Study Of Group Decision Making Theory And Methods Based On Fuzzy Number Preference Relations With Consistency

Recently, with the socio-ecomomic development, one has paid much attention totheory and the methodologies of group decision-making (GDM). These methodologies arevery important for GDM and imply a potence of the rule. It is the basis for developing agroup decision support system. The GDM is one of the most important parts of moderndecision science and has broad topics to study. Some new materials will be presented inthis dissertation about GDM. That is, when experts express their comparison values informats of intervals or triangular fuzzy numbers, some methodologies and applications ofGDM with intervals or triangular fuzzy or multiplicative preference relations are deeplystudied. Consistent interval, triangular fuzzy and implicative preference relations aredefned and the corresponding properties are studied in detail. Some transformationmethods between interval fuzzy and multiplicative preference relations are given to unifythe experts’ opinions. Based on the consistency of preference relations and the idea ofoptimization, programming models are shown to estimate incomplete preference valuesof interval preference relations. Some operators are extended to address the methods ofgiving the priority weights. The sufcient conditions of acceptable consistent collectivepreference relations are given by proving a controversial theorem in the literature, andthe corresponding GDM problem is dealt with.This dissertation will enrich the methodologies of GDM, has contributions to Saaty’sanalytic hierarchy process, and provide the theoretical basis for experts. The main resultsof the paper are summarized as follows:1. A defciency of the defnition of consistent triangular multiplicative preferencerelations proposed by Buckley [Fuzzy Sets and Systems,1985,17:233–247] is pointedout. Based on the fuzzy logic relation, a new defnition of consistent triangular multi-plicative preference relations is given. And its consistent properties are further studied.It is noted that Dubois has pointed out how to defne the consistency of preference re-lations with fuzzy number in [Dubois, Fuzzy Sets and Systems,2011,184:3–28]. Thenew defnition is a feasible solution of the given question. Moreover, the defnitions ofadditive consistent interval fuzzy preference relations, multiplicative consistent intervaland triangular fuzzy preference relations are proposed respectively. The correspondingproperties of the consistent preference relations are further studied. Based on the con-sistency, the construction method of obtaining consistent triangular fuzzy and reciprocalpreference relations are proposed and the corresponding GDM problems are addressed. 2. When individual interval multiplicative preference relations are acceptably con-sistent, it is proved that the collective interval multiplicative preference relation is alsoacceptably consistent by using the eigenvector priorization method. It is shown thatthe GCI of the collective interval multiplicative preference relation is no bigger than themaximum of the GCIs of individual interval multiplicative preference relations. It isconfrmed that the results of Xu [Xu, European Journal of Operational Research,2000,160:683–687] about the acceptance consistency of the weighted geometric mean complexjudgement matrix (WGMCJM) are right. The suggestion in Lin et al.[Lin et al. Euro-pean Journal of Operational Research,2008,190:672–678] about ignoring Xu’s resultsfor four or more alternatives should be ignored. The observation in Escobar et al.[Eu-ropean Journal of Operational Research，2004，153：318–322] that the inconsistency ofthe group is smaller than the largest individual inconsistency according to GCI, is provedagain from a new viewpoint.3. A model is proposed to solve GDM problems with incomplete interval fuzzy ormultiplicative preference relations. Based on the consistency and optimization theory,some goal programming methods for evaluating incomplete information are proposed.Some operators are extended to generate a collective preference relation, then one canrank the alternatives or select the best alternative. As compared to that given by Herrera-Viedma et al.[IEEE Transactions on Fuzzy Systems,2007,15:863–877; IEEE Transac-tions on Systems, Man, and Cybernetics—Part B: Cybernetics,2007,37:176–189], thismodel is simpler. It is attributed to the fact that the proposed method is estimating theincomplete values from one way, not from three diferent ways.4. A GDM model with interval fuzzy and multiplicative preference relations isproposed. Based on the additive consistency and the multiplicative consistency, twotransformation methods between interval fuzzy and multiplicative preference relationsare presented to unify the decision makers’interval preference relations, respectively.After aggregating individual interval preference relations by utilizing the induced or-dered weighted averaging (IOWA) operator, we analyze the acceptable consistency of thecollective preference relation. In the end, the GDM problem with interval fuzzy andmultiplicative preference relations is addressed perfectly.5. A preferential model based on consistent interval multiplicative reciprocal matri-ces is given to to partnership selection by using n1pairwise comparisons. This paperfrstly point out that when n1pairwise comparisons is used to construct a completepreference relation, they should form a complete comparison chain. These pairwise val-ues are not a simple means of expression, which perfects a consistent fuzzy preferenceprogramming method Based on only n1pairwise comparisons proposed by Wang andChen [Wang and Chen, Omega,2007,35:384–388]. Additionally, the model doesn’t need two transformations given by Wang and Chen. Furthermore, the modifed methodis extended to address the problem where the number of pairwise comparisons given bya decision maker is between n1and n(n1)/2, which is a extension of the method ofWang and Chen and a fuzzy preference programming method of Mikhailov [Mikhailov,Omega,2002,30:393–401].