8. (7 points) Doc Nix Productions sells specialty T-Shirts that are sold at a single basketball game each. Doc Nix is trying to decide how many to buy for an upcoming game. During the game itself, which lasts one evening, Doc can sell T-Shirts for $10 apiece. However, when the game ends unsold T-shirts have no value. Due to a licensing agreement, Doc needs to make sure that leftover shirts are destroyed using a method that ends up costing $0.50 for each unsold T- Shirt. It costs Doc $6.50 to buy a specialty T-Shirt from the supplier. The supplier he uses has a cost per shirt of $2. Doc estimates that demand is distributed as shown below. Demand Probability 300 350 400 450 500 550 0.05 0.15 0.20 0.10 0.30 0.20 a. (2 points) How many specialty T-Shirts should Doc Nix buy from the supplier to maximize his profits? Demand 300 350 400 450 500 550 300 1050 700 350 0 -350 -700 350 1050 1225 875 525 175 -175 400 1050 1225 1400 1050 700 350 450 1050 1225 1400 1575 1225 875 500 1050 1225 1400 1575 1750 1400 550 1050 1225 1400 1575 1750 1925 Expected Profit 1050 1199 1269 1234 1076 901 Cost of Purchase $6.50 Selling Price $10 Salvage Price -0.5 Underage Cost (C) Cost of Overage (C.) Critical Fractile F(z) Selling Price - Cost of Purchase = $10 - $6.5 = $3.5 Cost of Purchase - Salvage Value = $6.5 - (-$0.5) = $7 C/C + C = 3.5/3.5+ 7 = 0.33 Doc Nix should buy 400 t-shirts in order to maximize his profit to $1,269.