Inventory Optimization with Chance-Constrained Programming Under Demand Uncertainty

Document Type : Research Paper


Department of Industrial Engineering, Haliç University, Eyüpsultan, Istanbul, Turkey


Uncertainty and variability in demand and supply processes make it difficult for companies to make inventory management decisions. In this study, a model is developed that will provide the maximum service level of a pharmaceutical warehouse under the budget constraint, taking into account stochastic demand. Due to stochastic demand, the chance constraint programming approach is used to achieve the desired service level at different levels. In this study, the problem of a pharmaceutical warehouse that supplies medicines to pharmacies and hospitals is considered as a real-world problem. The model is designed as a dynamic programming model based on periods. Since there are thousands of drugs in the pharmaceutical warehouse, as the number of products increases, it becomes difficult to find the appropriate solution in an acceptable time. The model is first solved as a mixed integer linear programming model in Lingo. A genetic algorithm (GA) approach is then proposed for large-scale problems. The simulation optimization method also applied to the problem and compared with the optimization method and GA. The GA approach yields better results in the shortest time as the number of periods increases. The developed integrated model demonstrated a numerical example in a pharmaceutical warehouse and was solved using three different approaches. This study is of great importance in terms of providing results that will enable managers to decide the amount of items they should keep in their warehouses by using their budgets in the most efficient way. Nine different scenarios have been derived with various chance constraint risk factors and budget values. Scenario analysis has revealed that the budget has a significant impact on the results at a 95% confidence level. If a pharmaceutical warehouse increases its budget by 10%, it can reduce its total annual inventory carrying costs by 70%.


