A common-weights approach for efficiency evaluation in fuzzy data envelopment analysis with undesirable outputs: Application in banking industry

Abstract

Data Envelopment Analysis (DEA) is a non-parametric approach based on linear programming for evaluating the performance of decision making units (DMUs) with multiple inputs and multiple outputs. The lack of the ability to generate the actual weights, not considering the impact of undesirable outputs in the evaluation process and the measuring of efficiencies of DMUs based upon precise observations are three main drawbacks of the conventional DEA models. This paper proposes a novel approach for finding the common set of weights (CSW) to compute efficiencies in DEA model with undesirable outputs when the data are represented by fuzzy numbers. The proposed approach is based on fuzzy arithmetic which formulates the fuzzy additive DEA model as a linear programing problem and gives fuzzy efficiencies of all DMUs based on resulting CSW. We demonstrate the applicability of the proposed model with a simple numerical example. Finally, in the context of performance management, an application of banking industry in Iran is presented for analyzing the influence of fuzzy data and depicting the impact of undesirable outputs over the efficiency results.

Keywords

Data envelopment analysis undesirable outputs fuzzy numbers common set of weights

1 Introduction

Data envelopment analysis (DEA) technique is one of the most important and widely used approaches for evaluating the performance of decision-making units (DMUs). DEA is a linear programming technique for measuring the relative efficiency of homogeneous DMUs with multiple inputs and multiple outputs, first performed by Charnes et al. [9]. In crisp DEA, the efficiency measuring of a DMU can be done from different point of views. Amirteimoori et al. [1] presented an additive model in the presence of shared resources in network DEA model. Aparicio et al. [2] compared the well-known weighted additive DEA model with the distance function structure in DEA and presented the weighted additive distance function.

1.1 Common weights

In DEA, the ratio of the weighted sum of outputs to the weighted sum of inputs is used to measure the efficiency of each DMU. The way that DEA calculates the weights for input and output data allows each DMU to evaluate its efficiency with the most favorable weights. In this regard, one linear programming (LP) problem is solved for each DMU to adopt the full weight flexibility that sets itself in the most favorable light against the other DMUs. Hence, the resulting efficiencies are not compared and ranked on the same basis. To overcome this shortcoming, the common set of weights (CSW) models in DEA have been proposed by researchers that calculate only common weight for each of input and output data and eliminate inter-unit weight flexibility completely [5]. Hence, various models with different approaches have been introduced using CSW. Cook et al. [10] presented a common set of weights to measure the efficiency of DMUS for the first time. That model then was developed by Roll et al. [33]. Kao and Hung [23] have proposed a compromise solution approach within the DEA framework. In this approach, the efficiency scores calculated from the standard DEA model are considered as the ideal solution to achieve the CSW of DMUs. Saati [34] proposed a new technique for finding CSW based on a two-step approach. Liu and Peng [28] presented a CWS in order to achieve the best efficiency score from a group of efficient DMUs. Sun et al. [43] evaluated the performance units considering ideal and anti-ideal DMU with common set of weights. Tavana et al. [44] presented a stochastic DEA model using a common set of weights based on the ideal point concept. Salahi et al. [37] presented an optimistic robust optimization approach to determine the common set of weights in DEA. Lam [27] provided a common set of weights of input and output for ranking DMUs. Jahanshahloo et al. [22] proposed two methods for allocating fixed costs using DEA based on the efficiency invariance and common set of weights principles. Mamizadeh-Chatghayeh and Noura [29] presented a two-step ranking process using a common set of weights in DEA. Hatami-Marbini and Saati [18] presented an aggregate DEA model for calculating the most desirable common set of weight to determine the efficiency of all production systems and their processes simultaneously.

1.2 Undesirable outputs

DEA models were presented initially based on desirable inputs and outputs. But in real-world issues, not considering the undesirable nature for evaluating the efficiency of decision making units will not be acceptable. Considering undesirable outputs for performance evaluation was initially suggested by Seiford and Zhu [38]. Homayounfar et al. [20] introduced a new DEA-based model in a centralized decision-making environment. It shares input-output changes between all units in a way that does not reduce individual efficiencies. Seyed-Esmaeili and Rostamy-Malkhalifeh [41] presented a DEA model in the present of undesirable negative inputs and outputs. Halkos et al. [16] examined the main methods for dealing with undesirable outputs in DEA. Zhou et al. [48] provided new transformation to reflect the exponential relationship between desirable and undesirable outputs. Toloo and Hančlová [45] developed a directional output distance method to deal with multi-valued measures in the presence of undesirable outputs.

1.3 Fuzzy DEA models

In addition, traditional DEA models evaluate each DMU with crisp input and output. But in real-world applications, this is not always possible, and in many cases the data is uncertain and ambiguous. One way to deal with the uncertain input and output data is to use of fuzzy numbers and in this case fuzzy DEA (FDEA) models are appeared in a natural way. FDEA model was first introduced by Sengupta [39, 40]. Khodabakhshi et al. [26] used an additive model by presenting an alternative method for estimating returns to scale in stochastic and fuzzy DEA models. Hatami-Marbini et al. [19] presented an additive DEA model for evaluating the efficiency of a set of DMUs with fuzzy inputs and outputs. This method uses the α-cut approach introduced by Saati et al. [36]. Kazemi Matin et al. [25] presented an additive DEA model for assessing the technical inefficiency of DMUs under imprecise data. Puri and Yadav [31] presented a concept of fuzzy input mix-efficiency in DEA and evaluated fuzzy input mix-efficiency using α-cutting approach. Puri and Yadav [32] presented a fuzzy DEA model with undesirable outputs that can be solved as crisp linear program for each α in (0, 1] using α-cut approach. Hatami-Marbini et al. [17] presented a novel fully fuzzified DEA approach that, in addition to input and output data, all the variables were considered fuzzy. Arya and Yadav [4] developed intuitionistic fuzzy DEA and dual intuitionistic fuzzy DEA models to evaluate the performance of DMUs. Peykani et al. [30] developed an adjustable and flexible fuzzy DEA model to consider DMUs’ preferences. Bagheri et al. [6] have used the fuzzy arithmetic approach in fuzzy DEA for solving fuzzy multi objective transportation problem. Shi et al. [42] investigated the problem of efficiency measurement for parallel systems with two components, while some inputs/outputs are fuzzy numbers. Arana-Jimenez et al. [3] proposed a new radial, input-oriented and fully fuzzy DEA approach for assessing the relative efficiency of a set of DMUs.

1.4 Contribution

Various approaches have been proposed to solve fuzzy DEA models. One of these approaches is fuzzy arithmetic approach. The basic idea of this approach is based on the principle that any FDEA fractional model cannot transform to fuzzy LP problem by traditional methods. Accordingly, Wang et al. [47] have introduced a fuzzy arithmetic approach that works on the FDEA fractional model. Bhardwaj et al. [7] mentioned the drawbacks of the method of Wang et al. [47] and proposed a modified model to address those problems. Azar et al. [5] developed the proposed model of Wang et al. [47] to obtain a common set of weights for solving the FDEA model. With this study, an additive model based on the common set of weights is developed for measuring the fuzzy efficiency of DMUs in the present of undesirable outputs. The main characteristics of the proposed approach are summarized as follows. (1) The proposed approach is not only beneficial from computation perspective by solving one LP model for all DMUs, but also equitably evaluate the efficiencies of all DMUs on the same scale by considering common set of weights. (2) In addition, to illustrate the validity of the proposed model, a simple numerical example and an application to Bank industry in Iran are presented to demonstrate the impact of non-performing assets/loans of undesirable outputs on the performance of banks and the influence of CSW and uncertainty in input and output data over the efficiency. (3) The proposed model has less computational complexity for calculating efficiency of DMUs compared to the Wang et al. [47] model. (4) In the proposed approach, one model is solved for all DMUs, while according to approach proposed by Wang et al. [47] for each DMU, three linear models are solved. (5) Also, the proposed model in contrast to Azar et al.’s [5] model considers the impact of undesirable outputs over the fuzzy efficiencies. (6) Finally, according to the proposed approach because of using fuzzy additive DEA model and finding CSW, less LP models are solved in comparison with the proposed approach by Puri and Yadav [32] on order to full ranking of DMUs.

Table 1 lists the characteristics of this method compared with other methods and analyzes the advantages and disadvantages.

