score can be aggregated with these self-appraisal and peer appraisal scores for each individual DMU [35–41]. The DEA cross-efficiency method has some preferable characteristics, such as satisfied discrimination power between good and poor performances [42], a full ranking of all DMUs [43], and more realistic weights attached to various inputs and outputs [44]. The literature has witnessed numerous studies on various cross-efficiency evaluation approaches for many kinds of real applications [45–47]. Although the cross-efficiency method considers several sets of weights to measure the performance, it is insufficient to involve all performance possibilities. Moreover, the determination of nonunique weights from each DMU’s perspective will also reduce the applicability of cross-efficiency methods, as does the aggregation of individual cross-efficiencies [41, 48–50].The recent research by Salo and Punkka [21] suggests developing ratio-based efficiency analysis over all sets of feasible weights. Salo and Punkka [21] build ranking intervals, dominance relations, and efficiency bounds to show how the DMUs’ efficiency ratios relate to each other for all sets of feasible weights rather than for some sets of weights that are typically used in classic DEA studies. More specifically, Tang et al. [51] propose a novel efficiency probability dominance model and develop the dominance efficiency probability over all sets of feasible weights, but their approach is based on a radial model under the constant returns to scale (CRS) assumption, and only traditional desirable outputs are considered by ignoring undesirable outputs. Shi [52] extends the Salo and Punkka [21] model over sets of all feasible weights to a more common and practical case considering the internal two-stage production structure. The proposed approach calculates each DMU’s efficiency bounds for the overall system as well as the efficiency bounds for each subsystem. Li et al. [53] build the efficiency ranking interval for two-stage production systems and calculate each DMU’s ranking interval for the overall system, as well as for each substage. Li et al. [54] propose a fixed cost allocation approach based on the efficiency ranking concept, which addresses the performance and efficiency ranking interval by considering all relative weights. It is of vital significance to consider all sets of feasible weights because doing so can address all possibilities from a data analytics perspective and provide evaluation results that are more logical and fairer.In this paper, we will develop a dominance probability approach with undesirable outputs to assess the ratio performance based on DEA models, and the proposed approach is illustrated with Chinese listed banks. From a data-driven analytics perspective, we will consider all sets of feasible weights that are used for estimating the ratio performance. Towards that purpose, we first build a performance evaluation model to address the banking activities in the classic DEA framework by considering only a set of optimal weights. Since banking operations inevitably yield some undesirable by-products, such as bad debts that are jointly produced with incomes, an extended eco-inefficiency model is developed. Furthermore, the eco-inefficiency model is used to develop the pair dominance structure taking all sets of weights into account based on Tang et al. [51]. The performance dominance probability is proposed to calculate the average probability of a certain DMU’s performance dominating all DMUs. A larger performance dominance probability indicates that it is easier for that DMU to dominate all other DMUs, implying that its inefficiency score is more likely to be smaller than that of other DMUs in the sense of data-driven analytics. Finally, the proposed approach is applied to a four-year dataset of 32 Chinese listed banks, and the empirical results show that (1) the inefficiency dominance probability is largely different from inefficiency scores; (2) these state-owned commercial banks are more likely to have a better dominance performance, while local rural commercial banks might have very promising inefficiency performances in some extreme situations but are likely to have poor performance in the sense of inefficiency dominance probability; and (3) China Construction Bank, Industrial and Commercial Bank of China, and Industrial Bank are the top three listed banks based on operations analytics, and on the contrary, Suzhou Rural Commercial Bank, Rural Commercial Bank of Zhangjiagang, and Jiangyin Rural Commercial Bank are the lowest three banks with inefficiency dominance probability. This paper contributes to the literature in at least the following aspects. First, this paper extends a new approach in the DEA framework by taking all sets of feasible weights into account, whereas previous studies have considered only one or several sets of weights. Second, this paper establishes a pairwise performance dominance concept, which can help decision-makers analyze the performance relation in any context with price information (i.e., weights or multipliers). Third, this paper analyzes the performance of Chinese listed banks and provides some empirical findings, which can facilitate the banking industries in China.The remainder of this paper is organized as follows. Section2 develops the mathematical methodology of an extended eco-inefficiency model with undesirable outputs and performance dominance structure using inefficiency scores. Afterwards, the proposed approach is used to study the empirical performance analytics of 32 listed banks in Section 3. Finally, Section 4 concludes and summarizes this paper.


## 2. Mathematical Modeling

We first propose an extended eco-inefficiency model to address banking activities and possible relative performances in Section2.1. Furthermore, we build a performance dominance structure considering all sets of feasible weights in Section 2.2.


### 2.1. An Extended Eco-Inefficiency Model

Suppose a set ofn peer banks, with each bank using m inputs for the sake of producing s traditional desirable outputs as well as q undesirable outputs, such as bad debt in the illustrative application. A bad debt or nonperforming loan is a jointly produced and unavoidable by-product in the banking industry. Without loss of generality, we consider each bank as a homogeneous decision-making unit (DMU) in the DEA framework. Furthermore, DMUjj=1,…,n consume inputs Xj=x1j,…,xmj to produce desirable outputs Yj=y1j,…,ysj and undesirable outputs Bj=b1j,…,bqj, respectively. Before constructing the modeling, a core task is to identify appropriate methods to handle undesirable outputs. It is clear that the strong and weak disposability assumption of undesirable outputs and desirable outputs are the two most common and natural methods in the literature [55–57]. A significant feature between the strong and weak disposability assumption is whether undesirable outputs can be produced without damage or subsequent cost to desirable outputs [58]. If undesirable outputs can be freely generated without damage or subsequent cost, implying that both inputs and outputs can change unilaterally without compromising each other, then undesirable outputs are assumed to be strongly disposable. In contrast, if the production of undesirable outputs indeed has some damage or subsequent cost to inputs or desirable outputs, implying that a reduction in undesirable outputs would result in a reduction of desirable outputs simultaneously [59, 60], then undesirable outputs are assumed to be weak disposable. Here, we consider the weak disposability assumption because it is more suitable for the real world and, more specifically, for the empirical application of banking operations, where it is very difficult to freely reduce bad debts without affecting incomes and changing banking operations. To this end, the production possibility set (PPS) under the variable returns to scale (VRS) assumption can be formulated as follows:(1)PPS=xi,yr,bp∑j=1nλjxij≤xi,i=1,...,mρj∑j=1nλjyrj≥yr,r=1,...,sρj∑j=1nλjbpj=bp,p=1,...,q∑j=1nλj=10≤ρj≤1,λj≥0,j=1,...,n.The above formula is nonlinear since both the reduction factorρj and intensity variable λj are unknown. Furthermore, we can equivalently change formula (1) into a linear version, which is presented in the following formula:(2)PPS=xi,yr,bp∑j=1nλj+ηjxij≤xi,i=1,...,m∑j=1nλjyrj≥yr,r=1,...,s∑j=1nλjbpj=bp,p=1,...,q∑j=1nλj+ηj=1,λj,ηj≥0,j=1,...,n.Based on formula (2), both desirable and undesirable outputs are weighted by nondisposed intensity variables λj, whereas the inputs are weighted by the sum of nondisposed intensity variables λj and disposed intensity variables ηj. In addition, the VRS assumption is ensured by summing the total nondisposed intensity variables λj and disposed intensity variables ηj to 1, i.e., ∑j=1nλj+ηj=1.Based on the above PPS in formula (2), we can build mathematical models to compute the relative performance of DMUs. In this paper, we follow the practice of Chen and Delmas [58] to develop an extended eco-inefficiency model for performance evaluation since the eco-inefficiency model has some advantages in modeling activities with undesirable outputs compared with four well-established models in the literature (undesirable outputs as inputs, transformation of undesirable outputs, directional distance function, and hyperbolic efficiency model. Readers can refer Chen and Delmas [58] for details on the comparison). The extended eco-inefficiency model is formulated as follows:(3)inEd∗=Max∑i=1mgix/xid+∑r=1sgry/yrd+∑p=1qgpb/bpdm+s+qs.t.∑j=1nλj+ηjxij≤xid−gim,i=1,…,m∑j=1nλjyrj≥yrd+gry,r=1,…,s∑j=1nλjbpj=bpd−gpb,p=1,…,q∑j=1nλj+ηj=1λj,ηj,gim,gry,gpb≥0,j=1,…,n;r=1,…,s;p=1,…,q.In model (3), the variables gix, gry, and gpb represent the amount of potential improvements in inputs, desirable outputs, and undesirable outputs, respectively, that the evaluated DMU can make relative to its current input usage and output production to reach its ideal benchmark target on the efficiency frontier. The potential improvements reflect input reduction potentials and desirable output expansion potentials (or undesirable output reduction potentials) instead of actual input usage and output production [61, 62]. Model (3) uses a slack-based formula that is similar to the directional distance function (DDF) model to maximize the average additive inefficiency index across all input and output measures, where the inefficiency index represents potential improvements divided by observed inputs or outputs. More specifically, the optimal direction vector of model (3) can be endogenously obtained in a similar way as that of Arabi et al. [63]. For that purpose, suppose gix=β⋅g¯i, gry=β⋅g˜r, and gpb=β⋅g^p, where the direction vector is g=g¯i,g˜r,g^p. Solving model (3) determines an optimal solution gix∗,gry∗,gpb∗, then we have β=gix∗/g¯i=gry∗/g˜r=gpb∗/g^pi=1,…,m;r=1,…,s;p=1,…,q. Hence, we have a system of m+s+q+1 unknown variables (β,g¯i,g˜r,g^p) and m+s+q linearly independent equations g1x∗g¯2=g2x∗g¯1,…,gmx∗g˜1=g1y∗g¯m, g1y∗g˜2=g2y∗g˜1,…,gsy∗g^1=g1b∗g˜s, g1b∗g^2=g2b∗g^1,...,gq−1b∗g^q=gqb∗g^q−1, and gqb∗=β⋅gq. Together with another equation such as ∑i=1mg¯i+∑r=1sg˜r+∑p=1qg^p=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 independent equations. Therefore, this system has a unique solution and we can obtain a unique optimal direction.Model (3) is slightly different from that of Chen and Delmas [58] in several aspects: first, we also take the input improvement into account, while Chen and Delmas [58] studied only the output improvement; secondly, we assume the weak disposability assumption of desirable outputs and undesirable outputs, while the strong disposability assumption was modeled in Chen and Delmas [58]. In addition, the VRS assumption is considered in model (3) such that it is more suitable for real applications. Since the individual inefficiency index theoretically has a value ranging from zero to unity for inputs and undesirable outputs and a value from zero to infinity for desirable outputs, the overall average inefficiency of DMUd also takes a value from zero to infinity. The larger the average inefficiency index is, the more inefficient the evaluated DMU is. An average inefficiency index of zero value means that the considered DMU is on the efficiency frontier and has no slack for improvement, and hence, the DMU is efficient.Furthermore, model (4) is a dual formulation of the above model (3) in its multiplier formulation:(4)inEd∗=Min∑i=1mvixid−∑r=1suryrd+∑p=1qwpbpd+u0s.t.∑i=1mvixij−∑r=1suryrj+∑p=1qwpbpj+u0≥0,j=1,…,n∑i=1mvixij+u0≥0,j=1,…,nvi≥1/m+s+qxid,i=1,…,mur≥1/m+s+qyrd,r=1,…,swp≥1/m+s+qbpd,p=1,…,qvi,ur≥0,i=1,…,m;r=1,…,s;wp,u0are free,p=1,…,q.Model (4) computes the inefficient component (i.e., the difference between aggregated inputs and aggregated outputs) for the evaluated DMU, yet the inefficient component is nonnegative for all DMUs, and some constraints on multipliers are held. Solving model (4) for each DMUdd=1,…,n determines a series of inefficiency scores inEd∗ using a series of optimal solutions urd∗,vid∗,wpd∗,u0d∗. The inefficiency score can be used for a performance indicator among all DMUs, and the smaller the inefficiency score is, the better the performance of DMUd is.


