Dynamic lot sizing problem for a warm/cold process

We consider a dynamic lot sizing problem with finite capacity for a process that can be kept warm until the next production period at a unit variable cost ωt only if more than a threshold value has been produced and is cold, otherwise. That is, the setup cost in period t is Kt if xt-1 < Qt-1 and kt, otherwise (0 ≤ kt ≤ Kt). We develop a dynamic programming formulation of the problem, establish theoretical results on the structure of the optimal production plan and discuss its computational complexity in the presence of Wagner-Whitin-type cost structures. Based on our stuctural results, we present an optimal polynomial-time solution algorithm for kt = 0, and also show that an optimal linear-time solution algorithm exists for a special case. Our numerical study indicates that utilizing the undertime option (i.e., keeping the process warm via reduced production rates) results in significant cost savings, which has managerial implications for capacity planning and selection.