A single machine scheduling with generalized and periodic due dates to minimize total deviation

Abstract

We consider a single-machine scheduling problem such that the due date is assigned not to a specific job, but to a job position, with identical intervals between consecutive due dates. We refer to the job as a large job if its processing time is greater than the interval, and as a small job, otherwise. The objective is to minimize the sum of the earliness and tardiness of each job. We analyze how the computational complexity changes depending on the number of large jobs, denoted nl. First, we show that problems with nl∈{3,4,…,n−1} and nl∈{0,1,n} are NP-hard and polynomially solvable, respectively, where n is the number of jobs. Note that the computational complexity of the case with nl=2 is open. Then, we show that, when idle time is not allowed, problems with nl∈{1,2,…,n−1} and nl∈{0,n} are NP-hard and polynomially solvable, respectively. Furthermore, we develop a heuristic for the problem with no idle time and verify its extremely good performance through numerical experiments.