Multiple Attribute Group Decision-making Problems Research Articles

Analysis of Social Networks by Using Pythagorean Cubic Fuzzy Einstein Weighted Geometric Aggregation Operators

Pythagorean cubic set (PCFS) is the combination of the Pythagorean fuzzy set (PFS) and interval-valued Pythagorean fuzzy set (IVPFS). PCFS handle more uncertainties than PFS and IVPFS and thus are more extensive in their applications. The objective of this paper is under the PCFS to establish some novel operational laws and their corresponding Einstein weighted geometric aggregation operators. We describe some novel Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operators to handle multiple attribute group decision-making problems. The desirable relationship and the characteristics of the proposed operator are discussed in detail. Finally, a descriptive case is given to describe the practicality and the feasibility of the methodology established.

On a generalized entropy and dissimilarity measure in intuitionistic fuzzy environment with applications

Imprecision is an intrinsic characteristic of human behaviour. The concept of fuzziness is a well-recognized mechanism to quantify the imprecision and ambiguity of human judgement. In the last three decades, various extensions of fuzzy theory have been put forward. Intuitionistic fuzzy theory reveals some interesting aspects of approximate models. Intuitionistic fuzzy entropy measures the average amount of imprecision and ambiguity present in an intuitionistic fuzzy set. In this work, we propose flexible (generalized) measure of intuitionistic fuzzy (IF) entropy and establish its superiority over some existing IF-entropies. We introduce a generalized IF-dissimilarity measure and prove some of its properties. We also provide an improved TOPSIS method to handle multiple attribute group decision-making problem in intuitionistic fuzzy settings using our proposed IF-entropy, IF-dissimilarity and IF-correlation efficiency. We then investigates the performance of the proposed IF-dissimilarity in a pattern recognition problem and obtain encouraging results.

Multiple attribute group decision-making based on cubic linguistic Pythagorean fuzzy sets and power Hamy mean

The linguistic Pythagorean fuzzy sets (LPFSs), which employ linguistic terms to express membership and non-membership degrees, can effectively deal with decision makers’ complicated evaluation values in the process of multiple attribute group decision-making (MAGDM). To improve the ability of LPFSs in depicting fuzzy information, this paper generalized LPFSs to cubic LPFSs (CLPFSs) and studied CLPFSs-based MAGDM method. First, the definition, operational rules, comparison method and distance measure of CLPFSs are investigated. The CLPFSs fully adsorb the advantages of LPFSs and cubic fuzzy sets and hence they are suitable and flexible to depict attribute values in fuzzy and complicated decision-making environments. Second, based on the extension of power Hamy mean operator in CLPFSs, the cubic linguistic Pythagorean fuzzy power average operator, the cubic linguistic Pythagorean fuzzy power Hamy mean operator as well as their weighted forms were introduced. These aggregation operators can effectively and comprehensively aggregate attribute values in MAGDM problems. Besides, some important properties of these operators were studied. Finally, we presented a new MAGDM method based on CLPFSs and their aggregation operators. Illustrative examples and comparative analysis are provided to show the effectiveness and advantages of our proposed decision-making method.

A multi-attribute group decision making method considering both the correlation coefficient and hesitancy degrees under interval-valued intuitionistic fuzzy environment

In this paper, we propose a novel multiple attribute group decision making (MAGDM) method based on the correlation coefficient and hesitancy degrees under interval-valued intuitionistic fuzzy environment. Firstly, the conception of individual hesitancy degree and group hesitancy degree are defined to ensure the effective communication of information among group members and overcome the restrictions of the methodology of “majority rule” that some critical information of decision-making would be treated as conflicting opinions to be modified. Secondly, the correlation coefficients of interval-valued intuitionistic fuzzy set (IVIFS) are utilized to measure the similarity instead of distance functions in order to alleviate the drawbacks that do not consider the parameters of hesitancy. Thirdly, we extend the subjective assignment methods on utilizing the IVIFS to assign the weights of different decision makers in order to better address uncertainty in the MAGDM problem. Finally, TOPSIS and Linear programming optimization method are used to calculate the optimal attribute weight, which results in more accurate weights. A real-word application example has been presented to demonstrate the working of the proposed methodology. Moreover, a thorough comparison has been done with related existing works in order to show the validity of this methodology.

