T$-$shirts$-$R$-$us $($TRS$)$ is planning production for its five t$-$shirt styles that are popular in Spring. All five styles have demand that is normally distributed. The best$-$selling style has an expected demand of $38,000$ and a standard deviation of $6,500.$ Each of the other four styles has an expected demand of $9,000$ with a standard deviation of $3,000.$ Currently, all t$-$shirts are produced before the start of the season. Production cost is $$25$ per t$-$shirt, and they are sold for a wholesale price of $$39.$ Any unsold t$-$shirts at the end of the season are discounted to $$12,$ and they all sell at that price. It costs $$3$ to hold the t$-$shirt in inventory for the entire season if it does not sell.

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How many t$-$shirts of each type should TRS manufacture?

What is the expected profit from this policy?

How many t$-$shirts does TRS expect to sell at a discount?

TRS is considering the postponement of manufacturing using new machines. This will require the identical basic t$-$shirts to be made in advance and the final touches added later. This will increase production cost of each t$-$shirt by $$2\mathrm{.}50.$ How many t$-$shirts should TRS manufacture with postponement? What is the expected profit from this policy?

Another option is to produce the popular style without postponement and the other four styles using postponement. What is the expected profit under this policy?