Pythagorean Hesitant Fuzzy Set Research Articles

Evaluation of Spare Parts Support Capacity of Civil Aircrafts Based on Type-2 Hesitant Pythagorean Fuzzy Sets and Improved Technique for Order Preference by Similarity to Ideal Solution

To improve the spare parts support capacity of civil aircrafts and given the actual lack of evaluation methods at present, the evaluation problem of spare parts support capacity was solved in this study by proposing a multi-attribute decision method based on Type-2 hesitant Pythagorean fuzzy sets and an improved Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). First, the basic definitions of Type-2 hesitant Pythagorean fuzzy sets were given, which were further promoted to Type-n hesitant Pythagorean fuzzy sets, and the basic order relation criterion of Type-2 hesitant Pythagorean fuzzy sets was introduced. Second, a complete evaluation system for spare parts supply support capacity was established with the spare parts of civil aircrafts as the study objects, and each evaluation indicator was introduced in detail. Then, the spare parts support solutions were preferentially sorted using the correlation coefficient formula of Type-2 hesitant Pythagorean fuzzy sets and improved TOPSIS. Finally, the reliability and reasonability of the proposed method were verified through an example calculation and comparative analysis. The experimental results indicate that the proposed method can acquire the evaluation results of spare parts support capacity more scientifically and can be referenced by relevant studies.

Research on the Safety Evaluation Method for Quayside Container Cranes Based on the Best–Worst Method–Pythagorean Fuzzy VIKOR Approach

In the port domain, quayside container cranes are an indispensable component of maritime freight transport. These cranes are not only costly but also associated with safety accidents that often result in casualties and property loss, severely impacting port operations and the surrounding environment. Given their complex operational environment, rapid technological updates, high dependency on human factors, and the challenges of maintenance and inspection, the safety of quayside container cranes is a significant concern for port enterprises and managers. This paper, based on the operational modes and structural characteristics of the cranes, divides them into five main systems and identifies twenty-eight safety evaluation indicators, covering a comprehensive range of risk factors from equipment integrity to operator behavior, as well as environmental factors. However, numerous pain points exist in the safety risk evaluation process of quayside container cranes, such as fuzziness, uncertainty, and complex multi-criteria decision-making (MCDM) environments. These issues make traditional safety evaluation methods inadequate in accurately reflecting the actual safety conditions. Therefore, this paper proposes a safety evaluation method for quayside container cranes based on the Best–Worst Method (BWM) and Pythagorean hesitant fuzzy VIKOR. This method effectively overcomes the uncertainties and fuzziness of traditional safety evaluation methods by integrating the decision maker’s preference information from the BWM and the fuzzy handling capability of Pythagorean hesitant fuzzy sets, enhancing the accuracy and reliability of the evaluation results. A case study was conducted on a quayside container crane at a specific port. Through empirical analysis, the feasibility of the proposed method was validated. Overall, the safety evaluation method for quayside container cranes based on the BWM and Pythagorean hesitant fuzzy VIKOR proposed in this paper enriches the theoretical research on the safety risk assessment of quayside container cranes and offers a new approach and tool for port enterprises and managers in practice.

Site evaluation of subsea tunnels with sightseeing function based on dynamic complex MARCOS method

Subsea tunnels with sightseeing function (STSF) are valuable for integrating transportation across different locations, particularly in cross-sea regions. Given their technical complexity, significant capital commitment, and substantial social impact, site evaluation of STSF has become a critical priority. In this study, a dynamic complex measurement alternative and ranking according to the compromise solution (MARCOS) framework is proposed for STSF site evaluation in a Pythagorean hesitant fuzzy environment. A novel score function for a Pythagorean hesitant fuzzy set is defined to handle an incomparable problem. Moreover, a growth coefficient matrix is constructed to reflect the internal development potential of decision objects, considering changes in expert and criteria weights during the dynamic decision-making process. An STSF evaluation system is designed to satisfy stability, sociality, and economic developments. An empirical study is conducted in Zhejiang Province, China, to verify the robustness and effectiveness of the proposed evaluation framework by comparing its results with other methods.