Ranking of interval type 2 fuzzy numbers using correlation coefficient and Mellin transform

Ranking of fuzzy numbers exhibits a significant role to solve the multiple attributes group decision making (MAGDM) problems. One of the commonly used indices for MAGDM problems is the correlation coefficient. Type-2 fuzzy numbers (T2FNs) are considered to be more effective than type-1 fuzzy numbers (T1FNs) to handle uncertainties in MAGDM problems. Moreover, T2FNs allow additional freedom to reflect uncertainty. In this paper, we propose a new approach to rank the interval type-2 fuzzy numbers (IT2FNs) using the correlation coefficient, where the correlation coefficient is obtained using the Mellin transform. Initially, we define the correlation coefficient for interval type-2 trapezoidal fuzzy numbers (IT2TrFNs) using Mellin transform and prove some of its major features. Then a new group decision-making method has been recommended using the proposed ranking concept. Eventually, three examples are given to demonstrate the usefulness of the suggested method. Finally, a comparative study is conducted to depict the feasibility of the developed ranking technique.

Unbalanced probabilistic linguistic decision-making method for multi-attribute group decision-making problems with heterogeneous relationships and incomplete information

In group decision-making problems, decision makers prefer to use several linguistic terms to describe their own perception and knowledge, and give their preference intensity of each possible linguistic terms based on their own understanding and interpretation. Due to the nonlinearity of decision maker’s cognition, the gaps between adjacent linguistic terms are unbalanced. The unbalanced probabilistic linguistic term set (UPLTS) is proposed to present such situation. To this phenomenon, a resolution framework is constructed to analyze multiple attribute group decision-making problems under unbalanced probabilistic linguistic environment. Firstly, the integration model based on evidential reasoning theory is proposed to aggregate UPLTSs from different groups in view of incomplete probabilistic distributions in UPLTS. Secondly, the transformation function based on proportional 2 tuple is developed to transform UPLTS into probabilistic linguistic term set, making it easier for subsequent analysis and processing. Thirdly, Based on the multiple types of partitioned structure relationship among attributes, partitioned fuzzy measure is developed to globally capture these interactions among attributes. Then the probabilistic linguistic Choquet integral operator with partitioned fuzzy measure is proposed to obtain the comprehensive performances of alternatives. Lastly, the effectiveness and practicability of the proposed method is demonstrated using three numerical examples and comparing with other methods.

A New Order Function for Interval-Valued Intuitionistic Fuzzy Numbers and Its Application in Group Decision Making

Purpose: We introduce a multiple attribute group decision making (MAGDM) method under interval-valued intuitionistic fuzzy (IVIF) domain. In these MAGDM problems, attribute values and attribute weights are represented in terms of IVIF sets (IVIFSs). Methodology: The present methodology is divided into two parts. Firstly, an appropriate order function has been developed for IVIFSs. Secondly, an algorithm has been proposed for solving the MAGDM problems with IVIF decision information. Findings: To show the effectiveness of the proposed method, a theoretical comparative study with respect to some existing literature is provided. A real-life based numerical example is given to demonstrate the proposed solution procedure. The advantage of this method is also described. Values: Using proposed order function, interval-valued intuitionistic fuzzy hybrid geometric and ordered weighted averaging operators, the present method provides a most preferred alternative in six steps. On the basis of a comparative study, it can be observed that the present method simple and easy to implement for solving the IVIF MAGDM model occurrence in real-life decision making problems.

Some self-evaluation models of enterprise’s credit based on some probabilistic double hierarchy linguistic aggregation operators

Probabilistic double hierarchy linguistic term set (PDHLTS) can not only express the complex linguistic information that the probabilistic linguistic term set (PLTS) cannot express, but also reflect the frequency or importance of linguistic term set (LTS)that cannot be reflected by the double hierarchy linguistic term set (DHLTS). It is an effective tool to deal with multiple attribute group decision making (MAGDM) problems. Therefore, in this paper, we propose several aggregation operators which can aggregate PDHLTS information and apply them to MAGDM problems. Firstly, the basic notion of PDHLTS is reviewed, and the distance formula and algorithm of PDHLTS are defined; then, extant weighted averaging (WA) operator, weighted geometric(WG) operator and power weighted averaging (PWA) operator, power weighted geometric(PWG) operator to PDHLTS, and establish probability double hierarchy linguistic weighted averaging (PDHLWA) operator, probability double hierarchy linguistic weighted geometric (PDHLWG) operator, probability double hierarchy linguistic power weighted averaging (PDHLPWA) operator, probability double hierarchy linguistic power weighted geometric (PDHLPWG) operator; in addition, The idempotency, boundedness and monotonicity of these aggregation operators are studied; what’s more, those aggregation operators are proposed to establish the enterprise credit self-evaluation model; Finally, compared with the available probabilistic double hierarchy linguistic MAGDM methods, the defined model is proved to be scientific and effective.

