Scheduling jobs that change over time
arxiv(2023)
摘要
We consider a 1-machine scheduling problem where the temperature of a job
rises during processing, and cools down when not being processed according to
given linear heating and cooling rates. No job's temperature is allowed to rise
above a given threshold, and no job's temperature can cool below 0. Another
crucial property of our problem is that jobs can be preempted an arbitrary
number of times, and even more, we allow that a job is processed for an
infinitely small amount of time. We consider two objectives: minimize the
makespan, and minimize the sum of completion times. Our results are as follows.
We show how to compactly represent a solution. Further, we prove that the
problem of minimizing the sum of completion times can be solved in polynomial
time by formulating it as a linear program, and deriving a structural property.
This result can be extended to hold for any number of machines. Further, we
show that a minimum makespan can be found in $O(n)$ time, even when heating and
cooling rates are job-dependent.
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