This study considers a single item make-to-stock system with continuous-time production and inventory controls to meet bulk demand with an exponential inter-arrival time. A key issue in this system is the non-convex shortage cost consisting of fixed and variable expenditures when the demand is not fully satisfied. We propose a self-reservation policy by building a Markov Decision Process to minimize the overall cost. We find that the optimal production control is still a base stock policy, but the structure of the optimal self-reservation policy is very complicated. However, if the effective outstanding variable shortage cost is sufficiently large, the optimal self-reservation policy has an easy form of “Reserve All or Nothing.” Our numerical examples indicate the optimal policy may reduce the total average cost by 47% on average.
- Fixed and variable shortage cost
- Markov processes
- Production management
- Reservation policy