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Question $39$ Hint.

In order to calculate total annual inventory cost, you must first calulate the optimal order size. $D=$ annual demand. $C=$ annual carrying. cost Co cannualordering cost

The optimal order quantity oceus at the point in Egure $13\mathrm{.}3$ where the total cost cune is at a minimum, which coincides exactly with the point where the carrying cost curve intersects the ordering cost curve. This enables us to deternine the optimal ralue of $Q$ by cquating the tho cost functions and solving for $Q$ :

$\frac{C_{0}D}{Q}=\frac{C_{c}Q}{2}$

$Q^2=\frac{2C_{0}D}{O_{lon}}$

$Q_{\text{e}\text{c}\text{t}}=\sqrt[\phantom{2}]{\frac{2C_{0}D}{C_{lon}}}$

Alternativel$),$ the optimal value of $Q$ can be determined by differentiating the total cost curve with respect to $Q,$ setting the resulting function equal to zero $($the slope at the minimum point on the total cost curve$),$ and solving for $Q$ :

$TC=\frac{C_{0}D}{Q}+\frac{C_{lon}Q}{2}$

$\frac{\text{d}\text{e}\text{l}\text{T}\text{C}}{\text{d}\text{e}\text{l}\text{Q}}=-\frac{C_{0}D}{Q^2}+\frac{C_{lon}}{2}$

$0=-\frac{C_{0}D}{Q^2}+\frac{C_2}{2}$

$Q_{e1}=\sqrt[\phantom{2}]{\frac{2C_{0}D}{C_{lon}}}$

The total minimum cost is determined by substituting the value for the optimal order size, $Q_{\text{o}\text{p}\text{t}},$ into the total cost equation:

$T{C}_{ma}=\frac{C_{0}D}{Q_{ct}}+\frac{C_c{Q}_{0p}}{2}$

$$2,168\mathrm{.}23$

$$65\mathrm{.}98$

$3.$$$4\mathrm{.}71$

$$2,970\mathrm{.}17$