Journal of Optimization

Volume 2015 (2015), Article ID 594727, 10 pages

http://dx.doi.org/10.1155/2015/594727

## Ranking All DEA-Efficient DMUs Based on Cross Efficiency and Analytic Hierarchy Process Methods

Department of Mathematics, Islamic Azad University, Arak Branch, Arak, Iran

Received 15 August 2014; Accepted 20 December 2014

Academic Editor: Farhad Hosseinzadeh Lotfi

Copyright © 2015 Dariush Akbarian. 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.

#### Abstract

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.

#### 1. Introduction

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) [1] and extended by Banker, Charnes, and Cooper (BCC) [2]. DEA is used to establish a best practice group from among a set of observed units and to identify the units that are inefficient when compared to the best practice group [3–6]. One important issue in DEA which has been studied by many DEA researchers is to discriminate between efficient DMUs. Several authors have proposed methods for ranking the best performers ([7–14] among others). Perhaps the most widely known and applied ranking method is the super-efficiency DEA model. Developed by Andersen and Petersen [13], the main idea of this approach is to evaluate DMU after this performer itself is excluded from the reference set. The problem with super-efficiency DEA is that under certain conditions infeasibility occurs which limits the applicability of the technique (for details see [15]). Wu [4] utilized the concept of fuzzy for ranking of DMUs in the traditional DEA models. For this purpose, firstly the DMUs are evaluated by the CCR and the cross efficiency models. Secondly, a fuzzy preference relation is established. Finally, a priority vector of the preference is constructed and is used for ranking DMUs. Jahanshahloo et al. [8] proposed a ranking system based on changing reference set. In the proposed ranking system, the evaluation for efficient DMUs is dependent on the efficiency changes of all inefficient units due to its absence in the reference set while the estimate for inefficient DMUs depends on the influence of the exclusion of each efficient unit from the reference set. One popular method to rank multiple alternatives with respect to multiple criteria is analytical hierarchy process (AHP) method developed by Saaty [16]. The analytic hierarchy process is a multicriteria decision making technique and was used for complete ranking of DMUs by Sinuany-Stern et al. [17] and Jablonsky [18, 19] and references cited therein. Sinuany-Stern et al. [17] presented an AHP/DEA methodology for ranking DMUs, which integrates DEA and the AHP. In this AHP/DEA methodology, DEA was first run for each pair of DMUs separately to create a pairwise comparison matrix, which was then solved using the eigenvector method to produce a full ranking for all DMUs, efficient and inefficient. Their method has some problems, such as illogic comparing of two DMUs in a DEA model (see [20] for details). Alirezaee and Sani [20] extended Sinuany-Stern’s approach and proposed a new two-stage AHP/DEA methodology for ranking DMUs that removes these problems. They used together with the efficiency of units the influence rate of each unit to the other units in order to determine the elements of pairwise comparison matrix. Jablonsky [18] used an AHP model with interval pairwise comparisons for the evaluation and classification of efficient units and compares the results with super-efficiency DEA scores. In [19], Jablonsky presented an original procedure for ranking of DMUs in DEA models based on combination of AHP and DEA principles. He/she considered the DMUs as alternatives and the ratios of output/input as criteria. An extensive review of ranking models in DEA is given in Adler et al. [12]. The aim of this paper is to propose a method for ranking extreme and nonextreme efficient DMUs using cross efficiency and AHP methods. Our approach includes two stages. In the first step 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 step is utilized to rank scale of the units via the AHP. This paper is organized as follows. Section 2 presents some basic DEA models and reviews some ranking method. In Section 3 we briefly review the AHP. Section 4 gives a brief introduction to the cross efficiency evaluation. The new ranking method for DEA-efficient DMUs is developed in Section 5. Numerical examples are examined in Section 6. The paper is concluded in Section 7.

#### 2. Data Envelopment Analysis Models

Consider a set of DMUs which is associated with inputs and outputs. Particularly, () consumes amount of input and produces amount of output . Let in which and and in which and . The production possibility set (PPS) of CCR model is defined as follows: and similarly the production possibility set of BCC model is defined as follows:

The input-oriented BCC and input-oriented CCR models, corresponding to , , are given by (3) and (4), respectively: where is non-Archimedean small and positive number and , , , and , , , are called slack variables belonging to . Note that and represent input excesses; also and represent output shortfalls. Also and , , are real numbers and . Models (3) and (4) are called envelopment forms (with non-Archimedean number).

is said to be* strong efficient* (*CCR-efficient*) if and only if and and . Where the superscript indicates optimality. In similar manner the* BCC-efficient* DMUs can be defined.

The dual of model (4) (without , i.e., ), which is called multiplier form (corresponding to ), is as follows:

The AP model is as follows [13]: Jahanshahloo’s method corresponding inefficient is as follows [8]: The efficiency of strong efficiency will be denoted by and will be given by in which is the set of nonstrong efficiency DMUs and is the number of nonstrong efficiency DMUs.

Jahanshahloo et al. [21] used -norm in order to rank the extremely efficient DMUs in DEA models with constant and variable returns to scale, and the proposed method can remove the difficulties arising from AP and MAJ [11] models. Their proposed model is as follows: As mentioned earlier, Sinuany-Stern’s approach [17] is a two-stage model for fully ranking model. In the first stage Sinuany-Stern run the following CCR model for each pair of DMUs and : is the efficiency score of unit . The cross efficiency of DMU is as follows: in which and are the optimal weights of model (10). In the similar manner and can be computed. The pairwise comparison matrix () is constructed as follows: In the second stage using a one-step process (Step 3 in Section 3), the vector weights of the pairwise comparison matrix are computed. Based on these weights, the efficient and inefficient DMUs are ranked.

Alirezaee and Sani [20] proposed a two-stage AHP/DEA methodology for ranking DMUs. In the first stage they create the pairwise comparison matrix as follows.

First, using CCR model (4) the relative efficiency of each DMU is determined. Then, each DMU is evaluated by the following model: This model measures efficiency after excluding from PPS. The pairwise comparison matrix is constructed as follows: in which and are the objective function of (4) and (13) at optimality, respectively.

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.

#### 3. Analytical Hierarchy Process (AHP)

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 [16].

Application of AHP to a decision problem involves four steps [23, 24].

*Step 1 (structuring of the decision problem into a hierarchical model). *The first step is to decompose a decision problem into elements. In its simplest form (Figure 1), this structure comprises a goal at the topmost level, criteria (and subcriteria) at the intermediate levels, while the lowest level contains a set of the alternatives. A simple AHP model (Figure 1) has three levels (goal, criteria, and alternatives).