Problem $2\mathrm{.}3$

You are selling fish and can purchase fresh products at $$18$ per crate each morning from the Fish Market. During the day, you sell crates of fish to local restaurants for $$120$ each. Coupled with the perishable nature of your product, your integrity as a quality supplier requires you to dispose of each unsold crate at the end of the day. Your cost of disposal is $$2$ per crate. You have a problem, however, because you do not know how many crates your customers will order each day. To address this problem, you have collected several days' worth of demand data shown in the table below. You now want to determine the optimal number of crates you should purchase each morning.

$\backslash$table$[[$Demand$,4,5,6,7,8,9,10,11,12],[$Frequency$,3,2,5,1,6,7,8,5,1]]$

What is the under$-$stocking cost $Cu?$

What is the over$-$stocking cost Co$?$

What is the optimal initial order quantity?

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Problem $2\mathrm{.}4$

Each year the admissions committee at a top business school receives a large number of applications for admission to the MBA program and they have to decide on the number of offers to make. Since some of the admitted students may decide to pursue other opportunities, the committee typically admits more students than the ideal class size of $720$ students. You were asked to help the admissions committee estimate the appropriate number of people who should be offered admission. It is estimated that in the coming year the number of people who will not accept the admission offer is normally distributed with a mean $50$ and a standard deviation $21.$ Suppose for now that this school does not maintain a waiting list, that is$,$ all students are accepted or rejected.

$16.$ Suppose $750$ students are admitted. What is the probability that the class size will be at least $720$ students?

$17.$ There is a mutual agreement that it is about two times more expensive to have a student in excess of the ideal $720$ than to have fewer students in the class. What is the appropriate number of students to admit?

$18.$ A waiting list mitigates the problem of having too few students since at the very last moment there is an opportunity to admit some students from the waiting list. Hence, the admissions committee has revised its estimate: It claims that it is five times more expensive to have a student in excess of $720$ than to have fewer students accepted among the initial group of admitted students. What is your revised suggestion?

Part $3$: Inventory Management.

Problem $3\mathrm{.}1$

A mail$-$order firm has a warehouse. The weekly demand at the warehouse is $10,000$ units. The company purchases each unit of product at $$10.$ The annual holding cost of one unit of product is $25\%$ of its value. Each order incurs an ordering cost of $$1,000($primarily from fixed transportation costs$),$ and the lead time is $1$ week. Assume $50($NOT $52)$ working weeks in a year.

$19.$ What is the optimal order quantity $($EOQ$)$ for one single warehouse?

$20.$ What is the reorder point for a single warehouse?

$21.$ How much average inventory can the company now expect to hold?

$22.$ What is the annual inventory holding cost?

$23.$ What is the annual ordering cost?

$24.$ What is the total cost?

$25.$ On average, how long $-$ in weeks $-$ does a unit of product spend in the warehouse before being sold $($Hint: Little's Law$)?$

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