Interval-valued Hesitant Fuzzy Elements Research Articles

An Improved Interval-Valued Hesitant Fuzzy Weighted Geometric Operator for Multi-Criterion Decision-Making

In this paper, an improved interval-valued hesitant fuzzy weighted geometric (IIVHFWG) operator for multi-criterion decision-making is proposed. This operator is free of the limitations of the existing interval-valued hesitant fuzzy weighted average operator, interval-valued hesitant fuzzy weighted geometric operator, generalized interval-valued hesitant fuzzy weighted geometric operator, interval-valued hesitant fuzzy Hammer weighted average operator, and interval-valued hesitant fuzzy Hammer weighted geometric operator, which are prone to being influenced by extreme values. Based on the proposed IIVHFWG operator, a new method to solve the multi-criterion decisionmaking problems with interval-valued hesitant fuzzy elements is presented. Several numerical examples together with comparisons are introduced to demonstrate the effectiveness and advantages of this method.

Dombi Interval-Valued Hesitant Fuzzy Aggregation Operators for Information Security Risk Assessment

This paper studies the improved aggregation operator based on Dombi and its application in multiattribute decision making (MADM). Firstly, the axiomatic definition of interval-valued hesitant fuzzy elements (IVHFEs) based on Dombi t-norm and t-conorm is introduced. Secondly, we propose a series of IVHFE aggregation operators (IVHFDWA, IVHFDWG, IVHFDOWA, IVHFDOWG, IVHFDHA, and IVHFDHG). In order to get the weight information of attributes, based on the basic idea of traditional distance formula, a distance formula between IVHFEs is constructed, and the optimization model of attribute weight is established. Then, a case study for selection of information security platform is given to demonstrate the merits of the developed method. The parameters of aggregation operator are analyzed in detail, and the critical value of parameters affecting decision making is given. Finally, this method is compared with other methods, which proves the validity and applicability of this method.

A Prospect Theory Based Probabilistic Interval-Valued Hesitant Fuzzy Sets with Interval Reference Point and Incomplete Weight Information for Multi-Criteria Decision Making

In this paper, we introduce probabilistic interval-valued hesitant fuzzy sets (PIVHFS) and probabilistic interval-valued hesitant fuzzy elements (PIVHFE), then define correlation operations. At the same time, the hybrid distance measures of PIVHFE and interval fuzzy number are defined, and the hybrid distance measures of two PIVHFEs are also defined. Considering the psychological behavior of decision makers, we use interval fuzzy number decision makers' psychological interval reference points to calculate the gains and losses of evaluation value. Furthermore, based on incomplete weight information, the grey relational analysis (GRA) method is extended to calculate the optimal weight. Then, based on the traditional TOPSIS method, the decision-making method is proposed. Finally, the proposed method is applied to an example, and the sensitivity analysis and stability analysis of the proposed method are also discussed.

Weighted Interval-Valued Hesitant Fuzzy Sets and Its Application in Group Decision Making

The interval-valued hesitant fuzzy set, which allows decision makers to use several interval numbers to assess a variable, is a useful tool to deal with situations in which people are hesitant in providing their interval-valued assessments. In this paper, we introduce the concept of weighted interval-valued hesitant fuzzy set, in which different weights are designed to these possible membership degrees, and the weights indicate that the decision maker has different confidence in giving every possible assessment of the membership degree. Then we define some basic operations such as union, intersection, complement, multiplication and power operation of weighted interval-valued hesitant fuzzy sets and weighted interval-valued hesitant fuzzy elements, discuss their operation properties, and propose the score function of the weighted interval-valued hesitant fuzzy element to compare two weighted hesitant fuzzy elements. Furthermore, we introduce the concept of hesitance degree of weighted interval-valued hesitant fuzzy element, present four aggregation operators such as the weighted interval-valued hesitant fuzzy-weighted averaging operator, the weighted interval-valued hesitant fuzzy-weighted geometric operator, the generalized weighted interval-valued hesitant fuzzy-weighted averaging operator and the generalized weighted interval-valued hesitant fuzzy-weighted geometric operator to aggregate weighted interval-valued hesitant fuzzy information, and build the mathematical model of multi-criteria group decision making based on the expert weights (known and unknown). Finally, a numerical example is given to illustrate the effectiveness and feasibility of our proposed method.

