Integrated production and delivery on parallel batching machines

We consider the problem of scheduling a set of n jobs on m identical and parallel batching machines. The machines have identical capacities equal to K and the jobs have identical processing times equal to p. Job j has a size s_j, a due date d_j and a profit R_j. Several jobs can be batched together and processed by a machine, provided that the total size of the jobs in the batch does not exceed the machine capacity K. The company will earn a profit of R_j dollars if job j is delivered by time d_j; otherwise, it earns nothing. A third party logistic (3PL) provider will be used to deliver the jobs. The 3PL provider picks up the jobs at times T₁ < T₂ < ⋯ < T_z, and v_k (1 ≤ k ≤ z) vehicles will be provided for delivery at time T_k. The vehicles have identical capacities equal to C. The objective is to find a production and delivery schedule so as to maximize the total profit that the company can earn. We show that the problem is solvable in polynomial time if the jobs have identical sizes, but it becomes unary NP-hard if the jobs have different sizes. We propose heuristics for various NP-hard cases and analyze their performances.