An inequality approach for evaluating decision making units with a fuzzy output 1

Abstract

In this paper, a fuzzy evaluation approach is developed to evaluate the efficiency of decision making units (DMUs) with a fuzzy output. In the approach, the production possibility set is spanned by all the DMUs except the DMU under evaluation, and expressed in terms of fuzzy inequalities. In addition, the line segment joining the origin to the evaluated DMU is employed in the proposed approach, and its mathematical expression is expressed by fuzzy inequality. Fuzzy efficiency of the evaluated DMU is dependent upon the number of solutions of fuzzy inequalities, and the fuzzy inequalities consist of inequalities of the production possibility set and the line segment. Moreover, the partially ordered set and maximal element are introduced to distinguish the weak efficiency and efficiency. Finally, the use of the fuzzy approach is illustrated by means of an example.

Keywords

Fuzzy evaluation inequality approach data envelopment analysis decision making unit maximal element

1 Introduction

In practical evaluation problems, inputs and outputs which are collected from observation or investigation are often fluctuated and imprecise. Fuzzy Data envelopment analysis (DEA) models have been proposed to evaluate the imprecise performances of many different kinds of entities engaged in many different kind of activities in many different contexts. Until now, there are many fuzzy DEA literature which were proposed to evaluate fuzzy efficiency of a set of entities called Decision Making Units (DMUs). For example, Khoshfetrat and Daneshvar [12] proposed fuzzy CCR model with the lower bounds of fuzzy data for inputs and outputs. and the weak efficiency frontiers of the corresponding production possibility set were improved. In the next year, Angiz et al. [2] introduced an alternative linear programming model, in which some uncertainty information with the α-cut approach was included. A few years later, Based on Free Disposal Hull (FDH) method which is basically DEA without the assumption of convexity, Hougaard and Baležentis [11] extended the crisp FDH-method to fuzzy data sets by mimicking the calculation of efficiency indexes for interval data (for each α-level set of triangular fuzzy data). Han et al. [8] proposed an efficiency analysis method based on fuzzy DEA cross-model for ethylene production systems in chemical industry, the proposed method has better objectivity and resolving power for the decision-making. Meng and Shi [19] proposed an extended DEA model with more general fuzzy data based upon the centroid formula. In the model, input and output data may be non-convex/convex and abnormal/normal. The centroid formula is introduced to deal with the fuzzy data. For more fuzzy DEA models, see [7 , 22–26].

Different from the fuzzy mathematical programming model, this paper proposed a fuzzy inequality evaluation approach to evaluate the efficiency of DMUs with a fuzzy output. Fuzzy efficiency of DEA models are obtained by the optimal objective value and the optimal weight vectors. However, fuzzy efficiency of this paper is dependent upon the number of solutions of the fuzzy inequalities, moreover, the maximal element is introduced in the approach to distinguish the fuzzy weak efficiency and fuzzy efficiency.

The rest of the paper is unfolded as follows. The initial DEA model, fuzzy DEA model and maximal element are reviewed in Section 2. In Section 3, the fuzzy inequality evaluation approach is proposed. In Section 4, an example is given to illustrate the presented approach. Discussion of the proposed approach is in Section 5. The paper is concluded in Section 6.

2 Preliminaries

2.1 DEA model

As an extremely common DEA model, the C²R model [4] supposes that there are n DMUs, and each DMU consumes the same type of inputs and produces the same type of outputs. Let m, r be the numbers of inputs and outputs, respectively. All inputs and outputs are assumed to be nonnegative, and at least one input and one output are positive. The multiple inputs and outputs of each DMU are aggregated into a single virtual input and a single virtual output. The efficiency of the evaluated DMU is obtained as a ratio of its virtual output to its virtual input, and is subject to the condition that the ratio for each DMU is not greater than 1. The corresponding model is as follows: $(C^{2} R) {\begin{matrix} \max \frac{u^{T} y_{0}}{v^{T} x_{0}}, \\ s . t . \\ \frac{u^{T} y_{j}}{v^{T} x_{j}} \leq 1, j = 1, \dots, n, \\ u \geq 0, u \neq 0, \\ v \geq 0, v \neq 0 . \end{matrix}$ (1) where x_j = (x_j1, ⋯ , x_jm) ^T and y_j = (y_j1, ⋯ , y_jr) ^T are the input and output vectors of the j-th DMU, and j₀ is the DMU under evaluation (usually denoted by DMU₀). u and v are the weight column vectors of output and input, respectively. u ≥ 0, u ≠ 0 represents the vector whose elements are not less than zero but at least one element is a positive value.

