An Enhanced Bidirectional Projection Approach for Fuzzy Credibility Multi-Attribute Group Decision Making: Application in Sports Equipment Supplier Selection

An Enhanced Bidirectional Projection Approach for Fuzzy Credibility Multi-Attribute Group Decision Making: Application in Sports Equipment Supplier Selection | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article An Enhanced Bidirectional Projection Approach for Fuzzy Credibility Multi-Attribute Group Decision Making: Application in Sports Equipment Supplier Selection Longfei Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5962615/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 18 Nov, 2025 Read the published version in Scientific Reports → Version 1 posted 11 You are reading this latest preprint version Abstract This article will introduce the fuzzy credibility bidirectional projection (BP) method and some important basic theories of fuzzy credibility set (FCS). Firstly, we reviewed the definition, operation rules, score function, and comparison methods of fuzzy credibility numbers (FCNs). In addition, in order to better integrate the fuzzy credibility evaluation information of multiple experts, we introduced two aggregation operators, such as FCNWAA and FCNWGA. Then, we proposed the concept of the module of FCS and introduced it into the BP method. Incorporating bounded rationality of decision-makers, we applied cumulative prospect theory (CPT) to this multi-attribute group decision-making (MAGDM) method. We employed the minimum identification information principle (MIIP) to integrate the entropy weight method with the method based on the removal effects of criteria (MEREC), alongside subjective weights, to determine decision attribute weights. This approach balances the subjective input of decision-makers with the significance of objective data, making it more applicable to real-world scenarios. Detailed calculation steps for this MAGDM technique are provided. Finally, we propose the MAGDM technique based on fuzzy credibility bidirectional projection method to selection of sports equipment suppliers (SSES) issue with fuzzy credibility information. This paper evaluates the effectiveness of the MAGDM technique by comparing it with the FCNWAA, FCNWGA, and FCDWG operators. Physical sciences/Mathematics and computing Physical sciences/Mathematics and computing/Applied mathematics Fuzzy credibility set Cumulative prospect theory Bidirectional projection method MAGDM selection of sports equipment suppliers Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 18 Nov, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 30 May, 2025 Reviews received at journal 21 May, 2025 Reviewers agreed at journal 22 Apr, 2025 Reviews received at journal 23 Feb, 2025 Reviewers agreed at journal 13 Feb, 2025 Reviewers agreed at journal 13 Feb, 2025 Reviewers invited by journal 13 Feb, 2025 Editor assigned by journal 10 Feb, 2025 Editor invited by journal 10 Feb, 2025 Submission checks completed at journal 10 Feb, 2025 First submitted to journal 05 Feb, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5962615","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":414376866,"identity":"3defe659-fdf8-4d84-b62d-bae0dddb9802","order_by":0,"name":"Longfei Liu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtklEQVRIiWNgGAWjYBACPjBZYMHPz8x8+AFRWtjApIGE5Mx2tjQD0rRsOM+jIEGcFun2x595DCQkjA/zMBgw1NhEE9Yic8ZMGqTF7DDvgQcMx9JyGwhqkchhYwZqqTM7zJdgwNhwmBgt6VCHNQNJIrUkGIAdZsBMtBagXyTnALVIHAYGcgIxfuEHhtiHNxU2Evz9hw8/+FBjQ1gLAzAumHhgnASCyqFaGH8QpXIUjIJRMApGLAAAGTIzFb1bEY0AAAAASUVORK5CYII=","orcid":"","institution":"Liupanshui Normal University","correspondingAuthor":true,"prefix":"","firstName":"Longfei","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2025-02-05 05:53:51","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5962615/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5962615/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-24279-y","type":"published","date":"2025-11-18T15:57:33+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":96651310,"identity":"14dc7cf5-6e72-4d0e-91d4-8f723390f7d9","added_by":"auto","created_at":"2025-11-24 16:14:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":474657,"visible":true,"origin":"","legend":"","description":"","filename":"AnEnhancedBidirectionalProjectionApproachforFuzzyCredibilityMultiAttributeGroupDecisionMakingApplicationinSportsEquipmentSupplierSelection.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5962615/v1_covered_804032b6-5569-4811-9e5b-f49fe19139b1.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An Enhanced Bidirectional Projection Approach for Fuzzy Credibility Multi-Attribute Group Decision Making: Application in Sports Equipment Supplier Selection","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Fuzzy credibility set, Cumulative prospect theory, Bidirectional projection method, MAGDM, selection of sports equipment suppliers","lastPublishedDoi":"10.21203/rs.3.rs-5962615/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5962615/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis article will introduce the fuzzy credibility bidirectional projection (BP) method and some important basic theories of fuzzy credibility set (FCS). 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