We propose a procedure for ranking decision making units in data envelopment analysis, based on ideal and anti-ideal points in the production possibility set. Moreover, a model has been introduced to compute the performance of a decision making unit for these two points through using common set of weights. One of the best privileges of this method is that we can make ranking for all decision making units by solving only three programs, and also solving these programs is not related to numbers of decision making units. One of the other advantages of this procedure is to rank all the extreme and nonextreme efficient decision making units. In other words, the suggested ranking method tends to seek a set of common weights for all units to make them fully ranked. Finally, it was applied for different sets holding real data, and then it can be compared with other procedures.
Data envelopment analysis (DEA) is a nonparametric method to define the relative efficiency of a group of decision making units (DMUs) that use multiple inputs to produce multiple outputs. Methodology of DEA pioneered by Farrell [
When DEA models are applied to calculate the performance of DMUs, usually several DMUs yield with the same efficiencies, that are all equal to one. Therefore, it is necessary to suggest a model to differentiate between these units. Otherwise, we are not able to rank them accordingly. Numerous models have been proposed to reduce the number of efficient units so far: Andersen and Petersen (AP) [
One of the most important and practical procedures in ranking is benchmarking methods, which are suggested by Torgesen et al. [
Assume that there is a set of
Extra flexibility to choose weights mostly brings several DMUs with relative efficient DMUs. However, to remove this problem, many attempts have been explored further restricting weights in DEA. One of the most important ones is the common weights method in DEA, which at first initiated by Cook et al. [
Several methods have been proposed to solve the aforementioned MOFP problem. One of them is goal programming (GP). Based on the GP method, model (
On the first step we are going to introduce ideal and anti-ideal points.
An ideal point is a point that can consume the least inputs to produce the most outputs.
An anti-ideal point is a point that uses the most inputs only to generate the least outputs.
Due to mentioned definitions we can show the inputs and outputs of ideal point with
Here we assume that
If
Model (
For evaluating DMUs, we can use model (
After including slack variables
Hence, only
Let
In this section, we provide three numerical examples that all involve a significant number of DEA efficient units. Then we compare them to other methods to show the potential usage of the proposed ranking methodology in the complete ranking of DMUs.
Consider the 12 flexible manufacturing systems (FMSs) given in Table
However, the main deficiency in Kao and Hung [
Data and CCR efficiency for Example
FMS | Inputs | Outputs | CCR efficiency | ||||
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1 | 17.02 | 5 | 42 | 45.3 | 14.2 | 30.1 | 1 |
2 | 16.46 | 4.5 | 39 | 40.1 | 13 | 29.8 | 1 |
3 | 11.76 | 6 | 26 | 39.6 | 13.8 | 24.5 | 0.9682 |
4 | 10.52 | 4 | 22 | 36 | 11.3 | 25 | 1 |
5 | 9.50 | 3.8 | 21 | 34.2 | 12 | 20.4 | 1 |
6 | 4.79 | 5.4 | 10 | 20.1 | 5 | 16.5 | 1 |
7 | 6.21 | 6.2 | 14 | 26.5 | 7 | 19.7 | 1 |
8 | 11.12 | 6 | 25 | 35.9 | 9 | 24.7 | 0.9614 |
9 | 3.67 | 8 | 4 | 17.4 | 0.1 | 18.1 | 1 |
10 | 8.93 | 7 | 16 | 34.3 | 6.5 | 20.6 | 0.9536 |
11 | 17.74 | 7.1 | 43 | 45.6 | 14 | 31.1 | 0.9831 |
12 | 14.85 | 6.2 | 27 | 38.7 | 13.8 | 25.4 | 0.8012 |
The results of Example
FMS | Kao and Hung models [ |
Wang et al. models [ |
Proposed method | |||
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Common model (1) | Common model (2) | Common model (3) | Model (5) | Model (6) | ||
