EOQ Calculator

The Economic Order Quantity balances two costs that move in opposite directions as order size changes. Ordering cost per year falls as the batch grows, because annual demand D divided by order size Q gives fewer purchase events, each costing S. Holding cost per year rises with batch size, because average on-hand stock under a sawtooth inventory pattern is Q divided by 2, carried at H dollars per unit per year. Setting the derivative of the sum to zero yields Q equal to the square root of 2DS divided by H, the point where annual ordering cost and annual holding cost are exactly equal - a useful check on any EOQ result. The square root also explains why the answer is forgiving: total cost near the optimum is flat. The penalty for ordering a quantity Q instead of the optimum Q* is the ratio 0.5 times (Q/Q* + Q*/Q), so ordering 20 percent above the optimum costs 1.7 percent more than the minimum and ordering 20 percent below costs 2.5 percent more; the curve is slightly steeper on the low side. Inputs need care. H is normally derived as the unit value multiplied by a carrying rate, and that rate should be assembled from a firm's own figures - cost of capital applied to working capital, measured storage and handling cost, insurance, and an obsolescence allowance drawn from actual write-offs - rather than taken from a published rule of thumb, since it moves with interest rates and with how perishable or fashion-sensitive the item is. The cost of capital is usually the largest single component. S should include only costs that genuinely recur per order - requisition handling, receiving, inspection, invoice processing - not allocated overhead that would exist regardless. The classic model assumes level known demand, instant replenishment, no quantity breaks, no stockouts and one item in isolation, so it usually serves as a baseline that is then adjusted for price breaks, shared truckloads, shelf-life limits or minimum order quantities imposed by suppliers.

The formula was first published by Ford Whitman Harris, an engineer at Westinghouse, in a 1913 article in Factory: The Magazine of Management titled How Many Parts to Make at Once. Harris was addressing a manufacturing setup problem rather than a purchasing one: how large a production lot should be when each changeover carries a fixed cost. His derivation already contained the square-root structure used today. The paper was obscure for decades and the original citation was widely garbled, until Donald Erlenkotter tracked down the source and documented its provenance in the 1980s and 1990s. The result reached a wide business audience through R. H. Wilson, a consultant who promoted lot-size analysis in the 1930s, notably in a 1934 Harvard Business Review article, which is why the expression is still called the Wilson formula or the Wilson lot size in parts of Europe. Wartime operations research and the post-war management science literature then embedded it in a broader theory of inventory, alongside the reorder point and safety stock work of the 1950s. Extensions followed quickly: quantity discount variants, the economic production quantity for finite replenishment rates, models with planned backorders, and joint replenishment for families of items. From the 1970s onward the model was coded into MRP and ERP packages as a standard lot-sizing rule, which is how most practitioners encounter it now.