An extended DEA with more general fuzzy data based upon the centroid formula 1

Abstract

The conventional data envelopment analysis (DEA) models for evaluating the relative efficiencies of a set of entities called decision making units (DMUs) converting multiple inputs into multiple outputs are limited to crisp data. In more general cases, the data of inputs and outputs, which are collected from observation or investigation, are often fluctuated and imprecise. There are many fuzzy DEA models with convex and normal fuzzy data, and those fuzzy data were transformed into crisp data by α-cuts or the lower and upper bounds to evaluate the efficiencies of DMUs. However, in many practical evaluation problems, the fuzzy data of inputs and outputs may be non-convex/convex, abnormal/normal. In such a case, the notion of fuzzy set, whose membership function is continuous and support is an interval, is presented. Based on the fundamental CCR model, a new fuzzy DEA model with the fuzzy set is proposed. The centroid formula of fuzzy set is established and introduced to evaluate DMUs. In order to illustrate the model more fully, the relationship between production frontier and fuzzy DEA efficiency is given. Finally, three numerical examples are provided to illustrate the proposed model.

Keywords

Data envelopment analysis (DEA)fuzzy set the centroid formula

1 Introduction

Data envelopment analysis (DEA) is a relative evaluation approach for evaluating the efficiency of peer decision making units (DMUs) which convert multiple inputs into multiple outputs. In traditional DEA models, such as CCR model [7] and BCC model [4], the inputs and outputs are assumed to be precise. But in more general cases, the data of inputs and outputs, which are collected from observation or investigation, are often fluctuated and imprecise.

Until now, there are many fuzzy DEA literature, such as, Meada et al. [23] proposed two fuzzy DEA models to evaluate DMUs with interval efficiency, the first model gave an upper limit (best case) efficiency, and another provided a lower limit (worst case) efficiency. A few years later, Kao and Liu [16] applied the α-cuts and Zadeh’s extension principle to transform the fuzzy DEA model to a series of conventional crisp DEA models, the idea is similar to that proposed by Meada et al. In the next year, Guo and Tanaka [12] represented fluctuating data as linguistic variables characterized by symmetric triangular fuzzy numbers for reflecting a kind of general feeling or experience of experts. Moreover, different efficiencies were provided in different h-levels and parameter t, and center efficiency value was first proposed in the paper. In 2002, Saati et al. [25] developed the fuzzy CCR model with asymmetrical triangular fuzzy number, and ranked the DMUs according to the given α-cuts. Similar to Guo and Tanaka’s method, the basic idea was to transform the fuzzy CCR model into a crisp linear programming problem using an alternative α-cuts technique. In the year 2003, Triantis et al. presented indices that captured the degree to which pair-wise dominance occured between two fuzzy production plans, and the fuzzy inputs and outputs were represented by right-directed and left-directed fuzzy numbers [28]. Lertworasirikul et al. [18] developed fuzzy DEA models by fuzzy sets and transformed them into possibility DEA models by using the possibility approach in which constraints were treated as fuzzy events. In the same year, Lertworasirikul et al. [19] gave fuzzy BCC model with normal fuzzy number by credibility approach. and León et al. [20] introduced LR-fuzzy numbers to express the data of production processes which cannot be precisely measured, and then developed some fuzzy versions of the classical DEA models by using some ranking methods based on the comparison of α-cuts. And then, Wang et al. [29] studied a minimax regret-based approach to compare and rank the efficiency interval of DMUs. In the next few years, Jahanshahloo et al. [15] studied fuzzy DEA model with a generalized L-R fuzzy number, and proposed a ranking method with l₁-norm. Wang et al. [30] proposed fuzzy DEA model based upon fuzzy arithmetic, and provided upper, middle and lower efficiency value for the evaluated DMUs. In 2010, Wen et al. [33] defined a fuzzy comparison of fuzzy variables, and extended CCR model into fuzzy CCR model based on credibility measure. In order to aggregate preference rankings, Angiz et al. [2] proposed a four-stage fuzzy DEA approach to aggregate preference rankings. In the following years, Khoshfetrat and Daneshvar [17] 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. Angiz et al. [1] introduced an alternative linear programming model, in which some uncertainty information with the α-cut approach was included. Based on Free Disposal Hull (FDH) method which is basically DEA without the assumption of convexity, Hougaard and Baležentis [14] 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). Muren et al. [24] firstly revealed the necessity of sample DMU model and improved the α-cut technique, the types of fuzzy number and the selected special point when evaluating fuzzy DEA model. For more fuzzy DEA models, see [11 , 32].

