# How does the EBQ formular come out?

I figured out how the economic order quantity (EOQ) formula comes out. I know it is quite similar with economic batch quantity (EBQ), sometimes termed economic production quantity (EPQ). I just don't understand why they are different. What makes the difference in holding cost part?

The models underlying the economic order quantity (EOQ) and economic production quantity (EPQ) formulae assume (realistically in many circumstances) that an order for $Q$ units of a good is delivered all at the same time, whereas production of a batch of $Q$ units occurs over a period.

In the case of ordering, immediately after an order has been delivered the stock is $Q$, and the stock then falls gradually to zero when (assuming timely ordering) the next order is delivered. The average stock is therefore simply $Q/2$.

In the case of production, however, the stock rises during production of a batch, and falls once production of the batch has been completed. Suppose annual demand for use of the good is $D$ and annual production (if it were continuous) is $P$, with $P>D$. Immediately after completion of a batch the stock will be $Q(1-D/P)$. It won't just be $Q$ because (assuming uniform demand), units of stock will be withdrawn for use not only after the batch is completed but also while it is being produced.

Therefore, writing $K$ for the fixed order cost and $h$ for the unit holding cost per year, to find the EOQ we find $Q$ that minimises total annual cost $C$ where:

$$C = \frac{DK}{Q} + \frac{hQ}{2}$$

The first term on the right is the total of the fixed costs of placing orders, $D/Q$ being the number of orders in a year, and the second term is the total holding cost over a year.

To find the EPQ, however, we need to minimise (interpreting $K$ now as the fixed production setup cost):

$$C = \frac{DK}{Q} + \frac{hQ}{2}\Bigg(1-\frac{D}{P}\Bigg)$$

The respective economic quantities may then be found by differential calculus.

In a way EBQ (also known as EPQ) EPQ is an extension of the EOQ model, the difference being

• In EPQ the company will produce its own parts
• In EOQ parts are produced by another company

Note that if you produce stuff you need to store it somewhere, which comes at a cost. This is explicitly included in the EPQ model via the holding cost variable