A Soft Constraint Optimization Approach for Clustering in Large-Scale Group Decision Making Based on Probabilistic Hesitant Fuzzy Information

The paper aims to present an integrated approach to solve the decision-making problem under the probabilistic hesitant fuzzy information (PHFI) features, which is an extension of the hesitant fuzzy set. The considered PHFI not only allows multiple opinions, but also associates occurrence probability to each opinion, which increases the reliability of the information. Motivated by these features of PHFI, an approach is presented to solve the decision problem with partial known information about the attribute and expert weights. In addition, an algorithm for finding some missing values in the preference information is presented and stated their properties. Afterward, the Hamy mean operator has been used to aggregate the different collective information into a single one. Also, we presented a COPRAS method to the PHFI for ranking the given alternatives. The presented algorithm has been demonstrated through a case study of cloud vendor selection and its validity has been revealed by comparing the approach results with the several existing algorithm results.

Some uncertainty measures for probabilistic hesitant fuzzy information

TODIM is a well-known multiple-criteria decision-making (MCDM) which considers the bounded rationality of decision makers (DMs) based on prospect theory (PT). However, in the classical TODIM, the perceived probability weighting function and the difference of the risk attitudes for gains and losses are not consistent with the original idea of PT. Moreover, probabilistic hesitant fuzzy information shows its superiority in handling the situation that the DMs hesitate among several possible values with different possibilities. Hence, a novel TODIM with probabilistic hesitant fuzzy information is proposed in this paper to simulate the perceptions of the DMs in PT. To show the advantages of the proposed method, a novel TODIM is combined with hesitant fuzzy information. Finally, a case study is carried out to demonstrate the feasibility of the proposed method, and a series of comparative analyses and the sensitivity analyses are used to show the stability of the proposed method.

In the realm of strategic decision making, particularly within interactive game-based decision contexts, the rational evaluation and prioritization of expected payoffs pose significant challenges. Decision making processes are often confronted with dual uncertainties, stemming from individual cognitive hesitancy and the inherent fuzziness of strategic options. To address these complexities, probabilistic hesitant fuzzy elements (PHFEs) serve as an effective mathematical framework. Furthermore, a comprehensive entropy measure for probabilistic hesitant fuzzy information is developed and successfully applied to game decision-making processes. Firstly, a new entropy measure method for PHFE is proposed in terms of dual-dimensional characterization of fuzziness and non-specificity. Then, a novel score function for PHFE is introduced by utilizing the proposed entropy foundation, and integrated it into a fuzzy bi-matrix game model. The model can better solve complex game decision making problems, such as ecological management. The framework demonstrates the superiority of strategy selection in complex fuzziness information.

This study presents a multi-criteria decision-making method that considers expected multiplicative consistency and a consensus reaching process with probabilistic hesitant fuzzy information. The concept of expected multiplicative consistent probabilistic hesitant fuzzy preference relation (PHFPR) is defined on the basis of multiplicative transitivity, and a theorem is developed to obtain the score values of complete expected multiplicative consistent PHFPR. Subsequently, a consistency index of individual PHFPR is proposed by using the distance between individual PHFPR and the score values of its complete expected multiplicative consistent PHFPR. When the individual PHFPR consistency level does not meet the expected value, an iteration algorithm is designed to improve its consistency level and obtain an acceptable one. Furthermore, a group consensus index is proposed according to the distance between the individual acceptable multiplicative consistent PHFPR and the score values of collective PHFPR. An iteration algorithm is designed to improve the consensus level when it does not reach the threshold value. Several numerical examples are provided to demonstrate the effectiveness of the proposed method, and a comparative study involving other methods is conducted with the same numerical examples.

Correlation coefficient is one of the broadly use indexes in multi-criteria decision-making (MCDM) processes. However, some important issues related to correlation coefficient utilization within probabilistic hesitant fuzzy environments remain to be addressed. The purpose of this study is introduced a MCDM method based on correlation coefficients utilize probabilistic hesitant fuzzy information. First, the covariance and correlation coefficient between two PHFEs is introduced, the properties of the proposed covariance and correlation coefficient are discussed. In addition, the northwest corner rule to obtain the expected mean related to the multiply of two PHFEs is introduced. Second, the weighted correlation coefficient is proposed to make the proposed MCDM method more applicable. And the properties of the proposed weighted correlation coefficient are also discussed. Finally, an illustrative example is demonstrated the practicality and effectiveness of the proposed method. An illustrative example is presented to demonstrate the correlation coefficient propose in this paper lies in the interval [−1, 1], which not only consider the strength of relationship between the PHFEs but also whether the PHFEs are positively or negatively related. The advantage of this method is it can avoid the inconsistency of the decision-making result due to the loss of information.

Probabilistic hesitant fuzzy sets (PHFSs) are currently attracting the attention of numerous scholars due to their efficiency in representing uncertain and fuzzy information. PHFSs are more convenient than hesitant fuzzy sets for decision makers to provide their preference information. However, several important issues in PHFSs utilization remain to be addressed. The shortcomings of the operations in PHFSs are discussed in this paper. Probabilistic hesitant fuzzy prioritized weighted average (PHFPWA) and probabilistic hesitant fuzzy prioritized weighted geometric (PHFPWG) operators are developed on the basis of idea of prioritized aggregation operators, and their properties are described. The relationship between PHFPWA and PHFPWG operators is investigated. A multi-attribute decision-making method based on the proposed operators is presented. Two practical examples are provided to illustrate the practicality and effectiveness of the proposed method. Comparison analysis and discussion with other aggregation operators based on the same examples are provided.

