Portfolio selection and scheduling on a single machine environment

We are concerned in this article with a decision problem of a company producing to order. The company receives commands with specific required delivery dates from different clients, and has to decide which commands should be accepted and which ones should be rejected. Each accepted command results in a benefit for the company, but this benefit is reduced by a financial penalty in case of tardy delivery. We study the static case of a portfolio of commands in a single resource environment. The objective is to determine the subset of the jobs to accept together with a schedule for them to maximize the total profit of the company, taking into account tardiness penalties. We present results on two cases of penalty functions, fixed and linear. For fixed penalties, we show that the problem can be seen as 1 ||Σω_iU_i problem. For linear penalties we propose a mixed integer formulation of the problem, and a fully polynomial time approximation scheme for a fixed sequence. We have also adapted 3 heuristics of the literature and give computational results.