A Variational Inequality Framework forMulti-Echelon Supply Chains

A Variational Inequality Framework forMulti-Echelon Supply Chains | 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 A Variational Inequality Framework forMulti-Echelon Supply Chains Gabriella Colajanni, Patrizia Daniele, Rebecca Fontanarosa This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9495163/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 6 You are reading this latest preprint version Abstract This paper proposes a profit-maximization model for a multi-echelon supply chain, integrating transportation and inventory decisions within a unified framework. The system is formulated as a Variational Inequality, ensuring equilibrium conditions under convex costs and network constraints. By incorporating the Economic Order Quantity policy, inventory variables are eliminated, reducing model complexity. A projection method is used for computation, and numerical results illustrate the effectiveness of the approach in capturing realistic logistics behaviors. multi-echelon supply chain variational inequality network optimization EOQ Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 30 Apr, 2026 Reviewers agreed at journal 29 Apr, 2026 Reviewers invited by journal 29 Apr, 2026 Editor assigned by journal 24 Apr, 2026 Submission checks completed at journal 24 Apr, 2026 First submitted to journal 22 Apr, 2026 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-9495163","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":632487447,"identity":"e1be15d3-4225-4cdd-81d9-12580f69cbce","order_by":0,"name":"Gabriella Colajanni","email":"","orcid":"","institution":"University of Catania","correspondingAuthor":false,"prefix":"","firstName":"Gabriella","middleName":"","lastName":"Colajanni","suffix":""},{"id":632487448,"identity":"77a38e84-7b9c-431a-9612-b12e13ba8386","order_by":1,"name":"Patrizia Daniele","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCUlEQVRIie2Qv2rDMBCHLxjqReD1FyjNKygYnARc/CoShmZxO5YOpdGULoaufo9CZplCpkBWjwZDpgyZiguhVC4p/QNK1g76kI4Tuo87icjh+K8Iwmc0a0yBSbXZzF7v/VJAfdUpgpjd8b57GUBcHw42ZeSvl7t6Px6Qn27q+g4XYXVd6reWzhOLMslTrxAMQ8U2Iy5WCKPqRpT5kcG4Tj0SQE9BRJBzyEWV8Zdjb+HrxigcicL0FfIds+filFJ1XQSkQhZ10egnlEnRhCQ00jnb3kIsMSxWW17mV2BMW34skE2v3T9cPvnTRb+9jwfBYxbu2jhOfGUZ7Cs5+3MBy1g/FIfD4XBY+QAtVFDLIwVe4gAAAABJRU5ErkJggg==","orcid":"","institution":"University of Catania","correspondingAuthor":true,"prefix":"","firstName":"Patrizia","middleName":"","lastName":"Daniele","suffix":""},{"id":632487449,"identity":"6450f9ba-28b3-49bb-8121-6448c2bb7261","order_by":2,"name":"Rebecca Fontanarosa","email":"","orcid":"","institution":"University of Catania","correspondingAuthor":false,"prefix":"","firstName":"Rebecca","middleName":"","lastName":"Fontanarosa","suffix":""}],"badges":[],"createdAt":"2026-04-22 10:54:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9495163/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9495163/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108805895,"identity":"d5393884-6b5f-4c3b-8fc6-e09f228bb65b","added_by":"auto","created_at":"2026-05-08 15:27:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":366448,"visible":true,"origin":"","legend":"","description":"","filename":"snNandSE.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9495163/v1_covered_5addf3cf-de33-4963-b7ce-b17ada188d59.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Variational Inequality Framework forMulti-Echelon Supply Chains","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"networks-and-spatial-economics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nets","sideBox":"Learn more about [Networks and Spatial Economics](http://link.springer.com/journal/11067)","snPcode":"11067","submissionUrl":"https://submission.nature.com/new-submission/11067/3","title":"Networks and Spatial Economics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"multi-echelon supply chain, variational inequality, network optimization, EOQ","lastPublishedDoi":"10.21203/rs.3.rs-9495163/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9495163/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper proposes a profit-maximization model for a multi-echelon supply chain, integrating transportation and inventory decisions within a unified framework. 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