Numerical schemes for the optimal input flow of a supply-chain

TL;DR

This paper introduces a novel numerical method using generalized tangent vectors to optimize input flow in supply chains modeled by conservation laws, aiming to minimize inventory costs and match desired outflows.

Contribution

It presents a new approach employing generalized tangent vectors for convergence analysis in supply chain control problems modeled by PDEs and ODEs.

Findings

Convergence results for the proposed numerical method.

Error estimates for the steepest descent algorithm.

Numerical simulations demonstrating effectiveness.

Abstract

An innovative numerical technique is presented to adjust the inflow to a supply chain in order to achieve a desired outflow, reducing the costs of inventory, or the goods timing in warehouses. The supply chain is modelled by a conservation law for the density of processed parts coupled to an ODE for the queue buffer occupancy. The control problem is stated as the minimization of a cost functional J measuring the queue size and the quadratic difference between the outflow and the expected one. The main novelty is the extensive use of generalized tangent vectors to a piecewise constant control, which represent time shifts of discontinuity points. Such method allows convergence results and error estimates for an Upwind- Euler steepest descent algorithm, which is also tested by numerical simulations.