Fuzzy stochastic undesirable two-stage data envelopment analysis models with application to banking industry

Abstract

This paper organizes a two-stage DEA models by taking into account undesirable output with fuzzy stochastic data. A normal distribution with fuzzy component adopted for inputs, intermediate outputs, desirable and undesirable outputs. We propose, finally, a linear and feasible model in deterministic form. To achieve this aim, a possibility-probability approach is applied on a reform of two-stage DEA models occupied with undesirable outputs. A case study in the banking industry is presented to exhibit the efficacy of the procedures and demonstrate the applicability of the proposed model.

AMS Mathematics Subject Classification (2000): 62A86, 90C70

Keywords

Data envelopment analysis Two-stage Undesirable output Fuzzy random variable Efficiency score

1. Introduction

Traditional Data Envelopment Analysis (DEA), initially introduced by Charnes et al. [2], requires crisp input and output data, whereas real-life decisions are usually made in the state of uncertainty. In such situations, we often face the uncertain programming in DEA model, where in the data could possess randomness and fuzziness. On the other hand, some extensions of crisp DEA model in complex system are founded such as two-stage model and undesirable output model. These two progres in crisp and uncertainty DEA models need to handel together. In some circumstances, since the parameter distribution embraces both randomness and fuzziness. Hence, one variable apparently is not the best way to handle this sort of hybrid or twofold uncertainty. Two typical approaches namely probability-theoretic approach and fuzzy-theoretic approach can be used for DEA models involving uncertainty. In this paper, we propose a new DEA model, probability-possiblity approche, for solving CCR models in which the input and output data are assumed to be characterized by fuzzy random variables. We accomplish this task by converting the non-linear models formulated in these approaches to linear programming models. A case study for Bank Melli Iran (BMI) is presented to illustrate the features and applicability of the proposed DEA models. The reminder of the paper is organized as follows: In the next section, an overview of such inexact optimization techniques to tackle uncertainties in DEA models is provided. Section 3 presents our proposed two-stage model equiped with CCR-UO model. Section 4 gives the possibility-probablity approach based on chance constraint programming to solve the fuzzy stochastic model proposed in this section. In section 5, the results of the case conducted for the banking industry to evaluate the efficiency of 31 branches. Section 6 presents our conclusions and future research directions.

2. Literature review

Kwon and Lee [14] proposes a new approach to model a two-stage production process supported by using data from large U.S. banks. Halkos et al. (2014) provided a unified calssification of two-stage DEA model. Liu et al. [21] proposed a two-stage DEA models with undesirable inputâŁ“intermediate-outputs. Nasseri et al. [22] suggested a new ranking method based on the extension of PPS by virtual units named as relative similar units. Wu et al. [35] introduced a cross-efficiency approach based on pareto optimality which can be generated by only a common set of weights. Hanafizade et al. [8] used neuoral network DEA for measuring the efficiency of mutual funds. Hatami-Marbini et al. [9] classified the fuzzy DEA methods in the literature into five general groups, the tolerance approach [30], the level based approach, the fuzzy ranking approach (Guo and Tanaka 2011, Hatami-Marbini et al. 2011), the possibility approach [16], and the fuzzy arithmetic approach [34]. Saati et al. [28] proposed a fuzzy CCR model as a possibilistic programming problem by applying an alternative cut approach. Puri and Yadav [27] applied the suggested methodology by Saati et al. [28] to solve fuzzy DEA model with undesirable outputs. Khodabakhshi et al. [12] proposed a fuzzy DEA model with an optemistic and pessimistic performance and congestion analysis in fuzzy DEA. In random environments, the crips inputs and outputs of traditional DEA become random variable. Land et al. [15] extended the chance constrained DEA model. Olesen and Petersen [25] developed the chance constrained programming model for efficiency evaluation using a piecewise linear envelopment of confidence regions for observed stochastic multiple-input multiple-output combinations in DEA. Cooper et al. [4], Li [17], and Bruni et al. [1] utilized joint chance constraints to extend the concept of stochastic efficiency. Zhou et al. [38] proposed a stochastic centralized two-stage network DEA model converted to linear models under some assumptions. A review of stochastic DEA models can be found in a recent work by Olesen and Petersen (2015). In many cases, the stochastic programming techniques are not as suitable to cope with the optimization problems. If we impose the farfetched stochastic programming as the approach to the problem with randomness and fuzziness simultaneously, then we have to ignore the fuzziness to make an uncertainty reduction. Kwakernaak [13] introduced the concept of fuzzy random variable, and then this idea enhanced by a number of researchers in the literature [6 , 20](Qin and Liu 2010). Tavana et al. [33] also introduced three different fuzzy stochastic DEA models consisting of probability-possibility, probability-necessity and probability-credibility constraints in which input and output data entailed fuzziness and randomness at the same time. After that, Tavana et al. (2014) proposed DEA models with birandom input-output. Nasseri et al. [23] proposed a fuzzy stochastic DEA models. They formulated a linear and feasible model with an extension of normal distrubtion to deal with fuzzy random data. This approach overcomes to the shortcomings of linearity and normal efficiency score relative to corresponding approaches. Nasseri and Khatir [24] introduced a three-stage DEA models by taking into account undesirable output with fuzzy stochastic data.

This study try to incorporate fuzzy random inputs and outputs in two-stage model with undesiarable output. We apply probablity-possibility measure to deal with the fuzzy random environments. This measure was also applied by Tavana et al. [33] for stanadard CCR-DEA model. Their approach leads to a non-linear (quadratic) programming. However, there is no any analytic result on feasibility of the their proposed models. Hence, in this work, we attempt to overcome these shortcomings by applying a new version of deterministic DEA model. To sum up with all the above aspects, the achivment of the present study is three cases: (1) to formulate a new version of two-stage DEA model equiped undesirable output (2) to formulate a linear and feasible model with the efficiency scores of DUMs with the range of zero to one for solving fuzzy stochastic two-stage DEA model with undesirable output, and (3) to demonstrate the applicability of the proposed model using a case study for the banking industry.

