Outsourcing non-preemptive operations to a third-party facility

We consider the problem where a group of manufacturers outsource non-preemptive operations to a third-party, where non-preemptive operations means processing operations can not be interrupted when it starts, such as mould making, blast furnace iron making. The manufacturers book time windows announced by third-party with prices and schedule operations in the booked time window. The cost for each manufacturer consists of booking and work-in-progress costs. We address some optimal decision for manufacturers to book time windows and schedule their jobs in the booked time windows so as to minimize their individual cost.

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