Hesitant Fuzzy Preference Relations Research Articles

A new group decision making approach based on incomplete probabilistic dual hesitant fuzzy preference relations

As an enhanced version of probabilistic hesitant fuzzy sets and dual hesitant fuzzy sets, probabilistic dual hesitant fuzzy sets (PDHFSs) combine probabilistic information with the membership degree and non-membership degree, which can describe decision making information more reasonably and comprehensively. Based on PDHFSs, this paper investigates the approach to group decision making (GDM) based on incomplete probabilistic dual hesitant fuzzy preference relations (PDHFPRs). First, the definitions of order consistency and multiplicative consistency of PDHFPRs are given. Then, for the problem that decision makers (DMs) cannot provide the reasonable associated probabilities of probabilistic dual hesitant fuzzy elements (PDHFEs), the calculation method of the associated probability is given by using an optimal programming model. Furthermore, the consistency level for PDHFPRs is tested according to the weighted consistency index defined by the risk attitude of DMs. In addition, a convergent iterative algorithm is proposed to enhance the unacceptable consistent PDHFPRs’ consistency level. Finally, a GDM approach with incomplete PDHFPRs is established to obtain the ranking of the alternatives. The availability and rationality of the proposed decision making approach are demonstrated by analyzing the impact factors of haze weather.

Consensus-based group decision-making methods with probabilistic dual hesitant fuzzy preference relations and their applications

Probabilistic dual hesitant fuzzy sets (PDHFSs) have good flexibility and integrity in expressing fuzzy and uncertain information. However, some crucial problems related to PDHFSs remain unsolved, such as how to define probabilistic dual hesitant fuzzy preference relations (PDHFPRs) and solve group decision-making (GDM) problems with PDHFPRs. This paper establishes the concept of PDHFPRs and investigates consensus-based GDM methods with PDHFPRs. First, a new distance measure is proposed to quantify the difference between two PDHFPRs, which does not increase the virtual elements of membership and non-membership degrees, and can contain all distance combination of membership and non-membership elements. Therefore, the distance calculation results are not affected by the subjectivity of decision-makers (DMs). Second, the consensus measures for PDHFPRs are proposed, which are effective tool to measure the consensus level among DMs. Moreover, two consensus-based GDM methods are proposed, which can improve the group consensus level for PDHFPRs by changing the PDHFPR with the worst consensus level or modifying the weights of DMs. Finally, the proposed methods are applied to the location selection of large-scale industrial solid waste treatment facilities. The comparison with existing methods illustrates the validity and feasibility of the proposed methods.

A new method for deriving priority from dual hesitant fuzzy preference relations

Dual hesitant fuzzy elements (DHFEs) are suitable to express hesitant possible preferred and nonpreferred judgments of decision makers. Preference relation is an important tool in decision making that only needs the decision makers to compare a pair of objects at one time. This study focuses on decision making with dual hesitant fuzzy preference relations (DHFPRs). Considering the consistency, an additive consistency concept is defined. Meanwhile, the property of the new concept is studied. Using this consistency concept, a method for assessing the additive consistency of DHFPRs is offered. To extend the application of DHFPRs, a programming model to determine the missing DHFEs in incomplete DHFPRs is built, which have the highest additive consistency level for the known ones. Two equivalent methods to calculate the priority vector are offered. One method obtains the probabilistic dual hesitant fuzzy priority vector, and the other derives the intuitionistic fuzzy priority vector. Furthermore, a consensus index is defined to measure the consensus of individual opinions in group decision making (GDM), and an interactive method for increasing the consensus level is offered. On the basis of the additive consistency and consensus, an algorithm to GDM with DHFPRs is offered that can address inconsistent and incomplete cases. Finally, a practical example about evaluating color TV is provided to demonstrate the usefulness of the new procedure.

Approaches for multicriteria decision-making based on the hesitant fuzzy best\u2013worst method

Hesitant fuzzy preference relations (HFPRs) have been widely applied in multicriteria decision-making (MCDM) for their ability to efficiently express hesitant information. To address the situation where HFPRs are necessary, this paper develops several decision-making models integrating HFPRs with the best worst method (BWM). First, consistency measures from the perspectives of additive/multiplicative consistent hesitant fuzzy best worst preference relations (HFBWPRs) are introduced. Second, several decision-making models are developed in view of the proposed additive/multiplicatively consistent HFBWPRs. The main characteristic of the constructed models is that they consider all the values included in the HFBWPRs and consider the same and different compromise limit constraints. Third, an absolute programming model is developed to obtain the decision-makers’ objective weights utilizing the information of optimal priority weight vectors and provides the calculation of decision-makers’ comprehensive weights. Finally, a framework of the MCDM procedure based on hesitant fuzzy BWM is introduced, and an illustrative example in conjunction with comparative analysis is provided to demonstrate the feasibility and efficiency of the proposed models.

Group decision making based on the modified probability calculation method and DEA cross-efficiency with probabilistic hesitant fuzzy preference relations

In this paper, we propose a novel group decision making (GDM) method with probabilistic hesitant fuzzy preference relations (PHFPRs) based on the modified probability calculation method and data envelopment analysis (DEA) cross-efficiency. The primary advantage of the method is that it can reduce information loss and ensure the reliability of the decision-making result. A modified probability calculation method for hesitant fuzzy preference relations (HFPRs) is first proposed to obtain complete and normalized PHFPRs. Then, for the consistent additive preference relation (APR), we construct an output-oriented DEA model to derive the priority vector, in which each alternative is regarded as a decision-making unit (DMU). A DEA cross-efficiency model is further proposed to yield the priority vector, irrespective of whether the desired consistency level is achieved. Furthermore, a stochastic analysis method is developed to obtain the expected priority vector for any PHFPR, and a mathematical programming model is established to obtain the weight vector of decision makers based on group consensus. Finally, the procedure of the GDM method based on the modified probability calculation and the DEA cross-efficiency is designed. We also show the applicability and effectiveness of the proposed GDM method using illustrative examples.

Expected loss utility for natural hazards and its application in pricing property insurance products

Due to climatic hazards and extreme weather events, the pricing of property insurance products is increasingly attracting the attention of policyholders, insurance companies, and governments. Pricing based on market-oriented methods has to consider the affecting factors from policyholders’ perceived value. Pricing strategy design generates the need for natural hazards risk assessments. A natural hazard risk assessment is closely related to the human factors of a disaster-bearing body. In response to this need, we design an extension of the expected utility that is inconsistent with the additive expected utility, considering the human factors of policyholders, which is referred to as the expected loss utility (ELU). The ELU presents two improvements of the currently used utility. First, subjective probability, which is derived from individual predictions over acts, is applied to the ELU function to overcome the disadvantage that objective probability attaches to uncertainty does not reflect the uncertainty of human factors. Policyholders’ risk attitudes are reflected by the interpretation of interactions among uncertain events. Second, the hesitant fuzzy linguistic preference relation (HFLPR) is employed as the assessment of individual loss evaluation to reflect a policyholder’s hesitation. We apply the techniques of fuzzy linguistic term aggregation and perform a comparison to simplify our loss utility function. A detailed process of expected loss assessment is proposed due to variations in natural environment factors, local social characteristics, and disaster-bearing body factors. An illustrative example is given to perform a comparison with cumulative prospect theory to show the merits of the ELU. This study quantifies policyholder’s cognition of uncertain event and the cognition’s influence on risk assessment which can guide pricing strategy of property insurance products.

Logical efficiency decomposition for general two-stage systems in view of cross efficiency

Most of the previous cross efficiency evaluation methods consider the evaluated system as a “black box” system. This probably leads to misleading results. For example, the system is efficient but all of its divisions are inefficient. This paper firstly investigates the cross efficiency for a general two-stage system. In such system, some outputs of one stage are taken as inputs of the other stage, and some other outputs directly become the final products. Since the optimal weight vector for the self-evaluation of a decision making unit (DMU) may be non-unique, we apply the leader-follower method for the decomposition of the system's efficiency and then adopt the optimal weights to obtain the peer-evaluation. Then, multiplicative hesitant fuzzy elements (MHFEs) are applied to denote the ratio of all possible cross efficiencies of any two DMUs. Based on MHFEs, multiplicative hesitant fuzzy preference relations (MHFPRs) are used to express these relationships comprehensively. Furthermore, the optimization models for deriving acceptably consistent MHFPRs with the minimum total adjustment and the minimum number of adjusted elements are built when the consistency of MHFPRs is unacceptable. An algorithm for decomposing the overall efficiency according to the cross efficiency and acceptably consistent MHFPRs in the setting of the general two-stage system is provided. Finally, our method is applied to the analysis of nine top universities in China.

