(181d) Safety Stocks Revisited: Terminal Constraints for Closed-Loop Scheduling
We generalize our methods from simple network structures to the more complex ones. First, we focus on the single-unit, multi-product problem. By solving an auxiliary linear programming model, we obtain the frequencies of tasks in a campaign mode. Using these frequency numbers, we formulate the linear terminal constraints, and we present two different formulations. We also prove that these constraints ensure feasibility for the next optimization problem. Second, we generalize the first network to the single-stage, multi-unit, multi-product network, in which both homogeneous and heterogeneous units are considered. Third, we study the multi-stage single-product network. Due to the integrality requirement of batches, the original terminal constraints have round-down operators leading to nonlinearities. We show different options to linearize the constraints, either by introducing auxiliary integer variables or by writing stronger terminal constraints. Fourth, based on the logic of the aforementioned simpler networks, we discuss how to formulate terminal constraints for the multi-stage, multi-product network. For the simpler networks, we rigorously prove that the terminal constraints are valid, i.e., the constrained inventory levels ensure the feasibility for future horizons. For the complex networks, we present results from a case-study to show the effectiveness of the proposed terminal constraints. By recursively solving the scheduling model starting from different initial inventory levels and checking the feasibility, we characterize the true feasible region of the terminal inventory levels. Comparing the true feasible region to the feasible region from terminal constraints, we show that most, if not all, of the inventory levels satisfying the terminal constraints lead to recursive feasibility.
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