Economic Order Quantity
EOQ · HARRIS (1913)
Logistics · Inventory · TacticalThe Economic Order Quantity is the oldest closed-form result in operations research: given a constant demand rate $D$, a fixed cost $S$ per order, and a holding cost $h$ per unit per unit time, what order quantity $Q$ minimises the total annual cost of ordering plus holding? Ford Whitman Harris (1913) derived the square-root formula $Q^* = \sqrt{2DS/h}$ — arguably the first quantitative inventory result. R. H. Wilson (1934) popularised it, and it has been the starting point of every inventory-management textbook since. This page presents the derivation, the optimal formula, an interactive cost-curve visualiser, and the principal extensions (quantity discounts, backorders, EPQ).
Why EOQ matters
The first closed-form in OR
Derivation & formula
From cost function to Q*
Assumptions
Classical EOQ assumes: constant deterministic demand $D$ (units/year); a fixed setup or ordering cost $S$ per order (independent of order size); a constant holding cost $h$ per unit per year; instantaneous replenishment; no stockouts; infinite horizon; no quantity discounts. Under these assumptions, an optimal policy orders a fixed quantity $Q$ whenever inventory reaches a reorder point (here, zero — zero lead time and no safety stock).
Notation
| Symbol | Meaning |
|---|---|
| $D$ | annual demand rate (units / year) |
| $S$ | fixed ordering / setup cost per order ($) |
| $h$ | holding cost per unit per year ($) |
| $Q$ | decision variable: quantity ordered per cycle |
| $TC(Q)$ | total annual cost as a function of $Q$ |
Total annual cost
Orders per year $= D/Q$, so annual ordering cost is $S \cdot D/Q$. Average on-hand inventory between replenishments is $Q/2$, so annual holding cost is $h \cdot Q/2$. Ignoring the (constant) purchase cost:
Optimal order quantity
$TC(Q)$ is convex; the first-order condition $dTC/dQ = 0$ gives:
Robustness near the optimum
The cost function is remarkably flat near $Q^*$: ordering $1.5 Q^*$ or $0.67 Q^*$ only raises total cost by ~8%. This is why EOQ survives real-world mis-estimation of $S$ and $h$.
Principal extensions
- Economic Production Quantity (EPQ) — finite production rate $P > D$: $Q^* = \sqrt{2DS / (h (1 - D/P))}$.
- EOQ with planned backorders — shortage cost $b$ per unit-year: $Q^* = \sqrt{2DS/h} \cdot \sqrt{(h+b)/b}$ with optimal stockout fraction $b/(h+b)$.
- Quantity discounts — piecewise-constant unit price; evaluate $TC$ at each discount break and at the in-interval $Q^*$.
- Stochastic demand — $(Q, r)$ continuous-review and $(s, S)$ periodic-review policies (Arrow-Harris-Marschak 1951; Scarf 1960).
- Time-varying demand — Wagner & Whitin (1958) dynamic-programming lot sizing.
Interactive calculator
Slide $D$, $S$, $h$ · watch $Q^*$ and the cost curve update
Key references
DOIs & permanent URLs