# (44d) Mixed-Integer Models for Simultaneous Optimization of Safety Stock and Inventory Policies in Supply Chain Planning

Authors:
Johnson & Johnson
University of Buffalo
Linde plc
Carnegie Mellon University
Demand forecasts are prone to error. Holding safety stock is necessary to serve demands larger than forecasted while waiting on replenishment to arrive. Inventory policies provide a simple and effective strategy to manage inventories while taking into account the uncertainties of daily operation. In this paper we analyze four mathematical programming formulations for incorporating safety stock in supply chain planning models and extend them to consider dynamic demand and multi-echelon systems. Additionally, novel formulations to represent inventory policies are derived using disjunctive programming. The safety stock formulations considered are: 1) proportional to throughput, 2) proportional to throughput with risk-pooling, 3) explicit risk-pooling, and 4) guaranteed service time. The last one is especially suited for multi-echelon systems. Formulations 1 and 2 give rise to LP or MILP models, while formulations 3 and 4 yield NLP or MINLP models. The inventory policies considered are the (r,Q) and (s,S) policies. In the (r,Q) inventory policy an order for Q units is placed every time the inventory level reaches level r, whereas in the (s,S) policy inventory is reviewed in predefined intervals. If the inventory is found to be below level s, an order is placed to bring the level back to level S. Traditionally, these parameters are obtained from single echelon analysis, while in this paper they are optimization variables in supply chain models. The models proposed allow simultaneous optimization of safety, reserve, and base stock levels together with material flows in supply chain planning. The service level achieved with each of the formulations was evaluated using discrete event simulation. Real-world examples are included to illustrate the proposed approaches.

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