Optimal Control for Inventory System Under Uncertainty on Demand and Delivery Using Robust Linear Quadratic Control Approach

Document Type: Research Paper


Department of Mathematics, Faculty of Science and Mathematics, Universitas Diponegoro, Semarang, Indonesia



The supply chain management comprises many uncertain parameters such as the demand value and delivered product rate as the result of an imperfect delivery process. In this article, therefore, a mathematical model in a linear dynamical state-space equation is formulated for an inventory system with uncertain demand value and imperfect delivery process developed from the existing classical model. The new model is used to determine the optimal decision for this inventory system i.e. to calculate the optimal amount of product that should be ordered from the supplier. Moreover, the optimal decision is calculated for the purpose to control the inventory level as the decision-maker wanted to, in this paper, the inventory level is brought to a set point. The robust linear quadratic control, which is an existing model, is employed to this system with a numerical experiment performed to illustrate the controlling responses. From the obtained results, it achieved the optimal decision with the proper control of the inventory level based on the performed set-point control problem. In addition, the performed computational experiment is compared to some related existing works. The analysis showed that the achieved optimal decision is well enough and is not worse than the other results. In conclusion, the proposed model and the method performed in this research are implementable and therefore can be used by practitioners especially in the supply chain management field.


Abdel-Aal, M. A. M., & Selim, S. Z. (2019). Robust optimization for selective newsvendor problem with uncertain demand. Computers and Industrial Engineering, Vol. 135(June), pp. 838–854. https://doi.org/10.1016/j.cie.2019.06.047

Abdellatif, H., & Graham, S. (2019). Green Supply Chain Management Practices in Developing Countries – A Case Study from Jordan. International Journal of Simulation Systems, Science & Technology, Vol. 20(S1), pp. 10.1-10.6. https://doi.org/10.5013/IJSSST.a.20.S1.10

Abdul-Hussin, M. H. (2019). Simulation and Control of Siphon Petri Nets for Manufacturing Systems. International Journal of Simulation Systems, Science & Technology, Vol. 20(4), 1–7. https://doi.org/10.5013/IJSSST.a.20.04.12

Araya-Sassi, C., Paredes-Belmar, G., & Gutiérrez-Jarpa, G. (2020). Multi-commodity inventory-location problem with two different review inventory control policies and modular stochastic capacity constraints. Computers and Industrial Engineering, Vol. 143(June 2019), 106410. https://doi.org/10.1016/j.cie.2020.106410

Bagshaw, K. B. (2015). Lead time uncertainties, average inventory and scheduling practice on manufacturing firms in Nigeria. International Review of Management and Business Research, Vol. 4(4), pp. 28–45.

Barman, H., Pervin, M., Roy, S. K., & Weber, G.-W. (n.d.). Back-ordered inventory model with inflation in a cloudy-fuzzy environment. Journal of Industrial & Management Optimization, Vol. 13. https://doi.org/10.3934/jimo.2020052

Ben-Tal, A., Golany, B., & Shtern, S. (2009). Robust multi-echelon multi-period inventory control. European Journal of Operational Research, Vol. 199(3), pp. 922–935. https://doi.org/10.1016/J.EJOR.2009.01.058

Bhatia, N., Gülpınar, N., & Aydın, N. (2020). Dynamic production-pricing strategies for multi-generation products under uncertainty. International Journal of Production Economics, Vol. 230(June), 107851. https://doi.org/10.1016/j.ijpe.2020.107851

Chawla, I., Chopra, V., & Singla, A. (2019). Robust LQR Based ANFIS Control of x-z Inverted Pendulum. Proceedings - 2019 Amity International Conference on Artificial Intelligence, AICAI 2019, pp. 818–823. https://doi.org/10.1109/AICAI.2019.8701333

Darmawan, A., Wong, H., & Thorstenson, A. (2021a). Supply chain network design with coordinated inventory control. Transportation Research Part E: Logistics and Transportation Review, Vol. 145(2), 102168. https://doi.org/10.1016/j.tre.2020.102168

