Designing an Optimal Control Model of a Capacity, Multi-stage Continuous MRP System Considering Delay in Production Lead Time and the Possibility of Reworking and Recycling

Document Type : Research Article

Authors

• Mahdi Miri ¹
• Alireza Pooya ¹
• Morteza Pakdaman ²

¹ Department of Management, Faculty of Economics and Administrative sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

² Atmospheric Science and Meteorological Research Center (ASMERC), Climate Research Institute (CRI), Mashhad, Iran.

10.22067/jstinp.2025.94678.1164

Abstract

In this paper, we propose a multi-stage continuous-MRP system using an optimal control model, considering the production lead time. In the proposed model, the lead time is specified for ordering work in process during the second stage and for final product manufacturing. Also, the intended dynamical system is a multi-stage production-inventory system that follows a linear-quadratic optimal control model with a time delay for the state variables. In the proposed system, inventories are considered state variables, and the delivery levels and orders are control variables. The return stage is considered for items whose production has been defective. According to their situation, there are three destinations: the reworking stage, the recycling stage, or disposal. The amount of shipment to the next stage of production is based on their BOM utilization coefficient and the inventory one. This stage will consume all sent items at any time and will not create a surplus inventory. In this paper, time is considered a continuous parameter proportional to the constant production processes. For validation, the proposed optimal control model was simulated in a real study in the polymer industry.

Keywords

• Optimal Control
• Continuous Material Requirements Planning (CMRP)
• Deteriorating Items
• Lead Time
• Reworking
• Recycling

---

References

AL-Khazraji, H., Cole, C. and Guo, W., 2021. Optimization and simulation of dynamic performance of production–inventory systems with multivariable controls. Mathematics, 9(5), p.568. https://doi.org/10.3390/math9050568.

Benkherouf, L., Skouri, K. and Konstantaras, I., 2016. Optimal control of production, remanufacturing and refurbishing activities in a finite planning horizon inventory system. Journal of Optimization Theory and Applications, 168(2), pp.677-698. https://doi.org/10.1007/s10957-015-0741-9.

Dhaiban, A.K., 2022. An Integer Optimal Control Model of Production-Inventory System. Statistics, Optimization & Information Computing, 10(3), pp.762-774. https://doi.org/10.19139/soic-2310-5070-747 .

Dizbin, N.M. and Tan, B., 2020. Optimal control of production-inventory systems with correlated demand inter-arrival and processing times. International Journal of Production Economics, 228, p.107692. https://doi.org/10.1016/j.ijpe.2020.107692.

Dong, M., He, F. and Wu, Z., 2011. Optimal control of a continuum model for single-product re-entrant manufacturing systems. International journal of production research, 49(21), pp.6363-6385. https://doi.org/10.1080/00207543.2010.519736

ElHafsi, M., Fang, J. and Hamouda, E., 2021. Optimal production and inventory control of multi-class mixed backorder and lost sales demand class models. European Journal of Operational Research, 291(1), pp.147-161. https://doi.org/10.1016/j.ejor.2020.09.009.

Foul, A. and Tadj, L., 2007. Optimal control of a hybrid periodic-review production inventory system with disposal. International Journal of Operational Research, 2(4), pp.481-494. https://doi.org/10.1504/IJOR.2007.014175.

Gao, Y., 2022. Operator splitting method for the stochastic production–inventory model equation. Computers & Industrial Engineering, 174, p.108712. https://doi.org/10.1016/j.cie.2022.108712.

Grubbstrom, R.W. and Tang, O., 2012. The space of solution alternatives in the optimal lotsizing problem for general assembly systems applying MRP theory. International Journal of Production Economics, 140(2), pp.765-777.

Grubbstrom, R.W., Bogataj, M. and Bogataj, L., 2010. Optimal Lotsizing within MRP Theory. IFAC Proceedings Volumes, 42(4), pp.15-30. https://doi.org/10.3182/20090603-3-RU-2001.0562.

Hatami-Marbini, A., Sajadi, S.M. and Malekpour, H., 2020. Optimal control and simulation for production planning of network failure-prone manufacturing systems with perishable goods. Computers & Industrial Engineering, 146, p.106614. https://doi.org/10.1016/j.cie.2020.106614.

Hedjar, R., Tadj, L. and Abid, C., 2012. Optimal Control of Integrated Production–Forecasting. Decision Support Systems, p.117.

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

Ignaciuk, P., 2022. Linear-quadratic optimal control of multi-modal distribution systems with imperfect channels. International Journal of Production Research, 60(18), pp.5523-5538. https://doi.org/10.1080/00207543.2021.1963876.

Le Thi, H.A. and Tran, D.Q., 2014. Optimizing a multi-stage production/inventory system by DC programming based approaches. Computational Optimization and Applications, 57(2), pp.441-468. https://doi.org/10.1007/s10589-013-9600-5.

Lee, C., Liu, X., Liu, Y. and Zhang, L., 2021. Optimal control of a time-varying double-ended production queueing model. Stochastic Systems, 11(2), pp.140-173. https://doi.org/10.1287/stsy.2019.0066.