Although numerous fuzzy DEA models with fuzzy inputs and outputs have been proposed, there are still shortcomings existed. In practical evaluation problems the fuzzy data of inputs and outputs may be convex/non-convex and normal/abnormal. For example, the customer satisfaction is viewed as one of the outputs in the evaluation of travel agencies, the customer satisfaction is obtained by the degrees of satisfaction of customers in different age groups, then it may be a fuzzy set $\tilde{A}$ or $\tilde{B}$ shown in the Fig. 1. Therefore, it is a worth considering problem to evaluate the efficiency of DMU with fuzzy set in this setting.

Fig.1

The fuzzy sets.

In this work, the fuzzy data of inputs and outputs for DMUs are expressed by fuzzy sets, whose membership functions are continuous and supports are intervals. Then a new fuzzy DEA model with fuzzy set is proposed to evaluate the efficiencies of the DMUs. Subsequently, the centroid formula, which is adopted in [3 , 31], is introduced in this paper as a way to solve the fuzzy DEA model and evaluate the efficiencies of DMUs from the viewpoint of global perspective. Moreover, the relationship between fuzzy DEA efficiency and production frontier is illustrated schematically. Finally, three numerical examples are provided to illustrate the proposed model.

2 Preliminaries

In this section, we briefly review CCR model, some basic concepts of fuzzy number and the centroid formula.

2.1 CCR model

As an extremely common DEA model, the CCR model supposes that there are n DMUs, each DMU consumes the same input type and produces the same output type. 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 multiple 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 subject to the condition that the ratio for each DMU is not greater than 1. The corresponding model is: $(CCR) {\begin{matrix} \max \frac{u^{T} y_{0}}{v^{T} x_{0}} \\ s . t . \\ \frac{u^{T} y_{k}}{v^{T} x_{k}} \leq 1 \\ k = 1, \dots, n \\ u \geq 0, u \neq 0 \\ v \geq 0, v \neq 0 \end{matrix}$ (1) where x₀ = (x₀₁, ⋯ , x_0m) ^T and y₀ = (y₀₁, ⋯ , y_0r) ^T are the input and output vectors of the evaluated DMU (usually denoted by DMU₀). x_k = (x_k1, …, x_km) ^T and y_k = (y_k1, …, y_kr) ^T are the input and output vectors of the k-th DMU, u and v are the weight column vectors of output and input, respectively. The optimal objective value of $\frac{u^{T} y_{0}}{v^{T} x_{0}}$ falls in the range of (0, 1].

2.2 Fuzzy number

The concept of fuzzy number can be defined as follows [10].

Definition 1. A real fuzzy number $\tilde{F}$ is described as any fuzzy subset of the real line R, whose membership function $μ_{\tilde{F}}$ satisfies the following conditions:

A continuous mapping from R to the closed interval [0, 1];

Constant on (- ∞ , c]: $μ_{\tilde{F}} (x) = 0$ , ∀x ∈ (- ∞ , c];

Strictly increasing on [c, a];

Constant on [a, b]: $μ_{\tilde{F}} (x) = 1$ , ∀x ∈ [a, b];

Strictly decreasing on [b, d];

Constant on [d, + ∞): $μ_{\tilde{F}} (x) = 0$ , ∀x ∈ [d, + ∞).

a, b, c and d are real numbers. We may let c = -∞, or a = b, or c = a, or b = d, or d =+ ∞. By the definition, we can see that $\tilde{F}$ is a convex and normal fuzzy set.

For convenience, the fuzzy number in Definition 1 can be denoted by $\tilde{F} = (c, a, b, d; 1)$ . The membership function μ of $\tilde{F}$ can be expressed as $μ_{\tilde{F}} (x) = {\begin{matrix} μ_{\tilde{F}}^{L} (x), c \leq x \leq a, \\ 1, a \leq x \leq b, \\ μ_{\tilde{F}}^{R} (x), b \leq x \leq d, \\ 0, otherwise, \end{matrix}$ (2) where $μ_{\tilde{F}}^{L} : [c, a] \to [0, 1]$ and $μ_{\tilde{F}}^{R} : [b, d] \to [0, 1]$ .