A game theoretic approach to solve multiple group decision making problems with interval-valued intuitionistic fuzzy decision matrices

ABSTRACT The objective of this paper is to develop a two person zero-sum game theoretic solution approach for multiple attribute group decision making (MAGDM) problems under interval-valued intuitionistic fuzzy environment. In this way, a new order function is introduced to defuzzify the interval-valued intuitionistic fuzzy numbers (IVIFNs) and discuss its properties. In complicated and uncertain group decision-making problems, decision makers face a problem for assessing the group weights for alternatives and attributes. In this paper, the weights of present MAGDM problem with interval-valued intuitionistic fuzzy decision matrices (matrices in which entries are represented by IVIFNs) are calculated by converting it into a traditional two person zero-sum game problem. Then, the expected score for each alternative is calculated using the optimal solution of this game problem. On the basis of the calculated expected score values, the prespecified alternatives are ranked in order to find the best alternative. The validity and applicability of the developed approach are given by a numerical example.

TODIM method for multiple attribute group decision making based on cumulative prospect theory with 2‐tuple linguistic neutrosophic sets

The 2-tuple linguistic neutrosophic set (2TLNS) does a good job in coping with vague and uncertain environments and has been widely used in numerous fields. In this article, we come up with a new solution of multiple attribute group decision-making (MAGDM) problems, named as the modified TODIM method based on cumulative prospect theory and 2-tuple linguistic neutrosophic sets (CPT-2TLN-TODIM method). To ensure relatively objective attribute weights, the CRITIC is integrated into this new model in this paper. The CPT-2TLN-TODIM method fully considers the psychological state of decision makers that makes it more realistic. What's more, we introduce the application of CPT-2TLN-TODIM method in the evaluation of network security assessment team for verifying the practicability of the CPT-2TLN-TODIM method. And finally, the reliability of CPT-2TLN-TODIM method is confirmed by comparing with other existed methods. In the immediate future, we will further explore the application of this method in other fields and keep trying to put forward more scientific and reasonable methods to solve MAGDM problems.

Exponential operational laws and new aggregation operators for intuitionistic multiplicative set in multiple-attribute group decision making process

The intuitionistic multiplicative preference set is one of the replacements to the intuitionistic fuzzy preference set, where the preferences related to the object is asymmetrical distribution about 1. In it, Saaty’s 1–9 scale has been used to represent the uncertain and imprecise information. Meanwhile, an aggregation operator by using general operational laws for some fuzzy sets is an important task to aggregate the different numbers. Motivated by these primary characteristics, it is interesting to present the concept of exponential operational laws, which differs from the traditional laws by the way, in which bases are real numbers while exponents are the intuitionistic multiplicative numbers. In this paper, we develop a methodto solve the Multiple Attribute Group Decision Making (MAGDM) problem under the Intuitionistic Multiplicative Sets (IMS) environment. To do it, firstly, we define some new exponential operational laws and a score function for IMS and studied their properties. Secondly, based on this, we develop some averaging and geometric aggregation operators and characterize their various properties. Thirdly, a novel approach is promoted to solve MAGDM problems with IMS information. Finally, some numerical illustrations are given with a comparative study to verify the approach.

Novel classes of coverings based multigranulation fuzzy rough sets and corresponding applications to multiple attribute group decision-making

The notion of covering based multigranulation fuzzy rough set (CMGFRS) models is a generalization of both granular computing and covering based fuzzy rough sets. Therefore it has become a powerful tool for coping with vague and multigranular information in cognition. In this paper we introduce three kinds of CMGFRS models by means of fuzzy β-neighborhoods and fuzzy complementary β-neighborhoods, and we investigate their axiomatic properties. We investigate three respective types of coverings based CMGFRS models, namely, optimistic, pessimistic and variable precision setups. In particular, by using multigranulation fuzzy measure degrees and multigranulation fuzzy complementary measure degrees, we derive three types of coverings based γ-optimistic (γ-pessimistic) CMGFRSs and E (F, G)-optimistic and E (F, G)-pessimistic CMGFRSs, respectively. We discuss the interrelationships among these three types of CMGFRS models and covering based Zhan-CMGFRS models. In view of the theoretical analysis for these three types of CMGFRS models, we put forward a novel methodology to multiple attribute group decision-making problem with evaluation of fuzzy information. An effective example is fully developed, hence concluding the applicability of the proposed methodology.

Research on the urban resilience evaluation with hybrid multiple attribute TOPSIS method: an example in China.

