Interval-valued Hesitant Fuzzy Sets Research Articles

A generalized TODIM evaluation approach based on the novel score function and trust network under interval-valued hesitant fuzzy environment

In multi-attribute group decision-making (MAGDM), the social relationships among group members, professional backgrounds and psychological behaviors of decision makers are important factors for achieving high-quality group decisions. However, realizing the consideration of these factors in fuzzy and complex group decision-making scenarios is an important and urgent issue. In MAGDM problems, interval-valued hesitant fuzzy sets (IVHFSs) can better show the hesitant and fuzzy uncertainty information because they are an extension of fuzzy sets. This paper focuses on the score function of IVHFSs and develops a generalized TODIM evaluation method utilizing the novel score functions of IVHFEs and trust networks. The developed method effectively realizes the application of these factors in fuzzy and complex group decision making scenarios. In this contribution, a new deviation function for IVHFSs is created in this paper. It takes into account two types of deviation that are common in IVHFSs: the ambiguity of interval width and the hesitation and uncertainty of multiple intervals. In the sequel, a novel score function for IVHFSs is developed with help of the novel deviation function and the traditional score function. Concerning the perspective of group consistency, a model for determining expert weights is constructed using quality parameters, which combine the novel score function with the trust network. A subjective and objective comprehensive weighting model is made to get attribute weights based on how certain decision information is and the experts’ subjective experience. This is done because the more uncertain something is, the higher the risk. Eventually, we develop a generalized TODIM technique for MAGDM based on the novel score function and trust network in IVHFS environment. The developed approach is demonstrated with a preventive engineering decision-making problem. Sensitivity and comparative analyses are implemented to demonstrate the superiority, stability, and validity of the proposed method in this paper. According to the findings of this study, we conclude that the proposed methodology offers a comprehensive and systematic approach to evaluate MAGDM in the context of IVHFS with trust networks.

Cloud model-based multi-stage multi-attribute decision-making method under probabilistic interval-valued hesitant fuzzy environment

Due to the complexity of multi-stage multi-attribute decision-making (MADM) problems, decision-makers (DMs) frequently provide incomplete and hesitant evaluation information. A crucial difficulty in multi-stage MADM problems is how to make a valid decision under uncertain environment. Firstly, we employ probabilistic interval-valued hesitant fuzzy set (PIVHFS) to characterize the hesitancy of DMs, and then build an optimization model to obtain the probability information based on the score function, deviation function of probabilistic interval-valued hesitant fuzzy element (PIVHFE) and information entropy. Secondly, cloud model is introduced to optimize the calculation process and an optimization model is developed to realize the transformation from PIVHFS to cloud model. Thirdly, a method to solve the stage weights is proposed by synthesizing the within-stage and between-stage decision information. Similarly, the attribute weights are obtained by constructing a mathematical model based on the within-attribute and between-attribute decision information. The noted method can utilize decision information to its fullest. Then, the regret theory is applied to a multi-stage MADM problem. Finally, the applicability and effectiveness of the method are illustrated through case analysis and methods comparison.

Some Information Measures for Interval‐Valued Hesitant Fuzzy Sets in Multiple Criteria Decision‐Making

