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The successful diagnosis of nuclear power equipment failures plays a vital role in guaranteeing the safe operation of nuclear power systems. Failure mode and effect analysis (FMEA) is one of the most commonly used methods for identifying potential failures. However, several shortcomings associated with the conventional FMEA method limit its further application. This paper develops an extended FMEA approach based on hesitant fuzzy linguistic Z-numbers (HFLZNs). Firstly, the concept of HFLZNs is proposed to describe the evaluation information, which inherits the prominent features of the hesitant fuzzy linguistic term set and linguistic Z-numbers (LZNs). Secondly, an HFLZN assessment method is developed to determine the weights of risk factors, and the weights of experts are measured based on hesitation degree. Subsequently, considering the psychological characteristics of decision makers, Tomada de Decisão Iterativa Multicritério and LZNs are integrated to obtain the risk ranking of failure modes. Finally, the practicability of the extended FMEA method is proven by an illustrative example concerning the risk evaluation of a nuclear main pump bearing, and its robustness is verified by indepth analysis.

In the era of big data, the research on clustering technologies is a popular topic because they can discover the structure of complex data sets with minimal prior knowledge. Among the existing soft clustering technologies, as an extension of fuzzy c-means (FCM) algorithm, the intuitionistic FCM (IFCM) algorithm has been widely used due to its superiority in reducing the effects of outliers/noise and improving the clustering accuracy. In the existing IFCM algorithm, the measurement of proximity degree between a pair of objects and the determination of parameters are two critical problems, which have considerable effects on the clustering results. Therefore, we propose an improved IFCM clustering technique in this paper. Firstly, a novel weighted proximity measure, which aggregates weighted similarity and correlation measures, is proposed to evaluate not only the closeness degree but also the linear relationship between two objects. Subsequently, genetic algorithms are utilized for identifying the optimal parameters. Lastly, experiments on the proposed IFCM technique are conducted on synthetic and UCI data sets. Comparisons with other approaches in cluster evaluation indexes indicate the effectiveness and superiority of our method.

The probabilistic linguistic term set (PLTS) is a powerful tool for describing linguistic evaluations derived from expert teams and has adequate capability to identify preferences among different evaluations. Due to the practicability of PLTSs, probabilistic linguistic decision making problems have been widely investigated in recent years. However, no study on probabilistic linguistic outranking relations has been conducted. This study aims to explore effective processing for the complex two-dimension structure of PLTSs and formulate probabilistic linguistic dominance and opposition relations for multi-criteria sorting decision making. Linguistic scale functions, which can generate different semantics for linguistic variables under different decision making environments, are introduced to deal with the linguistic terms in PLTSs. In this way, the probabilistic linguistic dominance degree, concordance and discordance indices are defined by systematically comparing the probabilities of PLTSs. Then, two kinds of outranking relations with dominance and opposition for PLTSs are formulated based on the defined outranking indices. Subsequently, an innovative sorting decision making framework is constructed by exploring the outranking relations between alternatives and characteristic actions under multiple criteria and implementing the outranking aggregation and exploitation. Finally, this framework is demonstrated using an illustrative example with result analyses and comparison discussions.

In this article, used Dombi t-norm (TN) and t-conorm (TCN) which can generate more complex and more flexible operation rules by adjusting a parameter to combine with prioritized aggregation operators (AOs) in bipolar fuzzy (BF) environment. In this study, introduced bipolar fuzzy Dombi prioritized AOs, namely bipolar fuzzy Dombi prioritized averaging operator, bipolar fuzzy Dombi geometric operator, bipolar fuzzy Dombi prioritized weighted averaging operator and bipolar fuzzy Dombi prioritized weighted geometric operator as these operators along with proofs to aggregate various preferences of the decision makers. In this purpose, we designed a multiple attribute decision-making technique for the proposed study. Finally, an illustrative example is given to demonstrate proposed approach under BF environment and a sensitivity analysis is considered for the working parameter on the ordering of the alternatives. A comparative study is provided for the choice best decision of the proposed approach with the existing problems. Finally, it is concluded that the proposed approach gives a more practical nature to aggregate the information process during the data analysis, and hence they take an alternative way for solving decision-making problems.

Under the double pressure of global energy consumption and climate change, nuclear power has become a low-carbon alternative energy source that could transform the energy structure of the globe. In the nuclear power industry, selecting suitable suppliers plays a significant role in improving the overall performance of nuclear power projects. Along with this symmetrical impact, this paper aims to develop a multistage decision-support framework to determine the optimal nuclear power equipment supplier, which is constructed in the context of Z-number information. Concretely, the Analytic Network Process (ANP) and Tomada de Decis&atilde, o Iterativa Multicrit&eacute, rio (TODIM) are extended by Z-numbers symmetrically&mdash, namely, Z-ANP and Z-TODIM. Z-ANP is first applied to analyze the symmetrical interdependence of criteria, so as to accurately determine the criterion weights. Further, the ranking of alternatives is obtained by Z-TODIM, which sufficiently considers the risk preference and psychological states of decision-makers. Finally, a practical case of nuclear-grade cable procurement in the Karachi 2-3 international nuclear power project is performed to illustrate the practicality of the proposed method, and its robustness and superiority are proven by comparing it with current representative approaches.