3. Proposed undeasirable two-stage model

Consider a two-stage process. We have n DMUs that each DM U_j (j = 1, 2, n) has m inputs x_j = (x_1j, x_2j, ⋯ , x_mj) and D outputs z_j = (x_1j, x_2j, ⋯ , x_Dj) to the first stage. These D outputs known as intermediate measure then are consumed to the second stage. The outputs from the second stage are y_j = (y_1j, y_2j, ⋯ , y_sj). Kao and Hwang (2008) proposed the following two-stage model based on input-oreinted CCR model.

$\begin{matrix} E_{k} = Max \sum_{r = 1}^{s} u_{r} y_{rk} \\ s . t . \sum_{i = 1}^{m} v_{i} x_{ik} = 1 \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{d = 1}^{D} w_{d} z_{dj} \leq 0, j = 1, 2, \dots, n \\ \sum_{d = 1}^{D} w_{d} z_{dj} - \sum_{i = 1}^{m} v_{j} x_{ij} \leq 0, j = 1, 2, \dots, n \\ u_{r} \geq 0, v_{i} \geq 0, w_{d} \geq 0, r = 1, 2, \dots, s; \\ i = 1, 2, \dots, m; d = 1, 2, \dots, D \end{matrix}$ (1) where v_i, u_r, w_d, and are the non-negative weights. In performance measurement problems with undesirable outputs, DMUs with more desirable outputs and less undesirable outputs (relative to less input resources) are considered efficient. In order to solve this, various methods have been proposed to incorporate undesirable outputs in DEA (DEA-UO). In this paper, we apply a direct approach to deal with undesirable outputs initially proposed by Korhonen and Luptacik (2004) and then modified by Puri and Yadav [27]. To this, consider outputs in model (1) in two classes with outputs $y_{j}^{g} = (y_{1 j}^{g}, y_{2 j}^{g}, \dots, y_{s_{1} j}^{g})$ are desirable (good) and outputs $y_{j}^{b} = (y_{1 j}^{b}, y_{2 j}^{b}, \dots, y_{s_{2} j}^{b})$ are undesirable (bad) such that s = s₁ + s₂. Finally, using the notations in model (1) and new notations on (desirable)undesirable outputs, undesiable two-stage CCR-model will be as follows:

$\begin{matrix} E_{k} = Max \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rk}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pk}^{b} \\ s . t . \sum_{i = 1}^{m} v_{i} x_{ik} = 1 \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{b} - \sum_{d = 1}^{D} w_{d} z_{dj} \leq 0, \\ j = 1, 2, \dots, n \\ \sum_{d = 1}^{D} w_{d} z_{dj} - \sum_{i = 1}^{m} v_{i} x_{rj} \leq 0, j = 1, 2, \dots, n \\ u_{r}^{g} \geq 0, u_{p}^{b} \geq 0, v_{i} \geq 0, w_{d} \geq 0, \\ r = 1, 2, \dots, s_{1}; p = 1, 2, \dots, s_{2}; \\ i = 1, 2, \dots, m; d = 1, 2, \dots, D \end{matrix}$ (2) where $u_{r}^{g}$ and $u_{p}^{b}$ are the non-negative weights corresponding to the desirable and undesirable outputs, respectively. The DMU_k is (technically) efficient if E_k = 1, otherwise if 0 < E_k < 1, it is (technically) inefficient.

Substituting ${\hat{x}}_{ij} = v_{i} x_{ij}$ , ${\hat{y}}_{rj} = u_{r}^{g} y_{rj}^{g}$ , and ${\hat{z}}_{dj} = w_{d} z_{dj}$ into Model (2), the following equivalent model is obtained:

$\begin{matrix} E_{k} = Max \sum_{r = 1}^{s_{1}} {\hat{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pk}^{b} \\ s . t . \sum_{i = 1}^{m} {\hat{x}}_{ik} = 1 \\ \sum_{r = 1}^{s_{1}} {\hat{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pj}^{b} - \sum_{d = 1}^{D} {\hat{z}}_{dj} \leq 0, \\ j = 1, 2, \dots, n \\ \sum_{d = 1}^{D} {\hat{z}}_{dj} - \sum_{i = 1}^{m} {\hat{x}}_{rj} \leq 0, j = 1, 2, \dots, n \\ u_{r}^{g} y_{rj}^{g} \leq {\hat{y}}_{rj}^{g} \leq u_{r}^{g} y_{rj}^{g}, \forall r, j \\ u_{p}^{b} y_{pj}^{b} \leq {\hat{y}}_{pj}^{b} \leq u_{p}^{b} y_{pj}^{b}, \forall p, j \\ u_{d} z_{dj} \leq {\hat{z}}_{dj} \leq u_{d} z_{dj}, \forall d, j \\ v_{i} x_{ij} \leq {\hat{x}}_{ij} \leq v_{i} x_{ij}, \forall i, j \\ \sum_{d = 1}^{D} w_{d} z_{dj} - \sum_{i = 1}^{m} v_{i} x_{rj} \leq 0, j = 1, 2, \dots, n, \\ u_{r}^{g} \geq 0, u_{p}^{b} \geq 0, v_{i} \geq 0, w_{d} \geq 0, φ \geq 0 \\ r = 1, 2, \dots, s_{1}; p = 1, 2, \dots, s_{2}; \\ i = 1, 2, \dots, m; d = 1, 2, \dots, D \end{matrix}$ (3)

4. Fuzzy stochastic undesirable two-stage CCR model: An probability-possibility approach

The aim of this section is to propose a two-stage DEA-based method for evaluating the efficiencies of DMUs with fuzzy stochastic inputs and fuzzy stochastic desirable/undesirable outputs.