2.2 Fuzzy DEA model

The C²R model with fuzzy input and output data is given as follows. $({FC}^{2} R) {\begin{matrix} \max \frac{u^{T} {\tilde{y}}_{0}}{v^{T} {\tilde{x}}_{0}}, \\ s . t . \\ \frac{u^{T} {\tilde{y}}_{j}}{v^{T} {\tilde{x}}_{j}} \leq 1, j = 1, \dots, n, \\ u \geq 0, u \neq 0, \\ v \geq 0, v \neq 0 . \end{matrix}$ (2) where ${\tilde{x}}_{j} = ({\tilde{x}}_{j 1}, \dots, {\tilde{x}}_{jm})^{T}$ and ${\tilde{y}}_{j} = ({\tilde{y}}_{j 1}, \dots, {\tilde{y}}_{jr})^{T}$ are the fuzzy input and fuzzy output column vectors of the j-th DMU. By applying the Charnes-Cooper transformation [3] in the model (2), the following equivalent linear model is obtained: $({FP}_{C^{2} R}) {\begin{matrix} \max μ^{T} {\tilde{y}}_{0}, \\ s . t . \\ ω^{T} {\tilde{x}}_{j} - μ^{T} {\tilde{y}}_{j} \geq 0, \\ j = 1, \dots, n, \\ ω^{T} {\tilde{x}}_{0} = 1, \\ ω \geq 0, ω \neq 0, \\ μ \geq 0, μ \neq 0 . \end{matrix}$ (3)

The essence of the fuzzy model above is to find the fuzzy weight vector to maximizing its fuzzy weighted output of the evaluated DMU, and the fuzzy weighted output is not greater than the fuzzy weighted input for every DMU. Moreover, the fuzzy optimal objective values of DEA models vary in (0, 1]. The relationship between fuzzy efficiency and the optimal objective value can be obtained as follows.

Definition 1. If the optimal objective value of the evaluated DMU is equal to 1 and there is at least one optimal solution in which the optimal weight vectors are greater than 0, then the evaluated DMU is fuzzy efficient.

Definition 2. If the optimal objective value of the evaluated DMU is equal to 1 and there is not any optimal solution in which the optimal weight vectors of inputs and outputs are greater than 0, then the evaluated DMU is fuzzy weak efficient.

Definition 3. If the optimal objective value of the evaluated DMU is less than 1, then the evaluated DMU is fuzzy inefficient.

2.3 The partially ordered set and maximal element

Let P be a nonempty set, any subset of the cartesian product set P × P = {(x, y) |x, y ∈ P} is called a binary relation, denoted by R. a, b ∈ P, aRb if and only if (a, b) ∈ R [6].

Definition 4. A relation R is called a partial ordering on P such that, for all x, y, z ∈ P,

Reflective, x ≤ x,

Antisymmetric, x ≤ y and y ≤ x imply x = y,

Transitive, x ≤ y and y ≤ z imply x ≤ z.

A nonempty set P equipped with a partial order relation is said to be a partially ordered set, or poset for short. A partial order R is traditionally replaced by “≤”, i.e., we usually replace xRy by x ≤ y which is read as ‘x is less than or equal to y’.

Definition 5. P is a partially ordered set and Q ⊆ P, a ∈ Q is called a maximal element of Q if x ≥ a and x ∈ Q imply a = x.

For any nonempty finite subset S ⊆ P, there exists at least one maximal element x ∈ S.

3 The fuzzy inequality evaluation approach

In this section, the fuzzy inequality evaluation approach is proposed. It should be noted that all DMUs used from this section onwards are in the form of a single output.

3.1 The proposed approach

We assume that there are n DMUs with m fuzzy inputs and one fuzzy output. ${\tilde{x}}_{j} = ({\tilde{x}}_{j 1}, \dots, {\tilde{x}}_{jm})^{T}$ denotes the fuzzy input vector of the j-th DMU with j ranging from 1 to n. Without loss of generality, ${\tilde{y}}_{j} = ({\tilde{y}}_{j})^{T}$ is the fuzzy output vector with a single element for the j-th DMU. The fuzzy inequality evaluation approach for the evaluated DMU j₀ is as follows: ${\begin{matrix} x \geq \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} {\tilde{x}}_{j}, \\ y \leq \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} {\tilde{y}}_{j}, \\ \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} \geq 1, \\ x = λ_{0} {\tilde{x}}_{0}, \\ y = λ_{0} {\tilde{y}}_{0}, \\ λ_{0} \in (0, 1], \\ λ_{j} \geq 0 . \end{matrix}$ (4) where, ${\tilde{x}}_{0}$ and ${\tilde{y}}_{0}$ are the fuzzy input and output vectors of the evaluated DMU. x = (x₁, ⋯ , x_m) ^T and y = (y) ^T are unknown input and output vectors. In order to distinguish weak efficiency and efficiency, the partially ordered set is introduced in the paper.

Definition 6. The efficiency of DMU_k is higher or equal to the efficiency of DMU_l (i.e., DMU_l≤ DMU_k) if the following conditions hold:

${\tilde{x}}_{ki} \leq c {\tilde{x}}_{li}$ (i = 1, ⋯, m),

${\tilde{y}}_{l} \leq c {\tilde{y}}_{k} (c is a positive number)$ .

Definition 7. The efficiency of DMU_k is equal to the efficiency of DMU_l if and only if ${\tilde{x}}_{ki} = c {\tilde{x}}_{li}$ , and ${\tilde{y}}_{k} = c {\tilde{y}}_{l}$ (c is a positive number).

The relationship between fuzzy efficiency and the solution of the inequality evaluation approach can be obtained as follows.

Theorem 1. If there is no solution of the fuzzy inequalities (4), then the evaluated DMU is fuzzy efficient.

Theorem 2. If there is exactly one solution of the fuzzy inequalities (4), and the evaluated DMU is not a maximal element of all the DMUs equipped with an order relation ≤, then the evaluated DMU is fuzzy weak efficient, otherwise it is fuzzy efficient.

Theorem 3. If the number of solutions of the fuzzy inequalities (4) is greater than one, then the evaluated DMU is fuzzy inefficient.

3.2 The geometric interpretation of the proposed model

In the most existing fuzzy DEA models, efficiencies are evaluated through transforming the fuzzy DEA models into crisp DEA models by applying an alternative α-cut technique. To better show the proposed approach, we consider the case of two fuzzy inputs and one fuzzy output. Then we change the inputs and output of each DMU in the same proportion until output data of all the DMUs are equal at an α-level set. Next, the coordinate system is established with input 1 and input 2 as the x and y coordinate axes. For the DMU under evaluation, the closer it gets to the coordinate origin, the higher production efficiency will be.

The production possibility set T_{j
₀} which is spanned by all the DMUs without the j₀-th evaluated DMU, and T_{j
₀} is given by the following formula: $\begin{matrix} T_{j_{0}} & = & {(x, y) | x \geq \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} x_{j}, \\ y \leq \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} y_{j}, λ_{j} \geq 0} . \end{matrix}$

The expression of line segment joining the origin to the evaluated DMU j₀ is as follows. ${\begin{matrix} x_{1} = λ x_{j_{0} 1}, \\ x_{2} = λ x_{j_{0} 2}, \\ ⋮ \\ x_{m} = λ x_{j_{0} m}, \\ λ \in (0, 1] . \end{matrix}$

The inequalities (4) consist of inequalities of the production possibility set T_{j
₀} and the line segment joining origin to the evaluated DMU j₀.

In this section, there are five DMUs (i.e., DMUs A, B, C, D and E), and DMU E is the evaluated DMU, then the production possibility set is spanned by DMU A, B, C and D. As shown in Fig. 1, the solid line segments connecting points A, B, C and D constitute an isoquant that represents the different input amounts to produce the same output amount. Since it is impossible to reduce the amount of one of the inputs without increasing another input amount if one is to stay on this isoquant, and the solid line segments represent the efficient production frontier [21, 28] of the production possibility set T_E.

Fig.1

Efficiency analysis of the evaluated DMU.

From Fig. 1-1 we can see that the line segment OE and the production possibility set T_E are disjoint. There exist the optimal weight vectors of inputs and output such that the production efficiency of DMU E is higher than DMUs A, B, C and D, then DMU E is efficient [1]. That is to say, if there is no solution of the inequalities (4), then the evaluated DMU E is efficient.

In Fig. 1-2, the line segment OE meets on the production possibility set T_E at a point E, and there is exactly one solution of the inequalities of the production possibility set T_E and the line segment OE. There are two cases of interest: (1) The evaluated DMU E on the efficient production frontier is efficient. (2) The evaluated DMU E on the weak efficient production frontier is weak efficient [5, 27]. It is important to stress here that at least one input of the weak efficient DMU is strictly greater than that of an efficient DMU. Moreover, the order relation ≤ for DMUs A, B, C, D and E is reflective, antisymmetric and transitive in the coordinate system, then the set {A, B, C, D, E} is a partially ordered set. If an evaluated DMU is located on the production frontier, its efficiency is dependent on whether the evaluated DMU is a maximal element or not. If there is exactly one solution of the inequalities (4), and the evaluated DMU is maximal element, then the evaluated DMU is efficient. If there is exactly one solution of the inequalities (4), and the evaluated DMU is not maximal element, then the evaluated DMU is weak efficient.

Refer to Fig. 1-3, the line segment OE meets on the production possibility set T_E at more than one point, DMU E is located in the production possibility set T_E, thus DMU E is inefficient. If the solution of inequalities (4) is not unique, then the evaluated DMU is inefficient.

4 Numerical experiment

In this section, an example is given to illustrate the practical relevance of the presented approach. There are eight DMUs (DMUs A, B, C, D, E, F, G and H) with two fuzzy inputs and one fuzzy output. Fuzzy input and output data are symmetrical triangular numbers listed in Table 1. (c, ω) is a symmetrical triangular number, c is the center, and ω is the spread.

Table 1
DMUs with two fuzzy inputs and one fuzzy output

DMU A B C D E F G H

Input 1 (2.0, 0.1) (1.0, 0.01) (3.0, 0.2) (5.0, 0.3) (9.0, 0.5) (4.0, 0.2) (7.0, 0.5) (11.0, 1.0)

Input 2 (21.6, 0.5) (8.0, 0.4) (10.0, 0.2) (6.0, 0.3) (6.8, 0.5) (0.8, 0.1) (0.8, 0.1) (13.0, 1.0)

Output (2.0, 0.3) (1.0, 0.07) (2.0, 0.2) (2.0, 0.3) (0.3, 0.2) (1.0, 0.2) (1.0, 0.1) (2.0, 0.2)

DMU	A	B	C	D	E	F	G	H
Input 1	(2.0, 0.1)	(1.0, 0.01)	(3.0, 0.2)	(5.0, 0.3)	(9.0, 0.5)	(4.0, 0.2)	(7.0, 0.5)	(11.0, 1.0)
Input 2	(21.6, 0.5)	(8.0, 0.4)	(10.0, 0.2)	(6.0, 0.3)	(6.8, 0.5)	(0.8, 0.1)	(0.8, 0.1)	(13.0, 1.0)
Output	(2.0, 0.3)	(1.0, 0.07)	(2.0, 0.2)	(2.0, 0.3)	(0.3, 0.2)	(1.0, 0.2)	(1.0, 0.1)	(2.0, 0.2)

At first, DMUs are evaluated by Meng and Shi [19] in which an extended DEA model with more general fuzzy data based upon the centroid formula was proposed. The results are provided in Table 2.

Table 2

Fuzzy efficiencies of DMUs in Meng and Shi [21]

DMU	A	B	C	D	E	F	G	H
Optimal value	1	1	1	1	1	1	1	0.45767
Efficiency	weak efficient	efficient	efficient	efficient	efficient	efficient	weak efficient	inefficient

Next, The DMUs will be estimated by the proposed approach. By using the model (4), we can get the following inequalities.

${\begin{matrix} x \geq \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} C ({\tilde{x}}_{j}), \\ y \leq \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} C ({\tilde{y}}_{j}), \\ \sum_{j = 1, j \neq j_{0}}^{n} λ_{j} \geq 1, \\ x = λ_{0} C ({\tilde{x}}_{0}), \\ y = λ_{0} C ({\tilde{y}}_{0}), \\ λ_{0} \in (0, 1], \\ λ_{j} \geq 0 . \end{matrix}$ (5)

It’s worthy to note that $C (\tilde{x})$ and $C (\tilde{y})$ denote the centroids of fuzzy input $\tilde{x}$ and fuzzy output $\tilde{y}$ . $C (\tilde{x})$ is represented as follows. $C (\tilde{x}) = \frac{\int_{supp \tilde{x}} t μ_{\tilde{x}} (t) dt}{\int_{supp \tilde{x}} μ_{\tilde{x}} (t) dt}$ where $supp \tilde{x}$ is the support of $\tilde{x}$ . $μ_{\tilde{x}} (t)$ denotes the degree of membership of point t, and $t \in supp \tilde{A}$ [19]. The results of the proposed model are listed in Table 3.

