Finding strong defining hyperplanes of production possibility ... - ispacs

Journal of Data Envelopment Analysis and Decision Science 2016 No. 1 (2016) 15-22

Available online at www.ispacs.com/dea Volume 2016, Issue 1, Year 2016 Article ID: dea-00123, 8 pages doi:10.5899/2016/dea-00123 Research Article

Data Envelopment Analysis and Decision Science

Finding strong defining hyperplanes of production possibility set with fuzzy data Mehdi Amiri1*, Mohsen Rostami Malkhalifeh1 (1) Department of Mathematics, College of science, Tehran Branch, Islamic Azad university, Tehran, Iran. Copyright 2016 © Mehdi Amiri and Mohsen Rostami Malkhalifeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract The production possibility set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs. In data envelopment analysis (DEA), identification of the strong defining hyperplanes of the empirical production possibility set (PPS) is important, because they can be used for determining rates of change of outputs with change in inputs. Also, efficient hyperplanes determine the nature of returns to scale, and also is important for defining a suitable pattern for inefficient DMUs. As we know, fuzzy data are one of the different kinds of data that show some uncertainty in inputs and outputs. Therefore we apply an algorithm for transforming fuzzy models in to linear models using Production Possibility Set (PPS). In this paper, we deal with the problem of finding the strong defining hyperplanes of the PPS with fuzzy data. A numerical example shows the reasonability of our method. Keywords: Production possibility set (PPS), Data envelopment analysis (DEA), Defining hyperplanes, Fuzzy data.

1 Introduction Data envelopment analysis (DEA) is defined based on observed units and by finding the distance of each unit to the border of estimated production possibility set (PPS). It dichotomies the decision making unit (DMU): efficient or inefficient. One of the most frequently studied subjects in the DEA context is the identification of efficient hyperplanes of the PPS. As far as we are aware, only few DEA-based papers have been published regarding the subject of efficient hyperplanes. Yu et al. [10] studied the construction of DEA efficient surfaces under the generalized DEA model.An alternative approach for determining these hyperplanes was proposed by Jahanshahloo et al. [10]. DEA was proposed by Charnes, Cooper, and Rhodes [1] to measure the relative efficiency of a set of decision-making units (DMUs) by utilizing multiple inputs to produce multiple outputs. The set of relatively efficient DMUs constitutes the efficient frontier, and any deviation from the frontier is treated as in efficiency.

* Corresponding Author. Email address: [email protected] 15

Journal of Data Envelopment Analysis and Decision Science 2016 No.1 (2016) 15-22 http://www.ispacs.com/journals/dea/2016/dea-00123/

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The initial DEA model is referred to as the CCR model, named after Charnes et al. [1], who assumed constant returns to scale (CRS) of the production function. Banker, Charnes and Cooper also proposed a modified model, named BCC in 1984 [1], which assumed variable returns to scale (VRS).The estimate of efficiency obtained from CCR is referred to as technical efficiency, whereas the estimate from BCC is pure technical efficiency[11].The original DEA models assume that the levels of inputs and outputs are known exactly. Efficient surfaces are useful in analyzing DEA efficient DMUs and incorporating preference information to find reference DMUs in multi-objective programming. In this paper strong defining hyperplanes of PPS in the presence of fuzzy data are obtained. The current article proceeds as follows: In section 3, we give the concept of fuzzy data and some basic definitions and models. In section 4, we try to find all Strong Defining Hyerplanes of Production Possibility Set (SDHP) with fuzzy data. In section 5, a numerical example is considered and section 5 gives our conclusive remarks. 2 Preliminaries and notations 2.1. Production possibility set (PPS) We will call a pair of input x∈ Rm and output y ∈ 𝑅 𝑠 an activity and express them by the notation (x,y). The set of feasible activities is called the production possibility set (PPS) and is denoted by P. we postulate the following properties of P: 1. The observed activities (xj , yj)∈P ; J=1,…,n 2. If an activity (xj , yj) ∈ P, then the activity (tx,tY) ∈ P for all t> 0. 3. If an activity (xj , yj) ∈ P, then (𝑥̅ , 𝑦̅) ∈ 𝑃 if 𝑥̅ ≥x and 𝑌̅ ≤y. 4. If an activity (xj , yj) ∈ P, then (𝑥̅ , 𝑦̅) ∈ 𝑃, then (𝜆𝑥 + (1 − 𝜆)𝑋̅, 𝜆𝑦 + (1 − 𝜆)𝑌̅ ∈ 𝑃P for all λ∈ [0,1] We show the set P as follow: 𝑛

𝑛

𝑃 = {(𝑥, 𝑦)|𝑥 ≥ ∑ 𝜆𝑗 𝑥𝑗 , 𝑦 ≤ ∑ 𝜆𝑗 𝑦𝑗 , 𝜆𝑗 ≥ 0, 𝑗 = 1, … , 𝑛} 𝑗=1

𝑗=1

2.2. Input oriented CCR model In order to evaluate the relative efficiency of DMUo (O ∈ {1,…,n} ) the input-oriented model (2.1) can be applied. min 𝑠. 𝑡 .

