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QUESTION

The stock holding cost for a product
 is \pounds45 per item per annum, and the cost of placing an order for
a replenishment is \pounds200.  Demand is steady and the annual
demand is 2000 items.  Shortages must not occur.  The purchase
cost depends on the number of items $Q$ in the order: the cost per
item is \pounds$c$, where $$c =\cases{50-Q/100&for $Q\le 500$;\cr
45&for $500 \le Q < 1000$;\cr 40&for $Q\ge 1000$.\cr}$$ Determine
the optimal order quantity.



ANSWER

The cost per annum is

$$K(Q_=\frac{sd}{Q}+\frac{1}{2}hQ+dc$$

First, find the optimal value of $Q$ in the range $0\leq
Q\leq500$.

$$K(Q)=\frac{sd}{Q}+\frac{1}{2}hQ+d\left[50-\frac{Q}{100}\right]$$

$\frac{dK(Q)}{dQ}=0$ gives
$\frac{-sd}{Q^2}+\frac{1}{2}h-\frac{d}{100}=0$, and

$$Q=\sqrt{\frac{2sd}{h-\frac{d}{50}}}$$

Substituting $s=200,\ d=2000,\ h=45$ we obtain

$$Q=\sqrt{\frac{2.200.2000}{45-40}}=400$$

Thus, for $0\leq Q\leq 500,\ k$ is minimized when $Q=400$, and

$$K(400)=\frac{200.2000}{400}+\frac{1}{2}.45.400+2000(50-4)=102000$$

The ECQ value is $Q+\sqrt{\frac{2sd}{h}}=133.33$. This shows that
for $500\leq Q<1000$, and for $Q\geq1000,\ K(Q)$ is minimized when
$Q=500$; for $Q\geq1000,\ K(Q)$ is minimized for $Q=1000$. We
already know that $K(500>K(400)$, so it remains to evaluate
$K(1000)$.

$$K(1000)=\frac{200.2000}{1000}+\frac{1}{2}45.1000+2000.40=102900$$

Thus the optimal order quantity is $Q=400$.





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