1 | 1.0000 (1) | 0.9654 (4) | 0.9111 (6) | 1.0101 (5) | 1.0051 (5) | 0.8680 (7) |
2 | 0.9766 (5) | 0.9616 (6) | 0.9026 (7) | 0.9867 (7) | 0.9818 (7) | 0.8370 (9) |
3 | 0.9488 (9) | 0.9132 (9) | 0.9021 (8) | 1.0185 (4) | 1.0134 (4) | 0.9256 (5) |
4 | 1.0000 (1) | 1.0000 (1) | 1.0000 (1) | 1.0702 (2) | 1.0649 (2) | 0.9783 (3) |
5 | 1.0000 (1) | 0.9641 (5) | 0.9663 (4) | 1.0893 (1) | 1.0893 (1) | 1.0000 (1) |
6 | 0.9624 (6) | 0.9866 (3) | 0.9872 (3) | 1.0058 (6) | 1.0008 (6) | 0.9481 (4) |
7 | 1.0000 (1) | 1.0000 (1) | 1.0000 (1) | 1.0481 (3) | 1.0429 (3) | 0.9853 (2) |
8 | 0.9614 (7) | 0.9423 (7) | 0.9203 (5) | 0.9752 (8) | 0.9704 (8) | 0.8832 (6) |
9 | 0.7528 (12) | 0.8462 (10) | 0.8760 (9) | 0.7190 (12) | 0.7155 (12) | 0.7462 (12) |
10 | 0.8334 (10) | 0.8041 (11) | 0.8295 (11) | 0.8521 (10) | 0.8478 (10) | 0.8374 (8) |
11 | 0.9507 (8) | 0.9160 (8) | 0.8591 (10) | 0.9528 (9) | 0.9481 (9) | 0.8173 (10) |
12 | 0.7943 (11) | 0.7750 (12) | 0.7602 (12) | 0.8501 (11) | 0.8460 (11) | 0.7618 (11) |
Consider the example investigated by Jahanshahloo et al. [
Inputs and Outputs and ranking by AP, MAJ, and new proposed ranking models.
DMU | Input 1 | Input 2 | Output 1 | Output 2 | CCR efficiency |
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A | 2 | 3 | 4 | 5 | 0.9254 (4) | 0.9286 (4) | 0.9524 (4) | 0.8667 (3) |
B | 3 | 4 | 5 | 6 | 0.7602 (5) | 0.7619 (5) | 0.7619 (5) | 0.7173 (5) |
C | 2 | 2 | 4 | 4 | 0.9900 (3) | 1.0000 (3) | 1.0000 (3) | 0.8571 (4) |
D | 2 | 3 | 5 | 4 | 1.0000 (1) | 1.2500 (2) | 1.1667 (2) | 0.9333 (2) |
E | 3 | 4 | 4 | 5 | 0.6174 (7) | 0.6100 (7) | 0.6190 (7) | 0.5909 (7) |
F | 2 | 2 | 4 | 6 | 1.0000 (1) | 1.5000 (1) | 1.2857 (1) | 1.0000 (1) |
G | 3 | 4 | 5 | 4 | 0.7093 (6) | 0.7143 (6) | 0.7143 (6) | 0.6364 (6) |
Result of correlation analysis.
Proposed method | AP method | MAJ method | |
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Proposed method | 1 | 0.945 | 0.982 |
AP method | / | 1 | 0.986 |
MAJ method | / | / | 1 |
Correlation is significant at the 0.01 level (two-tailed).
Consider the problem of measuring the performances of five DMUs, where each DMU has two inputs and one output. The data set is shown in Table
Data for Example
DMU | Input 1 | Input 2 | Output 1 |
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A | 1 | 8 | 1 | 3.0000 (1) | 1.2222 (2) | 1.0000 (1) |
B | 3 | 6 | 1 | 1.0000 (4) | 1.0000 (4) | 0.9048 (2) |
C | 5 | 4 | 1 | 1.0000 (4) | 1.0000 (4) | 0.8261 (3) |
D | 6 | 3 | 1 | 1.0833 (3) | 1.0400 (3) | 0.7917 (4) |
E | 9 | 2 | 1 | 1.5000 (2) | 1.1250 (1) | 0.6129 (5) |
Farrell frontier for five DMUs.
In the current paper, we have developed a new mixed integer programming based on ideal and anti-ideal points. In this procedure, firstly we must compute ideal and anti-ideal points to rank all DMUs. Then their efficiency scores could be obtained. Through using the proposed model, all DMUs can be ranked, whereas most of ranking methods cannot do it. One of the prominent features of this model compared to the others is that it is always feasible. On the other hand, traditional DEA models cannot define a DMU with the best performance. However, it can be easily conducted by the proposed model here. The other advantage of this new model is that we are able to rank all the extreme and nonextreme efficient units by solving only three programs. Three numerical examples have been tested and examined by applying the suggested ranking method. The proposed model complies with crisp data. It can be examined further in the future researches in accordance with interval or fuzzy data.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their thanks to Professor Farhad Hosseinzadeh Lotfi for his help in initiating and developing the basic idea of this article. We are grateful to the anonymous referees for their valuable comments.