It is proposed to apply nonlinear dynamics methods from dynamical systems theory and fluid dynamics to three problems in industrial mathematics: 1) The organization of production lines with flexible workers. A dynamical systems model will be derived and analyzed. The goal is to determine whether chaos or other instabilities are present, to find out whether the production line is still self-organizing, and to determine the resulting throughput. 2) A switched arrival system for three parallel machines has been shown to be chaotic. Increasingly realistic production scenarios will be added to this model, and the resulting dynamics will be determined. Networks of such systems will be studied to determine whether synchronization or partial synchronization can occur. Such systems are hybrid dynamical systems that require new tools to study their dependence on changing parameters. Such tools will be developed as extensions of standard dynamical systems methods. 3) A hierarchy of fluid-dynamical models to describe the flow of products through factories based on conservation laws will be developed. These models are based on the methods of gas dynamics and fluid averaging and promise to allow fast and accurate simulations for supply networks.
This project is based on an ongoing collaboration between researchers from Mathematics, Industrial Engineering, and Intel Corporation. We expect that our results have a significant impact on the understanding of the relationship between policies and management decisions and the resulting performance of factories and whole business networks. The three major areas of inquiry will be: i) The management concept of bucket brigades: The organizational principle of bucket brigades will be studied for workers with different skill levels along the production line as well as changing skill levels due to learning. ii) There are theoretical results stating that certain work allocation rules for parallel machines will lead to chaotic behavior. These results are based on highly abstracted models. We will study more realistic models and larger networks of production machines. iii) Supply chain modeling and simulation: We will derive models that allow fast scalable simulations of production flows in a supply chain. The long-term goal is to optimize production across the whole supply chain, an intermediate goal is to generate simulation tools that allow to explore business questions and to pose "what if" questions on these simulations.