In the fields of information science and artificial intelligence, dealing with uncertainty, fuzziness, and complexity has always been a hot and difficult research topic. Especially in modern society, with the continuous development of science and technology, people are facing more and more complex problems. Interval‐valued hesitant fuzzy set (IVHFS) is an extended form of a fuzzy set. It can more flexibly express and handle uncertainty and fuzziness in decision‐making processes. However, in the practical application of the IVHFS, its information measurement is crucial, which is directly related to the application value of the IVHFS in various fields. Therefore, studying the information measurement of the IVHFS has important theoretical significance and practical value for the fields of information science and artificial intelligence. In spite of significant advances, entropy and similarity as the well‐known information measures for interval‐valued hesitant fuzzy information have not yet been thoroughly researched. In this contribution, we investigate information measures in the IVHFS, including nonprobabilistic entropy, similarity, and cross‐entropy. We first analyze the change law of hesitating uncertainty and fuzzy uncertainty in geometric space, and a nonprobabilistic entropy measurement method and its axiomatic definition for IVHFS are further developed. Then, a novel similarity measurement formula for IVHFS and its axiomatic requirements are proposed on the basis of the two nonfuzzy elements ( and ). Furthermore, the novel similarity measure is used to construct the cross‐entropy measure for IVHFS and its axiomatic requirements based on the association between the similarity and the cross‐entropy. Lastly, a MAGDM method is proposed by using the developed three information measures, and the efficacy of the proposed method is demonstrated by a numerical example of emergency communication support capacity evaluation. Comparative analysis and computational cost analysis are implemented to demonstrate the superiority and validity of the proposed information measures.

The extended ELECTRE III group decision making method based on regret theory under probabilistic interval-valued hesitant fuzzy environments

The probabilistic interval-valued hesitant fuzzy set (PIVHFS), being able to generalize interval-valued hesitant fuzzy sets embedded with the decision makers (DMs)’ preferences for different elements, can depict uncertain decision information completely and precisely. Therefore, with this study, we propose an extended Elimination and Choice Translating Reality (ELECTRE) III method based on regret theory and Borda rule under a PIVHFS environment, which has the advantage of considering the regret psychology of the DMs and subordinate ranking orders. Using the proposed methodology, a fuzzy bidirectional projection measure is developed to overcome the shortcoming of unidirectional projection. The proposed method is validated by two numerical examples, and the effectiveness of the method is demonstrated through sensitivity and comparative analyses.

An efficiency-based interactive dynamic technique with interval-valued hesitant fuzzy constraint cone for rescue route planning

This paper develops an efficiency-based interactive dynamic technique to design the post-disaster rescue route by combining the data envelopment analysis (DEA), traveling salesman problem (TSP) and interval-valued hesitant fuzzy constraint cone (IVHFCC). The traditional DEA potentially presumes that the weights of the input (output) criteria are unrestricted, resulting in its inapplicability to some complex decision problems. To restrict the weights of input (output) criteria legitimately, this paper constructs an IVHFCC based on expert’s subjective preferences in the form of interval-valued hesitant fuzzy sets. This paper defines the additive consistent normal interval-valued hesitant fuzzy preference relation (NIVHFPR) and analyses its desirable properties. Based on these properties, three linear programming models are proposed to repair the incomplete NIVHFPR, judge the additive consistency of the repaired NIVHFPR and generate the additive consistent NIVHFPR. By incorporating the TSP, DEA and the constructed IVHFCC, an efficiency-based interactive dynamic model is originally proposed to generate the optimal rescue route for the rescue team. Some desirable properties of this efficiency-based interactive dynamic model are analyzed. The validity of this model is demonstrated with a real post-disaster rescue route planning problem of Wenchuan earthquake. Some comparative analyses are provided to show the superiorities of the proposed technique.

Two-Stage Multiobjective Decision-Making Method Based on Agricultural Water-Energy-Food Nexus: Case Study in Hetao Irrigation District, China

Due to the complex feedback cycle relationship among water resources, energy, and food, the agricultural water management mode and concept with a single object or two objects as the core can no longer meet the inherent requirements of the coordinated, sustained, and healthy development of economy and society. A two-stage multiobjective decision-making method based on a θ-dominance evolutionary algorithm (θ-DEA) is proposed for the optimization of agricultural systems with a water-energy-food nexus. First, the Pareto noninferior solution set of the model is obtained by the evolutionary algorithm based on θ-DEA. Then, the economic benefit and carbon emission of each solution are evaluated, and the optimal solution is selected by using the multiobjective decision-making method based on interval-valued hesitant fuzzy sets and the technique for order preference by similarity to ideal solution (TOPSIS). Through a case study of the Hetao Irrigation District (HID), the best optimization results are obtained when the carbon footprint is 4.16×109 kg CO2 eq, and the economic benefit is 8.09×109 yuan. The example shows that the model can well select the optimal allocation solution of water resources, which not only has high economic benefits and low pollution discharge but also meets the subjective wishes of decision-makers. The model has good practicability and stability.

