Efficient Exact Minimization of Total Tardiness in Tight-Tardy Progressive Single Machine Scheduling With Idling-Free Preemptions of Equal-Length Jobs

Abstract

Background. A schedule ensuring the exactly minimal total tardiness can be found with the respective integer linear programming problem. An open question is whether the exact schedule computation time changes if the job release dates are input to the model in reverse order. Objective. The goal is to ascertain whether the job order in tight-tardy progressive single machine scheduling with idling-free preemptions of equal-length jobs influences the speed of computing the exact solution. The Boolean linear programming model provided for finding schedules with the minimal total tardiness is used. Methods. To achieve the said goal, a computational study is carried out with a purpose to estimate the averaged computation time for both ascending and descending orders of job release dates. Instances of the job scheduling problem are generated so, that schedules which can be obtained trivially, without the exact model, are excluded. Results. Based on the three-dimensional barred plot of the relative difference between the averaged computation times, it has been shown that a possibility exists to find schedules more efficiently by manipulating the job order. For instance, schedules of 5 jobs consisting of two processing periods each are found on average by 14.67 % faster for the descending job order. In another example of 7 three-parted jobs, an optimal schedule is found on average in 69.51 seconds by the ascending job order, whereas the descending job order takes just 36.52 seconds to find it, saving thus 32.99 seconds. Conclusions. Scheduling a fewer jobs divided into a fewer job parts is executed on average faster by the descending job order. As the number of jobs increases along with increasing the number of their processing periods, the ascending job order becomes more efficient. However, the computation time efficiency by both job orders tends to be irregular.

Tight-Tardy Progressive Idling-Free 1-Machine Preemptive Scheduling by Heuristic’s Efficient Job Order Input

“…with a pseudorandom number  drawn from the standard normal distribution (with zero mean and unit variance), and function ( )   returning the integer part of number  (e. g., see [1,6]). In particular, the release dates can be given in ascending order as follows:…”

“…Thus, due dates (8) are not given in non-descending order if the job lengths have been occasionally generated in non-descending order. This is done so because in the case of when all inequalities (7) are simultaneously true, a schedule ensuring the exactly minimal total tardiness is found trivially, without resorting to any algorithm or model (see Theorem 1 in [8]).…”

“…Thus, due dates (8) are not given in non-descending order if the job lengths have been occasionally generated in non-descending order. This is done so because in the case of when all inequalities (7) are simultaneously true, a schedule ensuring the exactly minimal total tardiness is found trivially, without resorting to any algorithm or model (see Theorem 1 in [8]). By symmetrical reasoning, due dates (10) are not given in non-ascending order if the job lengths have been occasionally generated in non-ascending order: if all inequalities (9) are simultaneously true, an optimal schedule is found trivially as well owing to Theorem 2 in [8].…”

“…So, the job length is randomly generated between 2 and 17 [6]. When job lengths (19) and due date shifts (2) are properly generated by some N for the ascending job order input, i. e. inequality (6) holds and at least one of the inequalities in 7is violated, then an ascending order schedule by job lengths (8) for the ascending job order input. However, the version with generating the ascending job order input and descending job order input separately, independently of each other, is to be studied also.…”

“…The exact minimization of total tardiness is possible just for a few jobs whose processing periods are not very long [1,2]. Heuristics are the only means which capable of scheduling hundreds and thousands of jobs [3,4].…”