Some issues on consistency of fuzzy preference relations (original) (raw)
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Reciprocity and consistency of fuzzy preference relations
2003
Summary. Preference relations are the most common representation structures of information used in decision making problems because they are useful tool in modelling decision processes, above all when we want to aggregate experts' preferences into group preferences. Therefore, to establish rationality properties to be verified by preference relations is very important in the design of good decision making models.
Some problems on the definition of fuzzy preference relations
Fuzzy Sets and Systems, 1986
In this paper we deal with decision-making problems over an unfirzxy set of alternatives. On one hand, we propose the problem of finding a max-min transitive relation as near as possible to a given initial preference relation, under the least-squares criterion and such that it does not introduce deep qualitative changes. On the other hand, we define a linear extension of the initial preference relation between alternatives to a preference relation between lotteries.
Information Sciences, 2016
Triangular fuzzy numbers are effective in modeling imprecise and uncertain information, and have been widely applied in decision making. This paper uses a cross-ratio-expressed triplet to characterize a positive triangular fuzzy number, and introduces notions of crossratio-expressed triangular fuzzy numbers (CRETFNs) and triangular fuzzy additive reciprocal preference relations (TFARPRs). We present transformation methods between TFARPRs and triangular fuzzy multiplicative reciprocal preference relations, and develop operational laws of CRETFNs, such as complement, addition, multiplication and power. A crossratio-expressed triangular fuzzy multiplication based transitivity equation is established to define multiplicative consistency of TFARPRs. The new consistency captures Tanino's multiplicative consistency among the cross-ratio-expressed modal values, and geometric consistency of the interval fuzzy preference relation constructed from lower and upper support values of cross-ratio-expressed triangular fuzzy judgments. Some desirable properties are furnished for multiplicatively consistent TFARPRs. We propose a cross-ratio-expressed triangular fuzzy weighted geometric operator to aggregate CRETFNs, and extend it to fuse TFARPRs. Score and uncertainty index functions are defined and employed to devise a novel comparison method for CRETFNs. A detailed procedure is put forward to solve group decision making problems with TFARPRs. Six numerical examples are provided to illustrate the validity and applicability of the proposed models.
Methods for fuzzy complementary preference relations based on multiplicative consistency
Computers & Industrial Engineering, 2011
To improve the consistency of a preference relation is a hot topic in decision making. Wang and Chen (2008) gave a simple method to construct the complete fuzzy complementary preference relation from only n À 1 pairwise comparisons. However, some values may not be in the defined scope and need to be transformed, and thus some original information may be lost in the transformation process. In this paper, we propose a new method to avoid this issue based on the multiplicative consistency of the fuzzy complementary preference relation and apply it to fuzzy Analytic Hierarchy Process (AHP). An example is further given to illustrate our method.
Information Sciences, 2015
Triangular fuzzy preference relation (TFPR) is an effective framework to model pairwise 27 estimations with imprecision and vagueness. In order to obtain a reliable and rational deci-28 sion result, it is important to investigate consistency and priority derivation of TFPRs. The 29 paper analyzes existing definitions and properties of consistent TFPRs, and illustrates that 30 they have no invariance with respect to permutations of decision alternatives. A new 31 triangular fuzzy arithmetic based transitivity equation is introduced to define consistent 32 TFPRs. The new transitivity equation reflects multiplicative consistency of modal values 33 and multiplicative consistency of geometric means of triangular fuzzy estimations. Some 34 properties are presented for consistent TFPRs, and a notion of acceptable consistency is 35 put forward for TFPRs. Geometric mean and uncertainty ratio based transformation formu-36 lae are devised to convert normalized triangular fuzzy multiplicative weights into consis-37 tent TFPRs. A logarithmic least square model is further established for deriving a 38 normalized triangular fuzzy multiplicative weight vector from a TFPR with acceptable con-39 sistency. A geometric mean based method is developed to compare and rank triangular 40 fuzzy multiplicative weights. Three numerical examples including a group decision making 41 problem are examined to demonstrate validity and advantages of the proposed models. 42
Some properties of extended fuzzy preference relations using modalities
Information Sciences, 1992
A fuzzy preference relation is extended to six kinds of relations between fuzzy sets. The properties of the six extended relations are investigated. Especially, a sufficient condition for reducing the extended relations to four kinds is discussed when fuzzy sets are bounded and normal, and some examples are given to illustrate it. Further, a sufficient condition for the extended negative relations to coincide with the extended inverse relations is described.
