Coordination in a two‐stage production system: Complexity, conflict and cooperation

We study the benefits of coordination between two adjacent stages in a production system. A set of jobs specific to the customers’ demand are processed in a planning horizon. Each job is characterized by its processing times in both stages and a due date. Once a job is processed in the first stage it enters a buffer where it may be moved to a specific location; when all jobs are processed in the buffer, the second stage commences its operations. The objective of the first stage is to minimize the inventory cost measured by the sum of completion time, and that of the second stage is to minimize the tardiness cost and the resequencing cost in the buffer. The stages prefer schedules that minimize their respective objective functions. The performance of the system is measured by the sum of the costs associated at both stages. We evaluate each stage's cost of conflict as the relative increase in its cost as a result of using the other stage's optimal schedule. We analyze the computational tractability of the individual stage's problems and the system problem, demonstrating that the system problem is NP‐hard and that individual problems can be solved in polynomial time. We designed a Genetic Algorithm based on the idea of nondominated sorting for solving the system problem. The GA provided ideal solutions quickly, and numerical studies reveal that the cost saving provided by the coordinated schedule between both stages is usually significant. Coordination is possible between stages due to this cost saving. Finally, we briefly discuss the implications on our work for how both stages negotiate, coordinate, and implement their supply chain schedules in practice.