With the development of the urbanization process, the demand for water resources has increased significantly, but the pollution of water resources has caused serious problems. These changes pose a potential threat to water resource carrying capacity in many regions. However, how to determine the areas of highest risk in water resource carrying capacity is an urgent problem which remains to be solved. Resounding to these circumstances, this study establishes a TODIM-PROMETHEE Ⅱ (An acronym in Portuguese for interactive and multiple attribute decision making- preference ranking organization method for enrichment evaluation Ⅱ) based decision support framework to address this issue for the regions of intensive governance, thereby providing support. In this framework, a novel theoretical concept, namely probabilistic linguistic Z-numbers, is proposed to describe group decision information. The related knowledge of probabilistic linguistic Z-numbers is developed, including a comparison method, distance, and operational rules. Subsequently, a case study involving the evaluation of water resource carrying capacity is conducted to demonstrate the feasibility of the decision support model, followed by sensitivity analysis, comparison analysis, and discussion. The findings demonstrate that the constructed framework demonstrates great performance to address this issue.

This paper develops a novel multi-criteria group decision-making method for resolving single-valued neutrosophic group decision-making problems with unknown weight information. Firstly, considering the different knowledge backgrounds of decision-makers (DMs), two distance-based methods are developed to obtain the weights of DMs and criteria. One of these methods focuses on the closeness between the decision matrix given by an individual DM and that of the group judgment, and the proximity between the decision matrix given by an individual DM and the matrices given by all the other DMs to determine the weights of DMs with respect to each criterion. The other method extends the maximizing deviation method to objectively obtain the weights of criteria. Secondly, an extended ELECTRE (ELimination Et Choix Traduisant la REalite) II is investigated to effectively circumvent the issue that criteria cannot compensate for each other. In this method, three levels of concordance and discordance sets are constructed to fully denote the underlying preference information and specify the outranking degrees of alternatives. Finally, two illustrative examples are presented to demonstrate the feasibility of the proposed approach, followed by a sensitivity analysis and a comparative analysis to test its robustness and effectiveness.

A multi-criteria decision-making (MCDM) method with single-valued neutrosophic information is developed based on the Partitioned Heronian Mean (PHM) operator and the Shapley fuzzy measure, which recognizes correlation among the selection criteria. Motivated by the PHM operator and Shapley fuzzy measure, two new aggregation operators, namely the single-valued neutrosophic PHM operator and the weighted single-valued neutrosophic Shapley PHM operator, are defined, and their corresponding properties and some special cases are investigated. An MCDM model is applied to solve the single-valued neutrosophic problem where weight information is not completely known. An example is provided to validate the proposed method.

People make decisions based on their cognitive information about the objective world. Zadeh’s Z-number allows people to better express their cognition of the real world by considering the fuzzy restriction and reliability restriction of information. However, the Z-number is a complex construct, and some important issues must be discussed in its study. Here, a computationally simple method of ranking Z-numbers for multi-criteria decision-making (MCDM) problems is proposed, and a comprehensive possibility degree of Z-numbers is defined, as inspired by the possibility degree concept of interval numbers. The outranking relations of Z-numbers are also discussed on the basis of the proposed method. Then, a weight acquisition algorithm relative to the possibility degree of Z-numbers is presented. Finally, an extended Preference Ranking Organization Method for Enrichment Evaluation II (PROMETHEE II) based on the possibility degree of Z-numbers is developed for the MCDM problem under Z-evaluation, and a numerical example about the selection of travel plans is used to illustrate the validity of the proposed method. The applicability and superiority of the proposed method is demonstrated through sensitivity and comparative analyses along with other existing methods.

Information reliability has a remarkable effect on decision-making outcome. Zadeh’s Z-number considers information fuzziness and reliability, and can, therefore, help the decision-maker to manage complicated problems. However, due to its complex construct, several issues concerning the computation of Z-numbers require further research. This study develops a novel likelihood-based method of comparing two Z-numbers to solve multi-criteria decision-making (MCDM) problems. Four main parts can be outlined. First, the likelihood of fuzzy restriction of Z-numbers is defined based on the conversion method of Z-number. Second, the likelihood of underlying probability distributions of Z-numbers is also proposed to compare the difference of randomness of Z-information. Third, by adding to a risk preference parameter, this study constructs a comprehensive weighted likelihood of Z-numbers. Finally, a likelihood-based qualitative flexible approach is extended to address the MCDM problems under Z-evaluation. In addition, a numerical example of the selection of ERP systems for ABC enterprise is placed to illustrate the applicability, validity, and effectiveness of the proposed method.

