Minimizing the number of late jobs in a stochastic setting using a chance constraint

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. 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 present a new model to deal with the stochastic completion times, which is based on using a chance constraint to define whether a job is on time or late: a job is 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 have studied this problem for four classes of stochastic processing times. The jobs in the first three classes have processing times that follow: (i) A gamma distribution with shape parameter p j and scale parameter β, where β is common to all jobs; (ii) A negative binomial distribution with parameters p j and r, where r is the same for each job; (iii) A normal distribution with parameters p j and σ 2 j . The jobs in the fourth class have equally disturbed processing times, that is, the processing times consist of a deterministic part and a random component that is independently, identically distributed for each job. We show that the first two cases have a common characteristic that makes it possible to solve these problems in O(n log n) -mail: slam@cs.uu.nl tion that the due dates and the minimum success probabilities are agreeable. We show that under this assumption the third problem is N P-hard in the ordinary sense, whereas the fourth problem is solvable by Moore and Hodgson's algorithm.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 and when we require that the probability that a job is on time amounts to at least some given lower bound.

Combining two-stage stochastic programming and recoverable robustness to minimize the number of late jobs in the case of uncertain processing times

“…Lin and Wang (2007) and Hoogeveen and T'kindt (2012) present an O(n 2 ) algorithm, called the SPT-based algorithm, in which the jobs are added one by one to the on time set in shortest processing time order; if adding a job results in an infeasible set, then the job added last is removed. van den Akker and Hoogeveen (2004) present an overview of variants of the deterministic 1|| U j problem including release dates, deadlines, etc. Only a very limited amount of work has been published on the stochastic variant of the problem.…”

“…Only a very limited amount of work has been published on the stochastic variant of the problem. van den Akker and Hoogeveen (2008) and Trietsch and Baker (2008) consider the variant where the processing times follow some given probability distribution. Van den Akker and Hoogeveen show that when this probability distribution possesses some nice characteristics, then the problem of maximizing the number of jobs that are stochastically on time is solvable through Moore-Hodgson's algorithm; a job is stochastically on time if it satisfies the chance-constraint that the probability that it is on time in a given schedule is greater than or equal to a given minimum success probability.…”

Stable scheduling of single machine with probabilistic parameters

“…THEOREM 7. If the task completion times are independent random variables with erlang distribution E(α i , λ ), i ∈ J, then the expected value of tardiness (27) …”

Multi-criteria scheduling: an agent-based approach for expert knowledge integration

“…The first set has to be scheduled according to EDD to guarantee that L max remains less or equal zero while the second set can be scheduled arbitrarily. The more detailed review of this approach by van den Akker and Hoogeven [1] reveals that in fact Moore's algorithm is a dynamic algorithm with a special structure and that the EDD ordering is in any case crucial in the design of the algorithm.…”

“…Table 6 Properties of J 2 as problem instance for P m |d j |γ 1 0 4024 10 9 3948 19 25 2 1 4015 11 10 3940 20 28 3 2 4007 12 12 3934 21 31 4 3 3999 13 14 3927 22 34 5 4 3990 14 16 3921 23 37 6 5 3982 15 18 3915 24 40 7 6 3974 16 19 3914 25 43 8 7 3965 17 20 3908 26 47 9 8 3957 18 22 3902 27 52 Table 7 Maximum lateness (L max ) and total completion time (∑C j ) of all 34 Pareto-optimal solutions for job set J 1 .…”