Cost efficiency measurement with fuzzy data in DEA

Abstract

Being a nonparametric method, data envelopment analysis (DEA) is confined to measuring the efficiency of a collection of decision making units (DMUs) consuming multiple crisp inputs to produce multiple crisp outputs. Since not all data in the real world have determined values, and input and output values for DMUs are often subject to fluctuation, the concept of fuzziness has been introduced to deal with such imprecise data. This study intends to evaluate the cost efficiency of DMUs in three different scenarios, with one distinct model proposed for each scenario. The main idea is to use the alpha-cut method and the extension principle to convert the fuzzy cost efficiency model into a family of conventional crisp DEA models by obtaining one lower bound and one upper bound for the cost efficiency score of a DMU for any α varying between 0 and 1. When the lower and upper bounds are invertible with respect to α, the membership function of the fuzzy efficiency of a DMU— which falls within the scope of parametric programming problems— can be obtained by finding the inverse of the lower and upper bounds as well as employing the extension principle. In this case, the value of fuzzy cost efficiency varies between 0 and 1. Otherwise, the cost efficiency scores of DMUs can be specified as intervals, which are actually α-cuts of the fuzzy membership function, by collecting results for different α values. Furthermore, we demonstrate that Farrell’s decomposition also holds for cost efficiency with fuzzy data. In addition, all DMUs are divided into three classes in each scenario, with cost-efficient and cost-inefficient DMUs falling into independent classes. In other words, units which are cost-inefficient in the upper bound for any α ranging between 0 and 1 will definitely be cost-inefficient in the lower bound, too, and the units will be cost-efficient in the upper bound if they are cost-efficient in the lower bound for a specific α varying between 0 and 1. Moreover, the upper bound of units that are cost-inefficient in the lower bound could not be judged in terms of cost efficiency. Finally, a practical example dealing with data on all branches of the National Bank of Iran across Ardabil Province, Iran, during 2012–2014 is provided to demonstrate the applicability of the proposed method.

Keywords

Data envelopment analysis cost efficiency membership function parametric programming α -cut fuzzy

1 Introduction

One of the major topics in data envelopment analysis is cost efficiency measurement, on which so many studies have been conducted so far (See Subsection 1-1). The cost efficiency model, in fact, seeks to find a unit which consumes the minimum cost for purchasing inputs not greater than those to the unit under evaluation to produce outputs equal to those of the unit under evaluation. Thus, cost efficiency is defined as the ratio of minimum costs to observed costs [13]. It is necessary in conventional cost efficiency models that all inputs, outputs, and input prices have precise values, whereas it is difficult in the real world to determine their precise values. Hence, cost efficiency under data uncertainty is discussed in this paper, giving rise to the following two viewpoints: cost efficiency based on fuzzy data envelopment analysis (FDEA) and cost efficiency based on imprecise data envelopment analysis (IDEA).

Emrouznejad et al. [11] provided a review of the fuzzy DEA concepts and models. Most of the proposed methods for solving it are based on α-cut. For the first time, Cooper et al. [7] incorporated the concept of imprecise data into DEA. In general, imprecise data can be expressed in interval and ordinal numbers.

Herein is presented a method to extend the cost efficiency of DMUs to any of the following circumstances: input and output data are fuzzy numbers and the price vector is crisp, input and output data are crisp numbers and the price vector is fuzzy, or input and output data as well as the price vector are all fuzzy in nature. The presented model relies on the α-cut approach for finding the lower and upper bounds of cost efficiency, and on the extension principle for converting fuzzy DEA models into crisp ones. The fuzzy cost efficiency membership function, which is in the form of a parametric programming problem, can be obtained if the lower and upper bounds are invertible with respect to α. In this case, the membership function for cost efficiency is defined as the inverse of the lower and upper bounds. Thus, the cost efficiency score is a fuzzy number and varies within the interval [0,1]. On the other hand, if the lower and upper bounds are not invertible with respect to α, fuzzy data can be converted into crisp ones, using the α-cut approach. Hence, one upper bound and one lower bound are obtained for cost efficiency score in each case. In fact, the cost efficiency score changes over an interval, where the fuzzy membership function of DMUs can be obtained through plotting the figures on a graph. It is worth mentioning that there is no one-to-one correspondence between the membership function of cost efficiency scores and fuzzy data. In other words, that the price vector or input and output values are fuzzy in nature does not necessarily imply the cost efficiency score is undoubtedly not crisp; neither could it be inferred from trapezoidal or triangular fuzzy values for input and output data or the price vector that the cost efficiency score is definitely a triangular or trapezoidal fuzzy number. Moreover, units which are cost-efficient in the lower bound for a specific α belonging to [0,1] are also cost-efficient in the upper bound; besides, if they are cost-inefficient for any α ∈ [0, 1] in the upper bound, they will definitely be cost-inefficient in the lower bound, too. Units which are cost-inefficient in the lower bound, however, could be cost-efficient or cost-inefficient in the upper bound. Thus, units can generally be divided into three different classes based upon α values varying within [0,1] Furthermore, Farrell’s decomposition for cost efficiency with fuzzy data as well as the relationship between technical efficiency and allocative efficiency is graphically explained through an illustrative example.

