Incomplete Hesitant Fuzzy Preference Relations Research Articles

Two-stage group decision making methodology with hesitant fuzzy preference relations under social network: Multiplicative consistency determination and personalized feedback

Some social network group decision making (SNGDM) researches may overlook two issues: (1) the impact of consistency on decision reliability and decision makers’ (DMs’) status in social network, and (2) DMs’ personalization during consensus reaching process. In response to these two issues, a two-stage SNGDM model is proposed for hesitant fuzzy preference relations (HFPRs). In the first stage, multiplicative consistency determination and improvement of HFPRs are achieved. For complete and incomplete HFPRs, several programming models are constructed to classify consistency types. When it comes to inconsistent HFPRs, an inconsistency fixing method is designed to ensure the original HFPRs as much as possible. With the improvement of DMs’ consistency, their status in social network will be dynamically enhanced. In the second stage, two programming models are devised to elicit the reasonable individual priority weight vector in consistent HFPR. Further, minimum adjustment cost-based personalized feedback mechanism is invented to attain the optimal personalized adjustment coefficient and recommendation, and achieve the harmonious unity of individual personalization and group consensus. Finally, the superiority of our proposal is verified by comparative analysis and illustrative example.

Decision-making models based on satisfaction degree with incomplete hesitant fuzzy preference relation

To address the situation where incomplete hesitant fuzzy preference relation (IHFPR) is necessary, this paper develops decision-making models taking into account decision makers’ satisfaction degree. First, the consistency measures, respectively, from the perspectives of additive and multiplicative consistent IHFPR are defined, which is based on the relationships of the IHPFRs and their corresponding priority weight vector. Second, two decision-making models are developed, respectively, in view of the proposed additive and multiplicative consistency measures. The main characteristic of the constructed models is they taking into account the decision makers’ satisfaction degree. The objective functions of the models are developed by maximizing the parameter of the satisfaction degree. Third, a square programming model is developed to obtain the decision makers’ weights by utilizing the optimal priority weight vectors information, the solution of the model is obtained by solving the partial derivatives of Lagrange function. Finally, a procedure for multi-criteria decision-making (MCDM) problems with IHFPRs is given, and an illustrative example in conjunction with comparative analysis is used to demonstrate the proposed models are feasible and efficiency for practical MCDM problems.

Social network clustering and consensus-based distrust behaviors management for large-scale group decision-making with incomplete hesitant fuzzy preference relations

With the development of social network platforms, large-scale group decision-making in social network (LSGDM-SN) has been formed. As decision makers (DMs) come from different fields and have complex individual backgrounds, which leads to their distrust in the moderator. Moreover, in LSGDM-SN, since DMs can hardly grasp all the information about the decision problem, the hesitant fuzzy preference relations (HFPRs) they have expressed may be incomplete. However, in current LSGDM-SN issues, the distrust behaviors and incomplete HFPRs have never been discussed simultaneously. In this context, this paper aims to propose a method to estimate incomplete values in HFPRs, and develop a consensus management process which considers distrust behaviors. This paper focuses on LSGDM-SN on the basis of social network clustering and consensus-based distrust behaviors management with incomplete HFPRs. In this paper, a social network clustering method based on grey clustering algorithm is proposed to classify the DMs with similar social clustering degree into a subset. Afterwards, a method including two situations is developed to estimate incomplete values in HFPRs. Furthermore, an identification mechanism is presented to detect the DMs’ distrust behaviors, and three modification strategies are provided for managing different types of distrust behaviors. In addition, a case study is given to illustrate the feasibility of the proposed method. Finally, comparative analysis and discussion are explored to verify the advantages of the proposed LSGDM-SN with incomplete HFPRs.