Cubic Pythagorean Hesitant Fuzzy Linear Spaces and Its Relevance in Multi Criteria Decision Making

Pythagorean fuzzy sets and interval valued Pythagorean fuzzy sets have an important role in decision making techniques. Pythagorean hesitant fuzzy sets are time and again used in dealing with uncertain and vague data. The motive of this paper is to introduce the notion cubic Pythagorean hesitant fuzzy linear spaces. We also present the notion of P(R)-intersection, P(R)-union of cubic Pythagorean hesitant fuzzy linear spaces with examples. Secondly, a series of operators like cubic Pythagorean hesitant fuzzy weighted averaging aggregation operators, cubic Pythagorean hesitant fuzzy order weighted averaging aggregation operators and cubic Pythagorean hesitant fuzzy hybrid order weighted averaging aggregation operators are developed. Then, these aggregation operators are further extended to cubic Pythagorean hesitant fuzzy prioritized weighted averaging aggregation operators by assigning priorities to the criteria. A real life MCDM problem has been illustrated and the effectiveness of the results are compared with those solved using cubic picture hesitant fuzzy prioritized weighted averaging aggregation operators.

Hybridizations of Archimedean copula and generalized MSM operators and their applications in interactive decision-making with q-rung probabilistic dual hesitant fuzzy environment

The q-rung probabilistic dual hesitant fuzzy sets (qRPDHFSs), which outperform dual hesitant fuzzy sets, probabilistic dual hesitant fuzzy sets, and probabilistic dual hesitant Pythagorean fuzzy sets, are used in this research to develop an interactive group decision-making approach. We first suggest the Archimedean Copula-based operations on q-rung probabilistic dual hesitant fuzzy (qRPDHF) components and investigate their key features before constructing the approach. We then create some new aggregation operators (AOs) in light of these operations, including the qRPDHF generalized Maclaurin symmetric mean (MSM) operator, qRPDHF geometric generalized MSM operator, qRPDHF weighted generalized MSM operator, and qRPDHF weighted generalized geometric generalized MSM operator. These aggregation operators are better than current operators on qRPDHF because they can take into account the interactions between a large number of criteria and probability distributions. The evaluation findings are distorted since the present methodologies do not take expert involvement into account in order to achieve the required consistency level. We employ the idea of interaction, consistency, resemblance, and consensus-building among the decision-makers in our method to get around this. We create an optimization model based on the cross-entropy of the qRPDHF components to estimate the weights of the criterion. We provide contextual research on the choice of open-source software LMS in order to demonstrate the relevance of the recommended AOs. Likewise, we ran a sensitivity test on the weights of the criterion to make sure that our model is consistent. The comparison investigation has demonstrated that the suggested approach can overcome the challenges of previous works.

A novel multi-criteria group decision-making approach using aggregation operators and weight determination method for supplier selection problem in hesitant Pythagorean fuzzy environment

Uncertainty is an important factor in the decision-making process. Hesitant Pythagorean fuzzy sets (HPFS), a combination of Pythagorean and hesitant fuzzy sets, proved as a significant tool to handle uncertainty. Well-defined operational laws and attribute weights play an important role in decision-making. Thus, the paper aims to develop new Trigonometric Operational Laws, a weight determination method, and a novel score function for group decision-making (GDM) problems in the HPF environment. The approach is presented in three phases. The first phase defines new operational laws with sine trigonometric function incorporating its special properties like periodicity, symmetricity, and restricted range hence compared with previously defined aggregation operators they are more likely to satisfy the decision maker preferences. Properties of trigonometric operational laws (TOL) are studied and various aggregation operators are defined. To measure the relationship between arguments, the operators are combined with the Generalized Heronian Mean operator. The flexibility of operators is increased by the use of a real parameter λ to express the risk preference of experts. The second phase defines a novel weight determination method, which separately considers the membership and non-membership degrees hence reducing the information loss and the third phase conquers the shortcomings of previously defined score functions by defining a novel score function in HPFS. To further increase the flexibility of defined operators they are extended in the environment with unknown or incomplete attribute weights. The effectiveness of the GDM model is verified with the help of a supplier selection problem. A detailed comparative analysis demonstrates the superiority, and sensitivity analysis verifies the stability of the proposed approach.

