Scheduling a single parallel-batching machine with non-identical job sizes and incompatible job families

We study the scheduling of jobs on a single parallel-batching machine with non-identical job sizes and incompatible families. Jobs from same family have processing time can be loaded into batch, as long batch size respects capacity. The objective is to minimize total weighted completion time; this common equivalent minimization in-process inventory when all release date. Our problem combines two classic combinatorial problems, namely bin packing scheduling. develop three new mixed-integer linear-programming formulations, an assignment-based formulation, time-indexed formulation (TIF), set-partitioning (SPF). also propose column generation (CG) algorithm for SPF, branch-and-price (B&P) CG-based heuristic. A heuristic framework based proximity search developed using TIF. examine how reduce (number variables) formulations in preprocessing step. SPF B&P solve instances non-unit unit durations optimality up 80 150 within reasonable runtime limits, respectively. proposed heuristics perform better than methods available literature problem.