A Model justification

In recent years, there has been a lot of research in an attempt to provide a deeper understanding of optimal control models from a theoretical as well as an application point of view. In particular, an attractive feature of these models is that they provide good scheduling, production and inventory policies in a variety of settings. Furthermore, they approximate well the underlying stochasticity of problems in a deterministic way. Fluid models arise in applications as diverse as routing and communication systems as well as queueing, supply chain and transportation systems. A continuous time approach has the advantage of not introducing any approximation to the real setting: it provides the exact solution of the system. When taking a discrete time approach, one has to decide what a reasonable time step should be, and to allow price and production changes only at those times. In reality, in some settings a supplier may need more flexibility. In order to avoid being too restrictive, the time step needs be very small, and if the time horizon is large the size of the problem may become exceedingly large, in terms of number of variables and number of constraints. Therefore, the problem size usually implies significant delays in obtaining good solutions. Examples of supply chain industries where continuous-time optimal control models of the type we discuss in this paper are relevant, include industries with a high volume of throughput and data on costs and demand that change a lot. The hardware as well as the semiconductor industries are such examples. Moreover, we believe that a similar approach can be applied to problems in areas other than dynamic pricing and inventory control, where the evolution of the system evolves dynamically and justify a continuous time approach. We believe that the techniques presented in this paper may be helpful to those areas as well.