Doc$$s supplier is contemplating $$cutting out the middleman$($i$.$e$.,$ Doc Nix$)$ and offering its products only through a direct channel. The cost and demand information are the same as in question $8.$ In this case, please answer the following:

a$.$ How many specialty T$-$Shirts should the supplier produce to maximize profits?

Response: T$-$shirts

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b$.$ What is the supplier$$s expected profit?

Response: $ profit

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$2.$ Doc$$s supplier is also considering offering him a buyback contract with the same wholesale price.

c$.$ What is the optimal buyback amount $($i$.$e$.,$ b$)?$

Response: the optimal payback amount is $

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d$.$ What is Doc$$s expected profit with the buyback contract?

Response: the expected profit is $ profit

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e$.$ What is the supplier$$s expected profit with the buyback contract?

Response: the expected profit for the supplier is $

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Provide answer with detailed calculations