Developing a new formulation and exact solution approach for continuous-time optimal control model: application to coordinating a supplier-manufacturer supply chain

Document Type : Research Paper

Author

Fouman Faculty of Engineering, College of Engineering, University of Tehran

10.22070/jqepo.2023.16182.1234

Abstract

Supply chain coordination deals with the joint efforts of supply chain parties and making optimal global decisions, which in turn can improve the overall performance and efficiency of the entire supply chain. In many cases, the supply chain coordination problem leads to the formulation of a continuous-time optimal control model, where the optimal response is often calculated from numerical methods. Therefore, in this paper, a novel approach to optimal control problems is proposed by expanding a modern formulation supported by progressive concepts of differential geometry and Poisson geometry. This approach leads to providing an analytical answer to solve the optimal control problem in such a way that the Hamilton-Jacobi-Bellmann partial differential equation (PDE) can be converted into a reduced Hamiltonian system. To check the effectiveness of the proposed formulation, the problem of coordination of supplier development plans in a two-level supply chain including a single supplier and a manufacturing firm is investigated. The application of this approach is well illustrated by re-examining a numerical example. The unique advantages of the proposed approach lead to its efficiency in finding the exact solution of optimal control models in various optimization problems. The developed method provides further insights into analytical methods for solving supply chain coordination problems and is supported by advanced geometric concepts and structured instructions.

