Evaluating the performance of fuzzy group units in DEA using credibility measure

The purpose of this paper is to analyze the group performance in the presence of fuzzy data. In this paper, an extended form of multiplier CCR method is recommended for calculating group performance that in addition to its efficiency, it can calculate the efficiency of units in each group and the impact of performance within each group on the whole group. Then the proposed model is discussed in fuzzy environment. To solve the model by fuzzy data, credibility measure is applied on the model to its main advantages over other methods for solving fuzzy models. Finally, the proposed method is used to choose projects at NASA.


Introduction
Data Envelopment Analysis presented by Charnes et al. [2] to calculate performance was used to measure different economic concepts.Some of these concepts include ranking, returns to scale, productivity, efficiency, cost performance, group performance or Cook and Seiford [3] program.Group performance or effectiveness of the program is to evaluate the performance of applications running on a system that includes a set of decision making units with similar activities.The faculties of a university can be decision-making groups in which DMUs are the departments that do similar work.Suppose that there are a number of projects with some indices for their performance and we have a number of decision makers regarding the value of these indices.In such circumstances group performance can be used to select a project.Comanho and Dyson [5] used Malmquist productivity growth index to compare the pairs of groups.Another method by Cook and Zhu [4] to compare the performance and efficiency of the units was using common weights.Bagherzadeh Valami [1] in an article titled "Evaluation of the group performance" presented an index benchmark for comparison and ranking groups under evaluation and used this index to compare the performance of bank branches under different supervisions.Payan [9] presented a model to evaluate the performance of groups with common weights that has the ability to calculate the comparison of each group and the units within it compared with other groups and performance distribution.In this paper, a method for evaluating the DEA performance of a group that has the ability to choose projects as well as the importance of each project according to each decision-maker's idea.The proposed method expands further in fuzzy environment.Tavana et al. [10] suggested several DEA models for calculating group performance in the presence of fuzzy data.They used α-cut method to solve their model.Fasanghari et al. [6] used credibility theory to calculate the group performance to solve their proposed model.One of the most important issues in fuzzy environment is to choose the solution to solve fuzzy model.These methods are divided into three major categories; 1) Ranking, 2 α-cut, 3) measurement theory Although ranking fuzzy numbers is very simple and understandable, but since it provides a precise match for imprecise model, it does not keep the nature of model and this is the major weakness of this method.αcut does not suffer from this problem and for each α-cut provides a number range as the optimal solution, therefore, using decomposition theorem [11] through the aggregate of the number range, it is possible to obtain a membership function for the optimal value.Although in this method a fuzzy answer is presented for fuzzy model but for each α-cut two optimization model should be solved which contains a long problem calculation process.But among the fuzzy solution methods, the measurement theory has been considered by fuzzy theory researchers because of maintaining the fuzzy nature and having appropriate computational period.This theory has two basic ways called possibility-necessity measure and credibility measure.Possibility measure was initially introduced first by Zadeh [12] to measure the possibility of an event.This measure has a fundamental flaw and that is the failure to secure self-dual property for this measure.To illustrate, suppose we say a proposition is true with  possibility it seems rational to say that this proposition is wrong with the possibility of 1 −  which is not true about the measure of possibility.In other words, the proposition may be false with a possibility more or less than 1 − .To fix this flaw the necessity measure was introduced.So in possibility theory in order to use a concept we need two measure functions and this is one of the flaws of possibility theory.For example, when we want to solve linear programming problems with fuzzy data, we need to use possibility measure to solve the primal problem and apply necessity measure to solve dual problem.Liu and Liu [8] introduced the concept of credibility.Accordingly a theory and credibility measure was obtained that has all properties of theory and credibility measure which is dual at the same time.In order to analyze credibility theory and its properties to solve fuzzy programming problems further it is possible to refer to Liu [7].Accordingly, in this paper theory of credibility is applied to solve the proposed model.Accordingly, sections of this article are as follows: In the second part a method for calculating group performance will be presented.In the next section, the model is solved in the presence of fuzzy data.Then a functional example of choosing project in NASA will be solved by the proposed method and the results will be compared with the results obtained by other methods.Finally the results will be presented.

