The Fundamental Trade-Off
Every business faces the inventory dilemma: order too little and you pay frequent setup and shipping costs; order too much and capital sits idle on warehouse shelves. The Economic Order Quantity model, first formalized by Ford Harris in 1913, finds the sweet spot. It reveals that total cost follows a U-shaped curve — ordering cost falls as batch size increases while holding cost rises, and the minimum sits exactly at Q* = sqrt(2DS/H).
The Square Root Formula
The EOQ formula's square root is not a mathematical curiosity — it has profound practical implications. Because optimal quantity scales as the square root of demand, doubling your sales does not require doubling your inventory; it only needs a 41% increase. This sublinear scaling is why large retailers and warehouses achieve economies of scale in inventory management, and it explains the consolidation advantage in supply chains.
Sawtooth Dynamics
The classic inventory profile resembles a sawtooth wave: inventory jumps to Q* at each replenishment and linearly depletes to zero (or the reorder point) before the next delivery arrives. This simulation visualizes the sawtooth pattern over time, showing how order quantity, demand rate, and lead time interact. The reorder point determines when to trigger the next order so stock arrives just as inventory hits safety levels.
Beyond Classic EOQ
Modern inventory management extends EOQ in many directions: stochastic demand requires safety stock and service-level targets, quantity discounts create piecewise cost functions, multi-echelon supply chains optimize inventory at each stage simultaneously, and perishable goods add spoilage constraints. Yet the basic EOQ insight — that the optimal balances a per-order fixed cost against a per-unit-time holding cost — remains the cornerstone of inventory theory.