Stochastic multiple attribute decision making with Pythagorean hesitant fuzzy set based on regret theory.

The objective of this paper is to present an extended approach to address the stochastic multi-attribute decision-making problem. The novelty of this study is to consider the regret behavior of decision makers under a Pythagorean hesitant fuzzy environment. First, the group satisfaction degree of decision-making matrices is used to consider the different preferences of decision-makers. Second, the nonlinear programming model under different statues is provided to compute the weights of attributes. Then, based on the regret theory, a regret value matrix and a rejoice value matrix are constructed. Furthermore, the feasibility and superiority of the developed approach is proven by an illustrative example of selecting an air fighter. Eventually, a comparative analysis with other methods shows the advantages of the proposed methods.

Interval Valued q- Rung Orthopair Hesitant Fuzzy Choquet Aggregating Operators in Multi-Criteria Decision Making Problems

Abstract: In this paper, we introduce Interval valued q- Rung Orthopair Hesitant fuzzy sets (IVq-ROHFS) with motivation of Interval valued pythagorean Hesitant fuzzy sets [44] as a new concept. Then, we give some basic operations as complement, union, intersection, addition, scalar multiplication, scalar power. Also, we combine to (IVq-ROHFS) and choquet integral , together with aggregating operators, and develop to Interval valued q- rung orthopair hesitant fuzzy Choquet averaging operator (IVq-ROHCA) and Interval valued q- rung orthopair hesitant fuzzy Choquet geometric operator (IVq-ROHCG). Then, we offer to indicate soft approach of proposed IVq-ROHCA and IVq-ROHCG an example adopted from Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS (IVIHCI) [43]. The obtained results are agreement with IVIHCI but presented IVq-ROHCA and IVq-ROHCG have more advantages than existing structures as Interval-valued intuitionistic hesitant fuzzy sets (IVIHFS), interval-valued Pythagorean Hesitant fuzzy sets (IVPHFS) with reasons changing according to need, requirement, prefer of decision makers and moreover, being IVIHFS and IVPHFS are special cases of IVq-ROHFS. It is open from comparative analysis that the while some of offered approaches are giving no solution for some values, our operators present to needed results.

How do enterprises determine which breakthrough invention should be commercialized? A multiple attribute group decision-making-based method

The problem of selection of breakthrough inventions to be commercialized (SBIC) is significantly important for innovative enterprises. As the processes of commercializing breakthrough inventions are time-consuming and high-cost, enterprises have to tradeoff among all possible inventions and determine which inventions should be commercialized. SBIC refers to a process in which multiple experts are invited to evaluate the potential of all inventions from multiple aspects and based on their assessments, the breakthrough inventions to be commercialized are chosen. Hence, the essence of SBIC is consistent with multiple attribute group decision-making (MAGDM). This paper proposes a MAGDM-based model to help enterprises conduct SBIC. In our model, probabilistic dual Pythagorean hesitant fuzzy sets (PDPHFSs) are utilized to represent DMs’ evaluation information, which can express DMs’ evaluation information comprehensively. Then, an attributes’ weights determination method is proposed to handle MAGDM with unknown weights information. Afterwards, we extend the classical projection model into PDPHFSs and introduce a novel MAGDM method and its detailed calculation process is illustrated. Finally, our MAGDM method is applied to an SBIC problem to show its effectiveness. Comparison analysis is conducted to show the advantages of our method.

