An Integrated Framework for Inventory Control of Perishable Goods with Preservation, Financing, and Sustainability with Brownian Demand

Abstract

Effective management of perishable inventories requires balancing uncertain demand, product deterioration, financial constraints, and sustainability considerations. This study develops a unified stochastic inventory framework for perishable goods, where demand follows a Brownian motion process modeled through stochastic differential equations (SDEs). The model simultaneously optimizes the order-up-to level, replenishment cycle, and investment in preservation technology, while incorporating time-dependent deterioration, partial backlogging, hybrid advance-delay payment schemes, and carbon taxation. Closed-form expressions for inventory dynamics across freshness, deterioration, and shortage phases are derived using Itô calculus, and optimal policies are identified through a simulation-optimization approach combining Monte Carlo methods with a two-stage grid search. Numerical experiments demonstrate that while the optimal decision variables under stochastic demand remain consistent with deterministic cases, the average total cost increases by 11%–13%, highlighting the “stochastic premium” of explicitly accounting for uncertainty. Sensitivity analysis reveals that the average demand rate and purchase cost are dominant drivers of profitability, offering actionable managerial insights. Furthermore, the model’s compatibility with IoT and AI-enabled data analytics underscores its practical applicability for dynamic, data-driven decision-making. By jointly addressing operational, financial, and environmental dimensions, this framework advances beyond traditional deterministic models and provides a robust decision-support tool for industries such as fresh produce, pharmaceuticals, and cold-chain logistics.

Keywords

Stochastic differential equations / perishable inventory / Brownian demand / preservation technology / hybrid financing / carbon tax

References

[1] Albeverio S, Röckner M. Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probability Theory and Related Fields, 1991, 893347386.

[2] Ali H, Shaikh AA, Hazaimeh HM. Investigation to optimise preservation cost in an imperfect production systems under carbon emission tax: The role of warranty period, green level and price dependent demand. Production Engineering, 2025

[3] Aljazzar SM, Jaber MY, Moussawi-Haidar L. Coordination of a three-level supply chain (suppliermanufacturer- retailer) with permissible delay-inpayments and price discounts. Applied Mathematical Modelling, 2017, 48: 289-302.

[4] Antic S, Djordjevic ML, Lisec A. Dynamic discrete inventory control model with deterministic and stochastic demand in pharmaceutical distribution. Applied Sciences, 2022, 1231536.

[5] Bakker M, Riezebos J, Teunter RH. Review of inventory systems with deterioration since 2001. European Journal of Operational Research, 2012, 2212275284.

[6] Barman H, Pervin M, Roy SK. Impacts of green and preservation technology investments on a sustainable EPQ model during COVID-19 pandemic. RAIRO - Operations Research, 2022, 56422452275.

[7] Boujnah I, Tlili M, Korbaa O. A modified particle swarm optimization algorithm in a rolling horizon framework for the aggregate production planning problem: Pharmaceutical industry case. Annals of Operations Research, 2025, 355330813098.

[8] Cachon GP, Feldman P. Is advance selling desirable with competition?. Marketing Science, 2017, 362214231.

[9] Cai X, Chen J, Xiao Y, Xu X. Optimization and coordination of fresh product supply chains with freshness keeping effort. Production and Operations Management, 2010, 193261278.

[10] Chan LM, Simchi-Levi D, Swann J. Pricing, production, and inventory policies for manufacturing with stochastic demand and discretionary sales. Manufacturing & Service Operations Management, 2006, 82149168.

[11] Chang CC, Lu CJ, Yang CT. Multistage supply chain production-inventory model with collaborative preservation technology investment. Scientia Iranica, 2022, 29420992114

[12] Chang CT, Cheng MC, Ouyang LY. Deteriorating inventory model with advance-cash-credit payment schemes and partial backlogging. Soft Computing, 2025, 29: 2279-2295.

[13] Chaudhari RH, Gor AS, Narsingani FJ. Statistical model for inventory optimization using genetic approach. AIP Conference Proceedings, 2023, 27281050007

[14] Chen SC, Cárdenas-Barrón LE, Teng JT. Retailer’s economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity. International Journal of Production Economics, 2014, 155: 284-291.

[15] Chen YR, Dye C. Application of particle swarm optimisation for solving deteriorating inventory model with fluctuating demand and controllable deterioration rate. International Journal of Production Research, 2013, 51619101921

[16] Covert RP, Philip GC. An EOQ model for items with Weibull distribution deterioration. AIIE Transactions, 2007, 54323326.

[17] Daş GS, Yeşilkaya M, Birgören B. A two-stage stochastic model for an industrial symbiosis network under uncertain demand. Applied Mathematical Modelling, 2024, 125: 444-462.

