Interval‐valued probabilistic linguistic term sets in multi‐criteria group decision making

The theory of probabilistic linguistic term sets (PLTSs) is very useful in objectively dealing with the multi‐criteria group decision making (MCGDM) problems in which there is hesitancy in providing linguistic assessments; and PLTSs allow experts to express their preferences on one linguistic term over another. In order to reflect the uncertainty and inconsistency of decision‐makers and handle incomplete linguistic information, we propose a new PLTS called interval‐valued probabilistic linguistic term set (IVPLTS). In addition, the existing approaches associated with PLTSs are limited or highly complex in real applications. Therefore, new operations, comparison laws, and aggregation operators are developed for IVPLTS. Furthermore, we establish an efficient framework for MCGDM problems based on the proposed comparison method and the fuzzy preference relation. Then we apply it to a real‐life case under linguistic environment. The extended TOPSIS methods combined with PLTSs by using different operational laws are also included for comparison. The final results demonstrate the efficiency and practicality of the new framework.     

A novel two‐stage model for cloud service trustworthiness evaluation

In this paper, we address the cloud service trustworthiness evaluation problem, which in essence is a multi‐attribute decision‐making problem, by proposing a novel evaluation model based on the fuzzy gap measurement and the evidential reasoning approach. There are many sources of uncertainties in the process of cloud service trustworthiness evaluation. In addition to the intrinsic uncertainties, cloud service providers face the problem of discrepant evaluation information given by different users from different perspectives. To address these problems, we develop a novel fuzzy gap evaluation approach to assess cloud service trustworthiness and to provide evaluation values from different perspectives. From the evaluation values, the perception–importance, delivery–importance, and perception–delivery gaps are generated. These three gaps reflect the discrepancy evaluation of cloud service trustworthiness in terms of perception utility, delivery utility, and importance utility, respectively. Finally, the gap measurement of each perspective is represented by a belief structure and aggregated using the evidential reasoning approach to generate final evaluation results for informative and robust decision making. From this hybrid two‐stage evaluation process, cloud service providers can get improvement suggestions from intermediate information derived from the gap measurement, which is the main advantage of this evaluation process.     

Algorithms for complex interval‐valued q‐rung orthopair fuzzy sets in decision making based on aggregation operators,AHP, and TOPSIS

The interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS) and complex fuzzy set (CFS) are two generalizations of the fuzzy set (FS) to cope with uncertain information in real decision making problems. The aim of the present work is to develop the concept of complex interval‐valued q‐rung orthopair fuzzy set (CIVq‐ROFS) as a generalization of interval‐valued complex fuzzy set (IVCFS) and q‐rung orthopair fuzzy set (q‐ROFS), which can better express the time‐periodic problems and two‐dimensional information in a single set. In this article not only basic properties of CIVq‐ROFSs are discussed but also averaging aggregation operator (AAO) and geometric aggregation operator (GAO) with some desirable properties and operations on CIVq‐ROFSs are discussed. The proposed operations are the extension of the operations of IVq‐ROFS, q‐ROFS, interval‐valued Pythagorean fuzzy, Pythagorean fuzzy (PF), interval‐valued intuitionistic fuzzy, intuitionistic fuzzy, complex q‐ROFS, complex PF, and complex intuitionistic fuzzy theories. Further, the Analytic hierarchy process (AHP) and technique for order preference by similarity to ideal solution (TOPSIS) method are also examine based on CIVq‐ROFS to explore the reliability and proficiency of the work. Moreover, we discussed the advantages of CIVq‐ROFS and showed that the concepts of IVCFS and q‐ROFS are the special cases of CIVq‐ROFS. Moreover, the flexibility of proposed averaging aggregation operator and geometric aggregation operator in a multi‐attribute decision making (MADM) problem are also discussed. Finally, a comparative study of CIVq‐ROFSs with pre‐existing work is discussed in detail.     

Evaluating the quality of airport service using the fuzzy multi‐criteria decision‐making method: a case study of Taiwanese airports

In today's highly competitive air transportation environment, the quality of airport service is an important issue. In this paper, we present a fuzzy multi‐criteria decision‐making (MCDM) method for the evaluation of the quality of airport service. The data are collected from the passengers who depart from two international airports in Taiwan. The fuzzy MCDM model is defuzzied using the inverse function arithmetic representation method. The results show that the average quality of the Taoyuan International Airport service is better than that of the Kaohsiung International Airport service. This paper also analyses the quality of airport service using the fuzzy expert system to improve the performance of airport service. The analysis results show that the Kaohsiung International Airport and the Taoyuan International Airports' improvement efforts should be concentrated on seven and five specific service items, respectively. The findings of this paper are to help the decision‐makers and the managers of airports improve the quality of airport service.     