Green Logistics Partner Selection Based on Pythagorean Hesitant Fuzzy Set and Multiobjective Optimization

With the deepening of the uncertainty and fuzziness of people’s understanding of things, it becomes more convenient to use fuzzy decision making in evaluating schemes. These schemes not only include quantitative data but also consider more and more uncertainty evaluation information. In fuzzy theory, the Pythagorean fuzzy set is extended, both of which are less than or equal to 1, which increases the scope of application. This paper mainly studies the selection of green logistics. Based on the original TOPSIS principle, this paper introduces a Pythagorean fuzzy symmetric cross entropy to describe the difference between two Pythagorean fuzzy numbers and proves its rationality. The model is applied to the selection of green logistics and compared with the results obtained by other different methods, which also shows the effectiveness and practicability of the model. It extends the use of fuzzy sets and can solve some problems that cannot be solved by intuitionistic fuzzy sets before.

Novel measures for linguistic hesitant Pythagorean fuzzy sets and improved TOPSIS method with application to contributions of system-of-systems

Assessment of the contribution rates of weapon system-of-systems is an extremely uncertain and complex problem that should be addressed by domain experts. Addressing these problems, which is often hesitant, fuzzy, and linguistic, requires evaluation of multiple indices of different types. Hesitant Pythagorean fuzzy sets combined with linguistic term sets are powerful tools to handle this kind of problem. In this paper, recent studies on hesitant Pythagorean fuzzy sets (HPFSs) are first reviewed and linguistic HPFSs (LHPFSs) are proposed. To efficiently solve the evaluation problem, the axiomatic properties of entropies for HPFSs and LHPFSs are defined. According to these properties, novel entropy measures for HPFSs and LHPFSs are introduced. A novel method that can better reveal the characteristics of HPFSs and LHPFS to represent the HPF and LHPF numbers is also introduced. On this foundation, the definitions of distance measures for HPFSs and LPFSs are proposed, and a set of distance measures for HPFSs and LHPFSs are developed. Moreover, the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method is modified through hierarchical thought by using the presented entropy and distance measures for LHPFSs; the improved TOPSIS method is applied preferably to the case of contribution of weapon system-of-systems and shows better validity and distinguishability than other methods under the LHPFS environment in the examples. The hierarchical TOPSIS method can better solve the problem of weight allocation in case of a lot of indices.

An integrated ELECTRE-I approach for risk evaluation with hesitant Pythagorean fuzzy information

Hesitant Pythagorean fuzzy sets represent a successful generalization of Pythagorean fuzzy sets that embeds the benefits of hesitation. As such they enable the practitioner to incorporate vague data and information very effectively. We take advantage of this feature in the procedure of risk evaluation in failure modes and effects analysis (FMEA). This technique has demonstrated to be a powerful and valuable framework for the identification of possible failures and the reduction of their occurrence. The advantages of hesitant Pythagorean fuzzy sets motivate us to extend the existing methods of risk assessment in FMEA to the case where data are in that format. Therefore, we propose an ELimination and Choice Translating REality-I (ELECTRE-I) approach under the hesitant Pythagorean fuzzy (HPF) environment to measure and rank risks. To assess the failures, the HPF ELECTRE-I technique creates hesitant Pythagorean fuzzy concordance/discordance matrices. Then, we use an outranking decision graph to obtain the most crucial failure mode. We illustrate the structure of this procedure through a flow chart. Furthermore, the model is validated through the study of real-life applications, such as the prevention of infant abduction and risk evaluation of health care failure modes in blood transfusion. Finally, to showcase the superiority and effectiveness of the suggested method, we provide a comprehensive comparison of the HPF ELECTRE-I approach with existing techniques.

A linguistic Pythagorean hesitant fuzzy MULTIMOORA method for third-party reverse logistics provider selection of electric vehicle power battery recycling