### 2.2. Dominance Probability Based on Inefficiency Scores

It is notable that the inefficiency scoreinEd∗ obtained previously can be used to analyze the performance of Chinese listed banks, but there are two main concerns. On the one hand, it is calculated by only considering the optimal weight plan urd∗,vid∗,wpd∗,u0d∗ while ignoring any other feasible weights, and thus, the obtained inefficiency score inEd∗ might be too extreme or even unrealistic. On the other hand, the inefficiency score inEd∗ is calculated separately for each DMUdd=1,…,n, and different weights will be preferred by different DMUs; thus, the results are not completely comparable.Since the resulting inefficiency indexes can change relative to different sets of weights, it is important to explore the performance associated with all sets of feasible input/output weights. To this end, we focus on the efficiency dominance concept of Salo and Punkka [21]. As well defined and discussed in Salo and Punkka [21] and Tang et al. [51], the dominance relation is determined through a pairwise efficiency comparison among DMUs. Furthermore, a certain DMU dominates another DMU if and only if its efficiency score is as large as that of the other for all sets of feasible input/output weights and is larger for at least some sets of feasible input/output weights. Here, we follow the same idea of Salo and Punkka [21] and Tang et al. [51] to determine the dominance relation of Chinese listed banks. For comparison, we take inevitable undesirable outputs in banking operations, such as bad debts, into account. Furthermore, we follow Chen and Delmas [58] in focusing on eco-inefficiency scores rather than efficiency scores through a nonradial directional distance function model. To this end, we first build the inefficiency dominance concept as follows.Definition 1.
DMU
d dominates DMUk (denoted as DMUd≻DMUk) if and only if DMUd always has a smaller inefficiency score relative to DMUk for all sets of feasible input and output weights.
The dominance relation based on inefficiency scores is determined if the inefficiency score for a certainDMUd is as small as that of DMUk for all sets of feasible input/output weights and is smaller for at least some sets of feasible input/output weights. The dominance relation between DMUd and DMUk is determined through their inefficiency comparison. By rethinking the idea of model (3) and model (4), which calculate the maximal inefficiency score, we can formulate model (5) to calculate the inefficiency range of DMUd when DMUk is fixed with a prespecified inefficiency level of zero by requiring the constraint that inEk=∑i=1mvixik−∑r=1suryrk+∑p=1qwpbpk+u0=0. In fact, the inefficiency level of DMUk can be set to any nonnegative value M, and by substituting u0 with u0−M, we can get the same dominance probability as given in Definition 2 and Definition 3; hence, we immediately set the inefficiency level of DMUk to zero for simplification in the following model:(5)inEd˜kmaxinEd˜kmin=MinMax∑i=1mvixid−∑r=1suryrd+∑p=1qwpbpd+u0s.t∑i=1mvi+∑r=1sur+∑p=1qwp=1∑i=1mvixik−∑r=1suryrk+∑p=1qwpbpk+u0=0∑i=1mvixij+u0≥0,j=d,kvi≥1m+s+qxid,i=1,…,mur≥1m+s+qyrd,r=1,…,swp≥1m+s+qbpd,p=1,…,qvi,ur≥0,i=1,…,m;r=1,…,s;wp,u0are free,p=1,…,q.
An additional constraint that∑i=1mvi+∑r=1sur+∑p=1qwp=1 is inserted into model (5) to make the feasible weight space closed and bounded. The optimal objective function for model (5) shows the lower and upper bounds on how much different DMUd’s inefficiency score can be relative to DMUk across all sets of feasible input/output weights. Using model (5), the dominance structure can be determined for any pairwise DMUs. More specifically, if the inefficiency level of DMUk is fixed to zero and if inEd˜kmax<0, which means that DMUd will always have a smaller inefficiency score compared with DMUk, then DMUd dominates DMUk. In contrast, if inEd˜kmin>0, which means that DMUd will always have a larger inefficiency score than DMUk (which has an inefficiency level of zero), then DMUd is dominated by DMUk. For more general cases, however, we cannot obtain the complete dominance relation, which is usually true in practice. Therefore, we propose determining the performance dominance probability. To this end, we follow the work of Tang et al. [51] in building the inefficiency dominance probability concept, as given in Definition 2.Definition 2.
WhenDMUk has an inefficiency score of inEk, the probability that DMUd dominates DMUk over all sets of feasible input and output weights is calculated by Pd˜≻k=inEk−inEd˜kmin/inEd˜kmax−inEd˜kmin.
It is clear that, ifinEk≤inEd˜kmin, which means that DMUd will always have a larger inefficiency by remaining the inefficiency inEk for DMUk, the inefficiency dominance probability of DMUd relative to DMUk would take a nonpositive value. That is, it is impossible for DMUd to dominate DMUk. For the sake of a bounded range, we assume that Pd˜≻k=0 if inEk≤inEd˜kmin. For a more general case in which inEd˜kmin≤inEk≤inEd˜kmax, the probability that DMUd dominates DMUk takes a value from zero to unity. A larger value of Pd˜≻k means that it is more likely for DMUd to have a smaller inefficiency score compared with DMUk over all sets of feasible input/output weights.
Note in addition that Definition2 fixes the inefficiency level of DMUk to calculate the inefficiency range of DMUd and further computes the inefficiency dominance probability of DMUd relative to DMUk. In contrast, we can also compute the inefficiency dominance probability of DMUd relative to DMUk by fixing the inefficiency level of DMUd and calculating the inefficiency range of DMUk. The above idea is formulated in model (6), which is very similar to model (5) but substitutes DMUk for DMUd:(6)inEk˜dmaxinEk˜dmin=MinMax∑i=1mvixik−∑r=1suryrk+∑p=1qwpbpk+u0s.t.∑i=1mvi+∑r=1sur+∑p=1qwp=1∑i=1mvixid−∑r=1suryrd+∑p=1qwpbpd+u0=0∑i=1mvixij+u0≥0,j=d,kvi≥1m+s+qxik,i=1,…,mur≥1m+s+qyrk,r=1,…,swp≥1m+s+qbpk,p=1,…,qvi,ur≥0,i=1,…,m;r=1,…,s;wp,u0are free,p=1,…,q.
An alternative inefficiency dominance probability ofDMUd relative to DMUk by fixing the inefficiency level of DMUd is given in Definition 3.Definition 3.
WhenDMUd has an inefficiency score of inEd, the probability that DMUd dominates DMUk over all sets of feasible input and output weights is calculated by Pd≻k˜=inEk˜dmax−inEd/inEk˜dmax−inEk˜dmin.
Furthermore, the overall probability that the inefficiency score ofDMUd dominates that of DMUk is the average of the two probabilities by fixing the inefficiency level of DMUd and DMUk. More specifically, Definition 4 gives the pairwise performance dominance probability with regard to inefficiency scores.Definition 4.
The pairwise performance dominance probability ofDMUd relative to DMUk over all sets of feasible input and output weights is Pd≻k=Pd˜≻k+Pd≻k˜/2.
The classic DEA methods allow each DMU to generate a set of relative weights to maximize its ratio of aggregated weighted outputs to aggregated weighted inputs while ensuring that the same ratio is no more than one for all DMUs, and the maximum ratio is considered the efficiency score for the evaluated DMU. By taking all sets of feasible weights and the inefficiency dominance structure into account, the overall inefficiency dominance probability for a certainDMUd among all DMUs can be calculated by the average of pair inefficiency dominance probabilities across all DMUs. The above idea is given in Definition 5.Definition 5.
The performance dominance probability ofDMUd across all DMUs over all sets of feasible input and output weights is Pd=∑k=1nPd≻k/n.
The classic DEA approaches use deterministic (in)efficiency scores to measure the relative performance, while the performance dominance probability is an alternative performance indicator from a data analytics perspective that considers all sets of feasible weights that are stochastic to determine the relative performance. It is rather remarkable that different sets of weights will cause different performance measures, and the performance dominance probability involves all possibilities over all sets of feasible input and output weights. The performance dominance probability ofDMUd calculates the probability of its performance dominating the set of all DMUs. A larger performance dominance probability indicates that it is much easier for that DMU to dominate other DMUs, implying that its inefficiency score is more likely to be smaller than that of other DMUs.


## 2.1. An Extended Eco-Inefficiency Model

Suppose a set ofn peer banks, with each bank using m inputs for the sake of producing s traditional desirable outputs as well as q undesirable outputs, such as bad debt in the illustrative application. A bad debt or nonperforming loan is a jointly produced and unavoidable by-product in the banking industry. Without loss of generality, we consider each bank as a homogeneous decision-making unit (DMU) in the DEA framework. Furthermore, DMUjj=1,…,n consume inputs Xj=x1j,…,xmj to produce desirable outputs Yj=y1j,…,ysj and undesirable outputs Bj=b1j,…,bqj, respectively. Before constructing the modeling, a core task is to identify appropriate methods to handle undesirable outputs. It is clear that the strong and weak disposability assumption of undesirable outputs and desirable outputs are the two most common and natural methods in the literature [55–57]. A significant feature between the strong and weak disposability assumption is whether undesirable outputs can be produced without damage or subsequent cost to desirable outputs [58]. If undesirable outputs can be freely generated without damage or subsequent cost, implying that both inputs and outputs can change unilaterally without compromising each other, then undesirable outputs are assumed to be strongly disposable. In contrast, if the production of undesirable outputs indeed has some damage or subsequent cost to inputs or desirable outputs, implying that a reduction in undesirable outputs would result in a reduction of desirable outputs simultaneously [59, 60], then undesirable outputs are assumed to be weak disposable. Here, we consider the weak disposability assumption because it is more suitable for the real world and, more specifically, for the empirical application of banking operations, where it is very difficult to freely reduce bad debts without affecting incomes and changing banking operations. To this end, the production possibility set (PPS) under the variable returns to scale (VRS) assumption can be formulated as follows:(1)PPS=xi,yr,bp∑j=1nλjxij≤xi,i=1,...,mρj∑j=1nλjyrj≥yr,r=1,...,sρj∑j=1nλjbpj=bp,p=1,...,q∑j=1nλj=10≤ρj≤1,λj≥0,j=1,...,n.The above formula is nonlinear since both the reduction factorρj and intensity variable λj are unknown. Furthermore, we can equivalently change formula (1) into a linear version, which is presented in the following formula:(2)PPS=xi,yr,bp∑j=1nλj+ηjxij≤xi,i=1,...,m∑j=1nλjyrj≥yr,r=1,...,s∑j=1nλjbpj=bp,p=1,...,q∑j=1nλj+ηj=1,λj,ηj≥0,j=1,...,n.Based on formula (2), both desirable and undesirable outputs are weighted by nondisposed intensity variables λj, whereas the inputs are weighted by the sum of nondisposed intensity variables λj and disposed intensity variables ηj. In addition, the VRS assumption is ensured by summing the total nondisposed intensity variables λj and disposed intensity variables ηj to 1, i.e., ∑j=1nλj+ηj=1.Based on the above PPS in formula (2), we can build mathematical models to compute the relative performance of DMUs. In this paper, we follow the practice of Chen and Delmas [58] to develop an extended eco-inefficiency model for performance evaluation since the eco-inefficiency model has some advantages in modeling activities with undesirable outputs compared with four well-established models in the literature (undesirable outputs as inputs, transformation of undesirable outputs, directional distance function, and hyperbolic efficiency model. Readers can refer Chen and Delmas [58] for details on the comparison). The extended eco-inefficiency model is formulated as follows:(3)inEd∗=Max∑i=1mgix/xid+∑r=1sgry/yrd+∑p=1qgpb/bpdm+s+qs.t.∑j=1nλj+ηjxij≤xid−gim,i=1,…,m∑j=1nλjyrj≥yrd+gry,r=1,…,s∑j=1nλjbpj=bpd−gpb,p=1,…,q∑j=1nλj+ηj=1λj,ηj,gim,gry,gpb≥0,j=1,…,n;r=1,…,s;p=1,…,q.In model (3), the variables gix, gry, and gpb represent the amount of potential improvements in inputs, desirable outputs, and undesirable outputs, respectively, that the evaluated DMU can make relative to its current input usage and output production to reach its ideal benchmark target on the efficiency frontier. The potential improvements reflect input reduction potentials and desirable output expansion potentials (or undesirable output reduction potentials) instead of actual input usage and output production [61, 62]. Model (3) uses a slack-based formula that is similar to the directional distance function (DDF) model to maximize the average additive inefficiency index across all input and output measures, where the inefficiency index represents potential improvements divided by observed inputs or outputs. More specifically, the optimal direction vector of model (3) can be endogenously obtained in a similar way as that of Arabi et al. [63]. For that purpose, suppose gix=β⋅g¯i, gry=β⋅g˜r, and gpb=β⋅g^p, where the direction vector is g=g¯i,g˜r,g^p. Solving model (3) determines an optimal solution gix∗,gry∗,gpb∗, then we have β=gix∗/g¯i=gry∗/g˜r=gpb∗/g^pi=1,…,m;r=1,…,s;p=1,…,q. Hence, we have a system of m+s+q+1 unknown variables (β,g¯i,g˜r,g^p) and m+s+q linearly independent equations g1x∗g¯2=g2x∗g¯1,…,gmx∗g˜1=g1y∗g¯m, g1y∗g˜2=g2y∗g˜1,…,gsy∗g^1=g1b∗g˜s, g1b∗g^2=g2b∗g^1,...,gq−1b∗g^q=gqb∗g^q−1, and gqb∗=β⋅gq. Together with another equation such as ∑i=1mg¯i+∑r=1sg˜r+∑p=1qg^p=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 independent equations. Therefore, this system has a unique solution and we can obtain a unique optimal direction.Model (3) is slightly different from that of Chen and Delmas [58] in several aspects: first, we also take the input improvement into account, while Chen and Delmas [58] studied only the output improvement; secondly, we assume the weak disposability assumption of desirable outputs and undesirable outputs, while the strong disposability assumption was modeled in Chen and Delmas [58]. In addition, the VRS assumption is considered in model (3) such that it is more suitable for real applications. Since the individual inefficiency index theoretically has a value ranging from zero to unity for inputs and undesirable outputs and a value from zero to infinity for desirable outputs, the overall average inefficiency of DMUd also takes a value from zero to infinity. The larger the average inefficiency index is, the more inefficient the evaluated DMU is. An average inefficiency index of zero value means that the considered DMU is on the efficiency frontier and has no slack for improvement, and hence, the DMU is efficient.Furthermore, model (4) is a dual formulation of the above model (3) in its multiplier formulation:(4)inEd∗=Min∑i=1mvixid−∑r=1suryrd+∑p=1qwpbpd+u0s.t.∑i=1mvixij−∑r=1suryrj+∑p=1qwpbpj+u0≥0,j=1,…,n∑i=1mvixij+u0≥0,j=1,…,nvi≥1/m+s+qxid,i=1,…,mur≥1/m+s+qyrd,r=1,…,swp≥1/m+s+qbpd,p=1,…,qvi,ur≥0,i=1,…,m;r=1,…,s;wp,u0are free,p=1,…,q.Model (4) computes the inefficient component (i.e., the difference between aggregated inputs and aggregated outputs) for the evaluated DMU, yet the inefficient component is nonnegative for all DMUs, and some constraints on multipliers are held. Solving model (4) for each DMUdd=1,…,n determines a series of inefficiency scores inEd∗ using a series of optimal solutions urd∗,vid∗,wpd∗,u0d∗. The inefficiency score can be used for a performance indicator among all DMUs, and the smaller the inefficiency score is, the better the performance of DMUd is.