Darmawan, A., Wong, H., & Thorstenson, A. (2021b). Supply chain network design with coordinated inventory control. Transportation Research Part E: Logistics and Transportation Review, Vol. 145, 102168. https://doi.org/10.1016/J.TRE.2020.102168

Dey, B. K., Pareek, S., Tayyab, M., & Sarkar, B. (2021). Autonomation policy to control work-in-process inventory in a smart production system. International Journal of Production Research, Vol. 59(4), pp. 1258–1280. https://doi.org/10.1080/00207543.2020.1722325

Dey, B. K., Sarkar, B., Sarkar, M., & Pareek, S. (2019). An integrated inventory model involving discrete setup cost reduction, variable safety factor, selling price dependent demand, and investment. RAIRO - Operations Research, Vol. 53(1), pp. 39–57. https://doi.org/10.1051/ro/2018009

Dezhi, T., & Xiaojun, T. (2017). Design of UAV attitude controller based on improved robust LQR control. Proceedings - 2017 32nd Youth Academic Annual Conference of Chinese Association of Automation, YAC 2017, pp. 004–1009. https://doi.org/10.1109/YAC.2017.7967557

Doliya, D., & Bhandari, M. (2018). An LMI approach for robust LQR control of PWM buck converter with parasitics. Proceedings - 7th International Conference on Communication Systems and Network Technologies, CSNT 2017, pp. 103–108. https://doi.org/10.1109/CSNT.2017.8418519

Escárate, P., Agüero, J. C., Zúñiga, S., Castro, M., & Garcés, J. (2017). Linear quadratic regulator for laser beam shaping. Optics and Lasers in Engineering, Vol. 94(2017), pp. 90–96. https://doi.org/10.1016/j.optlaseng.2017.02.009

Hassibi, B., Sayed, A. H., & Kailath, T. (1999). Indefinite-Quadratic Estimation and Control--A Unified Approach to H^2 and H^inf Theories. PA:SIAM.

Huang, J., Ma, J., Qian, F., Chen, C., & Song, X. (2017). Research on linear quadratic robust control algorithm for non-stochastic uncertain system. Proceedings of the 29th Chinese Control and Decision Conference, CCDC 2017, pp. 52–56. https://doi.org/10.1109/CCDC.2017.7978065

Humaidi, A. J., Yousif, K. Y., Hameed, A. H., & Ibraheem, I. K. (2019). Optimal Robust Controller Design for Uncertain Linear Glucose System. 2019 6th International Conference on Control, Decision and Information Technologies (CoDIT), pp. 663–668. https://doi.org/10.1109/codit.2019.8820570

Ignaciuk, P. (2014). Robust Inventory Management under Uncertain Demand and Unreliable Delivery Channels. Advances in Intelligent Systems and Computing, Vol. 240, pp. 67–76. https://doi.org/10.1007/978-3-319-01857-7_7

Ignaciuk, P., & Bartoszewicz, A. (2010a). Automatica Linear – quadratic optimal control strategy for periodic-review inventory. Automatica, Vol. 46(12), pp. 1982–1993. https://doi.org/10.1016/j.automatica.2010.09.010

Ignaciuk, P., & Bartoszewicz, A. (2010b). Linearquadratic optimal control strategy for periodic-review inventory systems. Automatica, Vol. 46(12), pp. 1982–1993. https://doi.org/10.1016/j.automatica.2010.09.010

Ignaciuk, P., & Bartoszewicz, A. (2012). Linear-Quadratic Optimal Control of Periodic-Review Perishable Inventory Systems. IEEE Transactions on Control Systems Technology, Vol. 20(5), 1400–1407. https://doi.org/10.1109/TCST.2011.2161086

Ignaciuk, P., & Bartoszewicz, A. (2011). Sliding-mode inventory control in systems with fixed order quantity and uncertain demand. Proceedings of the 2011 12th International Carpathian Control Conference, ICCC’2011, Vol. 3, pp. 152–155. https://doi.org/10.1109/CarpathianCC.2011.5945837

Kosorukov, O. A., Maslov, S. E., & Sviridova, O. A. (2020). Accounting for the uncertainty of delivery terms in inventory management models. Euroasian Journal o BioCiences, Vol. 14, pp. 6323–6328.

