▁ ▁Operation ▁ ▁must ▁be ▁processed ▁for ▁ ▁units ▁on ▁machine ▁. ▁ ▁J ▁ ▁Job ▁shop ▁problem . ▁Every ▁job ▁ ▁consists ▁of ▁ ▁operations ▁ ▁for ▁, ▁to ▁be ▁scheduled ▁in ▁that ▁order . ▁ ▁Operation ▁ ▁must ▁be ▁processed ▁for ▁ ▁units ▁on ▁a ▁dedicated ▁machine ▁ ▁with ▁ ▁for ▁. ▁ ▁Job ▁characteristics ▁▁ ▁The ▁processing ▁time ▁may ▁be ▁equal ▁for ▁all ▁jobs ▁( , ▁or ▁) ▁or ▁even ▁of ▁unit ▁length ▁( , ▁or ▁). ▁All ▁processing ▁times ▁are ▁assumed ▁to ▁be ▁integers . ▁In ▁some ▁older ▁research ▁papers ▁however ▁they ▁are ▁assumed ▁to ▁be ▁rational s . ▁▁▁▁ ▁for ▁each ▁job ▁a ▁release ▁time ▁is ▁given ▁before ▁which ▁it ▁cannot ▁be ▁scheduled , ▁default ▁is ▁ 0 . ▁▁▁ ▁This ▁is ▁an ▁online ▁problem . ▁Job s ▁are ▁revealed ▁at ▁their ▁release ▁times . ▁ ▁In ▁this ▁context ▁the ▁performance ▁of ▁an ▁algorithm ▁is ▁measured ▁by ▁its ▁compet itive ▁ratio . ▁▁▁ ▁for ▁each ▁job ▁a ▁due ▁date ▁is ▁given . ▁▁ ▁The ▁idea ▁is ▁that ▁every ▁job ▁should ▁complete ▁before ▁its ▁due ▁date ▁and ▁there ▁is ▁some ▁penalty ▁for ▁jobs ▁that ▁complete ▁late . ▁ ▁This ▁penalty ▁is ▁denoted ▁in ▁the ▁objective ▁value . ▁ ▁The ▁presence ▁of ▁the ▁job ▁characteristic ▁ ▁is ▁implicitly ▁assumed ▁and ▁not ▁denoted ▁in ▁the ▁problem ▁name , ▁unless ▁there ▁are ▁some ▁restrictions ▁as ▁for ▁example ▁, ▁assuming ▁that ▁all ▁due ▁dates ▁are ▁equal ▁to ▁some ▁given ▁date . ▁▁▁ ▁for ▁each ▁job ▁a ▁strict ▁dead line ▁is ▁given . ▁