## 2.2. Dominance Probability Based on Inefficiency Scores

It is notable that the inefficiency scoreinEd∗ obtained previously can be used to analyze the performance of Chinese listed banks, but there are two main concerns. On the one hand, it is calculated by only considering the optimal weight plan urd∗,vid∗,wpd∗,u0d∗ while ignoring any other feasible weights, and thus, the obtained inefficiency score inEd∗ might be too extreme or even unrealistic. On the other hand, the inefficiency score inEd∗ is calculated separately for each DMUdd=1,…,n, and different weights will be preferred by different DMUs; thus, the results are not completely comparable.Since the resulting inefficiency indexes can change relative to different sets of weights, it is important to explore the performance associated with all sets of feasible input/output weights. To this end, we focus on the efficiency dominance concept of Salo and Punkka [21]. As well defined and discussed in Salo and Punkka [21] and Tang et al. [51], the dominance relation is determined through a pairwise efficiency comparison among DMUs. Furthermore, a certain DMU dominates another DMU if and only if its efficiency score is as large as that of the other for all sets of feasible input/output weights and is larger for at least some sets of feasible input/output weights. Here, we follow the same idea of Salo and Punkka [21] and Tang et al. [51] to determine the dominance relation of Chinese listed banks. For comparison, we take inevitable undesirable outputs in banking operations, such as bad debts, into account. Furthermore, we follow Chen and Delmas [58] in focusing on eco-inefficiency scores rather than efficiency scores through a nonradial directional distance function model. To this end, we first build the inefficiency dominance concept as follows.Definition 1.
DMU
d dominates DMUk (denoted as DMUd≻DMUk) if and only if DMUd always has a smaller inefficiency score relative to DMUk for all sets of feasible input and output weights.
The dominance relation based on inefficiency scores is determined if the inefficiency score for a certainDMUd is as small as that of DMUk for all sets of feasible input/output weights and is smaller for at least some sets of feasible input/output weights. The dominance relation between DMUd and DMUk is determined through their inefficiency comparison. By rethinking the idea of model (3) and model (4), which calculate the maximal inefficiency score, we can formulate model (5) to calculate the inefficiency range of DMUd when DMUk is fixed with a prespecified inefficiency level of zero by requiring the constraint that inEk=∑i=1mvixik−∑r=1suryrk+∑p=1qwpbpk+u0=0. In fact, the inefficiency level of DMUk can be set to any nonnegative value M, and by substituting u0 with u0−M, we can get the same dominance probability as given in Definition 2 and Definition 3; hence, we immediately set the inefficiency level of DMUk to zero for simplification in the following model:(5)inEd˜kmaxinEd˜kmin=MinMax∑i=1mvixid−∑r=1suryrd+∑p=1qwpbpd+u0s.t∑i=1mvi+∑r=1sur+∑p=1qwp=1∑i=1mvixik−∑r=1suryrk+∑p=1qwpbpk+u0=0∑i=1mvixij+u0≥0,j=d,kvi≥1m+s+qxid,i=1,…,mur≥1m+s+qyrd,r=1,…,swp≥1m+s+qbpd,p=1,…,qvi,ur≥0,i=1,…,m;r=1,…,s;wp,u0are free,p=1,…,q.
An additional constraint that∑i=1mvi+∑r=1sur+∑p=1qwp=1 is inserted into model (5) to make the feasible weight space closed and bounded. The optimal objective function for model (5) shows the lower and upper bounds on how much different DMUd’s inefficiency score can be relative to DMUk across all sets of feasible input/output weights. Using model (5), the dominance structure can be determined for any pairwise DMUs. More specifically, if the inefficiency level of DMUk is fixed to zero and if inEd˜kmax<0, which means that DMUd will always have a smaller inefficiency score compared with DMUk, then DMUd dominates DMUk. In contrast, if inEd˜kmin>0, which means that DMUd will always have a larger inefficiency score than DMUk (which has an inefficiency level of zero), then DMUd is dominated by DMUk. For more general cases, however, we cannot obtain the complete dominance relation, which is usually true in practice. Therefore, we propose determining the performance dominance probability. To this end, we follow the work of Tang et al. [51] in building the inefficiency dominance probability concept, as given in Definition 2.Definition 2.
WhenDMUk has an inefficiency score of inEk, the probability that DMUd dominates DMUk over all sets of feasible input and output weights is calculated by Pd˜≻k=inEk−inEd˜kmin/inEd˜kmax−inEd˜kmin.
It is clear that, ifinEk≤inEd˜kmin, which means that DMUd will always have a larger inefficiency by remaining the inefficiency inEk for DMUk, the inefficiency dominance probability of DMUd relative to DMUk would take a nonpositive value. That is, it is impossible for DMUd to dominate DMUk. For the sake of a bounded range, we assume that Pd˜≻k=0 if inEk≤inEd˜kmin. For a more general case in which inEd˜kmin≤inEk≤inEd˜kmax, the probability that DMUd dominates DMUk takes a value from zero to unity. A larger value of Pd˜≻k means that it is more likely for DMUd to have a smaller inefficiency score compared with DMUk over all sets of feasible input/output weights.
Note in addition that Definition2 fixes the inefficiency level of DMUk to calculate the inefficiency range of DMUd and further computes the inefficiency dominance probability of DMUd relative to DMUk. In contrast, we can also compute the inefficiency dominance probability of DMUd relative to DMUk by fixing the inefficiency level of DMUd and calculating the inefficiency range of DMUk. The above idea is formulated in model (6), which is very similar to model (5) but substitutes DMUk for DMUd:(6)inEk˜dmaxinEk˜dmin=MinMax∑i=1mvixik−∑r=1suryrk+∑p=1qwpbpk+u0s.t.∑i=1mvi+∑r=1sur+∑p=1qwp=1∑i=1mvixid−∑r=1suryrd+∑p=1qwpbpd+u0=0∑i=1mvixij+u0≥0,j=d,kvi≥1m+s+qxik,i=1,…,mur≥1m+s+qyrk,r=1,…,swp≥1m+s+qbpk,p=1,…,qvi,ur≥0,i=1,…,m;r=1,…,s;wp,u0are free,p=1,…,q.
An alternative inefficiency dominance probability ofDMUd relative to DMUk by fixing the inefficiency level of DMUd is given in Definition 3.Definition 3.
WhenDMUd has an inefficiency score of inEd, the probability that DMUd dominates DMUk over all sets of feasible input and output weights is calculated by Pd≻k˜=inEk˜dmax−inEd/inEk˜dmax−inEk˜dmin.
Furthermore, the overall probability that the inefficiency score ofDMUd dominates that of DMUk is the average of the two probabilities by fixing the inefficiency level of DMUd and DMUk. More specifically, Definition 4 gives the pairwise performance dominance probability with regard to inefficiency scores.Definition 4.
The pairwise performance dominance probability ofDMUd relative to DMUk over all sets of feasible input and output weights is Pd≻k=Pd˜≻k+Pd≻k˜/2.
The classic DEA methods allow each DMU to generate a set of relative weights to maximize its ratio of aggregated weighted outputs to aggregated weighted inputs while ensuring that the same ratio is no more than one for all DMUs, and the maximum ratio is considered the efficiency score for the evaluated DMU. By taking all sets of feasible weights and the inefficiency dominance structure into account, the overall inefficiency dominance probability for a certainDMUd among all DMUs can be calculated by the average of pair inefficiency dominance probabilities across all DMUs. The above idea is given in Definition 5.Definition 5.
The performance dominance probability ofDMUd across all DMUs over all sets of feasible input and output weights is Pd=∑k=1nPd≻k/n.
The classic DEA approaches use deterministic (in)efficiency scores to measure the relative performance, while the performance dominance probability is an alternative performance indicator from a data analytics perspective that considers all sets of feasible weights that are stochastic to determine the relative performance. It is rather remarkable that different sets of weights will cause different performance measures, and the performance dominance probability involves all possibilities over all sets of feasible input and output weights. The performance dominance probability ofDMUd calculates the probability of its performance dominating the set of all DMUs. A larger performance dominance probability indicates that it is much easier for that DMU to dominate other DMUs, implying that its inefficiency score is more likely to be smaller than that of other DMUs.


## 3. Illustrative Application of Chinese Listed Banks

In this section, we illustrate the proposed approach using empirical performance analytics for 32 Chinese listed banks. Since the proposed approach considers all sets of feasible weights, the performance relations and ratio index results are more comprehensive and reasonable.


