A Multiobjective Programming Method for Ranking All Units Based on Compensatory DEA Model

In order to rank all decision making units (DMUs) on the same basis, this paper proposes a multiobjective programming (MOP) model based on a compensatory data envelopment analysis (DEA) model to derive a common set of weights that can be used for the full ranking of all DMUs. We first revisit a compensatory DEA model for ranking all units, point out the existing problem for solving the model, and present an improved algorithm for which an approximate global optimal solution of the model can be obtained by solving a sequence of linear programming. Then, we applied the key idea of the compensatory DEA model to develop the MOP model in which the objectives are to simultaneously maximize all common weights under constraints that the sum of efficiency values of all DMUs is equal to unity and the sum of all common weights is also equal to unity. In order to solve the MOP model, we transform it into a single objective programming (SOP) model using a fuzzy programming method and solve the SOP model using the proposed approximation algorithm. To illustrate the ranking method using the proposed method, two numerical examples are solved.


Introduction
As a nonparametric method, data envelopment analysis (DEA) is proposed by [1] to evaluate the relative efficiency of a set of decision making units (DMUs) with the multiple inputs and outputs.For each DMU, the optimal weights for calculating efficiency value are obtained by solving a linear programming (LP) problem.By using DEA, DMUs can be divided into two categories: efficient DMUs and inefficient DMUs [2].However, there are always more than one DMU to be evaluated as efficient, which would cause the problem that all DMUs cannot be fully discriminated [3].Moreover, the efficiencies of different DMUs obtained by different sets of weights may be unable to be compared and ranked on the same basis [2][3][4][5].
To deal with the above-mentioned problems, many methods have been developed to rank all DMUs under the framework of DEA.The common weights DEA introduced by [6][7][8] is known as one of the popular methods in which all DMUs can be evaluated by a common set of weights (CSW).Since then, DEA models with CSW have been rapidly applied in many researches.For example, Sinuany-Stern and Friedman [9] developed a DR/DEA to provide the best CSW for given inputs and outputs in order to rank all DMUs on the same scale.Jahanshahloo et al. [10] presented a method to obtain the CSW of DMUs by solving only one problem, in order to measure the efficiency and to rank the efficient DMUs.Kao and Hung [11] proposed a compromise solution approach for generating a CSW which produces the vector of efficiency scores for the DMUs.Their approach is able to not only differentiate efficient DMUs but also detect abnormal efficiency scores on a common base.Amin and Toloo [12] presented an improved integrated DEA model in order to detect the most efficient DMUs.The method in their model is able to determine a CSW for all DMUs by solving a LP problem.Liu and Peng [13] introduced a DEA method to determine a CSW for the ranking of only DEA efficient DMUs.For the decision maker (DM), this ranking is based on the optimization of the group's efficiency.Jahanshahloo et al. [2] proposed two methods to rank DMUs concerning an

The Improved Method for Solving Compensatory DEA Model
In this section, we first revisit the compensatory DEA model proposed by [19] for ranking all DMUs.After pointing out the existing problem of their solution method, we will propose an approximation algorithm by improving it so that the model can be solved correctly.To demonstrate the improved algorithm, a numerical example is also presented in this section.
Let DMU  be a DMU under evaluation; then the following model is used to determine the minimum and maximum efficiency values of DMU  : (M1) min and max , V  ,   ≥ 0, ∀, , . ( Model (M1) must be run twice.The minimum/maximum value of   is determined by minimizing/maximizing the objective function of model (M1).Model (M1) is a nonlinear programming.Using the transformation   := V    , we can replace model (M1) by the following LP problem: Let  min  and  max  be the minimum and maximum values of   , respectively, which can be obtained by solving model (M2).Then the following convex combinations are used to determine the efficiency values for each DMU: where Considering ( 4) and the value of  obtained in (5) together, the values of   ( = 1, 2, . . ., ) are determined.Now, the DMUs are fully ranked with respect to their efficiency scores.In other words, a DMU has a better rank if it has a greater efficiency score.