Group performance
There are  groups and each group has  units.All units in different groups are homogeneous.The unit  from each group consumes  inputs   ℎ ( = 1, … , ) to produce  outputs   ℎ ( = 1, … , ).
Suppose  = ( 1 , … ,   ) is the input weight vector and  = ( 1 , … ,   ) is the output weight vector.Then the performance of unit  in group ℎ is obtained based on the following equation: The whole group efficiency is defined as a convex combination of efficiency of units within the group, so: In other words, the performance of group ℎ is the outcome of activities of all units within that group.From a statistical standpoint   ℎ is the weight average of the performance of the units within the group h and from an economic perspective,   ℎ is the share of unit  in group ℎ to calculate the performance of group ℎ.
The following non-linear programming model is recommended for calculating performance of the groups under evaluation: This model can be written as follows: ) Then: There are analyses: 1.   ℎ is total weighted output to input that is identical to the concept of efficiency in data envelopment analysis the output of which is the total outputs of all units within it and its inputs are the total inputs of all units within it.

𝑤 𝑗
ℎ Which is the share of unit  ( = 1, … , ) from group ℎ (ℎ = 1, … , ) in the performance of the whole group is obtained based on the result of the problem.The denominator in   ℎ is similar for all units of each group and the difference in the share of the units is the total weight of the inputs per unit.As this value becomes lower for a unit, it indicates the greater share of the unit to calculate group performance.We know that the sum of the weighted input is unit cost in the economic sense.So as higher unit cost leads to higher share in group performance and this is significant from an economic perspective.
Accordingly, the nonlinear programming problem (2.5) can be converted to a linear fractional programming problem as follows: Linear form of the above problem is as follows: Suppose  * and  * are the weights obtained by solving the above model.Performance of group , the units within it and the value of performance of each unit in group  is obtained using the following equations: ) ) Also after solving the above model for all units, the performance of each unit based on the total ideas of DMUs can be obtained by the following equation: )

Credibility measure
Suppose  is a nonempty set.The characteristic function of each subset  from  is a function of  to {0,1} which is defined as follows: 1  ∈  0  ∉  If the range of the function is developed from the two member set {0,1} to range [0,1] we will have a function that relates a number from range [0,1] to each  from .The function that is displayed as   () is called the membership function of the set .In this case the set  is called a fuzzy subset presented as  ̃.When we say a proposition is true with possibility of  it seems rational to say that this proposition is wrong with the possibility of 1 − .This is called self dual property.Possibility measure lacks such property.To fix this flaw necessity measure was introduced.Therefore in order to use a concept we need two measure functions and this is one of the flaw of possibility theory.For example, when we want to solve linear programming problems with fuzzy data, we need to use possibility measure to solve the primal problem and apply necessity measure to solve dual problem.Liu and Liu [8] introduced the concept of credibility.They considered fuzzy event  ̃ credibility as the average of possibility and necessity of fuzzy event ̃.Accordingly the theory and credibility measure were introduces that contain all characteristics of theory and possibility measure which is dual at the same time.
For real number  the necessity and possibility measures of the event  ≤  are as follows: Using the definition of credulity measure, ( ≤ ) is defined as follows: ≤ 1} ≥  if and only if: If we define  as a trapezoidal fuzzy number by: and consider  = 0, for  ≥ 0.

Example
Consider 20 projects in NASA each project has two inputs and four outputs.The values of these indices are determined by five decision makers.Data are collected by Tavana et al. [10].Project performance using the information obtained by each DM, for four credibility levels α = 0.1, α = 0.4, α = 0.7 and α = 1.0 can be calculated by solving models (4.17) and (4.18).The results of ranking units are reported in Figure 1.As can be seen from the figures the ranking obtained for α = 0.1 and α = 0.4 are close.Also ranking obtained for α = 1.0 and α = 0.7 are similar as well.As a comparison, top rankings for all levels of credit are almost the same.

Figure 1 :
Figure 1: Ranking projects in different credit levels of α

4 Group performance with fuzzy data
Suppose the data related to  group under trapezoidal fuzzy numbers evaluation are as follows: ≤  ≤   (  +  −   ) ((  −   )) ⁄   ≤  ≤      ≤  Considering objective function data are part of constraint data, therefore it is possible to determine objective function data behavior by analyzing constraints' behavior and replace equivalent terms in objective function.
5 we have:According to the above theorem the fuzzy input and outputs' behavior  ̃ ℎ and  ̃ ℎ