1.1 Related literature

In today’s world, where all daily affairs revolve around economy, empirical and theoretical sciences, with all their abilities and capabilities, play a key role in economy. Accordingly, DEA— as a mathematical tool in this combined effort— makes useful contributions to economic sciences by addressing costs, incomes, and economic revenues. In 1957, Farrell [13] expressed his concerns about applying the accurate measurement of prices to cost efficiency. What he said in his paper became a basic principle in the theory of cost efficiency measurement with certain and constant prices for each decision making unit. He broke down cost efficiency to technical efficiency and allocative efficiency; this useful decomposition was later known as Farrell’s Decomposition. Although the proposed model by Farrell was basically used for cost efficiency measurement, incorrect pricing tables and assessments led Charnes and Cooper [5] to emphasize technical efficiency score and the necessity of using it. For the first time, Fare and Grosskopf [14] presented a practical method of cost efficiency and revenue efficiency by means of DEA. Then, many researchers considered the cost efficiency model as a DEA model to assess cost efficiency of DMUs. In this model, the production of outputs for a target DMU is evaluated by the DMU’s input prices. In 1997, Sueyoshi [31] analyzed the concept of productivity with respect to production and cost. Jahanshaloo et al. [20] presented a model for cost efficiency measurement that caused a substantial reduction in the volume of calculations through decreasing the number of constraints and variables. Kuosmanen et al. [24] conducted an industry and firm-level profit efficiency analysis using shadow prices. Mozaffari et al. [25] presented cost and revenue efficiency in DEA-Rmodels. In an attempt to analyze the efficiency of DMUs, Despic et al. [8] introduced a ratio-based comparative efficiency model known as the DEA-R model. Wei et al. [36] used the DEA-R model for the analysis of efficiency in healthcare systems. Agarwal [1] measured the efficiency of DMUs by a fuzzy data envelopment analysis (FDEA) model. Sokic [29] used a stochastic frontier approach to investigate the differences in the cost efficiency of the banking industry in Serbia and Montenegro. Ray [26] employed a centralized resource allocation model to measure cost efficiency in an Indian bank branch network, with the aim of determining optimal branches. Färe et al. [16] proposed two cost decompositions based on efficiency (inefficiency) score.

Economic instability in the real world, specifically the fluctuations in global prices, renders precise information on input and output prices inaccessible. In other words, it is impossible to precisely determine the numerical values of prices, and this result in uncertain prices. Uncertain prices could be fuzzy, ordinal, or interval numbers. Most of the studies conducted to expand DEA dealt with certain data, while in real-world problems, decisions rely on qualitative as well as quantitative data; that is to say, imprecise data are also likely to occur in practice. The term uncertain data envelopment analysis was introduced for the first time by Cooper et al. [7]. This term refers to models obtained through by incorporating interval and ordinal data into the classical DEA models. Toloo et al. [33] proposed a method for calculating overall profit efficiency with interval prices. Salehpour and Aghayi [28] calculated the most revenue efficiency with price uncertainty. Emrouznejad et al. [12] proposed two novel methods for measuring the overall profit Malmquist Productivity Index when the inputs, outputs, and price vectors are fuzzy or vary in intervals. Rostamy-Malkhalifeh and Aghayi [27] measured overall profit efficiency with fuzzy data. Bellman and Zadeh [4] and Zadeh [38] introduced the notion of fuzziness to address imprecise data from a quantitative point of view. They converted the DEA model, which is a linear programming model, into a fuzzy linear programming model when some of the constraints and objective functions are fuzzy in nature. Sengupta [30] conducted research on the use of fuzzy set theory. Using the membership function, he transformed the DEA model into a fuzzy linear programming model. Kao and Liu [23] utilized the α-cut method for the efficiency measurement of the BCC model with fuzzy data. They transformed fuzzy DEA models into a family of crisp ones with different α-level sets. Wang et al. [35] investigated the fuzzy DEA model based on fuzzy logic. They measured the lower, middle, and upper bounds for the efficiency score of DMUs. Wen and Li [37] evaluated fuzzy DEA models by using concave membership functions. Triantis and Girod [32] fuzzified non-radial DEA models in order to handle imprecise data. Jahanshahloo et al. [21] developed some of the concepts of DEA with fuzzy data. They extended the classic CE model, where prices are constant and known for each DMU, to a model with fuzzy prices. In 2009, Bagherzadeh [3] examined cost efficiency with triangular fuzzy input prices. He introduced a method for comparing the production cost of the DMU under evaluation with the minimum cost fuzzy set. Fang and Li [15] calculated cost efficiency in DEA under the law of one price (LoOP). They presented models to calculate the upper and lower bounds of cost efficiency for each firm in the case of non-unique LoOP prices. Taking account of convex and non-convex DEA technologies, Fukuyama and Khanjani [17] were able to achieve cost-effectiveness measures. Chen et al. [6] evaluated bank branches in Taiwan by Fuzzy SBM model. Using the α-cut approach, they were able to estimate lower and upper bounds for input and output data. Han et al. [19] proposed an efficiency analysis method based on fuzzy DEA cross-model with fuzzy data. Dotoli et al. [9] presented a cross-efficiency fuzzy DEA technique for performance evaluation of DMUs under uncertainty. They also provided a ranking of the DMUs by assigning fuzzy triangular numbers to them. Wanke et al. [34] used fuzzy DEA to assess productive efficiency in Nigerian airports. In an attempt to overcome uncertainty impacts of input–output life cycle assessment models on eco-efficiency, Egilmez et al. [10] coupled a fuzzy DEA model with an input–output-based life cycle assessment approach and used the resulting framework for the performance assessment of 33 food manufacturing sectors in the United States. Aghayi et al. [2] presented a robust efficiency measurement with common set of weights under varying degrees of conservatism and datauncertainty.

