Rationality of Fuzzy Choice Functions under an S-Implication–S-Residuum Framework

The theory of fuzzy sets has been applied to social choice primarily in the contexts where one is given a set of individual fuzzy preference relations and the aim is to find a non-fuzzy choice set of winners or best alternatives. We discuss the problem of composing multi-member deliberative bodies starting from a set of individual fuzzy preference relations. We outline methods of aggregating these relations into a measure of how well each candidate represents each voter in terms of the latter's preferences. Our main goal is to show how the considerations discussed in the context of individual non-fuzzy complete and transitive preference relations can be extended into the domain of fuzzy preference relations.

There are alternative ways of decomposing a given fuzzy weak preference relation into its antisymmetric and symmetric components. In this paper I have provided support to one among these alternative specifications. It is shown that on this specification the fuzzy analogue of the General Possibility Theorem is valid even when the transitivity restrictions on the individual and the social preference relations are relatively weak. In the special case where the individual preference relations are exact but the social preference relation is permitted to be fuzzy it is possible to distinguish between different degrees of power of the dictator. This power increases with the strength of the transitivity requirement.

A fuzzy multi-criteria decision-making model by associating technique for order preference by similarity to ideal solution with relative preference relation

This is a first approach to the study of the connection between fuzzy preference relations and fuzzy choice functions. In particular we depart from a fuzzy preference relation and we study the conditions it must satisfy in order to get a fuzzy choice function from it. We are particulary interested in one function: G-rationalization. We discuss the relevance of the completeness condition on the departing preference relation. We prove that not every non-complete fuzzy preference relation leads to a choice function.Keywordsfuzzy preference relation G-rational choice functioncomplete fuzzy relation

Some issues on consistency of fuzzy preference relations

On the basis of geometric consistency (GCI) of multiplicative preference relations and the transformation relation between multiplicative preference relations and fuzzy preference relation, inconsistency measure of fuzzy preference relation and the thresholds associated with it are studied. A new satisfactory consistency concept of fuzzy preference relation is defined. For a fuzzy preference relation, which does not possess satisfactory consistency, an improving consistency approach is provided. Then, programming models for estimating interval priority weights from fuzzy preference relations is established. Finally, an example is given to illustrate the validity and practicality of the proposed methods.

Study on Assessment Level of Experts based on Difference Preference Information

This article presents a goal programming approach to solve group decision making problems where decision-makers' judgments are provided as multiple incomplete preference relations, including incomplete fuzzy preference relations, incomplete multiplicative preference relations, incomplete interval multiplicative preference relations and incomplete linguistic preference relations. The proposed approach consists three steps:(1) transform the incomplete linguistic preference relations into the format of incomplete fuzzy preference relations (2) aggregate four different formats incomplete preference relations using a linear goal programming model and obtain the collective ranking values of alternative by solving the model, (3) rank or select the alternative based on the obtained collective ranking values of alternatives. A capability gaps evaluation is used to demonstrate the applicability of approach.

Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations

In each hesitant fuzzy linguistic preference relation, experts may express their opinions through comparison linguistic information combined with a discrete fuzzy number. In this paper, a hesitant fuzzy linguistic computational model based on discrete fuzzy numbers whose support is a subset of consecutive natural numbers is proposed, which enriches the flexibility of group decision‐making. First, some main concepts related to discrete fuzzy numbers and an aggregation function of individual subjective linguistic preference relations are introduced. Then, a consistency measure is presented to check and improve the consistency in a given matrix. Further, in order to achieve the predefined degree of consensus and to arrive at the final result, a consensus‐reaching process based on the interactive feedback mechanism is defined. Meanwhile, a revised formula is introduced to calculate the consistency and the degree of consensus in a preference relation matrix. Besides, an illustrative example and comparative analysis are conducted through the proposed calculation process and the optimization algorithm. Finally, the analysis on the threshold values is made to help the decision‐maker determine critical consensus level. The proposed method can address both consistency and consensus, and the results confirmed the effectiveness of the proposed method and its potential use in the qualitative decision‐making problems.

The breaking of additively reciprocal property of fuzzy preference relations and its implication to decision making under uncertainty

This paper proposes an interactive consensus reaching model in the group decision making for heterogeneous hesitant preference relations (i.e., hesitant fuzzy preference relations, hesitant multiplicative preference relations, hesitant fuzzy linguistic preference relations). First, the consistencies of three hesitant preference relations are defined, respectively. Then, based on their definitions, three optimization models are constructed to obtain the weight vector of alternatives, based on which an interactive consensus adjustment algorithm is established based on the direct consensus framework. This framework adopts feedback mechanism to facilitate the information correction of decision makers. After several rounds of adjustment, the decision results with satisfactory consensus level are achieved. Finally, the practicability and effectiveness of the model are illustrated through a case study of mine accident emergency decision making.

In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. The fuzzy preference relation is the reasonable representation of preference relations by a fuzzy membership function. This paper studies Nakamura’s and Kołodziejczyk’s preference relations. Eight cases, each representing different levels of overlap between two triangular fuzzy numbers are considered. We analyze the ranking behaviors of all possible combinations of the decomposition and intersection of two fuzzy numbers through eight extensive test cases. The results indicate that decomposition and intersection can affect the fuzzy preference relations, and thereby the final ranking of fuzzy numbers.

In the paper properties of interval-valued fuzzy preference relations are considered and preservation of a preference property by some operations, including lattice operations, the converse and the complement relations are studied. The concept of a preference relation presented here is a generalization of the concept of crisp preference relations. Moreover, weak properties of interval-valued fuzzy relations, namely reflexivity, irreflexivity, connectedness, asymmetry, antisymmetry, transitivity, and moderate transitivity are defined. Furthermore, the assumptions under which interval-valued fuzzy preference relations fulfil the mentioned properties are proposed.Keywordsinterval-valued relationspreference propertyproperties of interval-valued fuzzy relationsweak transitivitymoderate transitivity

In multiple-attribute decision making, fuzzy theory is very helpful for dealing with the fuzziness widely existing human judgments. Usually, decision makers obtain their subjective opinions by comparing each pair of decision alternatives and then constructing a rational (consistent) judgment matrix, known as a preference relation (PR) (Saaty, 1980; Xu, 1988) to deal with the ranking of alternatives X = {x 1, x 2 ,..., x n } .