The Existing Problem for Solving Compensatory DEA
Model.In compensatory DEA model, the optimal solutions of model (M1) can be found by solving model (M2).Since constraint   = V    is not considered in model (M2), model (M2) is a relaxation problem of model (M1), and two models are not completely equivalent.Hence, we may not obtain the optimal solutions of model (M1) by solving model (M2).
Let    and V   be the optimal solutions of model (M2) with a minimum form.If they are also the optimal solutions of model (M1), then they must satisfy the constraints of model (M1); that is ∑  =1    = 1, where the efficiency score    is determined by (1).However the following example shows that the equation ∑  =1    = 1 does not hold.For example, suppose we want to evaluate DMU 1 in the numerical example presented in Section 2.4.For a minimum problem, we solve model (M2) to find the optimal solutions:   1 = 0.000678,   2 = 0, V  1 = 0.000130, V  2 = 0.012707, and V  3 = 0.050998.Using (1) we can obtain that the efficiency scores    of twelve DMUs are 0.0454, 0.0637, 0.0631, 0.0680, 0.0928, 0.0439, 0.0604, 0.0707, 0.0828, 0.0427, 0.0276, and 0.0599, respectively.Computing the sum of those efficiency scores we have ∑ 12 =1    = 0.7210 < 1, which means that    and V   are not the optimal solutions of model (M1).Therefore, we cannot obtain the optimal solutions of model (M1) by solving model (M2).

The Proposed Approximation Algorithm for Solving Model (M1).
In this section, the minimum problems as an example are used to illustrate the approximation algorithm.However, the procedure can be easily extended to solve the maximum problems.
Unlike the standard DEA, model (M1) involves the sum of linear-fractional functions in constraints.So model (M1) may not be considered a convex optimization problem.To prove it we first rewrite these constraints of linear-fractional functions as follows: The Hessian matrix of function   , with the order of variables being (, V, ), can be derived as where   = ( 1 ,  2 , . . .,   ).Zeros in the matrix are of appropriate dimension.Since principal minors of order 2 are      (  ) 2      = − 2 1 < 0, that is, there exists one 2 × 2 principal minor in (  ) which is negative, it can be concluded that these constraints function is nondefinite.Therefore, model (M1) is not a convex optimization, which implies that an efficient algorithm may be required to determine a globally optimal solution.
Although the optimal solutions of model (M1) cannot be found by solving model (M2), the following shows that the optimal solutions of model (M1) can be approximated by solving a sequence of LP models on the base of the optimal solution of model (M2).
Since the optimal solutions of model (M2) may not be unique, a secondary goal must be introduced in it.For the minimum problem, the secondary goal is to maximize the sum of efficiency scores of the rest DMUs; that is, The fractional form of the secondary objective function can be transformed to the following linear form: Therefore, introducing a sufficiently small positive number , model (M2) can also be rewritten as follows: where ℵ 2 (, V) denotes the set of the feasible solution to model (M2).Since model (M2) or (M3) is a relaxation problem of model (M1), we have the following lemma.
Lemma 1.Let   and V  be the optimal solutions of model (M3); then one has where   is determined by using (1).
Proposition 2. If ∑  =1   = 1, then the optimal solutions   and V  of model (M3) are also the optimal solutions of model (M1).
Proof.If ∑  =1   = 1, then the optimal solutions   and V  of model (M3) are the feasible solutions of model (M1).Since the objectives of two models are equivalent, the optimal solutions   and V  of model (M3) are also the optimal solutions of model (M1).
If ∑  =1   < 1, then we have to proceed further to find the optimal solutions on the basis of it.Let constraint   = V    be replaced by   ≥ V    , then add it into model (M3) to formulate the following LP problem: It is obvious that model (M4) is a relaxation problem of model (M1) and model (M3) is a relaxation problem of model (M4).Therefore, we have the following lemma.Lemma 3. Let  (1)   and V (1)   be the optimal solutions of model (M4); then one has where  (1)   is determined by (1).
According to Proposition 2, if ∑  =1  (1)  = 1, then  (1)   and V (1)   are also the optimal solutions of model (M1); otherwise, we must also continue to find the optimal solutions of model (M1).Let  (1)   replace   in model (M4); then solve it to obtain the optimal solutions  (2)   and V (2)   .Using (1) we will get the efficiency score  (2)   of each DMU and then judge whether ∑  =1  (2)   is equal to 1 or not.The process is repeated until the sum of efficiency scores of all DMUs is approximately equal to 1 within given error.
According to the above analysis, the procedure of the proposed approximation algorithm to solve model (M1) can be described as follows.
Algorithm 4. Consider the following.
Step 2. Solve model (M3) to find the optimal solutions   and V  , and then calculate the efficiency score   of each DMU using (1).
Step 3. Compute the sum of the efficiency scores of all DMUs.If      ∑  =1   − 1      ≤ , then stop with the optimal solutions   and V  to model (M1); otherwise,  := 1; go to Step 4.
Step 4. Solve model (M4) to find optimal solutions  ()  and V ()  , and then calculate the efficiency score  ()  of each DMU using (1).
Step 5. Compute the sum of the efficiency scores of all DMUs.If      ∑  =1  ()  − 1      ≤ , then stop with the optimal solutions  ()  and V ()  to model (M1); otherwise go to Step 6.

