Simulation and optimization of supply chains and networks using a macroscopic approach

Abstract

The aim of thesis is to present some macroscopic models for supply chains and networks able to reproduce the goods dynamics, successively to show, via simulations, some phenomena appearing in planning and managing such systems and, nally, to deal with optimization problems. Depending on the observation scale supply networks modeling is charac- terized by di¤erent mathematical approaches: discrete event simulations and continuous models. Since discrete event models (Daganzo 2003) are based on considerations of individual parts, their main drawback is, however, an enor- mous computational e¤ort. Then a cost-e¤ective alternative to them is continu- ous models, described by some partial di¤erential equation. The rst proposed continuous models date back to the early 60 s and started with the work of Baumol (1970) and Forrester (1964), but the most signi cant in this direction was Daganzo (1997), where the authors, via a limit procedure on the number of parts and suppliers, have obtained a conservation law (Armbruster-Marthaler- Ringhofer 2004, Dafermos 1999), whose ux involves either the parts density or the maximal productive capacity. Then, in recent years continuous and homogenous product ow models have been introduced and they have been built in close connection to other transport problems like vehicular tra¢ c ow and queuing theory. Extensions on networks have been also treated. In this work, starting by the historical model of Armbruster - Degond - Ringhofer, we have compared two di¤erent macroscopic models, i.e. the Klar model, based on a di¤erential partial equation for density and an ordinary dif- ferential equation to capture the evolution of queues, and a continuum-discrete model, formed by a conservation law for the density and an evolution equa- tion for processing rate. Both the models can be applied for supply chains and networks. Moreover, an optimization problem of sequential supply chains modeled by the Klar approach has been treated. The aim is to nd the con guration of pro- duction according to the supply demand minimizing the queues length, i.e. the costs of inventory, and obtaining an expected pre-assigned out ow. The control problem is solved introducing and minimizing a cost functional which takes into account the nal ux of production and the queues representing the stores. The functional is not linear, so to nd its minimum, the vectors tangent method is introduced. This technique is based on the choice of an input ow which is a piecewise constant function, with a nite number of discontinuities. Considering on each of them an in nitesimal displacement which generates traveling tempo- ral shifts on processors and shifts on queues, we are able to compute numerically the value of the variation of functional respect to each discontinuities. Finally, we use the steepest-descent algorithm to nd, via simulations, the optimal con- guration of input ow, according to the pre- xed desired production. [edited by the author]