1.2 Organization

The rest of the paper is organized as follows. Section 2 provides the concepts of cost efficiency with crisp data. Basic fuzzy concepts are presented in Section 3. The cost efficiency model with fuzzy data is discussed in Section 4 under the constant returns-to-scale assumption; Farrell’s decomposition for cost efficiency, when data are fuzzy numbers, is also studied in this section. A practical example, where the branches of the National Bank of Iran are examined, is given in Section 5 to demonstrate how cost efficiency is calculated by the proposed model. Finally, conclusions are presented.

2 Cost efficiency measurement with crisp data

Suppose there are n DMUs with m inputs and s outputs as given below: $\begin{matrix} x_{j} & = & {(x_{1 j}, \dots, x_{mj})}^{T}, x_{j} \geq 0, x_{j} \neq 0, \\ y_{j} & = & {(y_{1 j}, \dots, y_{sj})}^{T}, y_{j} \geq 0, y_{j} \neq 0 . \end{matrix}$

Furthermore, the cost of purchasing inputs is given by c = (c₁, c₂, …, c_m) ^T, which is assumed to be constant across all DMUs. In addition, let DMU_k be the unit under evaluation.

In fact, cost efficiency seeks to find a unit which consumes the minimum cost for purchasing inputs not greater than those of the unit under evaluation to produce outputs equal to those of the unit under evaluation. The primary cost efficiency model for the assessment of DMUs is as follows:

$\begin{matrix} min \sum_{i = 1}^{m} c_{i} x_{i} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} x_{ij} \leq x_{i}, \forall i = 1, \dots, m, \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{rk}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{matrix}$ (1)

where λ_j (j = 1, ⋯ , n) and x_i (i = 1, ⋯ , m) are variables, and x_i represents the amount of inputs required to produce a given amount of outputs.

If (x^*, λ^*) is the optimal solution to the above problem, then the overall cost efficiency of the k-th unit is defined as follows: ${CE}_{k} = \frac{\sum_{i = 1}^{m} c_{i} x_{i}^{*}}{\sum_{i = 1}^{m} c_{i} x_{ik}} .$

The value of CE_k always varies within the interval [0,1] and DMU_k is cost-efficient if and only ifCE_k = 1.

3 Basic concepts of fuzzy logic

Each single element can either belong to a set A denoted by A ⊆ X or not belong to A written as A⊈X. According to the fuzzy set theory, however, each element of X belongs to A with some degree of membership.

Definition 1. If X is a collection of elements, denoted by x, then a fuzzy set $\tilde{A}$ in X is represented as $\tilde{A} = {(x, μ_{\tilde{A}} (x)) | x \in X}$ , where $μ_{\tilde{A}} (x)$ is called the membership function or degree of membership of x in $\tilde{A}$ . The range of the membership function is [0,1].

Definition 2. The α -cut of the fuzzy set $\tilde{A}$ is defined as follows:

${\tilde{A}}_{α} = {x \in X | μ_{\bar{A}} (x) \geq α},$

Where α is any scalar in [0,1].

4 Cost efficiency measurement with fuzzy data

In this section, we consider the following three scenarios for cost efficiency measurement: (1) input and output data are fuzzy numbers, while the input price vector has crisp values; (2) inputs and outputs have crisp values, and the input price vector is fuzzy in nature; (3) inputs, outputs, and the price vector are all fuzzy in nature. A model for cost efficiency measurement and a definition of a cost-efficient unit are presented in each scenario. In addition, we consider a simple case, where four DMUs with two inputs— which can take fuzzy values as well— and a fixed output equal to one are examined. We draw Farrell’s frontier for these DMUs in both upper and lower bounds, and calculate cost efficiency in upper and lower bounds with respect to α. Since the calculated values are invertible with respect to α, we obtain the membership function of its efficiency score. Moreover, we demonstrate that Farrell’s decomposition also holds for cost efficiency with fuzzy data.