Louly, M.A., Dolgui, A. and Al-Ahmari, A., 2012. Optimal MRP offsetting for assembly systems with stochastic lead times: POQ policy and service level constraint. Journal of Intelligent Manufacturing, 23(6), pp.2485-2495. https://doi.org/10.1007/s10845-011-0515-7.

Luo, Y., Sima, Q., Ji, T., Sun, F., Liu, H. and Zhang, J., 2024, May. Smooth computation without input delay: robust tube-based model predictive control for robot manipulator planning. In 2024 IEEE International Conference on Robotics and Automation (ICRA) (pp. 10429-10435). IEEE. https://doi.org/10.1109/ICRA57147.2024.10610952.

Megoze Pongha, P., Kibouka, G.R., Kenné, J.P. and Hof, L.A., 2022. Production, maintenance and quality inspection planning of a hybrid manufacturing/remanufacturing system under production rate-dependent deterioration. The International Journal of Advanced Manufacturing Technology, 121(1), pp.1289-1314. https://doi.org/10.1007/s00170-022-09078-3.

Mezghiche, A., Moulaï, M. and Tadj, L., 2015. Model predictive control of a forecasting production system with deteriorating items. International Journal of Operations Research and Information Systems (IJORIS), 6(4), pp.19-37. https://doi.org/10.4018/IJORIS.2015100102.

Milne, R.J., Mahapatra, S. and Wang, C.T., 2015. Optimizing planned lead times for enhancing performance of MRP systems. International Journal of Production Economics, 167, pp.220-231. https://doi.org/10.1016/j.ijpe.2015.05.013.

Mishra, U., 2016. An EOQ model with time dependent Weibull deterioration, quadratic demand and partial backlogging. International Journal of Applied and Computational Mathematics, 2(4), pp.545-563. https://doi.org/10.1007/s40819-015-0077-z.

Nakhaeinejad, M., Khademi Zare, H., Habibi, M. and Khodoomi, M.R., 2023. Improvement of multi-item order systems and inventory management models using optimal control theory. Transactions of the Institute of Measurement and Control, 45(1), pp.104-119. https://doi.org/10.1177/01423312221109724.

Öztürk, H., 2021. Optimal production run time for an imperfect production inventory system with rework, random breakdowns and inspection costs. Operational Research, 21(1), pp.167-204. https://doi.org/10.1007/s12351-018-0439-5.

Pooya, A. and Pakdaman, M., 2017. Analysing the solution of production-inventory optimal control systems by neural networks. RAIRO-Operations Research-Recherche Opérationnelle, 51(3), pp.577-590.

Pooya, A. and Pakdaman, M., 2018. A delayed optimal control model for multi-stage production-inventory system with production lead times. The International Journal of Advanced Manufacturing Technology, 94(1), pp.751-761. https://doi.org/10.1007/s00170-017-0942-5.

Pooya, A. and Pakdaman, M., 2019. Optimal control model for finite capacity continuous MRP with deteriorating items. Journal of Intelligent Manufacturing, 30(5), pp.2203-2215. https://doi.org/10.1007/s10845-017-1383-6.

Pooya, A. and Pakdaman, M., 2021. A new continuous time optimal control model for manpower planning with promotion from inside the system. Operational Research, 21(1), pp.349-364. https://doi.org/10.1007/s12351-018-0431-0.

Pooya, A., Pakdaman, M. and Tadj, L., 2019. Exact and approximate solution for optimal inventory control of two-stock with reworking and forecasting of demand. Operational Research, 19(2), pp.333-346. https://doi.org/10.1007/s12351-017-0297-6.

Rachih, H., Mhada, F. and Chiheb, R., 2022. Simulation optimization of an inventory control model for a reverse logistics system. Decision Science Letters, 11(1), pp.43-54.

Rossi, T., Pozzi, R., Pero, M. and Cigolini, R., 2017. Improving production planning through finite-capacity MRP. International journal of production research, 55(2), pp.377-391. https://doi.org/10.1080/00207543.2016.1177235.

Sadeghian, R., 2011. Continuous materials requirements planning (CMRP) approach when order type is lot for lot and safety stock is zero and its applications. Applied soft computing, 11(8), pp.5621-5629. https://doi.org/10.1016/j.asoc.2011.04.002.

Shen, J., Jin, Y., Liu, B., Lu, Z. and Chen, X., 2022. An uncertain production-inventory problem with deteriorating items. Advances in Continuous and Discrete Models, 2022(1), p.42. https://doi.org/10.1186/s13662-022-03714-8.

Singer, G. and Khmelnitsky, E., 2021. A production-inventory problem with price-sensitive demand. Applied Mathematical Modelling, 89, pp.688-699. https://doi.org/10.1016/j.apm.2020.06.072.

Vercraene, S. and Gayon, J.P., 2013. Optimal control of a production-inventory system with productreturns. International Journal of Production Economics, 142(2), pp.302-310. https://doi.org/10.1016/j.ijpe.2012.11.012Get rights and content.

Yan, S. and Sun, H., 2024. Constrained Optimal Fuel Consumption of HEV: Considering the Observational Perturbation. arXiv preprint arXiv:2410.20913. https://doi.org/10.48550/arXiv.2410.20913.