Since $μ_{\tilde{F}}^{L}$ and $μ_{\tilde{F}}^{R}$ are continuous, strictly increasing and decreasing, we can see that the inverse functions of $μ_{\tilde{F}}^{L}$ and $μ_{\tilde{F}}^{R}$ exist, and can be denoted by $g_{\tilde{F}}^{L} : [0, 1] \to [c, a]$ and $g_{\tilde{F}}^{R} : [0, 1] \to [b, d]$ , respectively. Since $μ_{\tilde{F}}^{L}$ is continuous and strictly increasing, $g_{\tilde{F}}^{L}$ is also continuous and strictly increasing. Similarly, $g_{\tilde{F}}^{R}$ is also continuous and strictly decreasing. $g_{\tilde{F}}^{L}$ and $g_{\tilde{F}}^{R}$ are continuous on [0,1], they are integrable on [0,1], i.e., both $\int_{0}^{1} g_{\tilde{F}}^{L} dy$ and $\int_{0}^{1} g_{\tilde{F}}^{R} dy$ exist.

2.3 The centroid formula

The centroid, which is widely applied in ranking fuzzy numbers [3 , 31], is the arithmetic mean position of all the points in the shape for a plane figure. Chu and Tsao [9] determined the centroid point ( ${\bar{x}}_{0}, {\bar{y}}_{0}$ ) of fuzzy number $\tilde{F}$ and proposed the following centroid formulae: $\begin{matrix} {\bar{x}}_{0} (\tilde{F}) & = & \frac{\int_{- \infty}^{\infty} x μ_{\tilde{F}} (x) dx}{\int_{- \infty}^{\infty} μ_{\tilde{F}} (x) dx} = \frac{\int_{c}^{a} \frac{1}{a - c} (x - c) xdx + \int_{a}^{b} xdx + \int_{b}^{d} \frac{1}{b - d} (x - d) xdx}{\int_{c}^{a} \frac{1}{a - c} (x - c) dx + \int_{a}^{b} dx + \int_{b}^{d} \frac{1}{b - d} (x - d) dx} \\ {\bar{y}}_{0} (\tilde{F}) & = & \frac{\int_{0}^{1} ({yg}_{\tilde{F}}^{L}) dy + \int_{0}^{1} ({yg}_{\tilde{F}}^{R}) dy}{\int_{0}^{1} (g_{\tilde{F}}^{L}) dy + \int_{0}^{1} (g_{\tilde{F}}^{R}) dy} \end{matrix}$

If c = a (or b = d), then $\int_{c}^{a} \frac{1}{a - c} (x - c) xdx = \int_{c}^{a} \frac{1}{a - c} (x - c) dx = 0$ (or $\int_{b}^{d} \frac{1}{b - d} (x - d) xdx = \int_{b}^{d} \frac{1}{b - d} (x - d) dx = 0$ ) and the function $g_{\tilde{F}}^{L}$ (or $g_{\tilde{F}}^{R}$ ) does not exist. If c = a and b = d, then ${\bar{y}}_{0} (\tilde{F}) = 1$ .

3 The new fuzzy DEA model with fuzzy set

3.1 Fuzzy set

In this section, the fuzzy inputs and outputs are expressed by fuzzy sets, let $\tilde{A}$ be the fuzzy set satisfying the following two conditions:

the support of $\tilde{A}$ (i.e., $supp \tilde{A}$ ) is an interval;

its membership function is continuous on $supp \tilde{A}$ .

It’s worthy to note that the fuzzy set will degenerate into a real number if its support is a real number, then its centroid is equal to itself. Moreover, the fuzzy set may be trapezoid fuzzy number, triangular fuzzy number, and even irregular fuzzy data, etc. Since the membership function of fuzzy set $\tilde{A}$ is continuous on its supports, $\tilde{A}$ are integral, that is to say, the corresponding centroids $C (\tilde{A})$ are existent and can be represented as follows. $C (\tilde{A}) = \frac{\int_{supp \tilde{A}} t μ_{\tilde{A}} (t) dt}{\int_{supp \tilde{A}} μ_{\tilde{A}} (t) dt}$ (3) where $μ_{\tilde{A}} (t)$ denotes the degree of membership of point t, and $t \in supp \tilde{A}$ .

