Abstract
This paper concerns networks of precedence constraints between tasks with random durations, known as stochastic task networks, often used to model uncertainty in real-world applications. In some applications, we must associate tasks with reliable start-times from which realized start-times will (most likely) not deviate too far. We examine a dispatching strategy according to which a task starts as early as precedence constraints allow, but not earlier than its corresponding planned release-time. As these release-times are spread farther apart on the time-axis, the randomness of realized start-times diminishes (i.e. stability increases). Effectively, task start-times becomes less sensitive to the outcome durations of their network predecessors. With increasing stability, however, performance deteriorates (e.g. expected makespan increases). Assuming a sample of the durations is given, we define an LP for finding release-times that minimize the performance penalty of reaching a desired level of stability. The resulting LP is costly to solve, so, targeting a specific part of the solution-space, we define an associated Simple Temporal Problem (STP) and show how optimal release-times can be constructed from its earliest-start-time solution. Exploiting the special structure of this STP, we present our main result, a dynamic programming algorithm that finds optimal release-times with considerable efficiency gains.
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Notes
- 1.
As in Bidot et al. [3], stability here refers to the extent that a predictive schedule (planned release-times in our case) is expected to remain close to the realized schedule.
- 2.
Knowing the distribution of D, we assume to be able to draw \(\mathcal {P}\).
- 3.
The reader can easily recognize the similarity of the proposed LP with a so-called Sample Average Approximation (SAA) of a stochastic optimization problem [14].
- 4.
Since \((s,t)\in \varLambda ^*\) implies \(\max _{p'} s_{jp'} - w = t_j\) for all j.
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Mountakis, K.S., Klos, T., Witteveen, C. (2017). Stochastic Task Networks. In: Salvagnin, D., Lombardi, M. (eds) Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2017. Lecture Notes in Computer Science(), vol 10335. Springer, Cham. https://doi.org/10.1007/978-3-319-59776-8_25
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