Fusions and preference relations based on probabilistic interval-valued hesitant fuzzy information in group decision making

This paper proposes probabilistic interval-valued hesitant fuzzy set (P-IVHFS) by integrating the probability distribution into IVHFS. The concept can make some vital but seemingly repetitive information [such as the same memberships of interval forms given by different decision makers (DMs)], which is an important basis for decision-making rather than being discarded. Moreover, some operations and aggregation operators of probabilistic interval-valued hesitant fuzzy elements (P-IVHFEs) and probabilistic interval-valued hesitant fuzzy preference relation (P-IVHFPR) are put forward, based on which, a decision-making model is proposed. It can keep almost all information given by the DMs so that the results calculated with the model will be much more reliable than those calculated with approaches based on IVHFSs. Finally, a practical case concerning the evaluation of intelligent transportation systems is given to verify the advantages of the proposed decision-making approach based on P-IVHFPRs.

An Approach to Interval-Valued Hesitant Fuzzy Multiattribute Group Decision Making Based on the Generalized Shapley-Choquet Integral

The purpose of this paper is to develop an approach to multiattribute group decision making under interval-valued hesitant fuzzy environment. To do this, this paper defines some new operations on interval-valued hesitant fuzzy elements, which eliminate the disadvantages of the existing operations. Considering the fact that elements in a set may be interdependent, two generalized interval-valued hesitant fuzzy operators based on the generalized Shapley function and the Choquet integral are defined. Then, some models for calculating the optimal fuzzy measures on the expert set and the ordered position set are established. Because fuzzy measures are defined on the power set, it makes the problem exponentially complex. To simplify the complexity of solving a fuzzy measure, models for the optimal 2-additive measures are constructed. Finally, an investment problem is offered to show the practicality and efficiency of the new method.

Solving group decision-making problems in manufacturing systems by an uncertain compromise ranking method

The purpose of this paper is to introduce a new modified compromise ranking method (VIKOR), known as sorting the possible alternatives and determining the compromise solution under interval-valued hesitant fuzzy sets (IVHFSs) for solving group decision-making problems in manufacturing systems. By using numerical measure and crisp data, the decision making is complex in real-life situations because the decision makers (DMs)' judgments are usually hesitant and vague. A more applicable and realistic approach based on interval-valued hesitant fuzzy elements (variables) is presented and defined as the inputs to the presented VIKOR method. The proposed interval-valued hesitant fuzzy modified VIKOR (IVHF-MVIKOR) method utilises the membership degrees to demonstrate the degrees of satisfiability for each possible alternative according to selected criteria for the manufacturing assessments. Then, a new index for relative importance of the DMs is introduced with the extended fuzzy sets. Also, some operations in the proposed IVHF-MVIKOR method are developed for manufacturing decision problems. Then, new indexes are presented with interval-valued hesitant fuzzy hamming distance measure for the purpose of rankings. Finally, by presenting a practical example in flexible manufacturing systems, the performance of IVHF-MVIKOR method is evaluated and compared with two well-known methods under interval-valued hesitant fuzzy information.

Solving group decision-making problems in manufacturing systems by an uncertain compromise ranking method

The purpose of this paper is to introduce a new modified compromise ranking method (VIKOR), known as sorting the possible alternatives and determining the compromise solution under interval-valued hesitant fuzzy sets (IVHFSs) for solving group decision-making problems in manufacturing systems. By using numerical measure and crisp data, the decision making is complex in real-life situations because the decision makers (DMs)' judgments are usually hesitant and vague. A more applicable and realistic approach based on interval-valued hesitant fuzzy elements (variables) is presented and defined as the inputs to the presented VIKOR method. The proposed interval-valued hesitant fuzzy modified VIKOR (IVHF-MVIKOR) method utilises the membership degrees to demonstrate the degrees of satisfiability for each possible alternative according to selected criteria for the manufacturing assessments. Then, a new index for relative importance of the DMs is introduced with the extended fuzzy sets. Also, some operations in the proposed IVHF-MVIKOR method are developed for manufacturing decision problems. Then, new indexes are presented with interval-valued hesitant fuzzy hamming distance measure for the purpose of rankings. Finally, by presenting a practical example in flexible manufacturing systems, the performance of IVHF-MVIKOR method is evaluated and compared with two well-known methods under interval-valued hesitant fuzzy information.

