Finding strong defining hyperplanes of production possibility set with stochastic 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, stochastic data are one of the different kinds of data that show some uncertainty in inputs and outputs. Therefore we apply an algorithm for transforming stochastic 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 stochastic 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. [20] 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 [2] 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.The initial DEA model is referred to as the CCR model, named after Charnes et al. [2], who assumed constant returns to scale (CRS) of the production function.Banker, Charnes and Cooper also proposed a modified model, named BCC in 1984 [2], 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 [5].The original DEA models assume that the levels of inputs and outputs are known exactly.Korhonen [13] tried to provide the Decision Maker (DM) an interactive method which allows him/her to incorporate performance information into the efficient frontier analysis by enabling him to make a free search on efficient frontier.Indeed, he wants to provide the DM all references of an inefficient DMU and enabling him/her to choose the most preferable unit as reference.But finding all references of an inefficient DMU is not an easy job.The DEA models are based on the production possibility set (PPS) finding the defining hyperplans of this PPS is one of the main concept of the DEA literatures.Some researches on sensitivity and stability are based on the form of efficient frontier (see [7]).Also, efficient surfaces are useful in analyzing DEA efficient DMUs and non-dominated solutions (Pareto Solutions) in multi-objective programming.Not many papers have been written on the subject of ''finding efficient frontier''.Jahanshahloo et al [8] proposed a method to obtain efficient frontier by using 0-1 integer programming.Yu et al [20] studied the structural properties of DEA efficient surfaces of the PPS under the generalized DEA model.However, uncertainty such as a measurement error should be incorporated in observed data.There are some types of data in DEA.One of them is stochastic data.Stochastic formulation of the original DEA models was introduced to incorporate possible uncertainty in inputs and/or outputs (e.g.Cooper et al [4]; Cooper et al [3].Huang and Li [6]; Khodabakhshi and Asgharian [14]; Land et al [17]; Olesen and Petersen, [19].Morita et al [18], studied the robustness of efficiency results when input and output data are subject to stochastic measurement error, while Jess et al [11] introduced a semi-infinite programming model in DEA to study a chemical engineering problem.More recently, stochastic input and output variations into DEA have been studied by; for example, Asgharian et al [1], Khodabakhshi [15], and Khodabakhshi and Asgharian [14].See, also, Kall [12] for discussions on linear programming programs [16].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 stochastic data are obtained.The current article proceeds as follows: In section 2, we give the concept of stochastic data and some basic definitions and models.In section 3, we try to find all Strong Defining Hyerplanes of Production Possibility Set (SDHP) with stochastic data.In section 4, a numerical example is considered and section 5 gives our conclusive remarks.

production possibility set (PPS)
We will call a pair of input x R m 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 (x j , y j ) P ; J=1,…,n 2. If an activity (x j , y j ) P, then the activity (tx,tY) P for all t 0. 3.If an activity (x j , y j ) P, then ( ̅ ̅) P if ̅ x and ̅ y. 4. If an activity (x j , y j ) P, then ( ̅ ̅) P, then ( ( ) ̅ ( ) ̅) P for all λ [0,1] We show the set P as follow: Where  is a semipositive vector in R n .The model is called ) and the optimal value of following linear programming is equal to zero.
For DMU o , we define its reference set E o , that is defined as follows: The DMUs in o E 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: ( , ) 0 uv  exists, then O DMU 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.

