Effectiveness and design of sparse process flexibilities

Abstract

The long chain has been an important concept in the design of flexible processes. This design concept, as well as other sparse flexibility structures, have been applied by the automotive and other industries as a way to increase flexibility in order to better match available capacities with variable demands. Numerous empirical studies have validated the effectiveness of these structures. However, there is little theory that explains the effectiveness of the long chain, except when the system size is large, i.e., by applying an asymptotic analysis. Our attempt in this thesis is to develop a theory that explains the effectiveness of long chain and other sparse flexibility structures for finite size systems. We study the sales of sparse flexibility structures under both stochastic and worst-case demands. From our analysis, we not only provide rigorous explanation to the effectiveness of the long chain, but also refine guidelines in designing other sparse flexibility structures. Under stochastic demand, we first develop two deterministic properties, supermodularity and decomposition of the long chain, that serve as important building blocks in our analysis. Applying the supermodularity property, we show that the marginal benefit, i.e., the increase in expected sales, increases as the long chain is constructed, and the largest benefit is always achieved when the chain is closed by adding the last arc to the system. Then, applying the decomposition property, we develop four important results for the long chain under IID demands: (i) an effective algorithm to compute the performance of long chain using only matrix multiplications; (ii) a proof on the optimality of the long chain among all 2-flexibility structures; (iii) a result that the gap between the fill rate of full flexibility and that of the long chain increases with system size, thus implying that the effectiveness of the long chain relative to full flexibility increases as the number of products decreases; (iv) a risk-pooling result implying that the fill rate of a long chain increases with the number of products, but this increase converges to zero exponentially fast. Under worst-case demand, we propose the plant cover index, an index defined by a constrained bipartite vertex cover problem associated with a given flexibility structure. We show that the plant cover index allows for a comparison between the worst-case performances of two flexibility structures based only on their structures and is independent of the choice of the uncertainty set or the choice of the performance measure. More precisely, we show that if all of the plant cover indices of one structure are greater than or equal to the plant cover indices of the other structure, then the first structure is more robust than the second one, i.e. performs better in worst-case under any symmetric uncertainty set and a large class of performance measures. Applying this relation, we demonstrate the effectiveness of the long chain in worst-case performances, and derive a general heuristic that generates sparse flexibility structures which are tested to be effective under both stochastic and worst-case demands. Finally, to understand the effect of process flexibility in reducing logistics cost, we study a model where the manufacturer is required to satisfy deterministic product demand at different distribution centers. Under this model, we prove that if the cost of satisfying product demands at distribution centers is independent of production plants or distribution centers, then there always exists a long chain that is optimal among 2-flexibility structures. Moreover, when all plants and distribution centers are located on a line, we provide a characterization for the optimal long chain that minimizes the total transportation cost. The characterization gives rise to a heuristic that finds effective sparse flexibility structures when plants and distribution centers are located on a 2-dimensional plane.