To calculate the centroid of membership degree $μ (\tilde{A})$ , we divide the membership function into M constant functions and N strictly monotone functions (M and N are integers). With this in mind, we consider the i-th (i = 1, …, N) strictly increasing or decreasing function, the function is continuous, then its inverse function exists and can be denoted by $g_{\tilde{A}}^{i}$ . Let m_i and M_i be the minimum and maximum values of the i-th strictly monotone interval, the centroid of membership degree $μ (\tilde{A})$ is $C_{μ} (\tilde{A}) = \frac{\sum_{i = 1}^{N} \int_{m_{i}}^{M_{i}} {tg}_{\tilde{A}}^{i} dt}{\sum_{i = 1}^{N} \int_{m_{i}}^{M_{i}} g_{\tilde{A}}^{i} dt} .$

3.2 The new fuzzy DEA model

In most existing fuzzy DEA models, the fuzzy inputs and outputs were always expressed by convex and normal fuzzy data, and those fuzzy data were transformed into crisp data by α-cut (0 ≤ α ≤ 1) or the lower and upper bounds to evaluate DMUs. In many real applications, the fuzzy data of inputs and outputs may be non-convex or abnormal, then the efficiency evaluation will be unsolvable or the computation complexity will increase by the existing methods.

Based on the fundamental CCR model, we develop a new fuzzy DEA model, in which the data of inputs and outputs are expressed by fuzzy sets mentioned above. The centroid formulae (3) are introduced in the model to deal with the fuzzy set. Efficiency of the evaluated DMU is obtained by the maximum ratio of weighted outputs to inputs, and the ratio is not greater than one for every DMU. The new fuzzy DEA model is as follows: ${\begin{matrix} \max \frac{\sum_{j = 1}^{r} u_{j} C (y_{0 j})}{\sum_{i = 1}^{m} v_{i} C (x_{0 i})} \\ s . t . \\ \frac{\sum_{j = 1}^{r} u_{j} C (y_{kj})}{\sum_{i = 1}^{m} v_{i} C (x_{ki})} \leq 1 \\ k = 1, \dots, n \\ v \geq 0, v \neq 0 \\ u \geq 0, u \neq 0 \end{matrix}$ (4) where, x_ki and y_kj presented by fuzzy sets are the i-th input and the j-th output of the k-th DMU. u = {u₁, …, u_j, …, u_r} and v = {v₁, …, v_i, …, v_m} are the weight vectors of outputs and inputs. $C (x_{ki})$ and $C (y_{kj})$ indicate the centroids of the i-th fuzzy input and the j-th fuzzy output of the k-th DMU. By applying the Charnes-Cooper transformation [6] in the model (4), the following equivalent linear model is obtained: ${\begin{matrix} \max \sum_{j = 1}^{r} {\bar{μ}}_{j} C (y_{0 j}) \\ s . t . \\ \sum_{i = 1}^{m} ϖ_{i} C (x_{0 i}) = 1 \\ \sum_{i = 1}^{m} ϖ_{i} C (x_{ki}) - \sum_{j = 1}^{r} {\bar{μ}}_{j} C (y_{kj}) \geq 0 \\ k = 1, \dots, n \\ ϖ \geq 0, ϖ \neq 0 \\ \bar{μ} \geq 0, \bar{μ} \neq 0 \end{matrix}$ (5) where, $\bar{μ} = ({\bar{μ}}_{1}, \dots, {\bar{μ}}_{r})$ and ϖ = (ϖ₁, ⋯ , ϖ_m) are the weight vector assigned to the outputs and inputs. $\bar{μ} \geq 0, \bar{μ} \neq 0$ represents the output vector whose elements are not smaller than zero but at least one element is positive value.

For convenience, the centroid of membership degree for the i-th input (and the j-th output) will be abbreviated to $C_{μ} (I_{i})$ (and $C_{μ} (O_{j})$ ). The centroid of membership degree for DMU is defined by $C_{μ} (DMU) = (⋁_{i = 1}^{m} C_{μ} (I_{i})) ⋀ (⋁_{j = 1}^{r} C_{μ} (O_{j}))$ where ⋁ and ⋀ denote max and min, respectively.

The relationship between fuzzy DEA efficiency and the optimal objective value can be obtained as follows.

Definition 2. The evaluated DMU is fuzzy DEA efficient, if the optimal objective value is equal to 1, and ${\bar{μ}}_{0} > 0$ , ϖ₀ > 0.

Definition 3. The evaluated DMU is fuzzy DEA weak efficient, if the optimal objective value is equal to 1.

Definition 4. The evaluated DMU is fuzzy DEA inefficient, if the optimal objective value is less than 1.

4 The relationship between the production frontier and fuzzy DEA efficiency

In this section, we consider the case of DMUs with two fuzzy inputs and a single fuzzy output to show the relationship between the production frontier and fuzzy DEA efficiency. Notice that DEA efficiency is independent of the changes of inputs and output by the same proportion, then we change the fuzzy inputs and output in the same proportion for each DMU until all output data of the centroids are equal. Then the coordinate system is established with input 1 and input 2 as the x and y coordinate axes separately.

