A new fuzzy additive model for determining the common set of weights in Data Envelopment Analysis

Abstract

The main advantage of Data Envelopment Analysis (DEA) is that it does not require any priori weights for inputs and outputs and allows individual DMUs to evaluate their efficiencies with the input and output weights that are only most favorable weights for calculating their efficiency. It can be argued that if DMUs are experiencing similar circumstances, then the pricing of inputs and outputs should apply uniformly across all DMUs. That is using of different weights for DMUs makes their efficiencies unable to be compared and not possible to rank them on the same basis. This is a significant drawback of DEA; however literature observed many solutions including the use of common set of weights (CSW). Besides, the conventional DEA methods require accurate measurement of both the inputs and outputs; however, crisp input and output data may not relevant be available in real world applications. This paper develops a new model for the calculation of CSW in fuzzy environments using fuzzy DEA. Further, a numerical example is used to show the validity and efficacy of the proposed model and to compare the results with previous models available in the literature.

Keywords

Data Envelopment Analysis common set of weights fuzzy DEA

1 Introduction

Data Envelopment Analysis (DEA) initially was proposed by Charnes et al. [4] (CCR model) is a widely used mathematical programming approach for assessing the relative efficiency of a set of homogeneous decision-making units (DMUs). The DMUs usually consume multi-inputs to produce multi-outputs. DEA evaluates the efficiency of each DMU by using a ratio of the weighted sum of outputs to the weighted sum of inputs. A significant advantage of using DEA is that it does not require the determination of any parametric specification of the representation of the production technology (i.e. production function) in order to extract the efficiency scores and it makes no assumptions concerning the internal operations of a DMU. The CCR model that considers as a constant returns to scale (CRS) was extended by Banker, Charnes, and Cooper [1] (BCC model) to evaluate the relative efficiency of DMUs under assumption of variable returns to scale (VRS).

One of the most important advantage at of DEA is the way that DEA calculates the weights for input and output indices since DEA does not require any priori weights for inputs and outputs and allows individual DMUs to evaluate their efficiencies with the input and output weights that are only most favorable weights, or multipliers, for calculating their efficiency. It can be argued that if DMUs are experiencing similar circumstances, then the pricing of inputs and outputs should apply uniformly across all DMUs and the use of different weights for any DMUs, makes their efficiencies unable to be compared and ranked on the same basis.

For example, to measure the performance of nations at the 2008 Beijing Olympic Games the Chiang et al. [7] proposed using DEA where they used number of medals in each category of Gold, Silver and Bronze as outputs, and population and Gross Domestic Product (GDP) of the country as inputs. It is obvious that the weights for outputs, i.e. the importance degree of Gold, Silver, and Bronze medals, should not be variant among nations, however, nations or their people always value Gold medals higher than Silver and then Bronze medals. A possible answer to deal with the problem of a nation’s efficiency in obtaining medals is to use of common set of weights (CSW).

There is huge support in the literature to use CSW since many researchers showed that is not logical to assign different weights for same indices when the DMUs are homogeneous. Common set of weights first proposed by cook et al. [8] and then completed by Roll et al. [24]. Briefly, purpose of CSW, is calculating only one weight for each of input and output indices, hence DMUs proceed on common basis toward calculation and comparison of their efficiency.

Recently DEA literature observed many more use of common weights; hence various models with different approaches have been introduced using CSW.

Beside the conventional DEA such as CCR and BBC models require accurate measurement of both inputs and outputs; however, crisp input and output data may not always be relevant in real world situations. The observed values of the input and output data in real-world problems are sometimes contains missing data, judgment data, or predictive data or in generally imprecise or vague data. One way to deal with the uncertain input and output data is to use of fuzzy numbers. Since the initial study of fuzzy DEA introduce by Sengupta [26, 27], there has been extent developments in this area (for example see: [9 , 14]). In recent years, many researchers have formulated different fuzzy DEA models to overcome with issue of uncertainty problem. One of the main features of fuzzy DEA is the representation of each of the participating DMUs in the best possible conditions, relative to the others. In this perspective, weights of input(s) and output(s) are allowed to vary freely within the general constraints in each run of the model. Weight flexibility in fuzzy DEA assessments is such that it can lead to some DMUs having all but their most favorable input(s) and output(s) ignored in their assessment. To overcome this problem, the paper suggests a method to find a CSW in fuzzy environments.

