# Economic order quantity

Abbreviated EOQ. This is the level of inventory to order that minimizes the sum of holding and ordering costs. The inventory function, $LaTeX: I(t)$, is the following periodic sawtooth function, where $LaTeX: T$ is the time between orders, and $LaTeX: Q$ is the ordering quantity:

$LaTeX: I(t) = Q - dt \;$ for $LaTeX: \; 0 \le t \le T$,

where $LaTeX: d$ is the rate of demand (inventory units per time units), and $LaTeX: I(t) = I(t-T)$ for $LaTeX: t > T$. The inventory becomes zero at $LaTeX: T = Q/d$, which requires a new order of $LaTeX: Q$ units. The model is thus:

$LaTeX: \min \; (1/2) hdT + K/T$,

where $LaTeX: h$ is the holding cost (currency per time units), so $LaTeX: (1/2) hdT$ is the average holding cost, and $LaTeX: K$ is the fixed cost of ordering, so $LaTeX: K/T$ is the average ordering cost. The solution is $LaTeX: T^* = (2K/hd)^{1/2}$, which yields the Economic Order Quantity (EOQ): $LaTeX: Q^* = (2Kd/h)^{1/2}$. See the more general production scheduling problem.