JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 919547 10.1155/2014/919547 919547 Research Article Modified Nonradial Supper Efficiency Models Jahanshahloo H. 1 Lotfi F. Hosseinzadeh 1 Khodabakhshi M. 2 Nasseri S. H. 1 Department of Mathematics Science and Research Branch Islamic Azad University Tehran Iran srbiau.ac.ir 2 Department of Mathematics Faculty of Mathematical Sciences Shahid Beheshti University, G.C. Tehran Iran sbu.ac.ir 2014 1722014 2014 25 11 2013 28 12 2013 17 2 2014 2014 Copyright © 2014 H. Jahanshahloo et al. 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.

Ranking Efficient Decision Making Units (DMUs) are an important issue in Data Envelopment Analysis (DEA). This is one of the main areas for the researcher. Different methods for this purpose have been suggested. Appearing nonzero slack in optimal solution makes the method problematic. In this paper, we modify the nonradial supper efficiency model to remove this difficulty. Some numerical examples are solved by modified model.

1. Introduction

Data Envelopment Analysis is a mathematical programming technique which evaluates the relative efficiency of DMUs . DEA classifies the DMUs into two different classes, called set of efficient DMUs and the set of inefficient DMUs.

Conventional DEA model cannot differentiate the efficient DMUs whose efficiency value is one. Toward this end, different methods are suggested; see . One of the important models is AP-Model which was proposed by Andersen and Petersen  and also see [5, 6].

This model is widely used and the results are almost satisfactory. The main deficiencies of AP model are being as follows:

infeasible for some kind of data,

unstable in the sense that a small variation in data causes big increase (degrease) in the result,

not taking into account the nonzero slacks which appear in optional solution (projection of omitted DMU is weak efficient in new PPS).

For removing these difficulties, many researches have suggested different models; for example, see .

Tone  suggested the nonradial supper efficiency model. This model fails to remove the 3rd deficiency. In this paper, we modify the nonradial supper efficiency model that takes into account nonzero slack that appears in optimal solution.

The rest of the paper is organized as follows. In Sections 2 and 3, AP-model and Tone’s model are discussed. Section 4 contains the modified model. In Sections 5 and 6, input and output oriented models are proposed. Discussion and conclusion cover Section 7.

2. Anderson Peterson (AP) Model

Consider n decision making units DMU j ( j = 1 , , n ) which consumes 0 x j 0 and x j R m ( j = 1 , , n ) vector as input to produce output vector 0 y j 0 and y j R s ( j = 1 , , n ) . The supper efficiency model may be written as follows: (1) θ p * = Min θ s . t . j = 1 , j p n λ j x i j + s i - = θ x i p i = 1 , , m 111 j = 1 , j p n λ j y r j - s r + = y r p r = 1 , , s 111 λ j 0 j = 1 , , n , j p 111 s i - 0 i = 1 , , m 111 s r + 0 r = 1 , , s .

The following example which has been taken from  shows the deficiency in case of having nonzero slacks in optimal solution.

Example 1.

Consider the data given by Table 1.

DMU A B C D E F
Input 1 2 2 5 10 10 3.5
Input 2 12 8 5 4 6 6.5
Output 1 1 1 1 1 1 1

By using AP-Model, (2) Sup A = θ A * = 1 S 2 - * = 4 Sup B = θ B * = 1.260 All slacks are zero Sup C = θ C * = 1.133 All slacks are zero Sup D = θ D * = 1.250 S 1 - * = 7.5 Sup E = θ E * = 0.750 Sup F = θ F * = 1 .

In other words, the supper efficiencies are as follows. (3) θ A * = 1 - 4 ɛ ( * ) θ B * = 1.260 θ C * = 1.133 θ D * = 1.25 - 7.5 ɛ ( * ) θ E * = 0.750 θ F * = 1 ( * ) .

Note. These values are not correct in , because the result shown in the paper has taken from original paper in which F is not included. From Figure 1 it can be seen that the radial supper Efficiency model is not able to rank F which is non-extreme efficient and also is not able to rank D and A , with nonzero slack in optimal solutions.

