Wilson Formula

The Wilson formula is the closed-form solution to the classical Economic Order Quantity (EOQ) problem: Q* = sqrt(2DS/H). It gives the cost-minimising order quantity when annual demand D is deterministic and constant, the setup (order) cost S per order is fixed, and the per-unit-year holding cost H is fixed; the total annual cost TC(Q) = (D/Q) × S + (Q/2) × H is convex (U-shaped) and the closed-form is obtained by setting dTC/dQ to zero. Foundational: Harris (1913) *Factory* magazine first derived the formula; Wilson (1934) *Harvard Business Review* brought it into practice and lent it his name (the formula is also called Andler-Formel in the DACH region). At the optimum the annual order count is N* = D/Q* = sqrt(DH/2S), average inventory is Q*/2, and the optimal annual total cost is TC* = sqrt(2DSH). A practically important property is cost-flatness: the cost curve is flat around Q*, so a ±20% deviation from Q* gives only 2-5% extra cost and a ±50% deviation 10-25% — making the formula robust under input perturbation.