Capacity acquisition, subcontracting, and lot sizing

The fundamental question encountered in acquiring capacity to meet nonstationary demand over a multiperiod horizon is how to balance the trade-off between having insufficient capacity in some periods and excess capacity in others. In the former situation, part of the demand is subcontracted while, in the latter, capacity that has been paid for is rendered idle. Capacity and subcontracting decisions arise in many economic activities ranging from production capacity planning in semiconductor fabs to leasing communication networks, from transportation contracts to staffing of call centers. In this paper, we investigate the trade-offs between acquiring capacity, subcontracting, production, and holding inventory to satisfy nonstationary demand over a finite horizon. We present capacity acquisition models with holding and without holding inventory and identify forecast-robust properties of the models that restrict the dependence of optimal capacity decisions on the demand forecasts. We develop algorithms for numerous practical cost structures involving variable and fixed charges and prove that they all have polynomial complexity. For models with inventory, we solve a sequence of constant capacity lot-sizing and subcontracting subproblems, which is also of independent interest.