Three-way decisions based on hesitant sets over three-way decision spaces

The concept of hesitant set was introduced by Hu, which purpose is to unify hesitant fuzzy sets, interval-valued hesitant fuzzy sets, shadowed sets etc. This paper discusses the construction of decision evaluation functions and three-way decisions based on hesitant sets over three-way decision spaces. First, this paper presents the construction methods of decision evaluation functions based on hesitant sets over semi-three-way decision spaces and quasi-three-way decision spaces. Second, this paper presents some transformation methods of decision evaluation functions based on hesitant sets from semi-three-way decision spaces and quasi-three-way decision spaces to three-way decision spaces. Third, this paper presents three-way decisions based on hesitant sets. Numerous examples are given to support this paper’s argument on the existence of semi-(quasi-)decision evaluation functions and significance of the transformation methods.

On Uncertainty Measures of the Interval-Valued Hesitant Fuzzy Set

Interval-valued hesitant fuzzy sets (IVHFS), as a kind of decision information presenting tool which is more complicated and more scientific and more elastic, have an important practical value in multiattribute decision-making. There is little research on the uncertainty of IVHFS. The existing uncertainty measure cannot distinguish different IVHFS in some contexts. In my opinion, for an IVHFS, there should exist two types of uncertainty: one is the fuzziness of an IVHFS and the other is the nonspecificity of the IVHFS. To the best of our knowledge, the existing index to measure the uncertainty of IVHFS all are single indexes, which could not consider the two facets of an IVHFS. First, a review is given on the entropy of the interval-valued hesitant fuzzy set, and the fact that existing research cannot distinguish different interval-valued hesitant fuzzy sets in some circumstances is pointed out. With regard to the uncertainty measures of the interval-valued hesitant fuzzy set, we propose a two-tuple index to measure it. One index is used to measure the fuzziness of the interval-valued hesitant fuzzy set, and the other index is used to measure the nonspecificity of it. The method to construct the index is also given. The proposed two-tuple index can make up the fault of the existing interval-valued hesitant fuzzy set’s entropy measure.

Interval Grey Hesitant Fuzzy Set and Its Applications in Decision-Making

The use of interval-valued hesitant fuzzy sets (IVHFS) can aid decision-makers in evaluating a variable using multiple interval numbers, making it a valuable tool for addressing decision-making problems. However, it fails to obtain information with greyness. The grey fuzzy set (GFS) can improve this problem but studies on it have lost the advantages of IVHFS. In order to improve the accuracy of decision-making and obtain more reasonable results, it is important to enhance the description of real-life information. We combined IVHFS and GFS and defined a novel fuzzy set named interval grey hesitant fuzzy set (IGHFS), in which possible degrees of grey numbers are designed to indicate the upper and lower limits of the interval number. Meanwhile, its basic operational laws, score function, entropy method, and distance measures are proposed. And then, a multicriteria decision-making (MCDM) model IGHFS-TOPSIS is developed based on them. Finally, an example of MOOC platform selection issues for teaching courses illustrates the effectiveness and feasibility of the decision model under the IGHFS.