The Information Principle

This paper investigates multiple attribute group decision making (MAGDM) with hesitant fuzzy preference information, which is a significantly import issue to be deeply studied in management and industrial engineering. Firstly, simultaneously considering optimistic and pessimistic attitudinal preference information, an improved distance measure for hesitant fuzzy set is defined. Then, utilizing the newly defined distance measure, a hesitant fuzzy multiple attribute group decision making approach based on TOPSIS method is constructed, which can effectively avoid high complexity of aggregating hesitant fuzzy information in traditional methods. Further, an application study on parts supplier selection has verified the practically and effectiveness of developed methods.

The conventional wisdom is that the concept of information is closely related to the concept of probability. In Shannon's information theory, information is equated to a reduction in entropy — a probabilistic concept. In this paper, a different view of information is put on the table. Information is equated to restriction. More concretely, a restriction is a limitation on the values which a variable can take. The concept of a restriction is more general than the concept of a constraint and the concept of a probability distribution. There are three principal kinds of restrictions: possibilistic, probabilistic and bimodal. A bimodal restriction is a combination of possibilistic and probabilistic restrictions. Underlying the restriction-centered approach to information is what may be called the Information Principle. Briefly stated, the Information Principle has two parts. (a) There are three principal types of information: possibilistic information, probabilistic information and bimodal information. Bimodal information is a combination of possibilistic information and probabilistic information. (b) Possibilistic information and probabilistic information are underivable (orthogonal), in the sense that neither is derivable from the other. Information is all around us. And yet, there is widespread unawareness of the existence of the Information Principle. In particular, what is not recognized is that possibilistic information and probabilistic information are underivable (orthogonal). An important empirical observation is that propositions in a natural language are carriers of predominantly fuzzy possibilistic and fuzzy bimodal information. Existing systems of reasoning and computation — other than fuzzy logic — do not have the capability to reason and compute with fuzzy bimodal information.

A large-scale consensus decision-making model for non-cooperative behavior based on incomplete probabilistic hesitant fuzzy information in social trust networks

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.

With the increasing application of (probabilistic) hesitant fuzzy sets in decision-making, the existing integrated methods of hesitant fuzzy information have become too complicated to meet the needs of increasingly complex practical decision-making problems. Therefore, this paper combines the related knowledge of probability theory to firstly introduce the concept of continuous hesitant fuzzy element (C-HFE). Subsequently, the concept of uniform hesitant fuzzy element (U-HFE) is proposed, and discrete (probabilistic) hesitant fuzzy information is transferred to continuous one, benefited from the connection between U-HFEs and C-HFEs with uniform distribution. After then, integration methods of C-HFEs based on mathematical derivation are developed, which lays a theoretical foundation for the continuity of hesitant fuzzy information. Further, facing the problem that the method of mathematical derivation is too tedious, based on computer simulation, this paper proposes another integration method of C-HFEs, which is more concise and easier to apply. Finally, an example of the evaluation of water resources emergency management plans is given to apply the above method to practical decision-making problems.

Multi-criteria decision making with hesitant fuzzy information is a new research topic since the hesitant fuzzy set was firstly proposed. This paper investigates a multi-criteria decision making problem where the weight information is partially known. We firstly propose the hesitant fuzzy positive ideal solution and the hesitant fuzzy negative ideal solution. Motivated by the TOPSIS (Technique for Order Preference by Similarity to an ideal Solution) method, we definite the satisfaction degree of an alternative, based on which several optimization models are derived to determinate the weights. Subsequently, in order to make a more reasonable decision, we introduce an interactive method based on some optimization models for multi-criteria decision making problems with hesitant fuzzy information. Finally, a practical example on evaluating the service quality of airlines is provided to illustrate our models and method.

Intrusion Detection System (IDS) plays an important role in computer network defence mechanism against malicious objects. The ability of IDS to detect new sophisticated attacks compared to traditional method such as firewall is important to secure the network. Machine Learning algorithm such as unsupervised learning and supervised learning is capable to solve the problem of classification in IDS. To achieve that, KDD Cup 99 dataset is used in experiments. This dataset contains 5 million instances with 5 different categories which are Normal, DOS, U2R, R2L and Probe. With such a large dataset, the learning process consumes a lot of processing times and resources. Clustering is unsupervised learning method that can be used for organizing data by grouping similar features into same group. In literature, many researchers used global clustering approach whereby all input will be combined and clustered to construct a codebook. However, there is an alternative technique namely local clustering approach whereby the input will be split into 5 different categories and clustered independently to construct 5 different codebooks. The main objective of this research is to compare the classification performance between the global and local clustering approaches. For this purpose, the soft assignment approach is used for indexing on KDD input and SVM for classification. In the soft assignment approach, the smallest distance values are used for attack description and RBF kernel for SVM to classify attack. The results show that the global clustering approach outperforms the local clustering approach for binary classification. It gives 83.0% of the KDD Cup 99 dataset. However, the local clustering approach outperforms the global clustering approach on multi-class classification problem. It gives 60.6% of the KDD Cup 99 dataset.