MILP Continuous-Time Production Scheduling Approaches for the Phosphate Fertilizer Industry

The fertilizer industry is becoming increasingly essential for the future of mankind as the world population continues to grow. In order for the food production rate to meet the increasing food demand, the available tillable areas need to be more productive, which can be achieved through the use of fertilizers. In this work, we present three MILP production scheduling approaches for addressing a typical phosphate fertilizer problem, comprising continuous tasks as well as short and lengthy batch tasks. The scheduling problem is also featured as presenting mixed storage policies and sequence-dependent changeover times. All proposed formulations are based on a continuous single-time grid. In the first approach, batch tasks are allowed to freely take place over multiple time slots. In the second approach, batch tasks span a pre-defined number of time slots, and in the third hybrid approach, short batch tasks are allowed to freely take place over multiple time slots, while lengthy batch tasks span a pre-defined number of time slots. In all approaches, continuous tasks are assigned to a single time slot. A discrete time-based formulation is also used as a baseline comparison. It is verified that the first approach quickly becomes intractable as the number of time slots is increased. The second approach requires trial-and-error regarding the fixed number of time slots used for short batch tasks despite the excellent computational performance. Ultimately, the hybrid approach enables finding the optimal production schedule for a month scheduling horizon in approximately 4 CPU minutes without having to run multiple problem instances.