Minimizing total weighted tardiness on a single machine with release dates and equal-length jobs

In this paper we study the problem of scheduling n jobs with release dates, due dates, weights, and equal processing times on a single machine. The objective is to minimize total weighted tardiness. We formulate the problem as a time-indexed ILP after which we solve the LP-relaxation. We show that for certain special cases (namely when either all due dates, all weights, or all release dates are equal, or when all due dates and release dates are equally ordered), the solution for the LP-relaxation is either integral or can be adjusted in polynomial time into an integral one. For the general case we present a branching rule that performs well. Furthermore we show that the same approach holds for the m identical, parallel machines variant of the problem. Finally we show that with a minor modification the same approach also holds for the single-machine problems of minimizing the sum of weighted late jobs (1|rj,pj=p|∑wjUj) and the sum of weighted late work (1|rj,pj=p|∑wjVj) as well as their respective variants with m identical, parallel machines. We further show how we can solve these problems by applying column generation when there is not sufficient memory available to apply the direct ILP-approach.