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=== Minimizing the average completion time ===
'''Bruno, Coffman and Sethi'''<ref>{{Cite journal|last=Bruno|first=J.|last2=Coffman|first2=E. G.|last3=Sethi|first3=R.|date=1974-07-01|title=Scheduling independent tasks to reduce mean finishing time|url=https://doi.org/10.1145/361011.361064|journal=Communications of the ACM|volume=17|issue=7|pages=382–387|doi=10.1145/361011.361064|issn=0001-0782}}</ref> present an algorithm, running in time <math>O(\max(m n^2,n^3))</math>, for minimizing the average completion time on ''unrelated'' machines, '''R||'''<math>\sum C_i</math>.
Minimizing the ''weighted average'' completion time is NP-hard even on ''identical'' machines, by reduction from the [[knapsack problem]].'''<ref name=":0">{{Cite journal|last=Horowitz|first=Ellis|last2=Sahni|first2=Sartaj|date=1976-04-01|title=Exact and Approximate Algorithms for Scheduling Nonidentical Processors|url=https://doi.org/10.1145/321941.321951|journal=Journal of the ACM|volume=23|issue=2|pages=317–327|doi=10.1145/321941.321951|issn=0004-5411}}</ref>''' It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the [[partition problem]].<ref name=":3">{{Cite journal|last=Sahni|first=Sartaj K.|date=1976-01-01|title=Algorithms for Scheduling Independent Tasks|url=https://doi.org/10.1145/321921.321934|journal=Journal of the ACM|volume=23|issue=1|pages=116–127|doi=10.1145/321921.321934|issn=0004-5411}}</ref>
Minimizing the ''maximum'' completion time is NP-hard even for ''identical'' machines, by reduction from the [[partition problem]].
'''Horowitz and Sahni'''<ref name=":0" />''' presented:
* Exact [[dynamic programming]] algorithms for minimizing the ''maximum'' completion time on both uniform and unrelated machines. These algorithms run in exponential time (recall that these problems are all NP-hard).
* [[Polynomial-time approximation scheme|Polynomial-time approximation schemes]], which for any ''ε''>0, attain at most (1+ε)OPT. For minimizing the ''maximum'' completion time on two ''uniform'' machines, their algorithm runs in time <math>O(10^{2l} n)</math>, where <math>l</math> is the smallest integer for which <math>\epsilon \geq 2\cdot 10^{-l}</math>. Therefore, the run-time is in <math>O( n / \epsilon^2)</math>, so it is an [[FPTAS]]. For minimizing the ''maximum'' completion time on two ''unrelated'' machines, the run-time is <math>O(10^{l} n^2)</math> = <math>O( n^2 / \epsilon)</math>. They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case.
'''Glass, Potts and Shade'''<ref>{{Cite journal|date=1994-07-01|title=Unrelated parallel machine scheduling using local search|url=https://www.sciencedirect.com/science/article/pii/0895717794902054|journal=Mathematical and Computer Modelling|language=en|volume=20|issue=2|pages=41–52|doi=10.1016/0895-7177(94)90205-4|issn=0895-7177|doi-access=free}}</ref> compare various [[Local search (optimization)|local search]] techniques for minimizing the makespan on unrelated machines. Using computerized simulations, they find that [[tabu search]] and [[simulated annealing]] perform much better than [[Genetic algorithm|genetic algorithms]].
Versache and Wiese<ref>{{Cite journal|last=Verschae|first=José|last2=Wiese|first2=Andreas|date=2013-11-10|title=On the configuration-LP for scheduling on unrelated machines|url=https://link-springer-com.mgs.ariel.ac.il/article/10.1007/s10951-013-0359-4|journal=Journal of Scheduling|volume=17|issue=4|pages=371–383|doi=10.1007/s10951-013-0359-4|issn=1094-6136}}</ref> present a 2-factor approximation algorithm.
=== Maximizing the profit ===
'''Bar-Noy, Bar-Yehuda, Freund, Naor and Schieber'''<ref>{{Cite journal|last=Bar-Noy|first=Amotz|last2=Bar-Yehuda|first2=Reuven|last3=Freund|first3=Ari|last4=(Seffi) Naor|first4=Joseph|last5=Schieber|first5=Baruch|author5-link=Baruch Schieber|date=2001-09-01|title=A unified approach to approximating resource allocation and scheduling|url=https://doi.org/10.1145/502102.502107|journal=Journal of the ACM|volume=48|issue=5|pages=1069–1090|doi=10.1145/502102.502107|issn=0004-5411}}</ref> consider a setting in which, for each job and machine, there is a ''profit'' for running this job on that machine. They present a 1/2 approximation for discrete input and (1-''ε'')/2 approximation for continuous input.
== Extensions ==
'''Kim, Kim, Jang and Chen'''<ref>{{Cite journal|date=2002-06-01|title=Unrelated parallel machine scheduling with setup times using simulated annealing|url=https://www.sciencedirect.com/science/article/abs/pii/S0736584502000133|journal=Robotics and Computer-Integrated Manufacturing|language=en|volume=18|issue=3-4|pages=223–231|doi=10.1016/S0736-5845(02)00013-3|issn=0736-5845}}</ref> extend the problem by allowing each job to have a setup time, which depends on the job but not on the machine. They present a solution using [[simulated annealing]]. '''Vallada and Ruiz''' present a solution using a [[genetic algorithm]].<ref>{{Cite journal|date=2011-06-16|title=A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times|url=https://www.sciencedirect.com/science/article/abs/pii/S0377221711000142|journal=European Journal of Operational Research|language=en|volume=211|issue=3|pages=612–622|doi=10.1016/j.ejor.2011.01.011|issn=0377-2217|hdl=10251/35412|hdl-access=free|hdl=10251/35412}}</ref> present a solution using a [[genetic algorithm]].
'''Caragiannis'''<ref>{{Cite journal|date=2011-06-16|title=A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times|url=https://www.sciencedirect.com/science/article/abs/pii/S0377221711000142|journal=European Journal of Operational Research|language=en|volume=211|issue=3|pages=612–622|doi=10.1016/j.ejor.2011.01.011|issn=0377-2217|hdl=10251/35412|hdl-access=free}}</ref> extends the problem in a different way, by assuming that the jobs are owned by selfish agents (see [[Truthful job scheduling]]).
== External links ==
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