Table 1
Advantages and disadvantages of different FDEA approaches

Article Approach Advantages Disadvantages

Saati et al. [36] Alpha Cut •Non considering undesirable factors

•Using classical ranking approach •High computational complexity

•Linearity of the model •Not finding CWS

•Not providing the exact membership function of fuzzy efficiency

Saati and Memariani [35] Alpha Cut •Non considering undesirable factors

•Finding CWS •High computational complexity

•Linearity of the model •Not providing the exact membership function of fuzzy efficiency

Puri and Yadav [32] Alpha Cut •High computational complexity

•Considering undesirable factors •Solving many LP problems

•Full Ranking of DMUs •Not Finding CWS

•Not providing the exact membership function of fuzzy efficiency

Bhardwaj et al [7] Fuzzy arithmetic •Non considering undesirable factors

•Linearity of the model •Solving three LP problems for each DMU

•Not Finding CWS

Azar et al. [5] Fuzzy arithmetic •Solving one LP for all DMUS

•Finding CWS •Non considering undesirable factors

•Less computational complexity •Using defuzzification approach

The Present Study Fuzzy arithmetic •Solving one LP for all DMUS

•Full Ranking of DMUs

•Less computational complexity

•Easy for implementation •Non considering uncontrollable factors

•Linearity and simplicity of the model

•Providing the exact membership function of fuzzy efficiency

Article	Approach	Advantages	Disadvantages
Saati et al. [36]	Alpha Cut		•Non considering undesirable factors
		•Using classical ranking approach	•High computational complexity
		•Linearity of the model	•Not finding CWS
			•Not providing the exact membership function of fuzzy efficiency
Saati and Memariani [35]	Alpha Cut		•Non considering undesirable factors
		•Finding CWS	•High computational complexity
		•Linearity of the model	•Not providing the exact membership function of fuzzy efficiency
Puri and Yadav [32]	Alpha Cut		•High computational complexity
		•Considering undesirable factors	•Solving many LP problems
		•Full Ranking of DMUs	•Not Finding CWS
			•Not providing the exact membership function of fuzzy efficiency
Bhardwaj et al [7]	Fuzzy arithmetic	•Non considering undesirable factors
		•Linearity of the model	•Solving three LP problems for each DMU
			•Not Finding CWS
Azar et al. [5]	Fuzzy arithmetic	•Solving one LP for all DMUS
		•Finding CWS	•Non considering undesirable factors
		•Less computational complexity	•Using defuzzification approach
The Present Study	Fuzzy arithmetic	•Solving one LP for all DMUS
		•Full Ranking of DMUs
		•Less computational complexity
		•Easy for implementation	•Non considering uncontrollable factors
		•Linearity and simplicity of the model
		•Providing the exact membership function of fuzzy efficiency

1.5 Organisation

The rest of the paper is organized as follows. Section 2 describes the additive DEA model with undesirable outputs where all input and output data are represented by fuzzy numbers. Section 3 discusses the common set of weight approach for solving FDEA models with undesirable outputs. In Section 4, a numerical example is provided to illustrate the applicability of the proposed technique.

Section 5 analyses the performance of 20 commercial banks in Iran for the period 2015–2017. Section 6 concludes and suggests future research directions.

2 Fuzzy additive DEA model with undesirable outputs

In this section, FDEA model is formulated in the presence of undesirable where all input and output data are represented by triangular fuzzy numbers [21, 32].

In real application of DEA models, the input and output data are estimated with crisp values. However, this is not always possible in the real world applications and in many cases the data is inaccurate and vague. For this reason, in this study, fuzzy set theory is used in DEA model and the input and output data are expressed in terms of fuzzy numbers. Without loss of generality, triangular fuzzy numbers are used here to represent fuzzy data [11 –15].

Definition 1. A fuzzy number $\tilde{A}$ , denoted by $\tilde{A} = (a^{L}, a^{M}, a^{U})$ , is called a triangular fuzzy number if its membership function is given as follows: $μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - a^{L}}{a^{M} - a^{L}}, a^{L} ⩽ x ⩽ a^{M}, \\ \frac{a^{U} - x}{a^{U} - a^{M}}, a^{M} ⩽ x ⩽ a^{U} . \end{matrix}$

Definition 2. A triangular fuzzy number $\tilde{A} = (a^{L}, a^{M}, a^{U})$ , is called a positive fuzzy number if a^L > 0.

Definition 3. The arithmetic operation on two positive fuzzy numbers $\tilde{A} = (a^{L}, a^{M}, a^{U})$ and $\tilde{B} = (b^{L}, b^{M}, b^{U})$ is defined follows: $\begin{matrix} \tilde{A} + \tilde{B} = (a^{L} + b^{L}, a^{M} + b^{M}, a^{U} + b^{U}), \\ \tilde{A} - \tilde{B} = (a^{L} - b^{U}, a^{M} - b^{M}, a^{U} - b^{L}), \\ t \tilde{A} = ({ta}^{L}, {ta}^{M}, {ta}^{U}), t ⩾ 0, \\ t \tilde{A} = ({ta}^{U}, {ta}^{M}, {ta}^{L}), t < 0, \\ \frac{\tilde{A}}{\tilde{B}} = (\frac{a^{L}}{b^{U}}, \frac{a^{M}}{b^{M}}, \frac{a^{U}}{b^{L}}), b^{L} > 0 . \end{matrix}$

Data envelopment analysis is a non-parametric programming technique to measure the relative efficiency of homogeneous decision making units that use multiple inputs to produce multiple outputs (desirable and undesirable). Let ${\tilde{X}}_{j} = ({\tilde{x}}_{1 j}, \dots, {\tilde{x}}_{ij}, \dots, {\tilde{x}}_{mj})^{T}$ and ${\tilde{Y}}_{j} = ({\tilde{Y}}_{j}^{g}, {\tilde{Y}}_{j}^{b})^{T}$ are the positive fuzzy input and fuzzy output data of the DMU_j (j = 1, …, n), respectively, where ${\tilde{Y}}_{j}^{g} = ({\tilde{y}}_{1 j}^{g}, \dots, {\tilde{y}}_{rj}^{g}, \dots, {\tilde{y}}_{s_{1} j}^{g})^{T}$ and ${\tilde{Y}}_{j}^{b} = ({\tilde{y}}_{1 j}^{b}, \dots, {\tilde{y}}_{pj}^{b}, \dots, {\tilde{y}}_{s_{2} j}^{b})^{T}$ (s₁ + s₂ = s) are the vector of the fuzzy desirable outputs and the vector of the fuzzy undesirable output, respectively. The fuzzy efficiency of DMU_k is defined as follows [32]: ${\tilde{E}}_{k} = \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} {\tilde{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} {\tilde{y}}_{pk}^{b}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ik}}$ (1) where $u_{r}^{g}$ (r = 1, . . . , s₁), $u_{p}^{b}$ (p = 1, . . . , s₂) and v_i (i = 1, . . . , m) are the weights for fuzzy desirable outputs, fuzzy undesirable outputs and fuzzy inputs, respectively. Puri and Yadav [32] proposed the following FDEA model to find the fuzzy efficiency of DMU_k: $\begin{matrix} max {\tilde{E}}_{k} = \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} {\tilde{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} {\tilde{y}}_{pk}^{b}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ik}} \\ s . t . 0 ⩽ E_{j} = \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} {\tilde{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} {\tilde{y}}_{pj}^{b}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij}} ⩽ 1, \forall j, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (2)

Here, ɛ is the non-Archimedean infinitesimal.

This model is called the fuzzy input-oriented CCR multiplier model with undesirable outputs. Based on input-oriented DEA model, an efficiency score is generated for a DMU by minimizing inputs with fixed outputs. Similarly, based on output-oriented DEA model, an efficiency score is generated for a DMU by maximizing inputs with fixed inputs. Charnes et al. [8] introduced the additive model which combines both the input-oriented and the output-oriented. By introducing fuzzy input–output data and considering undesirable outputs, to that model, the following model is formulated to measure the fuzzy efficiency of DMU_k: $\begin{matrix} max (\sum_{r = 1}^{s_{1}} u_{r}^{g} {\tilde{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} {\tilde{y}}_{pk}^{b}) - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ik} \\ s . t . 0 ⩽ \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} {\tilde{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} {\tilde{y}}_{pj}^{b}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij}} ⩽ 1, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (3)

3 Common-weights fuzzy additive DEA model

In this section, the common set of weights based approach proposed by Azar et al. [5] is extended to find fuzzy efficiency of all DMUs in the presence of undesirable outputs.

It is worthy to note that the aim of the fuzzy additive DEA model (3) is to allow each DMU to choose the full weight flexibility that sets itself in the most favorable light against the other DMUs. However, solving one separate model for each DMU is not only beneficial from computation perspective, but also using different sets of weights that their efficiencies unable to ranked on the same basis. Hence, by extension of the fuzzy additive DEA model (3), a new model is formulated to measure the fuzzy efficiency of all DMUs.