To this end, consider n DMUs, each unit consumes m fuzzy stochastic inputs, denoted by ${\tilde{\bar{X}}}_{j} = ({\tilde{X}}_{j}, X_{j}^{α}, X_{j}^{β})_{LR}$ and D intermediate outputs ${\tilde{\bar{Z}}}_{j} = ({\tilde{Z}}_{j}, Z_{j}^{α}, Z_{j}^{β})_{LR}$ to the first stage, and produces s fuzzy stochastic outputs, denoted by ${\tilde{\bar{Y}}}_{j} = ({\tilde{Y}}_{j}^{g}, Z_{j}^{g, α}, Z_{j}^{g, β})_{LR}$ as desirable outputs and ${\tilde{\bar{Y}}}_{j}^{b} = ({\tilde{Y}}_{j}^{b}, Y_{j}^{b, α}, Y_{j}^{b, β})_{LR}$ as undesirable outputs. Let ${\tilde{X}}_{j}$ , ${\tilde{Z}}_{j}$ ${\tilde{Y}}_{j}^{g}$ and ${\tilde{Y}}_{j}^{b}$ , be normally distributed by ${\tilde{X}}_{j} \sim N (X_{j}, σ_{j})$ , ${\tilde{Z}}_{j} \sim N (Z_{j}, σ_{j})$ and ${\tilde{Y}}_{j}^{g} \sim N (Y_{j}^{g}, σ_{j}^{g})$ , respectively.

The chance-constrained programming (CCP) developed by Cooper et al. (2002) is a stochastic optimization approach suitable for solving optimization problems with uncertain parameters. Building on CCP and possibility theory as the principal techniques, the following probability-possibility CCR model is proposed: $\begin{array}{l} E_{k} (δ, γ) = M a x \sum_{r = 1}^{s_{1}} {\hat{y}}_{r k}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{p k}^{b} \\ \begin{array}{l} \sum_{i = 1}^{m} {\hat{x}}_{i k} = 1 \\ s . t . \sum_{r = 1}^{s_{1}} {\hat{y}}_{r k}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{p k}^{b} - \sum_{d = 1}^{D} {\hat{z}}_{d j} \leq 0, \\ j = 1, 2, ..., n \\ \sum_{d = 1}^{D} {\hat{z}}_{d j} - \sum_{i = 1}^{m} {\hat{x}}_{r j} \leq 0, j = 1, 2, ..., n \end{array}} (i) \end{array}$

$\begin{array}{l} \begin{array}{l} Pr [Pos (\frac{{\hat{x}}_{i j}}{v_{i}} \leq^{≃} x_{i}_{j} \leq \frac{{\hat{x}}_{i j}}{v_{i}}) \geq δ] \geq γ, \forall i, j \\ Pr [Pos (\frac{{\hat{y}}_{r j}^{g}}{u_{r}^{g}} \leq^{≃} y_{r j}^{g} \leq \frac{{\hat{y}}_{r j}^{g}}{u_{r}^{g}}) \geq δ] \geq γ, \forall r, j \\ Pr [Pos (\frac{{\hat{y}}_{p j}^{b}}{u_{p}^{b}} \leq^{≃} y_{p j}^{b} \leq \frac{{\hat{y}}_{p j}^{b}}{u_{p}^{b}}) \geq δ] \geq γ, \forall p, j \\ Pr [Pos (\frac{{\hat{z}}_{d j}}{u_{d}} \leq^{≃} z_{d}_{j} \leq \frac{{\hat{z}}_{d j}}{u_{d}}) \geq δ] \geq γ, \forall d, j \end{array}} (i i) \\ u_{r}^{g} \geq 0, u_{p}^{b} \geq 0, v_{i} \geq 0, w_{d} \geq 0, φ \geq 0, \\ r = 1, 2, ..., s_{1}; p = 1, 2, ..., s_{2}; \\ i = 1, 2, ..., m; d = 1, 2, ..., D \end{array}$

into a linear programing model, where δ and γ ∈ [0, 1] in constraint (ii) and (iii) are the predetermined thresholds defined by the DM. Pos [·] and Pr [·] in Model (??) denote the possibility and the probabilityd of [·] event.

In what follows we show that the probability-possibility model (??) can be equivalently transformed into a linear programing model.

In order to solve the possibility constrained programming model (??), we utilize Theorem 4 and Lemma 4 proposed, respectively, by Liu and Liu (2003) and Sakawa (1993).

Theorem 4.1. Assume that ξ is a fuzzy random vector, and g_j are real-valued continuous functions for j = 1, 2, $\hat{a} L_{|}^{|}$ , n. We have:

The possibility Pos {g_j (ξ (w)) ≤ 0, j = 1, 2, ⋯ , n} is a random variable.

The necessity Nec {g_j (ξ (w)) ≤ 0, j = 1, 2, ⋯ , n} is a random variable.

The credibility Cr {g_j (ξ (w)) ≤ 0, j = 1, 2, ⋯ , n} is a random variable.

Lemma 4.2. Let ${\bar{λ}}_{1}$ and ${\bar{λ}}_{2}$ be two fuzzy numbers with continuous membership functions. For a given confidence level α ∈ [0, 1], $Pos {{\bar{λ}}_{1}, {\bar{λ}}_{2}} \geq α$ if and only if $λ_{1, α}^{R} \geq λ_{2, α}^{R}$ , $Nec {{\bar{λ}}_{1}, {\bar{λ}}_{2}} \geq α$ if and only if $λ_{1, 1 - α}^{L} \geq λ_{2, α}^{R}$ .

where $λ_{1, α}^{L}$ , $λ_{1, α}^{R}$ and $λ_{2, α}^{L}$ , $λ_{2, α}^{R}$ are the left and the right side extreme points of the α-level sets ${\bar{λ}}_{1}$ and ${\bar{λ}}_{2}$ , respectively, $Pos {{\bar{λ}}_{1}, {\bar{λ}}_{2}} \geq α$ and $Nec {{\bar{λ}}_{1}, {\bar{λ}}_{2}} \geq α$ present the degree of possibility and necessity, respectively.