Z -number is the general representation of real-life information with reliability, and it has adequate description power from the point of view of human perception. This study develops an innovative method for addressing multicriteria group decision-making (MCGDM) problems with Z -numbers under the condition that the weight information is completely unknown. Processing Z -numbers requires effective support of reliable tools. Then, the normal cloud model can be employed to analyze the Z -number construct. First, the potential information involved in Z -numbers is invoked, and a novel concept of normal Z +-value is proposed with the aid of the normal cloud model. The operations, distance measurement, and power aggregation operators of normal Z +-values are defined. Moreover, an MCGDM method is developed by incorporating the defined distance measurement and power aggregation operators into the MultiObjective Optimization by Ratio Analysis plus the Full Multiplicative Form. Finally, an illustrative example concerning air pollution potential evaluation is provided to demonstrate the proposed method. Its feasibility and validity are further verified by a sensitivity analysis and comparison with other existing methods.

Reliability of information can directly affect the accuracy of decision-making. Compared with classical fuzzy sets, Z-number takes into account the uncertainty in information generation process and reliability of information. Z-number can also be an intuitive and effective description form of decision information. Existing research on the distance measure of Z-number is rare, and most of the research works cannot reflect well the advantages of Z-information and the characteristics of Z-number. Given this research gap, this paper simultaneously considered the randomness and fuzziness of Z-number and defined the comprehensive weighted distance measure of Z-number. To extend classic VlseKriterijum-ska Optimizacija I Kompromisno Resenje (VIKOR) method to the Z-information environment, we suggested the Z-VIKOR method based on the proposed distance measure. This method is convenient and effective for the direct computation of Z-numbers. We also provided an example of multicriteria decision-making for selecting regional circular economy development plan to illustrate the feasibility and validity of our proposed method. We then verified the applicability and superiority of our method through comparative analyses with other existing methods.

This study focuses on consensus-reaching in social network group decision-making. Using Archimedean t-norms and t-conorms as the basis, this study introduces a family of interval-valued dual trust propagation (IVDTP) operators. A perfect knowledge degree-induced ordered weighted averaging (PKD-IOWA) aggregation operator is also put forward for obtaining the overall trust and distrust values. The proposed PKD-IOWA aggregation operator makes full use of the knowledge in each trust path, which can effectively overcome the limitations of existing trust aggregation operators. The weights of the experts can then be derived by combining the IVDTP operators and the PKD-IOWA aggregation operator. Three levels of consensus indices are defined to identify the incompatible experts in a group. A two-fold feedback mechanism comprising a judgment feedback mechanism (JFM) and a weight feedback mechanism (WFM), is designed to reach consensus. In each round of this interaction model, the most incompatible expert is advised to modify his/her judgments on the basis of the JFM. If the expert rejects the suggestion, then his/her weight will be reduced according to the WFM, which can act as an incentive or penalty to force him/her to accept the judgment feedback suggestion. Two illustrative examples are used to validate the feasibility and effectiveness of the proposed approach, followed by comparative analysis and discussion.

The operations of t-norm and t-conorm introduced by Dombi was known as Dombi operations can have a lead of good flexibility with the general parameter. The Dombi operations have so far not yet been applied for bipolar fuzzy sets. Motivated from Dombi operations, we have proposed bipolar fuzzy Dombi weighted averaging operator, bipolar fuzzy Dombi order weighted averaging operator, bipolar fuzzy Dombi hybrid weighted averaging operator, bipolar fuzzy Dombi weighted geometric operator , bipolar fuzzy Dombi order weighted geometric operator and bipolar fuzzy Dombi hybrid weighted geometric operator as these operators have good advantage of flexibility in operational behavior. Many properties of these operators are investigated. Then, we developed a model for multiple attribute decision-making problem for bipolar fuzzy Dombi aggregation operators under the bipolar fuzzy environment. Finally, a practical example for selection of investment alternatives is given to demonstrate for the utility and application of the proposed work.

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.

This study aims to deal with hesitant fuzzy linguistic multi-criteria group decision-making (MCGDM) problems with multi-granular unbalanced linguistic term sets. Firstly, a signed distance measure is developed as the support tool for hesitant fuzzy linguistic term sets. This measure is based on the ordinal semantics of linguistic terms and the possibility distribution method. In this manner, the signed distance measure can release the strict constraint of symmetric and uniform linguistic term sets in qualitative decision-making. Secondly, a signed distance-based transformation function is proposed to unify the multi-granular hesitant unbalanced linguistic assessments. Moreover, comprehensive consensus measures on three levels are proposed to measure the consensus degree. These measures simultaneously consider the consensus degree between individual and collective preference and between experts. Subsequently, a consensus reaching algorithm is designed and incorporated into the proposed MCGDM approach. Lastly, an illustrative example, followed by an in-depth comparative analysis and discussion is presented to demonstrate the proposed approach. Results show that the proposed approach can flexibly and effectively tackle MCGDM problems with multi-granular hesitant unbalanced linguistic information.