Scenario 1. Fuzzy input and output and crisp input price vector

Suppose there are n DMUs with m fuzzy inputs and s fuzzy outputs, whose corresponding vectors are ${\tilde{x}}_{j} = ({\tilde{x}}_{1 j}, \dots, {\tilde{x}}_{mj})^{T}, {\tilde{x}}_{j} \geq 0, {\tilde{x}}_{j} \neq 0$ and ${\tilde{y}}_{j} = ({\tilde{y}}_{1 j}, \dots, {\tilde{y}}_{sj})^{T}, {\tilde{y}}_{j} \geq 0, {\tilde{y}}_{j} \neq 0$ , respectively. Fuzzy inputs and outputs are represented by convex fuzzy sets with $μ_{{\tilde{x}}_{ij}}$ and $μ_{{\tilde{y}}_{rj}}$ membership functions, respectively. Besides, c = (c₁, c₂, …, c_m) ^T, c ≥ 0, c ≠ 0 is the vector of input prices, which are crisp and assumed to be constant across all DMUs. Let DMU_k be the unit under evaluation. Thus, the following model is presented for cost efficiencymeasurement:

$\begin{matrix} (C {\tilde{E}}_{k}) = min \frac{\sum_{i = 1}^{m} c_{i} x_{i}}{\sum_{i = 1}^{m} c_{i} x_{ik}} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} {\tilde{x}}_{ij} \leq x_{i}, \forall i = 1, \dots m, \\ \sum_{j = 1}^{n} λ_{j} {\tilde{y}}_{rj} \geq {\tilde{y}}_{rk}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{matrix}$ (2)

Model (2) is a fuzzy programming problem, thus the obtained cost efficiency scores from this model should also be fuzzy numbers. In order to obtain these scores, the membership function of cost efficiency needs to be found. Let $S ({\tilde{Y}}_{ij})$ and $S ({\tilde{X}}_{ij})$ be the supports of ${\tilde{Y}}_{ij}$ and ${\tilde{X}}_{ij}$ , respectively, then the -cuts of ${\tilde{X}}_{ij}$ and ${\tilde{Y}}_{ij}$ are defined as the following crispsets: $(X_{ij})_{α} = {x_{ij} \in S ({\tilde{X}}_{ij}) | μ_{{\tilde{X}}_{ij}} (x_{ij}) \geq α}, \forall i, j,$ $(Y_{rj})_{α} = {y_{ik} \in S ({\tilde{Y}}_{rj}) | μ_{{\tilde{X}}_{ik}} (y_{rj}) \geq α}, \forall r, j .$

Hence, the fuzzy DEA model is converted into a family of crisp DEA models with different α-cuts {(X_ij) _α|0 < α ≤ 1} and {(Y_rj) _α|0 < α ≤ 1}.

The α-level sets shown in the above relations are represented as follows: $\begin{matrix} (X_{ij})_{α} & = & [min_{x_{ij}} {x_{ij} \in S ({\tilde{X}}_{ij}) | μ_{{\tilde{X}}_{ij}} (x_{ij}) \geq α}, \\ max_{x_{ij}} {x_{ij} \in S ({\tilde{X}}_{ij}) | μ_{{\tilde{X}}_{ij}} (x_{ij}) \geq α}], \end{matrix}$ $\begin{matrix} (Y_{rj})_{α} & = & [min_{y_{rj}} {y_{rj} \in S ({\tilde{Y}}_{rj}) | μ_{{\tilde{Y}}_{rj}} (y_{rj}) \geq α}, \\ max_{y_{rj}} {y_{rj} \in S ({\tilde{Y}}_{rj}) | μ_{{\tilde{Y}}_{rj}} (y_{rj}) \geq α}] . \end{matrix}$

In this case, according to the extension principle, the membership function of the cost efficiency of DMU_k can be defined as follows: $\begin{matrix} μ_{{\tilde{E}}_{k}} (z) & = & sup_{x, y} min {μ_{{\tilde{x}}_{ij}} (x_{ij}), μ_{{\tilde{Y}}_{rj}} (y_{rj}), \\ \forall i, j, r | z = E_{k} (x, y)} . \end{matrix}$

The upper and lower bounds of the α-cut of $μ_{C {\tilde{E}}_{k}}$ are required to obtain the membership function of $μ_{C {\tilde{E}}_{k}}$ : $\begin{array}{l} {(C E_{k})}_{α}^{L} \\ = \min_{\begin{matrix} {(x_{i j})}_{α}^{L} \leq x_{i j} \leq {(x_{i j})}_{α}^{U} \\ {(y_{r j})}_{α}^{L} \leq y_{r j} \leq {(y_{r j})}_{α}^{U} \\ \forall i, j, r \end{matrix}} { \begin{array}{l} C E_{k} = \min \frac{\sum_{i = 1}^{m} c_{i} x_{i}}{\sum_{i = 1}^{m} c_{i} x_{i k}} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} x_{i j} \leq x_{i}, \forall i = 1, \dots m, \\ \sum_{j = 1}^{n} λ_{j} y_{r j} \geq y_{r k}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{array} \end{array}$ (3) $\begin{array}{l} {(C E_{k})}_{α}^{U} \\ = \max_{\begin{matrix} {(x_{i j})}_{α}^{L} \leq x_{i j} \leq {(x_{i j})}_{α}^{U} \\ {(y_{r j})}_{α}^{L} \leq y_{r j} \leq {(y_{r j})}_{α}^{U} \\ \forall i, j, r \end{matrix}} { \begin{array}{l} C E_{k} = \min \frac{\sum_{i = 1}^{m} c_{i} x_{i}}{\sum_{i = 1}^{m} c_{i} x_{i k}} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} x_{i j} \leq x_{i}, \forall i = 1, \dots m, \\ \sum_{j = 1}^{n} λ_{j} y_{r j} \geq y_{r k}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{array} \end{array}$ (4)

