Interval-valued Pythagorean Fuzzy Set Research Articles

Some distance measures for pythagorean cubic fuzzy sets: Application selection in optimal treatment for depression and anxiety

Pythagorean cubic fuzzy sets represent an advancement beyond conventional interval-valued Pythagorean sets, integrating the principles of Pythagorean fuzzy sets and interval-valued Pythagorean fuzzy sets. Given the critical significance of distance measures in real-world decision-making and pattern recognition tasks, it is noteworthy that there exists a notable gap in the literature regarding distance measures specifically tailored for Pythagorean cubic fuzzy sets. The objectives of this paper are:•To define novel generalized distance measures between Pythagorean cubic fuzzy sets (PCFSs) to tackle intricate decision-making challenges.•These novel distance measures are undergoing testing on a real-world scenario concerning the management of anxiety and depression to evaluate their effectiveness and practical application.•We have illustrated the boundedness and nonlinear characteristics inherent in these distance measures.In addition, we conduct comparative analyses with existing approaches to validate the proposed methodology, thereby providing insights into its advantages and potential applications.

A fair bed allocation during COVID-19 pandemic using TOPSIS technique based on correlation coefficient for interval-valued pythagorean fuzzy hypersoft set

The relationship between two variables is an essential factor in statistics, and the accuracy of the results depends on the data collected. However, the data collected for statistical analysis can be unclear and difficult to interpret. One way to predict how one variable will change about another is by using the correlation coefficient (CC), but this method is not commonly used in interval-valued Pythagorean fuzzy hypersoft set (IVPFHSS). The IVPFHSS is a more advanced and generalized form of the Pythagorean fuzzy hypersoft set (PFHSS), which allows for more precise and accurate analysis. In this research, we introduce the correlation coefficient (CC) and weighted correlation coefficient (WCC) for IVPFHSS and their essential properties. To demonstrate the applicability of these measures, we use the COVID-19 pandemic as an example and establish a prioritization technique for order preference by similarity to the ideal solution (TOPSIS) model. The technique is used to study the problem of optimizing the allocation of hospital beds during the pandemic. This study provides insights into the importance of utilizing correlation measures for decision-making in uncertain and complex situations like the COVID-19 pandemic. It is a robust multi-attribute decision-making (MADM) methodology with significant importance. Subsequently, it is planned to increase a dynamic bed allocation algorithm based on biogeography to accomplish the superlative decision-making system. Moreover, numerical investigations deliberate the best decision structures and deliver sensitivity analyses. The efficiency of our encouraged algorithm is more consistent than prevalent models, and it can effectively control and determine the optimal configurations for the study.

Optimized multiple-attribute group decision-making in uncertainty employing TODIM and EDAS technique and application to product styling design quality evaluation

With the improvement of the public’s aesthetic level, product appearance has become an important influencing factor for consumers to make purchasing decisions. Product styling design is based on this market demand, combining the aesthetic and functional aspects of the product to create a personalized product appearance, in order to better attract consumers, improve the competitiveness and added value of the product. Usually, product styling design involves multiple elements such as product form, color, proportion, etc. The quality evaluation of product styling design is a MAGDM problems. Recently, the TODIM and EDAS technique has been employed to manage MAGDM issues. The interval-valued Pythagorean fuzzy sets (IVPFSs) are employed as a tool for characterizing uncertain information during the quality evaluation of product styling design. In this paper, the interval-valued Pythagorean fuzzy TODIM-EDAS (IVPF-TODIM-EDAS) technique is construct to manage the MAGDM under IVPFSs. Finally, a numerical case study for quality evaluation of product styling design is employed to validate the proposed technique. The main contribution of this paper is managed: (1) the TODIM and EDAS technique was extended to IVPFSs; (2) Entropy technique is employed to manage the weight values under IVPFSs. (3) the IVPF-TODIM-EDAS technique is founded to manage the MAGDM under IVPFSs; (4) Algorithm analysis for quality evaluation of product styling design and comparison analysis are constructed based on one numerical example to verify the feasibility and effectiveness of the IVPF-TODIM-EDAS technique.