Analysis of acceptable additive consistency and consensus of group decision making with interval-valued hesitant fuzzy preference relations

Interval-valued hesitant fuzzy sets are efficient to denote the hesitant and uncertain judgments of decision makers. To further research the utilization of this type of fuzzy sets, this paper limits to study group decision making (GDM) with interval-valued hesitant fuzzy preference relations (IVHFPRs). First, an additive consistency index is defined for interval fuzzy preference relations, by which we derive a new acceptable additive consistency concept for IVHFPRs. Then, new models based on acceptable additive consistency are built for ascertaining unknown values, and new models for judging the acceptable additive consistency of IVHFPRs are constructed. When the consistency of IVHFPRs is unacceptable, new models based on consistency and uncertain degree are constructed for obtaining acceptable additive consistency IVHFPRs that can ensure the smallest total adjustment and the minimum number of adjusted intervals. For GDM case, a new consensus index is defined to measure the agreement degree of individual judgments. When the consensus level does not satisfy requirement, new models for reaching the additive consistency and consensus requirements are built. Furthermore, a new acceptable additive consistency- and consensus-based algorithm for GDM with incomplete and unacceptable additive consistency IVHFPRs is provided. Finally, an example is given to indicate the utilization of the new algorithm and comparison analysis is made.

Dual hesitant fuzzy decision making in optimization models

Considering the advantages of dual hesitant fuzzy elements (DHFEs) for describing the hesitant and intuitionistic judgments of experts and identifying the limitations of previous research about dual hesitant fuzzy decision making, this paper studies decision making with dual hesitant fuzzy preference relations (DHFPRs) and provides a new group decision making (GDM) method based on a series of built optimization models. A multiplicative consistency concept for DHFPRs is first introduced and analyzed. On the basis of this concept, optimization describing the multiplicative consistency of DHFPRs is realized. Meanwhile, stepwise optimization models for obtaining complete multiplicatively consistent DHFPRs are constructed, and a method for calculating the intuitionistic fuzzy priority vector is offered. Then, optimization models for determining unknown values in incomplete DHFPRs are formed in view of the multiplicative consistency. An index for measuring the consensus is defined and optimization models for reaching the consensus requirement of a group are established. Furthermore, consensus-based optimization models for determining the weights of experts are established. In view of the multiplicative consistency and consensus analysis, a detailed procedure for GDM with incomplete and inconsistent DHFPRs is provided. Finally, the new method is illustrated by a case study concerning an evaluation of the environmental performance of enterprises and associated comparative analyses are reported.

Sustainable Decision Making Using a Consensus Model for Consistent Hesitant Fuzzy Preference Relations—Water Allocation Management Case Study

This paper presents an improved consensus-based procedure to handle multi-person decision making (MPDM) using hesitant fuzzy preference relations (HFPRs) which are not in normal format. At the first level, we proposed a ukasiewicz transitivity (TL-transitivity) based scheme to get normalized hesitant fuzzy preference relations (NHFPRs), subject to which, a consensus-based model is established. Then, a transitive closure formula is defined to construct TL-consistent HFPRs and creates symmetrical matrices. Following this, consistency analysis is made to estimate the consistency degrees of the information provided by the decision-makers (DMs), and consequently, to assign the consistency weights to them. The final priority weights vector of DMs is calculated after the combination of consistency weights and predefined priority weights (if any). The consensus process concludes whether the aggregation of data and selection of the best alternative should be originated or not. The enhancement mechanism is indulged in improving the consensus measure among the DMs, after introducing an identifier used to locate the weak positions, in case of the poor consensus reached. In the end, a comparative example reflects the applicability and the efficiency of proposed scheme. The results show that the proposed method can offer useful comprehension into the MPDM process.

Multiplicative Consistency Adjustment Model and Data Envelopment Analysis-Driven Decision-Making Process with Probabilistic Hesitant Fuzzy Preference Relations

Unlike other fuzzy modellings, probabilistic fuzzy sets can reflect clearly the importance of different numerical values. In group decision-making (GDM) problems, it is quite common for decision-makers (DMs) to elicit their knowledge with probabilistic hesitant fuzzy preference relations (PHFPRs), in which consistency adjustment and alternatives’ weight vector determination play a key role in the decision-making process. This study aims at constructing a decision-making model with PHFPRs. First, several new concepts are introduced, including the multiplicative consistency and consistency index of PHFPRs. Then, we present a construction approach for the multiplicative consistent PHFPRs, and a convergent local consistency improvement process for PHFPRs is designed to detect and improve their consistency when the PHFPRs do not meet the consistency level. The local adjustment strategy is utilized to retain the preference evaluation of DMs as much as possible. Afterwards, based on the obtained efficiency score values, we propose a new data envelopment analysis model to derive the weight values of alternatives. Furthermore, we explore a decision-making method with PHFPRs to obtain the optimal selection from the alternatives. Finally, an applied case about logistics company assessment is presented, and the effectiveness and rationality of the explored method are verified by the comparison with the various approaches.

