The aim of this paper is to present an original approach for ranking of DEA-efficient DMUs based on the cross efficiency and analytic hierarchy process (AHP) methods. The approach includes two basic stages. In the first stage using DEA models the cross efficiency value of each DEA-efficiency DMU is specified. In the second stage, the pairwise comparison matrix generated in the first stage is utilized to rank scale of the units via the one-step process of AHP. The advantage of this proposed method is its capability of ranking extreme and nonextreme DEA-efficient DMUs. The numerical examples are presented in this paper and we compare our approach with some other approaches.
Data envelopment analysis (DEA) is a tool for evaluation and measuring of the efficiency of a set of decision making units (DMUs) that consume multiple inputs and produce multiple outputs, first introduced by Charnes, Cooper, and Rhodes (CCR) [
Consider a set of
The input-oriented BCC and input-oriented CCR models, corresponding to
The dual of model (
The AP model is as follows [
Jahanshahloo et al. [
Alirezaee and Sani [
First, using CCR model (
In the second stage, similar to AHP/DEA method, using a one-step process of AHP, the DMUs are ranked. In this paper we rank DMUs in the CCR model; in a similar way one can also rank DMUs in the BCC model.
AHP is a powerful tool for analysis of complex decision problems. AHP is employed for ranking a set of alternatives or for the selection of the best in a set of alternatives with respect to multiple criteria. In this section a brief discussion of AHP is provided. For more details see Saaty [
Application of AHP to a decision problem involves four steps [
The first step is to decompose a decision problem into elements. In its simplest form (Figure
Three-level hierarchy.
In this step, the elements of a particular level are compared with respect to a specific element in the immediate upper level. The more important (or more attractive) element receives higher rating (between 1 and 9) than another one that has the less important element [
In this step, local weights of the elements are calculated from the judgment matrices using the eigenvector method (EVM). The normalized eigenvector corresponding to the principal eigenvalue of the judgment matrix provides the weights of the corresponding elements. Though EVM is followed widely in traditional AHP computations, other methods are also suggested for calculating weights, including the logarithmic least-square technique (LLST) [
Random index.
Matrix order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
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RI | 0.00 | 0.00 | 0.58 | 0.9 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 |
If the pairwise comparisons matrix is as follows:
In order to obtain final weights of the decision alternatives (elements at the lowest level), the local weights of elements of different levels are aggregated. For example, the final weight of alternative
The cross efficiency evaluation was first proposed by Sexton et al. [
In general, the optimal weights obtained using classical DEA in the first stage are multiple solutions. Therefore, the values
Any convex linear combination of optimal solutions of the following LP is optimal too:
In this section we combine the cross efficiency method with AHP technique and provide a two-stage method for ranking all DEA-efficient DMUs using cross efficient approach and AHP technique. In the first stage, we obtain the CCR-efficient DMUs and the optimal solutions of the CCR model (
It is evident that
It should be noted that since we intend to rank CCR-efficient DMUs, it is enough to compute the values of
In this section, two numerical examples are examined to illustrate the proposed method. Comparisons with other existing procedures will also be made. In Example
We evaluated with our method the data of 20 branch banks of Iran. This data was previously analyzed by Amirteimoori and Kordrostami [
Example
Branch | Input | Output | ||||
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Staff | Computer terminals | Space m2 | Deposits | Loans | Charge | |
1 | 0.9503 | 0.70 | 0.1550 | 0.1900 | 0.5214 | 0.2926 |
2 | 0.7962 | 0.60 | 1.0000 | 0.2266 | 0.6274 | 0.4624 |
3 | 0.7982 | 0.75 | 0.5125 | 0.2283 | 0.9703 | 0.2606 |
4 | 0.8651 | 0.55 | 0.2100 | 0.1927 | 0.6324 | 1.0000 |
5 | 0.8151 | 0.85 | 0.2675 | 0.2333 | 0.7221 | 0.2463 |
6 | 0.8416 | 0.65 | 0.5000 | 0.2069 | 0.6025 | 0.5689 |
7 | 0.7189 | 0.60 | 0.3500 | 0.1824 | 0.9000 | 0.7158 |
8 | 0.7853 | 0.75 | 0.1200 | 0.1250 | 0.2340 | 0.2977 |
9 | 0.4756 | 0.60 | 0.1350 | 0.0801 | 0.3643 | 0.2439 |
10 | 0.6782 | 0.55 | 0.5100 | 0.0818 | 0.1835 | 0.0486 |
11 | 0.7112 | 1.00 | 0.3050 | 0.2117 | 0.3179 | 0.4031 |
12 | 0.8113 | 0.65 | 0.2550 | 0.1227 | 0.9225 | 0.6279 |
13 | 0.6586 | 0.85 | 0.3400 | 0.1755 | 0.6452 | 0.2605 |
14 | 0.9763 | 0.80 | 0.5400 | 0.1443 | 0.5143 | 0.2433 |
15 | 0.6845 | 0.95 | 0.4500 | 1.0000 | 0.2617 | 0.0982 |
16 | 0.6127 | 0.90 | 0.5250 | 0.1151 | 0.4021 | 0.4641 |
17 | 1.0000 | 0.60 | 0.2050 | 0.0900 | 1.0000 | 0.1614 |
18 | 0.6337 | 0.65 | 0.2350 | 0.0591 | 0.3492 | 0.0678 |
19 | 0.3715 | 0.70 | 0.2375 | 0.0385 | 0.1898 | 0.1112 |
20 | 0.5827 | 0.55 | 0.5000 | 0.1101 | 0.6145 | 0.7643 |
Example
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3007266838159896400 |
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20 | 65310366646698125 | 10656566179635625 | 70175639377223750 | 104635477353783125 | 5446944927631181.25 | 6814920193906181.25 |
Example
DMU | Target DMU | ||||||
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1 | 4 | 7 | 12 | 15 | 17 | 20 | |
1 | 1 | 0.49504029 | 0.53660363 | 0.67987789 | 0.50878889 | 0.74467050 | 0.36909990 |
4 | 0.95525399 | 1 | 0.81873718 | 0.84047040 | 0.89121676 | 0.92400072 | 0.94263179 |
7 | 0.68426795 | 0.80156597 | 1 | 0.98769702 | 0.85624403 | 0.85682199 | 0.88697995 |
12 | 0.83162917 | 0.71092883 | 0.87483547 | 1 | 0.75165252 | 0.91722504 | 0.68657179 |
15 | 0.97375065 | 0.91292479 | 0.92595846 | 0.89933688 | 1 | 0.84347305 | 0.93897325 |
17 | 0.94554521 | 0.41786480 | 0.78137007 | 0.97194231 | 0.54846678 | 1 | 0.25490089 |
20 | 0.33804338 | 0.65282037 | 0.79256331 | 0.62361928 | 0.68029207 | 0.49238820 | 1 |
Example
DMU1 | DMU4 | DMU7 | DMU12 | DMU15 | DMU17 | DMU20 | Weight | |
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DMU1 | 1 | 0.7646271 | 0.9123273 | 0.9171495 | 0.7644273 | 0.8967514 | 1.0232103 | 0.33493 |
DMU4 | 1.3078269 | 1 | 1.0095312 | 1.0757141 | 0.9886519 | 1.3569705 | 1.1753435 | 0.42144 |
DMU7 | 1.0960978 | 0.9905587 | 1 | 1.0601981 | 0.9638027 | 1.0423561 | 1.0526712 | 0.38601 |
DMU12 | 1.0903347 | 0.9296149 | 0.9432199 | 1 | 0.9222442 | 0.9722520 | 1.0387729 | 0.36912 |
DMU15 | 1.3081688 | 1.0114783 | 1.0375567 | 1.0843114 | 1 | 1.1905150 | 1.1539501 | 0.41546 |
DMU17 | 1.115136 | 0.7369356 | 0.9593650 | 1.0285398 | 0.8399725 | 1 | 0.8408676 | 0.34737 |
DMU20 | 0.9773161 | 0.8508150 | 0.9499641 | 0.9626742 | 0.8665885 | 1.1892478 | 1 | 0.36284 |
Example
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Proposed method | Model ( |
Model ( |
Model ( |
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Weight | Rank |
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Rank |
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Rank |
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Rank | |
1 | 0.33493 | 7 | 0.675 | 6 | 1.1009 | 7 | 0.0156 | 7 |
4 | 0.42144 | 1 | 0.708 | 2 | 1.9336 | 2 | 0.4802 | 2 |
7 | 0.38601 | 3 | 0.698 | 3 | 1.1737 | 5 | 0.1915 | 3 |
12 | 0.36912 | 4 | 0.677 | 5 | 1.1095 | 6 | 0.0589 | 6 |
15 | 0.41546 | 2 | 0.790 | 1 | 4.9139 | 1 | 0.7965 | 1 |
17 | 0.34737 | 6 | 0.677 | 5 | 1.3484 | 3 | 0.1760 | 4 |
20 | 0.36284 | 5 | 0.680 | 4 | 1.1849 | 4 | 0.1077 | 5 |
Consider a performance assessment problem investigated by Sinuany-Stern et al. [
Table
Example
DMU | Input | Output | CCR efficiency | ||
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A | 20 | 12 | 60 | 36 | 1 |
B | 10 | 15 | 30 | 45 | 1 |
C | 15 | 12 | 30 | 36 | 0.999 |
D | 5 | 70 | 15 | 80 | 1 |
E | 3 | 9 | 3 | 9 | 0.510 |
F | 9 | 18 | 1 | 18 | 0.402 |
G | 63 | 19 | 8 | 19 | 0.330 |
H | 22 | 73 | 1 | 3 | 0.022 |
Example
Sinuany’s method | Alirezaee’s method | Proposed method | |||
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B | 0.20994 | B | 0.1993 | A | 0.591675 |
A | 0.153001 | D | 0.1929 | B | 0.582900 |
D | 0.14140 | A | 0.1883 | D | 0.556908 |
In this paper we propose a two-stage method for ranking all extreme and nonextreme efficient DMUs based on cross efficiency and AHP technique. In the first stage, cross efficiency score of each DEA-efficient DMU is calculated. In general, the optimal input-output weights obtained by classical DEA are not unique; therefore, the cross efficiency scores depending on these weights are also not unique. To remove this problem we first obtain all alternative optimal solutions of DEA model (
The author declares that there is no conflict of interests regarding the publication of this paper.