Lee, H. I., Yoo, D. W., Lee, B. Y., Moon, G. H., Lee, D. Y., Tahk, M. J., & Shin, H. S. (2017). Parameter-robust linear quadratic Gaussian technique for multi-agent slung load transportation. Aerospace Science and Technology, Vol. 71, pp.  119–127. https://doi.org/10.1016/j.ast.2017.09.014

Li, B., Wang, H. W., Yang, J. B., Guo, M., & Qi, C. (2011). A belief-rule-based inventory control method under nonstationary and uncertain demand. Expert Systems with Applications, Vol. 38(12), pp. 14997–15008. https://doi.org/10.1016/J.ESWA.2011.05.047

Li, Q. K., Lin, H., Tan, X., & Du, S. (2020). H∞Consensus for Multiagent-Based Supply Chain Systems under Switching Topology and Uncertain Demands. IEEE Transactions on Systems, Man, and Cybernetics: Systems, Vol. 50(12), pp. 4905–4918. https://doi.org/10.1109/TSMC.2018.2884510

Li, Q., Li, Y., & Lin, H. (2018). H∞ Control of Two-Time-Scale Markovian Switching Production-Inventory Systems. IEEE Transactions on Control Systems Technology, Vol. 26(3), pp. 1065–1073. https://doi.org/https://doi.org/10.1109/TCST.2017.2692748

Li, Y., & Liu, F. (2021). Joint inventory and pricing control with lagged price responses. International Journal of Production Economics, Vol. 241, 108253. https://doi.org/10.1016/J.IJPE.2021.108253

Limansyah, T., & Lesmono, J. D. (2020). A Mathematical Model for Inventory and Price-Dependent Demand with All-Units Discount. Journal of Physics: Conference Series, Vol. 1490(1). https://doi.org/10.1088/1742-6596/1490/1/012051

Lun, Y. Z., D’Innocenzo, A., & Benedetto, M. D. Di. (2017). Robust LQR for time-inhomogeneous Markov jump switched linear systems. IFAC-PapersOnLine, Vol. 50(1), pp. 2199–2204. https://doi.org/10.1016/j.ifacol.2017.08.281

Luo, X., Liu, Z., & Wu, J. (2020). Dynamic pricing and optimal control for a stochastic inventory system with non-instantaneous deteriorating items and partial backlogging. Mathematics, Vol. 8(6), pp. 1–22. https://doi.org/10.3390/MATH8060906

Luthfi, M. F., Sutrisno, & Widowati. (2018). Stock Control of Single Product Inventory System with Imperfect Delivery by Using Robust Linear Quadratic Regulator. 2018 4th International Conference on Science and Technology (ICST), 1–4. https://doi.org/10.1109/ICSTC.2018.8528682

Malladi, S. S., Erera, A. L., & White, C. C. (2020). A dynamic mobile production capacity and inventory control problem. IISE Transactions, Vol. 52(8), pp. 926–943. https://doi.org/10.1080/24725854.2019.1693709

Mishra, U., Wu, J.-Z., & Sarkar, B. (2021). Optimum sustainable inventory management with backorder and deterioration under controllable carbon emissions. Journal of Cleaner Production, Vol. 279, 123699. https://doi.org/https://doi.org/10.1016/j.jclepro.2020.123699

Nahmias, S., & Pierskalla, W. P. (2006). A Two-Product Perishable/Nonperishable Inventory Problem. Http://Dx.Doi.Org/10.1137/0130045, Vol. 30(3), pp. 483–500. https://doi.org/10.1137/0130045

Niel, F., Seuret, A., Zaccarian, L., & Fagley, C. (2017). Robust LQR control for stall flutter suppression: A polytopic approach. IFAC-PapersOnLine, Vol. 50(1), pp. 11367–11372. https://doi.org/10.1016/j.ifacol.2017.08.2041

Nobil, A. H., Sedigh, A. H. A., & Cárdenas-Barrón, L. E. (2020). Reorder point for the EOQ inventory model with imperfect quality items. Ain Shams Engineering Journal, Vol. 11(4), pp. 1339–1343. https://doi.org/10.1016/j.asej.2020.03.004

