Elsevier

Operations Research Letters

Volume 10, Issue 8, November 1991, Pages 467-472
Operations Research Letters

Proof of a conjecture of Schrage about the completion time variance problem

https://doi.org/10.1016/0167-6377(91)90024-JGet rights and content

Abstract

The problem of minimizing the completion time variance of n jobs on a single machine has been studied by several authors. We prove the correctness of a 1975 conjecture due to L. Schrage about the position of the third longest job in an optimal schedule.

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  • An almost exact solution to the min completion time variance in a single machine

    2021, European Journal of Operational Research
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    In this respect, Schrage (1975) characterized the first position of the optimal sequence, by proving that the longest job is always scheduled first. Hall and Kubiak (1991) characterized the second and last positions of the optimal sequence, verifying that the second largest job must be placed last and the third largest job must be situated in the second position. Similarly, bounds for the position of the smallest job in the CTV problem are established by Manna and Prasad (1999).

  • Completion time variance minimisation on two identical parallel processors

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    Discussion: Since start of job processing on a machine can be delayed, an optimal job sequence on any machine can be obtained as if we were solving a (1||CTV) problem. Hence, the proof follows from Theorem 1 in Hall and Kubiak (1991) and Theorem K in Merten and Muller (1972). Discussion: As the two longest jobs have to be assigned to each of the two machines, this is a direct extension of Property 3.

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