Hence the primary purpose of this is paper is to develop a fuzzy DEA model for calculating CSW with uncertain data. The proposed model is derived from fuzzy arithmetic approach introduced by Wang et al. [29]. We then introduce two methods for ranking DMUs with imprecise data. Some main advantages of the proposed models are 1) the models are linear and simple to solve, 2) the high discriminating power of models is clear from the results, 4) possibility of weights management according to appropriateness of problem by adding constraints to the model and 4) they are applicable in a variety of applications.

The rest of the paper is organized as follows. Section 2 presents summary of DEA. Section 3 gives a brief review of fuzzy DEA. Section 4 presents summary of additive model. Details of the proposed model are given in Section 5 while Section 6 investigates the validity and efficacy of the proposed model. A comparison study is also presented in this section. Conclusion and future research direction are given in Sections 7.

2 Data envelopment analysis

Efficiency measurement is a managerial concept that has long history in various topics of management science. The efficiency shows that an organization to what extent uses its resources for production in the best way possible [30].

Data Envelopment Analysis (DEA) is a non-parametric performance assessment methodology for assessing relative efficiencies within a group of DMUs. The purpose of this technique is achieving to relative efficiency decision making units, with several similar input(s) and output(s). In mathematical terms, consider a set of n DMUs, in which x_ij (i = 1, 2, …, m) and y_rj (r = 1, 2, …, s) respectively are inputs and outputs of DMU_j (j = 1, 2, …, n). The basic form of DEA model for assessing DMU₀ is written as: $\begin{matrix} Max Z_{0} = \frac{\sum_{r = 1}^{s} u_{r} y_{r 0}}{\sum_{i = 1}^{m} v_{i} x_{i 0}} \\ s . t . \frac{\sum_{r = 1}^{s} u_{r} y_{rj}}{\sum_{i = 1}^{m} v_{i} x_{ij}} \leq 1; (j = 1, 2, \dots, n) \\ u_{r}, v_{i} \geq ɛ \end{matrix}$ (1)

In the above model, v_i and u_r are the input and output weights assigned to the ith input and rth output, ɛ is a non-Archimedean value designed to enforce strict positivity on the variables. The imposition of a strictly positive lower limit (ɛ > 0) was introduced in Charnes et al. (1981). The above fractional model is convertible into linear programming (LP) model using Charnes, Cooper (1962) fractional programming transformation.

3 Fuzzy DEA

The conventional DEA methods require accurate measurement of both the inputs and outputs; however, crisp input and output data may not always be relevant in real world applications and the observed values of the input and output data are often imprecise or vague. Therefore, traditional DEA cannot be used for such problems. To overcome with this issue, researchers developed fuzzy concepts with bounded intervals, ordinal data or fuzzy numbers to DEA

Sengupta [26, 27] incorporated fuzzy input and output data into a DEA model and defined tolerance levels for the objective function and constraint violations using fuzzy DEA. Guo and Tanaka [12, 13] and León et al. [19] considered the uncertainties in fuzzy objectives and fuzzy constraints. Lertworasirikul et al. [21] proposed a fuzzy DEA model using the credibility approach. Kao and Liu [17, 18] transformed fuzzy input and output data into intervals by using α-level [11, 25, 11, 25], Lertworasirikul et al. [20] suggested a fuzzy DEA model to produce crisp efficiencies. Further development in fuzzy DEA can be found in Liu [22] and Liu and Chuang (2009) who extended the α-level set approach by proposing the assurance region approach in the fuzzy DEA model. Hatami-Marbini et al. (2010a, b) and Zerafat Angiz [31 , 34] proposed various alternations to fuzzy DEA. However none of these studies addressed the issue of common set of weights which is the main subject of this paper.