Patriarca, R., Di Gravio, G., Costantino, F., & Tronci, M. (2020). EOQ inventory model for perishable products under uncertainty. Production Engineering, Vol. 14(5–6), pp. 601–612. https://doi.org/10.1007/s11740-020-00986-5

Pervin, M., Roy, S. K., & Weber, G. (2020). An integrated vendor-buyer model with quadratic demand under inspection policy and preservation technology. Hacettepe Journal of Mathematics and Statistics, Vol. 49, pp. 1168–1189. https://doi.org/10.15672/hujms.476056

Prak, D., Teunter, R., Babai, M. Z., Boylan, J. E., & Syntetos, A. (2021). Robust compound Poisson parameter estimation for inventory control. Omega, Vol. 104, 102481. https://doi.org/10.1016/J.OMEGA.2021.102481

Rahdar, M., Wang, L., & Hu, G. (2018a). A tri-level optimization model for inventory control with uncertain demand and lead time. International Journal of Production Economics, Vol. 195, pp. 96–105. https://doi.org/10.1016/j.ijpe.2017.10.011

Rahdar, M., Wang, L., & Hu, G. (2018b). A tri-level optimization model for inventory control with uncertain demand and lead time. International Journal of Production Economics, Vol. 195, pp. 96–105. https://doi.org/10.1016/J.IJPE.2017.10.011

Roy, S. K., Pervin, M., & Weber, G. W. (2020). A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy. Journal of Industrial & Management Optimization, Vol. 16(2), pp. 553–578. https://doi.org/10.3934/jimo.2018167

Roy, S. K., Pervin, M., Weber, G.-W. W., Barman, H., Pervin, M., Roy, S. K., & Weber, G.-W. W. (2020). DETERIORATING INVENTORY WITH PRESERVATION TECHNOLOGY UNDER PRICE- AND STOCK-SENSITIVE DEMAND. Journal of Industrial and Management Optimization, Vol. 16(4), pp. 1585–1612. https://doi.org/10.3934/jimo.2019019

Saputra, A., Widowati, & Sutrisno. (2017). Optimal strategy analysis based on robust predictive control for inventory system with random demand. AIP Conference Proceedings, Vol. 1913(020017), pp. 1–5. https://doi.org/https://doi.org/10.1063/1.5016651

Sarkar, B., Omair, M., & Kim, N. (2020). A cooperative advertising collaboration policy in supply chain management under uncertain conditions. Applied Soft Computing, Vol. 88, 105948. https://doi.org/https://doi.org/10.1016/j.asoc.2019.105948

Sett, B. K., Dey, B. K., & Sarkar, B. (2020). The Effect of O2O Retail Service Quality in Supply Chain Management. In Mathematics (Vol. 8, Issue 10). https://doi.org/10.3390/math8101743

Sini, S., Vivek, A., & Nandagopal, J. L. (2017). Trajectory tracking of 3-DOF lab helicopter by robust LQR. Proceedings of IEEE International Conference on Circuit, Power and Computing Technologies, ICCPCT 2017. https://doi.org/10.1109/ICCPCT.2017.8074217

Sun, Y., Lyu, J., Fang, J., Fu, Z., & Dong, D. (2019). Robust LQR Anti-Swing Control for Quay-Side Crane System with Variable Load. 8th Annual IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems, CYBER 2018, 796–801. https://doi.org/10.1109/CYBER.2018.8688150

Sutrisno, Widowati, & Heru Tjahjana, R. (2017). Expected value analysis for integrated supplier selection and inventory control of multi-product inventory system with fuzzy cost. AIP Conference Proceedings, Vol. 1913(020038), pp. 1–7. https://doi.org/10.1063/1.5016672

Sutrisno, Widowati, & Heru Tjahjana, R. (2019). Robust predictive control application and simulation of inventory controlling with imperfect delivery process. 2019 International Conference on Information and Communications Technology, ICOIACT 2019, pp. 639–642. https://doi.org/10.1109/ICOIACT46704.2019.8938536

