Inventory Model $-$ I

Symbols

$D=$ expected total demand over given time horizon;

$q=$ quantity of stock ordered;

$h=$ unit holding cost; $,R=$ unit reorder cost

$C(q)=$ total cost $=$ Total holding cost $+$ Total reorder cost

$Q=$ optimal order size for minimum $C(q).$

Results so far

$C(q)=\frac{hq}{2}+R\frac{D}{q},Q=\sqrt[\phantom{2}]{\frac{2RD}{h}}$

Tasks

Under the optimal condition:

a$.$ Obtain algebraic expressions for:

No$.$ of orders placed. $N=$

$C(Q)=$

$($algebraically simplified$)$

b$.$ What do you observe about the two cost components at the optimal condition?

c$.$ Show that $C(q)$ is a minimum at $q=Q.$

d$.$ Suppose that total demand over the time horizon actually is four times as large as expected. What would be the new values of $Q$ and $N?$ Call them $Q_{\text{n}\text{e}\text{w}},N_{\text{n}\text{e}\text{w}}.$