Derivation of economic order quantity

Economic order quantity is the order size that minimises the total annual inventory cost (ordering plus holding).

Assumptions

• Demand $D$ per year is known and constant.
• Lead time is constant and no shortage is allowed.
• Whole order of $Q$ units arrives at once.
• Ordering cost $S$ per order and holding cost $H$ per unit per year are constant.
• Price per unit does not depend on the quantity (no discount).

Derivation

Inventory falls uniformly from $Q$ to 0, so the average inventory is $Q/2$.

$$\text{Annual ordering cost} = \frac{D}{Q} S, \qquad \text{Annual holding cost} = \frac{Q}{2} H$$ $$TC = \frac{D}{Q}S + \frac{Q}{2}H$$

(The purchase cost $DC$ is constant and is left out of the minimisation.) For minimum cost, differentiate with respect to $Q$:

$$\frac{dTC}{dQ} = -\frac{DS}{Q^2} + \frac{H}{2} = 0 \;\Rightarrow\; Q^2 = \frac{2DS}{H}$$ $$\boxed{Q^* = \sqrt{\frac{2DS}{H}}}$$

The second derivative $2DS/Q^3$ is positive, so this is a minimum.

Ordering cost equals holding cost at EOQ

From $Q^{*2} = 2DS/H$, we get $\dfrac{D}{Q^*}S = \dfrac{Q^*}{2}H$ because both equal $\sqrt{DSH/2}$. Hence

$$TC_{min} = 2\sqrt{\frac{DSH}{2}} = \sqrt{2DSH}$$

Related results

Number of orders per year $N = D/Q^*$; time between orders $T = Q^*/D$; reorder point $= d \times L$ (daily demand times lead time). The total cost curve is flat near the EOQ, so small changes in $Q$ change cost very little.