𝜃 ∑𝑛𝑗=1 𝜆𝑗 𝑥𝑗 ∑𝑛𝑗=1 𝜆𝑗 𝑦𝑗

≤ 𝜃0 𝑥0 ≥ 𝑦0

(2.1)

𝜆 ∈∧ Where𝜆 is a semipositive vector in 𝑅𝑛 . The model is called CCR if ∧= {𝜆|𝜆 ≥ 0}and BCC if ∧= {𝜆|𝜆 ≥ 0, 𝑒𝜆 = 1} where e = (1,…,1) (a vector of ones ).The PPS of CCR model is c T and the PPS of BCC model is Tv . The model (2.1) is called envelopment form. DMU0 (O∈ {1, … , n}) is pareto efficient if and only if 𝜃 ∗ =1 in (2.1) and the optimal value of following linear programming is equal to zero. max 𝑒𝑠 − + 𝑒𝑠 + ∑𝑛𝑗=1 𝜆𝑗 𝑥𝑗 + 𝑠 − = 𝑥0 𝑠. 𝑡 ∑𝑛𝑗=1 𝜆𝑗 𝑦𝑗 − 𝑠 + = 𝑦0 −

(2.2)

+

𝜆 ∈∧ , 𝑠 , 𝑠 ≥ 0

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For DMUo, we define its reference set Eo, that is defined as follows: E0={𝑗|𝜆𝑗 ∗ ≥ 0 𝑖𝑛 𝑠𝑜𝑚𝑒𝑜𝑝𝑡𝑖𝑚𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 (2.1) } ⊆ {1, … , 𝑛} The DMUs in E0 are pareto efficient and any semipositive combination of them is pareto efficient as well As a result, references of a DMU are efficient DMUs that there is a combination of them dominates it.The dual of (2.1), which is in the following from, is called multiplier form: max 𝑠. 𝑡.

𝑢𝑡 𝑦0 + 𝑢0 𝑣 𝑡 𝑥0 = 1, 𝑢𝑡 𝑦 − 𝑣 𝑡 𝑥𝑗 + 𝑢0 ≤ 0, 𝑢, 𝑣 ≥ 0 𝑢0 free

𝑗 = 1, … , 𝑛

(2.3)

Note that if u0 = 0 in (2.3) the model is CCR, else it is BCC. DMU0 is strong efficient if in (2.3) 𝑢∗𝑡 𝑦0 + 𝑢0 ∗ = 1 and (𝑢∗ , 𝑣 ∗ ) > 0 for some optimal solutions. If 𝑢∗𝑡 𝑦0 + 𝑢0 ∗ = 1 and no (𝑢∗ , 𝑣 ∗ ) > 0 exists, then DMU0 is called weak efficient. This means that, weak efficiency occurs when the optimal objective of (2.3) is one and at least one component of each optimal solution is zero. Efficient Frontier is the set of all points (actual or virtual) with efficiency score equal to unity. 3 Fuzzy Production Possibility Set and Fuzzy CCR Model In this section, we are going to de_ne the FPPS by using Zadeh's extension principle in constant return to scale. Also, the FCCR model is introduced in input oriented. 3.1. FPPS Let the input and output data for DMUj (j = 1,…,n) are 𝑥̃𝑗 = (𝑥̃1𝑗 , … , 𝑥̃𝑚𝑗 )And 𝑦̃𝑗 = (𝑦̃1𝑗 , … , 𝑦̃𝑠𝑗 ) respectively, where 𝑥̃𝑖𝑗 (i = 1,…,m) and 𝑦̃𝑟𝑗 (r = 1,..,s), for j = 1,…,n are m + s fuzzy numbers. We call the set of feasible activities with fuzzy data FPPS and denote it by 𝑝̃. We define the FPPS as follows: 𝑃̃ = {((𝑋, 𝑌), 𝜇𝑃̃ (𝑋, 𝑌))|𝑋 ∈ 𝑅𝑚 , 𝑌 ∈ 𝑅 𝑠 } Where 𝑋 = (𝑥1 , … , 𝑥𝑚 ), (𝑥𝑖 , 𝜇𝑥𝑖 ̃𝑖 𝑖 = 1, … , 𝑚 ̃ (𝑥𝑖 )) ∈ 𝑥

