(181d) Safety Stocks Revisited: Terminal Constraints for Closed-Loop Scheduling

Safety stocks have been used in supply chain (SC) management to prevent stockouts, and it has been shown that they also play an important role in scheduling, especially closed-loop scheduling. In the optimal solution for a scheduling model without any safety stock as terminal constraints, inventory levels tend to be very low at the end of horizon, so that the production, transition and inventory holding cost can be minimized ^[1]. However, if such a solution is implemented, the problem may be infeasible for the following horizon, because the initial inventory is almost depleted, and the demand at the start of the following horizon cannot be satisfied. In the SC literature, a safety stock is generally required for the inventory level of each product at the end of the horizon, based on the statistics of the lead time of SC arcs and demand rate of SC nodes ^[2]. In some scheduling works, the terminal inventory levels are required to be equal to the initial value or to one of the values in a cyclic solution, and these would lead to recursive feasibility ^[3]. However, all of the aforementioned approaches neglect the relationship of the inventory levels among products, or among materials in different stages (echelons). For instance, in a single-stage, two-product network, if the terminal level of one product is higher, we can possibly decrease the other. Thus, in this work, we present methods to generate terminal constraints based on the specific network structure. The proposed methods account for the interdependencies among products or materials in different stages. These terminal constraints are linear, which can be easily incorporated in any mixed integer programming (MIP) model.

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.

References:

[1] Lima, R. M.; Grossmann, I. E.; Jiao, Y. Long-term scheduling of a single-unit multi-product continuous process to manufacture high performance glass. Comput. Chem. Eng. 2011, 35(3), 554â??574.

[2] Eppen, G.D.; Martin, R.K. Determining safety stock in the presence of stochastic lead time and demand. Manag. Sci. 1988, 34(11), 1380-1390.

[3] Subramanian, K.; Maravelias, C.T.; Rawlings, J.B. A state-space model for chemical production scheduling. Comput. Chem. Eng. 2012, 47, 97-110.