{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:59:55Z","timestamp":1776841195757,"version":"3.51.2"},"reference-count":44,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,12]],"date-time":"2024-12-12T00:00:00Z","timestamp":1733961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Prince Sattam bin Abdulaziz University","award":["PSAU\/2024\/R\/1446"],"award-info":[{"award-number":["PSAU\/2024\/R\/1446"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This paper formulates six inventory models for products with instantaneous deterioration, focusing on the impacts of full and partial advance payment structures. The demand function depends on both price and stock levels and accounts for shortages through full backlogging. The primary objective is to determine the optimal payment policy under varying trade facilities, analyzing six distinct payment scenarios commonly employed in business practice. Each model is presented with closed-form solutions and supported by mathematical formulations. For each case, algorithms and mathematical proofs are developed to determine the optimal cycle duration and corresponding unit cost. Numerical examples and 2D graphical representations generated using MATLAB are included to validate the proposed models. Additionally, a sensitivity analysis is conducted to examine the effects of each payment policy and parameter variation, providing key managerial insights into payment planning in inventory management.<\/jats:p>","DOI":"10.3390\/computation12120244","type":"journal-article","created":{"date-parts":[[2024,12,12]],"date-time":"2024-12-12T12:25:54Z","timestamp":1734006354000},"page":"244","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Inventory Model for Instantaneously Deteriorating Items with Multiple Trade Facilities, Stock- and Price-Dependent Demand, and Full Backlogging"],"prefix":"10.3390","volume":"12","author":[{"given":"Rabeya","family":"Sarker","sequence":"first","affiliation":[{"name":"Department of Mathematics, Jahangirnagar University, Dhaka 1342, Bangladesh"},{"name":"Department of Business Administration, Dhaka International University, Dhaka 1212, Bangladesh"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8314-9473","authenticated-orcid":false,"given":"Md. Sharif","family":"Uddin","sequence":"additional","affiliation":[{"name":"Industrial Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"}]},{"given":"Md Abu","family":"Helal","sequence":"additional","affiliation":[{"name":"Information Systems and Operations Management, University of Texas at Arlington, Arlington, TX 76019, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1728-8090","authenticated-orcid":false,"given":"Aminur Rahman","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Jahangirnagar University, Dhaka 1342, Bangladesh"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1538-6124","authenticated-orcid":false,"given":"Ali","family":"AlArjani","sequence":"additional","affiliation":[{"name":"Industrial Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2703-1969","authenticated-orcid":false,"given":"El-Awady","family":"Attia","sequence":"additional","affiliation":[{"name":"Industrial Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"},{"name":"Mechanical Engineering Department, Faculty of Engineering (Shoubra), Benha University, Cairo 11629, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1111\/itor.12632","article-title":"Price discount facility in an EOQ model for deteriorating items with stock-dependent demand and partial backlogging","volume":"26","author":"Shaikh","year":"2019","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"102840","DOI":"10.1016\/j.jksus.2023.102840","article-title":"Effects of a quantity-based discount frame in inventory planning under time-dependent demand: A case study of mango businesses in Bangladesh","volume":"35","author":"Rukonuzzaman","year":"2023","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"L\u00f3pez-Forni\u00e9s, I., and Sierra-P\u00e9rez, J. (2022). Ephemeral products: Opportunities for circularity based on ideation for reuse. An experience. Design Tools and Methods in Industrial Engineering II: Proceedings of the Second International Conference on Design Tools and Methods in Industrial Engineering, ADM 2021, Rome, Italy, 9\u201310 September 2021, Springer International Publishing.","DOI":"10.1007\/978-3-030-91234-5_37"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/j.aej.2023.04.059","article-title":"Advertising and pricing strategies of an inventory model with product freshness-related demand and expiration date-related deterioration","volume":"73","author":"Khan","year":"2023","journal-title":"Alex. Eng. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1287\/mnsc.20.4.464","article-title":"Inventory policy and trade credit financing","volume":"20","author":"Haley","year":"1973","journal-title":"Manag. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1057\/jors.1985.56","article-title":"Economic order quantity under conditions of permissible delay in payments","volume":"36","author":"Goyal","year":"1985","journal-title":"J. Oper. Res. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1057\/jors.1995.90","article-title":"Ordering policies of deteriorating items under permissible delay in payments","volume":"46","author":"Aggarwal","year":"1995","journal-title":"J. Oper. Res. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1057\/palgrave.jors.2600428","article-title":"An ordering policy for deteriorating items with allowable shortages and permissible delay in payment","volume":"48","author":"Jamal","year":"1997","journal-title":"J. Oper. Res. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1016\/j.mcm.2008.09.015","article-title":"An application of genetic algorithm in solving an inventory model with advance payment and interval valued inventory costs","volume":"49","author":"Gupta","year":"2009","journal-title":"Math. Comput. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2178","DOI":"10.3934\/jimo.2022038","article-title":"A production inventory model for high-tech products involving two production runs and a product variation","volume":"19","author":"Ruidas","year":"2023","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2433","DOI":"10.1016\/j.apm.2008.07.024","article-title":"Inventory model with stochastic lead-time and price dependent demand incorporating advance payment","volume":"33","author":"Maiti","year":"2009","journal-title":"Appl. Math. Model."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/j.ijpe.2012.03.030","article-title":"Optimal price discounting and lot-sizing policies for perishable items in a supply chain under advance payment scheme and two-echelon trade credits","volume":"139","author":"Thangam","year":"2012","journal-title":"Int. J. Prod. Econ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.ijpe.2014.01.023","article-title":"An EOQ model with partial backordering and advance payments for an evaporating item","volume":"155","author":"Taleizadeh","year":"2014","journal-title":"Int. J. Prod. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.tre.2015.07.008","article-title":"A lot-sizing model with backordering under hybrid linked-to-order multiple advance payments and delayed payment","volume":"82","author":"Zia","year":"2015","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"8605","DOI":"10.1016\/j.apm.2016.05.022","article-title":"Inventory lot-size policies for deteriorating items with expiration dates and advance payments","volume":"40","author":"Teng","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s10479-015-2100-5","article-title":"Partial up-stream advanced payment and partial down-stream delayed payment in a three-level supply chain","volume":"238","author":"Lashgari","year":"2016","journal-title":"Ann. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1111\/itor.12297","article-title":"Lot-sizing model with advance payment pricing and disruption in supply under planned partial backordering","volume":"24","author":"Taleizadeh","year":"2017","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1016\/j.jmsy.2017.04.005","article-title":"A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items","volume":"45","author":"Diabat","year":"2017","journal-title":"J. Manuf. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/13873954.2017.1324882","article-title":"Optimal replenishment time for retailer under partial upstream prepayment and partial downstream overdue payment for quadratic demand","volume":"24","author":"Shah","year":"2018","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5357","DOI":"10.1016\/j.apm.2014.02.014","article-title":"An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments","volume":"38","author":"Taleizadeh","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1111\/itor.12733","article-title":"The effect of advance payment with discount facility on supply decisions of deteriorating products whose demand is both price and stock dependent","volume":"27","author":"Khan","year":"2020","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"120117","DOI":"10.1016\/j.jclepro.2020.120117","article-title":"The effect of inspection on deterioration rate: An inventory model for deteriorating items with advanced payment","volume":"254","author":"Khakzad","year":"2020","journal-title":"J. Clean. Prod."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"126608","DOI":"10.1016\/j.jclepro.2021.126608","article-title":"A sustainable inventory model with controllable carbon emissions, deterioration and advance payments","volume":"296","author":"Mashud","year":"2021","journal-title":"J. Clean. Prod."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109","DOI":"10.2298\/YJOR150404007K","article-title":"Two warehouse inventory model for deteriorating item with exponential demand rate and permissible delay in payment","volume":"27","author":"Kaliraman","year":"2016","journal-title":"Yugosl. J. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5910","DOI":"10.1016\/j.apm.2014.04.040","article-title":"Economic order quantity under advance payment","volume":"38","author":"Zhang","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"122708","DOI":"10.1016\/j.eswa.2023.122708","article-title":"Optimizing retailer prepayment installment in inventory systems with link-to-order discounts and power demand patterns of price and time: An effective decision support framework","volume":"242","author":"Khan","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1016\/j.ejor.2020.07.035","article-title":"EOQ-based pricing and customer credit decisions under general supplier payments","volume":"289","author":"Li","year":"2021","journal-title":"Eur. J. Oper. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"102418","DOI":"10.1016\/j.omega.2021.102418","article-title":"An inventory model under linked-to-order hybrid partial advance payment, partial credit policy, all-units discount and partial backlogging with capacity constraint","volume":"103","author":"Khan","year":"2021","journal-title":"Omega"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"140642","DOI":"10.1016\/j.jclepro.2024.140642","article-title":"Inventory model for livestock farm under quantity discount, power demand, prepayment and carbon rules","volume":"441","author":"Khan","year":"2024","journal-title":"J. Clean. Prod."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1016\/j.aej.2021.06.070","article-title":"Advance and delay in payments with the price-discount inventory model for deteriorating items under capacity constraint and partially backlogged shortages","volume":"61","author":"Duary","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"8469","DOI":"10.1016\/j.aej.2022.02.006","article-title":"Inventory management with hybrid cash-advance payment for time-dependent demand, time-varying holding cost and non-instantaneous deterioration under backordering and non-terminating situations","volume":"61","author":"Khan","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"108951","DOI":"10.1016\/j.ijpe.2023.108951","article-title":"Installment for advance payment and pricing decisions for an inventory system under power demand pattern and all-units discount","volume":"265","author":"Khan","year":"2023","journal-title":"Int. J. Prod. Econ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"106225","DOI":"10.1016\/j.cor.2023.106225","article-title":"Integrating prepayment installment, pricing and replenishment decisions for growing items with power demand pattern and non-linear holding cost under carbon regulations","volume":"156","author":"Khan","year":"2023","journal-title":"Comput. Oper. Res."},{"key":"ref_34","first-page":"241","article-title":"A two warehouse deterministic inventory model for deteriorating items with a linear trend in time dependent demand over finite time horizon by elitist real-coded genetic algorithm","volume":"4","author":"Bhunia","year":"2013","journal-title":"Int. J. Ind. Eng. Comput."},{"key":"ref_35","first-page":"1125","article-title":"A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging","volume":"232","author":"Bhunia","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_36","first-page":"831","article-title":"An application of PSO in a two-warehouse inventory model for deteriorating item under permissible delay in payment with different inventory policies","volume":"256","author":"Bhunia","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1504\/IJOR.2020.108835","article-title":"Economic ordering policy for deteriorating items with inflation induced time dependent demand under infinite time horizon","volume":"39","author":"Geetha","year":"2020","journal-title":"Int. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1080\/13873954.2019.1677724","article-title":"An ordering policy for deteriorating items with price-dependent iso-elastic demand under permissible delay in payments and price inflation","volume":"25","author":"Mahata","year":"2019","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1504\/IJOR.2023.131493","article-title":"Multi-item EOQ model for deteriorating items having multivariate dependent demand with variable holding cost and trade credit","volume":"47","author":"Yadav","year":"2023","journal-title":"Int. J. Oper. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"6451","DOI":"10.3934\/jimo.2022222","article-title":"Pricing strategy in an interval-valued production inventory model for high-tech products under demand disruption and price revision","volume":"19","author":"Ruidas","year":"2023","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"109098","DOI":"10.1016\/j.ijpe.2023.109098","article-title":"A sustainable inventory model for deteriorating items with power demand and full backlogging under a carbon emission tax","volume":"268","author":"Sicilia","year":"2024","journal-title":"Int. J. Prod. Econ."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Jayasankari, C., and Uthayakumar, R. (2024). A two-phase production inventory model for imperfect quality items with multivariate demand, upstream and downstream credit period offers under a carbon tax, and green subsidy. J. Control. Decis., 1\u201316.","DOI":"10.1080\/23307706.2024.2349630"},{"key":"ref_43","first-page":"100289","article-title":"Effects of variable prepayment installments on pricing and inventory decisions with power demand pattern and non-linear holding cost under carbon cap-and-price regulation","volume":"12","author":"Khan","year":"2024","journal-title":"Oper. Res. Perspect."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"107804","DOI":"10.1016\/j.ijpe.2020.107804","article-title":"Inventory models for perishable items with advanced payment, linearly time-dependent holding cost and demand dependent on advertisement and selling price","volume":"230","author":"Khan","year":"2020","journal-title":"Int. J. Prod. Econ."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/12\/244\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:53:53Z","timestamp":1760115233000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/12\/244"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,12]]},"references-count":44,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["computation12120244"],"URL":"https:\/\/doi.org\/10.3390\/computation12120244","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,12]]}}}