Some models to manage additive consistency and derive priority weights from hesitant fuzzy preference relations

Consistency has a crucial influence on the rationality of preference information and even the final decision result. This paper presents two new definitions of additive consistency for hesitant fuzzy preference relations (HFPRs): completely additive consistency (CAC) and weakly additive consistency (WAC). To verify the CAC or WAC of HFPRs, some linear programming models and 0–1 mixed programming models are developed. The methods consider all the information given by the decision maker without changing the length of hesitant fuzzy elements (HFEs). Accordingly, a method of modifying an inconsistent HFPR into an additively consistent HFPR is proposed. The deviation between the original complementary matrix and the modified one is minimal. Then, several linear programming models are developed to obtain priority weights from an HFPR. From these, an integrated algorithm is designed to illustrate the process of consistency test, inconsistency modification and weights derivation for HFPRs. The proposed methods are also extended to deal with WAC and CAC of incomplete HFPRs. Finally, three numerical examples and comparative analysis are presented to illustrate the feasibility and effectiveness of the proposed method.

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.

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.

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.

Incomplete hesitant fuzzy preference relation

In the process of decision making, a decision maker may give her/his judgments using hesitant fuzzy preference relations for hesitancy and uncertainty. Limitations of the experts’ professional knowledge, experience, and lack of time may lead to preferences in hesitant fuzzy preference relation(HFPR) which are usually incomplete. Zhang et al. [28] introduced the concept of incomplete HFPR (I-HFPR)and additive consistency property of I-HFPR. The additive consistency property of HFPR defined by Zhang et al. [28] does not satisfy the property of hesitancy given by Zhu and Xu [30]. The study of their consistency is an important feature to keep away from the puzzling solution. This paper aims to develop a method with I-HFPRs. A new definition of additive consistency property of HFPR is given that preserves the property of hesitancy and is used to construct the complete HFPR from incomplete one. The significance of consistency measure for HFPR make sure that the DMs are neither arbitrary nor unreasonable. We develop a method to check consistency level of I-HFPR. A numerical example is given to illustrate the developed procedure. Group decision-making problem with I-HFPR is also considered.

Addressing consistency and consensus with group decision support model based on incomplete hesitant fuzzy linguistic preference relations

Addressing consistency and consensus with group decision support model based on incomplete hesitant fuzzy linguistic preference relations

Handling group decision-making model with incomplete hesitant fuzzy preference relations and its application in medical decision

On account of uncertainty and complexity of environments, it is more suitable to express their assessed value by means of hesitant fuzzy information for decision makers. In this paper, we establish a new group decision-making (GDM) model with incomplete hesitant fuzzy preference relations (HFPRs) based on mathematical programming approach. Firstly, based on the multiplicative consistency of incomplete HFPR, a mathematical programming model is established to obtain multiplicative consistent fuzzy preference relation (FPR) from a given incomplete HFPR. Following this, experts are assigned with weights according to their consistency degree. Subsequently, a group consensus reaching process algorithm is constructed based on the obtained multiplicative consistent FPRs. Correspondingly, a GDM model is further established. Finally, a medical decision application is studied to present the practicability and effectiveness of the proposed method.

Additive consistency analysis and improvement for hesitant fuzzy preference relations

Hesitant fuzzy preference relation (HFPR) is an effective tool to elicit decision makers’ hesitant preference information over alternatives, and consistency analysis is of great importance for an HFPR since inconsistent judgments may result in unreasonable results. In this paper, the best additive consistency index, the worst additive consistency index and the average additive consistency index are defined to measure the consistency level of an HFPR. To improve the additive consistency of an HFPR, some mixed 0–1 linear programming models which aim to minimize the overall adjustment amount and the number of the elements that need to be adjusted are established. Moreover, the proposed models are extended to improve the additive consistency and impute the missing elements for incomplete HFPRs. Some numerical examples are presented to show the characteristics of the proposed models. The results demonstrate that the proposed models can improve the consistency of an HFPR effectively.

Missing values estimation and consensus building for incomplete hesitant fuzzy preference relations with multiplicative consistency

This paper proposes a decision support process for incomplete hesitant fuzzy preference relations (HFPRs). First, we present a revised definition of HFPRs, in which the values are not ordered for the hesitant fuzzy element. Second, we propose a method to normalize the HFPRs and estimate the missing elements in incomplete HFPRs based on multiplicative consistency. Based on this, a consensus model with incomplete HFPR is developed. A feedback mechanism is proposed to obtain a best choice with desired consensus level. Multiplicative consistency induced ordered weighted averaging (MC-IOWA) operator is used to aggregate the individual HFPRs into a collective one. A score HFPR is proposed for collective HFPR, and then the hesitant quantifier-guided non-dominance degrees (HQGNDD) of alternatives by using an OWA operator are obtained to rank the alternatives. Finally, a case study for evaluate the qualification of supply chain enterprises is provided to illustrate its application.

