DOI: https://doi.org/10.20535/kpi-sn.2019.3.164804

### ACCURACY OF A HEURISTIC FOR TOTAL WEIGHTED COMPLETION TIME MINIMIZATION IN PREEMPTIVE SINGLE MACHINE SCHEDULING PROBLEM BY NO IDLE TIME INTERVALS

#### Abstract

**Background.** A special case of the job scheduling process is that when jobs are processed on a single machine, preemptions are allowed, and there are no idle time intervals. Despite the exact solution models are always much slower than the heuristics, laws of heuristic’s rapidness advantage and heuristic’s solution closeness to the exact solution are unknown. Such laws would be useful to estimate real benefits of solution approximation.

**Objective.** Issuing from the lack of knowledge in relationship between heuristics and exact solution models, the goal is to study statistical difference between them for the preemptive single machine scheduling problem by no idle time intervals.

**Methods.** The two well-known approaches are invoked – the rule of weighted shortest remaining processing period as a heuristic and the Boolean linear programming model as an exact model. The relative error of the heuristic is defined and then studied how it varies against increasing complexity of job scheduling problems. The heuristic’s rapidness gain is shown as well.

**Results.** The main issue with the heuristic’s accuracy can arise at a few jobs to be scheduled. Additionally to this, if a sequence of jobs is divided into the fewest parts, the heuristic’s accuracy becomes the lowest. The exception exists for the shortest sequences – when only two jobs are to be scheduled. As the number of jobs increases up off 6, the relative error expectedly decreases along with the dramatically growing heuristic’s rapidness advantage. Therefore, scheduling a long sequence of jobs is preferable. The top relative error of the heuristic can exceed 6 % for three to five jobs to be scheduled, when they are divided into the fewest parts.

**Conclusions.**Starting off six jobs, the heuristic’s accuracy averagely increases, by a fixed rate of randomness in processing periods, priority weights, and release dates, as the complexity of job scheduling problems increases. The rate of randomness influences inversely: if processing periods, priority weights, and release dates are more randomly scattered, the heuristic schedules more accurately. The exact approach is truly applicable for cases when three to five jobs are to be scheduled (in particular cases, when the number of job parts is constant and is 2, the upper number of jobs can be increased up to 10). For such cases, an approximate solution’s real loss (given by the heuristic) is the average relative error not exceeding 1.2 % for job scheduling problems with low rate of the randomness. If such a loss is not admissible, the exact approach will work instead.

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