EOQ Calculator (Economic Order Quantity)

EOQ comes from minimizing the total annual inventory cost TC(Q) = (D/Q) * S + (Q/2) * H, where D is annual demand, S is the fixed ordering cost per order, H is the holding cost per unit per year, and Q is the order size. The first term shrinks as Q grows (fewer orders mean less ordering cost) and the second grows linearly with Q (a larger order means a larger average inventory). Take the derivative dTC/dQ = -DS/Q^2 + H/2 and set it to zero. The marginal saving from one fewer order exactly equals the marginal cost of carrying one more unit. Solving for Q gives EOQ = sqrt(2 * D * S / H). The second derivative is positive, so this is a true minimum. A useful property of EOQ is that at the optimum, annual ordering cost equals annual holding cost.

The classic Wilson model is deliberately simple. It assumes demand is known, constant, and continuous over the year, lead time is zero (or deterministic and constant), the entire order arrives at once in a single batch, no quantity discounts or price breaks apply, no stockouts or backorders are allowed, and ordering and holding costs are stable and linear. In real warehouses none of these hold perfectly, yet EOQ is still useful because the total-cost curve is flat near the optimum. Picking an order size within 20% of EOQ in either direction usually leaves total cost within 2% of the minimum. That insensitivity is the reason EOQ has survived almost a century of much fancier inventory models.

Two extensions cover the most common real-world wrinkles. The EOQ-with-planned-backorders model (sometimes called the EOQ-B or EPB) allows stockouts at a per-unit-per-year backorder penalty B. The optimal order quantity is EOQ * sqrt((H + B) / B), and you intentionally run out for part of each cycle to lower holding cost. The quantity-discount model (Hadley and Whitin, 1963) checks every price break in turn: compute EOQ at the lowest unit cost, and if that quantity qualifies for the discount it is optimal; otherwise try the next price break up and order the smallest quantity that earns the discount, comparing total costs (now including the D * unitCost term) across all candidates. Both extensions preserve the square-root structure but require a comparison step the basic formula skips.

EOQ collapses whenever demand is not steady or shelf life is short. Lumpy or seasonal demand (think snow shovels or wedding dresses) breaks the constant-demand assumption, and a fixed Q is a poor fit for a chunky reorder pattern; use a periodic-review or MRP system instead. Perishables (food, vaccines, fashion) penalize carrying inventory at a non-linear rate as expiration approaches; the newsvendor model or a perishable-inventory model is the right tool. Very low or very high demand makes the square root brittle: EOQ may recommend ordering once every five years or several times a day, neither of which is operationally sensible. Items with strong supplier minimums, container or truckload constraints, or quantity discounts also need a constrained or stepped variant. Finally, in just-in-time and lean systems the goal is to drive S close to zero, which collapses EOQ to one-piece flow.

The square-root formula itself was first published by Ford W. Harris, a Westinghouse manufacturing engineer, in a 1913 article in Factory: The Magazine of Management titled How Many Parts to Make at Once. Harris wrote it as a tooling rule for setting batch sizes on machine shops. The formula stayed mostly buried until R. H. Wilson, a consultant who ran the inventory practice at A. T. Kearney, popularized it in a 1934 Harvard Business Review article as a general-purpose stock-control method. Wilson's article reached a mainstream business audience and tied the formula to broader inventory policy, so the modern label EOQ or Wilson formula stuck. Practitioners had been independently rediscovering the same square-root relationship for decades before either publication, but Harris and Wilson are the names that anchor it in the textbook.