Keywords

References

Adabi, H., & Mashreghi, H. (2019). Coordination and competition in a duopoly with two-manufacturer and two-retailer with wholesale-price contract and demand uncertainty, International Journal of Industrial Engineering & Production Research, 30(4), 465-476.
Arasteh, A. (2022).  Mathematical modeling of flexible production lines with different part types on unreliable machines by a priority rule, Journal of Quality Engineering and Production Optimization, 7(2), 34-59.
Bertsekas, D. P. (1999). Nonlinear programming, Massachusetts Institute of Technology.
Da Silva, A. C., & Weinstein, A. (1998). Geometric models for noncommutative algebras, University of California.
Dastyar, H., Rippel, D., & Freitag, M. (2020). Optimization of Supplier Development under Market Dynamics, Mathematical Problems in Engineering, 2020, 2912380, DOI: 10.1155/2020/2912380.
Eshaghnezhad, M., Mansoori, M., & Effati, S. (2023).  Optimal Control Problem: A Case Study on Production Planning in the Reverse Logistics System,  Journal of Systems Thinking in Practice, 2(1), 28-38.
Evans, L. C. (1991). Partial Differential equations, University of California, AMS.
Gutt, S., Rawnsley, J., & Sternheimer, D. (2005). Poisson geometry, deformation quantization and group representations, Cambridge University Press.
Hasan-Zadeh, A. (2017). A note on variational symmetries of the constrained variational problems, Advances in Differential Equations and Control Processes, 18(2), 69-76.
Hasan-Zadeh, A. (2019). Exact inertial manifolds for dynamical systems, Advances in Differential Equations and Control Processes, 21(1), 117-122.
Hasan-Zadeh, A. (2021). Geometric Classification of analytical solutions of Fitzhugh-Nagumo equation and its generalization as the reaction-diffusion equation, Advances in Differential Equations and Control Processes, 24(2), 167-174.
Hasan-Zadeh, A. (2021). Dynamic optimal control problems in Hamiltonian and Lagrangian systems, Advances in Differential Equations and Control Processes, 24(2), 175-185.
Hasan-Zadeh, A. (2021). Geometric investigation of nonlinear reaction-diffusion-convection equations: Fisher equation and its extensions, Advances in Differential Equations and Control Processes, 24(2), 145-151.
Hasan-Zadeh, A., & Mohammadi-khanaposhti, M. (2018). Inertial manifolds for Navier-Stokes equations in notions of Lie algebras, WSEAS Transactions on Heat and Mass Transfer, 13(1), 95-102.
Hosseini-Motlagh, S. M., Ebrahimi, S., & Zirakpourdehkordi, R. (2020a). Coordination of dual-function acquisition price and corporate social responsibility in a sustainable closed-loop supply chain, Journal of Cleaner Production, 251, 119629, DOI: 10.1016/j.jclepro.2019.119629.
Hosseini-Motlagh, S. M., Nouri-Harzvili, M., & Zirakpourdehkordi, R. (2019). Two-level supply chain quality improvement through a wholesale price coordination contract on pricing, quality and services, International Journal of Industrial Engineering & Production Research, 30(3), 287-312.
Hosseini-Motlagh, S. M., Nouri-Harzvilia, M., Johari, M., & Sarker, B. R. (2020b). Coordinating economic incentives, customer service and pricing decisions in a competitive closed-loop supply chain, Journal of Cleaner Production, 255, 120241, DOI: 10.1016/j.jclepro.2020.120241.
Hsieh, F. S. (2018). Dynamic configuration and collaborative scheduling in supply chainsbased on scalable multi-agent architecture, Journal of Industrial Engineering, DOI:  10.1007/s40092-018-0291-5.
Hu, H., Zhang, Z., Wu, Q., & Han, S. (2020). Manufacturer’s customer satisfaction incentive plan forduopoly retailers with Cournot or collusion games, Advances in Production Engineering & Management, 15(3), 345-357.
Ivanov, D., Dolgui, A., & Sokolov, B. (2016). Robust dynamic schedule coordination control in the supply chain, Computers and Industrial Engineering, 94(1), 18-31.
Jafarzadeh, J., Amoozad-Khalili, H., & Shoja, N. (2022). A Multi-Objective Mathematical Model for Dynamic Cellular Manufacturing System Design under Uncertainty: A Sustainable approach, Journal of Quality Engineering and Production Optimization, 7(1), 98-120.
Javadi, E., Babazadeh, R., Doniavi, A., & Javadi, H. (2022). An Application of Mixed-Integer Linear Programming Method in Production Planning of Pipe Industry, Journal of Quality Engineering and Production Optimization, 7(1), 13-24.
Jian, J., Zhang, Y., Jiang, L., & Su, J. (2020). Coordination of Supply Chains with Competing Manufacturers considering Fairness Concerns, Complexity, 4372603, 15 pages, DOI: 10.1155/2020/4372603.
Johari, M., & Hosseini-Motlagh, S. M. (2020). Coordination contract for a competitive pharmaceutical supply chain considering corporate social responsibility and pricing decisions, RAIRO-Operations Research, 54(5), 1515-1535.
Kar, S., Maiti, M., Maity, K., & Roul, J. N. (2015). Multi–item reliability dependent imperfect production inventory optimal control models with dynamic demand under uncertain resource constraint, International Journal of Production Research, 53(16), 4993-5016.
Keshmiry Zadeh, K., Harsej, F., sadeghpour, M., & Molani Aghdam, M. (2021). A multi-objective multi-echelon closed-loop supply chain with disruption in the centers, Journal of Quality Engineering and Production Optimization, 6(2), 31-58.
Kim, B. (2000). Coordinating an innovation in supply chain management, European Journal of Operational Research, 123(3), 568-584.
Kotzab, H., Darkow, I. L., Bäumler, I., & Georgi, C. (2019). Coordination, cooperation and collaboration in logistics and supply chains: a bibliometric analysis, Production, 29, DOI: 10.1590/0103-6513.20180088.
Li, S., Kang, M., & Haney, M. H. (2017). The effect of supplier development on outsourcing performance: the mediating roles of opportunism and flexibility, Production Planning & Control, The Management of Operations, 28, 6-8, DOI: 10.1080/09537287.2017.1309711.
Mohammed, J. K., & Khudair, A. R. (2023). A novel numerical method for solving optimal control problems using fourth- degree hat functions, Partial Differential Equations in Applied Mathematics, 7, 100507.
Nabavi, S. M., Vahdani, B., Afshar Nadjafi, B., &  Adibi, M. A. (2021). Robust planning for debris clearance and relief distribution with split delivery and fairness, Journal of Quality Engineering and Production Optimization, 6(2), 181-200.
Pham, T. H., & Doan, T. D. U. (2020). Supply chain relationship quality, environmental uncertainty, supply chain performance and financial performance of high-tech agribusinesses in Vietnam, Uncertain Supply Chain Management, 8, 663–674.
Proch, M., Worthmann, K., & Schluchtermann, J. (2017). A negotiation–based algorithm to coordinate supplier development in decentralized supply chains, European Journal of Operational Research, 256(2), 412-429.
Olver, P. J. (1993). Applications of Lie Groups to Differential Equations, Springer–Verlag.
Roth, L. C., Bhaya, A., & Kaszkurewicz, E. (2023). Inventory Management Through One Step Ahead Optimal Control Based on Linear Programming, RAIRO Operations Research, 57, 17-42.
Rudolph, G., & Schmidt, M. (2013). Differential geometry and mathematical physics, Part 1. Manifolds, Lie groups and Hamiltonian systems, Springer.
Sutrisno, S., Widowati, W., & Tjahjana, R. (2022). Optimal Control for Inventory System under Uncertainty on Demand and Delivery Using Robust Linear Quadratic Control Approach, International Journal of Supply and Operations Management, 9(1), 1-14.
Journal of Quality Engineering and Production Optimization
Volume 8, Issue 1
May 2023
Pages 69-86

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