An appropriate greenhouse gas (GHG) emissions’ institution largely affects long-term development of cement corporates. The primary objective of this paper is to investigate a scientific method for selecting an optimal institution for a cement corporate. The paper mainly makes the following contributions to effectively resolving this selection problem. Given the capacity of picture 2-tuple linguistic sets (P2TLSs) in depicting voting information, they are used as evaluation representation in this case. Another contribution is that this paper newly proposes a prospect theory-based score function to rank picture 2-tuple linguistic numbers (P2TLNs). The proposed score function effectively overcomes the limitations in the existing score function by considering the voting potential of those decision-makers (DMs) who hold “abstain” opinion. Meanwhile, this proposed score function also adequately considers the rational-bounded behavior of DMs, thereby making the evaluation more approximate to the reality. Furthermore, this paper investigates a hybrid outranking method to select institution in the context of P2TLSs. It takes different types of preference functions and the Elimination and Choice Translating Reality (ELECTRE) method into account simultaneously. Compared with other similar type methods, this hybrid method can flexibly choose criteria type and provide specific values to represent quantitative dominance relation rather than qualitative judgment. Finally, an illustrative example and comparative analyses are provided to certify the availability and reliability of the proposed method.

The Z-number provides an adequate and reliable description of cognitive information. The nature of Z-numbers is complex, however, and important issues in Z-number computation remain to be addressed. This study focuses on developing a computationally simple method with Z-numbers to address multicriteria decision-making (MCDM) problems. Processing Z-numbers requires the direct computation of fuzzy and probabilistic uncertainties. We used an effective method to analyze the Z-number construct. Next, we proposed some outranking relations of Z-numbers and defined the dominance degree of discrete Z-numbers. Also, after analyzing the characteristics of elimination and choice translating reality III (ELECTRE III) and qualitative flexible multiple criteria method (QUALIFLEX), we developed an improved outranking method. To demonstrate this method, we provided an illustrative example concerning job-satisfaction evaluation. We further verified the validity of the method by a criteria test and comparative analysis. The results demonstrate that the method can be successfully applied to real-world decision-making problems, and it can identify more reasonable outcomes than previous methods. This study overcomes the high computational complexity in existing Z-number computation frameworks by exploring the pairwise comparison of Z-numbers. The method inherits the merits of the classical outranking method and considers the non-compensability of criteria. Therefore, it has remarkable potential to address practical decision-making problems involving Z-information.

Outsourcing has become a common and important link for enterprises. When selecting an outsourcing provider,single-valued neutrosophic linguistic sets can successfully express qualitative and fuzzy information. Furthermore, the selection of an outsourcing provider is a multi-criteria decision-making problem that can betackled by the multi-attribute border approximation area comparison (MABAC) method. In MABAC, criteria are assumed to be compensatory. However, criteria may be non-compensatory in outsourcing provider selection. Thus, this paper introduces the main idea of the elimination and choice translating reality (ELECTRE) method. An MABAC–ELECTRE method is established under single-valued neutrosophic linguistic environments. This method uses the mean-squared deviation weight method to obtain the weights of criteria. Moreover, an illustrative example is conducted to explain the procedure of the MABAC–ELECTRE method. A comparative analysis verifies its feasibility in solving problems with non-compensatory criteria.

Keen competition drives hotel companies to enhance their position. One way to do this is to select a proper hotel location. However, hotel location selection is a complex problem. This study establishes a multi-criteria hotel location selection method. In this method, cognitive information is depicted by multi-hesitant fuzzy linguistic term sets (MHFLTSs). Moreover, the method considers the non-compensation of criteria. It introduces the elimination and choice translating reality (ELECTRE) method. Notably, the method utilizes projection to define concordance and discordance indices. A case study and comparative study are performed in this study. They exhibit the feasibility of the method. Results of the studies show that the method can solve such problems, and they reveal the method’s advantages. One theoretical contribution lies in the characterization of cognitive information. MHFLTSs can handle vacillation of decision-makers caused by their complex cognition, and they express both conformity and divergence of opinions during cognitive processes. Our method has the advantages of the ELECTRE method. In addition, the ELECTRE method is improved by introducing the projection. The proposed method is promising in hotel location selection. Moreover, it is a potential option to address cognitive computation.

