Scheduling jobs with equal processing times and time windows on identical parallel machines

Brucker, Peter; Kravchenko, Svetlana A.

doi:10.1007/s10951-008-0063-y

Cited by 41 publications

(20 citation statements)

References 3 publications

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“…2.1.10 In [10], the following linear programming formulation was proposed for problem P |r j , p j = p, D j |−:…”

Section: 16mentioning

confidence: 99%

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Parallel machine problems with equal processing times: a survey

Kravchenko

Werner

2011

J Sched

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The basic scheduling problem we are dealing with is the following. There are n jobs, each requiring an identical execution time. All jobs have to be processed on a set of parallel machines. Preemptions can be either allowed or forbidden. The aim is to construct a feasible schedule such that a given criterion is minimized. For a couple of problems of this type, recently the complexity status has been solved and several interesting results have been presented. In this paper, we survey existing approaches for the problem class considered.

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“…2.1.10 In [10], the following linear programming formulation was proposed for problem P |r j , p j = p, D j |−:…”

Section: 16mentioning

confidence: 99%

“…It has been shown in [10] that, to solve problem P |r j , p j = p, D j | C j , it is sufficient to solve the following linear programming problem: (2), (3), (4).…”

Section: 112mentioning

confidence: 99%

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Parallel machine problems with equal processing times: a survey

Kravchenko

Werner

2011

J Sched

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“…While this linear program is not totally unimodular, the authors of [6] were still able to round the fractional solution into an integer solution of the same cost.…”

Section: Previous Workmentioning

confidence: 99%

Finding Total Unimodularity in Optimization Problems Solved by Linear Programs

Dürr

Hurand

2009

Algorithmica

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A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally unimodular. Still, sometimes it is possible to build an integer solution with the same cost from the fractional solution. Examples are two scheduling problems [4,6] and the single disk prefetching/caching problem [3]. We show that problems such as the three previously mentioned can be separated into two subproblems: (1) finding an optimal feasible set of slots, and (2) assigning the jobs or pages to the slots. It is straigthforward to show that the latter can be solved greedily. We are able to solve the former with a totally unimodular linear program, from which we obtain simple combinatorial algorithms with improved worst case running time.

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“…In [1], a dynamic programming approach has been used. In [2,3], like in this paper, linear programming models have been used, however without the graph model applied in the following. In [5], the complexity of the algorithm given in [4] has been improved by using preprocessing.…”

Section: Introductionmentioning

confidence: 99%