A hybrid multi-stage decision-making method with probabilistic interval-valued hesitant fuzzy set for 3D printed composite material selection

The 3D printed composite material selection is of great interest due to its extensive application prospect and can be considered as a challenging multiple-criteria decision making (MCDM) issue. The hesitation and uncertainty of experts are difficult to measure, and the high degree of interaction among criteria is often overlooked in the decision process. In addition, composites are required to serve in harsh environments for various mechanical and industrial fields, resulting in the degradation of mechanical properties. In this study, a hybrid multi-stage decision-making method is developed to conduct 3D printed composite material selection in harsh environments. The theory of probabilistic interval-valued hesitant fuzzy set (PIVHFS) is proposed to characterize the decision-making information of experts, which can effectively quantify the assessment in uncertain environments. An integrated method that combines Choquet fuzzy integral and Shapley value is proposed to obtain the weight vector of criteria, which can reflect the mutual influence between criteria and their overall importance. The final decision-making result and the optimal alternative can be calculated by the PIVHFS-based Tomada de Decisão Interativa Multicritério and Technique for order preference by similarity to an ideal solution (TODIM-TOPSIS) method. An empirical application, i.e., 3D printed composites in the background of automotive chassis, is applied to validate the application of the proposed method. Comparative analysis, sensitivity analysis, and managerial implications are also conducted to illustrate the validity of the method. This paper provides a valuable tool for addressing the material selection issue of 3D printed composites from a multi-criteria perspective.

Selection of suitable biomass conservation process techniques: a versatile approach to normal wiggly interval-valued hesitant fuzzy set using multi-criteria decision making

A country that relies on developing industrialization and GDP requires a lot of energy. Biomass is emerging as one of the possible renewable energy resources that may be used to generate energy. Through the proper channels, such as chemical, biochemical, and thermochemical processes, it can be turned into electricity. In the context of India, the potential sources of biomass can be broken down into agricultural waste, tanning waste, sewage, vegetable waste, food, meat waste, and liquor waste. Each form of biomass energy so extracted has advantages and downsides, so determining which one is best is crucial to reaping the most benefits. The selection of biomass conversion methods is especially significant since it requires a careful study of multiple factors, which can be aided by fuzzy multi-criteria decision-making (MCDM) models. This paper proposes the normal wiggly interval-valued hesitant fuzzy-based decision-making trial and evaluation laboratory model (DEMATEL) and the Preference Ranking Organization METHod for Enrichment of Evaluations II (PROMETHEE) for assessing the problem of determining a workable biomass production technique. The proposed framework is used to assess the production processes under consideration based on parameters such as fuel cost, technical cost, environmental safety, and CO_2 emission levels. Bioethanol has been developed as a viable industrial option due to its low carbon footprint and environmental viability. Furthermore, the superiority of the suggested model is demonstrated by comparing the results to other current methodologies. According to comparative study, the suggested framework might be developed to handle complex scenarios with many variables.

Undergraduate Teaching Audit and Evaluation Using an Extended ORESTE Method with Interval-Valued Hesitant Fuzzy Linguistic Sets

Undergraduate teaching audit and evaluation (UTAE) plays a substantial role in the teaching quality assurance and monitoring of universities. It achieves the goal of selecting the best university for promoting the quality of higher education in China. Generally, the UTAE is a complex decision-making problem by considering competing evaluation criteria. Moreover, the evaluation information on the teaching quality of universities is often ambiguous and hesitant because of the vagueness existing in human judgments. Previous studies on UTAE have paid subtle attention towards the managing of linguistic expressions and the performance priority of universities. The interval-valued hesitant fuzzy linguistic sets (IVHFLSs) can effectively describe uncertainty, hesitancy, and inconsistency inherent in decision-making process. The ORESTE (organísation, rangement et Synthèse de données relarionnelles, in French) is a new outranking decision-making method which can show detailed distinctions between alternatives. Therefore, in this study, we propose a new UTAE approach based on the VHFLSs and ORESTE method to resolve the prioritization of universities for selecting the optimal university to benchmark. Specifically, the presented method handles the hesitant and uncertain linguistic expressions of experts by adopting the IVHFLSs and determines the ranking of universities with an extended ORESTE approach. Finally, a practical UTAE example illustrates the feasibility the proposed approach and a comparison analysis provides grounding for the superiority of the integrated approach. When the obtained results are evaluated, U2 has been determined as the best university. The results indicate the good performance of the proposed UTAE approach in evaluating and improving the teaching quality of universities.

