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

Featured Application Dynamic supply chains are suitable for the mathematical modeling, analysis and control of high-volume manufacturing supply chains in which low variability is present, considering a cause-effect relation in a deterministic context. The application of ordinary differential equations via the mixing problem context is addressed. 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.

Automated summary

This study utilizes a system dynamics model and ordinary differential equations to analyze and control dynamic supply chains, aiming to reduce inventory, maximize profits, and minimize costs. The relationship between production rate and inventory levels in the supply chain is described using a model based on the mixing problem. Stability and sensitivity analysis are conducted to understand the effects of parameter variations on the supply chain.