Generalized hesitant multiplicative preference relations and the analytic risk-network process

The hesitant fuzzy preference relation (HFPR) and the hesitant multiplicative preference relation (HMPR) are emerging techniques in decision-making research field. Clearly, an intuitionistic idea to improve the above preference relations is to synthesize the HFPR and the HMPR, which is also a primary contribution of this paper. Based on which, we propose the generalized HMPR (GHMPR) with full desired properties using transformation formulas that combine HFPR with HMPR and convert the HMPR into the GHMPR. Then, the GHMPR not only considers different risk appetites of a decision maker, but also preserves the advantages of both. Moreover, we design the corresponding consistency check algorithms, weight calculation methods and convergence algorithms for the GHMPR. Based on these new methods, we further propose a qualitative decision-making method under the GHMPR environment named as the analytic risk-network process (ARNP). Saaty’s well-known analytic network process case is then utilized as an illustrated example to verify the feasibility of the proposed methods via comparisons. The simulation results show our proposed methods are effective and useful.

Composite Decision Makers in the Graph Model for Conflict Resolution: Hesitant Fuzzy Preference Modeling

Hesitant fuzzy preference relations (HFPRs) are formally proposed to model the conflict situation in which each decision maker (DM) consists of multiple individuals and each individual has its own fuzzy preferences over the feasible states within the framework of the graph model for conflict resolution (GMCR). Based on HFPRs, new definitions for hesitant fuzzy Nash stability, hesitant fuzzy general metarationality, hesitant fuzzy symmetric metarationality, and hesitant fuzzy sequential stability permit stability analyses to be carried out. Moreover, a new option prioritization technique, called hesitant fuzzy option prioritization, is developed for modeling a DM’s HFPRs based on the DM’s priority sequence of preference statements, the DM’s fuzzy truth values and levels of confidence. The groundwater contamination conflict of Elmira, Ontario, Canada, is utilized as a case study to illustrate the usefulness and applicability of the hesitant fuzzy option prioritization technique and GMCR with HFPRs.

Water–Energy–Food nexus evaluation with a social network group decision making approach based on hesitant fuzzy preference relations

With the rapid increase and development of global population and economic, the Water–Energy–Food (WEF) nexus evaluation which is related to the sustainable development of human has become a hotspot. Whereas, the study of the WEF nexus evaluation from the perspectives of social network group decision making (SNGDM) is still a challenge. Hence, this paper aims to develop a trust-based SNGDM approach with hesitant fuzzy preference relations to the WEF nexus evaluation. In the proposed model, a new fuzzy adjacency matrix and trust score matrix based on expert’s self-confidence are defined to imply experts’ trust relationship and trust score, respectively. To improve the reliability of the final decision(s), an iterative algorithm is presented to improve the consistency of experts’ evaluations. Subsequently, the individual evaluation can be aggregated into a group one by using the trust score induced ordered weighted averaging operator while the trust scores of experts are the induced factors. Additionally, an algorithm is utilized to achieve a high level consensus in SNGDM of the WEF nexus evaluation. Finally, some comparison analyses and discussions show the feasibility and validity of the proposed method.

Group decision making based on acceptable consistency analysis of interval linguistic hesitant fuzzy preference relations

To represent decision makers’ qualitative uncertainty and hesitation judgments, interval linguistic hesitant fuzzy variables (ILHFVs) are efficient tools, which can be regarded as an expansion of interval linguistic variables (ILVs). Taking the merits of ILHFVs and preference relations, this paper focuses on group decision making (GDM) with interval linguistic hesitant fuzzy preference relations (ILHFPRs). By considering the consistency of ILHFPRs, a new definition of acceptable consistency is presented. Using the acceptable consistency index, some models are built to measure whether a given ILHFPR is acceptable consistent. If the consistency is unacceptable, some models are constructed to derive acceptable consistent ILHFPRs by considering the total adjustment and the number of adjusting ILVs. In order to cope with incomplete ILHFPRs, some models for obtaining the values of unknown ILVs are proposed. For GDM with ILHFPRs, an index for measuring the consensus degree of ILHFPRs is proposed. When ILHFPRs do not meet the requirement of the consensus, some models for enhancing the consensus degree are proposed. According to the analysis of acceptable additive consistency and consensus of ILHFPRs, a new method for GDM with ILHFPRs is proposed. In order to show the merits of the proposed GDM method, an application example is used.