Table 3

Fuzzy efficiencies of DMUs in the proposed model

DMU	A	B	C	D	E	F	G	H
Solution	1	0	0	0	1	0	1	>2
Maximal element	No	–	–	–	Yes	–	No	–
Efficiency	weak efficient	efficient	efficient	efficient	efficient	efficient	weak efficient	inefficient

Comparison of Tables 2 and 3, the results of the proposed paper are consistent with the results from Meng and Shi.

5 Discussion

In this paper, a fuzzy inequality evaluation approach is developed to evaluate fuzzy efficiency of DMUs with a single fuzzy output. Inequalities consist of the inequalities of the production possibility set and the line segment joining the origin to the evaluated DMU. Fuzzy efficiency is obtained by considering the number of solutions of fuzzy inequalities. The evaluated DMU is efficient if there is no solution, or there is exactly one solution and the evaluated DMU is a maximal element of all the DMUs equipped with an order relation ≤. The evaluated DMU is weak efficient if there is exactly one solution and the evaluated DMU is not a maximal element. The evaluated DMU is inefficient if the number of solutions is greater than one. It is worthy to note that the fuzzy inequality evaluation approach is equivalent to fuzzy DEA model with one output.

6 Conclusion

What we focus on in this paper is to evaluate DMUs with a fuzzy output. There are three interesting relationships between the evaluated DMU and the fuzzy production possibility set which is spanned by all DMUs except the DMU under evaluation, i.e., (1) the evaluated DMU is not located in the fuzzy production possibility set, then it is fuzzy efficient. (2) The evaluated DMU which is located in the fuzzy production possibility set, but not located on the fuzzy production frontier, then it is fuzzy inefficient. (3) The evaluated DMU is located on the fuzzy weak efficient production frontier, then it is fuzzy weak efficient. The evaluated DMU is located on the fuzzy efficient production frontier, then the evaluated DMU is fuzzy efficient. It is worthy to note that at least one input (or output) of the weak efficient DMU is strictly greater (or less) than that of an efficient DMU. The order relation ≤ for DMUs is reflective, antisymmetric and transitive, then the set which consists of all the DMUs is a partially ordered set, and the efficient DMU is a maximal element.

In this work, an (m + 1)-dimensional coordinate system is established with fuzzy inputs and output as the coordinate axes respectively. In the coordinate system, the production possibility set is spanned by all the DMUs without the evaluated DMU, and the line segment is connected from the origin to the evaluated DMU. Moreover, the production possibility set and the line segment are represented by fuzzy inequalities. Efficiency of the evaluated DMU is obtained from the number of intersection points of the production possibility set and the line segment (i.e., the number of solutions of fuzzy inequalities). Subsequently, the partially ordered set and maximal element are used to distinguish the fuzzy weak efficiency and fuzzy efficiency.

References

Andersen

and Petersen

N.C.

, A procedure for ranking efficient units in data envelopment analysis, Management Science 39 (1993), 1261–1264.

Angiz

L.M.Z.

, Emrouznejad

and Mustafa

, Fuzzy data envelopment analysis: A discrete approach, Expert Systems with Applications 39 (2012), 2263–2269.

Charnes

and Cooper

W.W.

, Programming with linear fractional functionals, Naval Research Logistics 9 (1962), 181–186.

Charnes

, Cooper

W.W.

and Rhodes

, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978), 429–444.

Charnes

, Cooper

W.W.

and Thrall

R.M.

, A structure for classifying and characterizing efficiency and inefficiency in data envelopment analysis, The Journal of Productivity Analysis 2 (1991), 197–237.

Davey

B.A.

and Priestley

H.A.

, Introduction to Lattices and Order, Cambridge University Press, 2002, pp. 1–18.

Geng

Z.Q.

, Wang

, Peng

C.L.

and Han

Y.M.

, A new fuzzy process capability estimation method based on Kernel function and FAHP, IEEE Transactions on Engineering Management 63 (2016), 177–188.

Han

Y.M.

, Geng

Z.Q.

