Decision SupportA new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process☆
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: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: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. . 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.
Section snippets
DEAHP
Letbe a pairwise comparison matrix with aii = 1 and aji = 1/aij for j ≠ i and W = (w1, … , wn)T be its priority vector. The DEAHP views each row of the matrix A as a DMU, each column as an output and assumes a dummy input value of one for all the DMUs. Each DMU has therefore n outputs and one dummy constant input, based on which the following input-oriented CCR model [2] is constructed to estimate the local priorities (weights) of the pairwise comparison matrix
A new DEA model for priority derivation in the AHP
The DEAHP defines the efficiency of each criterion or alternative as its local priority and leads to more than one criterion or alternative being DEA efficient. Differing from the DEAHP, we view as the score of the ith criterion or alternative and denoted it byWhat we are concerned about is not the score of each criterion or alternative, but its relative score, which is defined asWe view the relative score wi as the local priority of the ith
DEA model for group decision making in the AHP
Let be a pairwise comparison matrix provided by the kth decision maker (DMk) (k = 1, … , m), hk > 0 be its relative importance weight satisfying , and (i = 1, … , n;k = 1, … , m) be the score of the ith criterion or alternative generated from the kth pairwise comparison matrix, where m is the number of DMs for group decision making. Then, we haveThe overall score of each criterion or alternative can be expressed as
Aggregation of the best local priorities
For a hierarchical structure in the AHP as shown in Fig. 1, suppose the best local priorities for both criteria and alternatives have all been derived by the DEA methodology (LP model (9) or (17)). Let w1, … , wm be the best local priorities of the m decision criteria and w1j, … , wnj be the best local priorities of the n decision alternatives with respect to the jth criterion (j = 1, … , m), which are shown in Table 2 and form a decision matrix in the terminology of multiple criteria decision making
Further comparison with the DEAHP
The proposed DEA methodology differs from the DEAHP not only in their models, but also in their ways of aggregating local priorities. As discussed in the previous section, the proposed DEA methodology aggregates the best local priorities using the SAW method which is the traditional way utilized by the AHP, but there is no normalization to be performed for the best local priorities. The DEAHP, however, provides two ways for the aggregation of local priorities. The first way is to take no
Conclusions
In this paper, we started with two inconsistent pairwise comparison matrices to illustrate a significant drawback of the DEAHP, which derived illogical and counterintuitive priorities for inconsistent pairwise comparison matrices no matter whether they have satisfactory consistency or not. To overcome this drawback and derive logical priorities for pairwise comparison matrices, we proposed a new DEA methodology for priority determination and group decision making in the AHP. Two special DEA
Acknowledgement
The authors would like to thank two anonymous reviewers for their constructive comments which are very helpful in improving the paper.
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The work described in this paper is supported by the National Natural Science Foundation of China (NSFC) under the Grant No. 70771027, and also supported by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China, under the Grant No. CityU 111906.