A Novel Source Code Clone Detection Method Based on Dual-GCN and IVHFS

Source code clone detection, which can identify code fragments with similar functions, plays a significant role in software development and quality assurance. Existing methods either extract single syntactic or semantic information, or ignore the associated information between code statements in different structures. It is difficult for these methods to effectively detect clone pairs with similar functions. In this paper, we propose a new model based on a dual graph convolutional network (GCN) and interval-valued hesitant fuzzy set (IVHFS), which we named DG-IVHFS. Specifically, we simplified and grouped the abstract syntax tree (AST) of source code to obtain the group representations. The group representations of the AST, as well as the control flow graph (CFG) representations, were transformed into graph structures, and then we applied GCNs on them to learn dependencies between nodes. In addition, we introduced IVHFS into the model for a more comprehensive evaluation of similarity. Our experimental results demonstrated that the precision, recall, and F1-scores of DG-IVHFS on the BigCloneBench and GoogleCodeJam datasets reached 98, 97 and 97% and 98, 93 and 95%, respectively, exceeding current state-of-the-art models. Moreover, our model performed well in terms of time consumption.

Matrix games involving interval-valued hesitant fuzzy linguistic sets and its application to electric vehicles

Game theory has been successfully applied in a variety of domains to deal with competitive environments between individuals or groups. The matrix games involving fuzzy, interval fuzzy, and intuitionistic fuzzy numbers exclusively examine the numeric components of an issue. However, several researchers have also examined various extensions of conventional game theory, considering the ambiguous situations for payoffs and goals. In many real-life scenarios, qualitative information is often critical in expressing the payoffs of a matrix game. Thus, the present work contributes to the field of matrix games where the payoffs have been quantified via qualitative variables, termed interval-valued hesitant fuzzy linguistic sets. The mathematical formulation and solution concept for matrix games involving interval-valued hesitant fuzzy linguistic numbers is designed by utilizing an aggregation operator supported by linguistic scale function and solving them by employing score function. Finally, the proposed approach is validated by applying it to electric vehicle sales.

The Generalized Similarity Measures Based on Complex Interval-Valued Hesitant Fuzzy Sets and Their Applications in Pattern Recognition and Medical Diagnosis

The Generalized Similarity Measures Based on Complex Interval-Valued Hesitant Fuzzy Sets and Their Applications in Pattern Recognition and Medical Diagnosis

Consensus reaching process under interval-valued hesitant fuzzy environment

This paper introduces a consensus reaching process (CRP) able to smooth disagreements between the experts who participate in a multi-criteria group decision-making problem in order to obtain a solution in which experts agree. To deal with the lack of information and uncertainty inherent in this type of problems, the experts’ preferences are modeled by means of interval-valued hesitant fuzzy sets (IVHFS), which allow both representing uncertainty and any doubts experts may have when giving their assessments over the alternatives. The proposal makes use of an extension of the ordered weight averaging operator under an IVHFS which performs aggregations reasonably. In addition, the CRP incorporates a feedback mechanism that provides precise and individual suggestions to the experts for increasing the level of agreement within the group. Finally, the proposal is validated by solving a practical study related to the selection of renewable energies in China under several scenarios.

A New Approach to Correspondence Analysis Based on Interval-Valued Hesitant Fuzzy Sets

Correspondence Analysis (CA), a multivariate statistical technique, allows a visual representation of the association between categorical variables through a contingency table consisting of frequencies representing the existence of relationships. Despite being a widely used statistical technique, the classical CA is not able to demonstrate the uncertainty in real-life problems. To address this issue, a new Interval-valued Hesitant Fuzzy CA approach is proposed to represent the uncertainty caused by human doubt. Due to the nature of operations defined on Hesitant Fuzzy Sets, it is hard to integrate the fuzzy calculations directly into the classical CA. Thus, a new hesitant expected value method is proposed to reveal the independence between two categorical variables. As the output of the proposed approach, an interval-valued hesitant fuzzy correspondence map consisting of rectangles of different sizes representing the amount of the hesitancy is constructed. The applicability of the proposed approach is demonstrated by a simple but effective illustrative example.