Aggarwal, R. (2018). A chance constraint based low carbon footprint supply chain configuration for an FMCG product. Management of Environmental Quality: An International Journal, 29(6), 1002–1025.
Ahmadi, E., Masel, D. T., Hostetler, S., Maihami, R., and Ghalehkhondabi, I. (2020). A centralized stochastic inventory control model for perishable products considering age-dependent purchase price and lead time. In Top (Vol. 28, Issue 1). Springer Berlin Heidelberg.
Ahmadi, E., Masel, D. T., Metcalf, A. Y., and Schuller, K. (2019). Inventory management of surgical supplies and sterile instruments in hospitals: a literature review. Health Systems, 8(2), 134–151.
Ahmadi, E., Mosadegh, H., Maihami, R., Ghalehkhondabi, I., Sun, M., and Süer, G. A. (2022). Intelligent inventory management approaches for perishable pharmaceutical products in a healthcare supply chain. Computers and Operations Research, 147(June 2021), 105968.
Ali, S. S., Barman, H., Kaur, R., Tomaskova, H., and Roy, S. K. (2021). Multi-product multi echelon measurements of perishable supply chain: Fuzzy non-linear programming approach. Mathematics, 9(17), 1–27.
Armagan Tarim, S., and Kingsman, B. G. (2004). The stochastic dynamic production/inventory lot-sizing problem with service-level constraints. International Journal of Production Economics, 88(1), 105–119.
Barman, H., Pervin, M., and Roy, S. K. (2022). Impacts of green and preservation technology investments on a sustainable EPQ model during COVID-19 pandemic. RAIRO - Operations Research, 56(4), 2245–2275.
Barman, H., Pervin, M., Roy, S. K., and Weber, G. W. (2023). Analysis of a dual-channel green supply chain game-theoretical model under carbon policy. International Journal of Systems Science: Operations and Logistics, 10(1).
Barman, H., Roy, S. K., Sakalauskas, L., and Weber, G. W. (2023). Inventory model involving reworking of faulty products with three carbon policies under neutrosophic environment. Advanced Engineering Informatics, 57(June), 102081.
Charnes, A., and Cooper, W. W. (1959). Chance-Constrained Programming. August 2015.
Chen, Z., and Rossi, R. (2021). A dynamic ordering policy for a stochastic inventory problem with cash constraints. Omega (United Kingdom), 102.
Das, S. K., Yu, V. F., Roy, S. K., and Weber, G. W. (2024). Location–allocation problem for green efficient two-stage vehicle-based logistics system: A type-2 neutrosophic multi-objective modeling approach. Expert Systems with Applications, 238(PE), 122174.
Ekren, B. Y., and Arslan, B. (2020). Simulation-based lateral transshipment policy optimization for s, S inventory control problem in a single-echelon supply chain network. International Journal of Optimization and Control: Theories and Applications, 10(1), 9–16.
Gen, M., and Cheng, R. (1997). Genetic algorithms and engineering design. John Wiley and Sons.
Ghalebsaz-Jeddi, B., Shultes, B. C., and Haji, R. (2004). A multi-product continuous review inventory system with stochastic demand, backorders, and a budget constraint. European Journal of Operational Research, 158(2), 456–469.
Goldberg, D. E., and Samtani, M. P. (1986). Engineering Optimization Via Genetic Algorithm. John Wiley and Sons, Inc.
Gómez-Rocha, J. E., Hernández-Gress, E. S., and Rivera-Gómez, H. (2021). Production planning of a furniture manufacturing company with random demand and production capacity using stochastic programming. PLoS ONE, 16(6 June), 1–26.
Guerrero Campanur, A., Olivares-Benitez, E., Miranda, P. A., Perez-Loaiza, R. E., and Ablanedo-Rosas, J. H. (2018). Design of a Logistics Nonlinear System for a Complex, Multiechelon, Supply Chain Network with Uncertain Demands. Complexity, 2018.
Gürler, Ü., and Özkaya, B. Y. (2008). Analysis of the (s, S) policy for perishables with a random shelf life. IIE Transactions (Institute of Industrial Engineers), 40(8), 759–781.
Hiassat, A., Diabat, A., and Rahwan, I. (2017). A genetic algorithm approach for location-inventory-routing problem with perishable products. Journal of Manufacturing Systems, 42, 93–103.
Holland, J. H. (1973). Genetic algorithms and the optimal allocation of trials. SIAM Journal on Computing, 2(2), 88–105.
Hooshangi-Tabrizi, P., Hashemi Doulabi, H., Contreras, I., and Bhuiyan, N. (2022). Two-stage robust optimization for perishable inventory management with order modification. Expert Systems with Applications, 193(December 2021), 116346.
Kleijnen, J. P. C., and Wan, J. (2007). Optimization of simulated systems: OptQuest and alternatives. Simulation Modelling Practice and Theory, 15(3), 354–362.
Kundu, A., and Chakrabarti, T. (2012). A multi-product continuous review inventory system in stochastic environment with budget constraint. Optimization Letters, 6(2), 299–313.
Li, W., Ding, Y., Yang, Y., Sherratt, R. S., Park, J. H., and Wang, J. (2020). Parameterized algorithms of fundamental NP-hard problems: a survey. Human-Centric Computing and Information Sciences, 10(1).
Modibbo, U. M., Gupta, S., Ahmed, A., and Ali, I. (2022). An integrated multi-objective multi-product inventory managed production planning problem under uncertain environment. Annals of Operations Research.
Movahed, K. K., and Zhang, Z. H. (2015). Robust design of (s, S) inventory policy parameters in supply chains with demand and lead time uncertainties. International Journal of Systems Science, 46(12), 2258–2268.
Nahmias, S. (2008). Production and Operation Analysis. In McGraw-Hill/Irwin Series Operation and Decision Science.
Noordhoek, M., Dullaert, W., Lai, D. S. W., and de Leeuw, S. (2018). A simulation–optimization approach for a service-constrained multi-echelon distribution network. Transportation Research Part E: Logistics and Transportation Review, 114, 292–311.
Paul, A., Pervin, M., Roy, S. K., Weber, G. W., and Mirzazadeh, A. (2021). Effect of price-sensitive demand and default risk on optimal credit period and cycle time for a deteriorating inventory model. RAIRO - Operations Research, 55, S2575–S2592.
Perera, S. C., and Sethi, S. P. (2023). A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)-type policies—Discrete-time case. Production and Operations Management, 32(1), 131–153.
Pervin, M., Roy, S. K., Sannyashi, P., and Weber, G. W. (2023). Sustainable inventory model with environmental impact for non-instantaneous deteriorating items with composite demand. RAIRO - Operations Research, 57(1), 237–261.
Qiu, R., Sun, Y., and Sun, M. (2022). A robust optimization approach for multi-product inventory management in a dual-channel warehouse under demand uncertainties. Omega (United Kingdom), 109, 102591.
Report, G. (2021). Public spending on the rise? 2021.
Rossi, R., Tarim, S. A., Hnich, B., and Prestwich, S. (2008). A global chance-constraint for stochastic inventory systems under service level constraints. Constraints, 13(4), 490–517.
Saracoglu, I. (2023). Simulation Optimization for Multi-product (s, S) Inventory Policy with Stochastic Demand. Lecture Notes in Production Engineering, Part F1164, 523–534.
Shapiro, A., and Ruszczyn, A. (2003). Chapter 5 Probabilistic Programming.pdf. 10, 1–18.
Shaw, K., Irfan, M., Shankar, R., and Yadav, S. S. (2016). Low carbon chance constrained supply chain network design problem: a Benders decomposition based approach. Computers and Industrial Engineering, 98, 483–497.
Sivazlian, B. D. (1974). A continuous-review (s, S) inventory system with arbitrary interarrival distribution between unit demand. Operations Research, 22(1), 65–71.
Taha, H. A. (2007). Operations Research: An Introduction. Pearson Education, Inc.
Veinott, A. F. (1965). Optimal Policy for a Multi-Product, Dynamic, Nonstationary Inventory Problem. Management Science, 12(3), 206–222.
Xiang, M., Rossi, R., Martin-Barragan, B., and Tarim, S. A. (2018). Computing non-stationary (s, S) policies using mixed integer linear programming. European Journal of Operational Research, 271(2), 490–500.
Xiang, M., Rossi, R., Martin-Barragan, B., and Tarim, S. A. (2023). A mathematical programming-based solution method for the nonstationary inventory problem under correlated demand. European Journal of Operational Research, 304(2), 515–524.
Xu, G., Feng, J., Chen, F., Wang, H., and Wang, Z. (2019). Simulation-based optimization of control policy on multi-echelon inventory system for fresh agricultural products. 12(2), 184–194.
Žic, S., Žic, J., and Đukić, G. (2023). Efficient planning and optimization of inventory replenishments for sustainable supply chains operating under (R, s, S) policy. Sustainable Futures, 5(June 2022).