[18] Das SC, Ali H, Khan MAA, Shaikh AA, Alrasheedi AF. Inventory model for green products with payment strategy, selling price and green level dependent demand using teaching learning based optimization algorithm. Scientific Reports, 2024, 1413033.

[19] Desport P, Lardeux F, Lesaint D, Cairano-Gilfedder CD, Liret A, Owusu G. A combinatorial optimisation approach for closed-loop supply chain inventory planning with deterministic demand. European Journal of Industrial Engineering, 2017, 113303327.

[20] Duary A, Das S, Arif MG, Abualnaja KM, Khan MAA, Zakarya M, et al. Advance and delay in payments with the price-discount inventory model for deteriorating items under capacity constraints and partially backlogged shortages. Alexandria Engineering Journal, 2022, 61217351745.

[21] Dye CY, Hsieh TP. An optimal replenishment policy for deteriorating items with effective investment in preservation technology. European Journal of Operational Research, 2012, 2181106112.

[22] Giri BC, Mondal C, Maiti T. Analysing a closedloop supply chain with selling price, warranty period and green sensitive consumer demand under revenue sharing contract. Journal of Cleaner Production, 2018, 190: 822-837.

[23] Gong H, Yang X. A two-stage stochastic programming for the integrated emergency mobility facility allocation and road network design under uncertainty. Networks and Spatial Economics, 2025, 252445486.

[24] Gong M, Lian Z, Xiao H. Inventory control policy for perishable products under a buyback contract and Brownian demands. International Journal of Production Economics, 2022, 251: 108522.

[25] Goyal SK. Economic order quantity under the condition of permissible delay in payments. Journal of the Operational Research Society, 1985, 36: 335-338.

[26] Goyal SK, Giri BC. Recent trends in modeling of deteriorating inventory. European Journal of Operational Research, 2001, 1341116.

[27] Hsu P, Wee H, Teng H. Preservation technology investment for deteriorating inventory. International Journal of Production Economics, 2010, 1242388394.

[28] Ibe CB, Ogbeide DO. An inventory model for deteriorating items with exponential increasing demand and time-varying holding cost under partial backlogging. Centrepoint Journal (Science Edition), 2016, 2226975

[29] Jamal AMM, Sarker BR, Wang S. Optimal payment time for a retailer under permitted delay of payment by the wholesaler. International Journal of Production Economics, 2000, 6615966.

[30] Jamali MB, Rasti-Barzoki M. A game theoretic approach for green and non-green product pricing in chain-to-chain competitive sustainable and regular dualchannel supply chains. Journal of Cleaner Production, 2018, 170: 1029-1043.

[31] Jani MY, Patel HA, Bhadoriya A, Chaudhari U, Abbas M, Alqahtani MS. Deterioration control decision support system for the retailer during availability of trade credit and shortages. Mathematics, 2023, 113580.

[32] Janssen L, Claus T, Sauer J. Literature review of deteriorating inventory models by key topics from 2012 to 2015. International Journal of Production Economics, 2016, 182: 86-112.

[33] Jin W, Luo J, Zhang Q. Optimal ordering and financing decisions under advance payment and carbon tax in a supply chain with carbon tax supply chain. Journal of the Operational Research Society, 2018, 691219761993.

[34] Kanekiyo H, Agata S. Optimal control in an inventory management problem considering replenishment lead time based upon a non-diffusive stochastic differential equation. Journal of Advanced Mechanical Design, Systems, and Manufacturing, 2019, 131112.

[35] Kar S, Bhunia AK, Maiti M. Deterministic inventory model with two levels of storage, a linear trend in demand and a fixed time horizon. Computers & Operations Research, 2001, 281313151331.

[36] Khan MAA, Cardenas-Barron LE, Trevino-Garza G, Cespedes-Mota A. Integrating prepayment installment, pricing and replenishment decisions for growing items with power demand pattern and non-linear holding cost under carbon regulations. Computers & Operations Research, 2024, 156: 106225.

[37] Khan MAA, Cardenas-Barron LE, Trevino-Garza G, Cespedes-Mota A, Loera-Hernandez IDJ. Strategizing emissions reduction investment for a livestock production farmamid power demand pattern: A path to sustainable growth under the carbon cap environmental regulation. Journal of Cleaner Production, 2024, 441: 140642.

[38] Khan MAA, Halim MA, Alarjani A, Shaikh AA, Uddin MS. 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. Alexandria Engineering Journal, 2022, 611184698486.

[39] Khan MAA, Shaikh AA, Panda GC, Konstantaras I. Two-warehouse inventory model for deteriorating items with partial backlogging and advance payment scheme. RAIRO - Operations Research, 2019, 53516911708.