Fig.2

The production frontier of fuzzy DEA model.

In the new fuzzy DEA model, if DMUs are fuzzy DEA efficient, then their centroids are located on the production frontier, and others are located in the production possibility set. In Figure 2, the dots (•) denote the centroids of DMUs, all the fuzzy data of input 1 and input 2 are illustrated as rectangles, and the production frontier is represented by solid line. We can see that DMU₁, DMU₂, DMU₃ and DMU₇ are fuzzy DEA efficient and their centroids are located on the production frontier, on the contrary, the centroids of DMU₄, DMU₅, DMU₆ and DMU₈ are located in the production possibility set.

5 Illustrative examples

In this section, three numerical experiments are presented to illustrate the proposed model.

5.1 The first example

The first example with three fuzzy inputs and two fuzzy outputs for five DMUs is considered. The supports and membership functions of fuzzy inputs and outputs are listed in Tables 1 and 2 separately, t belongs to the corresponding support of fuzzy data. The supports of fuzzy data for inputs and outputs are intervals, and the corresponding membership functions are continuous, then the fuzzy inputs and outputs are fuzzy sets mentioned above, and fuzzy efficiencies obtained from the model (5) are listed in Table 4. Notice that the fuzzy input and output might be abnormal, non-convex but bounded, such as the fuzzy Input 2 of DMU D listed in Tables 1, 2 and 3.

Table 1
The supports of the fuzzy inputs and outputs

DMU Input 1 Input 2 Input 3 Output l Output 2

A [3, 3] [10.8, 13.2] [7, 7] [6.2, 7.8] [8.3, 8.7]

B [5, 5] [16,0, 18.2] [8.5, 8.5] [8,6, 9.6] (9.3, 9.7)

C [4, 4] [13.2, 16.0] [11, 11] [6.1, 7.9] (8.8, 9.6)

D [3.6, 3.6] [10.8, 13.2] [8, 8] [5.3, 5.7] (8.0, 8.8)

E [5, 5] [16.0, 18.6] [7, 7] [4.3, 5.7] [8.5, 89]

DMU	Input 1	Input 2	Input 3	Output l	Output 2
A	[3, 3]	[10.8, 13.2]	[7, 7]	[6.2, 7.8]	[8.3, 8.7]
B	[5, 5]	[16,0, 18.2]	[8.5, 8.5]	[8,6, 9.6]	(9.3, 9.7)
C	[4, 4]	[13.2, 16.0]	[11, 11]	[6.1, 7.9]	(8.8, 9.6)
D	[3.6, 3.6]	[10.8, 13.2]	[8, 8]	[5.3, 5.7]	(8.0, 8.8)
E	[5, 5]	[16.0, 18.6]	[7, 7]	[4.3, 5.7]	[8.5, 89]

Table 2

Membership function of the fuzzy inputs and outputs

DMU	Input 1	Input 2	Input 3	Output 1	Output 2
A	1, t = 3	$\frac{2}{3} e^{- \frac{(t - 12.01)^{2}}{5}}$	1, t = 7	${\begin{matrix} 2 t / 7 - 1, t \in [6.2, 7] \\ 3 - 2 t / 7, t \in [7, 7.8] \end{matrix}$	$1 - \| \frac{t - 8.5}{0.6} \|$
B	1, t = 5	1, t ∈ [16, 18.2]	1, t = 8.5	0.7, t ∈ [8.6, 9.6]	$1 - \| \frac{t - 9.5}{0.2} \|$
C	1, t = 4	$1 - \| \frac{t - 14.6}{1.6} \|$	1, t = 11	${\begin{matrix} 0.5 t - 2.5, t \in [6.1, 7.0) \\ 4.5 - 0.5 t, t \in (7.0, 7.9] \end{matrix}$	$1 - \| \frac{t - 9.2}{0.4} \|$
D	1, t = 3.6	$\frac{2}{3} (t - 12)^{2}$	1, t = 8	${\begin{matrix} (t - 4.6)^{3}, t \in [5.3, 5.5] \\ (6.4 - t)^{3}, t \in (5.5, 5.7] \end{matrix}$	$1 - \| \frac{t - 8.4}{0.4} \|$
E	1, t = 5	1, t ∈ [16, 18.6]	1, t = 7	1, t ∈ [4.3, 5.7]	$1 - \| \frac{t - 8.7}{0.5} \|$