### 3.1. Data Description

This section addresses the performance of listed banks in China. For simplification and research purposes, we consider only those banks that have been registered in China and that are listed in the mainland of China. More specifically, only banks that are owned by Chinese organizations and are listed on the Shenzhen Stock Exchange and Shanghai Stock Exchange are collected. In contrast, neither Chinese banks listed on other stock exchanges nor foreign banks listed on the Shenzhen Stock Exchange and Shanghai Stock Exchange are involved in this study. As a result, we have 32 listed banks. For the research purpose, we give these 32 banks and their corresponding codes in Table1.Table 1
Codes for 32 Chinese listed banks.
DMUs
Banks
Abbreviation
DMU1
Bank of Beijing
BOB
DMU2
Changshu Rural Commercial Bank
CRCB
DMU3
Bank of Chengdu
BCD
DMU4
Industrial and Commercial Bank of China
ICBC
DMU5
China Everbright Bank
CEB
DMU6
Bank of Guiyang
BOG
DMU7
Bank of Hangzhou
BOH
DMU8
Huaxia Bank
HB
DMU9
China Construction Bank
CCB
DMU10
Bank of Jiangsu
BOJ
DMU11
Jiangyin Rural Commercial Bank
JRCB
DMU12
Bank of Communications
BC
DMU13
China Minsheng Bank
CMSB
DMU14
Bank of Nanjing
BON
DMU15
Bank of Ningbo
BN
DMU16
Agricultural Bank of China
ABC
DMU17
Ping An Bank
PB
DMU18
Shanghai Pudong Development Bank
SPDB
DMU19
Bank of Qingdao
BOQ
DMU20
Qingdao Rural Commercial Bank
QRCB
DMU21
Bank of Shanghai
BOS
DMU22
Suzhou Rural Commercial Bank
SRCB
DMU23
Wuxi Rural Commercial Bank
WRCB
DMU24
Bank of Xian
BOX
DMU25
Industrial Bank
IB
DMU26
Rural Commercial Bank of Zhangjiagang
RCBZ
DMU27
Bank of Changsha
BCS
DMU28
China Merchants Bank
CMB
DMU29
Bank of Zhengzhou
BOZ
DMU30
Bank of China
BOC
DMU31
China CITIC Bank
CCB
DMU32
Zijin Rural Commercial Bank
ZRCBIn practice, each bank will consume multiple inputs to generate multiple outputs and more specifically mainly for profits. In this study, we follow similar studies such as Wang et al. [4], Zha et al. [1], Fukuyama and Matousek [64], Li et al. [65], and Zhu et al. [2] in taking employment referring to human resource investment and manpower, fixed asset referring to the asset value of physical capital that can be used for business activities, and operation cost as three inputs. Note in addition that the operation cost in this study excludes the expense of labor input that occurs in banking operations because the employment has already taken the labor input into account. Furthermore, we consider three different outputs generated in banking operations, with interest income and noninterest income being two desirable outputs and nonperforming bad loan percentage in the current year as an undesirable output. The interest income is derived directly from the gap between the interest paid on deposits and the interest earned from loans, while the noninterest income is primarily derived from commissions, securities investments, fees, and other business activity incomes. However, a bad loan is a jointly produced and unavoidable by-product that will greatly harm the bank. The inputs and outputs to be used in this study are summarized in Table 2.Table 2
Input and output variables.
Input/output
Variable
Notation
Unit
Inputs
Employment
x
1
Person count
Fixed assets
x
2
Million yuan
Operation costs
x
3
Million yuan
Desirable outputs
Interest income
y
1
Million yuan
Noninterest income
y
2
Million yuan
Undesirable outputs
Bad debt percentage
z
1
PercentageOur empirical study contains operation data for 32 Chinese listed banks over the 2014–2018 period, accounting for 160 observations. All data for these listed banks were collected from official sources of bank annual reports and the financial reports of banks in China Stock Market Accounting Research (CSMAR) during 2014–2018. Table3 shows the descriptive statistics of the inputs, desirable outputs, and undesirable outputs of these 160 observations. It can be found that both the average fixed asset and operation cost among the 32 listed banks are increasing year by year, but the average employment and bad debt percentage increased to a peak in 2016 and then decreased continuously. Furthermore, both interest income and noninterest income show an increasing trend but decreased in a year.Table 3
Descriptive statistics for input-output measures.
Year
Statistics
x
1
x
2
x
3
y
1
y
2
z
1
2014
Max
493583
196238
271132
849879
165370
2.7100
Min
1089
412.01
1151.41
3533.90
109.48
0.7500
Mean
67582
28499.15
48231.62
154112.69
25685.99
1.2531
SD
135316
55898.04
78151.35
235785.69
44163.22
0.4237
2015
Max
503082
221502
306395
871779
189780
2.3900
Min
1180
418.04
1245.26
3486.11
130.69
0.8300
Mean
69772
31335.97
57387.23
162109.41
30344.05
1.5308
SD
136444
60262.26
87459.17
243294.78
49419.92
0.4226
2016
Max
496698
243619
282712
791480
204045
2.4100
Min
1352
413.99
1235.88
3268.89
162.62
0.8700
Mean
70266
34778.86
58285.96
151790.21
35560.52
1.6013
SD
134489
65118.10
85168.72
220693.57
57189.80
0.3678
2017
Max
487307
245687
331518
861594
204424
2.3900
Min
1426
462.93
1260.89
3789.14
166.05
0.8200
Mean
69782
36383.26
60079.96
166927.92
35168.24
1.5431
SD
132071
66621.76
91681.04
239463.71
53045.31
0.3240
2018
Max
473691
253525
368966
948094
201271
2.4700
Min
1454
418.08
1733.55
4567.00
203.17
0.7800
Mean
69354
38380.34
67444.02
183050.26
39183.08
1.5034
SD
129697
70061.74
100964.72
261751.20
55000.40
0.3382


### 3.2. Result Analysis

To provide data analytics of bank performance, we first calculate the inefficiency scores using model (3) or model (4) in the nonradial DDF-based formulation under the VRS property. The inefficiencies of these 32 banks from 2014 to 2018 are given in Table 4. Since the inefficiency score represents potential improvements divided by the observed inputs or outputs, an inefficiency score of zero indicates that there will be no improvement potential. Table 4 shows that, by selecting the optimal set of weights to aggregate inputs and outputs, many listed banks will be extremely efficient without improvement potentials. There are always ten banks that have an inefficiency score larger than zero, but the average inefficiency score across these 32 listed banks fluctuates according to the year. Since the VRS analysis is adopted in this study, we may not draw any significant conclusion as to inefficiency changes year by year. Furthermore, some banks (DMU6, Bank of Guiyang; DMU8, Huaxia Bank; DMU15, Bank of Ningbo; and DMU32, Zijin Rural Commercial Bank) will always have a nonzero inefficiency, meaning that these banks always show very terrible performance compared with banks that have a zero-value inefficiency index.Table 4
Inefficiency scores of 32 listed banks from 2014 to 2018.
Banks
2014
2015
2016
2017
2018
DMU1
0.0000
0.0000
0.0000
0.0000
0.0000
DMU2
0.0000
0.3483
0.6442
0.5430
0.3483
DMU3
0.0000
0.0979
0.2730
0.0000
0.0979
DMU4
0.0000
0.0000
0.0000
0.0000
0.0000
DMU5
0.0000
0.0590
0.0000
0.1087
0.0590
DMU6
0.2345
0.3469
0.3273
0.5898
0.3469
DMU7
0.1798
0.0000
0.0000
0.0000
0.0000
DMU8
0.2880
0.3059
0.2288
0.2768
0.3059
DMU9
0.0000
0.0000
0.0000
0.0000
0.0000
DMU10
0.0000
0.0000
0.0000
0.0000
0.0000
DMU11
0.0000
0.0000
0.0000
0.0000
0.0000
DMU12
0.1449
0.0000
0.0000
0.0000
0.0000
DMU13
0.0000
0.0000
0.0000
0.0000
0.0000
DMU14
0.2088
0.1072
0.1415
0.0000
0.1072
DMU15
0.2793
0.2415
0.2018
0.2225
0.2415
DMU16
0.1926
0.0000
0.0000
0.0000
0.0000
DMU17
0.0000
0.0000
0.0000
0.0000
0.0000
DMU18
0.0567
0.0000
0.0000
0.0000
0.0000
DMU19
0.0000
0.0000
0.0827
0.0000
0.0000
DMU20
0.0000
0.0000
0.0000
0.0000
0.0000
DMU21
0.0000
0.0000
0.0000
0.0000
0.0000
DMU22
0.0000
0.0000
0.0000
0.2032
0.0000
DMU23
0.0000
0.0000
0.0000
0.0000
0.0000
DMU24
0.0000
0.0000
0.0000
0.0000
0.0000
DMU25
0.0000
0.0000
0.0000
0.0000
0.0000
DMU26
0.0000
0.0000
0.0000
0.0000
0.0000
DMU27
0.0000
0.2287
0.2670
0.6379
0.2287
DMU28
0.0000
0.0000
0.0000
0.0000
0.0000
DMU29
0.0000
0.0000
0.1373
0.0967
0.0000
DMU30
0.0000
0.0000
0.0000
0.0000
0.0000
DMU31
0.0634
0.0799
0.0000
0.0000
0.0799
DMU32
0.5982
0.2792
0.3526
0.5700
0.2792
Mean
0.0702
0.0655
0.0830
0.1015
0.0655Since the previous inefficiency scores are separately derived by considering only one optimal set of weights, the resulting performance information might be unrealistic and unreasonable. Therefore, we can use the proposed eco-inefficiency dominance probability approach in this paper to analyze the banking performance considering all sets of feasible weights. To this end, we first use model (5) to calculate the inefficiency score intervals and then use Definition 2 to calculate the first-level pairwise dominating probability in 2018. Since the pairwise dominating probability involves an n×n matrix that is hard to present in this paper, we arbitrarily take the Bank of Jiangsu (BOJ), Agricultural Bank of China (ABC), and Rural Commercial Bank of Zhangjiagang (RCBZ) for instance, and the first-level inefficiency dominating probability for these three banks across all 32 banks is listed in the second, third, and fourth columns of Table 5, namely, PBOJ∼≻k, PABC∼≻k, and PRCBZ∼≻k (it is also possible to consider any other banks as examples to show the calculation results). The results represent the probability of the considered bank having a smaller inefficiency index across any other bank, with different sets of weights being attached to inputs and outputs to ensure an efficient status for other banks. For example, the value of 0.2699 implies that, by fixing the inefficiency score of the Bank of Beijing (DMU1) to zero, the inefficiency score of the Bank of Jiangsu will be smaller with a probability of 0.2699 and larger with a probability of 0.7301 (1–0.2699). At the same time, we can use model (6) and Definition 3 to calculate another dominating probability, and the second-level results of Bank of Jiangsu (BOJ), Agricultural Bank of China (ABC), and Rural Commercial Bank of Zhangjiagang (RCBZ) in 2018 are given in the last three columns of Table 5.Table 5
Two kinds of dominating probabilities for BOJ, ABC, and RCBZ in 2018.
Banks
Definition2
Definition3
BOJ
ABC
RCBZ
BOJ
ABC
RCBZ
DMU1
0.2699
0.5964
0.0093
0.3515
0.9452
0.2111
DMU2
0.8578
0.6238
0.2563
0.9850
0.9959
0.4689
DMU3
0.8520
0.6207
0.0496
0.9662
0.9910
0.1966
DMU4
0.0288
0.3164
0.0016
0.3339
0.3822
0.3216
DMU5
0.1671
0.5897
0.0074
0.3446
0.9119
0.2663
DMU6
0.8592
0.6197
0.0386
0.9599
0.9887
0.1905
DMU7
0.8518
0.6159
0.0253
0.9337
0.9821
0.1961
DMU8
0.3056
0.6073
0.0106
0.4337
0.9419
0.2603
DMU9
0.0315
0.1127
0.0017
0.3121
0.1311
0.2989
DMU
10
1.0000
0.6052
0.0088
1.0000
0.9589
0.1523
DMU11
0.8514
0.6230
0.3883
0.9915
0.9977
0.4059
DMU12
0.0835
0.5315
0.0042
0.3563
0.7917
0.3073
DMU13
0.1096
0.5695
0.0054
0.3154
0.8709
0.2684
DMU14
0.8678
0.6123
0.0146
0.9135
0.9736
0.1657
DMU15
0.9666
0.6173
0.0286
0.9704
0.9791
0.2599
DMU
16
0.0411
1.0000
0.0023
0.3948
1.0000
0.3768
DMU17
0.2062
0.5856
0.0089
0.4109
0.9140
0.3081
DMU18
0.0921
0.5531
0.0046
0.3014
0.8476
0.2634
DMU19
0.8593
0.6219
0.0995
0.9780
0.9943
0.2510
DMU20
0.8454
0.6226
0.1170
0.9772
0.9942
0.2986
DMU21
0.5626
0.6048
0.0133
0.5695
0.9618
0.2139
DMU22
0.8492
0.6230
0.2009
0.9916
0.9978
0.2256
DMU23
0.8463
0.6225
0.1454
0.9886
0.9971
0.1876
DMU24
0.8473
0.6222
0.0825
0.9808
0.9951
0.1796
DMU25
0.0724
0.5560
0.0038
0.2395
0.8469
0.2161
DMU
26
0.8477
0.6232
1.0000
0.9912
0.9977
1.0000
DMU27
0.8790
0.6204
0.0512
0.9669
0.9892
0.2431
DMU28
0.1159
0.5628
0.0058
0.3840
0.8491
0.3310
DMU29
0.8588
0.6202
0.0662
0.9644
0.9908
0.2669
DMU30
0.0389
0.3718
0.0021
0.3274
0.4493
0.3107
DMU31
0.1246
0.5690
0.0060
0.3564
0.8716
0.2997
DMU32
0.8494
0.6222
0.1199
0.9848
0.9960
0.2093Without loss of generality, each DMU will always dominate itself, as these three banks will have a dominating probability of 1 to itself regardless of whether the first-level dominance probability or the second-level dominance probability is considered. From Table5, we can find that all three banks have a larger second-level dominance probability than the first-level dominance probability to other banks, and it is indeed also held for all banks. This difference is due to the definition style of pairwise dominances, and we cannot arbitrarily use one dominating probability to represent the performance assessment with the other being ignored. By averaging the two pairwise dominating probabilities, we can calculate the pairwise dominance probability as well as the dominated probability. Reconsidering the three banks in Table 5, we show the pairwise dominance probability as well as dominated probability for Bank of Jiangsu, Agricultural Bank of China, and Rural Commercial Bank of Zhangjiagang in Table 6. Taking the inefficiency dominance probability of Bank of Jiangsu to Bank of Beijing (DMU1), for example, the arithmetic mean of 0.2699 and 0.3515 by Definition 4 is exactly the pairwise inefficiency dominance probability of Bank of Jiangsu to Bank of Beijing, 0.3107. Since the pairwise dominance probability shows the probability of the inefficiency score of Bank of Jiangsu relative to any other banks, by considering all sets of feasible input and output weights, it means that the inefficiency score of Bank of Jiangsu is smaller than that of Bank of Beijing with a probability of 0.3107. In contrast, the inefficiency of Bank of Jiangsu will be dominated by Bank of Beijing with a probability of 0.6893.Table 6
Pairwise dominance probability for BOJ, ABC, and RCBZ in 2018.
Banks
Dominating
Dominated
BOJ
ABC
RCBZ
BOJ
ABC
RCBZ
DMU1
0.3107
0.7708
0.1102
0.6893
0.2292
0.8898
DMU2
0.9214
0.8098
0.3626
0.0786
0.1902
0.6374
DMU3
0.9091
0.8058
0.1231
0.0909
0.1942
0.8769
DMU4
0.1813
0.3493
0.1616
0.8187
0.6507
0.8384
DMU5
0.2559
0.7508
0.1369
0.7441
0.2492
0.8631
DMU6
0.9096
0.8042
0.1145
0.0904
0.1958
0.8855
DMU7
0.8928
0.7990
0.1107
0.1072
0.2010
0.8893
DMU8
0.3696
0.7746
0.1354
0.6304
0.2254
0.8646
DMU9
0.1718
0.1219
0.1503
0.8282
0.8781
0.8497
DMU
10
1.0000
0.7820
0.0805
1.0000
0.2180
0.9195
DMU11
0.9214
0.8104
0.3971
0.0786
0.1896
0.6029
DMU12
0.2199
0.6616
0.1557
0.7801
0.3384
0.8443
DMU13
0.2125
0.7202
0.1369
0.7875
0.2798
0.8631
DMU14
0.8906
0.7929
0.0902
0.1094
0.2071
0.9098
DMU15
0.9685
0.7982
0.1442
0.0315
0.2018
0.8558
DMU
16
0.2180
1.0000
0.1895
0.7820
1.0000
0.8105
DMU17
0.3086
0.7498
0.1585
0.6914
0.2502
0.8415
DMU18
0.1968
0.7004
0.1340
0.8032
0.2996
0.8660
DMU19
0.9187
0.8081
0.1752
0.0813
0.1919
0.8248
DMU20
0.9113
0.8084
0.2078
0.0887
0.1916
0.7922
DMU21
0.5661
0.7833
0.1136
0.4339
0.2167
0.8864
DMU22
0.9204
0.8104
0.2132
0.0796
0.1896
0.7868
DMU23
0.9174
0.8098
0.1665
0.0826
0.1902
0.8335
DMU24
0.9141
0.8086
0.1311
0.0859
0.1914
0.8689
DMU25
0.1559
0.7014
0.1099
0.8441
0.2986
0.8901
DMU
26
0.9195
0.8105
1.0000
0.0805
0.1895
1.0000
DMU27
0.9229
0.8048
0.1472
0.0771
0.1952
0.8528
DMU28
0.2499
0.7060
0.1684
0.7501
0.2940
0.8316
DMU29
0.9116
0.8055
0.1666
0.0884
0.1945
0.8334
DMU30
0.1832
0.4106
0.1564
0.8168
0.5894
0.8436
DMU31
0.2405
0.7203
0.1529
0.7595
0.2797
0.8471
DMU32
0.9171
0.8091
0.1646
0.0829
0.1909
0.8354
Mean
0.6096
0.7375
0.1864
0.4217
0.2938
0.8448Furthermore, by aggregating the pairwise inefficiency dominance probability across all banks, we can obtain the average performance dominance probability in terms of inefficiency scores for these three banks, as given in the last column of Table6. Furthermore, proceeding in the same manner as above, we can determine the inefficiency dominance probability for all 32 Chinese listed banks from 2014 to 2018, as shown in Table 7.Table 7
Inefficiency dominance probability of 32 listed banks from 2014 to 2018.
Banks
2014
2015
2016
2017
2018
Mean
DMU1
0.6175
0.6149
0.6142
0.6173
0.6235
0.6175
DMU2
0.2872
0.2633
0.2243
0.2184
0.2293
0.2445
DMU3
0.4257
0.4131
0.3831
0.3685
0.3971
0.3975
DMU4
0.7964
0.7936
0.7786
0.7920
0.8056
0.7932
DMU5
0.6770
0.6678
0.6790
0.6815
0.6411
0.6693
DMU6
0.3007
0.3382
0.3529
0.3928
0.4268
0.3623
DMU7
0.4582
0.4853
0.4794
0.4922
0.4908
0.4812
DMU8
0.6105
0.5885
0.5749
0.5640
0.5857
0.5847
DMU9
0.8039
0.8091
0.7945
0.8038
0.8132
0.8049
DMU
10
0.5439
0.5812
0.5893
0.6105
0.6096
0.5869
DMU11
0.2742
0.2571
0.2469
0.2246
0.2099
0.2425
DMU12
0.7623
0.7480
0.7216
0.7106
0.7032
0.7291
DMU13
0.7042
0.6892
0.6916
0.7231
0.6982
0.7013
DMU14
0.5130
0.5143
0.5269
0.5392
0.5413
0.5269
DMU15
0.4937
0.4849
0.4806
0.4678
0.4710
0.4796
DMU
16
0.7135
0.7209
0.7130
0.7232
0.7375
0.7216
DMU17
0.6272
0.6105
0.5922
0.6169
0.6155
0.6125
DMU18
0.7315
0.7255
0.7244
0.7324
0.7331
0.7294
DMU19
0.3614
0.3654
0.3772
0.3510
0.3212
0.3552
DMU20
0.2796
0.2947
0.3033
0.2929
0.3048
0.2951
DMU21
0.5859
0.5885
0.6007
0.5769
0.5712
0.5846
DMU22
0.1967
0.1937
0.1941
0.1786
0.2062
0.1939
DMU23
0.2875
0.3021
0.2897
0.2705
0.2559
0.2812
DMU24
0.2823
0.2992
0.3333
0.3240
0.3304
0.3138
DMU25
0.7851
0.7710
0.7642
0.7895
0.7747
0.7769
DMU
26
0.2405
0.2264
0.2289
0.2039
0.1864
0.2172
DMU27
0.3743
0.3810
0.4116
0.3980
0.4038
0.3938
DMU28
0.7060
0.6765
0.6573
0.6634
0.6801
0.6766
DMU29
0.4041
0.4260
0.4448
0.4405
0.3855
0.4202
DMU30
0.7721
0.7665
0.7577
0.7673
0.7795
0.7686
DMU31
0.7224
0.6996
0.7011
0.6738
0.6721
0.6938
DMU32
0.1613
0.2040
0.2686
0.2909
0.2956
0.2441As Table7 shows, these banks have very different inefficiency dominance probabilities over all sets of feasible weights relative to the inefficiency scores derived from only an optimal set of weights. For example, Huaxia Bank (DMU8) always has a positive inefficiency in each year, and that bank will be ranked after most banks. However, Huaxia Bank will have an inefficiency dominance probability of 0.6105, 0.5885, 0.5749, 0.5640, and 0.5857 in the 2014–2018 period, respectively. This phenomenon implies that Huaxia Bank is more likely to have an inefficiency index smaller than that of half of all banks. In contrast, by considering only the optimal set of weights, the Rural Commercial Bank of Zhangjiagang (DMU26) is extremely efficient with an inefficiency score of zero for all years, but it has a relatively small inefficiency dominance probability in each year (0.2405, 0.2264, 0.2289, 0.2039, and 0.1864, respectively), implying that the performance of Rural Commercial Bank of Zhangjiagang in the sense of inefficiency scores is at a disadvantage to other banks by addressing all weight possibilities.Furthermore, it can be found that all banks have a relatively stable inefficiency dominance probability in the period of 2014–2018, with the largest variation being 0.1343 (0.2956–0.1613) for Zijin Rural Commercial Bank (DMU32) and the smallest variation being 0.0087 (0.7331–0.7244) for Shanghai Pudong Development Bank (DMU18). All banks always have an eco-inefficiency dominance probability that is either larger than 0.50 or less than 0.50 in the five-year sample (0.5 is a threshold where the dominating probability is equal to the dominated probability), implying that all banks can be divided into two groups, one for superior banks and another for inferior banks. The categories are shown in Table 8. From the average sense, we find that China Construction Bank (DMU9), Industrial and Commercial Bank of China (DMU4), Industrial Bank (DMU25), Bank of China (DMU32), and Shanghai Pudong Development Bank (DMU18) are the top five listed banks for operation performance. In contrast, the lowest five banks are Suzhou Rural Commercial Bank (DMU22), Rural Commercial Bank of Zhangjiagang (DMU26), Jiangyin Rural Commercial Bank (DMU11), Zijin Rural Commercial Bank (DMU32), and Changshu Rural Commercial Bank (DMU2), all of which have an average inefficiency dominance probability of less than 0.2500.Table 8
Superior and inferior listed banks.
Category
Banks
Superior (>0.5000)
Bank of Beijing, Industrial and Commercial Bank of China, China Everbright Bank, Huaxia Bank, China Construction Bank, Bank of Jiangsu, Bank of Communications, China Minsheng Bank, Bank of Nanjing, Agricultural Bank of China, Ping An Bank, Shanghai Pudong Development Bank, Bank of Shanghai, Industrial Bank, China Merchants Bank, Bank of China, China CITIC Bank
Inferior (<0.5000)
Changshu Rural Commercial Bank, Bank of Chengdu, Bank of Guiyang, Bank of Hangzhou, Jiangyin Rural Commercial Bank, Bank of Ningbo, Bank of Qingdao, Qingdao Rural Commercial Bank, Suzhou Rural Commercial Bank, Wuxi Rural Commercial Bank, Bank of Xian, Rural Commercial Bank of Zhangjiagang, Bank of Changsha, Bank of Zhengzhou, Zijin Rural Commercial BankThe eco-inefficiency and eco-inefficiency dominance probabilities of these 32 listed banks are given in Tables4 and 7, respectively. It is clear that the proposed approach will give performance indexes that are different from those of previous approaches. Furthermore, we give the ranking comparison of eco-inefficiency and eco-inefficiency dominance probabilities in Table 9. It can be found from Table 9 that, on the one hand, the proposed approach will give performance rankings that are largely different from those of previous approaches. On the other hand, the traditional DEA model cannot discriminate all banks, and more seriously, more than twenty banks are ranked as the first based on inefficiency scores, while the eco-inefficiency dominance probability approach can indeed give a full ranking of all banks. From this perspective, the proposed approach can give a more reasonable and discriminating performance assessment.Table 9
Ranking of inefficiency and inefficiency dominance probabilities.
Banks
2014
2015
2016
2017
2018
DMU1
1
13
1
12
1
12
1
12
1
12
DMU2
1
26
32
28
32
31
29
30
32
29
DMU3
1
20
25
21
29
22
1
23
25
22
DMU4
1
2
1
2
1
2
1
2
1
2
DMU5
1
11
23
11
1
10
25
9
23
11
DMU6
29
24
31
24
30
24
31
22
31
20
DMU7
26
19
1
18
1
19
1
18
1
18
DMU8
31
14
30
15
27
16
28
16
30
15
DMU9
1
1
1
1
1
1
1
1
1
1
DMU10
1
16
1
16
1
15
1
14
1
14
DMU11
1
29
1
29
1
29
1
29
1
30
DMU12
25
5
1
5
1
6
1
8
1
7
DMU13
1
10
1
9
1
9
1
7
1
8
DMU14
28
17
26
17
25
17
1
17
26
17
DMU15
30
18
28
19
26
18
27
19
28
19
DMU16
27
8
1
7
1
7
1
6
1
5
DMU17
1
12
1
13
1
14
1
13
1
13
DMU18
23
6
1
6
1
5
1
5
1
6
DMU19
1
23
1
23
23
23
1
24
1
25
DMU20
1
28
1
27
1
26
1
26
1
26
DMU21
1
15
1
14
1
13
1
15
1
16
DMU22
1
31
1
32
1
32
26
32
1
31
DMU23
1
25
1
25
1
27
1
28
1
28
DMU24
1
27
1
26
1
25
1
25
1
24
DMU25
1
3
1
3
1
3
1
3
1
4
DMU26
1
30
1
30
1
30
1
31
1
32
DMU27
1
22
27
22
28
21
32
21
27
21
DMU28
1
9
1
10
1
11
1
11
1
9
DMU29
1
21
1
20
24
20
24
20
1
23
DMU30
1
4
1
4
1
4
1
4
1
3
DMU31
24
7
24
8
1
8
1
10
24
10
DMU32
32
32
29
31
31
28
30
27
29
27All 32 listed banks can be mainly categorized into four groups according to the ownership, namely, state-owned banks, joint-stock banks, city commercial banks, and rural commercial banks. Table10 shows the divisions, and Table 11 gives the average inefficiency dominance probabilities for different kinds of banks.Table 10
Categories of different listed banks.
Category
Banks
State-owned banks (5)
Industrial and Commercial Bank of China
China Construction Bank
Bank of Communications
Agricultural Bank of China
Bank of China
Joint-stock banks (8)
China Everbright Bank, Huaxia Bank
China Minsheng Bank, Ping An Bank
Shanghai Pudong Development Bank
Industrial Bank, China Merchants Bank
China CITIC Bank
City commercial banks (12)
Bank of Beijing, Bank of Chengdu
Bank of Guiyang, Bank of Hangzhou
Bank of Jiangsu, Bank of Nanjing
Bank of Ningbo, Bank of Qingdao
Bank of Shanghai, Bank of Xian
Bank of Changsha, Bank of Zhengzhou
Rural commercial banks (7)
Changshu Rural Commercial Bank
Jiangyin Rural Commercial Bank
Qingdao Rural Commercial Bank
Suzhou Rural Commercial Bank
Wuxi Rural Commercial Bank
Rural Commercial Bank of Zhangjiagang
Zijin Rural Commercial BankTable 11
Average inefficiency dominance probabilities for four kinds of listed banks.
Banks
2014
2015
2016
2017
2018
Mean
State-owned banks
0.7696
0.7676
0.7531
0.7594
0.7678
0.7635
Joint-stock banks
0.6955
0.6786
0.6731
0.6806
0.6751
0.6806
City commercial banks
0.4467
0.4577
0.4662
0.4649
0.4644
0.4600
Rural commercial banks
0.2467
0.2488
0.2508
0.2400
0.2412
0.2455It can be learned from Table11 that the four kinds of listed banks exhibit considerably different performance dominance probabilities. More specifically, those five state-owned banks have the highest average inefficiency dominance probability, which is almost three times that of the lowest rural commercial banks. This result shows that these state-owned banks are more likely to have better performance compared with other banks over all sets of weights. In contrast, those rural commercial banks are more likely to have worse performance relative to other banks. Furthermore, joint-stock banks are inferior to state-owned banks and superior to city commercial banks, which are further superior to rural commercial banks.By using the proposed inefficiency dominance probability approach, we can provide a performance analysis of 32 Chinese listed banks that is derived from the real world. Since the proposed approach considers all sets of feasible weights, which is different from classic DEA approaches that focus on only one or several optimal sets of weights, the resulting performance analytics are more reasonable due to taking full weights and all possibilities into account. Furthermore, by considering all sets of feasible weights, the performance dominance probability is largely different from traditional performance indexes that are obtained with some extreme weights, and the corresponding ranking orders are also changed considerably. Therefore, it makes sense for the proposed approach because it can provide a comprehensive performance assessment instead of only some extreme performances from a data analytics perspective.