Identifying the priority of risk factors through risk evaluation is a significant decision-making problem for construction project management team. In this study, a multi-criteria decision-making framework is constructed for risk evaluation of construction project with picture fuzzy information. First, several limitations of the extant picture fuzzy projection model are identified. Second, a picture fuzzy normalized projection (PFNP) model is proposed to overcome the limitations of the extant picture fuzzy projection model. Third, an entropy weight method of picture fuzzy sets is established to calculate the objective weight vector of the criteria. Furthermore, an integrated picture fuzzy normalized projection-based VIKOR method is constructed by integrate the PFNP model and VIKOR method under picture fuzzy environment. The proposed method not only considers the difference between two picture fuzzy numbers (PFNs), including distance and included angle, but also considers the compromise among the criteria. Finally, a case study, sensitivity analysis, and comparative analysis are conducted to interpret the practicality and validity of the proposed method.

Purpose Online reviews increasingly present the characteristic of bidirectional communication with the advent of Web 2.0 era and tend to be asymmetrical and individualized in linguistic information. The authors aim to develop a new linguistic conversion model that exploits the asymmetric and personalized information from online reviews to express such linguistic information. A new online recommendation approach is provided. Design/methodology/approach The necessity of new linguistic conversation model is elucidated, and a leverage factor is incorporated into the linguistic label of negative review to handle the asymmetry problems of linguistic scale. A possible value range of the leverage factor is studied. A new linguistic conversation model is accordingly established with an unbalanced linguistic label and a cloud model. The authors develop a new online recommendation approach based on several modules, such as initialization, conversion, user-clustering and recommendation models. Findings The unbalanced effect between negative and positive reviews is verified with real data and measured using indirect methods. A new online recommendation approach of electronic products is proposed and used as an illustrative example to prove the practicality, effectiveness and feasibility of the proposed approach. Research limitations/implications Due to the unavailable transaction information of customers, the limitation of this study is the effectiveness of the authors’ established recommendation system for platform or website cannot be verified. Originality/value In most existing studies, the influence of negative review is counterbalanced by positive review, and the unbalanced effect between negative and positive reviews is ignored. The negative review receives much attention from consumers and businesses. This study thus highlights the influence of negative review.

Linguistic hesitant fuzzy sets (LHFSs), which can be used to both represent decision-makers’ qualitative preferences and reflect their hesitancy and inconsistency, have attracted much attention due...

Hotel selection on tourism websites has been a prevailing trend in traveling in recent years, and selecting a suitable hotel is a high-risk decision for millions of tourists because of the intrinsic imperceptibility of the provided products. Decision support models for hotel selection have drawn the interest of numerous scholars. The problem is that some existing models cannot address substantial amounts of online information, and they neglect the influence of interrelationships among different criteria on tourists’ decisions. To cover these defects, an applicable hotel decision support model is developed for tourists utilizing online reviews on TripAdvisor.com. Considering a great deal of review information associated with hotels posted by numerous tourists on TripAdvisor.com, probabilistic linguistic term sets (PLTSs) are introduced to summarize this information statistically. Processing qualitative concepts requires effective support of reliable tools; then, a cloud model can be employed to deal with probabilistic linguistic information. First, PLTSs are converted, and a novel concept of probabilistic linguistic integrated cloud (PLIC) is proposed. Moreover, the essential algorithms and distance measure of PLICs are defined. Two information fusion tools are subsequently presented based on Heronian mean operator. Then, the hotel decision support model is established. Finally, a hotel selection problem on TripAdvisor.com is provided to demonstrate the model. Its stability and validity are further verified by a sensitivity analysis and comparison with other extant methods.

Cloud service reliability assessment is a vital decision-making activity for companies and individuals. In this assessment, the evaluation information can be represented by multi-valued neutrosophic numbers (MVNNs). MVNNs are regarded as an integration of single-valued neutrosophic numbers (SVNNs) and hesitant fuzzy numbers (HFNs); therefore, considering the defects of entropy and crossentropy measures for SVNNs and HFNs, we first define a framework of entropy measures and a family of cross-entropy measures for MVNNs in this paper. Second, a novel extended VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method based on entropy and cross-entropy measures is developed to address the decision-making problems when information about criteria weights is absolutely unknown. Finally, we apply the proposed method to evaluate cloud service reliability; also, a sensitivity analysis and a comparative analysis are made to interpret the practicality and effectiveness of it. The results of analyses verify that the proposed method based on cross-entropy is much better than the methods using general distance measures.

Safety risk assessment of metro construction is necessary to prevent catastrophic accidents that may cause heavy financial losses and casualties. In this paper, we introduce a comprehensive risk assessment framework that incorporates credal networks (CNs) and an improved evaluation based on distance from average solution (EDAS) method. First, the CN model can intuitively demonstrate dependencies between risk factors, capture the epistemic uncertainty of expert judgement in the form of imprecise probability, and propagate the uncertainty quickly and accurately with advanced inference algorithms. In addition, the improved EDAS method can be used to comprehensively identify critical risk factors with multiple indicators, including the nature of the risk itself and company capabilities, being taken into account. Moreover, to effectively aggregate expert opinions, a weighting method that integrates subjective weights and similarity-based objective weights is provided. A case study on the safety risk analysis of metro construction in China verified the feasibility of the framework. The proposed approach can be extended to other similar projects to help engineers systematically assess risks during the life cycle of the project to ensure the safety of metro construction.

