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

Goal programming approaches to deriving interval fuzzy preference relations

This article investigates the consistency of interval fuzzy preference relations based on interval arithmetic, and new definitions are introduced for additive consistent, multiplicative consistent and weakly transitive interval fuzzy preference relations. Transformation functions are put forward to convert normalized interval weights into consistent interval fuzzy preference relations. By analyzing the relationship between interval weights and consistent interval fuzzy preference relations, goal-programming-based models are developed for deriving interval weights from interval fuzzy preference relations for both individual and group decision-making situations. The proposed models are illustrated by a numerical example and an international exchange doctoral student selection problem

A multi-step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices

This article presents a goal programming framework to solve group decision making problems where decision-makers’ judgments are provided as incomplete interval additive reciprocal comparison matrices (IARCMs). New properties of multiplicative consistent IARCMs are put forward and used to define consistent incomplete IARCMs. A two-step goal programming method is developed to estimate missing values for an incomplete IARCM. The first step minimizes the inconsistency of the completed IARCMs and controls uncertainty ratios of the estimated judgments within an acceptable threshold, and the second step finds the most appropriate estimated missing values among the optimal solutions obtained from the previous step. A weighted geometric mean approach is proposed to aggregate individual IARCMs into a group IARCM by employing the lower bounds of the interval additive reciprocal judgments. A two-step procedure consisting of two goal programming models is established to derive interval weights from the group IARCM. The first model is devised to minimize the absolute difference between the logarithm of the group preference and that of the constructed multiplicative consistent judgment. The second model is developed to generate an interval-valued priority vector by maximizing the uncertainty ratio of the constructed consistent IARCM and incorporating the optimal objective value of the first model as a constraint. Two numerical examples are furnished to demonstrate validity and applicability of the proposed approach

An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and Fusion: Taxonomy and future directions

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.The reciprocal preference relation (RPR) is a powerful tool to represent decision makers’ preferences in decision making problems. In recent years, various types of RPRs have been reported and investigated, some of them being the ‘classical’ RPRs, interval-valued RPRs and hesitant RPRs. Additive consistency is one of the most commonly used property to measure the consistency of RPRs, with many methods developed to manage additive consistency of RPRs. To provide a clear perspective on additive consistency issues of RPRs, this paper reviews the consistency measurements of the different types of RPRs. Then, consistency-driven decision making and information fusion methods are also reviewed and classified into four main types: consistency improving methods; consistency-based methods to manage incomplete RPRs; consistency control in consensus decision making methods; and consistency-driven linguistic decision making methods. Finally, with respect to insights gained from prior researches, further directions for the research are proposed

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

Hesitant Fuzzy Linguistic Analytic Hierarchical Process With Prioritization, Consistency Checking, and Inconsistency Repairing

Analytic hierarchy process (AHP), as one of the most important methods to tackle multiple criteria decision-making problems, has achieved much success over the past several decades. Given that linguistic expressions are much closer than numerical values or single linguistic terms to a human way of thinking and cognition, this paper investigates the AHP with comparative linguistic expressions. After providing the snapshot of classical AHP and its fuzzy extensions, we propose the framework of hesitant fuzzy linguistic AHP, which shows how to yield a decision for qualitative decision-making problems with complex linguistic expressions. First, the comparative linguistic expressions over criteria or alternatives are transformed into hesitant fuzzy linguistic elements and then the hesitant fuzzy linguistic preference relations (HFLPRs) are constructed. Considering that HFLPRs may be inconsistent, we conduct consistency checking and improving processes after obtaining priorities from the HFLPRs based on a linear programming method. Regarding the consistency-improving process, we develop a new way to establish a perfectly consistent HFLPR. The procedure of the hesitant fuzzy linguistic AHP is given in stepwise. Finally, a numerical example concerning the used-car management in a lemon market is given to illustrate the ef ciency of the proposed hesitant fuzzy linguistic AHP method.This work was supported in part by the National Natural Science Foundation of China under Grant 71771156, in part by the 2019 Sichuan Planning Project of Social Science under Grant SC18A007, in part by the 2019 Soft Science Project of Sichuan Science and Technology Department under Grant 2019JDR0141, and in part by the Project of Innovation at Sichuan University under Grant 2018hhs-43

Isomorphic multiplicative transitivity for intuitionistic and interval-valued fuzzy preference relations and its application in deriving their priority vectors

Intuitionistic fuzzy preference relations (IFPRs) are used to deal with hesitation while interval-valued fuzzy preference relations (IVFPRs) are for uncertainty in multi-criteria decision making (MCDM). This article aims to explore the isomorphic multiplicative transitivity for IFPRs and IVFPRs, which builds the substantial relationship between hesitation and uncertainty in MCDM. To do that, the definition of the multiplicative transitivity property of IFPRs is established by combining the multiplication of intuitionistic fuzzy sets and Tanino's multiplicative transitivity property of fuzzy preference relations (FPRs). It is proved to be isomorphic to the multiplicative transitivity of IVFPRs derived via Zadeh's Extension Principle. The use of the multiplicative transitivity isomorphism is twofold: (1) to discover the substantial relationship between IFPRs and IVFPRs, which will bridge the gap between hesitation and uncertainty in MCDM problems; and (2) to strengthen the soundness of the multiplicative transitivity property of IFPRs and IVFPRs by supporting each other with two different reliable sources, respectively. Furthermore, based on the existing isomorphism, the concept of multiplicative consistency for IFPRs is defined through a strict mathematical process, and it is proved to satisfy the following several desirable properties: weak--transitivity, max-max--transitivity, and center-division--transitivity. A multiplicative consistency based multi-objective programming (MOP) model is investigated to derive the priority vector from an IFPR. This model has the advantage of not losing information as the priority vector representation coincides with that of the input information, which was not the case with existing methods where crisp priority vectors were derived as a consequence of modelling transitivity just for the intuitionistic membership function and not for the intuitionistic non-membership function. Finally, a numerical example concerning green supply selection is given to validate the efficiency and practicality of the proposed multiplicative consistency MOP model

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)