Numerical schemes for the optimal input flow of a supply chain. (English) Zbl 1282.90061
Summary: 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 modeled 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 a method allows convergence results and error estimates for an upwind-Euler steepest descent algorithm, which is also tested by numerical simulations.
MSC:
90B30 | Production models |
35Q90 | PDEs in connection with mathematical programming |
35L50 | Initial-boundary value problems for first-order hyperbolic systems |
35L65 | Hyperbolic conservation laws |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
90B05 | Inventory, storage, reservoirs |
65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |