A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

Abstract

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.

Introduction

How to derive a priority vector from a pairwise comparison matrix has being an important research topic in the analytic hierarchy process (AHP) and has been substantially investigated in the AHP literature. Apart from Saaty’s well-known eigenvector method (EM) [10], quite a number of alternative approaches have been suggested such as the weighted least-square method (WLSM) [3], the logarithmic least squares method (LLSM) [5], the geometric least squares method (GLSM) [7], the fuzzy programming method (FPM) [8], the gradient eigenweight method (GEM) [4], and so on.

Recently, Ramanathan [9] developed a DEAHP method for weight derivation and weight aggregation in the AHP. The DEAHP method views each criterion or alternative in a pairwise comparison matrix as a decision making entity, which is referred to as decision making unit (DMU) in data envelopment analysis (DEA) [2], the row elements of the pairwise comparison matrix as the outputs of the DMUs, and uses a dummy input that has a value of one for all the DMUs to build an input-oriented CCR model for each DMU to calculate its best possible relative efficiency. The best relative efficiencies are then served as the local priorities of the DMUs (decision criteria or alternatives). The DEAHP proved to be able to produce true weights for perfectly consistent pairwise comparison matrices and has been applied more recently to supplier selection by Sevkli et al. [12]. However, the DEAHP method was also found to have some drawbacks, which were illustrated with numerical examples in Wang et al. [13]. The most significant drawback is that there is no guarantee that the DEAHP method can produce rational weight vectors for inconsistent pairwise comparison matrices. Consider for example the following pairwise comparison matrix:

$$A=\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 1/4\hfill & \hfill 1\hfill & \hfill 7\hfill & \hfill 3\hfill & \hfill 1/5\hfill & \hfill 1\hfill \\ \hfill 1/3\hfill & \hfill 1/7\hfill & \hfill 1\hfill & \hfill 1/5\hfill & \hfill 1/5\hfill & \hfill 1/6\hfill \\ \hfill 1\hfill & \hfill 1/3\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1/3\hfill \\ \hfill 1/3\hfill & \hfill 5\hfill & \hfill 5\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 1/4\hfill & \hfill 1\hfill & \hfill 6\hfill & \hfill 3\hfill & \hfill 1/3\hfill & \hfill 1\hfill \end{array}\right]\text{,}$$

which is inconsistent (CR = 0.229 > 0.1). For this inconsistent pairwise comparison matrix, the DEAHP produces the priority vector as (1, 1, 1/3, 1, 1, 1), which makes no sense and is obviously illogical, counterintuitive and unacceptable because the pairwise comparisons in the first row of A clearly show that the first criterion or alternative is more important than others and cannot be as important as them. It is this flaw that seriously restricts the applications of the DEAHP.

The DEAHP not only produces counterintuitive results for highly inconsistent pairwise comparison matrices, but may also produce illogical results for those pairwise comparison matrices with satisfactory consistency, i.e. CR < 0.1. Consider for example the following pairwise comparison matrix:

$$B=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 1/3\hfill & \hfill 1/3\hfill & \hfill 1\hfill & \hfill 1/2\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 1/2\hfill & \hfill 1/2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 1/2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 1/2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]\text{,}$$

which has satisfactory and acceptable consistency (CR = 0.0999 < 0.1) and meets the requirement of CR < 0.1. The DEAHP, however, produces equal weights for all the five decision criteria or alternatives in this matrix, i.e. $W_{\text{DEAHP}}^{\ast }=(1\text{,}1\text{,}1\text{,}1\text{,}1{)}^{\text{T}}$. Such a priority vector also makes no sense and needs to be rejected.

In this paper, we propose a new DEA methodology to overcome the drawbacks of the DEAHP and extend it to the group AHP situation. The new DEA methodology employs two specially constructed DEA models to derive the best local priority vector from a pairwise comparison matrix or the best local priority vectors from a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent.

The paper is organized as follows. In Section 2 we briefly review the DEAHP and in Section 3 we propose the new DEA methodology for priority determination in the AHP. We then extend the new DEA methodology to the group AHP situation in Section 4. The aggregation approach of local priorities in a hierarchical structure is provided in Section 5. Comparisons between the new DEA methodology and the DEAHP are discussed in Section 6. The paper is concluded in Section 7. Numerical examples are provided and examined throughout the paper.