The　controllable　processing　times　(CPTs)　have　many　practical　applications,　enabling　the　length　of　the　job　processing　to　vary　within　an　interval　flexibly　with　additional　costs.　In　this　paper,　we　study　an　unrelated　parallel　machine　scheduling　problem　with　machine-　and　sequence-dependent　setup　times　and　a　special　case　of　CPTs　without　extra　costs.　The　objective　is　to　maximize　the　difference　between　the　sum　of　realized　state　of　processing　of　all　jobs　and　makespan.　To　solve　the　problem,　a　mixed　integer　programming　(MIP)　model　is　formulated　first　and　then　a　logic-based　Benders　decomposition　(LBBD)　method　is　developed,　in　which　the　master　problem　is　for　job　assignments　and　the　subproblem　is　furthered　decomposed　into　a　sequencing　problem　for　minimum　total　setup　times　on　each　machine　and　a　processing　time　determination　problem.　Several　LBBD-based　heuristics　are　also　employed　by　imposing　different　optimality　gaps　for　the　master　problem.　The　performance　of　the　MIP　formulation,　the　exact　LBBD　method　and　the　LBBD-based　heuristics　is　compared　through　extensive　computational　experiments.　The　results　demonstrate　that　the　exact　LBBD　method　with　the　preprocessing　procedure　which　generates　initial　feasible　solutions　and　cuts　is　effective　and　the　proposed　LBBD-based　heuristics　reduce　the　computation　time　significantly　at　the　expense　of　slightly　solution　quality　reduction.　©　2022　Elsevier　Ltd