Lagrangian relaxationwithcutgenerationforhybridflowshopschedulingproblemsto minimize thetotalweightedtardiness

In thispaper,weaddressanewLagrangianrelaxation(LR)methodforsolvingthehybridflowshop scheduling problemtominimizethetotalweightedtardiness.FortheconventionalLR,theproblemrelax- ing machinecapacityconstraintscanbedecomposedintoindividualjob-levelsubproblemswhichcanbe solved bydynamicprogramming.TheLagrangiandualproblemissolvedbythesubgradientmethod.In this paper,aLagrangianrelaxationwithcutgenerationisproposedtoimprovetheLagrangianboundsfor the conventionalLR.Thelowerboundisstrengthenedbyimposingadditionalconstraintsfortherelaxed problem. Thestatespacereductionsfordynamicprogrammingforsubproblemsarealsoincorporated. Computational resultsdemonstratethattheproposedmethodoutperformstheconventionalLRmethod without significantlyincreasingthetotalcomputingtime.