(3.4)

𝑌 = (𝑦1 , … , 𝑦𝑠 ), (𝑦𝑟 , 𝜇𝑦𝑟 ̃𝑟 𝑟 = 1, … , 𝑠 ̃ (𝑦𝑟 )) ∈ 𝑦

(3.6)

(3.5)

And 𝜇𝑃̃ (𝑋, 𝑌) = max min {𝜇𝑥1̃𝑗 (𝑥1𝑗 ), … , 𝜇𝑥𝑚 (𝑥𝑚𝑗 ), 𝜇𝑦1̃𝑗 (𝑦1𝑗 ), … , 𝜇𝑦𝑠̃𝑗 (𝑦𝑠𝑗 )} ̃𝑗 𝜆𝑗 >0

𝑠. 𝑡.

𝑋 𝑡 ≥ ∑𝑛𝑗=1 𝜆𝑗 𝑥𝑗 𝑡 𝑌 𝑡 ≥ ∑𝑛𝑗=1 𝜆𝑗 𝑌𝑗 𝑡 𝜆𝐽 ≥ 0 𝑗 = 1, … , 𝑛

(3.7)

such as (𝑋̃𝑗 , 𝑌̃𝑗 ) (j = 1, 2,…, n) is considered to be activities. 3.2. Fuzzy CCR Model (FCCR) Let us consider the DMU0. In CCR model with fuzzy data, the objective function seeks the minimum value of 𝜃 when the activity (𝜃𝑋̃0 , 𝑌̃0 ) is belong to 𝑃̃. While (𝑋0 , 𝜇𝑋̃0 (𝑋0 )) ∈ 𝑋̃0 any input vector X0 reduces radially to 𝜃 X0. Hence FCCR model proposes the following model:


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min 𝑠. 𝑡.

𝜃 𝜃𝑋 𝑡 0𝑙

≥

∑𝑛𝑗=1 𝜆𝑗 𝑥 𝑡𝑗 𝑙

𝜃𝑋 𝑡 0𝑚

≥

∑𝑛𝑗=1 𝜆𝑗 𝑥 𝑡𝑗 𝑚

𝜃𝑋 𝑡 0𝑟

≥

∑𝑛𝑗=1 𝜆𝑗 𝑥 𝑡𝑗 𝑟

𝑦 𝑡 0𝑙

≤

∑𝑛𝑗=1 𝜆𝑗 𝑦 𝑡 𝑗 𝑙

𝑦 𝑡 0𝑚

≤

∑𝑛𝑗=1 𝜆𝑗 𝑦 𝑡 𝑗 𝑚

𝑦 𝑡 0𝑟

≤

∑𝑛𝑗=1 𝜆𝑗 𝑦 𝑡 𝑗 𝑟

0 ≤ 𝜆𝑗 𝑗 = 1, … , 𝑛

(3.8)

Where 𝑋 𝑙𝑗 , 𝑋 𝑚𝑗 , 𝑋 𝑟𝑗 , 𝑦 𝑙 𝑗 , 𝑦 𝑚 𝑗 𝑎𝑛𝑑 𝑦 𝑟 𝑗 (𝑗 = 1, … , 𝑛)are the left, the mean and the right values of triangular fuzzy numbers 𝑥̃𝑗 and 𝑦̃𝑗 respectively.dual of model (5.11) following. max

𝑢𝑙 𝑦0 𝑙 + 𝑢𝑚 𝑦0 𝑚 + 𝑢𝑟 𝑦0 𝑟

s. t.

