{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T12:42:24Z","timestamp":1777293744273,"version":"3.51.4"},"reference-count":65,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T00:00:00Z","timestamp":1631577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article considers an inventory model for non-instantaneous deteriorating items with expiration dates, such as seasonal items, first-hand vegetables, and fruits. Interestingly, an inspection will be performed to manage the quality of the items during the state of no deterioration because it is difficult to purchase items with 100% perfection. Additionally, we assume that the upstream member has the power of controlling or influencing downstream members\u2019 decisions. That is, the supplier asks the retailer for a partial advance payment to avoid cancellation of orders and offers them a credit payment to stimulate sales; in turn, the customer must pay some cash when placing an order and pay the remainder in credit for the retailer. The goal of this article is to determine an optimal replenishment cycle and the total annual cost function, so we explore the functional properties of the total annual cost function and show that the total annual cost function is convex. Theoretical analysis of the optimal properties shows the existence and uniqueness of the optimal solution. Then, we obtain simple and easy solution procedures for the inventory system. Moreover, numerical analysis of the inventory model is conducted, and the corresponding examples are considered with a view to illustrating the application of the supply chain model that we have investigated in this article. Finally, in the concluding section, we have not only provided the motivation and the need for our usages of mathematical analytic solution procedures based upon the convexity, monotonicity (increasing and decreasing) and differentiability properties of the object function (that is, the total annual cost function), which involve some symmetry aspects of the object function, but we have also indicated the limitations and shortcomings in our investigation, which will naturally lead to some potential directions for further research on the supply chain model, which we have considered and mathematically analyzed in this article.<\/jats:p>","DOI":"10.3390\/sym13091695","type":"journal-article","created":{"date-parts":[[2021,9,14]],"date-time":"2021-09-14T23:34:11Z","timestamp":1631662451000},"page":"1695","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Inventory Models for Non-Instantaneous Deteriorating Items with Expiration Dates and Imperfect Quality under Hybrid Payment Policy in the Three-Level Supply Chain"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2088-307X","authenticated-orcid":false,"given":"Jui-Jung","family":"Liao","sequence":"first","affiliation":[{"name":"Department of Business Administration, Chihlee University of Technology, Banqiao District, New Taipei City 22050, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kun-Jen","family":"Chung","sequence":"additional","affiliation":[{"name":"College of Business, Chung Yuan Christian University, Chung Li 320314, Taiwan"},{"name":"Department of Industrial Management, National Taiwan University of Science and Technology, Taipei 10607, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shih-Fang","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 320314, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kuo-Nan","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Industrial Management and Business Administration, St. John\u2019s University, Tamsui District, New Taipei City 25135, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shy-Der","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Business Administration, Chung Yuan Christian University, Chung Li 320314, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/0305-0483(96)00023-0","article-title":"Optimal advance payment scheme involving fixed per-payment costs","volume":"24","author":"Zhang","year":"1996","journal-title":"Omega"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/j.mcm.2012.07.002","article-title":"An economic order quantity model with multiple partial prepayments and partial backordering","volume":"57","author":"Taleizadeh","year":"2013","journal-title":"Math. 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