TODIM-GRA based multiple-attribute group decision-making technique for risk assessment of intangible cultural heritage tourism development under interval-valued pythagorean fuzzy circumstances

Intangible cultural heritage can be said to be an important component of tourism resources. With the rapid development of society in today’s era, tourism development and intangible cultural heritage protection have gradually attracted attention from Chinese society, and in recent years, it has attracted high attention from relevant departments of the Chinese government. Tourism development has a “dual” impact on the protection of intangible cultural heritage, with both positive and negative impacts. The risk assessment of intangible cultural heritage tourism development is a MAGDM problems. Recently, the TODIM and GRA technique has been employed to manage MAGDM issues. The interval-valued Pythagorean fuzzy sets (IVPFSs) are employed as a tool for characterizing uncertain information during the risk assessment of intangible cultural heritage tourism development. In this paper, the interval-valued Pythagorean fuzzy TODIM-GRA (IVPF-TODIM-GRA) technique is construct to manage the MAGDM under IVPFSs. Finally, a numerical case study for risk assessment of intangible cultural heritage tourism development is employed to validate the proposed technique.

Aczel–Alsina T-norm based group decision-making technique for the evaluation of electric cars using generalized orthopair fuzzy aggregation information with unknown weights

Data management and finding precise outcomes from large amounts of information are among the biggest challenges for scientists. The technique of multi-attribute group decision-making (MAGDM) is a valuable tool for investigating fuzzy data precisely. The key objective of the paper is to redefine the q-rung orthopair (q-RO) fuzzy set (FS) (q-ROFS) in the term of interval-valued and proposed new aggregation operators (AOs) based on the Aczel-Alsina (AA) t-norm (TN) and t-conorm (TCN) operations. The AA operational laws are a generalized form of existing TNs and TCNs and give more reliable results because they can fluctuate in their parametric values. The concept of interval-valued enlarges the space of membership degree (MD) and non-membership degree (NMD) for decision-makers. By taking qth power, the interval-valued q-ROFS (IV-q-ROFS) structure. The IV-q-ROFS can handle the uncertainty and vagueness in data, then interval-valued intuitionistic FS (IV-IFS) and interval-valued Pythagorean FS (PyFS) (IV-PyFS) and provide accurate results. The thought of power AOs (PAOs) makes the relationship between weight vectors and reduces the chances of uncertainty in aggregated results. By taking advantage of PAOs, this article is devoted to introducing the interval-valued q-ROF Aczel-Alsina power-weighted averaging (IV-q-ROFAAPWA) and interval-valued q-ROF Aczel-Alsina power-weighted geometric (IV-q-ROFAAPWG) operators. The fundamental axioms of AOs, idempotency, boundedness, and monotonicity, are also discussed. To illustrate the importance of suggested AOs, the real-life problem of electric car selection was solved by applying the MAGDM method using the proposed IV-q-ROFAAPWA and IV-q-ROFAAPWG operators. The comparison of proposed AOs with currently present AOs is also part of the article. We finally constructed solid conclusions.

Evaluation of sustainable energy systems in smart cities using a Multi-Expert Pythagorean fuzzy BWM & TOPSIS methodology

Smart cities are technological settlements using the collected data to utilize resources and services effectively by combining information and communication technologies with various tools connected to the internet of things network. Sustainable energy systems in smart cities are systems can be evaluated by using multiple criteria decision making methods or methodologies based on several vague/imprecise evaluation criteria. In this paper, sustainable energy systems in smart cities are evaluated by interval-valued Pythagorean fuzzy (IVPF) sets with an integrated optimization based multi-expert fuzzy Best Worst Method (BWM) and TOPSIS methodology that can better handle uncertainty and vagueness in experts’ linguistic assessments than existing methodologies. The considered criteria are weighted by multi-expert IVPF Best Worst Method, which has become a popular weighting method in recent years. Later, the energy alternatives for a real case study are prioritized by multi-expert IVPF TOPSIS method. In the analysis, the most important criterion is found as Environmental sustainability (C1) with the defuzzified weight of 0.218 while the other weights are as initial investment (C2) with 0.196, operating expenses (C3) with 0.163, technical feasibility (C4) with 0.154, social acceptability (C5) with 0.140, and scalability (C6) with 0.129. The obtained results indicate that “Investing in advanced technologies” in a smart city with relative degree of closeness (RDC) value of 0.798, has been determined as the best alternative among the considered five alternatives. It is closely followed by “Developing a transportation system” with the RDC value of 0.681. Sensitivity analysis shows that the ranking results are quite robust and reliable. The comparative analysis with crisp BWM and TOPSIS methodology is applied to check the validity of the proposed methodology.