𝑢𝑙 𝑦𝑗 𝑙 + 𝑢𝑚 𝑦𝑗 𝑚 + 𝑢𝑟 𝑦𝑗 𝑟 − 𝑣 𝑙 𝑥𝑗 𝑙 − 𝑣 𝑚 𝑥𝑗 𝑚 − 𝑣 𝑟 𝑥𝑗 𝑟 ≤ 0 𝑣 𝑙 𝑥0 𝑙 + 𝑣 𝑚 𝑥0 𝑚 + 𝑣 𝑟 𝑥0 𝑟 = 1 𝑢𝑙 , 𝑢𝑚 , 𝑢𝑟 , 𝑣 𝑙 , 𝑣 𝑚 , 𝑣 𝑟 ≥ 0

(3.9)

4 Finding strong defining hyperplanes of PPS with fuzzy data At first glance, it seems that using multiplier form, all defining hyperplanes of PPS can be obtained.However in reality this is not true, since the structure of envelopment form imposes strong degeneracythen the multiplier form may produce alternative optimal solutions. For example see situation of DMUo in Figure 1.Using model (2.3), it is seen that there arealternative optimal solutions which define infinite number of hyperplanes passing through A, from which only two hyperplanes (H1 and H2) are defining hyperplanes.

Figure 1: Hyperplane H is not defining. H1 and H2 are defining.

To remove this drawback, the following method for finding Strong Defining Hyperplane is presented. Suppose among all DMUs, Q of them are strong efficient; then all of this DMUs are on some strong supporting hyperplane. Because of strong efficiency, there exist some (u*,v*)> 0 solution for (2.3); then the strong efficient DMU lies on 𝑢∗𝑡 𝑦0 − 𝑣 ∗𝑡 𝑥0 + 𝑢0 ∗ = 0and this hyperplane is strong, according to its


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definition. Using the following theorems it can be determined which DMUs lie on the same supporting hyperplane. Theorem 4.1. Let (xp ,yp ) and (xq ,yq ) be observed DMUs that lie on a strong supporting hyperplane, then each convex combination of them is on the same hyperplane. Proof. See [4]. Corollary 4.1. If (xp ,yp ) and (xq ,yq ) are two efficient DMUs (strong or weak) and lie on the samehyperplane then for 𝜇 ∈ (0,1), 𝜇(𝑥𝑝 , 𝑦𝑝 ) + (1 − 𝜇)(𝑥𝑞 , 𝑦𝑞 ) is efficient and lies on that hyperplane. Theorem 4.2. Consider (xp , yp ) and (xq , yq ) be two observed DMUs that lie on different hyperplanes(excluding their intersection, if it is not empty). Then every point (virtual DMU) which obtained by strict convex combination of them is an interior point of PPS. In other words this virtual DMU is radial inefficient. Proof. See [4]. Suppose that LDMUs are strong efficient. Without lose of generality we can assume that these efficient DMUs are DMU1,…, DMUl Consider the set F = {1,…,L}, we take a distinct pair DMUp and DMUq, where p and q are belong to F, and construct a virtual DMUk as follows: 1

1

2

2

DMUk = DMUp + DMUq Using the DEA models we can determine DMUk is efficient or inefficient. In the first case, (by Theorem 4.2), DMUp and DMUq are on the same hyperplane; in the second case they are not (by Theorem 4.1). For each member j (j = 1,…, L) of F a new set Fj will be constructed. Fj is a subset of F that its members are coplanar. This means there exist some hyperplane contains DMUj and some DMUs in Fj. It is obvious that j∈Fj. Definition 4.1. H is a strong defining hyperplane of PPS if and only if it is supporting, at least m + s strong efficient DMUs of PPS lie on H and in its gradient components corresponding with output vector are nonnegative and components corresponding with input vector are nonpositive. In the light of above definition, we can make following criterion. We choose an arbitrary m + s members of F such that none of them belongs to some others F. Again we note that when we deal with T 0, one of these m + s DMUs can be origin, therefore only m+s -1 members of F are needed. We call this set 𝐷 = {𝑗1 , … , 𝑗𝑚+𝑠 }Using D; a hyperplane can be constructed as follows:

𝑥1 − 𝑥1𝑗1 𝑥 | 12 − 𝑥1𝑗1 𝑥1𝑗𝑚+𝑠 − 𝑥1𝑗1

… 𝑥𝑚 − 𝑥𝑚𝑗1 … 𝑥𝑚𝑗2 − 𝑥𝑚𝑗1 … 𝑥𝑚𝑗𝑚+𝑠 − 𝑥𝑚𝑗1

𝑦1 − 𝑦1𝑗1 … 𝑦1𝑗2 − 𝑦1𝑗1 … 𝑦1𝑗𝑚+𝑠 − 𝑦1𝑗1 …

𝑦𝑠 − 𝑦𝑠𝑗1 𝑦𝑠𝑗2 − 𝑦𝑠𝑗1 | = 0 𝑦𝑠𝑗𝑚+𝑠 − 𝑦𝑠𝑗1

(4.10)

