Two-Machine Flow Shop Scheduling Problem Under Linear Constraints

Abstract

We introduce a two-machine flow shop scheduling problem under linear constraints (2-FLC problem in short), in which the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal is to determine the processing times of each job, and to schedule the jobs to the two-machine flow shop such that the makespan, i.e., the completion time of all the jobs is minimized. This problem can find applications in various areas, such as industrial production and advertising planning. We study the computational complexity and algorithms for the 2-FLC problem. Particularly, we show that although the two-machine flow shop scheduling problem can be solved in polynomial time, the 2-FLC problem is generally NP-hard in the strong sense. Then we consider the design and analysis of algorithms on various settings of the 2-FLC problem. In particular, we propose a polynomial time algorithm for the 2-FLC problem when there is a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial time approximation scheme (PTAS).