Adaptive Neutrosophic Goal Programming for SoH-Aware Tri-Objective Closed-Loop Supply Chain Optimisation under Battery Degradation Uncertainty

Adaptive Neutrosophic Goal Programming for SoH-Aware Tri-Objective Closed-Loop Supply Chain Optimisation under Battery Degradation Uncertainty | 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 Research Article Adaptive Neutrosophic Goal Programming for SoH-Aware Tri-Objective Closed-Loop Supply Chain Optimisation under Battery Degradation Uncertainty Vaitheeswaran Gnanaraj, Balakrishanan Vellaikannan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9613524/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Background. Battery-powered logistics networks face a structurally complex challenge: progressive State-of-Health (SoH) degradation simultaneously contracts fleet capacity, inflates operational cost, and elevates parameter estimation uncertainty in ways that classical deterministic optimisation models cannot represent. This paper addresses the gap by embedding SoH directly into the feasible region of a tri-objective closed-loop supply chain (CLSC) linear programme in which all cost and emission parameters are encoded as adaptive Single-Valued Triangular Neutrosophic Numbers (SVTNN). Methods. The multi-echelon CLSC spans three suppliers, two manufacturing plants, three SoH-degradable distribution centres (capacity Capₑ(k) = 280 × SoH), four customer zones, two collection centres, and two remanufacturing centres over three planning periods. All 26 arc-level cost and emission parameters are initialised as SVTNN triplets (a, b, c; α, β, γ) and updated adaptively each period via three rules driven by SoH, State-of-Charge (SoC), and acoustic emission signal intensity. Three complementary solution methods are deployed: (i) a weighted-sum LP sensitivity sweep across seven SoH levels (1.00– 0.73) establishing the cost–emission–shortage baseline; (ii) an epsilon-constraint Pareto frontier at SoH = 0.92 over 29 emission budget levels; and (iii) Neutrosophic Goal Programming (NGP) with corrected constrained aspiration levels — G₁* = min Z₁ | Z₃ ≤ Z₃_WS, G₂* = min Z₂ | Z₃ ≤ Z₃_WS, G₃* = min Z₃ — and neutrosophic deviation weights wₜ = 0.60, w i = 0.20, w_f = 0.20. The resulting NGP comprises 144 decision variables and 97 constraints per SoH level. Results. SoH degradation from 1.00 to 0.73 reduces Z₁ by 27.0% (₹9,780→₹7,139) and Z₂ by 27.0% (892→651 kg CO₂), while raising Z₃ from 975 to 1,052 units (83.6→ 90.2%). The invariant ratio Z₂/Z₁ = 0.0912 kg CO₂/₹ across all SoH levels enables carbon footprint estimation directly from cost accounts. The Pareto frontier reveals a near-constant exchange rate of 5.8 shortage units per 28 kg CO₂. The NGP yields LP-Optimal status at all seven SoH levels with d i ⁺ = d i ⁻ = 0 for all three objectives, confirming that the aspiration point is attainable with zero compromise. Additionally, Z₂_NGP lies 7.5–10.3 kg CO₂ below the weighted-sum LP emission at every degradation level — a routing-reallocation gain unavailable to the weighted-sum formulation. The adaptive SVTNN analysis identifies SoH = 0.80 as a diagnostic falsity-peak inflection warranting intensified battery management and SoH = 0.73 as the operational replacement threshold. Conclusions. The SVTNN–NGP framework is the first to jointly integrate battery degradation physics, adaptive neutrosophic uncertainty quantification, Neutrosophic Goal Programming with corrected aspiration levels, and an exact Pareto frontier into a multi-period closed-loop supply chain model. The derived SoH thresholds, emission–shortage exchange rate, cost–emission invariance rule, and NGP ideal-point attainability result are immediately deployable as fleet management and regulatory policy criteria without additional computational infrastructure. Operations Research Decision Sciences State of Health (SoH) Neutrosophic Goal Programming (NGP) Closed-Loop Supply Chain (CLSC) Single-Valued Triangular Neutrosophic Number (SVTNN) Epsilon-constraint Pareto frontier Battery degradation Multi-objective optimisation Adaptive membership update Green logistics Full Text Additional Declarations The authors declare no competing interests. 