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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?

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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 It won't just be $Q$ because (assuming uniform demand), units of stock will be withdrawn for use not only after the batch is completed but also while it is being produced.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.

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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

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