This　study　investigates　the　unrelated　parallel　machine　scheduling　problem　with　release　dates　to　minimise　the　makespan.　The　solution　to　this　problem　finds　wide　applications　in　manufacturing　and　logistics　systems.　Due　to　the　strong　NP-hardness　of　the　problem,　most　researchers　develop　heuristics,　and　the　largest　instances　they　consider　are　limited　to　400　jobs.　To　tackle　this　problem,　we　develop　a　novel　mixed-integer　linear　program　(MILP)　with　significantly　fewer　integer　variables　than　the　state-of-the-art　ones.　The　proposed　MILP　does　not　rely　on　a　binary　sequence　variable　usually　used　in　the　existing　models.　To　deal　with　large-sized　instances,　a　new　three-stage　matheuristic　algorithm　(TSMA)　is　proposed　to　obtain　scheduling　decisions.　It　uses　a　dispatching　rule　to　sequentially　schedule　jobs　on　machines.　Then　a　reassignment　procedure　is　performed　to　reduce　the　makespan.　Finally,　it　employs　a　re-optimisation　procedure　based　on　the　proposed　MILP　to　perform　job　moves　and　exchanges　between　two　selected　machines.　We　conduct　numerical　experiments　on　1440　instances　with　up　to　3000　jobs　and　20　machines.　Our　results　first　clearly　indicate　that　the　proposed　model　significantly　outperforms　existing　ones.　Moreover,　the　results　on　large-sized　instances　show　that　the　proposed　TSMA　can　obtain　high-quality　near-optimal　solutions　in　a　short　computation　time.　©　2021　Informa　UK　Limited,　trading　as　Taylor　&　Francis　Group.