In this view, Model (1) can only be used for cases where the data are crisp. Model (2) is formulated by introducing fuzzy input–output variables to Model (1). $\begin{matrix} Max Z_{0} = \frac{\sum_{r = 1}^{s} u_{r} {\tilde{y}}_{r 0}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{i 0}} \\ s . t . \frac{\sum_{r = 1}^{s} u_{r} {\tilde{y}}_{rj}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij}} \leq 1; (j = 1, 2, \dots, n) \\ u_{r}, v_{i} \geq ɛ \end{matrix}$ (2)where outputs and inputs ${\tilde{y}}_{rj}$ and ${\tilde{x}}_{ij}$ , respectively, are fuzzy outputs and fuzzy inputs. In this paper, we use basic fuzzy arithmetic operations on triangular fuzzy input and output data to construct the fuzzy DEA model.

4 The additive model

Charnes et al. [3] introduced the additive model which, to an extent, combines both the input-oriented and the output-oriented. There are several versions of the additive model, the most basic being given by the linear optimization problem (multiplier model) shown as (3). $\begin{matrix} Max \sum_{r} u_{r} y_{r 0} - \sum_{i} v_{i} x_{i 0} \\ s . t . \frac{\sum_{r} u_{r} y_{rj}}{\sum_{i} v_{i} x_{ij}} \leq 1, \forall j \\ \sum_{i} v_{i} \geq 1 \\ \sum_{r} u_{r} \geq 1 \\ u_{r}, v_{i} \geq ɛ, \forall r, i \end{matrix}$ (3)

5 A new model for common sets of weights in fuzzy DEA

Let’s clarify the issue of assigning different weights to inputs and outputs for each DMU by a simple example. Assume there are two DMUs with efficiency of 1 and 0.5 that are obtained with different weights for inputs and outputs. However, if the weights of the first DMU are applied to the second DMU, the result may not be equal to 0.5, and if the weights of the second DMU are applied to the first DMU, then it may become inefficient.

For this the CSW has been proposed but CSW works only with crisp data. This paper develops a new model for the calculation of CSW under fuzzy environments.

Suppose input (x_ij) and output (y_rj), are imprecise or vague and characterized by triangular fuzzy numbers, ${\tilde{x}}_{ij} = (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})$ and ${\tilde{y}}_{rj} = (y_{rj}^{L}, y_{rj}^{M}, y_{rj}^{U})$ , respectively, that are positive fuzzy numbers. Crisp data can be consider as a special case of triangular fuzzy input and output data, ${\tilde{x}}_{ij} = (x_{ij}^{L} = x_{ij}^{M} = x_{ij}^{U})$ and ${\tilde{y}}_{rj} = (y_{rj}^{L} = y_{rj}^{M} = y_{rj}^{U})$ . In this case the fuzzy efficiency of DMU_j is defined as: ${\tilde{θ}}_{j} = \frac{\sum_{r = 1}^{s} u_{r} {\tilde{y}}_{rj}}{\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij}}$ (4)

According to the additive Model (3) the objective function of proposed Model (5) is seeking to maximize the efficiency of all DMUs (by combining both the input- and output-oriented), with the condition that efficiency ratio for all units are being less than or equal to one. Hence our proposed model in the crisp situation is as follows (see Troutt [28] for further details on Maximin efficiency ratio): $\begin{matrix} Max \underset{j \in N}{Min} {\sum_{r} U_{r} Y_{r}^{j} - \sum_{i} V_{i} X_{i}^{j}} \\ s . t . \frac{\sum_{r} U_{r} Y_{r}^{j}}{\sum_{i} V_{i} X_{i}^{j}} \leq 1; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i . \end{matrix}$ (5)

Where ɛ is a non-Archimedean value designed to enforce strict positivity on the variables. U and V are 1 × s and 1 × m row vectors with nonnegative elements. X is a m × n matrix and Y is a s × n matrix; X^j and Y^j are jth column of matrices X and Y, respectively. In the problem setting, matrices X and Y represent the inputs and outputs of all DMU’s, and the decision variables (U_r, V_i) represent a set of weights among inputs and outputs. This set of weights also is named as a common set of weights (CSW).