Models (3) and (4) are used for measuring the lower and upper bounds of cost efficiency, respectively. Model (3) calculates the lower bound of cost efficiency, so the inputs of the unit under evaluation should lie in the upper bound, with the inputs of other DMUs lying in the lower one. Furthermore, the output of the unit under evaluation and the outputs of other DMUs lie in the lower and upper bounds, respectively. Likewise, model (4) calculates the upper bound of cost efficiency, so the inputs of the unit under evaluation should lie in the lower bound, with the inputs of other DMUs lying in the upper one. In addition, the output of the unit under evaluation and the outputs of other DMUs lie in the upper and lower bounds, respectively. Thus, models (3) and (4) can be replaced with linear programming (LP) models (5) and (6), respectively. $\begin{array}{l} {(C E_{k})}_{α}^{L} = \min \frac{\sum_{i = 1}^{m} c_{i} x_{i}}{\sum_{i = 1}^{m} c_{i} {(x_{i k})}_{α}^{U}} \\ s . t . \\ {λ_{k} (x_{i k})}_{α}^{u} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(x_{i j})}_{α}^{L} \leq x_{i}, \forall i = 1, \dots m, \\ {λ_{k} (y_{r k})}_{α}^{L} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(y_{r j})}_{α}^{U} \geq {(y_{r k})}_{α}^{L}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{array}$ (5) $\begin{array}{l} {(C E_{k})}_{α}^{U} = \min \frac{\sum_{i = 1}^{m} c_{i} x_{i}}{\sum_{i = 1}^{m} c_{i} {(x_{i k})}_{α}^{L}} \\ s . t . \\ λ_{k} {(x_{i k})}_{α}^{L} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(x_{i j})}_{α}^{U} \leq x_{i}, \forall i = 1, \dots m, \\ λ_{k} {(y_{r k})}_{α}^{U} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(y_{r j})}_{α}^{L} \geq {(y_{r k})}_{α}^{U}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{array}$ (6)

Since the input and output values of DMUs in models (5) and (6) vary for different values of α ∈ [0, 1], there will be a different cost efficiency score for each α. Hence, the above models are classified as parametric programming problems [18, 22].

If $({CE}_{k})_{α}^{L}$ and $({CE}_{k})_{α}^{U}$ are invertible with respect to α, then the membership function of cost efficiency can be obtained using $μ_{C {\tilde{E}}_{k}} (Z) = {\begin{matrix} L (Z) = [({CE}_{k})_{α}^{L}]^{- 1} & Z_{1} \leq Z \leq Z_{2}, \\ 1 & Z_{2} \leq Z \leq Z_{3}, \\ R (Z) = [({CE}_{k})_{α}^{U}]^{- 1} & Z_{3} \leq Z \leq Z_{4} . \end{matrix}$ (7)

Otherwise, the cost efficiency of DMU_k is obtained in an interval like ${[{({CE}_{k})}_{α}^{L}, {({CE}_{k})}_{α}^{U}] | α \in [0, 1]}$ , which actually reflects the membership function $μ_{C {\tilde{E}}_{k}}$ . That is to say, the corresponding α-cut of the cost efficiency of DMU_k for α ∈ [0, 1], is obtained.

When the objective function of model (5) is equal to one for a specific α varying in the range of [0,1], DMU_k is cost-efficient at that particular level. Otherwise, if $({CE}_{k})_{α}^{L} < 1$ , then DMU_k is cost-inefficient in the lower bound at level α. Similarly, if $({CE}_{k})_{α}^{U} = 1$ , then DMU_k is cost-efficient in the upper bound at level α; and if $({CE}_{k})_{α}^{U} < 1$ , then DMU_k is cost-inefficient in the upper bound at level α. Therefore, DMUs can be divided into the following classes at level α:

Class 1: Includes DMUs that are cost-efficient in the lower bound at level α. Since $(CE)_{α}^{L} \leq (CE)_{α}^{U}$ , these DMUs are also cost-efficient in the upper bound at level α, so ${(E)}_{α}^{+ +} = {{DMU}_{j} | {({CE}_{j})}_{α}^{L} = 1} .$

Class 2: Includes DMUs which are cost-inefficient in the upper bound at level α and, thus, are definitely cost-inefficient in the lower bound, too. Hence, ${(E)}_{α}^{-} = {{DMU}_{j} | {({CE}_{j})}_{α}^{U} < 1}$ .

Class 3: Includes DMUs that are cost-inefficient in the lower bound, but nothing could be said about the cost efficiency of these DMUs in the upper bound. Hence, ${(E)}_{α}^{+} = {{DMU}_{j} | {({CE}_{j})}_{α}^{L} < 1}$ .