New exponential operational laws and their aggregation operators for interval‐valued Pythagorean fuzzy multicriteria decision‐making

In this article, we define two new exponential operational laws about the interval‐valued Pythagorean fuzzy set (IVPFS) and their corresponding aggregation operators. However, the exponential parameters (weights) of all the existing operational laws of IVPFSs are crisp values in IVPFS decision‐making problems. As a supplement, this paper first introduces new exponential operational laws of IVPFS, where bases are crisp values or interval numbers and exponents are interval‐valued Pythagorean fuzzy numbers. The prominent characteristic of these proposed operations is studied. Based on these laws, we develop some new weighted aggregation operators, namely the interval‐valued Pythagorean fuzzy weighted exponential averaging operator and the dual interval‐valued Pythagorean fuzzy weighted exponential averaging. Finally, a decision‐making approach is presented based on these operators and illustrated with some numerical examples to validate the developed approach.     

Interval‐valued Pythagorean fuzzy GRA method for multiple‐attribute decision making with incomplete weight information

In this paper, the concept of multiple‐attribute group decision‐making (MAGDM) problems with interval‐valued Pythagorean fuzzy information is developed, in which the attribute values are interval‐valued Pythagorean fuzzy numbers and the information about the attribute weight is incomplete. Since the concept of interval‐valued Pythagorean fuzzy sets is the generalization of interval‐valued intuitionistic fuzzy set. Thus, due the this motivation in this paper, the concept of interval‐valued Pythagorean fuzzy Choquet integral average (IVPFCIA) operator is introduced by generalizing the concept of interval‐valued intuitionistic fuzzy Choquet integral average operator. To illustrate the developed operator, a numerical example is also investigated. Extended the concept of traditional GRA method, a new extension of GRA method based on interval‐valued Pythagorean fuzzy information is introduced. First, utilize IVPFCIA operator to aggregate all the interval‐valued Pythagorean fuzzy decision matrices. Then, an optimization model based on the basic ideal of traditional grey relational analysis (GRA) method is established, to get the weight vector of the attributes. Based on the traditional GRA method, calculation steps for solving interval‐valued Pythagorean fuzzy MAGDM problems with incompletely known weight information are given. The degree of grey relation between every alternative and positive‐ideal solution and negative‐ideal solution is calculated. To determine the ranking order of all alternatives, a relative relational degree is defined by calculating the degree of grey relation to both the positive‐ideal solution and negative ideal solution simultaneously. Finally, to illustrate the developed approach a numerical example is to demonstrate its practicality and effectiveness.     

Multi-criteria group decision making based on trapezoidal intuitionistic fuzzy information

With respect to multi-criteria group decision making (MCGDM) problems under trapezoidal intuitionistic fuzzy environment, a new MCGDM method is investigated. The proposed method can effectively avoid the failure caused by the use of inconsistent decision information and provides a decision-making idea for the case of “the truth be held in minority”. It consists of three interrelated modules: weight determining mechanism, group consistency analysis, and ranking and selection procedure. For the first module, distance measures, expected values and arithmetic averaging operator for trapezoidal intuitionistic fuzzy numbers are used to determine the weight values of criteria and decision makers. For the second module, a consistency analysis and correction procedure based on trapezoidal intuitionistic fuzzy weighted averaging operator and OWA operator is developed to reduce the influence of conflicting opinions prior to the ranking process. For the third module, a trapezoidal intuitionistic fuzzy TOPSIS is used for ranking and selection. Then a procedure for the proposed MCGDM method is developed. Finally, a numerical example further illustrates the practicality and efficiency of the proposed method.     

Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees

As an important component of group decision making, the hybrid multi-criteria group decision making (MCGDM) is very complex and interesting in real applications. The purpose of this paper is to develop a novel interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for hybrid MCGDM considering alternative comparisons with hesitancy degrees. The subjective preference relations between alternatives given by each decision maker (DM) are formulated as an IVIF set (IVIFS). The IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers (TrFNs), linguistic variables, intervals and real numbers are used to represent the multiple types of criteria values. The information of criteria weights is incomplete. The IVIFS-type consistency and inconsistency indices are defined through considering the fuzzy positive and negative ideal solutions simultaneously. To determine the criteria weights, we construct a novel bi-objective IVIF mathematical programming of minimizing the inconsistency index and meanwhile maximizing the consistency index, which is solved by the technically developed linear goal programming approach. The individual ranking order of alternatives furnished by each DM is subsequently obtained according to the comprehensive relative closeness degrees of alternatives to the fuzzy positive ideal solution. The collective ranking order of alternatives is derived through establishing a new multi-objective assignment model. A real example of critical infrastructure evaluation is provided to demonstrate the applicability and effectiveness of this method.     

A ranking function based on principal‐value Pythagorean fuzzy set in multicriteria decision making

Multicriteria decision making (MCDM) has been attracting attention in recent years. There are two essential directions in the research territory, one direction is the research of representation of evaluation information and another is the construction of ranking function. In this paper, we consider some nonstandard fuzzy sets, intuitionistic, and interval‐valued fuzzy sets. Especially, the Pythagorean membership grade and Pythagorean fuzzy set receive much attention. Then, to reflect the importance of principal value, we shall propose the principal‐value Pythagorean fuzzy number (p‐PFN) and principal‐value Pythagorean fuzzy set. Furthermore, a novel ranking function is constructed to select the ideal alternative(s) based on p‐PFNs in MCDM. Finally, an investment strategy decision‐making problem is proposed to reveal the availability and practicability of the function under the new environment.     

Interval‐valued Pythagorean fuzzy extended Bonferroni mean for dealing with heterogenous relationship among attributes

Owing to the information insufficiency, it might be difficult for decision makers to precisely evaluate their assessments in real decision‐making. As a new extension of the Pythagorean fuzzy sets, the interval‐valued Pythagorean fuzzy sets (IVPFSs) can availably provide enough input space for decision makers to evaluate their assessments with interval numbers. By extending the Bonferroni mean to model the heterogeneous interrelationship among attributes, the extended Bonferroni mean (EBM) was examined. Considering the partition structure of relationship among the attributes, we introduce the EBM into the interval‐valued Pythagorean fuzzy environment and develop two new aggregation operators, namely, interval‐valued Pythagorean fuzzy extended Bonferroni mean and weighted interval‐valued Pythagorean fuzzy extended Bonferroni mean (WIVPFEBM) operators. Meanwhile, some of their special cases and properties are also deeply discussed. Subsequently, by employing the WIVPFEBM operator, we propose an approach for multiple attribute decision making with IVPFSs. Finally, a practical illustration of the E‐commerce project selection problem is investigated by our proposed method, which successfully demonstrates the applicability of our results.     

A group decision making framework based on fuzzy VIKOR approach for machine tool selection with linguistic information

Computer numerical control (CNC) machines are used for repetitive, difficult and unsafe manufacturing tasks that require a high degree of accuracy. However, when selecting an appropriate CNC machine, multiple criteria need to be considered by multiple decision makers. In this study, a multi-criteria group decision making (MCGDM) technique based on the fuzzy VIKOR method is developed to solve a CNC machine tool selection problem. Linguistic variables represented by triangular fuzzy numbers are used to reflect decision maker preferences for the criteria importance weights and the performance ratings. After the individual preferences are aggregated or after the separation values are computed, they are then defuzzified. In this paper, two algorithms based on a fuzzy linguistic approach are developed. Based on these two algorithms and the VIKOR method, a general MCGDM framework is proposed. A CNC machine tool selection example illustrates the application of the proposed approach. A comparative study of the two algorithms using the above case study information highlighted the need to combine the ranking results, as both algorithms have distinct characteristics.     

New doctors ranking system based on VIKOR method

Nowadays, we can use different websites that help us make decisions about various aspects of our lives. However, privacy protection prevents websites from providing personalised guidelines to users. We propose a novel doctor‐ranking system (DRS) based on multi‐criteria group decision‐making (MCGDM) method to address the problems of privacy protection. The following aspects differentiate our proposed DRS model from previous works: (a) textual information reviews are used to identify user preferences and complementary criteria, (b) criteria weights are determined by term frequency inverse document frequency (TF‐IDF) instead of Delphi method or expert opinion, (c) intuitionistic fuzzy sets (IFSs) are used to replace sentiment analysis to express subjective user criteria, and (d) VIsekriterijumsko KOmpromisno Rangiranjie (VIKOR) method for MCGDM with IFSs is used to solve the doctor‐ranking problem. We apply our proposed model to datasets from Haodf.com to compare the performance of our method with that of sentiment analysis and technique for order performance by similarity to ideal solution (TOPSIS) methods. The experimental results show that our method provides accurate ranking and increases the reliability of DRS.     

The linear assignment method for multicriteria group decision making based on interval‐valued Pythagorean fuzzy Bonferroni mean

The interval‐valued Pythagorean fuzzy sets can easily handle uncertain information more flexibly in the process of decision making. Considering the interrelationship among the input arguments, we extend the Bonferroni mean and the geometric Bonferroni mean to the interval‐valued Pythagorean fuzzy environment and solve its practical application problems. First, we develop the interval‐valued Pythagorean fuzzy Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy Bonferroni mean (WIVPFBM) operators. The properties of these aggregation operators are investigated. Then, we also develop the interval‐valued Pythagorean fuzzy geometric Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy geometric Bonferroni mean (WIVPFGBM) operators and analyze their properties. Third, we utilize the WIVPFBM and WIVPFGBM operators to fuse the information in the interval‐valued Pythagorean fuzzy multicriteria group decision making (IVPFMCGDM) problem, which can obtain much more information in the process of group decision making. With the aid of the linear assignment method, we present its extension and further design a new algorithm for the application of IVPFMCGDM. Finally, an example is given to elaborate our proposed algorithm and validate its excellent performance.     

Consolidating SWOT analysis with nonhomogeneous uncertain preference information

SWOT analysis is an important support tool for decision-making, and is commonly used to systematically analyze organizations’ internal and external environments. However, one of its deficiencies is in the measurement and evaluation of prioritization of the factors and strategies. This paper is aimed to present a novel quantified SWOT analytical method based multiple criteria group decision-making (MCGDM) concept, in which the priorities of SWOT factors and groups are derived by multiple decision makers (DMs) with nonhomogeneous uncertain preference information (NUPI), such as interval multiplicative preference relations, interval fuzzy preference relations, and uncertain linguistic preference relations. In this method, the SWOT analysis provides a basic frame within which to perform analyses of decision situations, in turn, MCGDM methods assist in carrying out SWOT more analytically and in elaborating the results of the analyses so that SWOT factors and groups can be prioritized with respect to the entire SWOT. The uniform and aggregation of the NUPI and the derivation of priorities for SWOT groups and factors are investigated in detail. Finally, an example is to validate the procedure of the proposed method.     

A normal wiggly hesitant fuzzy MABAC method based on CCSD and prospect theory for multiple attribute decision making

Normal wiggly hesitant fuzzy set (NWHFS) is a new fuzzy information form to help decision makers (DMs) express their evaluations, which can further dig the potential uncertain information hidden in the original data given by the DMs. Firstly, we define a new distance measure and new operational laws of NWHFSs. Then, for the situation where attribute weights are completely unknown, we propose an extended CCSD method to produce them objectively, which comprehensively uses standard deviation (SD) and correlation coefficient (CC). What's more, we introduce the MABAC (multiattributive border approximation area comparison) method, which takes the distance between alternatives and the border approximation area (BAA) into consideration for handling the complex and uncertain decision‐making problems. Meanwhile, we combine the MABAC method with prospect theory (PT), which considers DMs' psychological behavior, and propose a new NWHF‐CCSD‐PT‐MABAC method to cope with the multi‐attribute decision making problems under normal wiggly hesitant fuzzy environment. Lastly, we illustrate the validity and advantages of the proposed method through an example of college book supplier selection.     

A new distance measure for interval valued intuitionistic fuzzy sets and its application to group decision making problems with incomplete weights information

The aim of this study is to introduce a novel generalized distance measure for interval valued intuitionistic fuzzy sets and to illustrate the applicability of the proposed distance measure to group decision making problems. Firstly, a generalized distance measure is proposed along with proofs satisfying its axioms. Then, a comparison between the proposed distance measure and well-known distance measures is performed in terms of counter-intuitive cases. Subsequently, the extension of TOPSIS method, in which the proposed distance measure is used to calculate separation measures, to an interval valued intuitionistic fuzzy (IVIF) environment is demonstrated to solve multi-criteria group decision making (MCGDM) problems using optimal criteria weights determined with linear programming model based on the concept of maximizing relative closeness coefficient. Finally, two illustrative examples are provided for proof-of-concept purposes and to demonstrate benefits of using the proposed distance measure over the existing ones in IVIF TOPSIS method for MCGDM problems.     

Signed distance‐based ORESTE for multicriteria group decision‐making with multigranular unbalanced hesitant fuzzy linguistic information

The objective of this study is to develop an integrated approach for solving multicriteria group decision‐making problems with multigranular unbalanced hesitant fuzzy linguistic term sets (HFLTSs). Firstly, a signed distance‐based transformation function is proposed to unify multigranular unbalanced hesitant fuzzy linguistic (HFL) assessments. Secondly, a mathematical programming model based on the maximum consensus is constructed to allocate decision‐makers (DMs)' weights objectively. Thirdly, a new signed distance‐based preference score function is defined to aggregate HFL assessments and determine the weak ranking of alternatives, and a novel preference, indifference, and incomparability test framework is constructed to identify the subtle relations among alternatives. On these bases, a signed distance‐based ORESTE (Organísation, rangement et Synthèse de données relarionnelles, in French) method, in which knowledge regarding criterion values and weights are expressed as multigranular unbalanced HFLTSs, is developed to obtain the ranking of alternatives. Finally, an illustrative example, followed by sensitivity and comparative analyses, is presented to verify the feasibility and effectiveness of the proposed approach.     

The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets

Pythagorean fuzzy sets (PFSs) as a new generalization of fuzzy sets (FSs) can handle uncertain information more flexibly in the process of decision making. In our real life, we also may encounter a hesitant fuzzy environment. In view of the effective tool of hesitant fuzzy sets (HFSs) for expressing the hesitant situation, we introduce HFSs into PFSs and extend the existing research work of PFSs. Concretely speaking, this paper considers that the membership degree and the non-membership degree of PFSs are expressed as hesitant fuzzy elements. First, we propose a new concept of hesitant Pythagorean fuzzy sets (HPFSs) by combining PFSs with HFSs. It provides a new semantic interpretation for our evaluation. Meanwhile, the properties and the operators of HPFSs are studied in detail. For the sake of application, we focus on investigating the normalization method and the distance measures of HPFSs in advance. Then, we explore the application of HPFSs to multi-criteria decision making (MCDM) by employing the technique for order preference by similarity to ideal solution (TOPSIS) method. A new extension of TOPSIS method is further designed in the context of MCDM with HPFSs. Finally, an example of the energy project selection is presented to elaborate on the performance of our approach.     

Uncertain linguistic hesitant fuzzy sets and their application in multi‐attribute decision making

To denote the quantitative and qualitative fuzzy information simultaneously, this paper introduces a new type of fuzzy sets called uncertain linguistic hesitant fuzzy sets, which are denoted by an uncertain linguistic variable with several possible interval membership degrees. Considering the application of this type of fuzzy sets, several basic operational laws are defined, and several properties are studied. Meanwhile, an ordered relationship is introduced. Then, two types of uncertain linguistic hesitant fuzzy aggregation operators are defined. One uses additive measures, and the other is based on λ‐fuzzy measures. Then, a similarity measure is presented, by which models for the optimal weight vector are constructed. After that, an approach to uncertain linguistic hesitant fuzzy multi‐attribute decision making is developed. Finally, an illustrative example for evaluating corporate environmental performance is offered to show the concrete practicality of the procedure.     

Development of a fuzzy decision support framework for complex multi‐attribute decision problems: A case study for the selection of skilful basketball players

The selection of skilful players is a complicated process due to the problem criteria consisting of both qualitative and quantitative attributes as well as vague linguistic terms. This study seeks to develop a decision support framework for the selection of candidates eligible to become basketball players through the use of a fuzzy multi‐attribute decision making (MADM) algorithm. The proposed model is based on fuzzy analytic hierarchy process (FAHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) methods. The model was employed in the Youth and Sports Center of Mugla, Turkey, with the participation of seven junior basketball players aged between 7 and 14. In the present study, physical fitness measurement values and observation values of technical skills were utilized. FAHP was used to determine the weights of the criteria and the observation values of technical skills by decision makers. Physical fitness measurement values were converted to fuzzy values by using a fuzzy set approach. Subsequently, the overall ranking of the candidate players was determined by the TOPSIS method. Results were compared with human experts’ opinions. It is observed that the developed model is more reliable to be used in decision making. The model architecture and experimental results along with illustrative examples are further demonstrated in the study.