Let’s use triangular fuzzy numbers ${\tilde{x}}_{ij} = (x_{ij}^{L},$ $x_{ij}^{M}, x_{ij}^{U})$ and ${\tilde{y}}_{rj} = (y_{rj}^{L}, y_{rj}^{M}, y_{rj}^{U})$ for input and output variables, respectively, hence the above model can be written as follows: $\begin{matrix} Max \underset{j \in N}{Min} {\sum_{r} U_{r} (y_{rj}^{L}, y_{rj}^{M}, y_{rj}^{U}) - \sum_{i} V_{i} (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})} \\ s . t . \frac{\sum_{r} U_{r} (y_{rj}^{L}, y_{rj}^{M}, y_{rj}^{U})}{\sum_{i} V_{i} (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})} \leq 1; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i . \end{matrix}$ (6)

Let $\tilde{A} = (a_{L}, a_{M}, a_{U})$ and $\tilde{B} = (b_{L}, b_{M}, b_{U})$ be two positive triangular fuzzy numbers. Then basic fuzzy arithmetic operations on these fuzzy numbers are defined as:

Addition: $\tilde{A} + \tilde{B} = (a_{L} + b_{L}, a_{M} + b_{M}, a_{U} + b_{U})$ ;

Subtraction: $\tilde{A} - \tilde{B} = (a_{L} - b_{U}, a_{M} - b_{M},$ a_U - b_L);

Multiplication: $\tilde{A} \times \tilde{B} \approx (a_{L} b_{L}, a_{M} b_{M}, a_{U} b_{U})$ ;

Division: $\tilde{A} / \tilde{B} = \frac{(a_{L}, a_{M}, a_{U})}{(b_{L}, b_{M}, b_{U})} \approx (\frac{a_{L}}{b_{U}}, \frac{a_{M}}{b_{M}}, \frac{a_{U}}{b_{L}}) .$

According to fuzzy arithmetic, Model (6) can be expressed as:

$\begin{matrix} Max \underset{j \in N}{Min} {[\sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U}, \sum_{r} {U_{r} y_{rj}}^{M} \\ - \sum_{i} V_{i} x_{ij}^{M}, \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L}]} \\ s . t . \frac{[\sum_{r} U_{r} y_{rj}^{L}, \sum_{r} U_{r} y_{rj}^{M}, \sum_{r} U_{r} y_{rj}^{U}]}{[\sum_{i} V_{i} x_{ij}^{L}, \sum_{i} V_{i} x_{ij}^{M}, \sum_{i} V_{i} x_{ij}^{U}]} \\ \leq 1; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i . \end{matrix}$ (7)

Hence, it is easy also to see that the above model can be simplified to: $\begin{matrix} Max \underset{j \in N}{Min} {[\sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U}, \sum_{r} {U_{r} y_{rj}}^{M} \\ - \sum_{i} V_{i} x_{ij}^{M}, \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L}]} \\ s . t . [\frac{\sum_{r} U_{r} y_{rj}^{L}}{\sum_{i} V_{i} x_{ij}^{U}}, \frac{\sum_{r} U_{r} y_{rj}^{M}}{\sum_{i} V_{i} x_{ij}^{M}}, \frac{\sum_{r} U_{r} y_{rj}^{U}}{\sum_{i} V_{i} x_{ij}^{L}}] \\ \leq 1; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \end{matrix}$ (8) $\begin{matrix} \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i . \end{matrix}$

In the Model (8), as long as upper bound $(\frac{\sum_{r} U_{r} y_{rj}^{U}}{\sum_{i} V_{i} x_{ij}^{L}})$ is kept to be less than or equal to one, then $\frac{\sum_{r} U_{r} y_{rj}^{M}}{\sum_{i} V_{i} x_{ij}^{M}}$ and $\frac{\sum_{r} U_{r} y_{rj}^{L}}{\sum_{i} V_{i} x_{ij}^{U}}$ will be spontaneously satisfied. Therefore, Model (8) can be simplified to:

$\begin{matrix} Max \underset{j \in N}{Min} {[\sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U}, \sum_{r} {U_{r} y_{rj}}^{M} \\ - \sum_{i} V_{i} x_{ij}^{M}, \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L}]} \\ s . t . \frac{\sum_{r} U_{r} y_{rj}^{U}}{\sum_{i} V_{i} x_{ij}^{L}} \leq 1; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i . \end{matrix}$ (9)

Thus Model (9) is simplified to the following model: $\begin{matrix} Max \underset{j \in N}{Min} {[\sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U}, \sum_{r} {U_{r} y_{rj}}^{M} \\ - \sum_{i} V_{i} x_{ij}^{M}, \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L}]} \\ s . t . \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L} \leq 0; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i . \end{matrix}$ (10)

To transform Model (10) into a linear programming model, we suppose that θ denotes the term of $Min {[\sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U}, \sum_{r} U_{r} y_{rj}^{M} - \sum_{i} V_{i} x_{ij}^{M},$ $\sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L}]}$ and therefore Model (10) can be simplified as:

$\begin{matrix} Max θ \\ s . t . θ = \underset{j \in N}{Min} {[\sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U}, \sum_{r} {U_{r} y_{rj}}^{M} \\ - \sum_{i} V_{i} x_{ij}^{M}, \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L}]} \\ \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L} \leq 0; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i \\ θ : Free in Sign . \end{matrix}$ (11)

The above model can be transformed into linear programming (LP) model as follows: $\begin{matrix} Max θ \\ s . t . \sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U} \geq θ; j = 1, 2, \dots, N \\ \sum_{r} U_{r} y_{rj}^{M} - \sum_{i} V_{i} x_{ij}^{M} \geq θ; j = 1, 2, \dots, N \\ \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L} \geq θ; j = 1, 2, \dots, N \\ \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L} \leq 0; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i \\ θ : Free in Sign . \end{matrix}$ (12)

For Model (12), as long as $\sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U} \geq θ$ ,the constrains of $\sum_{r} U_{r} y_{rj}^{M} - \sum_{i} V_{i} x_{ij}^{M} \geq θ$ and $\sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L} \geq θ$ will be constantly followed, so these constrains are unnecessary and can be remove from the model. Therefore final model for the calculation of CSW in fuzzy environments is summarized as follows:

$\begin{matrix} Max θ \\ s . t . \sum_{r} U_{r} y_{rj}^{L} - \sum_{i} V_{i} x_{ij}^{U} \geq θ; j = 1, 2, \dots, N \\ \sum_{r} U_{r} y_{rj}^{U} - \sum_{i} V_{i} x_{ij}^{L} \leq 0; j = 1, 2, \dots, N \\ \sum_{r} U_{r} \geq 1; \\ \sum_{i} V_{i} \geq 1; \\ U_{r}, V_{i} \geq ɛ, \forall r, i \\ θ : Free in Sign . \end{matrix}$ (13)

One of the main advantage of the proposed model in this paper is its less computational complexity. To calculate the efficiency using Wang et al. [29] we need to run the model N times, once for each DMU, while in the proposed model we need to solve the model once only. Further, the final model (13) is a linear programming which can easily be solved by any LP-solver.

6 Illustration with numerical example

To illustrate the CSW approach with fuzzy DEA models, we consider the same example presented Wang et al. [29]. Consider the performance assessment problem of Wang et al. [29] with eight DMUs and two inputs (X₁, X₂) and two outputs (Y₁, Y₂) (one fuzzy input (X₁) and two fuzzy outputs (Y₁, Y₂)) presented inTable 1.

Wang et al. [29] proposed two Fuzzy DEA models based upon fuzzy arithmetic [12].

The results of the obtained common set of weights (CSW) from our proposed model are shown as Table 2.