Group decision making based on acceptable multiplicative consistency of hesitant fuzzy preference relations

This paper deals with group decision making (GDM) with hesitant fuzzy preference relations (HFPRs) based on the acceptable multiplicative consistency and the consensus analysis. We first offer a multiplicative consistency index for fuzzy preference relations (FPRs) and then use the Monte Carlo simulation method to derive the average multiplicative consistency value. After that, a model-based interactive algorithm is offered to test acceptable multiplicative consistency of HFPRs, by which the concept of acceptable multiplicative consistency for HFPRs is obtained. Meanwhile, a model-based interactive algorithm for deriving acceptable multiplicative consistent HFPRs from unacceptable multiplicative consistent ones is provided, where both the total adjustment and the number of adjusted variables are considered. As for incomplete HFPRs, a model-based interactive algorithm for getting the values of missing preferences is provided. Furthermore, the weights of the decision makers are determined by the offered model and an algorithm of model-based adjustment for the consensus level is provided. Finally, a procedure for GDM with acceptable multiplicative consistent HFPRs is given, and a case study about selecting the most suitable project management information systems (PMISs) is provided to show the application of the proposed GDM method and to compare the proposed GDM method with the previous GDM methods.

New method for interval-valued hesitant fuzzy decision making based on preference relations

Interval-valued hesitant fuzzy preference relations (IVHFPRs) are powerful to express the judgments of decision makers (DMs) such as hesitancy and uncertainty. This paper continues to research decision making with IVHFPRs. To do this, it first defines a multiplicative consistency index for interval fuzzy preference relations (IFPRs). Then, it gives a new (acceptably) multiplicative consistency concept for IVHFPRs. After that, optimization models for judging the acceptably multiplicative consistency of IVHFPRs are built. When the multiplicative consistency is unacceptable, optimization models for improving the multiplicative consistency level and for minimizing the number of adjusted judgments are constructed, respectively. Based on the acceptably multiplicative consistency analysis, an algorithm for decision making with IVHFPRs is presented. Furthermore, formulae for determining the weights of the DMs and for measuring the consensus are offered. Based on the acceptably multiplicative consistency and consensus analysis, a new method for group decision making with IVHFPRs is proposed. Finally, an example about selecting the reservoir operation schemes is offered to show the efficiency of the new method and to compare with previous research.

Optimal Interaction Priority Calculation From Hesitant Fuzzy Preference Relations Based on the Monte Carlo Simulation Method for the Acceptable Consistency and Consensus.

To address the situation where the complete consistency is unnecessary, a stepwise optimization model-based method for testing the acceptably additive consistency (AAC) of hesitant fuzzy preference relations (HFPRs) is introduced. Then, an AAC concept for HFPRs is defined. Meanwhile, incomplete HFPRs (iHFPRs) are discussed and a series of optimization models to acquire complete HFPRs is constructed. If the consistency is unacceptable, an optimization model for revising unacceptably consistent HFPRs under the conditions of the AAC and maximizing the ordinal consistency (OC) is offered. Subsequently, a model for minimizing the number of adjusted variables is presented. Considering the weighting information and the consensus for group decision making (GDM), the weights of fuzzy preference relations (FPRs) obtained from each individual HFPR and the decision makers (DMs) are determined using the distance measure. With regard to the consensus, two models for reaching the consensus requirement and minimizing the amount of revised variables are separately constructed, which are both based on the analysis of maximizing the OC. Furthermore, the thresholds of the additive consistency and the consensus are studied using the Monte Carlo simulation method. A GDM algorithm with HFPRs is offered. Finally, an example and comparison are provided to show the efficiency of the new procedure.

Group decision making based on hesitant fuzzy ranking of hesitant fuzzy preference relations

Hesitant fuzzy sets (HFSs) play a dominant role in the decision making process. Different tools are developed to attract the decision makers (DMs) in making the effective decision, the hesitant fuzzy preference relation (HFPR) is one of the important implementation of them. Preference of an alterna tive over another alternative is a useful way to express the opinion of decision maker. In this paper, a hesitant fuzzy ranking (HFR) technique is established, constructed the hesitant fuzzy ranking from the HFPR in the group decision making situations. Secondly, a correlation between the alternatives is developed by using Spearman ranked correlation coefficient formula, which helps the DMs to identify the better alternative. The novelty of the proposed strategy is that it evades the need to compute the cooperative preference relations and approvals are generated for the individuals in their original domains.

The consistency improvement of probabilistic linguistic hesitant fuzzy preference relations and their application in venture capital group decision making

The consistency improvement of probabilistic linguistic hesitant fuzzy preference relations and their application in venture capital group decision making