A ski jacket manufacturer has outsourced production to a supplier in Asia. The electronics

manufacturer sells three types of jackets $$ a high$-$end, a mid$-$range and a low$-$end jacket. For the

upcoming winter season, the demand forecast for the high$-$end jacket is normally distributed, with

a mean of $600$ and a standard deviation of $400,$ the demand forecast for the mid$-$range jacket has a

mean of $1000$ and a standard deviation of $500$ and the demand forecast for the low$-$end jacket has

a mean of $2000$ and a standard deviation of $600.$ The high$-$end jacket has a sale price of $$300,$ a

production cost of $$120$ and a salvage value of $$100.$ The mid$-$range jacket has a sale price of

$$200,$ a production cost of $$100$ and a salvage value of $$85.$ The low$-$end jacket has a sale price of

$$110,$ a production cost of $$70$ and a salvage value of $$50.$

$($i$)$ How many jackets of each type should the manufacturer order, if there are no capacity

constraints?

$($ii$)$ The supplier has available production capacity of only $3600$ jackets. What is the

expected profit if the manufacturer orders $600$ high end jackets, $1000$ mid$-$range jackets

and $2000$ low end jackets?

$($iii$)$ How many jackets of each type should the manufacturer order if the available capacity

is $3600?$ What is the expected profit?