Cities are important barriers to protect people’s lives and property in the face of natural disasters, economic fluctuations and epidemic diseases. The evaluation of urban resilience is a hybrid multiple attribute group decision-making problem involving both crisp and fuzzy indicators. In order to evaluate the urban resilience reasonably and quantitatively, an urban resilience evaluation index system is established, including four primary indicators of ecological environment, municipal facilities, economic development and social development, and 28 secondary indicators. An evaluation model based on the theory of intuitionistic fuzzy set and TOPSIS method is proposed. The intuitionistic trapezoidal fuzzy number is used to quantify the fuzzy index and determine the weights of experts. The weight of each index is determined based on the maximizing deviation method. The relevant data of Dalian City from 2013 to 2017 are collected to evaluate the city resilience, and a sensitivity analysis is carried out based on the proposed model. The results may provide insights for the further urban resilience promotion.

A novel approach to multi-criteria group decision-making problems based on linguistic D numbers

The theory of D numbers (DNs) is an extension of Dempster–Shafer (D–S) evidence theory. Linguistic DNs (LDNs) combine the advantages of the linguistic terms (LTs) and DNs, and can express the uncertain and incomplete information more easily and precisely in decision-making. In addition, the Bonferroni mean (BM) operator and the TOPSIS method are both useful tools to address multiple-attribute group decision-making (MAGDM) problems. Then, in this paper, we extend the BM operator based on the new defined operational rules of LDNs to propose the linguistic D number arithmetic Bonferroni mean (LDABM) operator and the linguistic D number weighted arithmetic Bonferroni mean (LDWABM) operator. We also integrate the proposed LDWABM operator and the TOPSIS method, and further establish a novel method to address MAGDM problems. Finally, we use the proposed method to solve the practical problems of investment decision-making and materials supplier selection and then we prove the feasibility and the superiority of the proposed method by comparing with other methods.

Entropy-Based GLDS Method for Social Capital Selection of a PPP Project with q-Rung Orthopair Fuzzy Information.

The social capital selection of a public–private-partnership (PPP) project could be regarded as a classical multiple attribute group decision-making (MAGDM) issue. In this paper, based on the traditional gained and lost dominance score (GLDS) method, the q-rung orthopair fuzzy entropy-based GLDS method was used to solve MAGDM problems. First, some basic theories related to the q-rung orthopair fuzzy sets (q-ROFSs) are briefly reviewed. Then, to fuse the q-rung orthopair fuzzy information effectively, the q-rung orthopair fuzzy Hamacher weighting average (q-ROFHWA) operator and q-rung orthopair fuzzy Hamacher weighting geometric (q-ROFHWG) operator based on the Hamacher operation laws are proposed. Moreover, to determine the attribute weights, the q-rung orthopair fuzzy entropy (q-ROFE) is proposed and some significant merits of it are discussed. Next, based on the q-ROFHWA operator, q-ROFE, and the traditional GLDS method, a MAGDM model with q-rung orthopair fuzzy information is built. In the end, a numerical example for social capital selection of PPP projects is provided to testify the proposed method and deliver a comparative analysis.

A novel trigonometric operation-based q-rung orthopair fuzzy aggregation operator and its fundamental properties

The q-rung orthopair fuzzy sets (q-ROFSs) are a prominent idea to express the fuzzy information in the decision-making process and are the generalization of the existing intuitionistic fuzzy set and Pythagorean fuzzy set. The q-ROFSs can dynamically adapt the information by changing the parameter $$q\ge 1$$ based on the membership degree and therefore support more innumerable possibilities. Driven by these requisite characteristics, this paper aspires to present some sine trigonometric operations laws for q-ROFSs. The sine trigonometry function preserves the periodicity and symmetric about the origin, and hence, it satisfies the decision-maker preferences toward the evaluation of the objects. Associated with these laws, we define a series of new aggregation operators named as sine trigonometry weighted averaging and geometric operators to aggregate the q-rung orthopair fuzzy information. The fundamental relations between the proposed operators are also examined. Afterward, we present a group decision-making technique to solve the multiple attribute group decision-making problems based on proposed operators and illustrate with a numerical example to verify it. The superiors, as well as the advantages of the proposed operators, are also discussed in it. Lastly, the influence of the membership degrees on the operations has been investigated and found that when the parameter q increases from 2 to 4 and then from 4 to 7, then there is the certain change in the range of the score values.

