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

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

A variational approach for continuous supply chain networks

We consider a continuous supply chain network consisting of buffering queues and processors first proposed by [D. Armbruster, P. Degond, and C. Ringhofer, SIAM J. Appl. Math., 66 (2006), pp. 896–920] and subsequently analyzed by [D. Armbruster, P. Degond, and C. Ringhofer, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), pp. 433–460] and [D. Armbruster, C. De Beer, M. Fre- itag, T. Jagalski, and C. Ringhofer, Phys. A, 363 (2006), pp. 104–114]. A model was proposed for such a network by [S. G ̈ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559] using a system of coupling ordinary differential equations and partial differential equations. In this article, we propose an alternative approach based on a variational method to formulate the network dynamics. We also derive, based on the variational method, a computational algorithm that guarantees numerical stability, allows for rigorous error estimates, and facilitates efficient computations. A class of network flow optimization problems are formulated as mixed integer programs (MIPs). The proposed numerical algorithm and the corresponding MIP are compared theoretically and numerically with existing ones [A. Fu ̈genschuh, S. Go ̈ttlich, M. Herty, A. Klar, and A. Martin, SIAM J. Sci. Comput., 30 (2008), pp. 1490–1507; S. Go ̈ttlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559], which demonstrates the modeling and computational advantages of the variational approach

Negotiating and Sharing Capacities of Large Additive Manufacturing Networks

This paper focuses on dynamics of productive and demanding nodes for Scattered Manufacturing Networks within 3D Printings contexts. The various nodes issue orders or sell production slots in order to achieve their own aims. An orchestrator coordinates the dynamics along the network according to principles of sustainability, equated shared resources and transparency by managing communication activities among nodes. In particular, suitable tradeoffs occur by a unique framework that, with the aim of optimizing the overall costs, suggests either logistics paths along the network or negotiation policies among nodes in order to reallocate resources. Numerical examples present the proposed approach. Keywords: Industry 4.0, Additive Manufacturing, Sharing Capacities, Operation Models, Optimization of networks JEL Codes:  C02; O21 and P4

Negotiating and Sharing Capacities of Large Additive Manufacturing Networks

This paper focuses on dynamics of productive and demanding nodes for Scattered Manufacturing Networks within 3D Printings contexts. The various nodes issue orders or sell production slots in order to achieve their own aims. An orchestrator coordinates the dynamics along the network according to principles of sustainability, equated shared resources and transparency by managing communication activities among nodes. In particular, suitable tradeoffs occur by a unique framework that, with the aim of optimizing the overall costs, suggests either logistics paths along the network or negotiation policies among nodes in order to reallocate resources. Numerical examples present the proposed approach. Keywords: Industry 4.0, Additive Manufacturing, Sharing Capacities, Operation Models, Optimization of networks JEL Codes: C02; O21 and P4

On Optimization of a Highly Re-Entrant Production System

We discuss the optimal control problem stated as the minimization in the $L^2$-sense of the mismatch between the actual out-flux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the switch dispatch point (SDP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a quasi-pull policy. The main question we discuss in this paper is about the optimal choice of the input in-flux, push and quasi-pull constituents, and the position of SDP

On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms

We study one class of nonlinear fluid dynamic models with impulse source terms. The model consists of a system of two hyperbolic conservation laws: a nonlinear conservation law for the goods density and a linear evolution equation for the processing rate. We consider the case when influx-rates in the second equation take the form of impulse functions. Using the vanishing viscosity method and the so-called principle of fictitious controls, we show that entropy solutions to the original Cauchy problem can be approximated by optimal solutions of special optimization problems

Control Problems for Conservation Laws with Traffic Applications

Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks

Control Problems for Conservation Laws with Traffic Applications

Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks