Research Article  Open Access
Feng Li, Lunwen Wu, Qingyuan Zhu, Yanling Yu, Gang Kou, Yi Liao, "An EcoInefficiency Dominance Probability Approach for Chinese Banking Operations Based on Data Envelopment Analysis", Complexity, vol. 2020, Article ID 3780232, 14 pages, 2020. https://doi.org/10.1155/2020/3780232
An EcoInefficiency Dominance Probability Approach for Chinese Banking Operations Based on Data Envelopment Analysis
Abstract
Data envelopment analysis (DEA) has proven to be a powerful technique for assessing the relative performance of a set of homogeneous decisionmaking 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 ecoinefficiency model based on the DEA methodology to address banking activities and their possible relative performances. Since the ecoinefficiency 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 datadriven perspective. Then, we develop two pairwise ecoinefficiency dominance concepts and propose the inefficiency dominance probability model. Finally, we illustrate the ecoinefficiency dominance probability approach with 32 Chinese listed banks from 2014 to 2018 to demonstrate the usefulness and efficacy of the proposed method.
1. Introduction
Forty years have gone by since the great reform and 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 stateowned and listed banks, has played a great role in Chinese economic growth [1, 2]. Throughout the everincreasing 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 decisionmaking 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 inputoutput 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 DEAbased 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], RamazaniTarkhorani 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 crossefficiency 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 crossefficiency 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 selfappraisal and several smaller efficiency scores based on peer appraisal. The ultimate crossefficiency score can be aggregated with these selfappraisal and peer appraisal scores for each individual DMU [35–41]. The DEA crossefficiency 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 crossefficiency evaluation approaches for many kinds of real applications [45–47]. Although the crossefficiency 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 crossefficiency methods, as does the aggregation of individual crossefficiencies [41, 48–50].
The recent research by Salo and Punkka [21] suggests developing ratiobased 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 twostage 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 twostage 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 datadriven 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 byproducts, such as bad debts that are jointly produced with incomes, an extended ecoinefficiency model is developed. Furthermore, the ecoinefficiency 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 datadriven analytics. Finally, the proposed approach is applied to a fouryear 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 stateowned 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 decisionmakers 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. Section 2 develops the mathematical methodology of an extended ecoinefficiency 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 ecoinefficiency model to address banking activities and possible relative performances in Section 2.1. Furthermore, we build a performance dominance structure considering all sets of feasible weights in Section 2.2.
2.1. An Extended EcoInefficiency Model
Suppose a set of n 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 byproduct in the banking industry. Without loss of generality, we consider each bank as a homogeneous decisionmaking unit (DMU) in the DEA framework. Furthermore, consume inputs to produce desirable outputs and undesirable outputs , 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:
The above formula is nonlinear since both the reduction factor and intensity variable are unknown. Furthermore, we can equivalently change formula (1) into a linear version, which is presented in the following formula:
Based on formula (2), both desirable and undesirable outputs are weighted by nondisposed intensity variables , whereas the inputs are weighted by the sum of nondisposed intensity variables and disposed intensity variables . In addition, the VRS assumption is ensured by summing the total nondisposed intensity variables and disposed intensity variables to 1, i.e., .
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 ecoinefficiency model for performance evaluation since the ecoinefficiency model has some advantages in modeling activities with undesirable outputs compared with four wellestablished 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 ecoinefficiency model is formulated as follows:
In model (3), the variables , , and 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 slackbased 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 , , and , where the direction vector is . Solving model (3) determines an optimal solution , then we have . Hence, we have a system of unknown variables () and linearly independent equations , , , and . Together with another equation such as that is used to ensure a bounded and closed space, we then have a system of unknown variables and 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 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:
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 determines a series of inefficiency scores using a series of optimal solutions . 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 is.
2.2. Dominance Probability Based on Inefficiency Scores
It is notable that the inefficiency score 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 while ignoring any other feasible weights, and thus, the obtained inefficiency score might be too extreme or even unrealistic. On the other hand, the inefficiency score is calculated separately for each , 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 ecoinefficiency 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. dominates (denoted as ) if and only if always has a smaller inefficiency score relative to for all sets of feasible input and output weights.
The dominance relation based on inefficiency scores is determined if the inefficiency score for a certain is as small as that of 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 and 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 when is fixed with a prespecified inefficiency level of zero by requiring the constraint that . In fact, the inefficiency level of can be set to any nonnegative value M, and by substituting with , we can get the same dominance probability as given in Definition 2 and Definition 3; hence, we immediately set the inefficiency level of to zero for simplification in the following model:An additional constraint that 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 DMU_{d}’s inefficiency score can be relative to 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 is fixed to zero and if , which means that will always have a smaller inefficiency score compared with , then dominates . In contrast, if , which means that will always have a larger inefficiency score than (which has an inefficiency level of zero), then is dominated by . 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. When has an inefficiency score of , the probability that dominates over all sets of feasible input and output weights is calculated by .
It is clear that, if , which means that will always have a larger inefficiency by remaining the inefficiency for , the inefficiency dominance probability of relative to would take a nonpositive value. That is, it is impossible for to dominate . For the sake of a bounded range, we assume that if . For a more general case in which , the probability that dominates takes a value from zero to unity. A larger value of means that it is more likely for to have a smaller inefficiency score compared with over all sets of feasible input/output weights.
Note in addition that Definition 2 fixes the inefficiency level of to calculate the inefficiency range of and further computes the inefficiency dominance probability of relative to . In contrast, we can also compute the inefficiency dominance probability of relative to by fixing the inefficiency level of and calculating the inefficiency range of . The above idea is formulated in model (6), which is very similar to model (5) but substitutes for :An alternative inefficiency dominance probability of relative to by fixing the inefficiency level of is given in Definition 3.
Definition 3. When has an inefficiency score of , the probability that dominates over all sets of feasible input and output weights is calculated by .
Furthermore, the overall probability that the inefficiency score of dominates that of is the average of the two probabilities by fixing the inefficiency level of and . More specifically, Definition 4 gives the pairwise performance dominance probability with regard to inefficiency scores.
Definition 4. The pairwise performance dominance probability of relative to over all sets of feasible input and output weights is .
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 certain 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 of across all DMUs over all sets of feasible input and output weights is .
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 of 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 Table 1.

In 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 byproduct that will greatly harm the bank. The inputs and outputs to be used in this study are summarized in Table 2.

Our 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. Table 3 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.

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 DDFbased 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 (DMU_{6}, Bank of Guiyang; DMU_{8}, Huaxia Bank; DMU_{15}, Bank of Ningbo; and DMU_{32}, 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 zerovalue inefficiency index.

Since 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 ecoinefficiency 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 firstlevel pairwise dominating probability in 2018. Since the pairwise dominating probability involves an 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 firstlevel inefficiency dominating probability for these three banks across all 32 banks is listed in the second, third, and fourth columns of Table 5, namely, , , and (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 (DMU_{1}) 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 secondlevel 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.

Without 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 firstlevel dominance probability or the secondlevel dominance probability is considered. From Table 5, we can find that all three banks have a larger secondlevel dominance probability than the firstlevel 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 (DMU_{1}), 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.

Furthermore, 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 Table 6. 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.

As Table 7 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 (DMU_{8}) 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 (DMU_{26}) 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 (DMU_{32}) and the smallest variation being 0.0087 (0.7331–0.7244) for Shanghai Pudong Development Bank (DMU_{18}). All banks always have an ecoinefficiency dominance probability that is either larger than 0.50 or less than 0.50 in the fiveyear 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 (DMU_{9}), Industrial and Commercial Bank of China (DMU_{4}), Industrial Bank (DMU_{25}), Bank of China (DMU_{32}), and Shanghai Pudong Development Bank (DMU_{18}) are the top five listed banks for operation performance. In contrast, the lowest five banks are Suzhou Rural Commercial Bank (DMU_{22}), Rural Commercial Bank of Zhangjiagang (DMU_{26}), Jiangyin Rural Commercial Bank (DMU_{11}), Zijin Rural Commercial Bank (DMU_{32}), and Changshu Rural Commercial Bank (DMU_{2}), all of which have an average inefficiency dominance probability of less than 0.2500.

The ecoinefficiency and ecoinefficiency dominance probabilities of these 32 listed banks are given in Tables 4 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 ecoinefficiency and ecoinefficiency 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 ecoinefficiency 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.

All 32 listed banks can be mainly categorized into four groups according to the ownership, namely, stateowned banks, jointstock banks, city commercial banks, and rural commercial banks. Table 10 shows the divisions, and Table 11 gives the average inefficiency dominance probabilities for different kinds of banks.


It can be learned from Table 11 that the four kinds of listed banks exhibit considerably different performance dominance probabilities. More specifically, those five stateowned 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 stateowned 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, jointstock banks are inferior to stateowned 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 DEAbased 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 datadriven decisionmaking 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 ecoinefficiency 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 datadriven 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.
Data Availability
The illustration data used to support the findings of this study are collected from annual financial reports that are publicly produced by these listed banks in Shenzhen Stock Exchange and Shanghai Stock Exchange in China. In addition, the data are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (nos. 71901178, 71904084, 71910107002, and 71725001), the Natural Science Foundation of Jiangsu Province (no. BK20190427), the Social Science Foundation of Jiangsu Province (no. 19GLC017), the Fundamental Research Funds for the Central Universities at Southwestern University of Finance and Economics (nos. JBK2003021, JBK2001020, and JBK190504) and at Nanjing University of Aeronautics and Astronautics (no. NR2019003), and the Innovation and Entrepreneurship Foundation for Doctor of Jiangsu Province, China.
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Copyright © 2020 Feng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.