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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-9613524","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":634487516,"identity":"ae480c67-0c2c-45fc-a84c-3c21c88e9c27","order_by":0,"name":"Vaitheeswaran Gnanaraj","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVRIiWNgGAWjYFCCA1CavQFIGFiQooUHxDCQIMU2iQQwSViheePZoxu/7rDL45/5/OqGHwUSDPzt3Ql4tcgcOJd2W/ZMcrHE7Zyymz1Ah0mcObsBv3MYzpjdlmxjTmy4nZN2gweoxUAilygt9Ynzb55Ju/mHWC03P7YdTtxwg/3YbSJtAfqFse144sYzOWy3ZQwkeAj7ReLssZs/26oT5x0//uzmmz82cvztvfi1AAOIgZkHzOIxAJP4lYMAfw8D4w8wi/0BYdWjYBSMglEwIgEAEtJO9SqVxSMAAAAASUVORK5CYII=","orcid":"","institution":"Thiagarajar College of Engineering","correspondingAuthor":true,"prefix":"","firstName":"Vaitheeswaran","middleName":"","lastName":"Gnanaraj","suffix":""},{"id":634487535,"identity":"e2c993d3-6a96-49df-8344-ccdc369d594a","order_by":1,"name":"Balakrishanan Vellaikannan","email":"","orcid":"","institution":"Thiagarajar College of Engineering","correspondingAuthor":false,"prefix":"","firstName":"Balakrishanan","middleName":"","lastName":"Vellaikannan","suffix":""}],"badges":[],"createdAt":"2026-05-05 03:55:39","currentVersionCode":1,"declarations":{"humanSubjects":true,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":true,"humanSubjectConsent":true,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9613524/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9613524/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108805243,"identity":"4dba535a-8711-4904-96c6-66778411b1e3","added_by":"auto","created_at":"2026-05-08 15:25:20","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":913949,"visible":true,"origin":"","legend":"","description":"","filename":"BMSDigitalTwinMultiObjectiveV1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9613524/v1_covered_42397960-4e6e-42da-a68a-dfa4a3dfd78d.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eAdaptive Neutrosophic Goal Programming for SoH-Aware\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTri-Objective Closed-Loop Supply Chain Optimisation\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eunder Battery Degradation Uncertainty\u003c/strong\u003e\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Thiagarajar College of engineering","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"State of Health (SoH), Neutrosophic Goal Programming (NGP), Closed-Loop Supply Chain (CLSC), Single-Valued Triangular Neutrosophic Number (SVTNN), Epsilon-constraint Pareto frontier, Battery degradation, Multi-objective optimisation, Adaptive membership update, Green logistics","lastPublishedDoi":"10.21203/rs.3.rs-9613524/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9613524/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBackground. Battery-powered logistics networks face a structurally complex challenge: progressive State-of-Health (SoH) degradation simultaneously contracts fleet capacity, inflates operational cost, and elevates parameter estimation uncertainty in ways that classical deterministic optimisation models cannot represent. This paper addresses the gap by embedding SoH directly into the feasible region of a tri-objective closed-loop supply chain (CLSC) linear programme in which all cost and emission parameters are encoded as adaptive Single-Valued Triangular Neutrosophic Numbers (SVTNN).\u003c/p\u003e \u003cp\u003e \u003cb\u003eMethods.