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European Journal of Operational Research 190 (2008) 672–678 www.elsevier.com/locate/ejor
Decision Support
Note on group consistency in analytic hierarchy process Robert Lin
b
a,*
, Jennifer Shu-Jen Lin b, Jason Chang c, Didos Tang d, Henry Chao e, Peter C Julian e
a Department of General Education Center, Oriental Institute of Technology, Taiwan, ROC Institute of Technological & Vocational Education, National Taipei University of Technology, Taiwan, ROC c Department of Civil Engineering, National Taiwan University, Taiwan, ROC d Army Command and Staff College, National Defense University, Taiwan, ROC e Department of Traffic Science, Central Police University, Taiwan, ROC
Received 21 September 2006; accepted 6 July 2007 Available online 14 July 2007
Abstract We study the paper of Xu [Z. Xu, On consistency of the weighted geometric mean complex judgement matrix in AHP, European Journal of Operational Research 126 (2000) 683–687] for the group consistency in analytic hierarchy process of multicriteria decision-making. The purpose of this note is threefold. First, we point out the questionable results in this paper. Second, for three by three comparison matrices, we provide a patchwork for his method. Third, we constructed a counter example to show that in general his method is wrong. Numerical examples are provided to illustrate our findings. If there are four or more alternatives, then we may advise researchers to ignore his results to avoid questionable estimation of group consistency. 2007 Elsevier B.V. All rights reserved. Keywords: Analytic hierarchy process (AHP); Consistency
1. Introduction Saaty’s analytic hierarchy process (AHP) provides an adequate tool for multicriteria decision-making quantitatively supports the evaluation of best alternative with regard to quantitative and qualitative criteria [13]. The AHP model is widely and successfully used in many fields, for example, Zanakis, et al. [19] studied over 100 applications of AHP within the service, social/manpower, natural resource/energy, education, and government sectors. In spite of this, some researchers still question its suitability and completeness. For example, just to mention a few, Apostolou and Hassell [2] considered that comparison matrices with consistency ratio >0.1 could be accepted. Bernhard and Canada [4] suggested that the incremental benefit/cost ratios should be compared with a cutoff ratio instead of the benefit/cost ratios of Saaty [13,15]. Finan and Hurley [9]
*
Corresponding author. Tel.: +886 2 77380145 216. E-mail address:
[email protected] (R. Lin).
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.07.007
R. Lin et al. / European Journal of Operational Research 190 (2008) 672–678
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constructed a diagonal procedure to construct a rank-order consistent matrix. On the other hand, some researchers have tried to revise those improvements. For example, Chu and Liu [7] illustrated problems in Apostolou and Hassell [2]. Yang et al. [18] demonstrated that the method of Bernhard and Canada [4] was incomplete and then modified it. Chao et al. [5] explained that the diagonal procedure of Finan and Hurley [9] did not pass the consistency test of Saaty [13]. Following this trend, we consider the paper of Xu [17] to study the group consistency of AHP. He used the weighted geometric mean method (WGMM) as the aggregation method for decision makers in the group. He showed that if the comparison matrices for all member of the group pass the consistency test then the group comparison matrix would pass the consistency test. In the following research, Xu had quoted Xu [17] ten times. There are six other papers, namely Escobar et al. [8], Hongwei et al. [10], Li et al. [11], Zhu et al. [20], Aull-Hyde et al. [3], and Regan et al. [12], that have referred to Xu [17] in their references. According to some citations of Xu’s papers, it may be worthwhile to provide a deep examination of his results. However, none of these 16 papers had discovered the questionable results that we will point out in this note. We will try to provide a further discussion to point out the inherent problem for the eigenvector of the group comparison matrix proposed by Xu [17]. We will first explain the eigenvector of the group implicitly assumed by him. Secondly, we will prove that his prediction is valid for three by three comparison matrices. Thirdly, we will provide a counter example to illustrate his prediction is false for a four by four comparison matrix. It will indicate that the estimation for the group consistency in Xu [17] contains questionable results so that we may advise researchers do not apply his approach to avoid mistakes. 2. Review of his results ½k Xu [17] assumed that there are s decision makers, with comparison matrices, Ak ¼ aij
, for k =
nn
1, 2, . . . , s and the coefficient for Weighted Geometric Mean Complex Judgment Matrix (WGMCJM), say A ¼ ð aij Þnn , satisfies A ¼ Ak11 Ak22 Aks s where means the Hardmard product (componentwise product) k1 k2 ks Ps ½1 ½2 ½s aij aij and k¼1 kk ¼ 1, kk > 0 for k = 1, 2, . . . , s, where kk is relative weight of with aij ¼ aij the kth decision maker for WGMM. For a comparison matrix, A = (ai j)n·n with its maximum eigenvalue, kmax and eigenvector (w1, w2, . . . , wn). w Xu [17] assumed that eij ¼ aij wji for 1 6 i, j 6 n and we denote that E = (ei j). He derived that the consistency index (CI) of Saaty [13] as X 1 CI ¼ ðeij þ eji 2Þ: ð1Þ nðn 1Þ 16i