Innovative exponential TODIM-CoCoSo framework for uncertain multiple-attribute group decision-making in the effectiveness evaluation of short video marketing strategies

In recent years, e-commerce live streaming and short video marketing supported by big data and artificial intelligence technology have flourished, adding new sales models for e-commerce products to mass consumption, promoting the multimodal development of the e-commerce industry, giving new impetus and connotation to economic and social development, and being an effective means to achieve high-quality development in the new era. The effectiveness evaluation of short video marketing strategies is a multiple-attribute group decision-making (MAGDM) problem. Recently, the Exponential TODIM technique and Combined Compromise Solution (CoCoSo) technique has been employed to cope with MAGDM issues. The interval-valued Pythagorean fuzzy sets (IVPFSs) are employed as a tool for characterizing uncertain information during the effectiveness evaluation of short video marketing strategies. In this paper, the interval-valued Pythagorean fuzzy Exponential TODIM (ExpTODIM) (IVPF-ExpTODIM-CoCoSo) technique is constructed to solve the MAGDM under IVPFSs. In the end, a numerical case study for effectiveness evaluation of short video marketing strategies is given to validate the proposed technique. The main contributions of this paper are outlined: (1) the Exp-TODIM and CoCoSo technique has been extended to IVPFSs; (2) Information Entropy is employed to manage the weight values under IVPFSs. (3) the IVPF-ExpTODIM-CoCoSo technique is founded to implement the MAGDM under IVPFSs; (4) a numerical case study for effectiveness evaluation of short video marketing strategies and some comparative analysis is supplied to verify the IVPF-ExpTODIM-CoCoSo technique.

Extended TODIM technique based on TOPSIS for county preschool education resource allocation level evaluation under interval-valued Pythagorean fuzzy sets

Optimizing the allocation of preschool education resources and improving the efficiency of resource allocation is of great strategic significance for the universal and inclusive development of preschool education and the realization of “education for young children". In recent years, the shift from high-speed development to high-quality development of the social economy has significantly improved the balanced development level of China’s preschool education industry. However, preschool education remains the weakest link in China’s education system and the most unfavorable aspect of educational resource allocation. Problems such as shortage of preschool education resources, insufficient investment, uneven regional development, imbalanced supply and demand structure, low resource allocation efficiency, and “difficult to enter, expensive to enter” are still prominent. How to optimize resource allocation and improve resource utilization efficiency in the limited resources of preschool education is the key to achieving balanced, fair, coordinated, and high-quality development of preschool education. The county preschool education resource allocation level evaluation is MAGDM problems. Recently, the TODIM and TOPSIS technique was employed to cope with MAGDM issues. The interval-valued Pythagorean fuzzy sets (IVPFSs) are employed as a tool for characterizing uncertain information during the county preschool education resource allocation level evaluation. In this manuscript, the interval-valued Pythagorean fuzzy TODIM-TOPSIS (IVPF-TODIM-TOPSIS) technique is built to solve the MAGDM under IVPFSs. Finally, a numerical case study for county preschool education resource allocation level evaluation is given to validate the proposed technique. The main contribution of this paper is managed: (1) the TODIM and TOPSIS technique was extended to IVPFSs; (2) Information Entropy is employed to manage the weight values under IVPFSs. (3) the IVPF-TODIM-TOPSIS technique is founded to manage the MAGDM under IVPFSs; (4) Algorithm analysis for county preschool education resource allocation level evaluation and comparison analysis are constructed based on one numerical example to verify the feasibility and effectiveness of the IVPF-TODIM-TOPSIS technique.