To this end, the following model is formulated to maximize the fuzzy efficiency of all DMUs (for more details in crisp environment see [46]): $\begin{matrix} max min_{j = 1, 2, . . ., n} {(\sum_{r = 1}^{s_{1}} u_{r}^{g} {\tilde{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} {\tilde{y}}_{pj}^{b}) - \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij}} \\ s . t . 0 ⩽ \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} {\tilde{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} {\tilde{y}}_{pj}^{b}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij}} ⩽ 1, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (4) Suppose ${\tilde{x}}_{ij} = (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})$ , ${\tilde{y}}_{rj}^{g} = (y_{rj}^{gL}, y_{rj}^{gM}, y_{rj}^{gU})$ and ${\tilde{y}}_{pj}^{b} = (y_{pj}^{bL}, y_{pj}^{bM}, y_{pj}^{bU})$ are triangular fuzzy numbers corresponding to fuzzy inputs, fuzzy desirable outputs, and fuzzy undesirable outputs, respectively. In this case, the fuzzy model (4) is changed to the following model: $\begin{matrix} max min_{j = 1, 2, . . ., n} {(\sum_{r = 1}^{s_{1}} u_{r}^{g} (y_{rj}^{gL}, y_{rj}^{gM}, y_{rj}^{gU}) - \sum_{p = 1}^{s_{2}} u_{p}^{b} (y_{pj}^{bL}, y_{pj}^{bM}, y_{pj}^{bU})) - \sum_{i = 1}^{m} v_{i} (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})} \\ s . t . \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} (y_{rj}^{gL}, y_{rj}^{gM}, y_{rj}^{gU}) - \sum_{p = 1}^{s_{2}} u_{p}^{b} (y_{pj}^{bL}, y_{pj}^{bM}, y_{pj}^{bU})}{\sum_{i = 1}^{m} v_{i} (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})} ⩽ 1, \forall j, \\ \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} (y_{rj}^{gL}, y_{rj}^{gM}, y_{rj}^{gU}) - \sum_{p = 1}^{s_{2}} u_{p}^{b} (y_{pj}^{bL}, y_{pj}^{bM}, y_{pj}^{bU})}{\sum_{i = 1}^{m} v_{i} (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})} ⩾ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (5) Regarding Definition 3, the fuzzy model (5) is reformulated as follows: $\begin{matrix} max min_{j = 1, 2, . . ., n} {\begin{matrix} (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} - \sum_{i = 1}^{m} v_{i} x_{ij}^{U}), (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM} - \sum_{i = 1}^{m} v_{i} x_{ij}^{M}), \\ (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L}) \end{matrix}} \\ s . t . \\ (\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{U}}, \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{M}}, \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{L}}) ⩽ 1, \forall j, \\ (\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{U}}, \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{M}}, \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{L}}) ⩾ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (6) For model (6), as long as $\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{L}}$ is kept to be less than or equal to one, then $\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{M}}$ and $\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{U}}$ will be automatically satisfied. In a similar way, as long as $\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{U}}$ is kept to be more than or equal to zero, then $\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{L}}$ and $\frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{M}}$ will be spontaneously satisfied. Hence, model (6) is simplified into the following model: $\begin{matrix} max min_{j = 1, 2, . . ., n} {\begin{matrix} (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} - \sum_{i = 1}^{m} v_{i} x_{ij}^{U}), (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM} - \sum_{i = 1}^{m} v_{i} x_{ij}^{M}), \\ (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L}) \end{matrix}} \\ s . t . \\ \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{L}} ⩽ 1, \forall j, \\ \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU}}{\sum_{i = 1}^{m} v_{i} x_{ij}^{U}} ⩾ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (7) Therefore, model (7) is converted to the following model: $\begin{matrix} max min_{j = 1, 2, . . ., n} {\begin{matrix} (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} - \sum_{i = 1}^{m} v_{i} x_{ij}^{U}), (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM} - \sum_{i = 1}^{m} v_{i} x_{ij}^{M}), \\ (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L}) \end{matrix}} \\ s . t . \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L} ⩽ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} ⩾ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (8)

To convert model (8) into a linear programing problem, let $θ = min_{j = 1, 2, . . ., n} {\begin{matrix} (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} - \sum_{i = 1}^{m} v_{i} x_{ij}^{U}), (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM} - \sum_{i = 1}^{m} v_{i} x_{ij}^{M}), \\ (\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L}) \end{matrix}}$ (9)

Hence, model (8) is converted to the following linear model: $\begin{matrix} max θ \\ s . t . \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} - \sum_{i = 1}^{m} v_{i} x_{ij}^{U} ⩾ θ, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM} - \sum_{i = 1}^{m} v_{i} x_{ij}^{M} ⩾ θ, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L} ⩾ θ, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L} ⩽ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} ⩾ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (10)

In model (10), as long as $\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} - \sum_{i = 1}^{m} v_{i} x_{ij}^{U}$ is kept to be more than or equal to θ, then $\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gM} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bM} - \sum_{i = 1}^{m} v_{i} x_{ij}^{M} ⩾ θ$ and $\sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L} ⩾ θ$ will be spontaneously satisfied. Hence, model (10) is simplified into the following model: $\begin{matrix} max θ \\ s . t . \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} - \sum_{i = 1}^{m} v_{i} x_{ij}^{U} ⩾ θ, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gU} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bL} - \sum_{i = 1}^{m} v_{i} x_{ij}^{L} ⩽ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{gL} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{bU} ⩾ 0, \forall j, \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} ⩾ 1, \\ \sum_{p = 1}^{s_{2}} u_{p}^{b} ⩾ 1, \\ \sum_{i = 1}^{m} v_{i} ⩾ 1, \\ u_{r}^{g}, u_{p}^{b}, v_{i} ⩾ ɛ, \forall r, p, i . \end{matrix}$ (11)

The optimal solution of model (11), i.e., $v_{i}^{*}$ , $u_{r}^{g *}$ and $u_{p}^{b *}$ is the CSW and is used to find the fuzzy efficiency of all DMUs. In this case, the fuzzy efficacy of each DMU is obtained based on the following formulation: ${\tilde{E}}_{j}^{*} = \frac{\sum_{r = 1}^{s_{1}} u_{r}^{g *} {\tilde{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b *} {\tilde{y}}_{pj}^{b}}{\sum_{i = 1}^{m} v_{i}^{*} {\tilde{x}}_{ij}}$ (12)

After computing the fuzzy efficiency of all DMUs based on formula (12), the ranking of each DMU is given based on the ranking of fuzzy efficiency values proposed by Wang et al. [47]. According to this approach, first, a preference degree is defined based on the ranking approach for comparing the fuzzy efficiency of decision making units. Then, the matrix of preference degree is formed using the obtained degrees leading an algorithm for ranking of all DMUs. Figure 1 sums up the proposed procedure in this study using three structured successive phases.

Fig. 1

The proposed method.

It is worthy to note that to compute the efficiency using Wang et al. [47] we need to run the model 3n times, three times for each DMU, while according to our proposed approach only one model is solved. Moreover, our proposed approach considers both undesirable and desirable outputs in the evaluation process, while in Wang et al.’s [47] models does not consider the impact of undesirable outputs over the fuzzy efficiency. Hence, our proposed approach is preferred to the Wang et al.’s [47] approach from the computational point of view. Also, for calculating the fuzzy efficiency using Puri and Yadav’s [32] approach, it is needed to run the model nα times, α times for each DMU at a given α- cut, while in the proposed model we need to solve the model once only. Hence, the proposed approach needs less computational effort in comparison with Puri and Yadav’s [32] approach. Finally, the approach proposed in this paper, in contrast to the approaches proposed by Puri and Yadav [32] and Wang et al. [47], uses a common set of weight to solve fuzzy data envelopment analysis model in the presence of undesirable outputs. The results of the proposed approach are therefore more acceptable to decision makers. Hence, the main contributions of the proposed approach and the advantageous of the proposed method over the compared methods can also be summarized as follows:

The proposed approach is not only beneficial from computation perspective by solving one LP model for all DMUs, but also equitably evaluate the efficiencies of all DMUs on the same scale by considering common set of weights.

The proposed model has less computational complexity for calculating efficiency of DMUs compared to the existing models [47].

In the proposed approach, one model is solved for all DMUs, while according to approach proposed by Wang et al. [47] for each DMU, three linear models are solved.

The proposed model in contrast to Azar et al.’s [5] model considers the impact of undesirable outputs over the fuzzy efficiencies.

According to the proposed approach due to using fuzzy additive DEA model and finding CSW less linear programming models are solved in comparison with Puri and Yadav’s method [32] in order to full ranking of DMUs.

Fuzzy efficiency measures which provide ranges of flexibility to decision maker(s) can be provided by using the proposed approach instead of obtaining crisp efficiency measures.

Proposed method does not require the solution of a non-linear program (As in the method of Kao and Liu [24]) since the objective function and constraints of the proposed model are linear functions.

Kao and Liu’s [24] method needs to solve two distinct mathematical programming models for each DMU at a given α- cut. In sum, this method solves 2nα models to find the efficiency measures of all DMUs. However, in our proposed approach, one can obtain the fuzzy efficiency measures of all DMUs by solving a single LP model.