## 3.1. Data Description

This section addresses the performance of listed banks in China. For simplification and research purposes, we consider only those banks that have been registered in China and that are listed in the mainland of China. More specifically, only banks that are owned by Chinese organizations and are listed on the Shenzhen Stock Exchange and Shanghai Stock Exchange are collected. In contrast, neither Chinese banks listed on other stock exchanges nor foreign banks listed on the Shenzhen Stock Exchange and Shanghai Stock Exchange are involved in this study. As a result, we have 32 listed banks. For the research purpose, we give these 32 banks and their corresponding codes in Table1.Table 1
Codes for 32 Chinese listed banks.
DMUs
Banks
Abbreviation
DMU1
Bank of Beijing
BOB
DMU2
Changshu Rural Commercial Bank
CRCB
DMU3
Bank of Chengdu
BCD
DMU4
Industrial and Commercial Bank of China
ICBC
DMU5
China Everbright Bank
CEB
DMU6
Bank of Guiyang
BOG
DMU7
Bank of Hangzhou
BOH
DMU8
Huaxia Bank
HB
DMU9
China Construction Bank
CCB
DMU10
Bank of Jiangsu
BOJ
DMU11
Jiangyin Rural Commercial Bank
JRCB
DMU12
Bank of Communications
BC
DMU13
China Minsheng Bank
CMSB
DMU14
Bank of Nanjing
BON
DMU15
Bank of Ningbo
BN
DMU16
Agricultural Bank of China
ABC
DMU17
Ping An Bank
PB
DMU18
Shanghai Pudong Development Bank
SPDB
DMU19
Bank of Qingdao
BOQ
DMU20
Qingdao Rural Commercial Bank
QRCB
DMU21
Bank of Shanghai
BOS
DMU22
Suzhou Rural Commercial Bank
SRCB
DMU23
Wuxi Rural Commercial Bank
WRCB
DMU24
Bank of Xian
BOX
DMU25
Industrial Bank
IB
DMU26
Rural Commercial Bank of Zhangjiagang
RCBZ
DMU27
Bank of Changsha
BCS
DMU28
China Merchants Bank
CMB
DMU29
Bank of Zhengzhou
BOZ
DMU30
Bank of China
BOC
DMU31
China CITIC Bank
CCB
DMU32
Zijin Rural Commercial Bank
ZRCBIn practice, each bank will consume multiple inputs to generate multiple outputs and more specifically mainly for profits. In this study, we follow similar studies such as Wang et al. [4], Zha et al. [1], Fukuyama and Matousek [64], Li et al. [65], and Zhu et al. [2] in taking employment referring to human resource investment and manpower, fixed asset referring to the asset value of physical capital that can be used for business activities, and operation cost as three inputs. Note in addition that the operation cost in this study excludes the expense of labor input that occurs in banking operations because the employment has already taken the labor input into account. Furthermore, we consider three different outputs generated in banking operations, with interest income and noninterest income being two desirable outputs and nonperforming bad loan percentage in the current year as an undesirable output. The interest income is derived directly from the gap between the interest paid on deposits and the interest earned from loans, while the noninterest income is primarily derived from commissions, securities investments, fees, and other business activity incomes. However, a bad loan is a jointly produced and unavoidable by-product that will greatly harm the bank. The inputs and outputs to be used in this study are summarized in Table 2.Table 2
Input and output variables.
Input/output
Variable
Notation
Unit
Inputs
Employment
x
1
Person count
Fixed assets
x
2
Million yuan
Operation costs
x
3
Million yuan
Desirable outputs
Interest income
y
1
Million yuan
Noninterest income
y
2
Million yuan
Undesirable outputs
Bad debt percentage
z
1
PercentageOur empirical study contains operation data for 32 Chinese listed banks over the 2014–2018 period, accounting for 160 observations. All data for these listed banks were collected from official sources of bank annual reports and the financial reports of banks in China Stock Market Accounting Research (CSMAR) during 2014–2018. Table3 shows the descriptive statistics of the inputs, desirable outputs, and undesirable outputs of these 160 observations. It can be found that both the average fixed asset and operation cost among the 32 listed banks are increasing year by year, but the average employment and bad debt percentage increased to a peak in 2016 and then decreased continuously. Furthermore, both interest income and noninterest income show an increasing trend but decreased in a year.Table 3
Descriptive statistics for input-output measures.
Year
Statistics
x
1
x
2
x
3
y
1
y
2
z
1
2014
Max
493583
196238
271132
849879
165370
2.7100
Min
1089
412.01
1151.41
3533.90
109.48
0.7500
Mean
67582
28499.15
48231.62
154112.69
25685.99
1.2531
SD
135316
55898.04
78151.35
235785.69
44163.22
0.4237
2015
Max
503082
221502
306395
871779
189780
2.3900
Min
1180
418.04
1245.26
3486.11
130.69
0.8300
Mean
69772
31335.97
57387.23
162109.41
30344.05
1.5308
SD
136444
60262.26
87459.17
243294.78
49419.92
0.4226
2016
Max
496698
243619
282712
791480
204045
2.4100
Min
1352
413.99
1235.88
3268.89
162.62
0.8700
Mean
70266
34778.86
58285.96
151790.21
35560.52
1.6013
SD
134489
65118.10
85168.72
220693.57
57189.80
0.3678
2017
Max
487307
245687
331518
861594
204424
2.3900
Min
1426
462.93
1260.89
3789.14
166.05
0.8200
Mean
69782
36383.26
60079.96
166927.92
35168.24
1.5431
SD
132071
66621.76
91681.04
239463.71
53045.31
0.3240
2018
Max
473691
253525
368966
948094
201271
2.4700
Min
1454
418.08
1733.55
4567.00
203.17
0.7800
Mean
69354
38380.34
67444.02
183050.26
39183.08
1.5034
SD
129697
70061.74
100964.72
261751.20
55000.40
0.3382