The first constraint of (ii) in Model (2), can be transformed into the following two constraints: $\begin{matrix} \Pr [Pos (v_{i} {\tilde{\bar{x}}}_{ij} \leq {\hat{x}}_{ij}) \geq δ] \geq γ, \\ \Pr [Pos ({\hat{x}}_{ij} \leq v_{i} {\tilde{\bar{x}}}_{ij}) \geq δ] \geq γ . \end{matrix}$

These constraints can be rewritten as the following constraints based on Lemma 4: $\begin{matrix} \Pr [Pos ({\tilde{\bar{x}}}_{ij} \leq \frac{{\hat{x}}_{ij}}{v_{i}}) \geq δ] \geq γ \Leftrightarrow \Pr [{({\tilde{\bar{x}}}_{ij})}_{δ}^{L} \leq \frac{{\hat{x}}_{ij}}{v_{i}}] \geq γ \\ \Leftrightarrow \Pr [{\tilde{x}}_{ij} - L^{- 1} (δ) x_{ij}^{α} \leq \frac{{\hat{x}}_{ij}}{v_{i}}] \geq γ \\ \Pr [Pos (\frac{{\hat{x}}_{ij}}{v_{i}} \leq {\tilde{\bar{x}}}_{ij}) \geq δ] \geq γ \Leftrightarrow \Pr [{(\frac{{\hat{x}}_{ij}}{v_{i}} \leq {\tilde{\bar{x}}}_{ij})}_{δ}^{R}] \geq γ \\ \Leftrightarrow \Pr [\frac{{\hat{x}}_{ij}}{v_{i}} \leq {\tilde{x}}_{ij} + R^{- 1} (δ) x_{rj}^{β}] \geq γ \end{matrix}$

In a similar way, the other constraints of (ii) in Model (??), can be rewritten. Therefore, Model (??) can be reformulated as follows:

$\begin{matrix} Max φ \\ s . t . φ \leq \sum_{r = 1}^{s_{1}} {\hat{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pk}^{b} \\ (i) {\begin{matrix} \sum_{i = 1}^{m} {\hat{x}}_{ik} = 1 \\ s . t . \sum_{r = 1}^{s_{1}} {\hat{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pk}^{b} - \sum_{d = 1}^{D} {\hat{z}}_{dj} \leq 0, j = 1, 2, \dots, n \\ \sum_{d = 1}^{D} {\hat{z}}_{dj} - \sum_{i = 1}^{m} {\hat{x}}_{ij} \leq 0, j = 1, 2, \dots, n \end{matrix}} \end{matrix}$

$\begin{matrix} \Pr [\frac{{\hat{x}}_{ij}}{v_{i}} \leq {\tilde{x}}_{ij} + R^{- 1} (δ) x_{ij}^{β}] \geq γ, \Pr [{\tilde{x}}_{ij} - L^{- 1} (δ) x_{ij}^{α} \leq \frac{{\hat{x}}_{ij}}{v_{i}}] \geq γ, \forall i, j \\ \Pr [\frac{{\hat{y}}_{rj}^{g}}{u_{r}^{g}} \leq {\tilde{y}}_{rj}^{g} + R^{- 1} (δ) y_{rj}^{g, β}] \geq γ, \Pr [{\tilde{y}}_{rj}^{g} - L^{- 1} (δ) y_{rj}^{g, α} \leq \frac{{\hat{y}}_{rj}^{g}}{u_{r}^{g}}] \geq γ, \forall r, j \\ \Pr [\frac{{\hat{y}}_{pj}^{b}}{u_{p}^{b}} \leq {\tilde{y}}_{pj}^{b} + R^{- 1} (δ) y_{pj}^{b, β}] \geq γ, \Pr [{\tilde{y}}_{pj}^{b} - L^{- 1} (δ) y_{pj}^{b, α} \leq \frac{{\hat{y}}_{pj}^{b}}{u_{p}^{b}}] \geq γ, \forall p, j \\ \Pr [\frac{{\hat{z}}_{dj}}{w_{d}} \leq {\tilde{z}}_{dj} + R^{- 1} (δ) z_{dj}^{β}] \geq γ, \Pr [{\tilde{z}}_{dj} - L^{- 1} (δ) z_{dj}^{α} \leq \frac{{\hat{z}}_{dj}}{w_{d}}] \geq γ, \forall d, j \\ u_{r}^{g} \geq 0, u_{p}^{b} \geq 0, v_{i} \geq 0, w_{d} \geq 0, φ \geq 0 \\ r = 1, 2, \dots, s_{1}; p = 1, 2, \dots, s_{2}; i = 1, 2, \dots, m; d = 1, 2, \dots, D \end{matrix}$ (4)

By the help of standardized normal distribution, (see, e.g., Cooper et al. 2004), Model (4) can be transformed into a deterministic linear programming model. Consequently, let us consider the first probabilistic constraint in Model (4) as where ${\tilde{x}}_{ij} + R^{- 1} (δ) x_{ij}^{β} - \frac{{\hat{x}}_{ij}}{v_{i}} \geq 0$ . Due to the normal distribution of ${\hat{x}}_{ij}$ , $\hat{h}$ also has normal distribution with the following mean and variance:

$\begin{matrix} E (\hat{h}) = E [{\hat{x}}_{ij} + R^{- 1} (δ) x_{ij}^{β} - \frac{{\hat{x}}_{ij}}{v_{i}}] = x_{ij} + R^{- 1} (δ) x_{ij}^{β} - \frac{{\hat{x}}_{ij}}{v_{i}} \\ Var (\hat{h}) = Var [{\hat{x}}_{ij} + R^{- 1} (δ) x_{ij}^{β} - \frac{{\hat{x}}_{ij}}{v_{i}}] = Var ({\tilde{x}}_{ij}) = σ_{ij}^{2} \end{matrix}$

By standardizing the normal distribution, $\Pr (\tilde{h} \geq 0) \geq γ$ is converted to $\Pr [z \geq \frac{- E (\tilde{h})}{\sqrt{Var (\tilde{h})}}] \geq γ$ where $Z = \frac{h - E (\tilde{h})}{\sqrt{Var (\tilde{h})}}$ is the standard normal random variable with zero mean and unit variance. The corresponding cumulative

distribution function is $Φ = [\frac{- E (\tilde{h})}{\sqrt{Var (\tilde{h})}}] \leq 1 - γ$ and it is equal to $\frac{{\hat{x}}_{ij}}{v_{i}} - x_{ij} - R^{- 1} (δ) x_{ij}^{β} \leq σ_{ij} Φ_{1 - γ}^{- 1}$ , where $Φ_{1 - γ}^{- 1}$ is the inverse of Φ at the level of 1 - γ. Finally, the deterministic version of constraint (i) in Model (4) will be as follows:

