A Pseudo-polynomial Algorithm for the Completion Time Variance Problem with Controllable Processing Times

Abstract

We investigate the problem of scheduling n independent jobs on a single machine, in which processing times p _i (i = 1,⋯ ,n) of the n jobs are decision variables whose values can be compressed between [l _i ,u _i ]. Past research mainly concerned with minimizing the sum of a first order function and a cost function for compression. In this paper, we consider a new model with the aim to minimize the sum of CTV (Completion Time Variance) and a linear cost for compression. The CTV problem, which is NP-hard, can be considered as a special case of our problem when l _i = u _i for all i. Up to now, there is no algorithm, except complete enumeration of all the schedules of the n processing times, to solve this problem. We show, in this paper, that if an agreeable condition is satisfied, then our problem can be solved by a pseudo-polynomial algorithm with time complexity bounded above by $O({n^2}{u_n}{U^2}),where U = {\Sigma _i}^n{ = _1}{u_i}.$