According to Kao and Liu’s [24] approach the exact function form of fuzzy efficiency is not known explicitly. However, our proposed approach provides the exact function form of the fuzzy efficiency measures of all DMUs explicitly.

4 Numerical example

In order to get the deep insight of the proposed methodology, consider a performance assessment problem of 12 DMUs with two fuzzy inputs, two fuzzy desirable (good) outputs and one fuzzy undesirable (bad) output. It should be noted that all inputs and outputs are represented by triangular fuzzy numbers as shown in Table 2.

Table 2
Input and output data for 12 DMUs

DMU I₁ I₂ O₁(good) O₂(good) O₁(bad)

1 (16,20,22) (150,151,152) (95,100,102) (87,90,94) (1,1,1)

2 (18,19,20) (130,131,132) (149,150,151) (46,50,52) (1,2,2.5)

3 (23,25,28) (158,160,162) (158,160,163) (53,55,56) (1.4,2,3)

4 (26,27,29) (165,168,169) (177,180,181) (70,72,75) (1,1,1)

5 (20,22,25) (155,158,162) (90,94,98) (63,66,68) (3,5,6)

6 (52,55,59) (250,255,259) (222,230,235) (83,90,95) (3,4,5)

7 (30,33,34) (234,235,236) (210,220,225) (81,88,90) (1,3,5)

8 (27,31,33) (202,206,208) (151152,155) (75,80,84) (4,5,6)

9 (26,30,35) (240,244,247) (188,190,193) (99,100,101) (2,5,7)

10 (47,50,54) (262,268,271) (246,250,252) (94,100,108) (1,3,4)

11 (50,53,56) (300,306,309) (255,260,264) (143,147,152) (4.5,5,5.6)

12 (30,38,42) (283,284,285) (246,250,254) (116,120,123) (2,3,4)

DMU	I₁	I₂	O₁(good)	O₂(good)	O₁(bad)
1	(16,20,22)	(150,151,152)	(95,100,102)	(87,90,94)	(1,1,1)
2	(18,19,20)	(130,131,132)	(149,150,151)	(46,50,52)	(1,2,2.5)
3	(23,25,28)	(158,160,162)	(158,160,163)	(53,55,56)	(1.4,2,3)
4	(26,27,29)	(165,168,169)	(177,180,181)	(70,72,75)	(1,1,1)
5	(20,22,25)	(155,158,162)	(90,94,98)	(63,66,68)	(3,5,6)
6	(52,55,59)	(250,255,259)	(222,230,235)	(83,90,95)	(3,4,5)
7	(30,33,34)	(234,235,236)	(210,220,225)	(81,88,90)	(1,3,5)
8	(27,31,33)	(202,206,208)	(151152,155)	(75,80,84)	(4,5,6)
9	(26,30,35)	(240,244,247)	(188,190,193)	(99,100,101)	(2,5,7)
10	(47,50,54)	(262,268,271)	(246,250,252)	(94,100,108)	(1,3,4)
11	(50,53,56)	(300,306,309)	(255,260,264)	(143,147,152)	(4.5,5,5.6)
12	(30,38,42)	(283,284,285)	(246,250,254)	(116,120,123)	(2,3,4)

According to the proposed approach, model (11) is solved to determine the CSW. The results are presented in Table 3.

Table 3

Common set of weights

	$v_{1}^{*}$	$v_{2}^{*}$	$u_{1}^{g *}$	$u_{2}^{g *}$	$u_{1}^{b *}$
CSW	0.406	0.594	0.326	0.674	1

The efficiency values evaluated using Equation (12) is shown in Table 4. Table 4 shows that DMU 1 and DMU 4 together define an efficient frontier, and DMU 4 is the one with best performance. The fuzzy efficiency of each decision making unit is shown graphically in Fig. 2.

Table 4

Fuzzy efficiencies of DMUs

DMUs	Fuzzy efficiency
1	(0.893, 0.943, 1.000)
2	(0.891, 0.942, 0.985)
3	(0.783, 0.829, 0.867)
4	(0.926, 0.959, 1.000)
5	(0.619, 0.682, 0.746)
6	(0.694, 0.757, 0.811)
7	(0.767, 0.837, 0.880)
8	(0.685, 0.730, 0.788)
9	(0.752, 0.791, 0.842)
10	(0.763, 0.812, 0.881)
11	(0.843, 0.880, 0.927)
12	(0.828, 0.866, 0.908)

Fig. 2

Fuzzy efficiencies of DMUs.

Table 5 shows the matrix of degree of preference calculated using the Wang et al.’s [47] method for the fuzzy efficiency presented in Table 4.

Table 5

Matrix of degree of preference for fuzzy efficiencies and their rankings

DMUs	1	2	3	4	5	6	7	8	9	10	11	12	Rank
1	–	0.5472	1.0000	0.3362	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9376	0.9936	2
2	0.4528	–	1.0000	0.2726	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9277	0.9821	3
3	0.0000	0.0000	–	0.0000	1.0000	0.9610	0.4700	0.9987	0.7938	0.5866	0.0457	0.1220	7
4	0.6639	0.7274	1.0000	–	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9999	1.0000	1
5	0. 0000	0.0000	0.0000	0.0000	–	0.8730	0.0000	0.1484	0.0000	0.0000	0.0000	0.0000	12
6	0.0000	0.0000	0.0415	0.0000	0.9127	–	0.0679	0.6681	0.1808	0.0952	0.0000	0.0000	10
7	0.0000	0.0000	0.5300	0.0000	0.9356	0.7378	–	0.8153	0.7710	0.5953	0.0869	0.1730	6
8	0.0000	0.0000	0.0013	0.0000	0.8516	0.3319	0.0160	–	0.0692	0.0264	0.0000	0.0000	11
9	0.0000	0.0000	0.2062	0.0000	1.0000	0.8192	0.2290	0.9308	–	0.3000	0.0000	0.0130	9
10	0.0000	0.0000	0.4134	0.0000	1.0000	0.9048	0.4047	0.9736	0.7000	–	0.0674	0.1326	8
11	0.6018	0.0743	0.9543	0.0001	1.0000	1.0000	0.9131	1.0000	1.0000	0.9326	–	0.6739	4
12	0.0131	0.1790	0.8780	0.0000	1.0000	1.0000	0.8270	1.0000	0.9855	0.8674	0.3261	–	5

From Table 5, 12 decision making units are fully ranked according to their fuzzy efficiency as $4 \overset{66.39 %}{≻} 1 \overset{54.72 %}{≻} 2 \overset{92.77 %}{≻} 11 \overset{67.39 %}{≻} 12 \overset{82.70 %}{≻} 7 \overset{53.00 %}{≻}$ $3 \overset{58.66 %}{≻} 10 \overset{70.00 %}{≻} 9 \overset{81.92 %}{≻} 6 \overset{66.81 %}{≻} 8 \overset{85.16 %}{≻} 5$ . Here, $4 \overset{66.39 %}{≻} 1$ means that DMU 4 performs better than DMU 1 to the extent of 66.39%. In a similar way, from $2 \overset{92.77 %}{≻} 11$ it can be concluded that DMU 2 performs better than DMU 11 to the extent of 92.77 %. Hence, based on this approach, not only the complete ranking of DMUs is given, but also the degree of preference of each DMU to other DMUs is determined.

For comparison purpose, we have also listed the ranking results of Puri and Yadav [32] in Table 6. Puri and Yadav [32] used the method of Saati [34] and the α-cut approach to calculate the efficiency of DMUs. They then ranked DMUs using the cross-efficiency technique. It can be seen that from Table 6 that the ranking results of the proposed method and Puri and Yadav [32] are somewhat similar. However, for several reasons the proposed approach is superior to Puri and Yadav’s [32] method. First, according to approach of Puri and Yadav [32], 120 models must be solved to compare and rank of given 12 DUMs. While based on the proposed approach, the efficiencies of all DMUs are given by solving only one linear programming model. Second, according to the method of Puri and Yadav [32] the values of efficiency are determined numerically at given cut levels. The means that for each given level a crisp value is obtained as the value of efficiency of DMU under consideration and the mathematical form of fuzzy efficiency cannot be obtained. But, based on the proposed approach in this study for each DMU the mathematical form of fuzzy efficiency is given by formula (12). Third, in contrast the approach proposed by Puri and Yadav [32], the proposed approach in this study uses a set of unique weights for evaluating all the DMUs.

Table 6

The comparison results

The proposed CWS approach		4>1>2>11>12>7>3>10>9>6>8>5
Puri and Yadav [32] for different levels	0.1	4>2>12>1>7>9>3>11>10>8>5>6
	0.2	4>2>12>7>1>9>3>11>10>8>5>6
	0.3	2>4>12>1>7>9>3>11>10>8>5>6
	0.4	2>4>12>1>7>9>11>3>10>8>6>5
	0.5	2>4>12>1>9>7>11>3>10>6>8>5
	0.6	2>4>12>1>7>9>11>3>10>6>8>5
	0.7	2>4>12>1>9>7>11>3>10>6>8>5
	0.8	2>4>1>12>9>11>7>3>10>6>8>5
	0.9	2>4>1>12>9>11>7>3>10>6>8>5
	1	2>4>1>12>9>11>7>3>10>6>8>5

5 Application in banking industry

In this section, we analyze the performance of 20 commercial banks in Iran for the period 2015–2017.