On priority weights and consistency for incomplete hesitant fuzzy preference relations

The hesitant fuzzy preference relation (HFPR) is a useful tool for decision makers to elicit their preference information over a set of alternatives. In this paper, it is first proposed an approach to deriving a priority weight vector from an incomplete HFPR using the logarithmic least squares method. Based on the priority weight vector, the consistency index of an incomplete HFPR is defined, which calculates the average deviation between the priority weight vector and all elements of the incomplete HFPR. For an incomplete HFPR which is unacceptably consistent, an automatic algorithm is developed to improve the consistency. These results are then extended to propose a new procedure for group analytic hierarchy process to deal with multi-criteria group decision making problems. The feasibility and effectiveness of the proposed approaches are demonstrated by some numerical examples.

A Natural Method for Ranking Objects from Hesitant Fuzzy Preference Relations

Hesitant fuzzy preference relations (HFPRs) are efficient tools to denoting the decision maker’s judgements that permit the decision makers to compare objects using several values in [0, 1], and the number of elements in different hesitant fuzzy elements may be different. After reviewing the previous researches about decision making with HFPRs, one can find that there are several limitations. To avoid these issues and to guarantee the reasonable ranking order, this paper introduces a new additive consistency concept for HFPRs. Different from the previous consistency concepts, the new concept neither needs to add values into hesitant fuzzy elements nor disregards any information offered by the decision makers. To measure the additive consistency of HFPRs, two 0-1 mixed programming models are constructed. Meanwhile, an additive consistency based 0-1 mixed programming model is established to determining the missing values in incomplete HFPRs that can address the situation where ignored objects exist. Then, an algorithm to obtaining the hesitant fuzzy priority weight vector from (incomplete) HFPRs is provided. Considering group decision making, a new group consensus index is defined, and an interactive approach to improving the group consensus level of individual HFPRs is offered. Furthermore, a probability distance measure between two HFPRs is defined to deriving the weights of the decision makers. According to the additive consistency and consensus analysis, an approach to group decision making with incomplete and inconsistent HFPRs is performed. Finally, two practical numerical examples are provided, and comparison analysis is offered.

A new approach for group decision making method with hesitant fuzzy preference relations

This paper focuses on group decision making with hesitant fuzzy preference relations (HFPRs). To derive the consistent ranking order, a new multiplicative consistency concept for HFPRs is introduced that considers all information offered by the decision makers. The main feature is that this concept neither adds values to hesitant fuzzy elements nor disregards any information provided by the decision makers. To judge the multiplicative consistency of HFPRs, 0–1 mixed programming models are constructed. According to the assumption that there is an independent uniform distribution on values in hesitant fuzzy elements, the hesitant fuzzy priority weight vector is derived from multiplicative consistent reciprocal preference relations and their probabilities. Meanwhile, several consistency based 0–1 mixed models to estimate missing values in incomplete HFPRs are constructed that can address the situation where ignored objects exist. Considering the consensus in group decision making, a distance measure based consensus index is defined, and a method for improving the group consensus is provided to address the situation where the consensus requirement is unsatisfied. Then, a distance measure between any two HFPRs is introduced that is used to define the weights of the decision makers. Furthermore, a multiplicative consistency and consensus based interactive algorithm for group decision making with HFPRs is developed. Finally, a multi-criteria group decision making problem with HFPRs is offered to show the concrete application of the procedure, and comparison analysis is also made.

Incomplete Hesitant Fuzzy Preference Relations in Group Decision Making

In this article, incomplete hesitant fuzzy preference relations are under consideration. In order to estimate expressible missing preferences, a hesitant upper bound condition (hubc) is defined for decision makers presenting incomplete information. With the help of this condition, the estimated preference intensities lie inside the defined domain and thus are expressible. An algorithm is proposed to revise minimal possible preferences so that the resultant satisfies property (hubc). Moreover, ranking rule, HF-Borda count, for hesitant fuzzy preference relations is defined. This method dissolves possible ties among alternatives.