${\hat{x}}_{ij} \leq v_{i} (x_{ij} - R^{- 1} (δ) x_{ij}^{β} + σ_{ij} Φ_{1 - γ}^{- 1}), i, j .$ A similar procedure adopted for remain constraints in Model (4) results in the following deterministic equivalent model (5): $\begin{matrix} E_{k} (δ, γ) = Max φ \\ s . t . φ \leq \sum_{r = 1}^{s_{1}} {\hat{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pk}^{b} \\ \sum_{i = 1}^{m} {\hat{x}}_{ik} = 1 \\ \sum_{r = 1}^{s_{1}} {\hat{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pj}^{b} - \sum_{d = 1}^{D} {\hat{z}}_{dj} \leq 0, j = 1, 2, \dots, n \\ \sum_{d = 1}^{D} {\hat{z}}_{dj} - \sum_{i = 1}^{m} {\hat{x}}_{ij} \leq 0, j = 1, 2, \dots, n \end{matrix}$

$\begin{matrix} u_{r}^{g} (y_{rj}^{g} - L^{- 1} (δ) y_{rj}^{α, g} - σ_{rj} Φ_{1 - \frac{γ}{2}}^{- 1}) \leq {\hat{y}}_{rj}^{g} \leq u_{r}^{g} (y_{rj}^{g} + R^{- 1} (δ) y_{rj}^{β, g} + σ_{rj} Φ_{1 - \frac{γ}{2}}^{- 1}), \forall r, j \\ u_{p}^{b} (y_{pj}^{b} - L^{- 1} (δ) y_{pj}^{α, b} - σ_{pj} Φ_{1 - \frac{γ}{2}}^{- 1}) \leq {\hat{y}}_{pj}^{b} \leq u_{p}^{b} (y_{pj}^{b} + R^{- 1} (δ) y_{pj}^{β, b} + σ_{pj} Φ_{1 - \frac{γ}{2}}^{- 1}), \forall p, j \\ w_{d} (z_{dj} - L^{- 1} (δ) z_{dj}^{α} - σ_{dj} Φ_{1 - \frac{γ}{2}}^{- 1}) \leq {\hat{z}}_{dj} \leq w_{d} (z_{dj} + R^{- 1} (δ) z_{dj}^{β} + σ_{dj} Φ_{1 - \frac{γ}{2}}^{- 1}), \forall d, j \\ v_{i} (x_{ij} - L^{- 1} (δ) x_{ij}^{α} - σ_{ij} Φ_{1 - \frac{γ}{2}}^{- 1}) \leq {\hat{x}}_{ij} \leq v_{i} (x_{ij} + R^{- 1} (δ) x_{ij}^{β} + σ_{ij} Φ_{1 - \frac{γ}{2}}^{- 1}), \forall i, j \\ u_{r}^{g} \geq 0, u_{p}^{b} \geq 0, v_{i} \geq 0, w_{d} \geq 0, φ \geq 0 \\ r = 1, 2, \dots, s_{1}; p = 1, 2, \dots, s_{2}; i = 1, 2, \dots, m; d = 1, 2, \dots, D \end{matrix}$ (5) The above model is obviously a linear program. This model is an extension of Nasseri et al. (2016)’s model to the two-stage CCR model when undesirable outputs are considered. However, the methodology used in two works are different; One is the extended probability approach and the other is probability-possibility approach. This extension of the Nasseri et al. (2016)’s model keeps some properties. Theorem 4 presents a similar important property, in this way, in which the objective function of Model (5), E_k (δ, γ), is monotonously decreasing related to the each of δ and γ level.

Theorem 4.3. If E_k (δ, γ) is the optimum objective function value of Model (5) then E_k (δ₁, γ)≥

E_k (δ₂, γ) and E_k (δ, γ₁) ≥ E_k (δ, γ₂) where δ ≤ δ₂ and γ₁ ≤ γ₂.

Proof. Denote the feasible space of Model (5) by S_δ,γ. We need to prove that S_δ₂,γ₂ ⊆ S_δ₁,γ₁.To this, consider the following constraint of Model (5) $\begin{matrix} v_{i} (x_{ij} - L^{- 1} (δ) x_{ij}^{α} - σ_{ij} Φ_{1 - γ}^{- 1}) \\ \leq {\hat{x}}_{ij} \leq v_{i} (x_{ij} + R^{- 1} (δ) x_{ij}^{β} + σ_{ij} Φ_{1 - γ}^{- 1}) \end{matrix}$ Let $Φ^{- 1} (γ) = Φ_{γ}^{- 1}$ . As Φ^-1 (1 - γ), L^-1 (γ) and R^-1 (δ) are decreasing function, the functions -Φ^-1 (1 - γ), -L^-1 (γ) and -R^-1 (δ) will be increasing. It is concluded that $\begin{matrix} [x_{ij} - L^{- 1} (δ_{2}) x_{ij}^{α} - σ_{ij} Φ^{- 1} (1 - γ_{2}), x_{ij} + R^{- 1} (δ_{2}) x_{ij}^{β} - σ_{ij} Φ^{- 1} (1 - γ_{2})] \subseteq \\ [x_{ij} - L^{- 1} (δ_{1}) x_{ij}^{α} - σ_{ij} Φ^{- 1} (1 - γ_{1}), x_{ij} + R^{- 1} (δ_{1}) x_{ij}^{β} - σ_{ij} Φ^{- 1} (1 - γ_{1})] \end{matrix}$ In a similar way, we can conclude that $\begin{matrix} [y_{rj} - L^{- 1} (δ_{2}) y_{rj}^{α} - σ_{rj} Φ^{- 1} (1 - γ_{2}), y_{rj} + R^{- 1} (δ_{2}) y_{rj}^{β} - σ_{rj} Φ^{- 1} (1 - γ_{2})] \subseteq \\ [y_{rj} - L^{- 1} (δ_{1}) y_{rj}^{α} - σ_{rj} Φ^{- 1} (1 - γ_{2}), y_{rj} + R^{- 1} (δ_{1}) y_{rj}^{β} - σ_{rj} Φ^{- 1} (1 - γ_{2})] \end{matrix}$ This completes the proof.