For evaluating the performance of these banks, personnel rate and total of deposits are considered as input data, received interest and fee are considered as desirable output data and nonperforming loans are undesirable output data. The concepts of two inputs used in this study are as follows:

Personnel rate: it is the weighted combination of personal qualifications, quantity, education and others.

Total of deposits (TDs): It is total deposits of current, short duration and long duration accounts.

Two desirable outputs used in our study are as given below:

Received Interest (RI): It is the interest received from loans paid by banks.

Other income (OI): It accounts for the income from off-balance sheet items such as commission, exchange and brokerage etc.

Moreover, the one undesirable output in this study is Non-performing assets/loans (NPAs) defined as follows:

Non-performing assets/loans (NPAs): it is the delay in delivering loans and other facilities.

This input–output data of each bank are available in crisp form. However, there always exist some degrees of uncertainty which may affect the efficiency results of the banks. Such uncertainty occurs due to the difference between the actual data and the available data and or risks factor behind loans. To deal with the above mentioned bank performance evaluation problem, there is a need for more realistic approach other than the crisp DEA. Therefore, in this study, crisp DEA is extend to fuzzy DEA in which the input/output data of the banks are represented by fuzzy numbers, in particular triangular fuzzy number.

Remark 1: In the present study, triangular membership functions are chosen to describe fuzziness of all input-output data. Triangular membership functions are the simplest to use in fuzzy computations and fuzzy arithmetic operations. In detail, usage of triangular membership functions may provide an appropriate representation of domain expert knowledge and also reduces the computational complexity. Moreover, triangular membership functions are more appropriate when the membership function of a fuzzy set is not known precisely.

Remark 2: The triangular membership functions of a real-world DEA data can be constructed by two common ways. At first, triangular fuzzy numbers related to the uncertain input-output data can be generated with the help of bank experts and managers. For instance, optimistic, most likely and pessimistic values of the fuzzy data can be obtained from bank experts based on their knowledge and experiences. Secondly, triangular fuzzy numbers for the uncertain input-output data can also be generated from actual numerical data by using several data driven approaches like; histogram analysis, clustering algorithms, decision tables, heuristic methods, neural networks, genetic algorithms etc.

Remark 3: In the present study, the collected data from banks are represented by a^M and the corresponding triangular fuzzy number is represented by (a^L, a^M, a^U). For inputs, a^L and a^U are calculated by subtracting 0.5 percent of a^M from a^M and by adding 0.75 percent of a^M to a^M, respectively. Further, for outputs, a^L and a^U are calculated by subtracting one percent of a^M from a^M and by adding 0.75 percent of a^M to a^M, respectively. It is worth noting that these values have been determined after discussions with banks managers, as well as other experts in the field.

The fuzzy data created based on the data of 2015, 2016 and 2017 to deal with imprecise is shown in Table 7.

Table 7
Input and output data of commercial banks for the period 2015–2017

DMU Year Input 1 Personnel rate Input 2 Desirable output 1 Desirable output 2 Undesirable output

Total of deposits Received interest Other income Non-performing loans

1 2015 (37.81,38,38.285) (2817685,2831844,2853083) (118303,119498,120394.2) (1051.38,1062,1069.965) (99325.71,100329,101081.5)

2016 (40.795,41,41.3075) (2958569.82,2973437,2995738) (130133.5,131448,132433.9) (1156.32,1168,1176.76) (114224.2,115378,116243.3)

2017 (42.785,43,43.3225) (3195254.45,3211311,3235396) (156159.6,157737,158920) (187.98,1402,1412.515) (119935.5,121147,122055.6)

2 2015 (18.905,19,19.1425) (355608,357395,360075.5) (35279.64,35636,35903.27) (331.65,335,337.5125) (11321.64,11436,11521.77)

2016 (19.9,20,20.15) (373388.675,375265,378079.5) (38807.01,39199,39492.99) (364.32,368,370.76) (13020.48,13152,13250.64)

2017 (20.895,21,21.1575) (403259.57,405286,408325.6) (46568.61,47039,47391.79) (437.58,442,445.315) (13670.91,13809,13912.57)

3 2015 (26.865,27,27.2025) (1073528,1078923,1087015) (105460.7,106526,107324.9) (2487.87,2513,2531.848) (54163.89,54711,55121.33)

2016 (28.855,29,29.2175) (1127204.66,1132869,1141366) (116007.2,117179,118057.8) (2736.36,2764,2784.73) (62287.83,62917,63388.88)

2017 (29.85,30,30.225) (1217380.51,1223498,1232674) (139208.9,140615,141669.6) (3283.83,3317,3341.878) (65402.37,66063,66558.47)

4 2015 (20.895,21,21.1575) (1196484,1202496,1211515) (35243.01,35599,35865.99) (225.72,228,229.71) (21963.15,21185,22351.39)

2016 (22.885,23,23.1725) (1256307.9,51262621,1272091) (37766.42,39158,39451.69) (248.49,251,252.8825) (25257.87,25513,25704.35)

2017 (23.88,24,24.18) (1356812.85,1363631,1373858) (46520.1,46990,47342.43) (297.99,301,303.2575) (26521.11,26789,26989.92)

5 2015 (14.925,15,15.1125) (408394.8,410447,413525.4) (13788.72,13928,14032.46) (68.31,69,69.5175) (3902.58,3942,3971.565)

2016 (15.92,16,16.12) (428814.155,430969,434201.3) (15167.79,15321,15435.91) (75.24,76,76.57) (4487.67,4533,4566.998)

2017 (15.92,16,16.12) (463119.765,465447,468937.9) (18202.14,18386,18523.9) (90.09,91,91.6825) (4712.4,4760,4795.7)

6 2015 (19.9,20,20.15) (1032332,1037520,1045301) (58360.5,58950,59392.13) (522.72,528,531.96) (99325.71,100329,101081.5)

2016 (21.89,22,22.165) (1083950.02,1089397,1097567) (64196.55,64845,65331.34) (574.2,580,584.35) (104508.4,105564,106355.7)

2017 (22.885,23,23.1725) (1170665.26,1176548,1185372) (77035.86,77814,78397.61) (690.03,697,702.2275) (109733.6,110842,111673.3)

7 2015 (28.855,29,29.2175) (120047.7,120651,121555.9) (44112.42,44558,44892.19) (192.06,194,195.455) (31680.99,32001,32241.01)

2016 (30.845,31,31.2325) (126049.585,126683,127633.1) (48523.86,49014,49381.61) (210.87,213,214.5975) (36432.99,36801,37077.01)

2017 (31.84,32,32.24) (136133.91,136818,137844.1) (58228.83,58817,59258.13) (253.44,256,257.92) (38254.59,38641,38930.81)

8 2015 (24.875,25,25.1875) (426428.1,428571,431785.3) (61267.14,61886,62350.15) (5068.8,5120,5158.4) (10325.7,10430,10508.23)

2016 (26.865,27,27.2025) (447749.005,449999,453374) (67393.26,68074,68584.56) (5575.68,5632,5674.24) (11875.05,11995,12084.96)

2017 (28.855,29,29.2175) (483569.005,485999,489644) (80872.11,81689,82301.67) (6690.42,6758,6808.685) (12468.06,12594,12688.46)

9 2015 (19.9,20,20.15) (381557.6,383475,386351.1) (200873,202902,204423.8) (637.56,644,648.83) (44869.77,45323,45662.92)

2016 (20.895,21,21.1575) (412082.235,414153,417259.1) (231003.6,233337,235087) (732.6,740,745.55) (49356.45,49855,50228.91)

2017 (21.89,22,22.165) (428565.405,430719,433949.4) (254104.3,256671,258596) (806.85,815,821.1125) (56759.67,5733,57763)

10 2015 (13.93,14,14.105) (753124.5,756909,762585.8) (754511.7,762133,767849) (66.33,67,67.5025) (9405.99,9501,9572.258)

2016 (14.925,15,15.1125) (813374.69,817462,823593) (867689.5,876454,883027.4) (76.23,77,77.5775) (10346.49,10451,10529.38)

2017 (14.925,15,15.1125) (845909.2,850160,856536.2) (954458,964099,971329.7) (84.15,85,85.6375) (11898.81,12019,12109.14)

11 2015 (19.9,20,20.15) (689548.9,693014,698211.6) (19554.44,20156,20307.17) (1218.69,1231,1240.233) (544.5,550,554.125)