Interval-valued Pythagorean fuzzy entropy and its application to multi-criterion group decision-making

<abstract><p>Entropy is an important tool of information measurement in the fuzzy set and its inference. The research on information measurement based on interval-valued Pythagorean fuzzy sets mostly involves the distance formula for interval-valued Pythagorean fuzzy numbers, but seldom involves the measurement of fuzziness. In view of this situation, we have aimed to propose new interval-valued Pythagorean fuzzy entropy and weighted exponential entropy schemes. Based on the interval-valued Pythagorean fuzzy weighted averaging operator, a strategy based on weighted exponential entropy is proposed to solve the multi-criteria group decision-making (MCGDM) problem in the interval-valued Pythagorean environment. Two examples illustrate that this paper provides a feasible new method to solve the MCGDM problem in an interval-valued Pythagorean fuzzy (IVPF) environment. Finally, by comparing with the existing methods, it is concluded that the entropy measure of IVPF schemes and the corresponding MCGDM method can select the optimal solution of the practical problem more precisely and accurately. Therefore, the comparative analysis shows that the proposed measurement method has the characteristics of flexibility and universality.</p></abstract>

A novel interval-valued Fermatean fuzzy three-way decision making method with probability dominance relations

Interval-valued Fermatean fuzzy sets (IVFFSs) as an extension of Fermatean fuzzy sets (FFSs) are a new powerful mathematical tool that apply interval value to describe uncertainty. Compared to interval-valued intuitionistic fuzzy sets (IVIFSs) and interval-valued Pythagorean fuzzy sets (IVPFSs), IVFFSs can handle more uncertain information. This paper presents a novel fuzzy three-way multi-attribute decision-making approach with the probabilistic dominance relation, based on the model of IVFSSs. The proposed method firstly uses the probabilistic dominance relation to calculate the conditional probability of each alternative. Subsequently, it builds loss functions from the perspective of the ideal solution. Our suggested method is able to rank and classify the alternatives into the positive region, the boundary region, and the negative region while the existing MADM approaches under IVFSSs only sort them without classification. Additionally, the suggested solution involves a much lower error rate than the existing methods. And our method’s conditional probability may be determined objectively and takes the relation between attributes and interactions among alternatives into account, which reduce the risk associated with subjectivity in the decision-making process of existing approaches. The effectiveness and suitability of the suggested approach are illustrated by two real-world applications such as the photovoltaic poverty alleviation project selection and the hotel location evaluation and experimental findings.

Approaches to multiple attribute group decision making under interval-valued pythagorean fuzzy sets and applications to environmental design majors teaching quality evaluation

The environmental design major has certain practical characteristics. In the teaching process, it is necessary to effectively combine practical methods and theoretical knowledge in order to improve the overall quality and efficiency of education to a certain extent. The most important training measure currently is project-based teaching method, which has also been recognized by many researchers. The environmental design majors teaching quality evaluation is a classical multiple-attribute group decision-making (MAGDM). Recently, the Taxonomy has been used to cope with MAGDM problems. The interval-valued Pythagorean fuzzy sets (IVPFSs) are useful tools to depict uncertainty in MAGDM. This paper use Taxonomy method to solve MAGDM with IVPFSs. Firstly, the concepts of IVPFSs are presented. Then, the classical Taxonomy method is used to solve MAGDM with IVPFSs. Finally, a numerical example for environmental design majors teaching quality evaluation is proposed and a comparative analysis is also given to show the developed approach. The main study motivations of built paper are outlined: (1) the Taxonomy method is extended to IVPFSs; (2) the score function of IVPFSs and information entropy method is used to derive the attribute weights; (3) the interval-valued Pythagorean fuzzy Taxonomy (IVPF-Taxonomy) method is built to deal with MAGDM; (4) a case study about environmental design majors teaching quality evaluation is given to show the built method; (5) some comparative analysis are given to show the rationality of IVPF-Taxonomy method.

Extension of correlation coefficient based TOPSIS technique for interval-valued Pythagorean fuzzy soft set: A case study in extract, transform, and load techniques.