Now, we can present the following defiition to define the efficiecy of each DMU.

Definition 4.4. For the given level δ and γ, we define $E_{k}^{T} (δ, γ) = E_{k} (δ, \frac{γ}{2})$ as probabilistic-possibilistic efficiency score of DMU_k in fuzzy random DEA Model.

The corresponding model with $E_{k}^{T} (δ, γ)$ , is called model (6), can be constructed. $\begin{matrix} E_{k} (δ, γ) = Max φ \\ (i) {\begin{matrix} s . t . φ \leq \sum_{r = 1}^{s_{1}} {\hat{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pk}^{b} \\ \sum_{i = 1}^{m} {\hat{x}}_{ik} = 1 \\ \sum_{r = 1}^{s_{1}} {\hat{y}}_{rj}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pj}^{b} - \sum_{d = 1}^{D} {\hat{z}}_{dj} \leq 0, j = 1, 2, \dots, n \\ \sum_{d = 1}^{D} {\hat{z}}_{dj} - \sum_{i = 1}^{m} {\hat{x}}_{ij} \leq 0, j = 1, 2, \dots, n \end{matrix}} \end{matrix}$

$\begin{matrix} u_{r}^{g} (y_{rj}^{g} - L^{- 1} (δ) y_{rj}^{α, g} - σ_{rj} Φ_{1 - γ}^{- 1}) \leq {\hat{y}}_{rj}^{g} \leq u_{r}^{g} (y_{rj}^{g} + R^{- 1} (δ) y_{rj}^{β, g} + σ_{rj} Φ_{1 - γ}^{- 1}), \forall r, j \\ u_{p}^{b} (y_{pj}^{b} - L^{- 1} (δ) y_{pj}^{α, b} - σ_{pj} Φ_{1 - γ}^{- 1}) \leq {\hat{y}}_{pj}^{b} \leq u_{p}^{b} (y_{pj}^{b} + R^{- 1} (δ) y_{pj}^{β, b} + σ_{pj} Φ_{1 - γ}^{- 1}), \forall p, j \\ w_{d} (z_{dj} - L^{- 1} (δ) z_{dj}^{α} - σ_{dj} Φ_{1 - γ}^{- 1}) \leq {\hat{z}}_{dj} \leq w_{d} (z_{dj} + R^{- 1} (δ) z_{dj}^{β} + σ_{dj} Φ_{1 - γ}^{- 1}), \forall d, j \\ v_{i} (x_{ij} - L^{- 1} (δ) x_{ij}^{α} - σ_{ij} Φ_{1 - γ}^{- 1}) \leq {\hat{x}}_{ij} \leq v_{i} (x_{ij} + R^{- 1} (δ) x_{ij}^{β} + σ_{ij} Φ_{1 - γ}^{- 1}), \forall i, j \\ u_{r}^{g} \geq 0, u_{p}^{b} \geq 0, v_{i} \geq 0, w_{d} \geq 0, φ \geq 0 \\ r = 1, 2, \dots, s_{1}; p = 1, 2, \dots, s_{2}; i = 1, 2, \dots, m; d = 1, 2, \dots, D \end{matrix}$ (6)

Theorem 4.5. Consider the optimum objective function value of Model (6) for DMU_k, then

$E_{k}^{T} (δ_{1}, γ) \geq E_{k}^{T} (δ_{2}, γ)$ and $E_{k}^{T} (δ, γ_{1}) \geq E_{k}^{T} (δ, γ_{2})$ where δ₁ ≤ δ₂ and γ₁ ≤ γ₂.

$0 < E_{j}^{T} (δ, γ) \leq 1 j = 1, 2, \dots, n$ .

Model (6) is feasible for any δ and γ.

Proof. a. It is straightforward using Theorem 4 and Definition 4. In assertion b, obviously, it is followed immediately from the restriction and four constraints in part (i) of model (6) as follows: $φ \leq \sum_{r = 1}^{s_{1}} {\hat{y}}_{rk}^{g} - \sum_{p = 1}^{s_{2}} {\hat{y}}_{pk}^{b} \leq \sum_{d = 1}^{D} {\hat{z}}_{dk} \leq \sum_{i = 1}^{m} {\hat{x}}_{ik} = 1$ To prove assertion c, Let δ = 1 and γ = 1, then L^-1 (1) = R^-1 (1) =0 and Φ^-1 (0.5) =0. Hence, we have ${\hat{x}}_{ij} = v_{i} x_{ij}$ , ${\hat{y}}_{rj}^{g} = u_{r}^{g} y_{rj}^{g}$ , ${\hat{y}}_{pj}^{b} = u_{p}^{b} y_{pj}^{b}$ and ${\hat{z}}_{dj} = w_{d} z_{dj}$ . In Model (6). Therefore, the correspondig model with will $E_{k}^{T} (1, 1)$ be as follows:

$\begin{matrix} E_{k}^{T} (1, 1) = \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rk}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pk}^{b} \\ s . t . \sum_{i = 1}^{m} v_{i} x_{ik} = 1 \\ \sum_{r = 1}^{s_{1}} u_{r}^{g} y_{rj}^{g} - \sum_{p = 1}^{s_{2}} u_{p}^{b} y_{pj}^{b} - \sum_{d = 1}^{D} w_{d} z_{dj} \leq 0, \\ j = 1, 2, \dots, n \\ \sum_{d = 1}^{D} w_{d} z_{dj} - \sum_{i = 1}^{m} v_{i} x_{ij} \leq 0, j = 1, 2, \dots, n \\ u_{r}^{g} \geq 0, u_{p}^{b} \geq 0, v_{i} \geq 0, w_{d} \geq 0, \\ r = 1, 2, \dots, s_{1}; p = 1, 2, \dots, s_{2}; \\ i = 1, 2, \dots, m; d = 1, 2, \dots, D \end{matrix}$ (7)

To prove assertion c, denote the feasible space of Model (6) by $S_{δ, γ}^{T}$ . According to the proof of Theorem 4, $S_{1, 1}^{T} \subseteq S_{δ, γ}^{T}$ . Therefore, it is sufficient to show that the feasible space $S_{1, 1}^{T}$ is nonempty. Suppose that $u_{p}^{b} = 0 (p = 1, 2, \dots, s_{2})$ , then $E_{k}^{T} (1, 1)$ is converted to the model (7) as a two-stage model. Chen et al. (2009) explicity showed the feasibiliy of model (7). This complets the proof of part (c).

