On the normalization of a priority vector associated with a reciprocal relation.

In this paper we show that the widely used normalization constraint SUM(i=1,n) wi = 1 does not apply to the priority vectors associated with reciprocal relations, whenever additive transitivity is involved. We show that misleading applications of this type of normalization may lead to unsatisfactory results and we give some examples from the literature. Then, we propose an alternative normalization procedure which is compatible with additive transitivity and leads to better results.reciprocal relation; fuzzy preference relation; priority vector; normalization

Incomplete pairwise comparison and consistency optimization

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.consistency, pairwise comparison matrices

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

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)

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

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

Risk assessment in project management by a graphtheory- based group decision making method with comprehensive linguistic preference information

Risk assessment is a vital part in project management. It is possible that experts may provide comprehensive linguistic preference information in distinct forms with respect to different aspects of the risk assessment problem in investment management. It is a challenge to model and deal with comprehensive linguistic preference assessments in multiple forms given by experts. In this regard, this paper defines the generalised probabilistic linguistic preference relation (GPLPR) to represent different forms of linguistic preference information in a unified structure. Then, a probability cutting method is proposed to simplify the representation of a GPLPR. Afterwards, a graph-theory-based method is developed to improve the consistency degree of a GPLPR. A group decision making method with GPLPRs is then proposed to carry on the risk assessment in project management. Discussions regarding the comparative analysis and managerial insights are given

Risk assessment in project management by a graph-theory-based group decision making method with comprehensive linguistic preference information

The work was supported by the National Natural Science Foundation of China (71971145, 71771156, 72171158), the Andalusian Government under Project P20-00673, and also by the Spanish State Research Agency under Project PID2019-103880RB-I00/AEI/10.13039/501100011033.Risk assessment is a vital part in project management. It is possible that experts may provide comprehensive linguistic preference information in distinct forms with respect to different aspects of the risk assessment problem in investment management. It is a challenge to model and deal with comprehensive linguistic preference assessments in multiple forms given by experts. In this regard, this paper defines the generalised probabilistic linguistic preference relation (GPLPR) to represent different forms of linguistic preference information in a unified structure. Then, a probability cutting method is proposed to simplify the representation of a GPLPR. Afterwards, a graph-theory-based method is developed to improve the consistency degree of a GPLPR. A group decision making method with GPLPRs is then proposed to carry on the risk assessment in project management. Discussions regarding the comparative analysis and managerial insights are given.National Natural Science Foundation of China (NSFC) 71971145 71771156 72171158Andalusian Government P20-00673Spanish Government PID2019-103880RB-I00/AEI/10.13039/50110001103