Probabilistic Hesitant Fuzzy Element Research Articles

Probabilistic hesitant fuzzy multi-attribute decision-making method based on improved distance measurement

In order to solve the problem of multi-attribute decision-making with unknown weights under probabilistic hesitant fuzzy information, considering the shortcomings of the existing probabilistic hesitant fuzzy distance measure, such as weak distinguishing ability, a probabilistic hesitant fuzzy multi-attribute decision-making method based on improved distance measures is proposed. Firstly, the hesitancy degree of probabilistic hesitant fuzzy element and the improved difference measure of probabilistic hesitant fuzzy element are defined, and an improved probabilistic hesitant fuzzy distance measure based on hesitancy degree, incompleteness degree and improved difference measure is proposed. Secondly, based on the improved distance measure, a mathematical programming model with the goal of minimizing the relative approach degree is con-structed to determine the attribute weights of evaluation indexes in multi-attribute decision making problems. Using it as a base, a new probabilistic hesitant fuzzy multi-attribute decision-making method is proposed by combining the improved probabilistic hesitant fuzzy distance measure with the compromise ratio method. Finally, the proposed method is applied to the problem of green supplier selection, and the feasibility and effectiveness of the proposed method are verified by case analysis and comparison with other methods.

Occurrence probability derivation considering different behavior strategies and decision making under the probabilistic hesitant fuzzy environment

Compared with the general fuzzy decision making, the calculation process under the probabilistic hesitant fuzzy environment involves two new issues that need further consideration, namely quantitative element probability and qualitative element preference. The contributions of this study are to address them in the same modeling framework. For the quantitative element probability, also element occurrence probability, we propose an optimization method by developing the probabilistic hesitant fuzzy envelopment rate and introducing six behavior strategies. This method deals with two complex scenarios, namely, the occurrence probabilities are completely unknown or partially missing. For the qualitative element preference given by decision makers, we design a fusion technique and propose the probabilistic hesitant fuzzy element preference fusion (PHFEF) model based on the above optimization method. Thus, the PHFEF model is used to address the above issues in the same modeling framework. To apply the above models, we further provide a step-by-step decision-making process to reasonably use the PHFEF model in the probabilistic hesitant fuzzy environment. Finally, an illustrative example is used to show the feasibility and further comparisons are given to present the rationality of the proposed methods.

A novel probabilistic hesitant fuzzy rough set based multi-criteria decision-making method

In this study, the fusion of probabilistic hesitant fuzzy sets (PHFSs) and rough sets is explored and applied to uncertain multi-criteria decision-making (MCDM). First, to reduce the huge amount of computation of probabilistic hesitant fuzzy operations in the MCDM process, we introduce an advanced method to obtain normalized PHFSs (NPHFSs), in which each probabilistic hesitant fuzzy element (PHFE) has the same length and probability distribution. Second, based on NPHFSs, a novel probabilistic hesitant fuzzy rough set (PHFRS) model is established to solve the information aggregation problem in MCDM. Subsequently, we propose the concept of fuzziness of a PHFRS and develop a fuzziness-based objective weight determination method. Further, we construct an extended PROMETHEE II method based on a PHFRS for MCDM problems. Finally, we analyze the effectiveness of our proposed method through two MCDM problems along with experimental comparisons. The results illustrate that the proposed method is useful in numerous uncertain application fields.

Probabilistic Hesitant Fuzzy Recognition Method Based on Comprehensive Characteristic Distance Measure

The probabilistic hesitant fuzzy set (PHFS) and probabilistic hesitant fuzzy element (PHFE) have drawn the attention of scholars in recent years and have been applied in several disciplines. However, existing PHFE distance measures have several shortcomings. Therefore, in this study, we propose a new PHFE multi-attribute decision-making (MADM) method, based on the comprehensive characteristic distance measure. First, we devise a new PHFE comparison method and then define the comprehensive characteristic distance measure, based on four characteristics. Finally, based on the traditional TODIM method and prospect theory, we propose a new PHFE recognition method. The comprehensive characteristic distance measure avoids the introduction of errors, including an unequal number of elements and order adjustment. Meanwhile, the four characteristics make the measurement results more comprehensive and reasonable, and applicable to a variety of situations while avoiding counterintuitive phenomena. Compared with traditional approaches, the method in this article selects appropriate parameters according to actual situations to obtain more objective conclusions, which results in better flexibility and operability. Besides, the simulation results verify the effectiveness of this recognition method.

A probabilistic hesitant fuzzy Choquet integral-based TODIM method for multi-attribute group decision-making

A probabilistic hesitant fuzzy set (PHFS) is a generalization of the hesitant fuzzy set (HFS) that adds the probability of occurrence to each possible value of the hesitant fuzzy element (HFE). It is a powerful tool for handling uncertain information, especially when there is no complete consensus or full dissensus among decision-makers. The TODIM (an acronym in Portuguese for interactive and multiple attribute decision-making) approach is a multi-attribute decision-making (MADM) technique based on prospect theory. In this paper, we developed a Choquet integral-based TODIM method under a probabilistic hesitant fuzzy environment. Firstly, we introduced both a geometric score function and a geometric variance function of the probabilistic hesitant fuzzy element (PHFE). Furthermore, we proposed a novel comparison law between two PHFEs, and we defined a new distance measure of PHFEs. Considering the interaction between the criteria, we calculated the degree of importance of the criteria with the Shapley value. Next, we introduced the principles and steps of the developed TODIM method. One of the prominent components of the TODIM method is its risk aversion coefficient. However, it has not been sufficiently addressed in the research literature. In this regard, a novel optimization-based method is proposed to calculate the attenuation factor of the TODIM method. Finally, we verify the effectiveness of the Choquet integral-based probabilistic hesitant fuzzy-TODIM (PHF-TODIM) method using a supplier selection problem in the dairy industry.

An approach to probabilistic hesitant fuzzy risky multiattribute decision making with unknown probability information

As a useful tool, probabilistic hesitant fuzzy set is an enhanced version for hesitant fuzzy set. It could be used to model the uncertainty very effectively. However, in probabilistic hesitant fuzzy risky multiple attribute decision making problems, the occurrence probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status are often difficult to obtain by subjective evaluation of a decision maker. This paper aims to propose two nonlinear programming models for calculating the probabilities of elements in a probabilistic hesitant fuzzy element and the probability of risk status respectively. First, a nonlinear programming model using maximum entropy principle is established for determining the probabilities of elements in a probabilistic hesitant fuzzy element. Second, by introducing the water-filling theory, we put forward its extension and design a novel mathematical programming model to determine the probability of risk status. Moreover, we have proved that both the two mathematical programming models are convex programming models and their global optimal solutions can be found. Thirdly, the collective overall expected values of alternatives are calculated and the ranking order can be derived. Then, the selection of investment project is investigated, and comparison analysis shows the superiority of the presented approach.

Probabilistic hesitant fuzzy TOPSIS emergency decision-making method based on the cumulative prospect theory

In order to fully consider the decision-maker’s limited rationality and attitude to risk, this paper constructs the probabilistic hesitant fuzzy TOPSIS emergency decision-making model based on the cumulative prospect theory under the probabilistic hesitant fuzzy environment. Aiming at the problem of missing probabilistic information in the probabilistic hesitant fuzzy element, a new complement scheme is proposed. In this scheme, the weighted average result of the original data information is used to complement, and the original data information is retained to a large extent. Then this paper proposes several probabilistic hesitant fuzzy distance measures based on Lance distance. The decision reference point is constructed by the probabilistic hesitant fuzzy Lance distance, which overcomes the influence of the extreme value on the decision-making result, and defines the value function based on the probabilistic hesitant fuzzy Lance distance. In view of the fact that the attribute weights are completely unknown, the probabilistic hesitant fuzzy exponential entropy is constructed by using the actual data, and the attribute weights of different prospect states are obtained. Aiming at the problem that attribute weights of different prospect states have different effects on the cumulative prospect value, the expression of the cumulative prospect value is improved. The improved closeness coefficient of the TOPSIS method is used to order the emergency schemes. Finally, the new method is applied to the emergency decision-making case of a sudden outbreak of epidemic respiratory disease. The results show that the contrast of the new method is obvious, which is conducive to distinguish different schemes. The new method is more suitable for the complex and changeable emergency decision-making field.

Developing the comparison techniques of probabilistic hesitant fuzzy elements in multiple criteria decision making

All the existing probabilistic hesitant fuzzy element (P-HFE) comparison techniques are directly based on the possible membership degree of an element together with its probability of occurring. Here, we are going to propose a class of P-HFE comparison techniques which are indirectly based on the ingredients of a P-HFE. Indeed, the proposed P-HFE comparison techniques are mainly related to the multiplying and exponential deformation formulas of each pair of possible membership degree and its associated probability. The proposed P-HFE comparison techniques are classified into three categories: (i) the element-based processes for comparing P-HFEs, (ii) the step-based processes for comparing P-HFEs, and (iii) the step-based processes for comparing P-HFEs. These indirect P-HFE comparison techniques provide us with more insights into the ways to increase the applicability of the P-HFE comparison algorithms, and enable us to deal with the comparison step in multiple criteria decision making problems from a different viewpoint.

Extended TODIM method for CCUS storage site selection under probabilistic hesitant fuzzy environment

Carbon capture, utilization and storage (CCUS) technologies are effective for urgently dealing with climate change and reducing carbon dioxide (CO2). The storage of CO2 in deep strata often leads to CO2 leakage due to geological and engineering reasons, which has a huge impact on humans and ecology. CO2 storage site selection can be regarded as a multi-criteria decision-making (MCDM) problem. Decision makers do not always show completely rational and may have the preference of bounded rational behavior, which may affect the selection of CO2 storage site. At the same time, criteria interaction is an interesting issue in multi-criteria decision-making. In this paper, we develop an extended novel TODIM method based on λ-fuzzy measure and Choquet integral to select the CO2 storage site using evaluation information given by decision makers which can take the form of a probabilistic hesitant fuzzy set, whereby λ-fuzzy measure and Choquet integral are used to calculate the weights of criteria. The hamming distance measure between two probabilistic hesitant fuzzy element (P-HFE) is calculated and the gain and loss matrices for every criterion are obtained. Further, the overall values of all alternatives can be calculated to get the ranking order of CO2 storage site. Decision makers can select a suitable CO2 storage site according to ranking results. Finally, a CO2 storage site selection example is used to describe the effectiveness of the proposed procedures. The sensitivity analysis also explores the influence of the loss aversion coefficient and the change of criteria weights on decision results.

Recommendation decision-making algorithm for sharing accommodation using probabilistic hesitant fuzzy sets and bipartite network projection

In recent years, with the uninterrupted development of sharing accommodation, it not only caters to the diversified accommodation of tourists, but also takes an active role in expanding employment and entrepreneurship channels, enhancing the income of urban and rural residents, and promoting the revitalization of rural areas. However, with the continuous expansion of the scale of sharing accommodation, it is fairly complicated for users to search appropriate services or information. The decision-making problems become more and more complicated. Hence, a probabilistic hesitant fuzzy recommendation decision-making algorithm based on bipartite network projection is proposed in this paper. First of all, combining the users’ decision-making information and the experts’ evaluation information, a bipartite graph connecting users and alternatives is established. Then, the satisfaction degree of probabilistic hesitant fuzzy element is defined. Besides, the recommended alternative is obtained by the allocation of resources. Finally, a numerical case of Airbnb users is given to illustrate the feasibility and effectiveness of the proposed method.

A consensus-based approach for multi-criteria decision making with probabilistic hesitant fuzzy information

As a generalized fuzzy number, probabilistic hesitant fuzzy element (PHFE) improves the flexibility for decision makers in expressing hesitant information, and it has been receiving increased attention. This study develops a multi-criteria decision-making (MCDM) approach that considers consensus reaching among decision makers with probabilistic hesitant fuzzy information. To obtain this aim, first, a new approach to derive normalized PHFE (NPHFE) is proposed to overcome the shortcomings in previous studies. Subsequently, a new Euclidean distance and some operations related to PHFEs are developed based on the new proposed NPHFEs. At the same time, the effectiveness and rationality of the new proposed approaches are discussed. Second, a consensus index of group with PHFEs is presented, which based on the proposed Euclidean distance of decision-makers’ evaluation information on all the criteria. Third, if the consensus level of the group does not reach the expect threshold value, an iteration algorithm is designed to improve its consensus level. Moreover, the proof of the convergence of the proposed algorithm is provided to verify its effectiveness, and a MCDM approach based on group consensus is proposed. Finally, the most comprehensive candidate selection problems are provided to demonstrate the effectiveness of the proposed MCDM approach. And a comparative study with other methods is conducted with the same illustrative example.

