$[$Q$5$: ROP$-$Q$]$ The supermarket that Mr$.$ SpongeBob goes to is facing a slightly different problem. Every day, the

supermarket will have to make a decision of how many apples to purchase from their supplier.

Ideally, they want to purchase exactly the number of apples they will be able to sell the next day. However,

demand for the next day is unknown. After some market research, they estimated that the demand of apple for

tomorrow is normally distributed with a mean of $200$ and a standard deviation of $60.$

Mr$.$ SpongeBob purchases apples at $$0\mathrm{.}60$ per apple. The selling price is $$1$ each apple; unsold apples are sold at

a discount price of $$0\mathrm{.}20$ per apple every evening. Some customers will choose to buy apples online if they come

into the supermarket for apples only to find out that they don$$t have any left. This results in a goodwill cost of

$$0\mathrm{.}20$ each apple.

a$)$ What are the overage and underage costs for apples?

b$)$ What is the optimal number of apples that the supermarket should order $($in order to maximize the total

expected profit$)?$

c$)$ The emerging online shopping channel is putting more pressure on the supermarket, which increases the

goodwill cost significantly. Does the optimal ordering quantity increase or decrease because of the increase

in goodwill costs? Explain why.

Wrong submission $=$ No credit. You may resubmit as late submission $($See policy in syllabus$).$

Combine$/$submit all your work in ONE FILE in the correct order to avoid $50\%$ penalty.