Consensus models for AHP group decision making under row geometric mean prioritization method

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Abstract

The consistency measure is a vital basis for consensus models of group decision making using preference relations, and includes two subproblems: individual consistency measure and consensus measure. In the analytic hierarchy process (AHP), the decision makers express their preferences using judgement matrices (i.e., multiplicative preference relations). Also, the geometric consistency index is suggested to measure the individual consistency of judgement matrices, when using row geometric mean prioritization method (RGMM), one of the most extended AHP prioritization procedures. This paper further defines the consensus indexes to measure consensus degree among judgement matrices (or decision makers) for the AHP group decision making using RGMM. By using Chiclana et al.'s consensus framework, and by extending Xu and Wei's individual consistency improving method, we present two AHP consensus models under RGMM. Simulation experiments show that the proposed two consensus models can improve the consensus indexes of judgement matrices to help AHP decision makers reach consensus. Moreover, our proposal has two desired features: (1) in reaching consensus, the adjusted judgement matrix has a better individual consistency index (i.e., geometric consistency index) than the corresponding original judgement matrix; (2) this proposal satisfies the Pareto principle of social choice theory.

Introduction

Preference relations are widely used in group decision making. According to element forms in preference relations, there are often two kinds of preference relations: linguistic preference relations [17], [27], [28] and numerical preference relations (i.e., multiplicative preference relations [26], [37], [42] and fuzzy preference relations [11], [12], [20], [22], [40]). In general, group decision making problems using preference relations are faced by applying two different models (or processes) before a final solution can be given [29], [30]: 1) the selection model and 2) the consensus model. The selection model obtains the final solution according to the preferences given by the decision makers. It involves two different steps: aggregation of individual preferences and exploitation of the collective preference.

The consensus model is an important aspect in group decision making [5], [6], [9], [10], [13], [16], [27], [29], [30], [44]. Classically, consensus is defined as the full and unanimous agreement of all the decision makers regarding all the possible alternatives. However, some researchers consider that complete agreement is not necessary in real life. This has led to the use of the consistency measure, which also is called “soft” consensus degree in [27], [29], [30]. The consistency measure is used to measure the difference among decision makers, and is a vital basis of consensus models. For group decision making using preference relations, the consistency measure itself includes two subproblems [31]:

  • 1.

    When can a decision maker, considered individually, be said to be consistent and

  • 2.

    When can a whole group of decision makers be considered consistent.

In this paper, we call the first subproblem individual consistency measure, and the second subproblem consensus measure. Generally, at the beginning for each group decision making problem, decision makers’ opinions may differ from each other substantially. And, consensus models are decision aid tools to help decision makers to reach consensus, based on the established consistency measure.

In the analytic hierarchy process (AHP) [37], multiplicative preference relations are called judgement matrices, and adopted to express the decision makers’ preferences. Many researchers [1], [7], [18], [23], [32], [34], [35], [36], [39], [41], [46] focus on the selection model in AHP group decision making (i.e., aggregation rules and prioritization methods). Two of the methods that have been found to be the most useful in AHP group decision making are the aggregation of individual judgments (AIJ) and the aggregation of individual priorities (AIP). AIJ follows the common resolution scheme of selection models, as mentioned above. In the aggregation phase of AIJ, the weighted geometric mean method is used to aggregating individual judgement matrices to obtain a collective judgement matrix. In the exploitation phase of AIJ, prioritization methods, such as eigenvalue method (EM) [38] and row geometric mean method (RGMM) [14], are used to derive a priority vector to order collective judgement matrix. AIP has differences in the common resolution scheme among other selection models. The weighted geometric mean method is employed to aggregate individual priorities derived using prioritization methods, to obtain the best alternative(s).

The consistency problem is also a critical step in AHP group decision making [2], [8], [19], [21], [24], [33], [34], [38], [43], [45]. A number of studies focus on individual consistency in AHP. Saaty [37] develops an individual consistency index based on the EM. Crawford and Williams [14] propose an individual consistency index based on RGMM. Aguarón et al. [2] call Crawford and Williams's consistency index the geometric consistency index, and provide the thresholds associated with it. Many approaches [8], [21], [24], [38], [45] have been developed to aid the AHP decision makers to revise the individual inconsistency in judgement matrices. There are fewer studies about the consensus building for AHP group decision making. Bryson [7], Moreno-Jiménez et al. [34] and Van den Honert [41] also introduce several interesting approaches for consensus building in AHP.

In general, research progress in group decision making using preference relations can benefit research in AHP. Recently, the consensus problem has become a hot topic in group decision making using preference relations. In particular, Chiclana et al. [13] present a framework for integrating individual consistency into consensus model. Inspired by Chiclana et al.'s study [13], this paper develops AHP consensus models under RGMM. This paper is organized as follows. In Section 2, we introduce some preliminary knowledge of AHP. In Section 3, we define the consensus indexes (i.e., the geometric cardinal consensus index and the geometric ordinal consensus index) for AHP, and export Chiclana et al.'s consensus framework to AHP. In Section 4, we develop two AHP consensus models under Chiclana et al's framework and RGMM, and show some desired properties of the proposed consensus models. In Section 5, an illustrative example is provided. Concluding remarks and future research agenda are presented in Section 6.