[40] Khedlekar UK, Kumar L. Mathematical modelling for convertible items with rework using particle swarm optimization. International Journal of Systems Science: Operations & Logistics, 2024, 1112306222

[41] Khedlekar UK, Kumar L, Keswani M. A stochastic inventory model with price-sensitive demand, restricted shortage and promotional efforts. Yugoslav Journal of Operations Research, 2023, 334613642.

[42] Li B, Zhu M, Jiang Y, Li Z. Pricing policies of a competitive dual-channel green supply chain. Journal of Cleaner Production, 2016, 112: 2029-2042.

[43] Li R, Lan H, Mawhinney JR. A review on deteriorating inventory study. Journal of Service Science and Management, 2010, 301117.

[44] Lin TY, Sarker BR. A pull system inventory model with carbon tax policies and imperfect quality items. Applied Mathematical Modelling, 2017, 50: 450-462.

[45] Liu Q, Van Ryzin GJ. Strategic capacity rationing to induce early purchases. Management Science, 2008, 54611151131.

[46] Maiti AK, Maiti MK, Maiti M. Inventory model with stochastic lead-time and price dependent demand incorporating advance payment. Applied Mathematical Modelling, 2009, 33524332443.

[47] Manna AK, Das B, Tiwari S. Impact of carbon emission on imperfect production inventory system with advance payment base free transportation. RAIRO - Operations Research, 2020, 54: 1103-1117.

[48] Mishra U, Wu JZ, Sarkar B. Optimum sustainable inventory management with backorder and deterioration under controllable carbon emissions. Journal of Cleaner Production, 2021, 279: 123699.

[49] Mishra U, Wu JZ, Tseng ML. Effects of a hybridprice- stock dependent demand on the optimal solutions of a deteriorating inventory system and trade credit policy on re-manufactured product. Journal of Cleaner Production, 2019, 241: 118282.

[50] Momena AF, Haque R, Rahaman M, Mondal SP. A two-storage inventory model with trade credit policy and time-varying holding cost under quantity discounts. Logistics, 2023, 7477.

[51] Nath BK, Bhattacharjee N, Sen N. A three-warehouse model for deteriorating items to study the effective preservation cost and green investment under partial substitution. International Journal of Applied and Computational Mathematics, 2024, 104129.

[52] Neri M, Guelpa E, Verda V. Two-stage stochastic programming for the design optimization of district cooling networks under demand and cost uncertainty. Applied Thermal Engineering, 2024, 236: 121594.

[53] Oksendal B. Stochastic Differential Equations: An Introduction with Applications, 20156ed

[54] Oprocha P, Czyżewska N, Klimczak K, Kusiak J, Morkisz P, Pietrzyk M, Szeliga D. A comparative study of deterministic and stochastic models of microstructure evolution during multi-step hot deformation of steels. Materials, 2023, 1693316.

[55] Ouyang LY, Chang CT. Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging. International Journal of Production Economics, 2013, 1442610617.

[56] Panja S, Mondal SK. Exploring a two-layer green supply chain game theoretic model with credit linked demand and mark-up under revenue sharing contract. Journal of Cleaner Production, 2020, 250: 119491.

[57] Paul A, Pervin M, Pinto RV, Roy SK, Maculan N, Weber GW. Effects of multiple prepayments and green investment on an EPQ model. Journal of Industrial and Management Optimization, 2023, 19966886704.

[58] Paul A, Pervin M, Roy SK, Maculan N, Weber GW. A green inventory model with the effect of carbon taxation. Annals of Operations Research, 2022, 3091233248.

[59] Purohit SK, Panigrahi S. Novel deterministic and probabilistic forecasting methods for crude oil price employing optimized deep learning, statistical and hybrid models. Information Sciences, 2024, 658: 120021.

[60] Raafat F. Survey of literature on continuously deteriorating inventory models. Journal of the Operational Research Society, 1991, 42: 27-37.

[61] Rahman MS, Khan MAA, Halim MA, Nofal TA, Shaikh AA, Mahmoud EE. Hybrid price and stock dependent inventory model for perishable goods with advance payment related discount facilities under preservation technology. Alexandria Engineering Journal, 2021, 60334553465.

[62] Rahman MS, Manna AK, Shaikh AA, Bhunia AK. An application of interval differential equation on a production inventory model with interval-valued demand via center-radius optimization technique and particle swarm optimization. International Journal of Intelligent Systems, 2020, 35812801326.

[63] Saha S, Nielsen I, Moon I. Optimal retailer investments in green operations and preservation technology for deteriorating items. Journal of Cleaner Production, 2017, 140: 1514-1527.