It is mentionable that if $(\tilde{λ}, {\tilde{x}}_{i})$ , $(λ^{*}, x_{i}^{*})$ , and $(\bar{λ}, {\bar{x}}_{i})$ are the optimal solutions to models (2), (5), and (6), respectively, then $({CE}_{k})_{α}^{L} \leq (C {\tilde{E}}_{k}) \leq ({CE}_{k})_{α}^{U} .$

Scenario 2. Crisp input and output and fuzzy input price vector

Suppose there are n DMUs with m inputs and s outputs, whose corresponding vectors are x_j = (x_1j, …, x_mj) ^T, x_j ≥ 0, x_j ≠ 0 and y_j = (y_1j, …, y_sj) ^T, y_i ≥ 0, y_i ≠ 0, respectively. Furthermore, fuzzy input prices, which are assumed to be constant across all DMUs, are represented by $\tilde{c} = ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{m})^{T}, c \geq 0, c \neq 0$ . Let DMU_k be the unit under evaluation. Thus, the cost efficiency model with fuzzy linear programming problem is given by

$\begin{matrix} (C {\tilde{E}}_{k}) = min \frac{\sum_{i = 1}^{m} {\tilde{c}}_{i} x_{i}}{\sum_{i = 1}^{m} {\tilde{c}}_{i} x_{ik}} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} x_{ij} \leq x_{i}, \forall i = 1, \dots m, \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{rk}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{matrix}$ (8)

Similar scenario 1, we can compute the lower and upper of cost efficiency score with linear programming models (9) and (10), respectively,

$\begin{matrix} ({CE}_{k})_{α}^{L} = min \frac{\sum_{j = 1}^{m} (c_{i})_{α}^{L} x_{i}}{\sum_{j = 1}^{m} (c_{i})_{α}^{L} (x_{ik})} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} x_{ij} \leq x_{i}, \forall i = 1, \dots m, \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{rk}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{matrix}$ (9)

$\begin{matrix} ({CE}_{k})_{α}^{U} = min \frac{\sum_{i = 1}^{m} (c_{i})_{α}^{U} x_{i}}{\sum_{i = 1}^{m} (c_{i})_{α}^{U} (x_{ik})} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} x_{ij} \leq x_{i}, \forall i = 1, \dots m, \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{rk}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{matrix}$ (10)

where λ_j ≥ 0 (j = 1, …, n) and x_i (i = 1, …, m) are variables, and x_i represents the amount of inputs required to produce a given amount of outputs. All DMUs can be divided into three classes like scenario 1.

Scenario 3. Fuzzy input, output, and input price vector

Suppose there are n DMUs with m fuzzy inputs and s fuzzy outputs. Moreover, let the input price vector, which is assumed to be constant across all DMUs, be fuzzy in nature. Then, the following model is presented to assess DMUs:

$\begin{matrix} (C {\tilde{E}}_{k}) = min \frac{\sum_{i = 1}^{m} {\tilde{c}}_{i} x_{i}}{\sum_{i = 1}^{m} {\tilde{c}}_{i} x_{ik}} \\ s . t . \\ \sum_{j = 1}^{n} λ_{j} {\tilde{x}}_{ij} \leq x_{i}, \forall i = 1, \dots m, \\ \sum_{j = 1}^{n} λ_{j} {\tilde{y}}_{rj} \geq {\tilde{y}}_{rk}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{matrix}$ (11) where ${\tilde{c}}_{i}$ , ${\tilde{y}}_{rj}$ and ${\tilde{x}}_{ij}$ represent the price of the i-th fuzzy input, the r-th fuzzy output of DMU_j, and the i-th fuzzy input to DMU_j, respectively.

As shown earlier in scenarios (1) and (2), LP models can be written as follows: $\begin{array}{l} {(C E_{k})}_{α}^{L} = \min \frac{\sum_{i = 1}^{m} {(c_{i})}_{α}^{L} x_{i}}{\sum_{i = 1}^{m} {(c_{i})}_{α}^{L} {(x_{i k})}_{α}^{U}} \\ s . t . \\ λ_{k} {(x_{i k})}_{α}^{U} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(x_{i j})}_{α}^{L} \leq x_{i}, \forall i = 1, \dots m, \\ λ_{k} {(y_{r k})}_{α}^{L} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(y_{r j})}_{α}^{U} \geq {(y_{r k})}_{α}^{L}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{array}$ (12) $\begin{array}{l} {(C E_{k})}_{α}^{U} = \min \frac{\sum_{i = 1}^{m} {(c_{i})}_{α}^{U} x_{i}}{\sum_{i = 1}^{m} {(c_{i})}_{α}^{U} {(x_{i k})}_{α}^{L}} \\ s . t . \\ λ_{k} {(x_{i k})}_{α}^{L} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(x_{i j})}_{α}^{U} \leq x_{i}, \forall i = 1, \dots m, \\ λ_{k} {(y_{r k})}_{α}^{U} + \sum_{\begin{matrix} j = 1 \\ j \neq k \end{matrix}}^{n} λ_{j} {(y_{r j})}_{α}^{L} \geq {(y_{r k})}_{α}^{U}, \forall r = 1, \dots, s, \\ λ_{j} \geq 0, \forall j = 1, \dots, n . \end{array}$ (13)

In this scenario, like the previous ones, DMUs can be divided into three classes.

Now, let’s consider a simple case where we deal with four DMUs with two inputs and one output, as shown in Table 1 and Fig. 1.

