Problem $1.(26$ points$)$ Newsvendor Inventory Model. A popular Halloween shop, SpookyShadows, is preparing for the Halloween season by stocking up on a highly sought$-$after costume, the "Ghostly Ghoul." Every year, demand for this costume surges in October but is uncertain and highly variable. The store can only order once and should order $6$ months before the Halloween season. The store must balance the risk of overstocking, which would lead to leftover inventory after Halloween, and understocking, which could result in missed sales. Each "Ghostly Ghoul" costume costs SpookyShadows $$25$ from the supplier. The selling price for each costume is $$80.$ After Halloween, any unsold costumes can only be sold at a $75\%$ markdown. The demand for the costume is normally distributed, with a mean of $1,200$ units and a standard deviation of $300$ units.

a$)(2$ points$)$ What is the cost of overage per costume?

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b$)(2$ points$)$ What is the cost of underage per costume?

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c$)(5$ points$)$ How many costumes should SpookyShadows order to maximize the store$$s expected profit for sales during Halloween season, i$.$e$.,$ what is the optimal order quantity?

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d$)(4$ points$)$ What is the expected inventory after the Halloween season ends if SpookyShadows order the optimal order quantity?

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e$)(3$ points$)$ What is the expected lost sales after the Halloween season ends if SpookyShadows order the optimal order quantity?

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f$)(4$ points$)$ What is the expected profit of SpookyShadows if they order the optimal order quantity?

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g$)(3$ points$)$ What is the maximum profit if there were no uncertainty in demand, i$.$e$.,$ demand can be matched with quantity perfectly?

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h$)(3$ points$)$ What is the mismatch cost because of uncertainty if SpookyShadows order the optimal order quantity?

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