Deriving the priority weights from incomplete hesitant fuzzy preference relations based on multiplicative consistency

In this paper, we define the concept of incomplete hesitant fuzzy preference relations to deal with the cases where the decision makers express their judgments by using hesitant fuzzy preference relations with incomplete information, and investigate the consistency of the incomplete hesitant fuzzy preference relations and obtain the reliable priority weights. We first establish a goal programming model for deriving the priority weights from incomplete hesitant fuzzy preference relations based the α -normalization. Then, we give the definition of multiplicative consistent incomplete hesitant fuzzy preference relations based on the β-normalization, and develop a method for complementing the acceptable incomplete hesitant fuzzy preference relations by using the multiplicative consistency property. Furthermore, utilizing a convex combination method, a new algorithm for obtaining the priority weights from complete or incomplete hesitant fuzzy preference relations is presented on the basis of the β-normalization. Finally, several numerical examples are provided to illustrate the validity and practicality of the proposed methods.

Deriving the priority weights from incomplete hesitant fuzzy preference relations in group decision making

The concept of hesitant fuzzy preference relation (HFPR) has been recently introduced to allow the decision makers (DMs) to provide several possible preference values over two alternatives. This paper introduces a new type of fuzzy preference structure, called incomplete HFPRs, to describe hesitant and incomplete evaluation information in the group decision making (GDM) process. Furthermore, we define the concept of multiplicative consistency incomplete HFPR and additive consistency incomplete HFPR, and then propose two goal programming models to derive the priority weights from an incomplete HFPR based on multiplicative consistency and additive consistency respectively. These two goal programming models are also extended to obtain the collective priority vector of several incomplete HFPRs. Finally, a numerical example and a practical application in strategy initiatives are provided to illustrate the validity and applicability of the proposed models.

Fusion of heterogeneous incomplete hesitant preference relations in group decision making

AbstractIn this paper, we focus on the fusion of heterogeneous incomplete hesitant preference relations (including hesitant fuzzy preference relations and hesitant multiplicative preference relations) under group decision making settings. First, some simple formulae are developed to derive a priority weight vector from an incomplete hesitant fuzzy preference relation or an incomplete hesitant multiplicative preference relation based on the logarithmic least squares method. Based on the priority weight vector, an induced fuzzy or multiplicative preference relation can be derived for an incomplete hesitant preference relation. Moreover, the consistency indices of hesitant fuzzy preference relations and hesitant multiplicative preference relations are defined. Afterwards, an approach to group decision making based on incomplete hesitant fuzzy preference relations and incomplete hesitant multiplicative preference relations is developed to deal with group decision making problems with multiple decision organiz...

Multi-criteria group decision making with incomplete hesitant fuzzy preference relations

In order to simulate the hesitancy and uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of hesitant fuzzy preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. This paper defines the concept of additive consistent hesitant fuzzy preference relations. The characterizations of additive consistent hesitant fuzzy preference relations are studied in detail. Owing to the limitations of the experts’ professional knowledge and experience, the provided preferences in a hesitant fuzzy preference relation are usually incomplete. Consequently, this paper introduces the concepts of incomplete hesitant fuzzy preference relation, acceptable incomplete hesitant fuzzy preference relation, and additive consistent incomplete hesitant fuzzy preference relation. Then, two estimation procedures are developed to estimate the missing information in an expert's incomplete hesitant fuzzy preference relation. The first procedure is used to construct an additive consistent hesitant fuzzy preference relation from the lowest possible number, (n−1), of pairwise comparisons. The second one is designed for the estimation of missing elements of the acceptable incomplete hesitant fuzzy preference relations with more known judgments. Moreover, an algorithm is given to solve the multi-criteria group decision making problem with incomplete hesitant fuzzy preference relations. Finally, a numerical example is provided to illustrate the solution processes of the developed algorithm and to verify its effectiveness and practicality.