## 3.2. Result Analysis

To provide data analytics of bank performance, we first calculate the inefficiency scores using model (3) or model (4) in the nonradial DDF-based formulation under the VRS property. The inefficiencies of these 32 banks from 2014 to 2018 are given in Table 4. Since the inefficiency score represents potential improvements divided by the observed inputs or outputs, an inefficiency score of zero indicates that there will be no improvement potential. Table 4 shows that, by selecting the optimal set of weights to aggregate inputs and outputs, many listed banks will be extremely efficient without improvement potentials. There are always ten banks that have an inefficiency score larger than zero, but the average inefficiency score across these 32 listed banks fluctuates according to the year. Since the VRS analysis is adopted in this study, we may not draw any significant conclusion as to inefficiency changes year by year. Furthermore, some banks (DMU6, Bank of Guiyang; DMU8, Huaxia Bank; DMU15, Bank of Ningbo; and DMU32, Zijin Rural Commercial Bank) will always have a nonzero inefficiency, meaning that these banks always show very terrible performance compared with banks that have a zero-value inefficiency index.Table 4
Inefficiency scores of 32 listed banks from 2014 to 2018.
Banks
2014
2015
2016
2017
2018
DMU1
0.0000
0.0000
0.0000
0.0000
0.0000
DMU2
0.0000
0.3483
0.6442
0.5430
0.3483
DMU3
0.0000
0.0979
0.2730
0.0000
0.0979
DMU4
0.0000
0.0000
0.0000
0.0000
0.0000
DMU5
0.0000
0.0590
0.0000
0.1087
0.0590
DMU6
0.2345
0.3469
0.3273
0.5898
0.3469
DMU7
0.1798
0.0000
0.0000
0.0000
0.0000
DMU8
0.2880
0.3059
0.2288
0.2768
0.3059
DMU9
0.0000
0.0000
0.0000
0.0000
0.0000
DMU10
0.0000
0.0000
0.0000
0.0000
0.0000
DMU11
0.0000
0.0000
0.0000
0.0000
0.0000
DMU12
0.1449
0.0000
0.0000
0.0000
0.0000
DMU13
0.0000
0.0000
0.0000
0.0000
0.0000
DMU14
0.2088
0.1072
0.1415
0.0000
0.1072
DMU15
0.2793
0.2415
0.2018
0.2225
0.2415
DMU16
0.1926
0.0000
0.0000
0.0000
0.0000
DMU17
0.0000
0.0000
0.0000
0.0000
0.0000
DMU18
0.0567
0.0000
0.0000
0.0000
0.0000
DMU19
0.0000
0.0000
0.0827
0.0000
0.0000
DMU20
0.0000
0.0000
0.0000
0.0000
0.0000
DMU21
0.0000
0.0000
0.0000
0.0000
0.0000
DMU22
0.0000
0.0000
0.0000
0.2032
0.0000
DMU23
0.0000
0.0000
0.0000
0.0000
0.0000
DMU24
0.0000
0.0000
0.0000
0.0000
0.0000
DMU25
0.0000
0.0000
0.0000
0.0000
0.0000
DMU26
0.0000
0.0000
0.0000
0.0000
0.0000
DMU27
0.0000
0.2287
0.2670
0.6379
0.2287
DMU28
0.0000
0.0000
0.0000
0.0000
0.0000
DMU29
0.0000
0.0000
0.1373
0.0967
0.0000
DMU30
0.0000
0.0000
0.0000
0.0000
0.0000
DMU31
0.0634
0.0799
0.0000
0.0000
0.0799
DMU32
0.5982
0.2792
0.3526
0.5700
0.2792
Mean
0.0702
0.0655
0.0830
0.1015
0.0655Since the previous inefficiency scores are separately derived by considering only one optimal set of weights, the resulting performance information might be unrealistic and unreasonable. Therefore, we can use the proposed eco-inefficiency dominance probability approach in this paper to analyze the banking performance considering all sets of feasible weights. To this end, we first use model (5) to calculate the inefficiency score intervals and then use Definition 2 to calculate the first-level pairwise dominating probability in 2018. Since the pairwise dominating probability involves an n×n matrix that is hard to present in this paper, we arbitrarily take the Bank of Jiangsu (BOJ), Agricultural Bank of China (ABC), and Rural Commercial Bank of Zhangjiagang (RCBZ) for instance, and the first-level inefficiency dominating probability for these three banks across all 32 banks is listed in the second, third, and fourth columns of Table 5, namely, PBOJ∼≻k, PABC∼≻k, and PRCBZ∼≻k (it is also possible to consider any other banks as examples to show the calculation results). The results represent the probability of the considered bank having a smaller inefficiency index across any other bank, with different sets of weights being attached to inputs and outputs to ensure an efficient status for other banks. For example, the value of 0.2699 implies that, by fixing the inefficiency score of the Bank of Beijing (DMU1) to zero, the inefficiency score of the Bank of Jiangsu will be smaller with a probability of 0.2699 and larger with a probability of 0.7301 (1–0.2699). At the same time, we can use model (6) and Definition 3 to calculate another dominating probability, and the second-level results of Bank of Jiangsu (BOJ), Agricultural Bank of China (ABC), and Rural Commercial Bank of Zhangjiagang (RCBZ) in 2018 are given in the last three columns of Table 5.Table 5
Two kinds of dominating probabilities for BOJ, ABC, and RCBZ in 2018.
Banks
Definition2
Definition3
BOJ
ABC
RCBZ
BOJ
ABC
RCBZ
DMU1
0.2699
0.5964
0.0093
0.3515
0.9452
0.2111
DMU2
0.8578
0.6238
0.2563
0.9850
0.9959
0.4689
DMU3
0.8520
0.6207
0.0496
0.9662
0.9910
0.1966
DMU4
0.0288
0.3164
0.0016
0.3339
0.3822
0.3216
DMU5
0.1671
0.5897
0.0074
0.3446
0.9119
0.2663
DMU6
0.8592
0.6197
0.0386
0.9599
0.9887
0.1905
DMU7
0.8518
0.6159
0.0253
0.9337
0.9821
0.1961
DMU8
0.3056
0.6073
0.0106
0.4337
0.9419
0.2603
DMU9
0.0315
0.1127
0.0017
0.3121
0.1311
0.2989
DMU
10
1.0000
0.6052
0.0088
1.0000
0.9589
0.1523
DMU11
0.8514
0.6230
0.3883
0.9915
0.9977
0.4059
DMU12
0.0835
0.5315
0.0042
0.3563
0.7917
0.3073
DMU13
0.1096
0.5695
0.0054
0.3154
0.8709
0.2684
DMU14
0.8678
0.6123
0.0146
0.9135
0.9736
0.1657
DMU15
0.9666
0.6173
0.0286
0.9704
0.9791
0.2599
DMU
16
0.0411
1.0000
0.0023
0.3948
1.0000
0.3768
DMU17
0.2062
0.5856
0.0089
0.4109
0.9140
0.3081
DMU18
0.0921
0.5531
0.0046
0.3014
0.8476
0.2634
DMU19
0.8593
0.6219
0.0995
0.9780
0.9943
0.2510
DMU20
0.8454
0.6226
0.1170
0.9772
0.9942
0.2986
DMU21
0.5626
0.6048
0.0133
0.5695
0.9618
0.2139
DMU22
0.8492
0.6230
0.2009
0.9916
0.9978
0.2256
DMU23
0.8463
0.6225
0.1454
0.9886
0.9971
0.1876
DMU24
0.8473
0.6222
0.0825
0.9808
0.9951
0.1796
DMU25
0.0724
0.5560
0.0038
0.2395
0.8469
0.2161
DMU
26
0.8477
0.6232
1.0000
0.9912
0.9977
1.0000
DMU27
0.8790
0.6204
0.0512
0.9669
0.9892
0.2431
DMU28
0.1159
0.5628
0.0058
0.3840
0.8491
0.3310
DMU29
0.8588
0.6202
0.0662
0.9644
0.9908
0.2669
DMU30
0.0389
0.3718
0.0021
0.3274
0.4493
0.3107
DMU31
0.1246
0.5690
0.0060
0.3564
0.8716
0.2997
DMU32
0.8494
0.6222
0.1199
0.9848
0.9960
0.2093Without loss of generality, each DMU will always dominate itself, as these three banks will have a dominating probability of 1 to itself regardless of whether the first-level dominance probability or the second-level dominance probability is considered. From Table5, we can find that all three banks have a larger second-level dominance probability than the first-level dominance probability to other banks, and it is indeed also held for all banks. This difference is due to the definition style of pairwise dominances, and we cannot arbitrarily use one dominating probability to represent the performance assessment with the other being ignored. By averaging the two pairwise dominating probabilities, we can calculate the pairwise dominance probability as well as the dominated probability. Reconsidering the three banks in Table 5, we show the pairwise dominance probability as well as dominated probability for Bank of Jiangsu, Agricultural Bank of China, and Rural Commercial Bank of Zhangjiagang in Table 6. Taking the inefficiency dominance probability of Bank of Jiangsu to Bank of Beijing (DMU1), for example, the arithmetic mean of 0.2699 and 0.3515 by Definition 4 is exactly the pairwise inefficiency dominance probability of Bank of Jiangsu to Bank of Beijing, 0.3107. Since the pairwise dominance probability shows the probability of the inefficiency score of Bank of Jiangsu relative to any other banks, by considering all sets of feasible input and output weights, it means that the inefficiency score of Bank of Jiangsu is smaller than that of Bank of Beijing with a probability of 0.3107. In contrast, the inefficiency of Bank of Jiangsu will be dominated by Bank of Beijing with a probability of 0.6893.Table 6
Pairwise dominance probability for BOJ, ABC, and RCBZ in 2018.
Banks
Dominating
Dominated
BOJ
ABC
RCBZ
BOJ
ABC
RCBZ
DMU1
0.3107
0.7708
0.1102
0.6893
0.2292
0.8898
DMU2
0.9214
0.8098
0.3626
0.0786
0.1902
0.6374
DMU3
0.9091
0.8058
0.1231
0.0909
0.1942
0.8769
DMU4
0.1813
0.3493
0.1616
0.8187
0.6507
0.8384
DMU5
0.2559
0.7508
0.1369
0.7441
0.2492
0.8631
DMU6
0.9096
0.8042
0.1145
0.0904
0.1958
0.8855
DMU7
0.8928
0.7990
0.1107
0.1072
0.2010
0.8893
DMU8
0.3696
0.7746
0.1354
0.6304
0.2254
0.8646
DMU9
0.1718
0.1219
0.1503
0.8282
0.8781
0.8497
DMU
10
1.0000
0.7820
0.0805
1.0000
0.2180
0.9195
DMU11
0.9214
0.8104
0.3971
0.0786
0.1896
0.6029
DMU12
0.2199
0.6616
0.1557
0.7801
0.3384
0.8443
DMU13
0.2125
0.7202
0.1369
0.7875
0.2798
0.8631
DMU14
0.8906
0.7929
0.0902
0.1094
0.2071
0.9098
DMU15
0.9685
0.7982
0.1442
0.0315
0.2018
0.8558
DMU
16
0.2180
1.0000
0.1895
0.7820
1.0000
0.8105
DMU17
0.3086
0.7498
0.1585
0.6914
0.2502
0.8415
DMU18
0.1968
0.7004
0.1340
0.8032
0.2996
0.8660
DMU19
0.9187
0.8081
0.1752
0.0813
0.1919
0.8248
DMU20
0.9113
0.8084
0.2078
0.0887
0.1916
0.7922
DMU21
0.5661
0.7833
0.1136
0.4339
0.2167
0.8864
DMU22
0.9204
0.8104
0.2132
0.0796
0.1896
0.7868
DMU23
0.9174
0.8098
0.1665
0.0826
0.1902
0.8335
DMU24
0.9141
0.8086
0.1311
0.0859
0.1914
0.8689
DMU25
0.1559
0.7014
0.1099
0.8441
0.2986
0.8901
DMU
26
0.9195
0.8105
1.0000
0.0805
0.1895
1.0000
DMU27
0.9229
0.8048
0.1472
0.0771
0.1952
0.8528
DMU28
0.2499
0.7060
0.1684
0.7501
0.2940
0.8316
DMU29
0.9116
0.8055
0.1666
0.0884
0.1945
0.8334
DMU30
0.1832
0.4106
0.1564
0.8168
0.5894
0.8436
DMU31
0.2405
0.7203
0.1529
0.7595
0.2797
0.8471
DMU32
0.9171
0.8091
0.1646
0.0829
0.1909
0.8354
Mean
0.6096
0.7375
0.1864
0.4217
0.2938
0.8448Furthermore, by aggregating the pairwise inefficiency dominance probability across all banks, we can obtain the average performance dominance probability in terms of inefficiency scores for these three banks, as given in the last column of Table6. Furthermore, proceeding in the same manner as above, we can determine the inefficiency dominance probability for all 32 Chinese listed banks from 2014 to 2018, as shown in Table 7.Table 7
Inefficiency dominance probability of 32 listed banks from 2014 to 2018.
Banks
2014
2015
2016
2017
2018
Mean
DMU1
0.6175
0.6149
0.6142
0.6173
0.6235
0.6175
DMU2
0.2872
0.2633
0.2243
0.2184
0.2293
0.2445
DMU3
0.4257
0.4131
0.3831
0.3685
0.3971
0.3975
DMU4
0.7964
0.7936
0.7786
0.7920
0.8056
0.7932
DMU5
0.6770
0.6678
0.6790
0.6815
0.6411
0.6693
DMU6
0.3007
0.3382
0.3529
0.3928
0.4268
0.3623
DMU7
0.4582
0.4853
0.4794
0.4922
0.4908
0.4812
DMU8
0.6105
0.5885
0.5749
0.5640
0.5857
0.5847
DMU9
0.8039
0.8091
0.7945
0.8038
0.8132
0.8049
DMU
10
0.5439
0.5812
0.5893
0.6105
0.6096
0.5869
DMU11
0.2742
0.2571
0.2469
0.2246
0.2099
0.2425
DMU12
0.7623
0.7480
0.7216
0.7106
0.7032
0.7291
DMU13
0.7042
0.6892
0.6916
0.7231
0.6982
0.7013
DMU14
0.5130
0.5143
0.5269
0.5392
0.5413
0.5269
DMU15
0.4937
0.4849
0.4806
0.4678
0.4710
0.4796
DMU
16
0.7135
0.7209
0.7130
0.7232
0.7375
0.7216
DMU17
0.6272
0.6105
0.5922
0.6169
0.6155
0.6125
DMU18
0.7315
0.7255
0.7244
0.7324
0.7331
0.7294
DMU19
0.3614
0.3654
0.3772
0.3510
0.3212
0.3552
DMU20
0.2796
0.2947
0.3033
0.2929
0.3048
0.2951
DMU21
0.5859
0.5885
0.6007
0.5769
0.5712
0.5846
DMU22
0.1967
0.1937
0.1941
0.1786
0.2062
0.1939
DMU23
0.2875
0.3021
0.2897
0.2705
0.2559
0.2812
DMU24
0.2823
0.2992
0.3333
0.3240
0.3304
0.3138
DMU25
0.7851
0.7710
0.7642
0.7895
0.7747
0.7769
DMU
26
0.2405
0.2264
0.2289
0.2039
0.1864
0.2172
DMU27
0.3743
0.3810
0.4116
0.3980
0.4038
0.3938
DMU28
0.7060
0.6765
0.6573
0.6634
0.6801
0.6766
DMU29
0.4041
0.4260
0.4448
0.4405
0.3855
0.4202
DMU30
0.7721
0.7665
0.7577
0.7673
0.7795
0.7686
DMU31
0.7224
0.6996
0.7011
0.6738
0.6721
0.6938
DMU32
0.1613
0.2040
0.2686
0.2909
0.2956
0.2441As Table7 shows, these banks have very different inefficiency dominance probabilities over all sets of feasible weights relative to the inefficiency scores derived from only an optimal set of weights. For example, Huaxia Bank (DMU8) always has a positive inefficiency in each year, and that bank will be ranked after most banks. However, Huaxia Bank will have an inefficiency dominance probability of 0.6105, 0.5885, 0.5749, 0.5640, and 0.5857 in the 2014–2018 period, respectively. This phenomenon implies that Huaxia Bank is more likely to have an inefficiency index smaller than that of half of all banks. In contrast, by considering only the optimal set of weights, the Rural Commercial Bank of Zhangjiagang (DMU26) is extremely efficient with an inefficiency score of zero for all years, but it has a relatively small inefficiency dominance probability in each year (0.2405, 0.2264, 0.2289, 0.2039, and 0.1864, respectively), implying that the performance of Rural Commercial Bank of Zhangjiagang in the sense of inefficiency scores is at a disadvantage to other banks by addressing all weight possibilities.Furthermore, it can be found that all banks have a relatively stable inefficiency dominance probability in the period of 2014–2018, with the largest variation being 0.1343 (0.2956–0.1613) for Zijin Rural Commercial Bank (DMU32) and the smallest variation being 0.0087 (0.7331–0.7244) for Shanghai Pudong Development Bank (DMU18). All banks always have an eco-inefficiency dominance probability that is either larger than 0.50 or less than 0.50 in the five-year sample (0.5 is a threshold where the dominating probability is equal to the dominated probability), implying that all banks can be divided into two groups, one for superior banks and another for inferior banks. The categories are shown in Table 8. From the average sense, we find that China Construction Bank (DMU9), Industrial and Commercial Bank of China (DMU4), Industrial Bank (DMU25), Bank of China (DMU32), and Shanghai Pudong Development Bank (DMU18) are the top five listed banks for operation performance. In contrast, the lowest five banks are Suzhou Rural Commercial Bank (DMU22), Rural Commercial Bank of Zhangjiagang (DMU26), Jiangyin Rural Commercial Bank (DMU11), Zijin Rural Commercial Bank (DMU32), and Changshu Rural Commercial Bank (DMU2), all of which have an average inefficiency dominance probability of less than 0.2500.Table 8
Superior and inferior listed banks.
Category
Banks
Superior (>0.5000)
Bank of Beijing, Industrial and Commercial Bank of China, China Everbright Bank, Huaxia Bank, China Construction Bank, Bank of Jiangsu, Bank of Communications, China Minsheng Bank, Bank of Nanjing, Agricultural Bank of China, Ping An Bank, Shanghai Pudong Development Bank, Bank of Shanghai, Industrial Bank, China Merchants Bank, Bank of China, China CITIC Bank
Inferior (<0.5000)
Changshu Rural Commercial Bank, Bank of Chengdu, Bank of Guiyang, Bank of Hangzhou, Jiangyin Rural Commercial Bank, Bank of Ningbo, Bank of Qingdao, Qingdao Rural Commercial Bank, Suzhou Rural Commercial Bank, Wuxi Rural Commercial Bank, Bank of Xian, Rural Commercial Bank of Zhangjiagang, Bank of Changsha, Bank of Zhengzhou, Zijin Rural Commercial BankThe eco-inefficiency and eco-inefficiency dominance probabilities of these 32 listed banks are given in Tables4 and 7, respectively. It is clear that the proposed approach will give performance indexes that are different from those of previous approaches. Furthermore, we give the ranking comparison of eco-inefficiency and eco-inefficiency dominance probabilities in Table 9. It can be found from Table 9 that, on the one hand, the proposed approach will give performance rankings that are largely different from those of previous approaches. On the other hand, the traditional DEA model cannot discriminate all banks, and more seriously, more than twenty banks are ranked as the first based on inefficiency scores, while the eco-inefficiency dominance probability approach can indeed give a full ranking of all banks. From this perspective, the proposed approach can give a more reasonable and discriminating performance assessment.Table 9
Ranking of inefficiency and inefficiency dominance probabilities.
Banks
2014
2015
2016
2017
2018
DMU1
1
13
1
12
1
12
1
12
1
12
DMU2
1
26
32
28
32
31
29
30
32
29
DMU3
1
20
25
21
29
22
1
23
25
22
DMU4
1
2
1
2
1
2
1
2
1
2
DMU5
1
11
23
11
1
10
25
9
23
11
DMU6
29
24
31
24
30
24
31
22
31
20
DMU7
26
19
1
18
1
19
1
18
1
18
DMU8
31
14
30
15
27
16
28
16
30
15
DMU9
1
1
1
1
1
1
1
1
1
1
DMU10
1
16
1
16
1
15
1
14
1
14
DMU11
1
29
1
29
1
29
1
29
1
30
DMU12
25
5
1
5
1
6
1
8
1
7
DMU13
1
10
1
9
1
9
1
7
1
8
DMU14
28
17
26
17
25
17
1
17
26
17
DMU15
30
18
28
19
26
18
27
19
28
19
DMU16
27
8
1
7
1
7
1
6
1
5
DMU17
1
12
1
13
1
14
1
13
1
13
DMU18
23
6
1
6
1
5
1
5
1
6
DMU19
1
23
1
23
23
23
1
24
1
25
DMU20
1
28
1
27
1
26
1
26
1
26
DMU21
1
15
1
14
1
13
1
15
1
16
DMU22
1
31
1
32
1
32
26
32
1
31
DMU23
1
25
1
25
1
27
1
28
1
28
DMU24
1
27
1
26
1
25
1
25
1
24
DMU25
1
3
1
3
1
3
1
3
1
4
DMU26
1
30
1
30
1
30
1
31
1
32
DMU27
1
22
27
22
28
21
32
21
27
21
DMU28
1
9
1
10
1
11
1
11
1
9
DMU29
1
21
1
20
24
20
24
20
1
23
DMU30
1
4
1
4
1
4
1
4
1
3
DMU31
24
7
24
8
1
8
1
10
24
10
DMU32
32
32
29
31
31
28
30
27
29
27All 32 listed banks can be mainly categorized into four groups according to the ownership, namely, state-owned banks, joint-stock banks, city commercial banks, and rural commercial banks. Table10 shows the divisions, and Table 11 gives the average inefficiency dominance probabilities for different kinds of banks.Table 10
Categories of different listed banks.
Category
Banks
State-owned banks (5)
Industrial and Commercial Bank of China
China Construction Bank
Bank of Communications
Agricultural Bank of China
Bank of China
Joint-stock banks (8)
China Everbright Bank, Huaxia Bank
China Minsheng Bank, Ping An Bank
Shanghai Pudong Development Bank
Industrial Bank, China Merchants Bank
China CITIC Bank
City commercial banks (12)
Bank of Beijing, Bank of Chengdu
Bank of Guiyang, Bank of Hangzhou
Bank of Jiangsu, Bank of Nanjing
Bank of Ningbo, Bank of Qingdao
Bank of Shanghai, Bank of Xian
Bank of Changsha, Bank of Zhengzhou
Rural commercial banks (7)
Changshu Rural Commercial Bank
Jiangyin Rural Commercial Bank
Qingdao Rural Commercial Bank
Suzhou Rural Commercial Bank
Wuxi Rural Commercial Bank
Rural Commercial Bank of Zhangjiagang
Zijin Rural Commercial BankTable 11
Average inefficiency dominance probabilities for four kinds of listed banks.
Banks
2014
2015
2016
2017
2018
Mean
State-owned banks
0.7696
0.7676
0.7531
0.7594
0.7678
0.7635
Joint-stock banks
0.6955
0.6786
0.6731
0.6806
0.6751
0.6806
City commercial banks
0.4467
0.4577
0.4662
0.4649
0.4644
0.4600
Rural commercial banks
0.2467
0.2488
0.2508
0.2400
0.2412
0.2455It can be learned from Table11 that the four kinds of listed banks exhibit considerably different performance dominance probabilities. More specifically, those five state-owned banks have the highest average inefficiency dominance probability, which is almost three times that of the lowest rural commercial banks. This result shows that these state-owned banks are more likely to have better performance compared with other banks over all sets of weights. In contrast, those rural commercial banks are more likely to have worse performance relative to other banks. Furthermore, joint-stock banks are inferior to state-owned banks and superior to city commercial banks, which are further superior to rural commercial banks.By using the proposed inefficiency dominance probability approach, we can provide a performance analysis of 32 Chinese listed banks that is derived from the real world. Since the proposed approach considers all sets of feasible weights, which is different from classic DEA approaches that focus on only one or several optimal sets of weights, the resulting performance analytics are more reasonable due to taking full weights and all possibilities into account. Furthermore, by considering all sets of feasible weights, the performance dominance probability is largely different from traditional performance indexes that are obtained with some extreme weights, and the corresponding ranking orders are also changed considerably. Therefore, it makes sense for the proposed approach because it can provide a comprehensive performance assessment instead of only some extreme performances from a data analytics perspective.


