Stochastic scheduling with optimal customer service
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Stochastic scheduling with optimal customer service V Portougal* and D Trietsch University of Auckland, Auckland, New Zealand Existing research in stochastic scheduling often ignores the need to achieve high service levels. Optimality is usually de®ned in terms of minimizing the expected makespan, with the intent to increase the expected utilization of the facility. We argue that this does not address the full rami®cations of stochastic variation. Instead, we should minimize our total cost, including losses due to the variation. This, we show, leads to focusing on optimal service level. Furthermore, we show how to compare the mean and the standard deviation of the makespan directly. While this method applies for any distribution, we demonstrate it speci®cally for the important special case where the makespan distribution is (at least approximately) normal. Finally, we show by simulation (i) that it is very important to take into account that high variation in individual operations causes increases both in the mean and the variance of the ®nal makespan; and (ii) that using the normal distribution results is a good approximation. Keywords: scheduling; stochastic processes
Introduction In this paper we address a jobshop or project scheduling problem for stochastic environments where jobs have random processing times. One commonly used objective function in the scheduling literature is to minimize the expected makespan. The explicit intent is to ®nish the jobs earlier. An associated implicit intent is to increase the expected utilization of the facility overtime. We argue that this does not address the full rami®cations of stochastic variation. Instead, we should minimize our total cost, including losses due to the variation. We propose a model that adds safety time to prevent frequent problems, and includes a penalty for delays that still occur. This, we show, implies that we have to maintain an optimal service level. Even with the optimal service level in place, high variation should be discouraged, and it is conceptually possible to compare variance reduction to mean makespan reduction. We show in detail how to do this. We also demonstrate, by simulation, that using the output of deterministic scheduling models with expected job durations as the input for planning purposes may cause low service level because it neglects part of the effect of job variation on the expected makespan: the expected makespan is higher than what such application of deterministic models suggests. Formally, we limit ourselves to pre-established (static) sequences that do not change based on actual performance and where each operation or task (ie part of a job) requires a single resource; that is, each schedule consists of a set of job sequences Ð one for each machine Ð and a planned *Correspondence: V Portougal, MSIS Department, University of Auckland, Private Bag 92019, Auckland, New Zealand. E-mail: [email protected]
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