2016 (20.895,21,21.1575) (744712.725,748455,754068.4) (22948.2,23180,23353.85) (1401.84,1416,1426.62) (598.95,605,609.5375)

2017 (21.89,22,22.165) (774501.035,778393,784230.9) (25243.02,25498,25689.24) (1541.43,1557,1568.678) (688.05,695,700.2125)

12 2015 (21.89,22,22.165) (10158376,10209423,10285994) (40709.79,41121,41429.41) (426.69,431,434.2325) (17584.38,17762,17895.22)

2016 (22.885,23,23.1725) (10971046.1,11026177,11108873) (46816.11,47289,47643.67) (491.04,496,499.72) (19343.61,19539,19685.54)

2017 (23.88,24,24.18) (11409888.9,11467225,11553229) (51497.82,52018,52408.14) (539.55,545,549.0875) (22244.31,22469,22637.52)

13 2015 (17.91,18,18.135) (594432.9,597420,601900.7) (51975,52500,52893.75) (498.96,504,507.78) (18690.21,18879,19020.59)

2016 (18.905,19,19.1425) (641987.93,645214,650053.1) (59771.25,60375,60827.81) (53.21,579,583.3425) (20559.33,20767,20922.75)

2017 (19.9,20,20.15) (667666.89,671022,676054.7) (65747.88,66412,66910.09) (630.63,637,641.7775) (23643.18,23882,24061.12)

14 2015 (21.89,22,22.165) (332770.8,334443,336951.3) (9271.35,9365,9435.238) (1066.23,1077,1085.078) (19511.91,19709,19856.82)

2016 (22.885,23,23.1725) (359393.005,361199,363908) (10662.3,10770,10850.78) (1225.62,1238,1247.285) (21463.2,21680,21842.6)

2017 (23.88,24,24.18) (373768.765,375647,378464.4) (11728.53,11847,11935.85) (1348.38,1362,1372.215) (24682.68,24932,25118.99)

15 2015 (39.8,40,40.3) (307494.8,309040,311357.8) (121589.8,12818,123739.1) (1370.16,1384,1394.38) (56172.6,56740,57165.55)

2016 (41.79,42,42.315) (332095.18,333764,336267.2) (139828.6,141241,142300.3) (1576.08,1592,1603.94) (61789.86,62414,62882.11)

2017 (43.78,44,44.33) (345378.43,347114,349717.4) (153811.4,155365,156530.2) (1733.49,1751,1764.133) (71058.24,71776,72314.32)

16 2015 (25.87,26,26.195) (332620.5,334292,336799.2) (106885.4,107965,108774.7) (237.6,240,241.8) (21802.77,22023,22188.17)

2016 (26.865,27,27.2025) (345925.68,347664,350271.5) (128262.4,129558,130529.7) (285.12,288,290.16) (22892.76,23124,23297.43)

2017 (28.855,29,29.2175) (363221.765,365047,367784.9) (147502.1,148992,150109.4) (327.69,331,333.4825) (25182.63,25437,25627.78)

17 2015 (27.86,28,28.21) (639614.9,642829,647650.2) (39247.56,39644,3941.33) (3303.63,3337,3362.028) (43617.42,44058,44388.44)

2016 (28.855,29,29.2175) (665199.29,668542,673556.1) (47096.28,47572,47928.79) (3963.96,4004,4034.03) (45797.4,46260,46606.95)

2017 (31.84,32,32.24) (698459.155,701969,707233.8) (54160.92,54708,55118.31) (4558.95,4605,4639.538) (50377.14,50886,51267.65)

18 2015 (12.935,13,13.0975) (256759.8,258050,259985.4) (11315.7,11430,11515.73) (285.12,288,290.16) (19377.27,19573,19719.8)

2016 (13.93,14,14.105) (267030.14,268372,270384.8) (13578.84,13716,13818.87) (341.55,345,347.5875) (20345.49,20551,20705.13)

2017 (14.925,15,15.1125) (280381.05,281790,283903.4) (15616.26,15774,15892.31) (393.03,397,3999.9775) (22379.94,22606,22775.55)

19 2015 (27.86,28,28.21) (663192.4,666525,671523.9) (78522.84,79316,79910.87) (1141.47,1153,1161.648) (20270.25,20475,20628.56)

2016 (28.855,29,29.2175) (689720.07,693186,698384.9) (94227.21,95179,95892.84) (1370.16,1384,1394.38) (21283.02,21498,21659.24)

2017 (31.84,32,32.24) (724205.775,727845,733303.8) (108361.4,109456,110276.9) (1576.08,1592,1603.94) (23411.52,23648,23825.36)

20 2015 (28.855,29,29.2175) (124121.3,124745,125680.6) (26125.11,26389,26586.92) (335.61,339,341.4525) (32952.15,33285,33534.64)

2016 (29.85,30,30.225) (129086.325,129735,130708) (31350.33,31667,31904.5) (402.93,407,410.0525) (34599.51,34949,35211.12)

2017 (32.835,33,33.2475) (135540.89,136222,137243.7) (36052.83,36417,36690.13) (463.32,468,471.41) (38059.56,38444,38732.33)

