An efficient algorithm for minimizing earliness, tardiness, and due-date costs for equal-sized jobs

Department of Logistics2008-2009 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Refined conditions for V-shaped optimal sequencing on a single machine to minimize total completion time under combined effects

We address single machine scheduling problems for which the actual processing times of jobs are subject to various effects, including a positional effect, a cumulative effect and their combination. We review the known results on the problems to minimize the makespan, the sum of the completion times and their combinations and identify the problems for which an optimal sequence cannot be found by simple priority rules such as SPT (Shortest Processing Time) and/or LPT (Longest Processing Time). Typically, these are problems to minimize the sum of the completion times under a deterioration effect, and we verify under which conditions for these problems an optimal permutation is V-shaped (an LPT subsequence followed by an SPT subsequence). We demonstrate that previously used techniques for proving that an optimal sequence is V- shaped are not properly justified. We use the corrected method to describe a wide range of problems with a pure positional effect and a combination of a cumulative effect with a positional effect for which an optimal sequence is V-shaped. On other hand, we show that even the refined approach has its limitations

Heuristics for Multimachine Scheduling Problems with Earliness and Tardiness Costs

We consider multimachine scheduling problems with earliness and tardiness costs. We first, analyze problems in which the cost of a job is given by a general nondecreasing, convex function F of the absolute deviation of its completion time from a (common) unrestrictive due-date, and the objective is to minimize the sum of the costs incurred for all N jobs. (A special case to which considerable attention is given is the completion time variance problem.) We derive an easily computable lower bound for the minimum cost value and a simple "Alternating Schedule" heuristic, both of which are computable in O(N log N) time. Under mild technical conditions with respect to F, we show that the worst case optimality (accuracy) gap of the heuristic (lower bound) is bounded by a constant as well as by a simple function of a single measure of the dispersion among the processing times. We also show that the heuristic (bound) is asymptotically optimal (accurate) and characterize the convergence rate as O(N -2 ) under very general conditions with respect to the function F. In addition, we report on a numerical study showing that the average gap is less than 1% even for problems with 30 jobs, and that it falls below 0.1% for problems with 90 or more jobs. This study also establishes that the empirical gap is almost perfectly proportional with N -2 , as verified by a regression analysis. Finally, we generalize the heuristic to settings with a possibly restrictive due date and general asymmetric, and possibly nonconvex, earliness and tardiness cost functions and demonstrate its excellent performance via a second numerical study.multimachine scheduling, earliness and tardiness costs, heuristics, worst case, asymptotic and probabilistic analysis

Coupled task scheduling with convex resource consumption functions

We study a single machine scheduling problem with coupled tasks under limited resource availability. Each job comprises of two tasks which are separated by an exact amount of delay. We assume that the processing time of the initial task and the duration of the delay period are equal and the same for all jobs. The processing time of the completion task, however, is job-dependent and modelled as a convex function of the amount of resource the job is allocated. The scheduling objective consists of minimizing the makespan, subject to an upper-bound on the resource availability. We provide several properties of an optimal solution and develop an O(n2) time algorithm for the problem