On apairofjob-machineassignmentproblemswithtwostages

We considertwoassignmentproblemsinwhichanumberof jobs are assignedtothesamenumberof machines that operateinparallel,butintwostages.Theyareknownasthe`2-stagetimeminimizing assignment problemʹandthe`bi-leveltimeminimizingassignmentproblemʹ.Theseproblemshavethe following characteristicsincommon: • Each ofthemachinesprocessesonejob(non-preemptively,withouthelpofothermachines). • The job-machineassignmentsarepartitionedintotwosuccessivestagesofparallelprocessing. • The objectiveistominimizethemakespan,thesumoftheprimaryandthesecondarycompletion time. Both problemscanbesolvedby(aseriesof)assignmentproblems.Weimprovetherelatedcomputational complexities byapplyingreoptimization.Undersomeconditionsaquadraticcomplexityisderived. We introduceaparameterweighingtherelativeimportanceoftheprimaryandthesecondarycostper time unit.Theparametricproblemscanbesolved,forallparametervaluessimultaneously,withinthe same reducedtimecomplexitybounds. Scope andpurpose As itisoftenimportanttosolveproblemsquickly,itisessentialtoreducethecom- putational complexityofavailablealgorithms,asfaraspossible.Weconsidertwoproblemswhicharise when parallelschedulingisdoneintwosuccessivestages;theycanbetackledbysolvingaseriesof linear assignmentproblems.Weshowthattheycanbesolvedmoreefficiently,usingpropertiesofthe classical linearassignmentproblem. In practice,thecostpertimeunitinthetwostagesneednotbeequal.Aparametercontrollingthe ratio betweenthesecostsdefinesaparametricversionofeachproblem.Thealgorithmsofreducedtime complexity canbeextendedtotheseparametricproblemversions.