Abstract
A job schedule ensuring the exact minimum of total weighted tardiness can be found with the respective integer linear programming problem model, in which the infinity showing that the respective states are either forbidden or impossible is substituted with a sufficiently great positive integer. An open question is whether the substitute can be selected so that the computation time would be decreased. Thus, it is ascertained that, whichever job lengths and its priority weights are, substituting the infinity with just “a sufficiently great positive integer” is never recommended. To decrease the computation time on average, it is strongly recommended to select the infinity substitute as multiple maximum over finite decision variable weights in the exact model. It is sufficient to take 2 to 5 such maxima as the infinity substitute. However, the shortened computation time is not guaranteed for solving a single or few scheduling problems. It is only an expected benefit, which builds up as a few hundred scheduling problems are solved at least.
Highlights
A job schedule ensuring the exact minimum of total weighted tardiness can be found with the respective integer linear programming problem model, in which the infinity showing that the respective states are either forbidden or impossible is substituted with a sufficiently great positive integer
When it is scheduled on a single machine (1-machine), the model of exact minimization of total weighted tardiness is rendered to solving an integer linear programming problem (ILPP) involving the branch-and-bound approach [1, 11]
The ILPP model to find the exact minimum of total weighted tardiness contains infinities which are intended to show that the respective states are either forbidden or impossible
Summary
Total weighted tardiness is a measure which indicates a cumulative lag in a job schedule executed on a single or multiple machines [13, 7]. One of the most practically valuable scheduling problems is the tight-tardy progressive 1-machine scheduling by idling-free preemptions [11, 12] In this problem, release dates are set at non-repeating integers from 1 through the total number of jobs, and due dates are tightly set after the respective release dates, a few jobs still can be completed without tardiness. In order to decrease the computation time, the paper [10] strongly recommended to select the infinity substitute as less as possible for equal-length job scheduling without weights
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