Exact Solution of the Optimal Control Problem of Coordinating a Supplier-manufacturer Supply Chain in Advanced Geometric Concepts

Abstract

Supply chain coordination deals with collaborative efforts of supply chain parties and making globally-optimal decisions that can improve overall performance and efficiency of the entire supply chain. In many situations, the problem of supply chain coordination requires formulation of a continuous time optimal control model, in which optimal solution is identified approximately through numerical estimation. Therefore, in this paper, a novel approach was presented for optimal control problem by developing a new formulation based on advanced ingredients of differential and Poisson geometry. Thus, the exact optimal solution of control problem can be obtained using an analytical methodology that converts the Hamilton-Jacobi-Bellman partial differential equation (PDE) into a reduced Hamiltonian system. The proposed approach was applied to the problem of coordinating supplier development programs in a two-echelon supply chain comprising of a single supplier and a manufacturing firm. For further illustrating applicability and efficiency of the proposed methodology, a numerical example was also provided. The proposed approach offers unique advantages and can be applied to find the exact solution of optimal control models in various optimization problems.