The Logic of Logistics

Abstract

Because of its effectiveness in responding to uncertainties, process flexibility has gained significant attention in several industries, in particular, in the automotive industry. Evidently, it is often too expensive to achieve a high degree of flexibility, for example, full flexibility, and as a result, sparse or partial flexibility is implemented instead. One set of sparse flexibility designs is the 2-flexibility designs. A flexibility design is a 2-flexibility design if each plant produces exactly two products and demand for each product can be satisfied from exactly two plants.Of course, there are many ways to implement sparse designs, and the challenge is to identify an effective one. An important concept analyzed in the literature and applied in practice by various companies is the concept of the long chain. The first to observe the power of the long chain were Jordan and Graves (1995), who, through empirical analysis, showed that the long-chain design can provide almost as much benefit as full flexibility. In particular, Jordan and Graves (1995) found that for randomly generated demand, the expected amount of demand that can be satisfied by a long-chain design is very close to that of a full flexibility design.Though the analysis and results can be extended to more general settings, for simplicity, we focus on balanced manufacturing systems, that is, manufacturing systems with an equal number of plants and products, and each plant has a equal capacity. Given a balanced manufacturing system, a flexibility design A is represented by the arc set of a directed bipartite graph, where an arc from plant node i to product node j implies that plant i is capable of producing product j. For example, if A is a dedicated design, then A has exactly n arcs such that each plant node is incident to one arc and each product node is incident to one arc. By contrast, if A is a full flexibility design, then A has arcs connecting every plant node to all product nodes.Because A is represented by a bipartite graph, applying standard graph theory notation, we define an undirected cycle in A to be a set of arcs that forms a cycle when the arc directions are ignored. A flexibility design A is a long chain if its arcs form exactly one undirected cycle containing all plant and product nodes (see Fig. 13.1 for an example). A closed chain is defined as an induced subgraph in A that forms an undirected cycle, while an open chain is an induced subgraph in A that forms an undirected line (one arc less than an undirected cycle). Figure 13.1 presents an example of an open and a closed chain. It can be seen that any 2-flexibility design, where each product/plant node is incident to two arcs, is the union of a number of closed chains.The results presented in this chapter are motivated by a few observations made in the literature regarding the effectiveness of the long-chain flexibility design. The first is an observation that has been made in (Graves 2008 and Hopp et al. 2004) regarding the performance of the long chain for a balanced syste...

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