An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects

Multiple attribute group decision-making (MAGDM) methods have a significant influence on decision-making in a variety of strategic fields, including science, business and real-life studies. The problem of evaluation in green building energy-saving design projects could be regarded as a type of MAGDM problem. The evaluation based on distance from average solution (EDAS) method is one of the MAGDM methods, which simplifies the traditional decision-making process. Symmetry among some attributes that are known and unknown as well as between pure attribute sets and fuzzy attribute membership sets can be an effective way to solve MAGDM problems. In this paper, the classical EDAS method is extended to intuitionistic fuzzy environments to solve some MAGDM issues. First, some concepts of intuitionistic fuzzy sets (IFSs) are briefly reviewed. Then, by integrating the EDAS method with IFSs, we establish an IF-EDAS method to solve the MAGDM issues and present all calculating procedures in detail. Finally, we provide an empirical application for evaluating green building energy-saving design projects to demonstrate this novel method. Some comparative analyses are also made to show the merits of the method.

A new approach to cubic q-rung orthopair fuzzy multiple attribute group decision-making based on power Muirhead mean

The q-rung orthopair fuzzy sets (q-ROFSs) have been proved to be an efficient tool in expressing decision makers’ (DMs) evaluation values in multiple attribute group decision-making (MAGDM) procedure. To more effectively represent DMs’ evaluation information in complicated MAGDM process, this paper proposes a new tool, called cubic q-rung orthopair fuzzy sets (Cq-ROFSs), based on the combination of q-ROFSs with interval-valued q-ROFSs. Then, we investigate MAGDM problems in which DMs’ preference information is given in terms of cubic q-rung orthopair fuzzy numbers. First, the definition, operations and comparison method of Cq-ROFSs are introduced. Second, to effectively aggregate cubic q-rung orthopair fuzzy information we propose the cubic q-rung orthopair fuzzy power average operator, the cubic q-rung orthopair fuzzy power Muirhead mean operator as well as their weighted forms. We illustrate the powerfulness and flexibility of the proposed operators in fusing cubic q-rung orthopair fuzzy decision-making information. Third, on the basis of the proposed operators we give the main steps of a novel cubic q-rung orthopair fuzzy MAGDM method. We utilize the method to solve real MAGDM problems to prove its effectiveness and validity. Finally, we explain why DMs should choose our proposed method rather than some others through comparison analysis.

A Generalized p-Norm Knowledge-Based Score Function for an Interval-Valued Intuitionistic Fuzzy Set in Decision Making

This paper addresses the commonly used methods for solving multiple attribute group decision-making problems under an interval-valued intuitionistic fuzzy environment. It is shown that either the most popular score function and its latterly improved versions or other related methods may lead to the undesirable results. Apart from these seemly heuristic methods, we propose a generalized p-norm knowledge-based score function for the interval-valued intuitionistic fuzzy set, which is a generalization of the score function for intuitionistic fuzzy sets as well. The proposed score function is defined as the average amount of knowledge of information depicted in terms of the interval-valued intuitionistic fuzzy sets, that regards their quantitative level of information referred to a reference level, fuzziness and qualitative significance of information. It is proved that the proposed score function has good algebraic properties, such as monotonicity under multiplication or aggregation operator, which are not preserved in the case of existing score functions. We show that the proposed score function is a simple, easily interpreted approach, which provides intuitive results in ranking the IvIFSs as well as IFSs, particularly. Several illustrative examples are performed to demonstrate the superiority of the proposed method in measuring and discriminating the IvIFSs. A comparative analysis with the preceding methods in solving decision-making problems is conducted to show the effectiveness of the proposed method in real-life applications.

Multiple attribute group decision‐making based on order‐α divergence and entropy measures under q‐rung orthopair fuzzy environment

The q-rung orthopair fuzzy set (qROPFS), proposed by Yager, is a more effective and proficient tool to represent uncertain or vague information in real-life situations. Divergence and entropy are two important measures, which have been extensively studied in different information environments, including fuzzy, intuitionistic fuzzy, interval-valued fuzzy, and Pythagorean fuzzy. In the present communication, we study the divergence and entropy measures under the q-rung orthopair fuzzy environment. First, the work defines two new order-α divergence measures for qROPFSs to quantify the information of discrimination between two qROPFSs. We also examine several mathematical properties associated with order-α qROPF divergence measures in detail. Second, the paper introduces two new parametric entropy functions called “order-α qROPF entropy measures” to measure the degree of fuzziness associated with a qROPFS. We show that the proposed order-α divergence and entropy measures include several existing divergence and entropy measures as their particular cases. Further, the paper develops a new decision-making approach to solve multiple attribute group decision-making problems under the qROPF environment where the information about the attribute weights is completely unknown or partially known. Finally, an example of selecting the best enterprise resource planning system is provided to illustrate the decision-making steps and effectiveness of the proposed approach.