Section snippets

Prioritization method

Let A = (aij)n × n, where aij > 0 and aij × aji = 1, be a judgement matrix. The prioritization method refers to the process of deriving a priority vector w = (w1,...,wn)T, where wi  0 and ∑i = 1nwi = 1, from the judgement matrix A. Two most commonly used prioritization methods (EM and RGMM) are listed below.

  • (1)

    The eigenvalue method

    Saaty [37], [38] proposes the principal eigenvector of A as the desired priority vector w, which can be obtained by solving the linear system:Aw=λw,eTw=1,where λ is the principal

Consistency measures in AHP

As indicated in previous sections, the problem of consistency measure itself includes two subproblems: individual consistency and consensus measure. For the RGMM, Crawford and Williams [14] and Aguarón et al. [2] have developed a consistency index to measure individual consistency, namely the geometric consistency index (GCI) (see Definition 1).

Definition 1

(Crawford and Williams [14]). Let A = (aij)n × n be a judgement matrix, and let w = (w1,w2,...,wn)T be the priority vector derived from A using the RGMM. The

Consensus models in AHP

Under Chiclana et al.'s framework, once judgement matrices are of acceptably individual consistency, we need to further apply consensus models to help AHP decision makers reach consensus. In this section, we propose two AHP consensus models under Chiclana et al.'s framework and RGMM.

Numerical examples

In order to show how the consensus models work in practice, let us consider the following example. Suppose we have a set of five decision makers providing the following judgement matrices {A(1),...,A(5)} on a set of four alternatives. Let w(k) = (w1(k),...,w4(k))T be the individual priority vector derived from judgment matrix A(k) using RGMM. A(k) and w(k) (k = 1,2,...,5) are listed below.A(1)=14671/41341/61/3121/71/41/21,w(1)={0.6145,0.2246,0.0985,0.0624}T.A(2)=15791/51461/71/4121/91/61/21,w(2)={

Conclusion

Consensus models have been a hot topic in group decision making using preference relations. In general, research progress in group decision making using preference relations can benefit research in AHP. This paper defines the consensus indexes (i.e., the geometric cardinal consensus index and the geometric ordinal consensus index) to measure consensus degree among judgement matrices (or decision makers). Inspired by Chiclana et al.'s consensus framework [13] and Xu and Wei's individual

Acknowledgements

Yucheng Dong and Yinfeng Xu acknowledge the financial support of grants (nos. 70801048 and 70525004) from NSF of China and a grant (no. 200806981067) form the Ph.D. Programs Foundation of Ministry of Education of China. Wei-Chiang Hong acknowledges the financial support of grants (nos. 98-2410-H-161-001 and 98-2811-H-161-001) from NSC of Taiwan.

Yucheng Dong is an Assistant Professor at the School of Management, Xi'an Jiaotong University, Xi'an, China. He is a research fellow of the Department of Information Management, Oriental Institute of Technology, Taipei, Taiwan. He received his Ph.D. degree in management from Xi'an Jiaotong University in 2008. His current research interests include group decision making, computing with words, and on-line algorithm. His research results have been published in the European Journal of Operational

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  • Cited by (0)

    Yucheng Dong is an Assistant Professor at the School of Management, Xi'an Jiaotong University, Xi'an, China. He is a research fellow of the Department of Information Management, Oriental Institute of Technology, Taipei, Taiwan. He received his Ph.D. degree in management from Xi'an Jiaotong University in 2008. His current research interests include group decision making, computing with words, and on-line algorithm. His research results have been published in the European Journal of Operational Research, IEEE Transactions on Fuzzy Systems, System Engineering and Electronics, and Fuzzy Sets and Systems, among others.

    Guiqing Zhang is a PhD candidate at the School of Management, Xi'an Jiaotong University, Xi'an, China. She received her B.S. degree from the Department of Information and Computation Science, Chongqing University in 2004, and her M.S. degree from the Department of Applied Mathematics, Chongqing University in 2007. Her research interests include group decision making and decision support systems.

    Wei-Chiang Hong is an Assistant Professor at the Department of Information Management, Oriental Institute of Technology, Taipei, Taiwan. He received his Ph.D. degree in management from Da Yeh University in 2008. He is the editor-in-chief of the International Journal of Applied Evolutionary Computation. His current research interests include decision analysis, evolutionary computation, and decision support systems. His research results have been published in Applied Mathematical Modelling, Electric Power Systems Research, International Journal of Electrical Power & Energy Systems, Energy Conversion and Management, and Applied Mathematics and Computation, among others.

    Yinfeng Xu is a Professor at the School of Management, Xi'an Jiaotong University, Xi'an, China. He received his Ph.D. degree in operational research from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, in 1992. His current research interests include combinatorial optimization, theoretical computer science and decision analysis. His research results have been published in Theoretical Computer Science, Journal of Global Optimization, Information Processing Letters, Journal of Combinatorial Optimization, Discrete & Computational Geometry, European Journal of Operational Research, and IEEE Transactions on Fuzzy Systems, among others.

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