Supply Chain Design in Perfect Competition

In this paper, we apply the theory of optimal control and theory of tra¢ c assignment for supply chain design in perfect competition. We model the time staging and generalized routing of input factors needed for production by a …rm. The production process will typically involve several stages, and as such is described by paths through a production network whose nodes are the various stages of production. We develop an algorithm and test it for a small numerical example. 1 Introduction In the last two decades, productions as well as consumptions has become global, …rms began to recognize the importance of supply chain optimization. In sequence, supply chain management and design has become a very important issue both in practice and in research. Among many other methodologies to give optimal policy to manage supply chain, mathematical modeling and analysis of supply chain is of particular interest in scienti…c communities: see Beamon (1998), Min and Zhou (2002) and references therein. In this paper, we apply the theory of optimal control (see Bryson and Ho, 1975) and theory of tra¢ c assignment (see Friesz and Mookherjee, 2006) to a supply chain design problem. For the applications of optimal control theory in management problems, see Bensoussan et al. (1974) and Sethi and Thompson (2000) for the general introduction and see Ortega and Lin (2004) for an extensive survey. We model the time staging and generalized routing of input factors needed for production by a …rm. To do so, we will employ a supply network which is connected to the production process of the …rm of interest at supply-intake nodes. For the supply network, origin nodes are the sources of factor supplies while destination nodes represent the location in time and space at which factors enter the production process. The production process will typically involve several stages, and as such is described by paths through a production network whose nodes are the various stages of production. Although this perspective is reminiscent of the well known critical path method, it is substantially more general. We assume that the supply-production network just described has associated demands for …nished goods that compel production activity that in turn compels the formation of supply chains within the