Proposition 5. The optimal solutions of model (M4) will be close to the optimal solutions of model (M1) by using Algorithm 4.
Proof.Let  ()   and V ()  ,  = 1, 2, . . ., , be the optimal solutions to model (M4) for tth solution; using (1) we will get the efficiency score  ()  of each DMU; then we have Since {∑  =1  ()  } increases monotonically and exists upper bound 1, we have ∑  =1  ()  → 1, which means that the optimal solutions  ()  and V ()  of model (M4) are close to the optimal solutions of model (M1) by repeatedly solving model (M4).
We have discussed the approximation algorithm with minimum problem.For a maximum problem, let constraint   ≥ V    be replaced by constraint   ≤ V    in model (M4).Using the above approximation algorithm, we can also obtain the optimal solutions to model (M1) with maximum form.

A Numerical Example.
In order to compare the proposed method with the method in [19], a numerical example used by them is presented in this subsection.There are twelve DMUs with three inputs ( 1 ,  2 , and  3 ) and two outputs ( 1 ,  2 ) as shown in Table 1.The minimum and maximum efficiency scores, the integrated score, and the rank of DMUs obtained by the method in [19] are exhibited in the seventh, the eighth, and the ninth column of Table 1, respectively.
In order to rank all DMUs by the proposed method, we first use the proposed approximation Algorithm 4 to calculate the minimum and maximum efficiency scores of each DMU and then use ( 4) and ( 5) to compute the integrated scores.Finally, the DMUs were ranked according to their integrated scores.
Using Algorithm 4, the procedure to calculate the minimum efficiency score of DMU 1 is described as follows.
Using (4) and ( 5), the integrated score   is determined and the results are exhibited in the eleventh column of Table 1.The twelfth column of Table 1 shows the results of rankings according to their integrated scores.
Table 1 shows that the rankings of DMUs obtained by proposed method are not exactly the same as that obtained by the method in [19].For instance, DMU 6 is ranked as eighth in our method and sixth in the method of [19], whereas DMU 10 is placed sixth in our method and eighth in the method of [19].

Proposed Multiobjective Programming
Model Based on Compensatory DEA

The Multiobjective Programming Model for Ranking All
Units.Khodabakhshi and Aryavash [19] contribute to a very interesting variant of the basic DEA model by supposing that the sum of efficiency values of all DMUs is equal to unity.
Using their method all DMUs can be fully ranked.However, their method does not provide more information about the weight used for calculating the efficiency scores of each DMU.
In this section, we aim to develop a new model to find a CSW, which can be used for the full ranking of all DMUs.To do it, here we propose a MOP model as follows: where V  ( = 1, 2 . . ., ) and   ( = 1, 2, . . ., ) are the common weights assigned to th input and th output, respectively.x ( = 1, 2, . . ., ) and ŷ ( = 1, 2, . . ., ) represent normalized input and output values of DMU  ( = 1, 2, . . ., ), respectively.In order to eliminate the impacts of measurement units on a CSW, we normalize all inputs and outputs values by using the following equations [23]: In model (M5), in order to avoid a choice of zero common weights and get them as big as possible, the objectives are to simultaneously maximize the common weights assigned to each input and output.The first constraint is supposing that the sum of efficiency values of all DMUs is equal to unity, which can avoid the problem that more than one DMU is evaluated as efficient in DEA.The second constraint ∑  =1   + ∑  =1 V  = 1 is used to avoid arbitrariness in determining common weights.Without it, there will be an infinite number of common weights that can meet the model (M5).

Fuzzy Programming Method for Solving the MOP Model (M5).
Many methods can be used to solve the MOP model, such as compromise programming method, goal programming method, and fuzzy programming method.In this paper, we use fuzzy programming methods to solve the MOP model (M5).
In the MOP model (M5), it is unlikely that all objectives simultaneously achieve their optimal solutions subject to the given constraints.So in practice the DM usually chooses a satisfying solution according to the aspiration level fixed for each objective [24,25].Let   ,  = 1, 2, . . ., , V  ,  = 1, 2, . . ., , be the aspiration level of objectives; then model (M5) can be expressed as follows [24,26,27]: To satisfy   ≥  ,  = 1, 2, . . ., , The expressions   ≥  and V  ≥V  are represented by a fuzzy set called fuzzy goal, which means that the DM would be satisfied even for objective value slightly less than   or V  within the value of allowed deviations.The symbols " ≥" denote the fuzzified versions of "⩾" and they can be read as "approximately greater than or equal to." The expression (  , V  ) ∈ ℵ 5 (, V) is system constraint in model (M5), and ℵ 5 (, V) denotes the set of the feasible solution of model (M5).
Model (M6) is a fuzzy multiobjective programming (FMOP) model.The procedure for finding the fuzzy efficient solution can be summarized as follows.Algorithm 6.Consider the following.