Correlation is an essential statistical concept for analyzing two dissimilar variables' relationships. Although the correlation coefficient is a well-known indicator, it has not been applied to interval-valued Pythagorean fuzzy soft sets (IVPFSS) data. IVPFSS is a generalized form of interval-valued intuitionistic fuzzy soft sets and a refined extension of Pythagorean fuzzy soft sets. In this study, we propose the correlation coefficient (CC) and weighted correlation coefficient (WCC) for IVPFSS and examine their necessary properties. Based on the proposed correlation measures, we develop a prioritization technique for order preference by similarity to the ideal solution (TOPSIS). We use the Extract, Transform, and Load (ETL) software selection as an example to demonstrate the application of these measures and construct a prioritization technique for order preference by similarity to the ideal solution (TOPSIS) model. The method investigates the challenge of optimizing ETL software selection for business intelligence (BI). This study offers to illuminate the significance of using correlation measures to make decisions in uncertain and complex settings. The multi-attribute decision-making (MADM) approach is a powerful instrument with many applications. This expansion is predicted to conclude in a more reliable decision-making structure. Using a sensitivity analysis, we contributed empirical studies to determine the most significant decision processes. The proposed algorithm's productivity is more consistent than prevalent models in controlling the adequate conformations of the anticipated study. Therefore, this research is expected to contribute significantly to statistics and decision-making.

Extended TODIM method based on VIKOR for quality evaluation of higher education scientific research management under interval-valued Pythagorean fuzzy sets

Scientific research management is an important component of university management and an important guarantee for the sustainable operation of universities. A mature scientific research management system is a sharp tool to promote the discipline construction and comprehensive strength improvement of universities. Therefore, colleges and universities should actively explore new concepts, models, and mechanisms for scientific research management while continuously improving their own educational quality. Based on the analysis of the current situation of university scientific research management in China, it is of great significance to explore effective ways to improve the quality of university scientific research management in the new era. The quality evaluation of higher education scientific research management in entrepreneurship undergraduate colleges is a classical MAGDM problems. Recently, the TODIM and VIKOR method has been used to cope with MAGDM issues. The interval-valued Pythagorean fuzzy sets (IVPFSs) are used as a tool for characterizing uncertain information during the quality evaluation of higher education scientific research management in entrepreneurship undergraduate colleges. In this manuscript, the interval-valued Pythagorean fuzzy TODIM-VIKOR (IVPF-TODIM-VIKOR) method is built to solve the MAGDM under IVPFSs. In the end, a numerical case study for quality evaluation of higher education scientific research management in entrepreneurship undergraduate colleges is given to validate the proposed method.

Evaluating the blockchain-based healthcare supply chain using interval-valued Pythagorean fuzzy entropy-based decision support system

In the current era, blockchain technology has emerged as a novel technique to maintain the operations of healthcare management systems. The global COVID-19 disease has led to increase in the utilization of technology for healthcare supply chain, patient data management, and claims settlement. Data management in healthcare industry is a complex procedure where multiple organizations offer appropriate supply chain facilities in everyday life. Inadequate data handling disrupts the supply chain, which has a longstanding impact on the healthcare region. Blockchain-based solutions are renowned for their ability to produce safe and traceable results and effective in the health sector for secure data retrieval and storage, resulting in more effectual product creation and tracking. These technologies can deliver data provenance, sponsors candid healthcare industry demands, and certifies the immutability of multi direction transactions. Due to involvement of multiple factors/criteria, the selection of an effective blockchain platform-based healthcare supply chain can be treated as a decision support system (DSS) problem. Thus, the objective of this paper is to present a novel interval-valued Pythagorean fuzzy DSS for evaluating the blockchain platforms for healthcare supply chain and selecting a suitable one with respect to multiple criteria and uncertainty. The proposed DSS framework is divided into three stages. First, some fairly aggregation operators are presented to combine the individual’s information and also discussed their desirable characteristics on interval-valued Pythagorean fuzzy sets (IVPFSs). Second, an integrated weighting tool is developed using entropy-based tool for objective weight of criteria and pivot pairwise relative criteria importance assessment (PIPRECIA) model for subjective weight of criteria with IVPFSs. For this purpose, new entropy is presented on IVPFSs. Third, the multi-attribute ideal-real comparative analysis (MAIRCA) model is developed with the combination of fairly aggregation operators, entropy-based weighting tool and PIPRECIA model with interval-valued Pythagorean fuzzy information. Further, the proposed DSS is used on a case study of blockchain platforms for healthcare supply chain evaluation, which confirms its applicability and usefulness. Sensitivity and comparative assessments are carried out to check the consistency, robustness and efficiency of the presented method.