Table 3

The maximum membership degrees of fuzzy inputs and outputs

DMU	Input 1	Input 2	Input S	Output 1	Output 2
A	1	2/3	1	1	1
B	1	1	1	0.7	1
C	1	1	1	1	1
D	1	0.96	1	0.729	1
E	1	1	1	1	1

Table 4

The optimal objective value for fuzzy DMUs

DMU	A	B	C	D	E
The optimal objective value	1	1	0.8896052	0.9882353	1

5.2 The second example

The second example with a single fuzzy input and a single fuzzy output is taken from Guo and Tanaka [12]. Table 5 provides the fuzzy data for the example, these fuzzy data are symmetrical triangular numbers. The support of each fuzzy input or output is an interval, and the corresponding membership function is continuous, then the fuzzy data of inputs and outputs are fuzzy sets mentioned above. Therefore, DMUs can be evaluated by the model (5), and the results of these calculations are given in Table 6.

Table 5
The supports and membership functions of the fuzzy inputs

DMU Support Membership Support Membership

of function of function

input of input output of output

A (1.5, 2.5) 1 - | (t - 2)/0.5| (0.7, 1.3) 1 - | (t - 1)/0.3|

B (2.5, 3.5) 1 - | (t - 3)/0.5| (2.3, 3.7) 1 - | (t - 3)/0.7|

C (2.4, 3.6) 1 - | (t - 3)/0.6| (1.6, 2.4) 1 - | (t - 2)/0.4|

D (4.0, 6.0) 1 - | (t - 5)/1.0| (3.0, 5.0) 1 - | (t - 4)/1.0|

E (4.5, 5.5) 1 - | (t - 5)/0.5| (1.8, 2.2) 1 - | (t - 2)/0.2|

DMU	Support	Membership	Support	Membership
A	(1.5, 2.5)	1 - \| (t - 2)/0.5\|	(0.7, 1.3)	1 - \| (t - 1)/0.3\|
B	(2.5, 3.5)	1 - \| (t - 3)/0.5\|	(2.3, 3.7)	1 - \| (t - 3)/0.7\|
C	(2.4, 3.6)	1 - \| (t - 3)/0.6\|	(1.6, 2.4)	1 - \| (t - 2)/0.4\|
D	(4.0, 6.0)	1 - \| (t - 5)/1.0\|	(3.0, 5.0)	1 - \| (t - 4)/1.0\|
E	(4.5, 5.5)	1 - \| (t - 5)/0.5\|	(1.8, 2.2)	1 - \| (t - 2)/0.2\|

Table 6

Fuzzy efficiencies of DMUs in the proposed model

DMU	A	B	C	D	E
The optimal objective value	0.5	1	0.667	0.8	0.4

Table 7 shows fuzzy efficiencies of DMUs with different h values for the method of Guo and Tanaka. E = (ω_l, η, ω_r) is a non-symmetrical triangular fuzzy number, ω_l, ω_r and η are the left, right spreads and the center of the fuzzy efficiency E, respectively. We can see that the result of this paper is consistent with the result of Guo and Tanaka.

Table 7

Fuzzy efficiencies of DMUs in the method of Guo and Tanaka [12]

DMU	A	B	C	D	E
h = 0	(0.21, 0.47, 0.35)	(0.32, 0.95, 0.45)	(0.21, 0.63, 0.32)	(0.28, 0.76, 0.43)	(0.07, 0.38, 0.08)
h = 0.5	(0.12, 0.49, 0.15)	(0.18, 0.97, 0.21)	(0.12, 0.65, 0.14)	(0.16, 0.78, 0.19)	(0.04, 0.39, 0.04)
h = 0.75	(0.06, 0.49, 0.07)	(0.09, 0.98, 0.10)	(0.06, 0.66, 0.07)	(0.08, 0.79, 0.09)	(0.02,0.39, 0.02)
h = 1	(0.0, 0.5, 0.0)	(0.0, 1.0, 0.0)	(0.0, 0.67, 0.0)	(0.0, 0.8, 0.0)	(0.0, 0.4, 0.0)

5.3 The third example

Changing market environment has challenged many companies to improve cost, quality and responsiveness to meet fierce competition. Flexible Manufacturing Systems (FMSs) present opportunities for manufacturers to improve their technology, competitiveness, and profitability through a highly efficient and focused approach to manufacturing effectiveness [5]. The primary reason for implementing FMS lies in its versatility. In general, increased flexibility enables a company to adjust more easily to changes in the market place and in customer requirements, while maintaining high quality standards for its products.