Developing an Enterprise Diagnostic Index System Based on Interval-Valued Hesitant Fuzzy Clustering

Global economic integration drives the development of dynamic competition. In a dynamic competitive environment, the ever-changing customer demands and technology directly affect the leadership of the core competence of enterprises. Therefore, assessing the performance of enterprises in a timely manner is necessary to adjust business activities and completely adapt to new changes. Enterprise diagnosis is a scientific tool for judging the development status of enterprises, and building a scientific and rational index system is the key to enterprise diagnosis. Considering the large number of enterprise diagnostic indicators and the high similarity among indicators, this study proposes a selection method for enterprise diagnostic indicators based on interval-valued hesitant fuzzy clustering by comparing the existing indicator systems. First, enterprise organizations are considered as the starting point. Through the key analysis of relevant indicators of domestic and foreign enterprise diagnosis, enterprise diagnosis candidate indicators are constructed from three aspects, namely enterprise performance, employee health, and social benefit. In view of the ambiguity and inconsistency of expert judgment, this study proposes an interval-valued hesitant fuzzy set based on the characteristics of hesitant fuzzy sets and interval-valued evaluation. For improving the interval-valued hesitant fuzzy entropy function, an interval-valued hesitant fuzzy similarity measurement formula considering information features is designed to avoid the problem of data length and improve the degree of identification among indicators. Then, the similarity, equivalence, and truncation matrices are constructed, and the interval-valued hesitant fuzzy clustering method is used to eliminate redundant indicators with repeated information. The availability of the proposed method is illustrated via an example, and the key indicators in the enterprise diagnostic index system are found. Finally, the advantages of the proposed method are discussed using comparative analysis with existing methods. A rational and comprehensive enterprise diagnostic index system was constructed. The system can be used as a scientific basis for diagnosing the development of enterprises and providing an objective and effective reference.

The Water, Energy, Food, and Sustainability Nexus Decision Environment: A Multistakeholder Transdisciplinary Approach

The water, energy, and food (WEF) nexus has gained particular attention in the sustainable development community. Making effective decisions in this environment is difficult. Entwinement from multiple sustainability dimensions and various stakeholder perspectives contribute to this difficulty. Stakeholders have differing goals, interests, and preferences for potential technological and developmental solutions that address WEF nexus concerns. A holistic sustainable management approach with supporting decision support can address these concerns. This article introduces such a holistic framework to address WEF-sustainability (WEFS) concerns. Using a joint neighborhood rough set, interval-valued hesitant fuzzy set, and regret theory technique, this article introduces a multistakeholder transdisciplinary method to support WEFS nexus decisions. An illustrative example integrates the multiple stakeholder and WEFS nexus factors. The illustrative example provides insights into the modeling effort and data requirements. Research, practical implications, along with limitations of the study—which are all discussed—provide a foundation for future research directions in this socio-environmentally important field.

Selection of an alternative based on interval-valued hesitant picture fuzzy sets

Motivated by interval-valued hesitant fuzzy sets (IVHFSs) and picture fuzzy sets (PcFSs), a notion of interval-valued hesitant picture fuzzy sets (IVHPcFSs) is presented in this article. The concept of IVHPcFSs is put forward and some operational rules are developed to deal with it. The cosine similarity measures (SMs) are modified for IVHPcFSs to deal with interval-valued hesitant picture fuzzy (IVHPcF) data and the linear programming (LP) methodology is used to find out the criteria’s weights. A multiple criteria decision making (MCDM) approach is then developed to tackle the vague and ambiguous information involved in MCDM problems under the framework of IVHPcFSs. For the validation and strengthen of the proposed MCDM approach a practical example is put forward to select the educational expert at the end.