Electric vehicle power battery recycling (EVPBR) is an effective way to utilize resources and reduce environmental damage. In order to ensure the safety and efficiency of electric vehicle batteries (EVBs) reuse and remanufacture, many EVB manufacturers are seeking cooperation with third-party reverse logistics (3PRL) providers to conduct the pre-treatment and transportation of post-used batteries. However, the selection of 3PRL provider is a matter of complex multi-criteria decision-making (MCDM) affected by numerous associated factors. The goal of this paper is to present a linguistic Pythagorean hesitant fuzzy MULTIMOORA method to investigate the selection of 3PRL providers for EVPBR. Firstly, given that the complexity of evaluation information, the linguistic Pythagorean hesitant fuzzy set (LPHFS) is defined in detail. On the basis of new evaluation representation tool, the weight determining models for expert panel and criteria set based on correlation consensus degrees and maximum deviation are derived, respectively. Then a MULTIMOORA method under the linguistic Pythagorean hesitant fuzzy environment is constructed. Later, the evaluation criteria system for the provider capability analysis is set up, and the proposed MCDM approach is applied to select the most suitable 3PRL provider for EVPBR in China. Finally, the sensitivity and comparative analysis results justify the robustness and feasibility of the proposed method.

Uncertainty evaluations through interval-valued Pythagorean hesitant fuzzy Archimedean aggregation operators in multicriteria decision making

In many cases, use of Pythagorean hesitant fuzzy sets may not be sufficient to characterize uncertain information associated with decision making problems. From that view point the concept of interval-valued Pythagorean hesitant fuzzy sets are introduced in this paper. Considering the flexibility with the general parameters, Archimedean t-conorms and t-norms are applied to develop several operational laws in interval-valued Pythagorean hesitant fuzzy environment. Some characteristics of the developed operators are presented. The newly developed operators are used to derive a methodology for solving multicriteria decision making problems with interval-valued Pythagorean hesitant fuzzy information. Finally, two illustrative examples are provided to establish the validity of the proposed approach and are compared with the existing technique to exhibit its flexibility and effectiveness.

Risk Assessment in Failure Mode and Effect Analysis: Improved ORESTE Method With Hesitant Pythagorean Fuzzy Information

As a system reliability analysis technology, failure mode and effect analysis (FMEA) has been widely used in various industries to reduce the failure risk of products, systems, and services. In recent years, many researchers have found that this technology has limitations in failure mode assessment and failure mode prioritization in complex uncertain environment. In order to solve this problem, this article proposes a new FMEA model, which combines hesitant Pythagorean fuzzy sets (HPFSs) and ORESTE method. Specifically, HPFSs are used to describe the risk assessment information of experts. The ORESTE method is extended to determine the risk priority of the failure modes in FMEA. Moreover, the PIR structure of ORESTE is improved to reduce the subjective influence of experts. Finally, an example of risk assessment of intelligent aquaculture system is given to verify the applicability and effectiveness of the proposed FMEA model. The results show that the proposed FMEA model can not only accurately describe the hesitation and uncertainty of experts’ risk assessment information, but also obtain more suitable risk ranking of failure modes, based on which managers can further take corresponding preventive measures.

Hesitant Pythagorean fuzzy ELECTRE-II method for multi-criteria decision-making problems

Hesitant Pythagorean fuzzy sets have remarkable theoretical and practical features, which derive from their role as a generalization of Pythagorean fuzzy sets that embeds the benefits of hesitation. Their advantages motivate us to extend existing decision-making methods to the case where data and information are in the form of several values. We propose an ELimination and Choice Translating REality-II (ELECTRE-II) technique under hesitant Pythagorean fuzzy (HPF) information to handle diverse opinions of decision experts. The main contribution of this work is the formulation of the basic structure of an HPF ELECTRE-II method, including three kinds of outranking sets (concordance, indifferent, and discordance), two types of outranking matrices (concordance and discordance), two kinds of outranking relations (weak and strong), and two types of outranking graphs (strong and weak graphs). Further, we discuss some quantitative applications to guarantee the applicability and flexibility of the presented framework. To endorse the advantages and accuracy of HPF ELECTRE-II technique, we provide a comprehensive comparative analysis with existing techniques, such as ELECTRE-II approach under hesitant fuzzy data, ELECTRE group decision-making method under fuzzy knowledge, PF ELECTRE-1, and HPF ELECTRE-II methods. Moreover, we state some important insights and discuss the limitations of the model here proposed.