In the example, the DMUs consist of twelve manufacturing systems. Capital and operating costs, floor space requirement are the input items. Improvements in qualitative factor, work-in-process (WIP), numbers of tardy jobs and yield are the output items. The fuzzy input and output data taken from Liu [21] is slightly modified from [26]. Since the fuzzy data are represented by convex fuzzy number as shown in Table 8, the fuzzy input and output data belong to the fuzzy set mentioned above. The flexible manufacturing systems can be evaluated by the proposed model (5), and the results displayed in Table 9 are consistent with the results of [21] and [26] displayed inTables 10 and 11.

Table 8
Data used to assess the relative efficiency of FMS alternatives

FMS Alternative Inputs Improvements in outputs

Capital and Operating Floor space Qualitative W1P No. of Yield

cost ($00000) needed (000 ft²) (%) (10) Tardy (%) (00)

1 (16.17,17.02,17.87) 5 42 (43.0,45.3,47.6) (13.5,14.2,14.9) (28.6,30.1,31.6)

2 (15.64,16.46,17.28) 4.5 39 (38.1,40.1,42.1) (12.4,13.0,13.7) (28.3,29.8,31.3)

3 (11.17,11.76,12.35) 6 26 (37.6,39.6,41.6) (13.1,13.8,14.5) (23.3,24.5,25.7)

4 (9.99,10.52,11.05) 4 22 (34.2,36.037.8) (10.7,11.3,11.9) (23.8,25.0,26.3)

5 (9.03,9.50,9.98) 3.8 21 (32.5,34.2,35.9) (11.4.12.0,12.6) (19.4,20.4,21.4)

6 (4.55,4.79,5.03) 5.4 10 (19.1.20.1,21.1) (4.8,5.0,5.3) (15.7,16.5,17.3)

7 (5.90,6.21,6.52) 6.2 14 (25.2,26.5,27.8) (6.7,7.0,7.4) (18.7,19.7,20.7)

8 (10.56,11.12,11.68) 6 25 (34.1,35.9,37.7) (8.6,9.0,9.5) (23.5,24.7,25.9)

9 (3.49,3.67,3.85) 8 4 (16.5,17.4,18.3) (0.1,0.1,0.1) (17.2,18.1,19.0)

10 (8.48,8.93,9.38) 7 16 (32.6.34.3,36.0) (6.2,6.5,6.8) (19.6,20.6,21,6)

11 (16.85,17.74,18.63) 7.1 43 (43.3,45.6,47.9) (13.3,14.0,14.7) (29.5,31.1,32.7)

12 (14.11,14.85,15.59) 6.2 27 (36.8,38.7,40.6) (13.1,13.8,14.5) (24.1,25.4,26.7)

FMS Alternative	Inputs	Improvements in outputs
1	(16.17,17.02,17.87)	5	42	(43.0,45.3,47.6)	(13.5,14.2,14.9)	(28.6,30.1,31.6)
2	(15.64,16.46,17.28)	4.5	39	(38.1,40.1,42.1)	(12.4,13.0,13.7)	(28.3,29.8,31.3)
3	(11.17,11.76,12.35)	6	26	(37.6,39.6,41.6)	(13.1,13.8,14.5)	(23.3,24.5,25.7)
4	(9.99,10.52,11.05)	4	22	(34.2,36.037.8)	(10.7,11.3,11.9)	(23.8,25.0,26.3)
5	(9.03,9.50,9.98)	3.8	21	(32.5,34.2,35.9)	(11.4.12.0,12.6)	(19.4,20.4,21.4)
6	(4.55,4.79,5.03)	5.4	10	(19.1.20.1,21.1)	(4.8,5.0,5.3)	(15.7,16.5,17.3)
7	(5.90,6.21,6.52)	6.2	14	(25.2,26.5,27.8)	(6.7,7.0,7.4)	(18.7,19.7,20.7)
8	(10.56,11.12,11.68)	6	25	(34.1,35.9,37.7)	(8.6,9.0,9.5)	(23.5,24.7,25.9)
9	(3.49,3.67,3.85)	8	4	(16.5,17.4,18.3)	(0.1,0.1,0.1)	(17.2,18.1,19.0)
10	(8.48,8.93,9.38)	7	16	(32.6.34.3,36.0)	(6.2,6.5,6.8)	(19.6,20.6,21,6)
11	(16.85,17.74,18.63)	7.1	43	(43.3,45.6,47.9)	(13.3,14.0,14.7)	(29.5,31.1,32.7)
12	(14.11,14.85,15.59)	6.2	27	(36.8,38.7,40.6)	(13.1,13.8,14.5)	(24.1,25.4,26.7)