This paper proposes two extended projection-based preference ranking organization methods for enrichment evaluations for multi-criteria decision making with hesitant fuzzy linguistic term sets (HFLTSs). We first utilize three linguistic scale functions to solve linguistic information transformation with differences in semantics. Correspondingly, we define projection and normalized projection measurements for HFLTSs. We then propose a projection-based difference measurement for HFLTSs while considering the risk appetite of decision-makers. The effectiveness and robustness of the proposals are demonstrated by their application to a practical example of green supply chain management problem, which is followed by sensitivity and comparative analyses. Finally, conclusions and future research directions are drawn.

Owing to the vagueness of real-world environments and limited knowledge of human beings, it is natural for decision makers to express their preferences by means of incomplete 2-tuple fuzzy linguistic preference relations (FLPRs). Besides, since different decision makers possess distinctive educational and academic background, they may select linguistic term sets with different granularities to represent their opinions. This study primarily focuses on multi-granular linguistic multi-criteria group decision-making (MCGDM) based on incomplete 2-tuple FLPRs. First, we introduce an extended four-way procedure for estimating unknown elements from an acceptable incomplete 2-tuple FLPR as well as a completion method for obtaining unknown linguistic 2-tuples in an unacceptable incomplete 2-tuple FLPR. Based on the four ways for estimating unknown values, a formula is then established to calculate the consistency index of an incomplete 2-tuple FLPR. Second, a 2-tuple linguistic induced generalized ordered weighted averaging operator is introduced and further analyzed from different aspects. Subsequently, by systematically fusing the aforementioned contributions, a novel approach is developed to address MCGDM problems with multi-granular incomplete 2-tuple FLPRs and unknown weight information. Specifically, the importance weights of decision makers with respect to different criteria are decided by the corresponding consistency indices. Finally, an investment problem is furnished to illustrate the application of our proposed approach.

Linguistic intuitionistic fuzzy numbers (LIFNs), characterized by a linguistic membership degree, linguistic non-membership degree, and linguistic indeterminacy degree, represent a helpful tool for depicting uncertain information under complex environments. This paper focuses on developing an innovative method to address multi-criteria decision-making (MCDM) problems with LIFNs in which the weight information is completely unknown. First, the distance of LIFNs is defined with the aid of linguistic scale functions (LSFs). Second, some extended outranking relationships between each pair of LIFNs are proposed based on the elicitation of the classic relation models. Moreover, a ranking method is constructed to deal with MCDM problems according to the proposed outranking relationships of LIFNs. Finally, an illustrative example concerning coal mine safety evaluation is provided to demonstrate the proposed method, and its feasibility and validity are further verified by a sensitivity analysis and comparison with other existing methods.

For decision makers, expressing their opinions through subintervals of [0, 1] is sometimes easier than using crisp numbers. This study defines some outranking relations derived by ELECTRE III for hesitant interval-valued fuzzy sets (HIVFSs). The properties of these outranking relations are discussed in detail. The concordance and discordance indexes of HIVFSs are then proposed. Ranking approach is developed based on the outranking relations of HIVFSs to handle multi-criteria decision-making problems. Finally, we provide practical examples, as well as sensitivity analysis and comparison with other techniques.

Treatment selection is a multi-criteria decision-making problem of significant concern in the medical field. In this study, a fuzzy decision-making framework is established for treatment selection. The framework mitigates information loss by introducing single-valued trapezoidal neutrosophic numbers to denote evaluation information. Treatment selection has multiple criteria that remarkably exceed the alternatives. In consideration of this characteristic, the framework utilises the idea of the qualitative flexible multiple criteria method. Furthermore, it considers the risk-averse behaviour of a decision maker by employing a concordance index based on TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method. A sensitivity analysis is performed to illustrate the robustness of the framework. Finally, a comparative analysis is conducted to compare the framework with several extant methods. Results indicate the advantages of the framework and its better performance comp...