An ORESTE approach for multi-criteria decision-making with probabilistic hesitant fuzzy information

As an important extension of fuzzy number, the probabilistic hesitant fuzzy element (PHFE) shows the flexibility of decision makers in expressing hesitant information in multi-criteria decision-making (MCDM) processes. Accordingly, numerous research findings have been obtained since PHFE introduction. However, a few important issues in PHFE utilization remain to be addressed. This study introduces the French organization Rangement Et Synthese De Ronnees Relationnelles’ (ORESTE) approach for MCDM with probabilistic hesitant fuzzy information. First, the limitations of normalized PHFE (NPHFE), Euclidean distance, and several operations in previous studies are discussed. Subsequently, an algorithm is designed to derive the new NPHFE. A new Euclidean distance and several operations are developed on the basis of the proposed NPHFE. Second, the ORESTE approach is extended to probabilistic hesitant fuzzy environments. Lastly, the problem of selecting best research topic is presented to demonstrate that the proposed approach is effective. A comparative study with other approaches is conducted with identical illustrative example.

Probabilistic hesitant fuzzy multiple attribute decision-making based on regret theory for the evaluation of venture capital projects

The selection of venture capital investment projects is one of the most important decision-making activities for venture capitalists. Due to the complexity of investment market and the limited cognition of people, most of the venture capital investment decision problems are highly uncertain and the venture capitalists are often bounded rational under uncertainty. To address such problems, this article presents an approach based on regret theory to probabilistic hesitant fuzzy multiple attribute decision-making. Firstly, when the information on the occurrence probabilities of all the elements in the probabilistic hesitant fuzzy element (P.H.F.E.) is unknown or partially known, two different mathematical programming models based on water-filling theory and the maximum entropy principle are provided to handle these complex situations. Secondly, to capture the psychological behaviours of venture capitalists, the regret theory is utilised to solve the problem of selection of venture capital investment projects. Finally, comparative analysis with the existing approaches is conducted to demonstrate the feasibility and applicability of the proposed method.

Fuzzy reliability appraisal of a system using probabilistic dual hesitant fuzzy element emphasizing score function

Fuzzy reliability appraisal of a system using probabilistic dual hesitant fuzzy element emphasizing score function

Dynamic Emergency Decision-Making Method With Probabilistic Hesitant Fuzzy Information Based on GM(1,1) and TOPSIS

Decision-makers (DMs) will face severe challenges when selecting an optimal alternative for an emergency response over multiple time periods. The aim of this paper is to develop a novel dynamic emergency decision-making method with probabilistic hesitant fuzzy information for handling emergencies. First, an approach based on the GM(1,1) model for predicting the decision-making information at the next stage is proposed. Second, a new probabilistic hesitant fuzzy distance measure based on the hesitant degree of the probabilistic hesitant fuzzy element is put forward, and a mathematical programming model to determine the stage weights is established. What is more, the closeness degree between each alternative and the ideal alternative is calculated, and the emergency alternatives are ranked on the strength of the technique for order preference by similarity to an ideal solution method. Moreover, a practical example is used to verify the feasibility and rationality of the proposed method.

New Correlation Coefficients Between Probabilistic Hesitant Fuzzy Sets and Their Applications in Cluster Analysis

The hesitant fuzzy set (HFS) is very significant in dealing with the multi-criteria decision-making problems when the decision makers have hesitancy in providing their assessments. However, with the deepening of the research, it may lose information in its applications. Hence, the probabilistic hesitant fuzzy set (P-HFS) has been proposed to improve the HFS, associating the probability with the HFS and remaining more information than the HFS. Considering the correlation coefficient is one of the most important tools in data analysis, we propose two new correlation coefficient formulas to measure the relationship between the P-HFSs, based on which, a new probabilistic hesitant fuzzy clustering algorithm is also developed. To do so, we define the mean of the probabilistic hesitant fuzzy element and the P-HFS, respectively. Furthermore, a practical case study is conducted to demonstrate practicability and validity of the proposed clustering algorithm.