3. Nonradial Supper Efficiency Model (Tone’s Model)

In 2002, Tone proposed the following nonradial supper efficiency model: (4) ρ p * = Min ( 1 / m ) i = 1 m ( x ¯ i / x i p ) ( 1 / s ) r = 1 s ( y ¯ r / y r p ) s . t . j = 1 , j p n λ j x i j x ¯ i i = 1 , , m 11 j = 1 , j p n λ j y r j y ¯ r r = 1 , , s 111 x ¯ i x i p i = 1 , , m 111 y ¯ r y r p r = 1 , , s 111 y ¯ r 0 r = 1 , , s 111 λ j 0 j = 1 , , n , j p . Based on the SBM model (4), the nonradial supper efficiency model should be as follows: (5) γ p * = Min 1 + ( 1 / m ) i = 1 m ( s i + / x i p ) 1 - ( 1 / s ) r = 1 s ( s r - / y r p ) s . t . j = 1 , j p n λ j x i j - s i + = x i p i = 1 , , m 111 j = 1 , j p n λ j y r j + s r - = y r p r = 1 , , s 111 λ j 0 j = 1 , , n , j p 111 s i + 0 i = 1 , , m 111 s r - 0 r = 1 , , s . We will show that (4) and (5) are not equivalent, in the sense that their optimal solutions are different.

Example 2.

Consider the eight Decision Making Units (DMU), with two inputs and one output (see Table 2).

DMU A B C D E F G H
Input 1 4 7 8 4 2 10 12 10
Input 2 3 3 1 2 4 1 1 1.5
Output 1 1 1 1 1 1 1 1 1

First the following are defined: (6) P P S = { ( x , y ) x j = 1 n λ j x j , y j = 1 n λ j y j , λ j 0 , j = 1 , , n } , P P S ¯ = { ( x , y ) x j = 1 , j p n λ j x j , y j = 1 , j p n λ j y j , λ j 0 , j = 1 , , n } , P P S ¯ ¯ = P P S ¯ { ( x , y ) x x p , y y p } . In above-mentioned example and shown in Figure 2, (7) P P S = R E D C K = S P P S ¯ = R E D C K = S ¯ R E K ¯ ¯ = S ¯ ¯ .

We can see that S ¯ ¯ is a proper subset of S ¯ and S ¯ is a proper subset of S ; that is, (8) S ¯ ¯ S ¯ S . In this example, the optimum value of objective function in (4) is (9) ρ E * = 1 2 ( 4 2 + 4 4 ) = 1.5 and the optimal value of objective function in (5) is (10) γ S ¯ * = 1 2 ( 4 2 + 2 4 ) = 1 2 ( 2 + 1 2 ) = 5 4 = 1.25 , so (4) and (5) are not equivalent. In other words, the constraints x ¯ x p and y ¯ y p and y ¯ 0 should be omitted from (4). Now we show that the nonradial supper efficiency model has the same difficulty as radial supper efficiency model in treating nonzero slacks in optimal solution. Using (5), the supper efficiency of E , D , C may be evaluated as follows: (11) γ E * = Min ( 1 + 1 2 i = 1 2 s i + x i E ) s . t . 4 λ A + 7 λ B + 8 λ C + 4 λ D + 10 λ F + 12 λ G + 10 λ H - s 1 + = 2 3 λ A + 3 λ B + λ C + 2 λ D + λ F + λ G + 1.5 λ H - s 2 + = 4 λ A + λ B + λ C + λ D + λ F + λ G + λ H + s - = 1 λ A , λ B , λ C , λ D , λ F , λ G , λ H , s 1 + , s 2 + , s - 0 , γ E * = 1 + 1 2 ( 2 2 + 0 4 ) = 1.5 . The projection of E is on weak frontier (12) γ D * = Min ( 1 + 1 2 i = 1 2 s i + x i D ) s . t . 4 λ A + 7 λ B + 8 λ C + 2 λ E + 10 λ F + 12 λ G + 10 λ H - s 1 + = 2 3 λ A + 3 λ B + λ C + 4 λ E + λ F + λ G + 1.5 λ H - s 2 + = 4 λ A + λ B + λ C + λ E + λ F + λ G + λ H + s - = 1 λ A , λ B , λ C , λ E , λ F , λ G , λ H , s 1 + , s 2 + , s - 0 , γ D * = 1 + 1 2 ( 2 4 + 0 3 ) = 1.25 .

The projection is on strong frontier (13) γ C * = Min ( 1 + 1 2 i = 1 2 s i + x i C ) s . t . 4 λ A + 7 λ B + 4 λ D + 2 λ E + 10 λ F + 12 λ G + 10 λ H - s 1 + = 8 3 λ A + 3 λ B + 2 λ D + 4 λ E + λ F + λ G + 1.5 λ H - s 2 + = 1 λ A + λ B + λ D + λ E + λ F + λ G + λ H + s - = 1 λ A , λ B , λ D , λ E , λ F , λ G , λ H , s 1 + , s 2 + , s - 0 , γ C * = 1 + 1 2 ( 2 8 ) = 1.125 . The projection is on strong frontier.

