Process Flexibility Revisited: The Graph Expander and Its Applications

We examine how a flexible process structure might be designed to allow the production system to better cope with fluctuating supply and demand, and to match supply with demand in a more effective manner. We argue that good flexible process structures are essentially highly connected graphs, and use the concept of graph expansion (a measure of graph connectivity) to achieve various insights into this design problem. A number of design guidelines are well known in the literature. Principles such as “a long chain performs better than many short chains,” and that one should “try to equalize the number of plants (resp. products), measured in total units of capacity (resp. mean demand), which each product (resp. plant) in the chain is directly connected to,” can now be interpreted from this new angle as a development of different ways in which the underlying network can achieve a good expansion ratio. The same principle extends to other new design guidelines trying to equalize the number of plants (measured in total number of units) assigned to each pair (or even triplet) of products, or vice versa, can also help the decision maker to arrive at a good process structure. We analyze the worst-case performance of the flexible design problem under a more general setting, which encompasses a large class of objective functions. We show that whenever demand and supply are balanced and symmetrical, the graph expander structure (a highly connected but sparse graph) is within ǫ optimality of the fully flexible system, for all demand scenarios, although it uses a far smaller number of links. Furthermore, the same graph expander structure works uniformly well for all objective functions in this class. Based on this insight, we develop a simple and easy-to-implement heuristic to design flexible process structure. Numerical results show that this heuristic performs well for a variety of numerical examples previously studied in the literature. We also use this idea on a set of real data obtained from a bread delivery system in Singapore, with the goal of minimizing the excess amounts of bread brought to each location.