5. Case Study

As the largest Iranian business bank, Bank Melli Iran has a comprehensive network of over 3,300 branches and 37.000 employees in Iran. Countrywide coverage in Iran, service quality and an experienced multi-lingual staff are important factors of their success. In this section, we apply the proposed approach in this study to some commercial bank branches in Mazandaran province. Here the data sources consist of the reports of 31 branches. The inputs for the first stage are fixed Personnel and budget. The intermediate measure is total of deposites (TDs) (of current, short duration and long duration accounts). The second stage’s outputs are profit and loans recovered as desirable outputs, and non-performing loans (delay in delivering loans and other facilities) as undesirable output. However, there always exist some degrees of uncertainty in the data which can be represented by fuzzy stochastic numbers. In banks, uncertainty occurs due to the difference between the actual data and the available data. For example, let actual budget for a bank be 1.68 8 billion (IRR) 1 and the possible data amounts of budget available at different places be 1.4 or 2 or 2.2 billion. Then the difference between actual data and possible data results into the occurrence of uncertainty in the data which further may affect. Therefore, in the present study, we fuzzify the data as TFNs. One input (budget), all desirable, undesirable, and intermedite outputs are taken as TFNs. The collected crisp data (except for personnel) in Table 1 are considered as the mean of TFNs. The left and right spreads are calculated by 1% of mean value. On the other hand, the inputs and outputs are supposed as random variables. By using goodness of fit tests, normal distributions have been fit on the random variables. The corresponding expected value is the observed inputs (outputs) data and the standard deviation is one. Hence, each DMU is considered as a fuzzy variable with randomized mean. This fuzzy random inputâŁ“output data of each bank are available in Table 1. Each input and output data are denoted by (N (m . σ) , α), where is the observed data as the expected value in normal distribution with σ = 1, and is the left and also the right spread.

