A Continuous-Time Strategic Capacity Planning Model Based on the Minimum-Cut Problem

The semiconductor industry has been one of the driving forces of the “new” economy. It boasts of the exponentially growing performance of semiconductor devices, coupled with rapidly decreasing chip prices; however, it faces highly volatile demands, and copes with astronomical fab costs, most of which can be attributed to tool costs. The leadtime for purchasing tools is between 6 and 18 months, upon which tools quickly become obsolete. Thus, semiconductor companies need to recover their capital investment in the tools over a short period of time. We develop models and algorithms for strategic capacity planning, which is to determine the sequence and timing of acquiring tools. Strategic planning decisions are made in the presence of high uncertainty. Uncertainty comes from factors such as technology, the market, and its products, and becomes amplified by long leadtimes. Although capacity planning decisions need to be made in the presence of high uncertainty, early research and even some current practices overlook the stochastic nature of planning, with the exception of simple case analyses. An extensive review of literature can be found in Çakanyildirim et al. (1999) and Roundy et al. (2000). Typical methods of stochastic optimization include stochastic-linear programming, stochastic-integer programming, and Markov decisions processes; yet, they have not been able to solve real-world capacity planning problems on the scale faced by the semiconductor industry. This paper takes the stochastic-optimization approach, which explicitly incorporates randomness in the model. We assume nonstationary stochastic demand, with the expected demand for product families increasing over time. We also assume lost sales and no finished good inventory. As in Çakanyildirim et al. (1999), we continue to explore alternative approaches based on continuous-time models. The time at which a machine is purchased becomes a continuous-decision variable. These models are more compact than traditional stochastic-programming methods based on discrete-time models. It is hoped that the small dimensionality of continuous-time models will make the strategic capacity planning problem computationally tractable. In this paper, we model multiple resource types used for multiple product families. The resulting problem is related to the continuous relaxation of the lot-sizing problem. We present an efficient divideand-conquer algorithm that will find a locally optimal solution of this problem. A subroutine to this algorithm is the parametric minimum-cut problem.