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This paper proposes a novel ranking model under probabilistic hesitant fuzzy set (PHFS) by extending the idea of evidence theory (ET). PHFS is a strong variant of hesitant fuzzy set that associates occurrence probabilities to multiple membership grades. This offers flexibility to experts during preference elicitation and aids proper handling of uncertainty. Evidence theory is also a powerful concept for managing uncertainty/hesitation. By integrating Bayesian approximation with ET, a flexible ranking model is developed that complements ET. Due to the partial availability of evidences for decision-making under uncertainty, an approximation strategy is combined. Previous studies on PHFS have not handled hesitation of experts better and to alleviate the issue, a new regret-rejoice approach is put forward that calculates weights of criteria by handling hesitation efficiently. Ranking values from each expert are obtained that are further fused by the Maclaurin operator to get the final ranking order. These approaches are integrated into a framework and its usefulness is exemplified using renewable energy technology selection for Tamil Nadu. Finally, comparative analysis with other approaches reveals the strengths and weaknesses of the proposed work. [ABSTRACT FROM AUTHOR]

Double hierarchy linguistic term set (DHLTS) is a powerful tool when expressing the real thoughts of experts and handling complex linguistic information considering that it divides complex linguistic information into two simple linguistic hierarchies in which the first hierarchy linguistic term set (LTS) is the main linguistic hierarchy and the second hierarchy LTS is the linguistic feature or detailed supplementary of each linguistic term in the first hierarchy LTS. Some extensions of DHLTS have been developed, such as the double hierarchy hesitant fuzzy LTS, the unbalanced DHLTS, the linguistic preference ordering, the double hierarchy linguistic preference relation, and the double hierarchy hesitant fuzzy linguistic preference relation. In recent years, DHLTS and its extensions have been researched by scholars in lots of fields, including the extended concepts, the operational laws, the comparative methods, the measure methods, the consistency and consensus methods, the decision‐making methods, the applications, and so forth. Therefore, the purpose of this paper is to review all the researches of DHLTS and its extensions, as well as proposing some challenges in the future. [ABSTRACT FROM AUTHOR]

Automatic reasoning based on propositional logic is considered as an important tool in machine learning, and intuitionistic fuzzy sets have turned out to deal with vague and uncertain information effectively in real world. In this paper, a novel plausible reasoning based on intuitionistic fuzzy propositional logic is proposed. On the basis of it, the categories of intuitionistic fuzzy logic proposition (IFLP) formula are discussed both considering the true degree and the false degree at the same time. Some basic operational laws and inference rules of IFLPs on the basis of closely-reasoned scientific proofs are put out. Then, we develop two classification methods of IFLPs, i.e., truth table and figure of equivalence, respectively. After that, the reasoning theory of IFLPs is introduced and three reasoning methods are further established including direct proof method, additional premise proof method and reduction to absurdity method, respectively. Finally, a case study about strategy initiatives of HBIS GROUP on Supply-side Structural Reform is presented and some discussions are provided to validate the proposed methods. As a result, the proposed methods based on the plausible reasoning offer sound structure and can improve the efficiency of logic programming. [ABSTRACT FROM AUTHOR]

Tracking a maneuvering target is an important technology. Due to complex environment and diversity of sensors, errors need to be optimized with respect to various motion states during the tracking process. In this paper, we first propose how to unify the coordinate system and data preprocessing in case of tracking using multiple sensors. We then combine fuzzy sets with a novel trace optimization method based on extended Kalman filter (EKF) with nested probabilistic-numerical linguistic information (NPN-EKFTO). We present a case study of trace optimization of an unknown maneuvering target in Sichuan province in China. We solve the case by using both the proposed method and the traditional EKF and offer comparative analysis to validate the proposed approach. [ABSTRACT FROM AUTHOR]

Uncertainty inference, which is used to infer the reasons corresponding to the results, has permeated through various fields, such as medicine, risk assessments, and marine management. Tackling ambiguity under fuzzy environment by using the Bayes and Dempster–Shafer theories has recently received special attention. In actual cases and experiments, we can estimate probabilities of occurrence and non-occurrence of events, which are expressed by intuitionistic fuzzy numbers (IFNs) or interval-valued intuitionistic fuzzy numbers (IVIFNs). To accomplish the multi-source uncertain inference in a fuzzy environment, this paper proposes the standard forms of Bayes formula under the intuitionistic fuzzy environment and the interval-valued intuitionistic fuzzy environment to solve grey models and decision-making problems with incomplete information. Firstly, we give some basic properties of IFN-probability and define its conditional probability based on the basic probability assignment of Dempster–Shafer theory. Subsequently, we also propose the independence of events, multiplication rule for the conditional probability of IFNs, and the law of total probability and the Bayes formula of IFNs. Due to the complexity and uncertainty of objective things, the indeterminacy degrees are often difficult to be expressed with crisp numbers. To address the challenge, interval values are used to describe the indeterminacy degrees instead of crisp numbers. Secondly, we give some basic properties of IVIFN-probability and define its conditional probability based on the IFN-probability theory. Afterward, we further propose the independence of events, multiplication rule for the conditional probability of IVIFNs, and the law of total probability and the Bayes formula of IVIFNs. From the angle of modeling and complexity, our approaches are more convenient and feasible than the interval probability method. Meanwhile, the practical applications show the feasibility of the proposed methods. [ABSTRACT FROM AUTHOR]