In summery, (14) γ C * = 1.125 γ D * = 1.25 γ E * = 1.5 .

In the case the projection of omitted DM U p lied on weak frontier, nonzero slacks appear in optimal solution and the following approach is suggested.

4. Modified Nonradial Supper Efficiency Model

First solve the following model: (15) γ p * = Min 1 + ( 1 / m ) i = 1 m ( s i + / x i p ) 1 - ( 1 / s ) r = 1 s ( s r - / y r p ) s . t . j = 1 , j p n λ j x i j - s i + = x i p i = 1 , , m j = 1 , j p n λ j y r j + s r - = y r p r = 1 , , s λ j 0 j = 1 , , n , j p s i + 0 i = 1 , , m s r - 0 r = 1 , , s . The model, (15), can be linearized by the suggested method in  and solve by the simplex method.

Suppose that (16) ( x ^ , y ^ ) = ( x p + s + * , y p - s - * ) is the projection of DM U p on the frontier of P P S ¯ and now solve the following model (17) W p * = Max ( 1 / m ) i = 1 m ( s ^ i - / x ^ i ) ( 1 / s ) r = 1 s ( s ^ r + / y ^ r ) s . t . j = 1 , j p n λ j x i j + s ^ i - = x ^ i i = 1 , , m 111 j = 1 , j p n λ j y r j - s ^ r + = y ^ r r = 1 , , s 111 s ^ r + 0 r = 1 , , s 111 s ^ i - 0 i = 1 , , m , 111 λ j 0 j = 1 , , n , j p . Set (18) μ p * = γ p * - W p * ( supper efficiency of DMU p ) . It is evident that if ( x ^ , y ^ ) is strongly efficient, then W p * = 0 and μ p * is supper efficiency score of DM U p .

5. Modified Input Oriented Nonradial Supper Efficiency Model

First the following model is solved: (19) ξ p * = Min ( 1 + 1 m i = 1 m s i + x i p ) s . t . j = 1 , j p n λ j x i j - s i + = x i p i = 1 , , m 1111111 j = 1 , j p n λ j y r j - s r + = y r p r = 1 , , s 1111111 s i + 0 i = 1 , , m 1111111 s r + 0 r = 1 , , s 1111111 λ j 0 j = 1 , , n , j p . Let (20) ( x ~ , y ~ ) = ( x p + s + * , y p + s + * ) . Now solve the following problem. (21) W p * = Max ( 1 / m ) i = 1 m ( t i - / x ~ i ) ( 1 / s ) r = 1 s ( t r + / y ~ r ) s . t . j = 1 , j p n λ j x i j + t i - = x ~ i i = 1 , , m 11111 j = 1 , j p n λ j y r j - t r + = y ~ r r = 1 , , s 11111 t i - 0 i = 1 , , m 11111 t r + 0 r = 1 , , s 11111 λ j 0 j = 1 , , n , j p . Let (22) μ p * = ξ p * - W p * .

6. Modified Output Oriented Nonradial Supper Efficiency Model

Consider the following model: (23) η p * = Min ( 1 1 - ( 1 / s ) i = 1 s ( s r - / y r p ) ) s . t . j = 1 , j p n λ j x i j + s i - = x i p i = 1 , , m j = 1 , j p n λ j y r j + s r - = y r p r = 1 , , s s i - 0 i = 1 , , m s r - 0 r = 1 , , s λ j 0 j = 1 , , n , j p . Let (24) DMU p    projection    = ( x ¯ , y ¯ ) = ( x p - s - * , y p - s - * ) .

Now solve the following problem: (25) W p * = Max ( 1 / m ) i = 1 m ( s ^ i - / x ¯ i ) ( 1 / s ) r = 1 s ( s ^ r + / y ¯ r ) s . t . j = 1 , j p n λ j x i j + s i - = x ¯ i i = 1 , , m j = 1 , j p n λ j y r j - s r + = y ¯ r r = 1 , , s s i - 0 i = 1 , , m s r + 0 r = 1 , , s λ j 0 j = 1 , , n , j p . Let (26) ρ p * = η p * - W p * .

7. Conclusion

In this paper, it has been shown that both AP-Model and nonradial supper efficiency model are not able to rank DMUs if the projection of omitted DMU is weak efficient in P P S ¯ . The new method removes this difficulties and the example which is solved by using the new method (modified method) confirms the validity.

The results for comparing the methods are shown in Table 3.

DMU C D E F G
AP 1.14 1.25 2.00 1.00 1.00
SBM 1.125 1.25 1.50 1.00 1.00
Modified SBM 1.125 1.25 1.25 0.90 0.73
Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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