In selecting logistics service providers, the evaluation criteria can be easily prioritized and possibly interrelated with each other, and the assessment of alternatives under qualitative criteria is usually accomplished by more than one decision maker. A novel multicriteria decision-making approach with multihesitant fuzzy linguistic term elements (MHFLTEs) based on the Heronian mean (HM) and prioritized average operators can effectively deal with the problems inherent in such a scenario. Multihesitant fuzzy linguistic term sets (MHFLTSs) were proposed on the basis of multihesitant fuzzy sets (MHFSs) and hesitant fuzzy linguistic sets (HFLSs), where each MHFLTE can contain nonconsecutive and repeated linguistic terms. Using MHFLTEs, one decision maker can provide one or several consecutive linguistic terms in evaluating an alternative under one specific criterion, different decision makers’ evaluation values can be collected, and the frequency of a linguistic term in the evaluation information can accord with reality. This paper revises the basic operations and comparison method for MHFLTEs on the basis of the originals and defines some multihesitant fuzzy linguistic HM operators for MHFLTEs to deal with problems in which weight information cannot be accurately established for criteria, but their priorities can be provided in groups or without groups. Finally, the validity and effectiveness of the proposed approach are demonstrated through an illustration of a logistics outsourcing problem and a comparison analysis.

In this study, comprehensive multi-criteria decision-making (MCDM) methods are investigated under bipolar neutrosophic environment. First, the operations of BNNs are redefined based on the Frank operations considering the extant operations of bipolar neutrosophic numbers (BNNs) lack flexibility and robustness. Subsequently, the Frank bipolar neutrosophic Choquet weighted Bonferroni mean operator and the Frank bipolar neutrosophic Choquet geometric Bonferroni mean operator are proposed based on the Frank operations of BNNs. The proposed operators simultaneously consider the interactions and interrelationships among the criteria by combining the Choquet integral operator and Bonferroni mean operators. Furthermore, MCDM methods are developed based on the proposed aggregation operators. A numerical example of plant location selection is conducted to explain the application of the proposed methods, and the influences of parameters are also discussed. Finally, the proposed methods are compared with several extant methods to verify their feasibility.

Z-numbers are a new concept considering both the description of cognitive information and the reliability of information. Linguistic terms are useful tools to adequately and effectively model real-life cognitive information, as well as to characterize the randomness of events. However, a form of Z-numbers, in which their two components are in the form of linguistic terms, is rarely studied, although it is common in decision-making problems. In terms of Z-numbers and linguistic term sets, we provided the definition of linguistic Z-numbers as a form of Z-numbers or a subclass of Z-numbers. Then, we defined some operations of linguistic Z-numbers and proposed a comparison method based on the score and accuracy functions of linguistic Z-numbers. We also presented the distance measure of linguistic Z-numbers. Next, we developed an extended TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method based on the Choquet integral for multi-criteria decision-making (MCDM) problems with linguistic Z-numbers. Finally, we provided an example concerning the selection of medical inquiry applications to demonstrate the feasibility of our proposed approach. We then verified the applicability and superiority of our approach through comparative analyses with other existing methods. Illustrative and comparative analyses indicated that the proposed approach was valid and feasible for different decision-makers and cognitive environments. Furthermore, the final ranking results of the proposed approach were closer to real decision-making processes. Linguistic Z-numbers can flexibly characterize real cognitive information as well as describe the reliability of information. This method not only is a more comprehensive reflection of the decision-makers’ cognition but also is more in line with expression habits. The proposed method inherited the merits of the classical TODIM method and considers the interactivity of criteria; therefore, the proposed method was effective for dealing with real-life MCDM problems. Consideration about bounded rational and the interactivity of criteria made final outcomes convincing and consistent with real decision-making.

Due to the defects of hesitant fuzzy sets (HFSs) in the actual decision-making process, it is necessary to add the probabilities corresponding to decision maker’s preferences to the values in HFSs. Hesitant probabilistic fuzzy sets (HPFSs) are suitable for presenting this kind of information and contribute positively to the efficiency of depicting decision maker’s preferences in practice. However, some important issues in HPFSs utilization remain to be addressed. In this paper, the qualitative flexible multiple criteria method (QUALIFLEX) and the preference ranking organization method for enrichment evaluations II (PROMETHEE II) are extended to HPFSs. First, we provide a comparison method for hesitant probabilistic fuzzy elements (HPFEs). Second, we propose a novel possibility degree depicting the relations between two HPFEs, and then, employ the possibility degree to extend the QUALIFLEX and PROMETHEE II methods to hesitant probabilistic fuzzy environments based on the proposed possibility degree. Third, an information integration method is introduced to simplify the processing of HPFE evaluation information. Finally, we provide an example to demonstrate the usefulness of the proposed methods. An illustrative example in conjunction with comparative analyses is employed to demonstrate that our proposed methods are feasible for practical multi-criteria decision-making (MCDM) problems, and the final ranking results show that the proposed methods are more accurate than the compared methods in an actual decision-making processes. HPFSs are more practical than HFSs due to their efficiency in comprehensively representing uncertain, vague, and probabilistic information. The proposed methods are effective for solving hesitant probabilistic MCDM problems and are expected to contribute to the solution of MCDM problems involving uncertain or vague information.