Using of the above CSW, the efficiency score for DMU is calculated and presented in the last column of Table 3a and 3b. For comparison purpose, we have also listed the results of Wang et al. [29] in this table. $\tilde{θ} = [\frac{{Uy}_{j}^{L}}{{Vx}_{j}^{U}}, \frac{{Uy}_{j}^{M}}{{Vx}_{j}^{M}}, \frac{{Uy}_{j}^{U}}{{Vx}_{j}^{L}}]$ (14)

The fair and consistent comparison of DMUs and better discrimination power compared to Wang et al. [29] models are the advantages of our model (in the proposed CSW model, efficiency upper bound of one DMU is equal to 1.0000 while, Wang et al. [29] models have further efficiency bound equal to one andmore.

Table 3b shows the normalized measures by sum; i.e. we used

$\begin{matrix} \tilde{A} = (a_{i}, b_{i}, c_{i}) \\ {\tilde{N}}_{i} = \frac{(a_{i}, b_{i}, c_{i})}{(\sum_{i = 1}^{n} a_{i}, \sum_{i = 1}^{n} b_{i}, \sum_{i = 1}^{n} c_{i})} = (\frac{a_{i}}{\sum_{i = 1}^{n} c_{i}}, \frac{b_{i}}{\sum_{i = 1}^{n} b_{i}}, \frac{c_{i}}{\sum_{i = 1}^{n} a_{i}}) \end{matrix}$

To provide a full ranking of all DMUs, the following procedure is recommended:

1. Defuzzification of fuzzy efficiency by beta functions as: $θ = \frac{{\tilde{θ}}_{L} + 4 {\tilde{θ}}_{M} + {\tilde{θ}}_{U}}{6}$ (15)

2. Using the average of possibility of dominance.

It should be noted that we have selected defuzzification in Equation 15 but other type of defuzzification can also be used, we believe that the defuzzification would not have much impact in the results.

Generally, if M₁ = (l₁, m₁, u₁) and M₂ = (l₂, m₂, u₂) are being two triangular fuzzy numbers, the possibility of dominance of M₁ to M₂ is definedas:

${\begin{matrix} V (M_{1} \geq M_{2}) = 1; m_{1} \geq m_{2} \\ V (M_{1} \geq M_{2}) = hgt (M_{1} \cap M_{2}) = \frac{u_{1} - l_{2}}{(u_{1} - l_{2}) + (m_{2} - m_{1})}; Otherwise \end{matrix}$ (16)

The average of possibility dominance of a triangular fuzzy number to k triangular fuzzy number obtained as follows:

$\begin{matrix} EV (M_{1} \geq M_{2}, \dots, M_{k}) \\ = \frac{V (M_{1} \geq M_{2}) + \dots + V (M_{1} \geq M_{k})}{k - 1} \end{matrix}$ (17)

The result of using above methods for the full ranking of obtained Fuzzy efficiencies with proposed CSW model is shown in Table 4. As can be seen in this table, both models rank the DMUs quite similarly.

Table 5a reveals the fact that the proposed CSW model produces narrower fuzzy efficiency than Wang et al. [29] models. In other word, the fuzzy efficiencies obtained with proposed CSW model have less fuzziness than fuzzy efficiencies obtained with Wang et al. [29] models. This is confirmed in Table 5a, where fuzziness is measured by the length of fuzzy interval (fuzziness = upper bound- lower bound).

Using similar normalization as explained above, Table 5b shows the normalized measures by sum;

7 Conclusions

Flexibility of weights and requiring crisp input and output data are two significant drawbacks of the standard DEA models and have aroused considerable research interest in the DEA literature. In this paper we have developed a new model based upon fuzzy arithmetic operations for the calculation of CSW in fuzzy environments and also two methods for full ranking DMUs, are presented.

Clear rationale, applicable in a variety of standard DEA models, linearity and simplicity of the model, the high discriminating power and possibility of weights management according to appropriateness of problem (by adding constraints to the model) are the main advantages of the proposed model that conventional DEA models do not have this feature. Researcher interested could develop CSW to other fuzzy DEA models.

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