Triangular bounded consistency of fuzzy preference relations

There are typically two types of consistency of fuzzy preference relations (FPR), namely additive and multiplicative consistency. They are defined based on the assumption that decision makers are rational and can provide strictly consistent FPRs. To take into consideration the bounded rationality of decision makers, the current study relaxes this assumption and proposes a new measure called triangular bounded consistency for judging the consistency of FPRs. To define triangular bounded consistency, a directed triangle is used to represent three FPRs among any three alternatives, with each directed edge representing an FPR. The condition of restricted max–max transitivity (RMMT) in the directed triangle is quantitatively examined. Under the assumption that the bounded rationality of a decision maker is characterized by their historical FPRs, which are represented by directed triangles that satisfy RMMT, triangular bounded consistency is determined using the historical FPRs. We then illustrate how triangular bounded consistency can be used to verify the consistency of FPRs that are newly provided by decision makers and how to estimate some missing FPRs that are not provided by decision makers. Finally, to demonstrate the application of triangular bounded consistency of FPRs in multi-attribute decision analysis, we investigate a problem that involves selecting areas to market products for a company

Incomplete interval fuzzy preference relations and their applications

This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n − 1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process

Characterisation of the consistent completion of AHP comparison matrices using graph theory

[EN] Decision-making is frequently affected by uncertainty and/or incomplete information, which turn decision-making into a complex task. It is often the case that some of the actors involved in decision-making are not sufficiently familiar with all of the issues to make the appropriate decisions. In this paper, we are concerned about missing information. Specifically, we deal with the problem of consistently completing an analytic hierarchy process comparison matrix and make use of graph theory to characterize such a completion. The characterization includes the degree of freedom of the set of solutions and a linear manifold and, in particular, characterizes the uniqueness of the solution, a result already known in the literature, for which we provide a completely independent proof. Additionally, in the case of nonuniqueness, we reduce the problem to the solution of nonsingular linear systems. 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Managing Incomplete Preference Relations in Decision Making: A Review and Future Trends

In decision making, situations where all experts are able to efficiently express their preferences over all the available options are the exception rather than the rule. Indeed, the above scenario requires all experts to possess a precise or sufficient level of knowledge of the whole problem to tackle, including the ability to discriminate the degree up to which some options are better than others. These assumptions can be seen unrealistic in many decision making situations, especially those involving a large number of alternatives to choose from and/or conflicting and dynamic sources of information. Some methodologies widely adopted in these situations are to discard or to rate more negatively those experts that provide preferences with missing values. However, incomplete information is not equivalent to low quality information, and consequently these methodologies could lead to biased or even bad solutions since useful information might not being taken properly into account in the decision process. Therefore, alternative approaches to manage incomplete preference relations that estimates the missing information in decision making are desirable and possible. This paper presents and analyses methods and processes developed on this area towards the estimation of missing preferences in decision making, and highlights some areas for future research

Consistency test and weight generation for additive interval fuzzy preference relations

Some simple yet pragmatic methods of consistency test are developed to check whether an interval fuzzy preference relation is consistent. Based on the definition of additive consistent fuzzy preference relations proposed by Tanino (Fuzzy Sets Syst 12:117–131, 1984), a study is carried out to examine the correspondence between the element and weight vector of a fuzzy preference relation. Then, a revised approach is proposed to obtain priority weights from a fuzzy preference relation. A revised definition is put forward for additive consistent interval fuzzy preference relations. Subsequently, linear programming models are established to generate interval priority weights for additive interval fuzzy preference relations. A practical procedure is proposed to solve group decision problems with additive interval fuzzy preference relations. Theoretic analysis and numerical examples demonstrate that the proposed methods are more accurate than those in Xu and Chen (Eur J Oper Res 184:266–280, 2008b)