\u003c/b\u003e The multi-echelon CLSC spans three suppliers, two manufacturing plants, three SoH-degradable distribution centres (capacity Capₑ(k)\u0026thinsp;=\u0026thinsp;280 \u0026times; SoH), four customer zones, two collection centres, and two remanufacturing centres over three planning periods. All 26 arc-level cost and emission parameters are initialised as SVTNN triplets (a, b, c; α, β, γ) and updated adaptively each period via three rules driven by SoH, State-of-Charge (SoC), and acoustic emission signal intensity. Three complementary solution methods are deployed: (i) a weighted-sum LP sensitivity sweep across seven SoH levels (1.00\u0026ndash; 0.73) establishing the cost\u0026ndash;emission\u0026ndash;shortage baseline; (ii) an epsilon-constraint Pareto frontier at SoH\u0026thinsp;=\u0026thinsp;0.92 over 29 emission budget levels; and (iii) Neutrosophic Goal Programming (NGP) with corrected constrained aspiration levels \u0026mdash; G₁* = min Z₁ | Z₃ \u0026le; Z₃_WS, G₂* = min Z₂ | Z₃ \u0026le; Z₃_WS, G₃* = min Z₃ \u0026mdash; and neutrosophic deviation weights wₜ = 0.60, w\u003csub\u003ei\u003c/sub\u003e = 0.20, w_f\u0026thinsp;=\u0026thinsp;0.20. The resulting NGP comprises 144 decision variables and 97 constraints per SoH level.\u003c/p\u003e \u003cp\u003e \u003cb\u003eResults.\u003c/b\u003e SoH degradation from 1.00 to 0.73 reduces Z₁ by 27.0% (₹9,780\u0026rarr;₹7,139) and Z₂ by 27.0% (892\u0026rarr;651 kg CO₂), while raising Z₃ from 975 to 1,052 units (83.6\u0026rarr; 90.2%). The invariant ratio Z₂/Z₁ = 0.0912 kg CO₂/₹ across all SoH levels enables carbon footprint estimation directly from cost accounts. The Pareto frontier reveals a near-constant exchange rate of 5.8 shortage units per 28 kg CO₂. The NGP yields LP-Optimal status at all seven SoH levels with d\u003csub\u003ei\u003c/sub\u003e⁺ = d\u003csub\u003ei\u003c/sub\u003e⁻ = 0 for all three objectives, confirming that the aspiration point is attainable with zero compromise. Additionally, Z₂_NGP lies 7.5\u0026ndash;10.3 kg CO₂ below the weighted-sum LP emission at every degradation level \u0026mdash; a routing-reallocation gain unavailable to the weighted-sum formulation. The adaptive SVTNN analysis identifies SoH\u0026thinsp;=\u0026thinsp;0.80 as a diagnostic falsity-peak inflection warranting intensified battery management and SoH\u0026thinsp;=\u0026thinsp;0.73 as the operational replacement threshold.\u003c/p\u003e \u003cp\u003e \u003cb\u003eConclusions.\u003c/b\u003e The SVTNN\u0026ndash;NGP framework is the first to jointly integrate battery degradation physics, adaptive neutrosophic uncertainty quantification, Neutrosophic Goal Programming with corrected aspiration levels, and an exact Pareto frontier into a multi-period closed-loop supply chain model. The derived SoH thresholds, emission\u0026ndash;shortage exchange rate, cost\u0026ndash;emission invariance rule, and NGP ideal-point attainability result are immediately deployable as fleet management and regulatory policy criteria without additional computational infrastructure.\u003c/p\u003e","manuscriptTitle":"Adaptive Neutrosophic Goal Programming for SoH-Aware\nTri-Objective Closed-Loop Supply Chain Optimisation\nunder Battery Degradation Uncertainty","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-06 07:11:29","doi":"10.21203/rs.3.rs-9613524/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"76273f4d-3838-4490-8527-973edf2cc546","owner":[],"postedDate":"May 6th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":67521662,"name":"Operations Research"},{"id":67521663,"name":"Decision Sciences"}],"tags":[],"updatedAt":"2026-05-06T07:11:30+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-06 07:11:29","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9613524","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9613524","identity":"rs-9613524","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}