Interval-valued Pythagorean fuzzy operational competitiveness rating model for assessing the metaverse integration options of sharing economy in transportation sector

The rise of the sharing economy (SE) has enabled new forms of urban mobility and had a substantial impact on the transportation sector. The applications of SE transportation lessen vehicle ownership and single-occupancy vehicles, which contributes to the sustainability of a region. Metaverse is a digital ecosystem that integrates SE applications with transportation sector and improves the sustainability of SE applications. The aim of this work is to assess and prioritize the Metaverse integration options of sharing economy from multiple criteria and uncertainty perspectives. In this regard, the proposed study develops a four stage decision-making method on interval-valued Pythagorean fuzzy (IVPF) setting. First, new score function is proposed to compare the interval-valued Pythagorean fuzzy numbers (IVPFNs) and further used to derive the weights of decision experts in the proposed method. Second, some IVPF interaction aggregation operators are proposed to aggregate the individual’s information in group decision-making. Third, an integrated weighting process is presented to obtain the objective weights of criteria using cross entropy-based formula and subjective weights by relative closeness coefficient-based model with IVPF information. For this purpose, a cross entropy is introduced for interval-valued Pythagorean fuzzy sets. Fourth, a hybrid operational competitiveness ratings assessment (OCRA) model is developed to rank the alternatives with IVPF information. Moreover, a case study of Metaverse integration options of sharing economy in transportation sector is taken to express the practicability and effectiveness of the presented method. Sensitivity analysis with respect to various values of parameter is discussed to reveal the stability of outcomes. It is also compared with some extant models to validate the superiority of developed method.

An investment decision framework for offshore wind-solar-seawater pumped storage power project under interval-valued Pythagorean fuzzy environment

Offshore wind-solar-seawater pumped storage (wind-PV-SPS) power system will be a very competitive offshore new energy project in the future because it can realize the complementarities of wind and photovoltaic resources in the dimensions of time and space, and reduce the waste of resources caused by voltage instability. In order to promote the sustainable development of offshore wind-PV-SPS system, it is particularly important to construct a scientific and reasonable investment decision-making framework. Firstly, a comprehensive criteria system is established in this paper, which includes economy, resource, environment, supporting conditions and market. Secondly, the interval-valued Pythagorean fuzzy set (IVPFS) is introduced to describe the uncertainty and fuzziness of information in the process of investment decision, and the expert weight correction model is established based on the similarity of IVPFSs. At the same time, IVPF-DEMATEL method is utilized to determine the weight of criteria, which takes the interaction between criteria into account. In addition, considering the decision-maker's risk preference psychology, we introduce the GOWUA-HARA operator into the PROMETHEE II method, and extend it to the interval-valued Pythagorean fuzzy environment to choose the optimal investment alternative. Finally, the proposed framework is applied to a case study. Then sensitivity analysis and comparative analysis are carried out to verify its feasibility and robustness.

Development and Application of an Interval-Valued Pythagorean Fuzzy Set-Based Grade Evaluation Method to Foreign Fiber Content

ABSTRACT To solve the problem of the grade evaluation of foreign fiber content in cotton, this paper divided the problem into multi-criteria decision-making field and proposed an interval-valued Pythagorean fuzzy set foreign fiber content grade evaluation method. Taking into account the various properties of foreign fibers, the multi-criteria evaluation of single foreign fiber content and comprehensive foreign fiber content is carried out to improve the quality of cotton textiles. Based on the Pythagorean fuzzy set, we used AHP and TOPSIS methods to determine the relevant weight information, combined with the grade evaluation index of foreign fiber content and the interval-valued Pythagorean fuzzy weighted average or geometric operators to construct a fiber multi-criteria decision model. Then, the evaluation index function is improved to make the score function more consistent with the facts, and the digitization and intelligence methods required for foreign fiber content detection and grade evaluation are developed. Further, an example is given to verify the feasibility and effectiveness of the model. In addition, the multi-criteria decision-making method adopted has high theoretical innovation and application value for the textile industry.