[64] Sana SS, Chaudhuri KS. A deterministic EOQ model with delays in payments and price-discount offers. European Journal of Operational Research, 2008, 1842509533.

[65] Sebatjane M. A sustainable inventory model for a two-echelon cold chain with green technology investments and stock-dependent demand under carbon emissions tax regulation. Cleaner Logistics and Supply Chain, 2024, 13: 100173.

[66] Sebatjane M. Three-echelon circular economic production-inventory model for deteriorating items with imperfect quality and carbon emissions considerations under various emissions policies. Expert Systems with Applications, 2024, 252: 124162.

[67] Sepehri A, Mishra U, Sarkar B. A sustainable production-inventory model with imperfect quality under preservation technology and quality improvement investment. Journal of Cleaner Production, 2021, 310: 127332.

[68] Shah NH, Shah Y. Literature survey on inventory models for deteriorating items. SID, 2000, 44: 221-237

[69] Shen Y, Shen K, Yang C. A production inventory model for deteriorating items with collaborative preservation technology investment under carbon tax. Sustainability, 2019, 11: 5027.

[70] Shugan SM, Xie J. Advance pricing of services and other implications of separating purchase and consumption. Journal of Service Research, 2000, 23227239.

[71] Su X, Zhang F. Strategic customer behavior, commitment, and supply chain performance. Management Science, 2008, 541017591773.

[72] Tadikamalla PR. An EOQ inventory model for items with gamma distributed deterioration. AIIE Transactions, 1978, 101100103.

[73] Taleizadeh AA, Niaki STA, Nikousokhan R. Constraint multiproduct joint replenishment problem under uncertain programming. Applied Soft Computing, 2011, 11: 5143-5154.

[74] Tang CS, Rajaram K, Alptekinoğlu A, Ou J. The benefits of advance booking discount programs: Model and analysis. Management Science, 2004, 504465478.

[75] Tarim SA, Kingsman BG. Modelling and computing (R, S) policies for inventory systems with non-stationary stochastic demand. European Journal of Operational Research, 2006, 1741581599.

[76] Teng JT, Cárdenas-Barrón LE, Chang HJ, Wu J, Hu Y. Inventory lot-size policies for deteriorating items with expiration dates and advance payments. Applied Mathematical Modelling, 2016, 40: 8605-8616.

[77] Tsao YC. Designing a supply chain network for deteriorating inventory under preservation effort and trade credits. International Journal of Production Research, 2016, 541338373851.

[78] Tsoularis A. Deterministic and stochastic optimal inventory control with logistic stock-dependent demand rate. International Journal of Mathematics in Operational Research, 2014, 614169.

[79] Tsoularis A. A stochastic differential equation inventory model. International Journal of Applied and Computational Mathematics, 2019, 58112

[80] Wei MM, Zhang F. Advance selling to strategic consumers: Preorder contingent production strategy with advance selling target. Production and Operations Management, 2018, 27712211235.

[81] Wu J, Teng JT, Chan YL. Inventory policies for perishable products with expiration dates and advance-cash-credit payment schemes. International Journal of Systems Science: Operations & Logistics, 2017, 54310326

[82] Xie J, Gerstner E. Service escape: Profiting from customer cancellations. Marketing Science, 2007, 2611830.

[83] Xie J, Shugan SM. Electronic tickets, smart cards, and online prepayments: When and how to advance sell. Marketing Science, 2001, 203219243.

[84] Xu G, Wu H, Liu Y, Wu CH, Tsai S 2020. A research on fresh-keeping strategies for fresh agricultural products from the perspective of green transportation. Discrete Dynamics in Nature and Society, 2020112

[85] Yang Y, Chi H, Zhou W, Fan T, Piramuthu S. Deterioration control decision support for perishable inventory management. Decision Support Systems, 2020, 134: 113308.

[86] Yu C, Qu Z, Archibald TW, Luan Z. An inventory model of a deteriorating product considering carbon emissions. Computers & Industrial Engineering, 2020, 148: 106694.

[87] Yuna F, Erkayman B, Yılmaz M. Inventory control model for intermittent demand: A comparison of metaheuristics. Soft Computing, 2023, 271064876505.

[88] Zeng C. Optimal advance selling strategy under price commitment. Pacific Economic Review, 2013, 182233258.

[89] Zhao X, Stecke KE. Pre-orders for new to-be-released products considering consumer loss aversion. Production and Operations Management, 2010, 192198215.

[90] Zhou YW, Zhong YG, Wahab MIM. How to make the replenishment and payment strategy under flexible two-part trade credit. Computers & Operations Research, 2013, 40: 1328-1338.

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