An exact analytical solution to the production inventory control problem

The full set of continuous, differential delay equations describing the inventory and orders for a typical industrial production control system are solved exactly for a step function surge in demand. The replenishment delay is explicitly included. The order rate is tuned with three parameters, which modulate the demand smoothing, the recovery of the inventory deficit, and the desired level of work-in-process. The analytical solutions are validated by comparison with numerical integration and confirm the instabilities and inventory deficits found elsewhere. Useful management strategies can be deduced, and the approach should be widely applicable in supply chains.