Interval-Valued Hesitant Fuzzy Multiattribute Group Decision Making Based on Improved Hamacher Aggregation Operators and Continuous Entropy

Under the interval-valued hesitant fuzzy information environment, we investigate a multiattribute group decision making (MAGDM) method with continuous entropy weights and improved Hamacher information aggregation operators. Firstly, we introduce the axiomatic definition of entropy for interval-valued hesitant fuzzy elements (IVHFEs) and construct a continuous entropy formula on the basis of the continuous ordered weighted averaging (COWA) operator. Then, based on the Hamachert-norm andt-conorm, the adjusted operational laws for IVHFEs are defined. In order to aggregate interval-valued hesitant fuzzy information, some new improved interval-valued hesitant fuzzy Hamacher aggregation operators are investigated, including the improved interval-valued hesitant fuzzy Hamacher ordered weighted averaging (I-IVHFHOWA) operator and the improved interval-valued hesitant fuzzy Hamacher ordered weighted geometric (I-IVHFHOWG) operator, the desirable properties of which are discussed. In addition, the relationship among these proposed operators is analyzed in detail. Applying the continuous entropy and the proposed operators, an approach to MAGDM is developed. Finally, a numerical example for emergency operating center (EOC) selection is provided, and comparative analyses with existing methods are performed to demonstrate that the proposed approach is both valid and practical to deal with group decision making problems.

A New Method of Multiattribute Decision-Making Based on Interval-Valued Hesitant Fuzzy Soft Sets and Its Application

Combining interval-valued hesitant fuzzy soft sets (IVHFSSs) and a new comparative law, we propose a new method, which can effectively solve multiattribute decision-making (MADM) problems. Firstly, a characteristic function of two interval values and a new comparative law of interval-valued hesitant fuzzy elements (IVHFEs) based on the possibility degree are proposed. Then, we define two important definitions of IVHFSSs including the interval-valued hesitant fuzzy soft quasi subset and soft quasi equal based on the new comparative law. Finally, an algorithm is presented to solve MADM problems. We also use the method proposed in this paper to evaluate the importance of major components of the well drilling mud pump.

Soft computing based on hierarchical evaluation approach and criteria interdependencies for energy decision-making problems: A case study

In numerous real-world energy decision problems, decision makers often encounter complex environments, in which existent imprecise data and uncertain information lead us to make an appropriate decision. In this paper, a new soft computing group decision-making approach is introduced based on novel compromise ranking method and interval-valued hesitant fuzzy sets (IVHFSs) for energy decision-making problems under multiple criteria. In the proposed approach, the assessment information is provided by energy experts or decision makers based on interval-valued hesitant fuzzy elements under incomplete criteria weights. In this respect, a new ranking index is presented respecting to interval-valued hesitant fuzzy Hamming distance measure to prioritize energy candidates, and criteria weights are computed based on an extended maximizing deviation method by considering the preferences experts' judgments about the relative importance of each criterion. Also, a decision making trial and evaluation laboratory (DEMATEL) method is extended under an IVHF-environment to compute the interdependencies between and within the selected criteria in the hierarchical structure. Accordingly, to demonstrate the applicability of the presented approach a case study and a practical example are provided regarding to hierarchical structure and criteria interdependencies relations for renewable energy and energy policy selection problems. Hence, the obtained computational results are compared with a fuzzy decision-making method from the recent literature based on some comparison parameters to show the advantages and constraints of the proposed approach. Finally, a sensitivity analysis is prepared to indicate effects of different criteria weights on ranking results to present the robustness or sensitiveness of the proposed soft computing approach versus the relative importance of criteria.

Multiple attribute group decision making based on interval-valued hesitant fuzzy information measures

Under the interval-valued hesitant fuzzy environment, we investigate a multiple attribute group decision making (MAGDM) method on the basis of some information measures. We first introduce three axiomatic definitions of information measures under interval-valued hesitant fuzzy environment, including the entropy, similarity measures and cross-entropy. Several information measure formulas for interval-valued hesitant fuzzy elements (IVHFEs) are further constructed, which is based on the continuous ordered weighted averaging (COWA) operator. Then, the relationship among the entropy, similarity measures and cross-entropy is discussed, from which we find that three information measures can be transformed by each other based on their axiomatic definitions. The programming model is established to determine optimal weight of attribute with the principle of minimum entropy and maximum cross-entropy. Furthermore, an approach to MAGDM is developed, in which the attribute values take the form of IVHFEs. Finally, a numerical example for emergency risk management (ERM) evaluation is provided to illustrate the application of the developed approach.