A design model of FBS based on interval-valued Pythagorean fuzzy sets

The FBS (Function-Behaviour-Structure) model is a research model that stimulates creative thinking of designers in the design process. In order to reduce the influence of user requirement ambiguity on design results in the product design process and improve the accuracy of user requirements in the function-behavior-structure (FBS) design model, this paper proposes an interval-valued Pythagorean fuzzy set-based FBS model integrating AHP and HOQ methods. Firstly, the design model will use IVPF-AHP method to study user requirements and use interval-valued Pythagorean linguistic terms to replace the traditional scoring method of AHP to get the weight of each user requirement. Secondly, the conversion between user requirements and functions will be realized by IVPF-HOQ method, which converts customer requirements into functional characteristics and calculates the weights of each functional characteristic. Finally, the design focus will be filtered according to the order of importance of the functional characteristics, which will be used as functions to guide the development of the FBS model. In this paper, the feasibility and effectiveness of the proposed method will be verified by an application example of a hand-held fluorescence spectrometer. The results show that the proposed FBS model can effectively reduce the subjectivity and ambiguity in the decision-making process, improve the accuracy and information richness of user requirements, and effectively highlight the focus of the design study. The innovation of the proposed method is to provide a more objective and accurate innovative design method for user requirements through the integration of AHP, HOQ and FBS to effectively explore and analyze user requirements. The use of IVPFS to deal with fuzzy information in the design process in a more flexible manner can reduce the ambiguity of requirements when user data is small, and effectively improve the limitations of the FBS design model which is more subjective.

Development of feature extraction method based on interval-valued Pythagorean fuzzy decision theory

Feature extraction is an important step in image processing and the accuracy of the extraction affects the result of subsequent target detection. There are many feature extraction algorithms, but there are few theories and methods about the accuracy of the extraction. Moreover, most of the existing extraction methods are computationally complex, and they will expose deficiencies once they encounter a large number of standard cases. In order to fill this gap, this study proposes a new soft computing multi-criteria group decision-making (MCGDM) method based on interval-valued Pythagorean fuzzy set information to deal with situations where multiple criteria are involved in complex decisions. Thus, the power average operator is used to eliminate the effects of unreasonable input arguments on decision-making results, and the Maclaurin symmetric mean operator is used to comprehensively consider the mutual relationship between input arguments in this article. Combining with the concept of IVPFS, the interval-valued Pythagorean fuzzy power Maclaurin symmetric mean (IVPFPMSM) operator and weighted IVPFPMSM operator are proposed. Then, the TOPSIS method is used to determine the weight vector of decision-making set, and a MCGDM method is developed under the interval-valued Pythagorean fuzzy framework. Next, aiming at the defects of the existing evaluation criteria, we consider the different ranges of membership degree and non-membership degree to propose a new evaluation criterion of MCGDM method. Finally, the proposed method is applied to the optimal selection of characteristic interval of foreign fibers to verify its practicability and effectiveness. The proposed method has certain guiding significance for the extension of decision-making fuzzy operators and the exploration of feature extraction methods.

Sustainability assessment of palm oil industry 4.0 technologies in a circular economy applications based on interval-valued Pythagorean fuzzy rough set-FWZIC and EDAS methods

The palm oil industry is one of the most competitive industries and must comply with industry standards for flexible supply chain operations, productivity, and sustainability. Many researchers have demonstrated that Industry 4.0 technologies in a circular economy and sustainability practices (I4.0-in-a-CE-and-SPs) present promising future research opportunities, particularly for an industry with sustainability challenges. Hence, determining the most sustainable I4.0-in-a-CE-and-SPs application is critical for the palm oil industry. However, this process is deemed as a multiple attributes decision-making (MADM) problem due to the presence of three issues, namely, multiple sustainability performance attributes, uncertainty of the attribute’s importance level, and data variation. Therefore, an MADM solution is necessary to address these issues. This study extended the fuzzy weighted with zero inconsistency (FWZIC) method with an interval-valued Pythagorean fuzzy rough set (IVPFRS) and integrated it with the evaluation based on distance from average solution (EDAS) method to rank I4.0-in-a-CE-and-SP applications. The research method starts with the creation of a decision matrix (DM) based on the intersection of 26 I4.0-in-a-CE-and-SPs applications and 14 sustainability performance attributes. The IVPFRS–FWZIC method is subsequently developed to determine the weights of the sustainability performance attributes. The EDAS method uses these weights and the constructed DM to rank I4.0-in-a-CE-and-SPs applications. The robustness of the proposed methods was evaluated using sensitivity analysis, the Spearman’s correlation coefficient test, and comparison analysis.