For simplicity, all DMUs are assumed to produce an output equal to 1. Furthermore, c = (3, 2) is the input price vector for all DMUs. Since the second input is a triangular fuzzy number for DMU_C, the corresponding α-cuts is (3 +5α, 13 - 5α). Thus, using this α-cuts, we can obtain the membership functions of DMU_C as follows: $μ_{{\tilde{X}}_{C}} (x) = {\begin{matrix} \frac{x - 3}{5} & 3 \leq x \leq 8 \\ \frac{13 - x}{5} & 8 \leq x \leq 13 \\ 0 & o . w \end{matrix}$ (14)

At this point, in order to calculate the fuzzy efficiency scores, we find the corresponding membership functions of $C {\tilde{E}}_{C}$ . To this end, the upper and lower bounds should be considered for DMU_C according to what was mentioned in scenario 1.

$\begin{matrix} ({CE}_{C})_{α}^{L} = min \frac{3 x_{1} + 2 x_{2}}{3 (40) + 2 (13 - 5 α)} \\ s . t . \\ 10 λ_{1} + 20 λ_{2} + 40 λ_{3} + 50 λ_{4} \leq x_{1}, \\ 40 λ_{1} + (10 + 5 α) λ_{2} + (13 - 5 α) λ_{3} + 3 λ_{4} \leq x_{2}, \\ λ_{1} + λ_{2} + λ_{3} + λ_{4} \geq 1, \\ λ_{j} \geq 0 . \end{matrix}$ (15)

$\begin{matrix} ({CE}_{C})_{α}^{U} = min \frac{3 x_{1} + 2 x_{2}}{3 (40) + 2 (3 + 5 α)} \\ s . t . \\ 10 λ_{1} + 20 λ_{2} + 40 λ_{3} + 50 λ_{4} \leq x_{1}, \\ 40 λ_{1} + (25 - 5 α) λ_{2} + (3 + 5 α) λ_{3} + 3 λ_{4} \leq x_{2}, \\ λ_{1} + λ_{2} + λ_{3} + λ_{4} \geq 1, \\ λ_{j} \geq 0 . \end{matrix}$ (16)

It should be noted that, according to Fig. 1, the lower and upper bounds for the cost efficiency of DMU_C are calculated for α ∈ [0, 1]. Cost efficiency scores at α = 0.1 and α = 0 in both upper and lower bounds can be obtained from the length of line segments in Fig. 1. $\begin{matrix} α = 0 \\ ({CE}_{C})_{α}^{L} & = & \frac{{OR}_{1}}{OC} = \sqrt{0.402} = 0.634 \\ ({CE}_{C})_{α}^{U} & = & \frac{{OR}_{1}^{'}}{OC} = \sqrt{0.567} = 0.753 \end{matrix}$ (17) $\begin{matrix} α = 0.1 \\ ({CE}_{C})_{α}^{L} & = & \frac{{OR}_{2}}{OC} = \sqrt{0.396} = 0.629 \\ ({CE}_{C})_{α}^{U} & = & \frac{{OR}_{2}^{'}}{OC} = \sqrt{0.575} = 0.758 \end{matrix}$ (18)

It is obvious that DMU_C is cost-inefficient in both lower and upper bounds for α ∈ [0, 1].

Now, we look at Farrell’s decomposition under this condition. Let us consider DMU_C. According to Fig. 1, OR′/ - OQ′ obviously gives the value of allocative efficiency, which is indeed equal to the cost efficiency of DMU_Q′, and OQ′/OC gives the value of technical efficiency. Thus, the allocative efficiency associated with the corresponding cost efficiency of DMU_C can be given as follows: ${AE}_{C} = \frac{{OE}_{C}}{{TE}_{C}}$ (19)

Where TE_C, AE_C, and OE_C represent the technical, allocative, and cost efficiency of DMU_C respectively.

Considering relation ${OE}_{C} = {AE}_{C} \times {TE}_{C} \Rightarrow \frac{{OR}^{'}}{{OC}^{'}} = \frac{{OR}^{'}}{{OQ}^{'}} \times \frac{{OQ}^{'}}{{OC}^{'}}$

In fact, based on the preceding relation, the result of multiplying allocative efficiency by technical efficiency is always equal to cost efficiency. Since this relation holds for any DMU, Farrell’s decomposition also holds in this case, where input data are fuzzy numbers.

5 A numerical example and case study

In this section, we put the proposed models into practice by providing a simple example including four DMUs, each with two inputs, two outputs, and two price vectors. The data set is shown in Table 2.

The first input and output, as well as the first input price vector, have fuzzy values, which can be written as intervals by using α-cut (Table 3).

Since it is difficult to find the membership function of cost efficiency for all DMUs, the cost efficiency scores of the DMUs are specified as intervals, with the lower and upper bounds given by models (16) and (17), respectively, for α ∈ [0, 1]. The resulting cost efficiency scores of the DMUs for different α values are shown in Table 4.