Soft computing-based new interval-valued hesitant fuzzy multi-criteria group assessment method with last aggregation to industrial decision problems

Recent studies have identified that multi-criteria group analysis methods should take the concepts of risk and uncertainty into account. In some real-life situations, determining the exact values for the potential alternatives' performances and criteria weights is so difficult. To overcome with these situations, their values should be regarded as fuzzy and fuzzy intervals. In this respect, interval-valued hesitant fuzzy sets (IVHFSs) as a suitable modern fuzzy sets theory can be considered because this theory allows decision makers (DMs) to assign some interval-values membership degrees for an alternative in terms of selected criteria under a set to margin of errors. Hence, this paper proposes a novel soft computing approach, namely IVHF-MCGA, based on new interval-valued hesitant fuzzy complex proportional assessment (IVHF-COPRAS) method that can be applied in solving the multi-criteria group decision-making (MCGDM) problems under uncertainty. In this approach, preference values of potential alternatives versus the selected criteria and weights of each criterion are expressed by linguistic variables and then are transformed to interval-valued hesitant fuzzy elements (IVHFEs). In addition, an interval-valued hesitant fuzzy entropy (IVHF-entropy) method is extended to determine the criteria weights by considering the DMs' opinions about the relative importance. Also, a new interval-valued hesitant fuzzy compromise solution (IVHF-CS) method is introduced to estimate the weight of each DM in the group decision-making process along with the last aggregation for the DMs' judgments to avoid the data loss. Then, three practical applications about the robot selection, industrial site selection and rapid prototyping process selection problems are considered to explain steps of the proposed IVHF-MCGA approach and to indicate its validity and applicability. Finally, a comparative analysis between the proposed approach and fuzzy group TOPSIS method is presented based on four comparison parameters, including adequacy to changes of alternatives and criteria, agility in decision process, influence of DMs' weights and impact of first and last aggregations, to show its suitability.

Correlation Coefficients of Interval-Valued Hesitant Fuzzy Sets and Their Application Based on the Shapley Function

Interval-valued hesitant fuzzy sets permit the membership degree of an element to have several possible interval values in [0, 1] rather than real numbers, which can well deal with inherent hesitancy and uncertainty in the human decision-making process. In this paper, we first point out the issue of the existing correlation coefficients of interval-valued hesitant fuzzy sets and then define several new ones, which do not have to consider the lengths of interval-valued hesitant fuzzy elements and the arrangement of their possible interval values. Since the assumption that the elements in a set are independent is usually violated, we further define several Shapley weighted correlation coefficients, which consider their interactions. To deal with the situations where the elements are correlative and the weight formation is incompletely known, models for the optimal fuzzy measures on a feature set and on an attribute set are established, respectively. Finally, a procedure to pattern recognition and multiattribute decision making with incomplete weight information and interactive conditions is developed. Meanwhile, the corresponding examples are provided to show the practicality and feasibility of the proposed procedures.

Interval-Valued Hesitant Fuzzy Prioritized Weighted Aggregation Operators for Multiple Attribute Decision Making

The paper proposes an interval-valued hesitant fuzzy prioritized weighted averaging (IVHFPWA) operator and an interval-valued hesitant fuzzy prioritized weighted geometric (IVHFPWG) operator for aggregating interval-valued hesitant fuzzy information and investigates some properties of the proposed aggregation operators. Then, the IVHFPWA and IVHFPWG operators are utilized to develop two approaches for dealing with the interval-valued hesitant fuzzy multiple attribute decision making problems in which the attributes are in different priority level. Through the IVHFPWA or IVHFPWG operator and the score function of interval-valued hesitant fuzzy elements (IVHFEs), the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example about talent introduction is given to demonstrate the practicality and effectiveness of the developed approaches.

