Maximizing the total weight value of just-in-time jobs in identical parallel machines with periodic time slots

We address the processing of jobs in an environment with periodic due dates. Every job must be completed exactly on a due date, the situation of which shall be referred to as just-in-time. Each job is associated with a weight which is non-increasing with time. The problem we address is to maximize the total weight of just-in-time jobs. We prove that this class of problem is NP-hard. The key idea is a reduction from the Hamiltonian path problem, known as strongly NP-hard. Moreover, we discuss some special cases where the problem is solvable. If no set-up times exist, there are cases where the problem is solvable, we then present a method to solve the problems. To achieve this, we derive partition and union procedures, and use network flow algorithms.