Single two-stage openshop and parallel two-stage openshops scheduling with optional job rejection under makespan constraint - Guo, Yanjie; Luo, Wenchang; Xu, Dehua (Computers & industrial engineering, 2026)
We consider a scheduling scenario on single two-stage openshop and parallel two-stage openshops where there is an upper bound on the makespan of the scheduled jobs. For each job, in the first version one will get a profit if the job is accepted to be processed while in the second version one will need to pay a cost if it is rejected to be processed. For the objective of maximizing the total profit of the accepted jobs, we first derive a polynomial time approximation scheme based on guessing strategy and linear program rounding technique for the case of single openshop and then we also propose a polynomial time approximation scheme for the case of a fixed number of openshops by more involved guessing strategy and linear program rounding technique. For the objective of minimizing the total cost of the rejected jobs, in the case of single openshop we address a polynomial time approximation scheme and in the case of a fixed number of openshops we develop a pseudo-polynomial time dynamic programming exact algorithm and present a super-optimal dual fully polynomial time approximation scheme. •The work is motivated by the outsourcing practice in make-to-order scenarios.•Both single openshop case and parallel openshops case are considered.•It focuses on profit maximizing and cost minimization rather than time oriented traditional criteria.•PTASes and super-optimal dual FPTAS on the considered problems are proposed.•It gives production managers more flexibility when handling job outsourcing issues.