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

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.