Modifying inconsistent comparison matrix in analytic hierarchy process: A heuristic approach

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Abstract

Test of consistency is a critical step in the AHP methodology. When a pairwise comparison matrix fails to satisfy the consistency requirement, a decision maker needs to make revisions. To aid the decision maker's revising process, several approaches identify changes to the consistency requirement with respect to changes to a single entry in the original inconsistent matrix [P.T. Harker, Derivatives of the Perron root of a positive reciprocal matrix: with application to the analytic hierarchy process, Applied Mathematics and Computation, 22, 217–232 (1987); T.L. Saaty, Decision-making with the AHP: Why is the principal eigenvector necessary, European Journal of Operational Research, 145, 85–91 (2003)]. Instead of revising single entries, Xu and Wei [Z. Xu and C. Wei, A consistency improving method in the analytic hierarchy process, European Journal of Operational Research, V.116, 443–449 (1999)] derived a consistent matrix by an auto-adaptive process based on the original inconsistent matrix. In this paper, we develop a heuristic approach that auto-generates a consistent matrix based on the original inconsistent matrix. Expressing the inconsistent matrix in terms of a deviation matrix, an iterative process adjusts the deviation matrix to improve the consistency ratio, while preserving most of the original comparison information. We show that the proposed method is able to preserve more original comparison information than Xu and Wei [Z. Xu and C. Wei, A consistency improving method in the analytic hierarchy process, European Journal of Operational Research, V.116, 443–449 (1999)]. It is also shown that the heuristic approach can be used to examine the effects of revising a sub-bloc as well as revising a single entry of the original matrix.

Introduction

Analytic Hierarchy Process (AHP) is a multi-attribute decision-making methodology widely used in many real-life problems [1], [6], [11]. Developed by Saaty [8], AHP uses pairwise comparisons to capture relative importance of attributes, which forms the basis of priority determination. But before a pairwise comparison matrix can be used, it needs to pass a consistency test.

The element aij of a pairwise comparison n × n matrix Ais the decision maker's relative preference of attribute i over attribute j, where aij > 0, and aij = 1 / aji. A pairwise comparison matrix is perfectly consistent if aij = aik akj holds for all element pairs. However, when performing pairwise comparisons involving intangibles, it is unrealistic to expect a pairwise comparison matrix that is perfectly consistent. Saaty [8] developed a consistency test that allows a certain level of acceptable deviations. The consistency test involves the use of a “consistency ratio”: CR=λmaxnn1/RI, where λmax is the maximum eigenvalue of the pairwise comparison matrix, and RI is a random index whose value depends on n. If CR > 0.1, the decision-maker is asked to revise his judgments until an acceptable level of consistency is reached. Further discussions on consistency ratio can be found in Lance and Verdini [5], Murphy [7], Saaty [9], and Finan and Hurley [2].

Several approaches have been developed to aid the revising process. Based on the sensitivity analysis of positive reciprocal matrices [4], Harker [3] identified an entry whose adjustment would result in the largest rate of change in the matrix's consistency level. Saaty [10] also discussed two similar approaches for revising a single entry, both of which use the relationship nλmaxn=i,j=1ijn(ɛij+1ɛij) where ɛij is determined by the Hadamard product relationship A=WE in which w = (w1wiwn)T is the principal eigenvector, W = [wi/wj] and E = (ɛij). The first approach identifies the ɛij that is farthest from one and a change of the corresponding aij would result in a new pairwise comparison matrix with a smaller eigenvalue. The second approach is to modify the aij with the largest λmax resulting from perturbing each single entry of the pairwise comparison matrix A. Both methods suggest that the decision maker can improve the consistency by iteratively modifying a single entry of the pairwise comparison matrix.

Instead of revising single entries, Xu and Wei [13] proposed to develop a consistent matrix by an auto-adaptive process based on the original inconsistent matrix. Here, the element aij of the entire inconsistent pairwise comparison matrix A is replaced by bij = aijα(wi / wj)1  α, where w = (w1wiwn)T is the priority vector derived from A, and α is a positive value less than but approaching 1.0. The new ratio is the weighted geometric mean of the pairwise comparison ratio and the prescribed ratio. The generated matrix B = [bij] has a reduced CR. This process is to be repeated many times, until a pairwise comparison matrix with CR  0.1 has been achieved. Once a consistent matrix is obtained, the decision maker can use this new matrix as a reference for revising the original inconsistent matrix.