Probability Calculation and Element Optimization of Probabilistic Hesitant Fuzzy Preference Relations Based on Expected Consistency

As a generalized fuzzy number, the hesitant fuzzy element (HFE) has received increasing attention. However, it is noted that the occurring probabilities of the elements in the HFE are equal, which is obviously problematic; thus, the preference relations on the HFEs can be inaccurate. To address this issue, this paper proposes the probabilistic hesitant fuzzy preference relations (PHFPRs) based on the probabilistic HFE (PHFE). It seems difficult to provide accurate probabilities that describe the occurring possibilities of the elements in the PHFPRs. This paper further demonstrates the probability calculation method for the PHFPEs based on a proposed variable, which is called the expected consistency. Moreover, the expected consistency index and the judgment principle are designed to evaluate the degree to which the PHFPRs are consistent. For the inconsistent PHFPRs, this paper presents an iterative optimization algorithm to improve their expected consistency by optimizing some elements in the PHFPRs. When the iteration terminates, the consistent PHFPRs and the priorities of the alternatives are identified. Finally, an example that selects a Ph.D. candidate is presented, and the results demonstrate the feasibility and effectiveness of using the PHFPRs and the consistency methods.

A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making

The probabilistic hesitant fuzzy element is a common tool to deal with multi-criteria decision-making problems when the decision makers are irresolute in providing their evaluations. The existing methods for ranking probabilistic hesitant fuzzy elements are limited and not reasonable in practical applications. The main purpose of this paper is to find a more precise and appropriate comparison method so that we can deal with multi-criteria decision-making problems more efficiently. We first propose a chart technique to analyze the structure of a probabilistic hesitant fuzzy element. After that, we propose a novel possibility degree formula to rank probabilistic hesitant fuzzy elements. Last but not least, we provide a useful process to solve the actual multi-criteria decision-making problems, and make a real case study which demonstrates that our method is feasible and reliable.

Consensus Building for Probabilistic Hesitant Fuzzy Preference Relations with Expected Additive Consistency

As an extension of hesitant fuzzy element, probabilistic hesitant fuzzy element (PHFE) has received increasing attention. However, some important issues in PHFE utilization remain to be addressed. This study aims to build a consensus among the decision makers for probabilistic hesitant fuzzy preference relations (PHFPRs) with expected additive consistency. First, several generalized operations that are suitable for PHFPRs are defined. Then, expected additive consistent PHFPRs are introduced, and a theorem is proposed to obtain them. Second, the consistency index is defined based on the Hausdorff distance of PHFEs to measure whether individual PHFPRs exhibit acceptable expected additive consistency. If not, then an automatic iterative algorithm is designed to obtain acceptable ones. Third, group PHFPR is obtained based on the proposed generalized operations, and then the consensus index is determined according to the Hausdorff distance. If at least one of the consensus levels of the decision makers is lower than a given threshold, then an automatic iterative algorithm is used to update the PHFPRs to reach a predefined consensus level. Finally, a numerical example is provided, and comparative analyses with existing methods are performed to demonstrate the validity of the proposed method in addressing group decision-making problems.

Expected hesitant VaR for tail decision making under probabilistic hesitant fuzzy environment

Generally, to reasonably make decision, all evaluation information should be aggregated, and thus, the ranking and the optimal alternative can be obtained. However, in some extreme cases, the decision maker (DM) can only focus on the tail information such as the big-loss or big-gain values and wants to ask the simple question “How bad can a thing become?” or “How good can a thing become?” To address this type of decision-making issue, this paper introduces the definition of value at risk (VaR), which is a famous term in the financial field, and the probabilistic hesitant fuzzy element (PHFE), which is a general hesitant fuzzy element (HFE) and has recently become a popular topic. Then, the hesitant VaR (HVaR) is defined, and its mathematical presentation is provided to measure the tail information of the PHFEs. It is found that the tail information calculated by the HVaR is segmentary, and only the boundary value is used. Therefore, this paper further develops the expected HVaR (EHVaR) to improve the HVaR, which can describe the entire tail information. Two simple examples are provided to show and compare the proposed HVaR and EHVaR. To apply the EHVaR into a group decision making that focuses on the tail information, this paper proposes a dynamic programming model to calculate the weights of the DMs based on the principle that the more accurate PHFE should be given a bigger weight. Then, the tail group decision making steps based on the EHVaR are presented. Finally, this paper provides an example of selecting the optimal stock for four newly listed stocks in China to demonstrate the effectiveness of the proposed approaches.