Sutrisno, Widowati, & Munawwaroh, D. A. (2016). Hybrid mathematical model of inventory system with piecewise holding cost and its optimal strategy. ICAMIMIA 2015 - International Conference on Advanced Mechatronics, Intelligent Manufacture, and Industrial Automation, Proceeding - In Conjunction with Industrial Mechatronics and Automation Exhibition, IMAE, pp. 29–33. https://doi.org/10.1109/ICAMIMIA.2015.7507996

Sutrisno, Widowati, & Tjahiana, R. H. (2019). Application of Robust Linear Quadratic Control for Inventory System with Unknown Demand: Single Product Case. International Conference on Informatics and Computational Sciences (ICICoS), pp. 174–178. https://doi.org/10.1109/ICICOS.2018.8621666

Sutrisno, Widowati, & Tjahjana, R. H. (2018). Single Product Inventory Control Considering Unknown Demand Using Linear Quadratic Gaussian. 2018 IEEE International Conference on Robotics, Biomimetics, and Intelligent Computational Systems (Robionetics ), pp. 17–20. https://doi.org/10.1109/ROBIONETICS.2018.8674677

Taparia, R., Janardhanan, S., & Gupta, R. (2019). Inventory control for nonperishable and perishable goods based on model predictive control. Https://Doi.Org/10.1080/23302674.2019.1600766, Vol. 7(4), pp. 361–373. https://doi.org/10.1080/23302674.2019.1600766

Terra, M. H., Cerri, J. P., & Ishihara, J. Y. (2014). Optimal Robust Linear Quadratic Regulator for Systems Subject to Uncertainties. IEEE Transactions on Automatic Control, Vol. 59(9), pp. 2586–2591. https://doi.org/10.1109/TAC.2014.2309282

Transchel, S., & Hansen, O. (2019). Supply planning and inventory control of perishable products under lead-time uncertainty and service level constraints. Procedia Manufacturing, Vol. 39(2019), pp. 1666–1672. https://doi.org/10.1016/j.promfg.2020.01.274

Vinodh Kumar, E., & Jerome, J. (2013). Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum. Procedia Engineering, Vol. 64, pp. 169–178. https://doi.org/10.1016/j.proeng.2013.09.088

Wang, Z., Montanaro, U., Fallah, S., Sorniotti, A., & Lenzo, B. (2018). A gain scheduled robust linear quadratic regulator for vehicle direct yaw moment Control. Mechatronics, Vol. 51(1), pp. 31–45. https://doi.org/10.1016/j.mechatronics.2018.01.013

Xie, L., & Soh, Y. C. (1993). Control of uncertain discete-time system with guaranteed cost. Proceedings of the 32nd IEEE Conference Decision and Control, Vol.  1, pp. 56–61. https://doi.org/https://doi.org/10.1109/CDC.1993.325190

Yang, Y., Chi, H., Zhou, W., Fan, T., & Piramuthu, S. (2020). Deterioration control decision support for perishable inventory management. Decision Support Systems, Vol. 134(March), 113308. https://doi.org/10.1016/j.dss.2020.113308

Zhang, H., Zhang, J., & Zhang, R. (2018). Managing Perishable Inventory Systems as Non-Perishable Ones. SSRN Electronic Journal. https://doi.org/10.2139/SSRN.3193331

Zhang, L., & Mu, Y. (2018). Parking Space Allocation with Uncertain Demand and Supply Consideration. 2018 15th International Conference on Service Systems and Service Management, ICSSSM 2018, 71772026, pp. 1–5. https://doi.org/10.1109/ICSSSM.2018.8464961

Zhang, Y., & Wang, Z. (2019). A Joint Ordering, Pricing, and Freshness-Keeping Policy for Perishable Inventory Systems with Random Demand over Infinite Horizon. IEEE Robotics and Automation Letters, Vol. 4(3), pp. 2707–2713. https://doi.org/10.1109/LRA.2019.2916471

Zhou, H., Xu, D., Shao, X., Ning, X., & Wang, T. (2019). A robust linear-quadratic-gaussian controller for the real-time hybrid simulation on a benchmark problem. Mechanical Systems and Signal Processing, Vol. 133, 106260. https://doi.org/10.1016/j.ymssp.2019.106260