Mathematical Modeling and Optimal Control for a Class of Dynamic Supply Chain: A Systems Theory Approach

Abstract

Dynamic supply chains (SC) are important to reduce inventory, enable the flow of materials, maximize profits, and minimize costs. This research work presents a capacity–inventory management model via system dynamics for a dynamic supply SC, applying model-based optimal control techniques. In the context of high-volume manufacturing (HVM) that present low variability and predictable demand, for mathematical modeling purposes, a set of coupled first-order ordinary differential equations, with an analogy from the mixing problem, is presented, which relates capacity and inventory levels, taking into account a production rate at each node of interaction. The application of ordinary differential equations via the mixing problem (or compartmental analysis) is important based on the idea of a balance between the influx and outflux of raw material along the supply chain. A proper literature review on optimal control for supply chains is analyzed. The mathematical model introduced is presented in a linear time-invariant (LTI) state-space formulation. Stability analysis for the dynamic serial SC is presented, and a sensitivity analysis is also conducted for the capacity and production rate parameters considering the effects of variations in parameters along the SC. An energy-based optimal control is also developed with proper simulations.