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• W1875341788 abstract "We consider the single-machine scheduling problem of minimizing the number of late jobs. We omit here one of the standard assumptions in scheduling theory, which is that the processing times are deterministic. Our main message is that in a number of cases the problem with stochastic processing times can be reformulated as a deterministic problem, which is solvable in polynomial time through the famous algorithm by Moore and Hodgson. We first review and reinterpret this algorithm as a dynamic programming algorithm. We then consider four problem classes with stochastic processing times. The first one has equally disturbed processing times, that is, the processing time consist of a deterministic part and a random component that is independently, identically distributed for each job. The jobs in the other three classes have processing times that follow: (i) A gamma distribution with shape parameter pj and scale parameter , where is common to all jobs; (ii) A negative binomial distribution with parameters pj and r, where r is the same for each job; (iii) A normal distribution with parameters pj and 2 j. In this scheduling environment, the completion times will be stochastic variables as well. Instead of looking at the expected number of on time jobs, we introduce the concept of a job being ‘stochastically on time’, that is, we qualify a job as being on time if the probability that it is completed by the deterministic due date is at least equal to a certain given minimum success probability. We show that in case of equally disturbed processing times we can solve the problem in O(nlogn) time through the algorithm by Moore and Hodgson, if we make the additional assumption that the due dates and the minimum success probabilities are agreeable, which encompasses the case of equal minimum success probabilities. The problems with processing times following a gamma or a negative binomial distribution can be solved in O(nlogn) time by Moore and Hodgson’s algorithm, even if the minimum success probabilities are arbitrary; based on these two examples, we characterize the properties that a distribution must possess to allow such a result. For the case with normally distributed processing times we need the additional assumption that the due dates and minimum success probabilities are agreeable. Under this assumption we present a pseudo-polynomial time algorithm, and we prove that this is the best we can hope for by establishing weak NPhardness. We also show that the problem of minimizing the weighted number of late jobs can be solved by an extension of the dynamic programming algorithm in all four cases; this takes pseudo-polynomial time. We further indicate how the problem of maximizing the expected number of on time jobs (with respect to the standard definition) can be tackled if we add the constraint that the on time jobs are sequenced in a given order." @default.
• W1875341788 created "2016-06-24" @default.
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• W1875341788 date "2005-01-01" @default.
• W1875341788 modified "2023-09-25" @default.
• W1875341788 title "Getting rid of stochasticity: applicable sometimes." @default.
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