, Zhu

Q.X.

and Qu

Y.X.

, Energy efficiency analysis method based on fuzzy DEA cross-model for ethylene production systems in chemical industry, Energy 83 (2015), 685–695.

Han

Y.M.

, Geng

Z.Q.

, Zhu

Q.X.

, Wang

and Cui

Y.F.

, Energy consumption analysis and evaluation of petrochemical industries using an improved fuzzy analytic hierarchy process approach, Journal of Intelligent & Fuzzy Systems 32 (2017), 4183–4195.

10.

Hatami-Marbini

, Emrouznejad

and Tavana

, A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making, European Journal of Operational Research 214 (2011), 457–472.

11.

Hougaard

J.L.

and Baležentis

, Fuzzy efficiency without convexity, Fuzzy Sets and Systems 255 (2014), 17–29.

12.

Khoshfetrat

and Daneshvar

, Improving weak efficiency frontiers in the fuzzy data envelopment analysis models, Applied Mathematical Modelling 35 (2011), 339–345.

13.

Lertworasirikul

, Fang

S.C.

, Joines

J.A.

and Nuttle

H.L.W.

, Fuzzy data envelopment analysis (DEA): A possibility approach, Fuzzy Sets and Systems 139 (2003), 379–394.

14.

Lertworasirikul

, Fang

S.C.

, Nuttle

H.L.W.

and Joines

J.A.

, Fuzzy BCC model for data envelopment analysis, Fuzzy Optimization and Decision Making 2 (2003), 337–358.

15.

León

, Liern

, Ruiz

J.L.

and Sirvent

, A fuzzy mathematical programming approach to the assessment of efficiency with DEA models, Fuzzy Sets and Systems 139 (2003), 407–419.

16.

Liu

S.T.

and Chuang

, Fuzzy efficiency measures in fuzzy DEA/AR with application to university libraries, Expert Systems with Applications 36 (2009), 1105–1113.

17.

Mardani

, Zavadskas

E.K.

, Streimikiene

, Jusoh

and Khoshnoudi

, A comprehensive review of data envelopment analysis (DEA) approach in energy efficiency, Renewable and Sustainable Energy Reviews 70 (2017), 1298–1322.

18.

Meada

, Entani

and Tanaka

, Fuzzy DEA with interval efficiency, 6th European Congress on Intelligent Techniques and Soft Computing 2 (1998), 1067–1071.

19.

Meng

X.L.

and Shi

F.G.

, An extended DEA with more general fuzzy data based upon the centroid formula, Journal of Intelligent & Fuzzy Systems 33 (2017), 457–465.

20.

Meng

X.L.

and Shi

F.G.

, A generalized fuzzy data envelopment analysis with restricted fuzzy sets and determined constraint condition, Journal of Intelligent & Fuzzy Systems 33 (2017), 1895–1905.

21.

Meng

X.L.

and Shi

F.G.

, An extended inequality approach for evaluating decision making units with a single output, Journal of Inequalities and Applications 2017 (2017), 199.

22.

Meng

X.L.

and Shi

F.G.

, An extended data envelopment analysis for the decision-making, Journal of Inequalities and Applications 2017 (2017), 240.

23.

Muren, Ma

Z.X.

and Cui

, Generalized fuzzy data envelopment analysis methods, Applied Soft Computing 19 (2014), 215–225.

24.

Saati

M.S.

, Memariani

and Jahanshahloo

G.R.

, Efficiency analysis and ranking of DMUs with fuzzy data, Fuzzy Optimization and Decision Making 1 (2002), 255–267.

25.

Triantis

, Sarangi

and Kuchta

, Fuzzy pair-wise dominance and fuzzy indices: An evaluation of productive performance, European Journal of Operational Research 144 (2003), 412–428.

26.

Wang

Y.M.

, Luo

and Liang

, Fuzzy data envelopment analysis based upon fuzzy arithmetic with an application to performance assessment of manufacturing enterprises, Expert Systems with Applications 36 (2009), 5205–5211.

27.

Wei

Q.L.

and Yan

, Characteristics and structures of weak efficient surfaces of production possibility sets, Journal of Mathematical Analysis and Applications 327 (2007), 1055–1074.

28.

, Wei

Q.L.

, Brockett

and Zhou

, Construction of all DEA efficient surfaces of the production possibility set under the generalized data envelopment analysis model, European Journal of Operational Research 95 (1996), 491–510.