Table 9

The relative efficiency of the proposed model

FMS	1	2	3	4	5	6	7	8	9	10	11	12
Efficiency	1	1	0.982369	1	1	1	1	0.961428	1	0.95355	0.983144	0.801173

Table 10

The α-cuts of the fuzzy efficiency scores of the FMS alternative at eleven distinctive α values

Firm		α = 0.0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 1.0
1	L	0.8614	0.8728	0.8844	0.8961	0.9080	0.9200	0.9322	0.9446	0.9571	0.9698	0.9827
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9827
2	L	0.8376	0.8487	0.8600	0.8714	0.8831	0.8948	0.9067	0.9188	0.9311	0.9435	0.9560
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9948	0.9817	0.9688	0.9650
3	L	0.8322	0.8436	0.8551	0.8668	0.8786	0.8906	0.9028	0.9151	0.9276	0.9402	0.9530
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9925	0.9792	0.9660	0.9530
4	L	0.8360	0.8753	0.8879	0.9008	0.9138	0.9270	0.9430	0.9538	0.9675	0.9813	0.9953
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9953
5	L	0.8899	0.9020	0.9144	0.9270	0.9397	0.9526	0.9659	0.9794	0.9931	1.0000	1.0000
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
6	L	0.8462	0.8487	0.8714	0.8842	0.8972	0.9105	0.9239	0.9375	0.9513	0.9653	0.9795
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9939	0.9795
7	L	0.9005	0.9132	0.9260	0.9390	0.9521	0.9655	0.9790	0.9927	1.0000	1.0000	1.0000
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
8	L	0.8208	0.8320	0.8432	0.8547	0.8663	0.8780	0.8899	0.9019	0.9141	0.9265	0.9390
	U	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9909	0.9775	0.9645	0.9517	0.9390
9	L	0.7169	0.7279	0.7390	0.7504	0.7619	0.7735	0.7853	0.7973	0.8095	0.8218	0.8343
	U	0.9698	0.9554	0.941 1	0.9271	0.9133	0.8996	0.8862	0.8729	0.8599	0.8470	0.8343
10	L	0.7199	0.7301	0.7405	0.7512	0.7621	0.7730	0.7842	0.7955	0.8069	0.8185	0.8303
	U	0.9568	0.9433	0.9301	0.9170	0.9042	0.8915	0.8789	0.8665	0.8542	0.8422	0.8303
11	L	0.8188	0.8295	0.8404	0.8514	0.8626	0.8739	0.8854	0.8970	0.9088	0.9207	0.9328
	U	1.0000	1.0000	1.0000	1.0000	1.0000	0.9969	0.9837	0.9708	0.9579	0.9453	0.9328
12	L	0.5686	0.5744	0.5864	0.5955	0.6047	0.6141	0.6235	0.6332	0.6429	0.6526	0.6625
	U	0.7696	0.7582	0.7469	0.7358	0.7249	0.7141	0.7035	0.6930	0.6827	0.6725	0.6625

L: lower bound, U: upper bound.

Table 11

The relative efficiency of FMSs in Shang and Sueyoshi [26]

FMS	1	2	3	4	5	6	7	8	9	l0	11	12
Efficiency	1	1	0.9824	1	1	1	1	0.9613	1	0.9536	0.9827	0.8011

6 Conclusion

In this paper, a new fuzzy DEA model is proposed to evaluate the efficiencies of DMUs with fuzzy inputs and outputs. The fuzzy data of inputs and outputs, which may be non-convex/convex, normal/abnormal in many real applications, are expressed by fuzzy sets. The fuzzy set is continuous and bounded, that is, its membership function is continuous and its support is an interval. In this case the maxima of membership functions of fuzzy inputs and outputs may be unequal, then the efficiency evaluation will be unsolvable or the computation complexity will increase by the most existing fuzzy DEA models. Furthermore, the centroid formula is introduced in the model to deal with the fuzzy set for the efficiency evaluation of DMUs. Moreover, the relationship between fuzzy DEA efficiency and production frontier is illustrated schematically. Finally, three examples are provided to illustrate the proposed model.

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