Cloud　computing　environments　impose　challenges　in　incorporating　the　maximum　degree　of　parallelism　constraints　into　Parallel　Batch　Machine　Scheduling　(PBMS).　Unlike　traditional　PBMS　that　considers　only　fixed　job　widths,　this　paper　studies　generalized　PBMS　with　malleable　jobs　that　allow　job　width　to　be　changed　during　the　job　execution,　provided　it　does　not　exceed　its　maximum　degree　of　parallelism.　We　propose　a　fast　O(nlogn)　approximation　algorithm　by　extending　the　state-of-the-art　PBMS　algorithm　by　setting　each　job’s　width　to　its　maximum　degree　of　parallelism,　where　n　is　the　number　of　jobs.　Due　to　the　unique　nature　of　malleable　jobs,　previous　ratio　proofs　are　inapplicable.　We　develop　new　proof　techniques　and　establish　that　our　algorithm　achieves　a　ratio　of　(4-2Bm),　where　B　and　m　denote　machine　capacities　and　numbers,　respectively.　Furthermore,　by　exploiting　the　relationship　between　maximum　job　demand　and　average　processing　capacity　per　processor,　we　refine　the　algorithm　and　achieve　an　improved　ratio　of　(4-4Bm)　while　preserving　O(nlogn)　runtime　complexity.　In　addition,　we　fine-tune　our　algorithm　to　achieve　a　ratio　of　(3-2Bm)　when　jobs　are　with　identical　release　times.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.　2024.