In this paper, the distance-based multi-criteria group decision-making (MCGDM) approaches using multi-hesitant fuzzy linguistic term sets (MHFLTSs) are proposed. MHFLTSs can contain nonconsecutive and repetitive linguistic terms, so as to deal with repeated linguistic values in group decision-making. A multi-hesitant fuzzy linguistic term element (MHFLTE) can be produced by collecting the evaluation values of several decision-makers or given by one person who has uncertainty in evaluation. The corresponding set operations and comparison rules are defined and the generalized hesitant fuzzy linguistic distance for MHFLTEs is given based on the linguistic scale function. Then this distance is embedded into the TOPSIS, VIKOR and TODIM approaches for the purpose of solving multi-criteria decision-making (MCDM) problems in the context of multi-hesitant fuzzy linguistic information. With increasing concerns about deterioration in environment, organizations are obliged to carry out more environmental sustainable activities than before, such as progressive practices in green supply chain management (GSCM). Therefore, with respect to the application of MHFLTSs in GSCM, two illustrations for evaluating the related alternatives are finally provided, together with the sensitivity and comparison analysis, to show the validity and effectiveness of our proposal.

For many companies, the selection of green suppliers has become a key strategic consideration due to regulatory requirements and market trends. In this paper, an extended qualitative flexible multiple (QUALIFLEX) method is used to solve problems regarding the selection of green suppliers using probability hesitant fuzzy information. First, a new Hausdorff distance between two probability hesitant fuzzy elements (PHFEs) is introduced, and the properties of the proposed distance are discussed. Second, a likelihood measurement for PHFEs is proposed based on the new Hausdorff distance; moreover, an extended QUALIFLEX is developed based on the likelihood measurement. Finally, the practicality and effectiveness of the proposed method is tested using a green supplier selection example; furthermore, a comparative analysis using probability hesitant fuzzy information is also performed.

Coal mine safety has been a pressing issue for many years, and it is a constant and non-negligible problem that must be addressed during any coal mining process. This paper focuses on developing an innovative multi-criteria decision-making (MCDM) method to address coal mine safety evaluation problems. Because lots of uncertain and fuzzy information exists in the process of evaluating coal mine safety, linguistic intuitionistic fuzzy numbers (LIFNs) are introduced to depict the evaluation information necessary to the process. Furthermore, the handling of qualitative information requires the effective support of quantitative tools, and the linguistic scale function (LSF) is therefore employed to deal with linguistic intuitionistic information. First, the distance, a valid ranking method, and Frank operations are proposed for LIFNs. Subsequently, the linguistic intuitionistic fuzzy Frank improved weighted Heronian mean (LIFFIWHM) operator is developed. Then, a linguistic intuitionistic MCDM method for coal mine safety evaluation is constructed based on the developed operator. Finally, an illustrative example is provided to demonstrate the proposed method, and its feasibility and validity are further verified by a sensitivity analysis and comparison with other existing methods.

Single-valued neutrosophic sets (SVNSs), which involve in truth-membership, indeterminacy-membership and falsity-membership, play a significant role in describing the decision-makers&rsquo, preference information. In this study, a single-valued neutrosophic multi-criteria decision-making (MCDM) approach is developed based on Shapley fuzzy measures and power aggregation operator that takes a correlative relationship among criteria into account and also simultaneously reduces the effects of abnormal preference information. Firstly, two aggregation operators, namely, generalized weighted single-valued neutrosophic power Shapley Choquet average (GWSVNPSCA) operator and generalized weighted single-valued neutrosophic power Shapley Choquet geometric (GWSVNPSCG) operator, are accordingly defined, and the corresponding properties are discussed as well. Secondly, based on the proposed aggregation operators, an integrated MCDM approach is proposed to effectively solve single-valued neutrosophic problems where the weight information is incompletely known. A programming model is constructed to obtain the optimal Shapley fuzzy measure. Next, the proposed operators are utilized to aggregate the decision-makers&rsquo, preference information. Finally, a theoretical example with tourism attraction selection is provided to examine the efficacy of the developed approach, in which the results is found reasonable and credible.

This study aims to introduce a series of novel distance measures of an intuitionistic fuzzy set and an extended Multi-attributive Border Approximation Area Comparison method to address multi-criteria group decision-making problems. To aggregate the intuitionistic fuzzy information, we propose two aggregation operators, namely, the intuitionistic fuzzy Dombi generalised $$ \lambda $$-Shapley Choquet arithmetical average operator and intuitionistic fuzzy Dombi generalised $$ \lambda $$-Shapley Choquet geometric average operator. These aggregation operators consider the importance of combinations and correlations among combinations of input arguments. Furthermore, an illustrative example of a human resource management problem is presented to verify the effectiveness of the proposed method, and sensitivity and comparison analyses are conducted to demonstrate the technique’s stability and advancements. Finally, conclusions are drawn.