In this paper, we develop a heuristic approach to arrive at a consistent matrix that can retain more information than Xu and Wei [13]. Firstly, the original matrix is decomposed as the Hadamard product of a consistent matrix and a reciprocal deviation matrix. Then a modified matrix is constructed via a convex combination of the reciprocal deviation matrix and a zero deviation matrix. We provide the conditions under which the modified matrix would have a reduced maximum eigenvalue, thus reducing the CR. An algorithm, which consists of auto-adaptive modifications using such convex combinations, is provided. This auto-generating process will converge to an acceptable CR. We illustrate the heuristic approach using numerical examples from Xu and Wei [13]. It is also shown that the heuristic approach can be used to examine the effects of revising a sub-bloc as well as revising a single entry of the original matrix.

Section snippets

Notations and definitions

The matrix A can be expressed as Hadamard product of two matrices:A=WDwhere A = [aij] is any n × n inconsistent reciprocal matrix; W = [wi / wj], w = (w1wiwn)T is the priority vector derived from A; D = [dij ] is a reciprocal positive matrix; and ° is the symbol of Hadamard product ( A = B ° C means aij = bij cij for i = 1,…,n, j = 1,,n). The matrix D has the maximum eigenvalue λmax(D) equal to λmax(A) [12].

When A is a consistent matrix, all dij = 1 and λmax(D) =λmax(A) = n. Otherwise, at least some dij  1, and λmax

Main theoretical results and an algorithm

Let λmax(D′) and λmax(A′) be the maximum eigenvalues of matrix D′ and A′ respectively; and wD′ = (wD′1wD′iwD′n)T, w′  = (w′1w′iw′n)T be the priority vector derived from matrices D′ and A′ , respectively.

By Definition 1, D′ is a positive reciprocal matrix. The following lemma gives the relationship between λmax(D′) and λmax(A′), as well as the relationship between λmax(A) and λmax(D).

Lemma 1

(Horn and Johnson [4], Saaty [8])1)λmax(A)=λmax(D);w=wwD2)λmax(A)=λmax(D).

The following theorem states that

A comparison

Here, a comparison is made between the proposed method and that of Xu and Wei [13]. We will first examine the situation when the parameters for the two methods are identical (i.e. α = γ). Letting α = γ merely serves as a reference point, as the two parameters have different meanings in their respective methods. We will have more discussion on this in Section 5. For Xu and Wei's method, the modified pairwise comparison matrix would be B = [bij] = [aijγ(wi / wj)1  γ], where 0  γ  < 1. Using the concept of

Numerical illustration and comparison

Consider a simple situation where a company is selecting a trucking company to ship its goods. The selection of a trucking company is based on the performance of the following eight attributes: punctuality, delivery time, temperature control, track and trace, error rate, service reputation, damage or loss, and GPS features.

Fig. 1 shows an AHP model where the overall goal depends on the attributes and each of the attributes will be rated against the respective trucking companies being

Discussion

Table 1, Table 2 summarize the results for these two cases; and, for comparison, we also include the results from Xu and Wei's approach (using identical parameter values respectively).

For both methods, it is quite obvious that the results obtained using 0.98 is significantly better than those using 0.5. That is, there is no reason to use the 0.5 parameter, other than for computational efficiency only. In the case of α = γ  = 0.5, it represents a one-step modification as both methods take one

Conclusions

This paper proposes a heuristic approach to derive a consistency matrix from an inconsistent one. The analytical results show that the proposed method is able to converge to a matrix that preserves more original comparison information than Xu and Wei when the parameter is close to 1.0. However, since the process stops at the requirement that CR  0.1, there is no guarantee that our result will be a superior one. It must be emphasized that such an approach should only be considered as a decision

Dr. Dong Cao is an Associate professor of the department of Industrial Engineering at the University of Québec at Trois-Rivières, Canada. He received his PhD in Operations Research from the University of Louvain, in Belgium.

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Dr. Dong Cao is an Associate professor of the department of Industrial Engineering at the University of Québec at Trois-Rivières, Canada. He received his PhD in Operations Research from the University of Louvain, in Belgium.

Dr. Lawrence C. Leung is a Professor of the department of Decision Sciences and Managerial Economics at the Chinese University of Hong Kong. He received his PhD in Industrial Engineering from Virginia Tech, USA.

Dr. Japhet S. Law is a Professor of the department of Decision Sciences and Managerial Economics at the Chinese University of Hong Kong. He received his PhD in Industrial Engineering from the University of Texas (Austin), USA.

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