Stochastic Inventory Systems with Minimum Order Quantity

Yao Zhao, PhD

Professor in

Supply Chain Management

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In practice, suppliers (e.g., manufacturers and distributors) often require their customers (e.g., retailers) to order either none or at least a certain amount of units Minimum Order Quantity (MOQ). While MOQ provides a way for the suppliers to achieve economies of scale in production/transportation, it presents a substantial challenge for the customers to manage inventories efficiently. Unlike fixed ordering costs and fixed batch sizes, MOQ received much less attention in the literature.

My research in this area focuses on identifying and computing efficient inventory replenishment policies for firms to handle MOQ.


Zhao, Y., M.N. Katehakis (2006). On the Structure of Optimal Ordering Policy of Stochastic Inventory Systems with Minimum Order QuantityProbability in the Engineering and Informational Sciences 20: 257-270.

Abstract: This paper analyzes a single-product inventory system over a finite time horizon, and models this type of problem as a stochastic dynamic program.  A new concept, dubbed M-increasing functions, is introduced, which allows us to  characterize the optimal inventory policies everywhere in the state-space outside of a given region for each time period, and obtains simple bounds for these regions. The complex structure of the optimal policies is exemplified for a number of cases.

Zhou, B., Y. Zhao, M.N. Katehakis (2007). Effective Control Policies for Stochastic Inventory Systems with a Minimum Order Quantity and Linear CostsInternational Journal of Production Economics 106: 523-531.

Abstract: This paper proposes and analyzes a new heuristic policy, called (s, t) policy, for single-product inventory systems with MOQ constraints. The paper demonstrates via a numerical study that the best (s, t)policy has a performance close to that of the optimal policy, and it always outperforms the best feasible (s, S)policies which are commonly used in practice. On average, the performance improvements are significant.

Zhou, B., M.N. Katehakis & Y. Zhao (2009). Managing stochastic inventory systems with free shipping optionEuropean Journal of Operational Research 196: 186-197.