Incomplete preference relations: An upper bound condition
In decision making, consistency in fuzzy preference relations is associated with the study of transitivity property. While using additive consistency property to complete incomplete preference relations, the preference values found may lie outside the interval [0, 1] or the resultant relation may itself be inconsistent. This paper proposes a method that avoids inconsistency and completes an incomplete preference relation using an upper bound condition. Additionally, the paper extends the upper bound condition for multiplicative reciprocal preference relations. The proposed methods ensure that if (n − 1) preference values are provided by an expert, such that they satisfy the upper bound condition, then the preference relation is completed such that the estimated values lie inside the unit interval [0, 1] in the case of preference relations and [1/9, 9] in the case of multiplicative preference relation. Moreover, the resultant preference relation obtained using the proposed method is transitive.
On Consensus in Group Decision Making Based on Fuzzy Preference Relations
Studies in Fuzziness and Soft Computing, 2011
In the process of decision making, the decision makers usually provide inconsistent fuzzy preference relations, and it is unreasonable to get the priority from an inconsistent preference relation. In this paper, we propose a method to derive the multiplicative consistent fuzzy preference relation from an inconsistent fuzzy preference relation. The fundamental characteristic of the method is that it can get a consistent fuzzy preference relation considering all the original preference values without translation. Then, we develop an algorithm to repair a fuzzy preference relation into the one with weak transitivity by using the original fuzzy preference relation and the constructed consistent one. After that, we propose an algorithm to help the decision makers reach an acceptable consensus in group decision making. It is worth pointing out that group fuzzy preference relation derived by using our method is also multiplicative consistent if all individual fuzzy preference relations are multiplicative consistent. Some examples are also given to illustrate our results.
International Journal of Information Technology & Decision Making, 2014
As we may have a set of possible values when comparing alternatives (or criteria), the hesitant fuzzy preference relation becomes a suitable and powerful technique to deal with this case. This paper mainly focuses on the consistency of the hesitant fuzzy preference relation. Before that, we explore some properties of the hesitant fuzzy preference relation and develop some new aggregation operators. Then we introduce the concepts of multiplicative consistency, perfect multiplicative consistency and acceptable multiplicative consistency for a hesitant fuzzy preference relation, based on which, two algorithms are given to improve the inconsistency level of a hesitant fuzzy preference relation. Furthermore, the consensus of group decision making is studied based on the hesitant fuzzy preference relations. Finally, several illustrative examples are given to demonstrate the practicality of our algorithms.
An axiomatic property based triangular fuzzy extension of Saaty’s consistency
Computers & Industrial Engineering, 2019
The fuzzy analytic hierarchy process (FAHP) is a widely applied fuzzy multi-criteria decision making method. In triangular FAHP, consistency of triangular fuzzy multiplicative preference relations (TFMPRs) plays a very important role for checking the quality of human fuzzy judgments. This paper introduces some indices to measure fuzziness of triangular fuzzy judgments and row fuzziness proportionalities in a TFMPR. An expected interval based possibility degree formula is developed to compare two triangular fuzzy judgments and used to judge whether the average-based intensity of a triangular fuzzy judgment is changed. The paper proposes four axiomatic properties aimed at characterizing multiplicative consistency of TFMPRs, and illustrates that four known consistency definitions fail to meet all the four axiomatic properties. By analyzing the axiomatic property of restriction on ambiguity of single judgments, a constrained triangular fuzzy multiplication based transitivity equation is devised and used to construct a triangular fuzzy matrix from a TFMPR. Based on the constructed fuzzy matrix, the paper establishes an ordinary triangular fuzzy multiplication based transitivity equation to define consistency of TFMPRs. Important properties of consistent TFMPRs are offered to demonstrate that all the four axiomatic properties are satisfied by the proposed consistency definition. Three numerical examples are given to illustrate the use of the proposed consistency model and the obtained properties for judging consistency of TFMPRs and determining unknown values of incomplete TFMPRs.