Table 1
The fuzzy random input and output data

DMU Personnel Budget Profit Loans Deposit Non-performing loans

1 7 1224 22818.54169 289229.8302 457623.6181 95077.361

2 8 1105 9385.069847 355478.9185 707985.4583 43173.45676

3 7 1145 28441.51261 305389.1359 314984.2195 25432.88114

4 9 1366 32793.02661 528388.7035 503629.5418 41459.1004

5 15 3287 30533.31539 681968.2244 769819.5554 88485.69166

6 10 1688 27477.77451 413802.3328 558069.2656 24996.09248

7 10 2460 23622.30016 266354.5353 327096.2443 8731.581823

8 10 1024 17652.97323 344910.1933 597479.4179 20787.94059

9 8 1125 25430.26916 352162.9103 407846.7061 10808.5889

10 7 880 15834.89596 372771.2556 544833.5966 3508.853844

11 8 1490 16612.652 282899.2299 505413.1373 14808.46909

12 8 1222 20600.5341 294264.7082 380570.5266 13472.00066

13 15 2715 37267.67625 432087.9694 509437.3264 65599.05887

14 7 1534 4956.54515 276544.5305 638379.9882 6948.951879

15 9 1153 28436.31893 453260.1648 541996.0451 21629.48307

16 16 1697 11907.47675 460784.1778 660375.4622 186254.9709

17 8 1066 17879.15234 197217.0512 356064.4698 33984.12774

18 14 2116 10484.2398 218695.1685 582385.9765 67370.44614

19 9 1814 12868.85036 259999.3131 528433.3344 34145.96418

20 9 2449 14214.74962 289323.5393 544372.3952 20245.61706

21 15 5733 2666.2637 372013.9297 911042.7117 24233.24085

22 9 1773 9777.445824 199972.0178 502660.7386 16489.289

23 10 1171 19026.2255 210177.8344 300715.1159 22042.81645

24 4 931 27339.82563 185545.625 221391.7296 2878.348634

25 12 1356 15409.81528 212394.7934 401986.5034 3274.169479

26 12 1815 12962.24022 152506.026 385425.7875 16486.01526

27 8 1790 28521.77609 414371.0683 553452.7902 12730.7546

28 8 1422 11471.67549 185890.3136 439790.572 4814.328501

29 15 2142 11001.82256 384419.0757 740570.3543 31609.19963

30 8 1410 15344.15833 253207.1981 398177.3066 18189.69114

31 8 1475 10706.33815 183051.2085 410958.9387 14468.96572

DMU	Personnel	Budget	Profit	Loans	Deposit	Non-performing loans
1	7	1224	22818.54169	289229.8302	457623.6181	95077.361
2	8	1105	9385.069847	355478.9185	707985.4583	43173.45676
3	7	1145	28441.51261	305389.1359	314984.2195	25432.88114
4	9	1366	32793.02661	528388.7035	503629.5418	41459.1004
5	15	3287	30533.31539	681968.2244	769819.5554	88485.69166
6	10	1688	27477.77451	413802.3328	558069.2656	24996.09248
7	10	2460	23622.30016	266354.5353	327096.2443	8731.581823
8	10	1024	17652.97323	344910.1933	597479.4179	20787.94059
9	8	1125	25430.26916	352162.9103	407846.7061	10808.5889
10	7	880	15834.89596	372771.2556	544833.5966	3508.853844
11	8	1490	16612.652	282899.2299	505413.1373	14808.46909
12	8	1222	20600.5341	294264.7082	380570.5266	13472.00066
13	15	2715	37267.67625	432087.9694	509437.3264	65599.05887
14	7	1534	4956.54515	276544.5305	638379.9882	6948.951879
15	9	1153	28436.31893	453260.1648	541996.0451	21629.48307
16	16	1697	11907.47675	460784.1778	660375.4622	186254.9709
17	8	1066	17879.15234	197217.0512	356064.4698	33984.12774
18	14	2116	10484.2398	218695.1685	582385.9765	67370.44614
19	9	1814	12868.85036	259999.3131	528433.3344	34145.96418
20	9	2449	14214.74962	289323.5393	544372.3952	20245.61706
21	15	5733	2666.2637	372013.9297	911042.7117	24233.24085
22	9	1773	9777.445824	199972.0178	502660.7386	16489.289
23	10	1171	19026.2255	210177.8344	300715.1159	22042.81645
24	4	931	27339.82563	185545.625	221391.7296	2878.348634
25	12	1356	15409.81528	212394.7934	401986.5034	3274.169479
26	12	1815	12962.24022	152506.026	385425.7875	16486.01526
27	8	1790	28521.77609	414371.0683	553452.7902	12730.7546
28	8	1422	11471.67549	185890.3136	439790.572	4814.328501
29	15	2142	11001.82256	384419.0757	740570.3543	31609.19963
30	8	1410	15344.15833	253207.1981	398177.3066	18189.69114
31	8	1475	10706.33815	183051.2085	410958.9387	14468.96572

Four different (δ, γ)-threshold levels of (γ = 0.9, δ = 0.7), (γ = 0.9, δ = 0.4), (γ = 0.7, δ = 0.7), (γ = 0.5, δ = 0.5), and finally overall efficiency. The results of model (6) for probability-possibility levels are reported in Table 2. As shown in this table, DMU 10 is probabilistic-possibilistic -efficient at four given levels. Generally from Table 2, we can see the applicability of Theorem 3. When the efficiency scores of the DMUs increase, the level decreases from (γ = 0.9, δ = 0.7) to (γ = 0.9, δ = 0.4) and the level decreases from (γ = 0.9, δ = 0.7) to (γ = 0.7, δ = 0.7) in the probabilistic-possibilistic model (6). It should be noted that the probabilistic-possibilistic model proposed in this study is feasible for any -threshold levels as another evident of Theorem 4. In addition, the proposed fuzzy stochastic DEA model in this paper, similarly to the conventional DEA models, provide the efficiency score for each DMU within the specified range (0, 1]. Finally, Table 2 presents the average efficiency scores and the final rankings of the 31 bank branches. As shown in Fig. 1 the proposed model (6) is stable in different levels. However, the average efficiency can be an appropriate overall index to indicate the efficiency variations.

Table 2

The fuzzy random efficiency scores and final ranking

DMU	(γ = 0.9, δ = 0.7)	(γ = 0.9, δ = 0.4)	(γ = 0.7, δ = 0.7)	(γ = 0.5, δ = 0.5)	Overall efficiency	Ranking
1	0.5603	0.5673	0.565	0.5744	0.5668	8
2	0.4843	0.4901	0.4843	0.4882	0.4867	15
3	0.6112	0.6151	0.6162	0.624	0.6166	7
4	0.7448	0.7495	0.75	0.7585	0.7507	4
5	0.556	0.5627	0.5593	0.5672	0.5613	10
6	0.5606	0.569	0.5643	0.5738	0.5669	9
7	0.3956	0.401	0.3984	0.4049	0.4	19
8	0.5495	0.561	0.5496	0.5572	0.5543	11
9	0.6791	0.6883	0.6842	0.6955	0.6868	6
10	1.0000	1.0000	1.0000	1.0000	1.0000	1
11	0.5005	0.5077	0.5045	0.5134	0.5065	14
12	0.5372	0.5449	0.5413	0.5505	0.5435	12
13	0.3558	0.3804	0.3779	0.3832	0.3743	20
14	0.5076	0.5142	0.5076	0.512	0.5103	13
15	0.7281	0.7385	0.7331	0.7452	0.7362	5
16	0.4117	0.4192	0.4117	0.4167	0.4148	18
17	0.3437	0.3479	0.3463	0.3517	0.3474	23
18	0.1937	0.1961	0.1948	0.1975	0.1955	30
19	0.3654	0.3698	0.3682	0.3739	0.3693	21
20	0.4218	0.4282	0.425	0.4325	0.4269	16
21	0.3204	0.3552	0.3223	0.3275	0.3313	24
22	0.281	0.2844	0.2832	0.2876	0.284	29
23	0.2927	0.298	0.2927	0.2962	0.2949	28
24	0.8176	0.8225	0.8283	0.8426	0.8278	2
25	0.3005	0.308	0.3005	0.3055	0.3036	26
26	0.1686	0.1707	0.1696	0.172	0.1702	31
27	0.7897	0.8002	0.7961	0.8096	0.7989	3
28	0.3604	0.3652	0.3632	0.3692	0.3645	22
29	0.3175	0.3214	0.3192	0.3236	0.3204	25
30	0.4175	0.4241	0.4207	0.4283	0.4227	17
31	0.293	0.2976	0.2953	0.3008	0.2967	27

6. Conclusions

This paper formulated a DEA model handled two-stage process and undesirable outputs in fuzzy random environment. Actually, the extended model depicts the influence of the presence of fuzziness and randomness in the data over the efficiency values. To do this, we have firstly incorporated an undesirable outputs in two-stage DEA model. The resulting model converted into a new model with some variable substitutions. Then, to solve the uncertainty part of model, we applied the possibility-probability measure. Our proposed model not only leads to a linear and always feasible program, as corresponding deterministic two-stage DEA models, but also it gives efficiency scores with the range of zero to one for all DMUs. In addition, a case study for banking industry is utilized to analyze the performance of some bank branches in Iran. For future study, a new measure in fuzzy stochastic programming can also be planned in chance constraint programming.

Footnotes

This paper takes Iranian Rial as currency rate.

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