Where 𝑥1 , … , 𝑥𝑚 , 𝑦1 , … , 𝑦𝑠 are variables, xpi , ( p =1,...,m;t =1,...,m+s is pth input of DMUjt and 𝑦𝑞𝑗𝑡 (𝑞 = 1, … , 𝑠: 𝑡 = 1, … , 𝑚 + 𝑠) is qth output of DMUj1. Suppose that the equation of the above mentioned hyperplanebe in the form of 0 Pt z + 𝛼 where 𝑧 = (𝑥1 … , 𝑥𝑚 , 𝑦1 , … , 𝑦𝑠 ), P is the gradient of the hyperplane and 𝛼 is a scalar.


02


Theorem 4.3. Consider 𝐻 = {𝑍|𝑃𝑇 𝑍 + 𝛼 = 0}that 𝑃𝑡 𝑧 + 𝛼 = 0 is constructed by (4.10). Suppose 𝑤 = (𝑥 𝑤 1 … , 𝑥 𝑤 𝑚 , 𝑦 𝑤 1 , … , 𝑦 𝑤 𝑠 ), is defined as follows: 𝑥 𝑤 𝑖 = max (𝑥𝑖𝑗 ) 𝑖 = 1, … , 𝑚. 𝑗=1,…𝑛

𝑦

𝑤

𝑟

= min (𝑦𝑟𝑗 ) 𝑟 = 1, … , 𝑠. 𝑗=1,…,𝑛

We call w Negative Ideal (NI). If 𝑃𝑡 𝑧𝑗 + 𝛼 = 0, 𝑗 ∈ 𝐷 𝑃𝑡 𝑧𝑗 + 𝛼 ≤ 0, 𝑗 ∈ 𝐹 − 𝐷 𝑃𝑡 𝑤 + 𝛼 < 0 that is all efficient DMUs and NI are in H- then H is supporting. Now we are in the position to put all together the ingredients of the method. Algorithm explanation: Step 1. Concider data with fuzzy inputs and outputs. Step 2. Evaluate n DMUs with fuzzy inputs and outputs using model (3.8) and (3.9). Put indexes of strong efficient DMUs in F. Let |F|= L. 1

1

2

2

Step 3. For each p,q ∈ F that p ≠q, evaluate DMUk = DMUp + DMUq If it is efficient, then setp∈ Fq and q∈ Fp. Step 4. For each j(j=1,…,L) F̅j = F - Fj Step 5. Choose an arbitrary m + s members of F such that none of them belongs to some others of F. When you deal with Tc, one of these m + s DMUs can be origin. Call this set D = {J1 , … , JM+S } Construct a hyperplane using Eq. (5.12). Suppose that the equation of hyperplane is in the form of 𝑃𝑡 𝑧 + 𝛼 = 0 Where z = (x1 … , xm , y1 , … , ys )y and 𝛼 is a scalar. Step 6. If P has any component less than or equal to zero go to 7, else let w = (x w1 … , x w m , y w1 , … , y w s ) is Defined as follows: x w i = max (xij ) i = 1, … , m. j=1,…n

y

w

r

= min (yrj ) r = 1, … , s. j=1,…,n

If P t zj + α = 0, j ∈ D P t zj + α ≤ 0, j ∈ F − D Pt w + α < 0 Then Pt z + α=0 is strong defining hyperplane. Step 7. If another subset of F with m + s members can be found go to 5, else stop. 5 Numerical examples For tow DMUs of one input and one output we get



𝑋̃1 = (1,2,3),

06

𝑌̃1 = (10,11,12)

𝑋̃2 = (3,4,5), 𝑌̃2 = (7,8,9) For evaluate 𝐷𝑀𝑈1 using the model (3.8) we get min 𝜃 𝑠. 𝑡. 𝜆1 + 3𝜆2 − 𝜃 + 𝑆1 = 0 2𝜆1 + 3𝜆2 − 𝜃 + 𝑆1 = 0 3𝜆1 + 5𝜆2 − 3𝜃 + 𝑆3 = 0 10𝜆1 + 7𝜆2 − 𝑠4 + 𝑡1 = 10 11𝜆1 + 8𝜆2 − 𝑠5 + 𝑡2 = 11 12𝜆1 + 9𝜆2 − 𝑠6 + 𝑡3 = 12 dual of model (5.11) the following max 10𝑢1 + 11𝑢2 + 12𝑢3 𝑠. 𝑡. 10𝑢1 + 11𝑢2 + 12𝑢3 − 𝑣1 − 2𝑣2 − 3𝑣3 ≤ 0 7𝑢1 + 8𝑢2 + 9𝑢3 − 3𝑣1 − 4𝑣2 − 5𝑣3 ≤ 0 𝑣1 + 𝑣2 + 𝑣3 = 1 𝑢, 𝑣 ≥ 0