## 4. Conclusion

This paper proposes a new DEA-based approach for assessing the operation performance of Chinese listed banks. Since the conventional DEA approaches consider only a set of optimal and extreme weights to measure the relative performance, the resulting performance indexes might be unreasonable and even unrealistic in practice. From a data-driven decision-making perspective, this paper proceeds to take all sets of feasible input/output weights into account rather than only some special weights. For that purpose, we first propose an extended eco-inefficiency model to address banking activities and build a pairwise performance dominance structure in terms of inefficiency scores. Furthermore, we calculate the overall inefficiency dominance probability based on all sets of feasible weights, and the inefficiency dominance probability can be used for data-driven performance analytics of those Chinese listed banks. The proposed approach can provide data analytics on relative performances instead of only some extreme possibilities, and it is further used for the empirical analytics of operation performances for 32 listed banks in China.This paper can be extended with regard to several aspects. First, this paper considers the possible inefficiency score range but ignores its associated possibility. That is, we consider each possible performance score coequally, but it is common that some performance scores are more likely than others. Therefore, future research can be developed to take the possibilities of various performances based on different sets of weights into account. Second, an important research avenue in the DEA field is how to address the internal production structure of DMUs, and thus, similar studies can be designed for situations with complex internal structures and linking connections. Third, similar approaches based on all sets of weights can also be developed for other purposes, such as fixed cost and resource allocation and target setting in real applications.



---

*Source: 3780232-2020-05-04.xml*