DMU	Year	Input 1 Personnel rate	Input 2	Desirable output 1	Desirable output 2	Undesirable output
1	2015	(37.81,38,38.285)	(2817685,2831844,2853083)	(118303,119498,120394.2)	(1051.38,1062,1069.965)	(99325.71,100329,101081.5)
	2016	(40.795,41,41.3075)	(2958569.82,2973437,2995738)	(130133.5,131448,132433.9)	(1156.32,1168,1176.76)	(114224.2,115378,116243.3)
	2017	(42.785,43,43.3225)	(3195254.45,3211311,3235396)	(156159.6,157737,158920)	(187.98,1402,1412.515)	(119935.5,121147,122055.6)
2	2015	(18.905,19,19.1425)	(355608,357395,360075.5)	(35279.64,35636,35903.27)	(331.65,335,337.5125)	(11321.64,11436,11521.77)
	2016	(19.9,20,20.15)	(373388.675,375265,378079.5)	(38807.01,39199,39492.99)	(364.32,368,370.76)	(13020.48,13152,13250.64)
	2017	(20.895,21,21.1575)	(403259.57,405286,408325.6)	(46568.61,47039,47391.79)	(437.58,442,445.315)	(13670.91,13809,13912.57)
3	2015	(26.865,27,27.2025)	(1073528,1078923,1087015)	(105460.7,106526,107324.9)	(2487.87,2513,2531.848)	(54163.89,54711,55121.33)
	2016	(28.855,29,29.2175)	(1127204.66,1132869,1141366)	(116007.2,117179,118057.8)	(2736.36,2764,2784.73)	(62287.83,62917,63388.88)
	2017	(29.85,30,30.225)	(1217380.51,1223498,1232674)	(139208.9,140615,141669.6)	(3283.83,3317,3341.878)	(65402.37,66063,66558.47)
4	2015	(20.895,21,21.1575)	(1196484,1202496,1211515)	(35243.01,35599,35865.99)	(225.72,228,229.71)	(21963.15,21185,22351.39)
	2016	(22.885,23,23.1725)	(1256307.9,51262621,1272091)	(37766.42,39158,39451.69)	(248.49,251,252.8825)	(25257.87,25513,25704.35)
	2017	(23.88,24,24.18)	(1356812.85,1363631,1373858)	(46520.1,46990,47342.43)	(297.99,301,303.2575)	(26521.11,26789,26989.92)
5	2015	(14.925,15,15.1125)	(408394.8,410447,413525.4)	(13788.72,13928,14032.46)	(68.31,69,69.5175)	(3902.58,3942,3971.565)
	2016	(15.92,16,16.12)	(428814.155,430969,434201.3)	(15167.79,15321,15435.91)	(75.24,76,76.57)	(4487.67,4533,4566.998)
	2017	(15.92,16,16.12)	(463119.765,465447,468937.9)	(18202.14,18386,18523.9)	(90.09,91,91.6825)	(4712.4,4760,4795.7)
6	2015	(19.9,20,20.15)	(1032332,1037520,1045301)	(58360.5,58950,59392.13)	(522.72,528,531.96)	(99325.71,100329,101081.5)
	2016	(21.89,22,22.165)	(1083950.02,1089397,1097567)	(64196.55,64845,65331.34)	(574.2,580,584.35)	(104508.4,105564,106355.7)
	2017	(22.885,23,23.1725)	(1170665.26,1176548,1185372)	(77035.86,77814,78397.61)	(690.03,697,702.2275)	(109733.6,110842,111673.3)
7	2015	(28.855,29,29.2175)	(120047.7,120651,121555.9)	(44112.42,44558,44892.19)	(192.06,194,195.455)	(31680.99,32001,32241.01)
	2016	(30.845,31,31.2325)	(126049.585,126683,127633.1)	(48523.86,49014,49381.61)	(210.87,213,214.5975)	(36432.99,36801,37077.01)
	2017	(31.84,32,32.24)	(136133.91,136818,137844.1)	(58228.83,58817,59258.13)	(253.44,256,257.92)	(38254.59,38641,38930.81)
8	2015	(24.875,25,25.1875)	(426428.1,428571,431785.3)	(61267.14,61886,62350.15)	(5068.8,5120,5158.4)	(10325.7,10430,10508.23)
	2016	(26.865,27,27.2025)	(447749.005,449999,453374)	(67393.26,68074,68584.56)	(5575.68,5632,5674.24)	(11875.05,11995,12084.96)
	2017	(28.855,29,29.2175)	(483569.005,485999,489644)	(80872.11,81689,82301.67)	(6690.42,6758,6808.685)	(12468.06,12594,12688.46)
9	2015	(19.9,20,20.15)	(381557.6,383475,386351.1)	(200873,202902,204423.8)	(637.56,644,648.83)	(44869.77,45323,45662.92)
	2016	(20.895,21,21.1575)	(412082.235,414153,417259.1)	(231003.6,233337,235087)	(732.6,740,745.55)	(49356.45,49855,50228.91)
	2017	(21.89,22,22.165)	(428565.405,430719,433949.4)	(254104.3,256671,258596)	(806.85,815,821.1125)	(56759.67,5733,57763)
10	2015	(13.93,14,14.105)	(753124.5,756909,762585.8)	(754511.7,762133,767849)	(66.33,67,67.5025)	(9405.99,9501,9572.258)
	2016	(14.925,15,15.1125)	(813374.69,817462,823593)	(867689.5,876454,883027.4)	(76.23,77,77.5775)	(10346.49,10451,10529.38)
	2017	(14.925,15,15.1125)	(845909.2,850160,856536.2)	(954458,964099,971329.7)	(84.15,85,85.6375)	(11898.81,12019,12109.14)
11	2015	(19.9,20,20.15)	(689548.9,693014,698211.6)	(19554.44,20156,20307.17)	(1218.69,1231,1240.233)	(544.5,550,554.125)
	2016	(20.895,21,21.1575)	(744712.725,748455,754068.4)	(22948.2,23180,23353.85)	(1401.84,1416,1426.62)	(598.95,605,609.5375)
	2017	(21.89,22,22.165)	(774501.035,778393,784230.9)	(25243.02,25498,25689.24)	(1541.43,1557,1568.678)	(688.05,695,700.2125)
12	2015	(21.89,22,22.165)	(10158376,10209423,10285994)	(40709.79,41121,41429.41)	(426.69,431,434.2325)	(17584.38,17762,17895.22)
	2016	(22.885,23,23.1725)	(10971046.1,11026177,11108873)	(46816.11,47289,47643.67)	(491.04,496,499.72)	(19343.61,19539,19685.54)
	2017	(23.88,24,24.18)	(11409888.9,11467225,11553229)	(51497.82,52018,52408.14)	(539.55,545,549.0875)	(22244.31,22469,22637.52)
13	2015	(17.91,18,18.135)	(594432.9,597420,601900.7)	(51975,52500,52893.75)	(498.96,504,507.78)	(18690.21,18879,19020.59)
	2016	(18.905,19,19.1425)	(641987.93,645214,650053.1)	(59771.25,60375,60827.81)	(53.21,579,583.3425)	(20559.33,20767,20922.75)
	2017	(19.9,20,20.15)	(667666.89,671022,676054.7)	(65747.88,66412,66910.09)	(630.63,637,641.7775)	(23643.18,23882,24061.12)
14	2015	(21.89,22,22.165)	(332770.8,334443,336951.3)	(9271.35,9365,9435.238)	(1066.23,1077,1085.078)	(19511.91,19709,19856.82)
	2016	(22.885,23,23.1725)	(359393.005,361199,363908)	(10662.3,10770,10850.78)	(1225.62,1238,1247.285)	(21463.2,21680,21842.6)
	2017	(23.88,24,24.18)	(373768.765,375647,378464.4)	(11728.53,11847,11935.85)	(1348.38,1362,1372.215)	(24682.68,24932,25118.99)
15	2015	(39.8,40,40.3)	(307494.8,309040,311357.8)	(121589.8,12818,123739.1)	(1370.16,1384,1394.38)	(56172.6,56740,57165.55)
	2016	(41.79,42,42.315)	(332095.18,333764,336267.2)	(139828.6,141241,142300.3)	(1576.08,1592,1603.94)	(61789.86,62414,62882.11)
	2017	(43.78,44,44.33)	(345378.43,347114,349717.4)	(153811.4,155365,156530.2)	(1733.49,1751,1764.133)	(71058.24,71776,72314.32)
16	2015	(25.87,26,26.195)	(332620.5,334292,336799.2)	(106885.4,107965,108774.7)	(237.6,240,241.8)	(21802.77,22023,22188.17)
	2016	(26.865,27,27.2025)	(345925.68,347664,350271.5)	(128262.4,129558,130529.7)	(285.12,288,290.16)	(22892.76,23124,23297.43)
	2017	(28.855,29,29.2175)	(363221.765,365047,367784.9)	(147502.1,148992,150109.4)	(327.69,331,333.4825)	(25182.63,25437,25627.78)
17	2015	(27.86,28,28.21)	(639614.9,642829,647650.2)	(39247.56,39644,3941.33)	(3303.63,3337,3362.028)	(43617.42,44058,44388.44)
	2016	(28.855,29,29.2175)	(665199.29,668542,673556.1)	(47096.28,47572,47928.79)	(3963.96,4004,4034.03)	(45797.4,46260,46606.95)
	2017	(31.84,32,32.24)	(698459.155,701969,707233.8)	(54160.92,54708,55118.31)	(4558.95,4605,4639.538)	(50377.14,50886,51267.65)
18	2015	(12.935,13,13.0975)	(256759.8,258050,259985.4)	(11315.7,11430,11515.73)	(285.12,288,290.16)	(19377.27,19573,19719.8)
	2016	(13.93,14,14.105)	(267030.14,268372,270384.8)	(13578.84,13716,13818.87)	(341.55,345,347.5875)	(20345.49,20551,20705.13)
	2017	(14.925,15,15.1125)	(280381.05,281790,283903.4)	(15616.26,15774,15892.31)	(393.03,397,3999.9775)	(22379.94,22606,22775.55)
19	2015	(27.86,28,28.21)	(663192.4,666525,671523.9)	(78522.84,79316,79910.87)	(1141.47,1153,1161.648)	(20270.25,20475,20628.56)
	2016	(28.855,29,29.2175)	(689720.07,693186,698384.9)	(94227.21,95179,95892.84)	(1370.16,1384,1394.38)	(21283.02,21498,21659.24)
	2017	(31.84,32,32.24)	(724205.775,727845,733303.8)	(108361.4,109456,110276.9)	(1576.08,1592,1603.94)	(23411.52,23648,23825.36)
20	2015	(28.855,29,29.2175)	(124121.3,124745,125680.6)	(26125.11,26389,26586.92)	(335.61,339,341.4525)	(32952.15,33285,33534.64)
	2016	(29.85,30,30.225)	(129086.325,129735,130708)	(31350.33,31667,31904.5)	(402.93,407,410.0525)	(34599.51,34949,35211.12)
	2017	(32.835,33,33.2475)	(135540.89,136222,137243.7)	(36052.83,36417,36690.13)	(463.32,468,471.41)	(38059.56,38444,38732.33)

The common set of weights obtained from model (11) for 20 commercial banks in the period 2015–2017 are shown in Table 8.

Table 8

The results of CSW obtained by solving model (11)

	Years	$v_{1}^{*}$	$v_{2}^{*}$	$u_{1}^{g *}$	$u_{2}^{g *}$	$u_{1}^{b *}$
	2015	28838.59	0.001	0.525	134.810	1.000
CSW	2016	28495.91	0.001	0.483	131.252	1.000
	2017	27676.83	0.001	0.428	114.018	1.000

Fuzzy efficiency results based on formula (12) for 20 commercial banks in the period 2015–2017 are shown in Table 9. The fuzzy efficiency of each bank for the period 2015–2017 is graphically shown in Fig. 3, 4 and 5. Also, the ranking of 20 units in the period 2015–2017 is presented in Table 10. We used the ranking approach of Wang et al. [47] to complete ranking of all units.

Table 9

The results of fuzzy efficiencies

DMUs	2015	2016	2017
1	(0.093,0.096,0.099)	(0,083,0.087,0.089)	(0.086,0.089,0.092)
2	(0.094,0.096,0.097)	(0.093,0.095,0.096)	(0.095,0.098,0.099)
3	(0.427,0.436,0,443)	(0.422,0.431,0.438)	(0.439,0.448,0.455)
4	(0.043,0.045,0.046)	(0.039,0.040,0.041)	(0.040,0.042,0.043)
5	(0.029,0.030,0.031)	(0.027,0.028,0.029)	(0.030,0.031,0.032)
6	(0.008,0.011,0.014)	(0.000,0.003,0.006)	(0.000,0.003,0.006)
7	(0.020,0.021,0.022)	(0.016,0.017,0.018)	(0.017,0.018,0.019)
8	(0.970,0.987,1.000)	(0.970,0.987,1.000)	(0.970,0.988,1.000)
9	(0.251,0.256,0.261)	(0.261,0.267,0.272)	(0.233,0.239,0.243)
10	(0.970,0.988,1.000)	(0.970,0.988,1.000)	(0.969,0.986,0.999)
11	(0.299,0.305,0.309)	(0.322,0.328,0.332)	(0.303,0.308,0312)
12	(0.094,0.096,0.098)	(0.100,0.103,0.104)	(0.089,0.092,0.093)
13	(0.144,0.147,0.150)	(0.152,0.156,0.158)	(0.136,0.139,0.142)
14	(0.201,0.205,0.209)	(0.218,0.223,0.226)	(0.200,0.204,0.207)
15	(0.165,0.168,0.171)	(0.175,0.179,0.183)	(0.156,0.160,0.163)
16	(0.087,0.089,0.091)	(0.098,0.100,0.102)	(0.093,0.095,0.096)
17	(0.518,0.528,0.536)	(0.596,0.607,0.616)	(0.551,0.561,0.569)
18	(0.065,0.067,0.069)	(0.076,0.079,0.080)	(0.069,0.071,0.073)
19	(0.214,0.219,0.222)	(0.244,0.249,0.253)	(0.226,0.231,0.234)
20	(0.030,0.031,0.033)	(0.038,0.039,0.041)	(0.032,0.033,0.035)

Fig. 3

Results of fuzzy efficiency of banks in 2015.

Fig. 4

Results of fuzzy efficiency of banks in 2016.

Fig. 5

Results of fuzzy efficiency of banks in 2017.

Table 10

The full ranking

Year	Ranking of the Banks
2015	10> 8> 17>3>11>9>19>14>15>13>12>1>2>16>18>4>20>5>7>6
2016	10> 8> 17>3>11>9>19>14>15>13>12>16>2>1>18>4>20>5>7>6
2017	8>10>17>3>11>9>19>14>15>13>2>16>12>1>18>4>20>5>7>6

Table 9 shows that DMU8 is efficient in the period 2015–2017. This means that NPAs has not had a negative effect on the performance of this branch, and this branch acts as a benchmark for other branches. Fig. 3, 4 and 5 also show that DMU8 is efficient in the period 2015–2017 and DMU10 is efficient in 2015 and 2016, but it is inefficient in 2017. This means that even a small degree of uncertainty can make a bank efficient or inefficient. In general, Figs. 3 –5 show the results of the proposed model in the presence of NPAs of undesirable output and uncertainty in the data. It also shows the efficiency results using common set of weights in the period 2015–2017, and this shows how close the efficiency results are for each decision making unit in the period 2015–2017, and this can help bank experts make the right decisions to increase the efficiency of banks. Descriptive statistics of efficiency according to Table 9 shows that DMU10 is on average 0.1% more efficient than DMU8 in 2015 and 2016, but in 2017, DMU8 is on average 0.1% more efficient than DMU10. Figures 6 and 7 show the fuzzy efficiency of branches 3 and 17, respectively, during the years 2015–2017. According to Table 9, the efficiency of Branch 3 has improved in 2015 compared to 2016, and its efficiency in 2017 is better than in 2015 and 2016. The efficiency of Branch 9 in 2016 is better than the efficiency of 2015, and its efficiency in 2017 has improved compared to 2015 and 2016. The efficiency of Branch 4 is better in 2015 than in 2016 and 2017, and 2017 is better than in 2016. The efficiency of Branch 10 has been stable in 2015 and 2016 and has decreased in 2017. The efficiency of Branch 8 has been stable in 2015 and 2016 and has improved in 2017. On the other hand, according to Table 10, the changes in the ranking in the years 2015–2017 are such that the ranking of 5% of branches in 2016 has improved compared to 2015 and the ranking of 5% of branches in 2016 compared to 2015 has worsened and the ranking of 90% of branches in 2015 and 2016 have remained fixed. Also, by examining the years 2016 and 2017, we find that the ranking of 10% of branches in 2017 has improved compared to 2016, and the ranking of 10% of branches has decreased and the ranking of 80% of branches has unchanged. In addition, the comparison of the results for 2017 with 2015 shows that the ranking of 15% of branches in 2017 is better than in 2015, the ranking of 15% of branches is worse and the ranking of 70% of branches have unchanged.

Fig. 6

Fuzzy efficiency of Branch 3 during the years 2015–2017

Fig. 7

Fuzzy efficiency of Branch 17 of bank during 2015–2017

Further, Table 10 shows that DMU 10 is ranked first in 2015 and 2016, and this branch is ranked second in 2017. Moreover, DMU 8 is ranked second in 2015 and 2016, while in 2017 it is ranked first. Branch 6 is ranked last in all years from 2015 to 2017. Branches 18, 4, 20, 5 and 7 are also in the last ranks. Branch 1 ranked twelfth in 2015, while in 2016 and 2017, it ranked 14th, down two places. Branch 2 is ranked 13th in 2015 and 2016 and 11th in 2017 with an increase of one step. Branch 12 is also ranked 11th in 2015 and 2016, while in 2017 it is ranked 13th with two steps down. The ranks of branches 17, 3, 11, 9, 19, 14, 15 and 13 in the periods 2015–2017 have remained stable and are ranked third to tenth, respectively.

Finally, from Tables 7 and 9, it can be seen that Branch 6, despite having much current, short-term and long-term deposits, has earned little interest due to its high overdue receivables. Therefore, Branch 6 in the period 2015–2017, according to the obtained results, had a very poor performance compared to other branches. Also, the high overdue receivables in Branch 1 in 2015 have reduced its efficiency and a negative impact on the profit receivable of this branch. This branch performed worse in 2016 and 2017 due to higher overdue receivables. Branch 14 performed better in 2016 than in 2015, but in 2017 the non-performing loans of this branch increased significantly, which led to a decrease in the efficiency of that branch despite the increase in interest receivables, current, short-term and long-term deposits compared to 2015 and 2016. Branches 8 and 10 in 2015–2017 had better performance than other branches. The increase in the non-performing loans of Branch 10 in 2017 has led to a decrease in the efficiency of this branch compared to Branch 8. Branch 8 has also achieved the highest efficiency in 2017 by controlling overdue receivables and growth in receivables and current deposits, short-term and long-term. Therefore, we can conclude that the results obtained from the proposed model are vigorous and valuable to analyze the impact of undesirable output (non-performing loans) on banks for the selected period.

Reducing non-performing loans means reducing abnormal customers in the bank, and this means increasing the effectiveness of the banks’ lending facilities. Giving loans and facilities to real and legal customers can be considered as efficiency, but if these facilities lead to solving the problem and prosperity of production and economy, if this happens, customer can return the received facilities to the bank in due time. It means increasing the effectiveness of lending facilities and the performance of the bank.

But in performance evaluation of banks providing a crisp efficiency is not easy if not impossible due to several uncertainties as discussed in the paper; therefore, it may always be desirable to provide fuzzy efficiencies by taking into consideration decision maker’s risk attitude in such occasions. Such a fuzzy efficiency will provide possible outcomes with certain degree of memberships to the decision maker. This is especially useful for strategic decisions where more uncertainty exists.

6 Conclusions

The classical DEA model determines the relative efficiency of each DMU under evaluation based on the most desirable weight. Such flexibility in the selection of weights causes assigning small weight values for some of the important criteria that neglect them in evaluating the system. Therefore, one of the major challenges of data envelopment analysis models is the control of weights. On the other hand, the real-world data are ambiguous and inaccurate that fuzzy data can be used to model such uncertainty in DEA models. For this purpose, in this paper, after formulating fuzzy DEA model in the presence of undesirable outputs, the common set of weight was computed to find the fuzzy efficiency of each DMU. The proposed approach not only is more acceptable to decision makers because of using the common set of weight, but also fewer models were utilized to evaluate the relative efficiency of DMUs compared to existing approaches. We illustrated the advantages, potentials and applications of the model proposed in this paper by a numerical example taken from the literature and an application example to evaluate the performance of 20 commercial banks in Iran in the periods 2015, 2016 and 2017. According to obtained results, in 2016, 5% of the branches improved their efficiencies compared to 2015, 10% of the branches in 2017 improved their efficiencies compared to 2016, and also 15% of the branches improved their efficiencies in 2017 compared to 2015. For future study, it would be compelling to develop the proposed approach to find the CSW for FDEA models in the presence of both undesirable outputs and uncontrollable factors.

Footnotes

Acknowledgments

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

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