An Approximation Algorithm for the Parallel-Machine Customer Order Scheduling with Delivery Time and Submodular Rejection Penalties

In this paper, we consider the parallel-machine customer order scheduling with delivery time and submodular rejection penalties. In this problem, we are given m dedicated machines in parallel and n customer orders. Each order has a delivery time and consists of m product types and each product type should be manufactured on a dedicated machine. An order is either rejected, in which case a rejection penalty has to be paid, or accepted and manufactured on the m dedicated machines. The objective is to find a solution to minimize the sum of the maximum delivery completion time of the accepted orders and the penalty of the rejected orders which is determined by a submodular function. We design an LP rounding algorithm with approximation ratio of n+1 for this problem.