Step 1 (determine the fuzzy aspiration level of each objective).
Since the DM usually knows little about the objectives, this paper uses the ideal solutions of model (M5) to determine the value of the fuzzy aspiration level of each objective.The ideal solutions are just the optimal solutions of the individual objective subject to the system constraint.That is, Model (M7) may not be also considered as a convex optimization problem like model (M1), and it usually has multiple local optimal solutions.Thus model (M7) is more difficult to be solved.In the following, a similar approximation algorithm with Algorithm 4 is used. Let then model (M7) can be rewritten as follows: Using the transformation   = V    [19], model (M8) can be rewritten by the following LP problem: Let   and V  be the optimal solutions of model (M9).Using (19), it is easy to compute the efficiency score   of each DMU.If ∑  =1   = 1, then   and V  are also the optimal solutions of model (M7); otherwise, that is, ∑  =1   ̸ = 1, we have to proceed further to find the optimal solutions on the basis of the optimal solution of model (M9).In the following, we only discuss the case that ∑  =1   > 1 for the sake of convenient description.However, the procedure can be easily extended to the case that ∑  =1   < 1.In order to find further the optimal solutions of model (M7) when ∑  =1   > 1, let constraints   = V    be replaced by constraints   ≤ V    , then add it into model (M9) to formulate the following LP problem: where ℵ 9 (, V) denotes the set of the feasible solutions to model (M9).
Let  (1)   and V (1)   be the optimal solutions of model (M10).Using (19), we will get the efficiency score = 1.Then  (1)   and V (1)   are the optimal solutions of model (M7); otherwise, we also have to proceed further to find the optimal solutions on the basis of it.

Let 𝜃
(1)  replace   in model (M10), and then solve it to find the optimal solutions  (2)   and V (2)  .Using (19) we will get the efficiency score (2)  of each DMU and then judge whether is equal to 1 or not.The process is repeated until that the sum of efficiency scores of all DMUs is approximately equal to 1 within given error.According to Algorithm 4, the optimal solutions of model (M10) will be close to the optimal solutions of model (M7) by repeatedly solving model (M10).
We have discussed the approximation algorithm in the case that ∑  =1   > 1.If ∑  =1   < 1, then let constraint   ≤ V    be replaced by constraint   ≥ V    in model (M10).According to the above analysis and Algorithm 4, we can also obtain the optimal solutions of model (M7).
Step 2 (construct membership functions).In the model (M6), each fuzzy goal is represented by fuzzy sets and defined by membership functions   (  ) and   (V  ) ( = 1, 2, . . ., ,  = 1, 2, . . ., ).Since linear membership functions are used more than other types of membership functions in the literature [28], we choose the linear membership functions.For each objective, the lower bound is zero, and the upper bound is the fuzzy aspiration level determined by solving model (M7).Hence, the membership functions for each objective may be defined as follows: Step 3 (build an auxiliary crisp model).Introducing the auxiliary variable , the FMOP model (M6) is transformed into the following auxiliary crisp model [27,29,30]: where  is a sufficiently small positive number.The optimal solution to model (M11) is a fuzzy efficient solution to the FMOP model (M6) and the Pareto optimal solution of MOP model (M5) [24,29,30].

Numerical Examples
In this section, we provide two numerical examples to illustrate our method and then compare the results with the results of the existing models to show the potential usage of the proposed method in the full ranking of DMUs.
Example 1.Consider the numerical example used in Section 2.4.There are twelve DMUs with three inputs ( 1 ,  2 , and  3 ) and two outputs ( 1 ,  2 ) as shown in Table 1.In order to rank all DMUs by using the proposed method, we first normalize all inputs and outputs values by using (16); the normalized values are shown in Table 2.The integrated score and the rank of DMUs obtained by the method in [19] are exhibited again in the seventh and the eighth column of Table 2, respectively.
Using the normalized values we can build a MOP model (M5), which is transformed into FMOP model (M6).We solve model (M6) to find the fuzzy efficient solution, which is also the Pareto optimal solution of MOP model (M5), using Algorithm 6.
The procedure to solve the model (M6) can be described as follows.
Step 1 (determine the fuzzy aspiration level of each objective by solving model (M7)).The algorithm for solving model (M7) is similar to the approximation algorithm, Algorithm 4.