Hence, the DMUs can be divided into the following three classes: $\begin{matrix} (E)_{α}^{+ +} & = & {{DMU}_{B}, {DMU}_{C}} \\ α = 0.7, α = 0.8, α = 0.9, α = 1 \\ (E)_{α}^{+ +} & = & {{DMU}_{C}} \\ α = 0, α = 0.1, α = 0.2, α = 0.3, \\ α = 0.4, α = 0.5, α = 0.6 \\ (E)_{α}^{_} & = & {{DMU}_{A}, {DMU}_{D}} \forall α \in [0, 1] \\ (E)_{α}^{+} & = & {{DMU}_{B}} \\ α = 0, α = 0.1, α = 0.2, α = 0.3, \\ α = 0.4, α = 0.5, α = 0.6 \end{matrix}$

Next, we will show that the obtained intervals for cost efficiency scores reflect the membership function of the DMUs.

As clearly shown in Fig. 2, DMU_C is cost-efficient, and DMU_A and DMU_D are cost-inefficient; besides, DMU_B is cost-efficient in the upper bound for α ∈ {0, …, 1} and cost-efficient in the lower bound for α ∈ {0.7, …, 1}. It should be noted that although the first input price vector for DMU_C is fuzzy, its corresponding cost efficiency score is crisp. In other words, the membership function of $C {\tilde{E}}_{C}$ equals one. In addition, it should be taken into consideration that there is no one-to-one correspondence between observations and the membership functions of cost efficiency scores. For example, the membership function of the first input, output, and input price vector for DMU_D is trapezoidal, while its cost efficiency score is an irregular shape rather than a trapezoid.

Furthermore, the membership function of the first input and input price vector for DMU_A is triangular, whereas its cost efficiency score is a trapezoidal fuzzy number.

Next, we run the proposed model with data taken from [23], including one fuzzy input and one fuzzy output, and compare the resulting cost efficiency scores with efficiency scores estimated by Agarwal [1]. For simplicity, we assume that the input price vector is crisp, constant across all DMUs, and equal to one. Efficiency scores and cost efficiency scores for different alpha values are presented inTable 5.

Different alpha values 0.1 apart, starting at 0 and increasing to 1, are given in the first column (on the left-hand side) of Table 5. Efficiency scores obtained from [1] and cost efficiency scores calculated by the proposed models are shown for each DMU in the next four columns. Since the input price vector is assumed to be equal to one, it is obvious that efficiency scores and cost efficiency scores are identical for each of the DMUs, as shown in Table 5. Obviously, DMU_A is efficient and cost-efficient both in the lower bound and in the upper bound for any α ∈ [0, 1], while other DMUs are inefficient and cost-inefficient.

Finally, a triangular fuzzy number dataset of 30 branches of the National Bank of Iran across Ardabil Province, Iran, for the years 2012–2014 is used to illustrate the applicability of the proposed models. Each branch is assumed to be a DMU that consumes non-performing loans and the sum of four major deposits as input variables to produce two outputs, i.e. profit and the balance of non-governmental deductions. As a result of the execution of models (16) and (17), cost efficiency scores of all branches are specified as intervals, the lower and upper bounds of which are shown in Tables 6 and 7, respectively, for different alpha values.

Tables 6 and 7 show that both lower and upper bounds of the cost efficiency score of DMU₅ are equal to one for any level of α. Hence, DMU₅ is cost-efficient, that is, ${(E)}_{α}^{+ +} = {{DMU}_{5}} \forall α \in [0, 1]$

As expected, there is at least one cost-efficient unit. All units that are cost-inefficient in the upper bound fall into class $(E)_{α}^{-}$ .

6 Conclusions

DEA is widely used to evaluate the efficiency of a set of DMUs consuming several inputs to produce several outputs. The existing DEA models are usually restricted to crisp data. In some cases, the input and output data of DMUs (e.g. service quality, resource quality, etc.) cannot be precisely measured. Thus, the uncertainty principle plays a key role in DEA. This paper aims to extend the conventional cost efficiency models to a fuzzy framework by proposing a new method for measuring the cost efficiency of a set of DMUs under the assumption of constant returns to scale in three different scenarios, with a distinct model offered for each scenario. The proposed model relies on the α-cut approach and the extension principle for converting the fuzzy DEA model into a crisp one. That is to say, the lower and upper bounds of cost efficiency scores can be obtained through the employment of α-cuts. If the upper and lower bounds are invertible with respect to α, the membership function of the fuzzy cost efficiency score of a DMU, which falls within the scope of the parametric programming problem, is obtained for α ∈ [0, 1]. Otherwise, fuzzy cost efficiency scores can be specified as intervals between 0 and 1, which are actually the corresponding α-cuts, by collecting results for different α values. This study reveals the fact that there is no one-to-one correspondence between the membership functions of cost efficiency scores and fuzzy data. In other words, fuzzy input and output data or fuzzy price vectors do not imply that the cost efficiency score is certainly not a crisp value; neither could it be concluded from trapezoidal or triangular fuzzy data that the cost efficiency score is definitely a trapezoidal or triangular fuzzy number. Furthermore, Farrell’s decomposition for cost efficiency when data are fuzzy numbers was investigated in a simple example, where cost efficiency proved to be the result of multiplying allocative efficiency by technical efficiency. In the future, the cost efficiency of DMUs with desirable and undesirable fuzzy input and output data can be evaluated.

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