A Decision Support Model for Hotel Recommendation Based on the Online Consumer Reviews Using Logarithmic Spherical Hesitant Fuzzy Information.

Spherical hesitant fuzzy sets have recently become more popular in various fields. It was proposed as a generalization of picture hesitant fuzzy sets and Pythagorean hesitant fuzzy sets in order to deal with uncertainty and fuzziness information. Technique of Aggregation is one of the beneficial tools to aggregate the information. It has many crucial application areas such as decision-making, data mining, medical diagnosis, and pattern recognition. Keeping in view the importance of logarithmic function and aggregation operators, we proposed a novel algorithm to tackle the multi-attribute decision-making (MADM) problems. First, novel logarithmic operational laws are developed based on the logarithmic, t-norm, and t-conorm functions. Using these operational laws, we developed a list of logarithmic spherical hesitant fuzzy weighted averaging/geometric aggregation operators to aggregate the spherical hesitant fuzzy information. Furthermore, we developed the spherical hesitant fuzzy entropy to determine the unknown attribute weight information. Finally, the design principles for the spherical hesitant fuzzy decision-making have been developed, and a practical case study of hotel recommendation based on the online consumer reviews has been taken to illustrate the validity and superiority of presented approach. Besides this, a validity test is conducted to reveal the advantages and effectiveness of developed approach. Results indicate that the proposed method is suitable and effective for the decision process to evaluate their best alternative.

Hesitant q-rung orthopair fuzzy aggregation operators with their applications in multi-criteria decision making

The aim of this manuscript is to present a new concept of hesitant q-rung orthopair fuzzy sets (Hq-ROFSs) by combining the concept of the q-ROFSs as well as Hesitant fuzzy sets. The proposed concept is the generalization of the fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and Pythagorean fuzzy sets as well as intuitionistic hesitant fuzzy sets (IHFSs) and hesitant Pythagorean fuzzy sets (HPFSs).Furthermore some basic operational laws of hesitant q-rung orthopair fuzzy have been investigated. The score and accuracy functions are defined which play a vital role in decision making process for making comparison between the hesitant q-rung orthopair fuzzy numbers (Hq-ROFNs). Under the Hq-ROF environment, Hq-ROF weighted averaging (Hq-ROFWA) and Hq-ROF weighted geometric (Hq-ROFWG) operators are introduced and various properties of these aggregation operators are studied. Additionaly, a numerical application shows that how the proposed operators are utilized to solve multi-criteria decision making (MCDM) problems in which experts added their optimistic and pessimistic preferences. Finally the analysis of proposed method with other methods is presented which show that the method presented in this paper is more flexible and superior than existing methods.

Correlation coefficients of Pythagorean hesitant fuzzy sets and their applicationto radar LPI performance evaluation

Correlation coefficients of Pythagorean hesitant fuzzy sets and their applicationto radar LPI performance evaluation

Interval-Valued Pythagorean Hesitant Fuzzy Set and Its Application to Multiattribute Group Decision-Making

Pythagorean hesitant fuzzy sets are widely watched because of their excellent ability to deal with uncertainty, imprecise and vague information. This paper extends Pythagorean hesitant fuzzy environments to interval-valued Pythagorean hesitant fuzzy environments and proposes the concept of interval-valued Pythagorean hesitant fuzzy set (IVPHFS), which allows the membership of each object to be a set of several pairs of possible interval-valued Pythagorean fuzzy elements. Furthermore, we develop a series of aggregation operators for interval-valued Pythagorean hesitant fuzzy information and apply them to multiattribute group decision-making (MAGDM) problems. Then, some desired operational laws and properties of IVPHFSs are studied. Especially, considering an interval-valued Pythagorean fuzzy element (IVPHFE) is formed by several pairs of interval values, this paper proposes the concepts of score function and accuracy function in the form of two interval numbers which can retain interval-valued Pythagorean fuzzy information as much as possible. Then, the relationship among these operators is discussed by comparing the interval numbers. Eventually, an illustrative example fully shows the feasibility, practicality, and effectiveness of the proposed approach.