Compatibility measure is a very well-known method to deal with decision making problems with uncertain preference relations. In this paper, on the basis of generalized distance measure of hesitant fuzzy linguistic elements, we define the compatibility measures for hesitant fuzzy linguistic preference relations (HFLPRs), including compatibility degree, compatibility index, and acceptable compatibility, and some important special properties are also discussed. Furthermore, based on the compatibility measures of HFLPRs, we propose a method for determining the experts' weights in group decision making (GDM), as well as establish an algorithm for solving the GDM problem with HFLPRs. A case study is set up to utilize the compatibility measures of HFLPRs for dealing with a practical GDM problem, which is to determine which aspect is optimal in the process of medical and health system reform. Finally, some comparisons and analyses are also discussed in detail. [ABSTRACT FROM AUTHOR]

Hesitant fuzzy linguistic term set (HFLTS), as a flexible tool to represent people's uncertain cognition, has attracted lots of scholars' research interests, and a series of methodologies have been proposed to deal with a variety of decision-making problems. In this paper, we develop a hesitant fuzzy linguistic possibility degree-based linear assignment (HFL-PDLA) method to tackle the multiple criteria decision-making (MCDM) problems under hesitant fuzzy linguistic environment. Firstly, we define the possibility degree of hesitant fuzzy linguistic element (HFLE). Additionally, some relevant concepts related to the HFL-PDLA method are proposed, such as the relative difference matrix, the rank contribution matrix, the optimal permutation matrix, etc. Furthermore, the algorithm of the HFL-PDLA method is given to deal with hesitant fuzzy linguistic MCDM problems. Moreover, we apply the HFL-PDLA method to deal with a practical case which is to select the optimal treatment technology for disposing the outspent or old medical apparatuses and instruments in West China Hospital (WCH). Finally, we show the advantages of the HFL-PDLA method by making some comparative analyses with the TOPSIS method, the VIKOR method the PROMETHEE method and the LINMAP method. [ABSTRACT FROM AUTHOR]

In recent years, hesitant fuzzy linguistic term sets (HFLTSs) have been studied by many scholars and are becoming gradually mature. However, some shortcomings of HFLTS also emerged. To describe the complex linguistic terms or linguistic term sets more accurately and reasonably, in this paper, we introduce the novel concepts named double hierarchy linguistic term set (DHLTS) and double hierarchy hesitant fuzzy linguistic term set (DHHFLTS). The operational laws and properties of the DHHFLTSs are developed as well. Afterwards, we investigate the multiple criteria decision making model with double hierarchy hesitant fuzzy linguistic information. We develop a double hierarchy hesitant fuzzy linguistic MULTIMOORA (DHHFL-MULTIMOORA) method to solve it. Furthermore, we apply the DHHFL-MULTIMOORA method to deal with a practical case about selecting the optimal city in China by evaluating the implementation status of haze controlling measures. Some comparisons between the DHHFL-MULTIMOORA method and the hesitant fuzzy linguistic TOPSIS method are provided to show the advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In recent decades, different extensional forms of fuzzy sets have been developed. However, these multitudinous fuzzy sets are unable to deal with quantitative information better. Motivated by fuzzy linguistic approach and hesitant fuzzy sets, the hesitant fuzzy linguistic term set was introduced and it is a more reasonable set to deal with quantitative information. During the process of multiple criteria decision making, it is necessary to propose some aggregation operators to handle hesitant fuzzy linguistic information. In this paper, two aggregation operators for hesitant fuzzy linguistic term sets are introduced, which are the hesitant fuzzy linguistic Bonferroni mean operator and the weighted hesitant fuzzy linguistic Bonferroni mean operator. Correspondingly, several properties of these two aggregation operators are discussed. Finally, a practical case is shown in order to express the application of these two aggregation operators. This case mainly discusses how to choose the best hospital about conducting the whole society resource management research included in a wisdom medical health system. [ABSTRACT FROM AUTHOR]

In practical decision-making processes, we can utilize various types of fuzzy sets to express the uncertain and ambiguous information. However, we may encounter such the situations: the sum of the support (membership) degree and the against (nonmembership) degree to which an alternative satisfies a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. The Pythagorean fuzzy sets (PFS), as the generalization of the fuzzy sets, can be used to effectively deal with this issue. Therefore, to enrich the theory of PFS, it is very necessary to investigate the fundamental properties of Pythagorean fuzzy information. In this paper, we first describe the change values of Pythagorean fuzzy numbers (PFNs), which are the basic components of PFSs, when considering them as variables. Then we divide all the change values into the eight regions by using the basic operations of PFNs. Finally, we develop several Pythagorean fuzzy functions and study their fundamental properties such as continuity, derivability, and differentiability in detail. [ABSTRACT FROM AUTHOR]

Highlights • Some issues of the Double Hierarchy Hesitant Fuzzy Linguistic Term Sets are fixed. • A new mathematical structure to deal with linguistic assessments is defined. • This structure enables the experts to be more precise by choosing their own scale. • A new approach of the TOPSIS methodology based on this structure is presented. • A case study based on tourism management in Barcelona is provided. Graphical abstract Abstract Hesitant fuzzy linguistic term sets have been an active field of research in recent times. Notwithstanding its usefulness to capture the human way of reasoning using linguistic expressions involving different levels of precision, in some situations they do not depict enough details. In this paper, we present a new kind of linguistic term sets, called free double hierarchy linguistic term sets, and their corresponding free double hierarchy hesitant fuzzy linguistic elements, in order to describe the complexity of linguistic expressions used by the decision makers in a more accurate and precise way. Furthermore, an order and a distance between free double hierarchy hesitant fuzzy linguistic elements are introduced to present an approach based on the TOPSIS method to rank alternatives with free double hierarchy hesitant fuzzy linguistic information by taking into consideration the opinions of a group of decision makers. A case study based on tourism management in Barcelona is also provided to illustrate the usefulness of the presented approach. [ABSTRACT FROM AUTHOR]