The aim of this paper is to present an original approach for ranking of DEAefficient 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 DEAefficiency 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 onestep process of AHP. The advantage of this proposed method is its capability of ranking extreme and nonextreme DEAefficient 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 inputoriented BCC and inputoriented 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 onestep 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
Threelevel 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 leastsquare technique (LLST) [
Random index.
Matrix order  1  2  3  4  5  6  7  8  9 

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 twostage method for ranking all DEAefficient DMUs using cross efficient approach and AHP technique. In the first stage, we obtain the CCRefficient DMUs and the optimal solutions of the CCR model (
It is evident that
It should be noted that since we intend to rank CCRefficient 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  

Staff  Computer terminals  Space m^{2}  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
DMU 







1 






4 






7 

3007266838159896400 




12 






15 






17 



217430715355767500 


20  65310366646698125  10656566179635625  70175639377223750  104635477353783125  5446944927631181.25  6814920193906181.25 
Example
DMU  Target DMU  

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
DMU_{1}  DMU_{4}  DMU_{7}  DMU_{12}  DMU_{15}  DMU_{17}  DMU_{20}  Weight  

DMU_{1}  1  0.7646271  0.9123273  0.9171495  0.7644273  0.8967514  1.0232103  0.33493 
DMU_{4}  1.3078269  1  1.0095312  1.0757141  0.9886519  1.3569705  1.1753435  0.42144 
DMU_{7}  1.0960978  0.9905587  1  1.0601981  0.9638027  1.0423561  1.0526712  0.38601 
DMU_{12}  1.0903347  0.9296149  0.9432199  1  0.9222442  0.9722520  1.0387729  0.36912 
DMU_{15}  1.3081688  1.0114783  1.0375567  1.0843114  1  1.1905150  1.1539501  0.41546 
DMU_{17}  1.115136  0.7369356  0.9593650  1.0285398  0.8399725  1  0.8408676  0.34737 
DMU_{20}  0.9773161  0.8508150  0.9499641  0.9626742  0.8665885  1.1892478  1  0.36284 
Example

Proposed method  Model ( 
Model ( 
Model ( 


Weight  Rank 

Rank 

Rank 

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 SinuanyStern et al. [
Table
Example
DMU  Input  Output  CCR efficiency  

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  

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 twostage 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 DEAefficient DMU is calculated. In general, the optimal inputoutput 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.