Question $3$B $(20$ Marks$)$

CWD sports center sells Mountain Bikes, which it orders from the USA. Because of shipping$$ and handling costs, each order must be for $20$ Bikes. Because of the time it takes to receive an$$ order, the company places an order every time the present stock drops to $10$ Bikes. It costs $$20$ to place an order. It costs the company $$100$ in lost sales when a customer asks for a Bike and$$ the warehouse is out of stock. It costs $$50$ to keep each Bike stored in the warehouse. If a$$ customer cannot purchase a Bike when it is requested, the customer will not wait until one$$ comes in but will go to a competitor. The following probability distribution for demand for$$ Bikes has been and the time required to receive an order once it is placed $($lead time$)$ has the$$ following probability distribution:$$

Lead time$$

$($weeks$)$

Probability

Demand$/$

week$$

Probability

$1$

$$

$0\mathrm{.}45$

$1$

$$

$0\mathrm{.}15$

$2$

$$

$0\mathrm{.}30$

$2$

$$

$0\mathrm{.}25$

$3$

$$

$0\mathrm{.}25$

$3$

$$

$0\mathrm{.}40$

$4$

$4$

$$

$0\mathrm{.}20$

The company has $15$ Bikes in stock. Orders are always received at the beginning of the week.$$ Note that a lead time of $2$ weeks imply that an order placed in week one will arrive in week $4.$

Required$$

a$)$ Construct the appropriate random number mappings for the random variables starting$$ with $\mathrm{.}00.(2\mathrm{.}5$ marks for demand and $1\mathrm{.}5$ mark for lead time$)$

b$)$ Simulate CWD$\text{'}$s ordering and sales policy for $20$ weeks. $(12$ marks$)$

c$)$ Compute the average cost of the policy$(4$ marks$)$

demand$$

$\mathrm{.}15$

$\mathrm{.}84$

$\mathrm{.}16$

$\mathrm{.}12$

$\mathrm{.}55$

$\mathrm{.}16$

$\mathrm{.}84$

$\mathrm{.}63$

$\mathrm{.}33$

$\mathrm{.}57$

$\mathrm{.}18$

$\mathrm{.}26$

$\mathrm{.}23$

$\mathrm{.}52$

$\mathrm{.}37$

$\mathrm{.}70$

$\mathrm{.}56$

$\mathrm{.}99$

$\mathrm{.}16$

$\mathrm{.}31$

lead time$$

$\mathrm{.}47$

$\mathrm{.}74$

$\mathrm{.}35$

$\mathrm{.}56$

$\mathrm{.}64$

$\mathrm{.}21$

$\mathrm{.}55$

$\mathrm{.}01$

$\mathrm{.}40$

Use the following headings$$

Month$$

OI$$

U R A I$$

RN$$

D$$

D F EI$$

SO$$

order$$

RN$$

lead$-$time IC$$

SOC OC TC$$