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QUESTION

The annual demand for a certain product is 6000 items. The stock
holding cost is \pounds30 per item per annum and the cost of
placing an order is \pounds2.25.  Each item costs \pounds50 to
purchase. However, a discount of 2.5\% is given if at least 500
items are purchased at the same time; alternatively, a discount of
5\% is given if at least 1200 items are purchased together.
Determine an optimal ordering policy.


ANSWER

We have $d=6,000,\ h=30,\ s=2.25$ and $c=50$. The ECQ value is

$$Q=\sqrt{\frac{2.\frac{9}{4}.6000}{30}}=30$$

The optimal order quantities are 30, 500 or 1200.

The annual cost is

$$K=\frac{sd}{Q}+\frac{1}{2}hQ+cd(1-\textrm{ discount })$$

For $Q=30,\
K=\frac{9}{4}.\frac{6000}{30}+\frac{1}{2}.30.30+50.6000=\pounds300,900.00$

For $Q=500,\
K=\frac{9}{4}.\frac{6000}{500}+\frac{1}{2}.30.500+300,000.\frac{97.5}{100}=\pounds300,027.00$

For $Q=1200,\
K=\frac{9}{4}.\frac{6000}{1200}+\frac{1}{2}.30.1200+300,000.\frac{95}{100}=\pounds303,011.25$

Thus $Q=500$ minimizes $K$.





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