Finding strong defining hyperplanes of production possibility set with fuzzy data

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.


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.

Data Envelopment Analysis and Decision Science
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.For DMUo, we define its reference set Eo, that is defined as follows: The DMUs in E 0 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: 3)  *   0 +  0 * = 1 and ( * ,  * ) > 0 for some optimal solutions.If  *   0 +  0 * = 1 and no ( * ,  * ) > 0 exists, then DMU 0 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.

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.

Fuzzy CCR Model (FCCR)
Let us consider the DMU0.In CCR model with fuzzy data, the objective function seeks the minimum value of  when the activity ( ̃0,  ̃0) is belong to  ̃.While ( 0 ,   ̃0( 0 )) ∈  ̃0any input vector X0 reduces radially to  X0.Hence FCCR model proposes the following model:

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.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  *   0 −  *   0 +  0 * = 0and this hyperplane is strong, according to its definition.Using the following theorems it can be determined which DMUs lie on the same supporting hyperplane.
Theorem 4.1.Let (x p ,y p ) and (x q ,y q ) be observed DMUs that lie on a strong supporting hyperplane, then each convex combination of them is on the same hyperplane.
Theorem 4.2.Consider (x p , y p ) and (x q , y q ) 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.
Suppose that LDMUs are strong efficient.Without lose of generality we can assume that these efficient DMUs are DMU 1 ,…, DMU l Consider the set F = {1,…,L}, we take a distinct pair DMU p and DMU q , where p and q are belong to F, and construct a virtual DMUk as follows:

DMU q
Using the DEA models we can determine DMU k is efficient or inefficient.In the first case, (by Theorem 4.2), DMU p and DMU q 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 F j .It is obvious that j∈F j .

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  = { 1 , … ,  + }Using D; a hyperplane can be constructed as follows: Where  1 , … ,   ,  1 , … ,   are variables, x pi , ( p =1,...,m;t =1,...,m+s is pth input of DMU jt and   ( = 1, … , :  = 1, … ,  + ) is qth output of DMU j1 .
Suppose that the equation of the above mentioned hyperplanebe in the form of 0 P t z +  where  = ( 1 … ,   ,  1 , … ,   ), P is the gradient of the hyperplane and  is a scalar.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.
Step 3.For each p,q ∈ F that p ≠q, evaluate DMU k = DMU q If it is efficient, then setp∈ F q and q∈ F p.
Step 4. For each j(j=1,…,L) F ̅ j = F -F j 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 = {J 1 , … , J M+S } Construct a hyperplane using Eq.(5.12).Suppose that the equation of hyperplane is in the form of    +  = 0 Where z = (x 1 … , x m , y 1 , … , y s )y and  is a scalar.
Step 6.If P has any component less than or equal to zero go to 7, else let w = (x w 1 … , x w m , y w 1 , … , y w s ) is Defined as follows: (y rj ) r = 1, … , s.
Step 7. If another subset of F with m + s members can be found go to 5, else stop.

Numerical examples
For tow DMUs of one input and one output we get And optimal value of (5.This hyperplan same is one constant of model (5.13)

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.

Figure 1 :
Figure 1: Hyperplane H is not defining.H1 and H2 are defining.
13) is one.now let  = {0,1} We arbitrary tow members of F it means  = {0,1} Now using the (4.10) )are the left, the mean and the right values of triangular fuzzy numbers  ̃ and  ̃ respectively.dual of model(5.11)following.