Stochastic models
In this section, we consider the stochastic model which allows for the possible presence of stochastic variability in the data.Following Cooper et al. [4] [16], where slack variables are all excluded from the objective function is as follows: Where  is a predetermined value between 0 and 1 which specifies the significance level.Since a solution with 00 1, We now the data arenot deterministic in this model but in this section, we exploit the normality assumption to introduce a deterministic equivalent to model (2.4).We assume that x ij and y rj are the means of the input and output variables, which are estimated in application by the average of observed values of the inputs and outputs for 4 years.The deterministic equivalent of (2.4) can be represented by model (2.5) as follows: Using the before mentioned property of normal distribution and replacing non-negative variables w , , , , , 0 We must have random inputs and outputs in order to using DEA stochastic models.Thus we try to make and output vector so that their values get randomly.It means that if

Finding strong defining hyperplanes of PPS with stochastic 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 degeneracy then the multiplier form may produce alternative optimal solutions.For example see situation of DMU o in Figure 1.Using model (2.3), it is seen that there are alternative optimal solutions which define infinite number of hyperplanes passing through A, from which only two hyperplanes (H1 and H2) are defining hyperplanes.x y x y Proof.See [9] Suppose that L DMUs are strong efficient.Without lose of generality we can assume that these efficient DMUs are 1 ,..., .
L DMU DMU Consider the set F = {1,…,L}, we take a distinct pair pq DMU and DMU , where p and q are belong to F, and construct a virtual DMU k as follows: 11 22 Using the DEA models we can determine k DMU is efficient or inefficient.In the first case, (by Theorem 3.2), pq DMU and DMU are on the same hyperplane; in the second case they are not (by Theorem 3.1).
For each member j (j = 1, . .., L) of F a new set F j will be constructed.F j is a subset of F that its members are coplanar.This means there exist some hyperplane contains DMU j and some DMUs in J F .It is obvious that j jF  .

Definition 3.1. 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 c T , one of these m + s DMUs can be origin, therefore only 1 ms  members of F are needed.We call this set   1 ,..., ms D j j   .Using D; a hyperplane can be constructed as follows: then H is supporting [21].Now we are in the position to put all together the ingredients of the method.Algorithm explanation: Step 1. Produce data with random inputs and outputs.
Step 2. Evaluate n DMUs with random inputs and outputs using model (2.6).Put indexes of strong efficient DMUs in F. Let FL  .
Step 3.For each , p q F  that p  q, evaluate 11 .22 If it is efficient, then set qp p F and q F  .
Step 4. For each j(j=1,…,L)   Therefore the hyperplane H is constructed on D 1 = {O, 1, 6} where O is the origin (see table 3).The equation of hyperplane H can be obtained by following determinant: After some Simplification, we have the following hyperplane; 10.09x 1 +292.38x 2 -97.92y=0

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.Stochastic 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 stochastic 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.Also they can be used for sensitivity analysis.Moreover, we think they can be used in analyzing Pareto solutions in multiobjective programming; which this is a problem for further research.The proposed approach is illustrated by a numerical example.
 and ** ( , ) 0 uv  for some optimal solutions.If ** 1 t oo u y u  and no **

Definition 2 . 1 .
the optimal value of objective function is less than or equal to 1. Stochastic efficiency with the model (2.4) can therefore be defined as below: (stochastic efficiency) O DMU is stochastically efficient if and only if the following conditions are satisfied: (i) * 1. o   (ii) Slack variables are zero in all alternative optimal solution.
following deterministic model which is a quadratic program: this paper, we follow the normal distribution with mean  and standard deviation σ.

Step 5 . 1 {
Choose an arbitrary m + s members of F such that none of them belongs to some others of F. When you deal with T c , one of these m + s DMUs can be origin.Call this set Construct a hyperplane using Eq.(3.1).Suppose that the equation of hyperplane is in the form of 0 t Pz   Where 11 ( ,..., , ,..., ) ms z x x y y  and  is a scalar.

Step 6 .
If P has any component less than or equal to zero go to 7, else letConsider the stochastic data setting of the table 1 containing 6 DMUs with two inputs and one output, where ̃ and ̃ show the inputs and ̃ shows the output.

Table 1 :
Data setting

Table 2 .
Data set with new virtual DMUs By using model (2.5), we can see that DMU7, DMU8 and DMU9, are efficient too.Therefore we have:

Table 3 :
DMUs on hyperplane H