(5.11)

(5.12)

The optimal solution of model (5.13) is (𝑢1 𝑡 , 𝑢2 𝑡 , 𝑢3 𝑡 , 𝑣1 𝑡 , 𝑣2 𝑡 , 𝑣3 𝑡 , 𝑠1 𝑡 , 𝑠2 𝑡 ) = (0,0,

1 12

1

11

3

12

, 0,0, , 0,

, 0)

(5.13)

And optimal value of (5.13) is one. now let 𝐹 = {0,1} We arbitrary tow members of F it means 𝐷 = {0,1} Now using the (4.10) 𝑥̃1 − 0 𝑦̃1 − 0 | |=0 (1,2,3) − (0,0,0) (10,11,12) − (0,0,0) That yields 10𝑥1 𝑙 + 11𝑥1 𝑚 + 12𝑥1 𝑟 − 𝑦1 𝑙 − 2𝑦1 𝑚 − 3𝑦1 𝑟 = 0 This hyperplan same is one constant of model (5.13) 6 Conclusion Data Envelopment Analysis (DEA) is a non-parametric method for measuring efficiency of a set of Decision Making Units and it is usually undertaken with absolute numerical data, which among other things reflect the size of the units. fuzzy data are one of the different kinds of data that show some uncertainty in inputs and outputs. In this paper a method for finding all Strong Defining Hyperplanes of PPS with fuzzy data is proposed. Although it seems that the subject is purely mathematical, but using these hyperplanes, all members of reference set of a DMU can be found.The proposed approach is illustrated by a numerical example. References [1] A. Charnes, W. W. Cooper, E. Rhodes, Measuring the efficiency of Decision Making Units, European Journal of Operational Research, 2 (6) (1978) 429-444. http://dx.doi.org/10.1016/0377-2217(78)90138-8



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[2] W. W. Cooper, L. M. Seiford, K. Tone, Data Envelopment Analysis, A comprehensive text with models, applications, references and DEA-solver software, Kluwer Academic Publisher, (2000). [3] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Zohrehbandian, Finding the piecewise linear frontier production function in Data Envelopment Analysis, Applied Mathematics and Computation, 163 (1) (2005) 483-488. http://dx.doi.org/10.1016/j.amc.2004.02.016 [4] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, H. ZhianiRezai, F. Rezai Balf, Finding strong defining hyperplanes of Production Possibility Set, European Journal of Operational Research, 177 (2007) 4254. http://dx.doi.org/10.1016/j.ejor.2005.11.031 [5] G. R. Jahanshahloo, A. Shirzadi, S. M. Mirdehghan, Finding strong defining hyperplanes of PPS using multiplier form, European Journal of Operational Research, 194 (2009) 933-938. http://dx.doi.org/10.1016/j.ejor.2008.01.053 [6] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Zohrehbandian, Finding the piecewise linear frontier production function in data envelopment analysis, Appl. Math. Comput, 163 (2005) 483-488. http://dx.doi.org/10.1016/j.amc.2004.02.016 [7] H. J. Zimmermann, Fuzzy Set Theory and its Application, Kluwer Academic Publishers, London, (1996). http://dx.doi.org/10.1007/978-94-015-8702-0 [8] F. Hosseinzadeh Lotfi, T. Allahviranloo, M. Alimardani Jondabeh, L. Alizadeh, Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution,Applied Mathematical Modelling, 33 (7) (2009) 3151-3156. http://dx.doi.org/10.1016/j.apm.2008.10.020 [9] T. Allahviranloo, F. Hosseinzadeh lotfi, M. Adabitabarfirozja, Efficiency in Fuzzy production possibility set, Iranian journal of Fuzzy system, (2012) 17-30. [10] G. Yu, Q. Wei, P. Brockett, L. 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 (3) (1996) 491-510. http://dx.doi.org/10.1016/0377-2217(95)00304-5 [11] H. H. Yang, Measuring the efficiencies of Asia–Pacific international airports – Parametric and nonparametric evidence, Computers & Industrial Engineering, 59 (2010) 697-702. http://dx.doi.org/10.1016/j.cie.2010.07.023