The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment

Hesitant fuzzy set (HFS) is used to deal with the situations in which the decision makers hesitate among several values to assess an indicator, alternative, variable, etc. Recently, the multi-criteria decision making (MCDM) problems with hesitant fuzzy information have received increasing attentions and many corresponding MCDM methods have also been developed, but none of them can be used to solve the MCDM problems in case of considering the decision maker’s psychological behavior. In this study, we extend the TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) method, which is based on prospect theory and can effectively capture the decision maker’s psychological behavior, to solve this type of problems under hesitant fuzzy environment. Firstly, we develop two novel measured functions for comparing the magnitude of hesitant fuzzy elements and interval-valued hesitant fuzzy elements, which are more reasonable and effective compared with the existing measured functions. Then, we calculate the dominance degree of each alternative related to the others based on novel measured functions and distance measures. By aggregating these dominance degrees, we can further obtain the overall value of each alternative and whereby rank the alternatives. Finally, a decision making problem that concerns the evaluation and ranking of the service quality among domestic airlines is used to illustrate the validity and applicability of the proposed method.

Some interval-valued hesitant fuzzy aggregation operators based on Archimedean t-norm and t-conorm with their application in multi-criteria decision making

As a generalization of hesitant fuzzy set (HFS), interval-valued hesitant fuzzy set (IVHFS) permits an element's membership degree to be a set of several possible interval values and can therefore be used as an efficient mathematical tool for modeling people's hesitancy and uncertainty in multi-criteria decision making (MCDM). The Archimedean t-conorm and t-norm provide a generalization of a variety of other t-conorms and t-norms that include the Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms as special cases. In this paper, we first present some new operational laws for interval-valued hesitant fuzzy elements (IVHFEs) based on the Archimedean t-conorm and t-norm. Then, based on these operational laws, we develop several Archimedean t-conorm-and t-norm-based interval-valued hesitant fuzzy aggregation operators, which provide a family of interval-valued hesitant fuzzy aggregation operators that include the existing interval-valued hesitant fuzzy aggregation operators based on the Algebraic and Einstein t-conorms and t-norms as special cases. Some desired properties and special cases of the developed operators are investigated in detail. Furthermore, we develop an approach to MCDM under interval-valued hesitant fuzzy environments. Finally, an illustrative example is provided to show the effectiveness and practicality of the proposed operators and approach.

Continuous hesitant fuzzy aggregation operators and their application to decision making under interval-valued hesitant fuzzy setting.

Interval-valued hesitant fuzzy set (IVHFS), which is the further generalization of hesitant fuzzy set, can overcome the barrier that the precise membership degrees are sometimes hard to be specified and permit the membership degrees of an element to a set to have a few different interval values. To efficiently and effectively aggregate the interval-valued hesitant fuzzy information, in this paper, we investigate the continuous hesitant fuzzy aggregation operators with the aid of continuous OWA operator; the C-HFOWA operator and C-HFOWG operator are presented and their essential properties are studied in detail. Then, we extend the C-HFOW operators to aggregate multiple interval-valued hesitant fuzzy elements and then develop the weighted C-HFOW (WC-HFOWA and WC-HFOWG) operators, the ordered weighted C-HFOW (OWC-HFOWA and OWC-HFOWG) operators, and the synergetic weighted C-HFOWA (SWC-HFOWA and SWC-HFOWG) operators; some properties are also discussed to support them. Furthermore, a SWC-HFOW operators-based approach for multicriteria decision making problem is developed. Finally, a practical example involving the evaluation of service quality of high-tech enterprises is carried out and some comparative analyses are performed to demonstrate the applicability and effectiveness of the developed approaches.

An Approach to Interval-Valued Hesitant Fuzzy Multi-Attribute Decision Making with Incomplete Weight Information Based on Hybrid Shapley Operators

With respect to interval-valued hesitant fuzzy multi-attribute decision making, this study first presents a new ranking method for interval-valued hesitant fuzzy elements. In order to obtain the comprehensive values of alternatives, two induced generalized interval-valued hesitant fuzzy hybrid operators based on the Shapley function are defined, which globally consider the importance of elements and their ordered positions as well as reflect the interactions between them. If the weight information is incompletely known, models for the optimal weight vectors on the attribute set and on the ordered set are respectively established. Furthermore, an approach to interval-valued hesitant fuzzy multi-attribute decision making with incomplete weight information and interactive characteristics is developed. Finally, an illustrative example is provided to show the concrete application of the proposed procedure.