Build-Up EOQ Model

Model Context

The Build-Up EOQ Model applies when inventory is replenished gradually over time rather than instantaneously. This scenario typically arises in production environments where items are manufactured at a constant rate $ p $, while demand depletes inventory at a constant rate $ d $.

$ Q $: Total order quantity
$ p $: Constant production rate
$ d $: Constant demand rate
$ t_p = \frac{Q}{p} $: Production time

Average Inventory Level

Inventory builds up during production and depletes simultaneously due to demand. The average inventory level is:

$$ \bar{Q} = \frac{p - d}{2} \cdot t_p = \frac{Q}{2} \left(1 - \frac{d}{p} \right) $$

Total Inventory Cost

The total annual inventory cost includes ordering and holding costs:

$$ T(Q) = \frac{D}{Q} \cdot A + \frac{Q}{2} \left(1 - \frac{d}{p} \right) \cdot h $$

Optimal Order Quantity

Minimizing $ T(Q) $ with respect to $ Q $ yields the optimal build-up EOQ:

$$ Q^* = \sqrt{ \frac{2AD}{h \left(1 - \frac{d}{p} \right)} } $$

This formula adjusts the EOQ to account for continuous production, reducing the effective holding cost.

Effective Holding Cost

Compared to the Simple EOQ Model, the effective holding cost in the build-up model becomes:

$$ h' = h \left(1 - \frac{d}{p} \right) $$

As $ d $ approaches $ p $, the effective holding cost decreases, reflecting reduced average inventory.