Step 3 (build an auxiliary crisp model (M11)).
Having determining the fuzzy aspiration level of each objective and the form of membership functions, we can get the auxiliary crisp model (M11) by introducing the auxiliary variable .
Using (19) we can compute the optimal efficiency score  *  of each DMU, and the results are shown in the ninth column of Table 2.The tenth column of Table 2 shows the results of ranking by our method.It shows that all DMUs are fully ranked with their optimal efficiency scores  *  .Table 2 shows that the rankings of DMUs using the proposed method are not entirely consistent with that of the method in [19].The main advantage of our method compared to the method in [19] is that a CSW for fully ranking DMUs is derived and all DMUs may be able to be compared and ranked on the same basis.Moreover, we improve their solution method and propose an approximation algorithm to solve their DEA model.
Example 2. Measure the performance of nations participating in the Olympic Games.Consider the example studied by Zhang et al. [20] and Azizi and Wang [21], in which 73 countries or areas are evaluated in terms of two inputs and three outputs defined as follows: 1 : gross domestic product (GDP),  2 : total population of the country or area,  1 : number of gold medals won by the country,  2 : number of silver medals won by the country,  3 : number of bronze medals won by the country.
The input and output data of the Athens 2004 Summer Olympic Games [20,21] are presented in Table 3.We use this example to compare the proposed method with Azizi and Wang's method [21] and Ramezani-Tarkhorani et al. 's method [22].
In Azizi and Wang's method [21], a pair of bounded DEA models was proposed to measure the interval efficiencies of DMUs.The lower bound of the interval efficiency is called the worst relative efficiency or pessimistic efficiency, and its value is determined using improved pessimistic DEA model [21].The upper bound of the interval efficiency is called the best relative efficiency or the optimistic efficiency, and its value is determined using the conventional DEA model.Using the  4 shows the rankings based on the efficiency intervals [21].Liu and Peng [13] proposed a common weights analysis methodology to generate a CSW for the performance indices of only DEA efficient DMUs.All DMUs are then ranked according to the efficiency scores weighted by the CSW.Ramezani-Tarkhorani et al. [22] pointed out the problem and proposed a new approach to rank all DMUs with common weights.From the second column of Table 4, it can be seen that, among 73 DMUs, 18 DMUs are DEA efficient.Using the data of these efficient DMUs, the CSW is determined as  1 = 0.01,  2 = 5.829,  3 = 3.235, V 1 = 0.01, and V 2 = 0.01.The efficiency scores of DMUs can then be calculated using the CSW and the results are shown in the fifth column of Table 4, and corresponding rankings are shown in the sixth column of Table 4.
In order to rank all DMUs by using the proposed method, we first normalize all inputs and outputs values by using (16) and then build a MOP model (M5), which is transformed into FMOP model (M6).We solve model (M6) to find the fuzzy efficient solution, which is also the Pareto optimal solution of MOP model (M5), using Algorithm 6.The procedure to solve model (M6) can be described as follows.
Step 2 (construct membership functions).The mathematical expression of membership functions are given in (23).For each objective, the lower bound is zero, and the upper bound is the fuzzy aspiration level determined by solving model (M7) in Step 1.
Step 3 (build an auxiliary crisp model (M11)).Having determining the fuzzy aspiration level of each objective and the form of membership functions, we can get the auxiliary crisp model (M11) by introducing the auxiliary variable .

Table 1 :
The input-output data and the rankings of DMUs.
Step 3. Computing the sum of the efficiency scores of all DMUs, we have ∑12=1   = 0.7335.Since      ∑ 12 =1   − 1      = 0.2665 > 0.001, we do not obtain the optimal solutions to model (M1), and we have to proceed further to find the optimal solutions.Consider  := 1; go to Step 4.
015451, and the objective function valve is  * 1 = 0.0573.Hence, the minimum efficiency value of DMU 1 is  min Similarly, using Algorithm 4, we can also obtain the minimum efficiency values of the rest DMUs.Let constraint   ≥ V    be replaced by constraint   ≤ V    in model (M4).Using Algorithm 4 we can also obtain the maximum efficiency values of all DMUs.The minimum and maximum efficiency scores of each DMU are shown in the tenth column of Table

Table 4 .
The efficiency intervals of the 73 DMUs according to Azizi and Wang's method are presented in the third column of Table4.The fourth column of Table

Table 4 :
Efficiencies of the countries participating in the Athens 2004 Olympic Games.