Question: Give an example of a real-world problem that fits the EOQ model. You can draw from your personal experience, previous job or internship, or an example you observed in the real world.

Part 1 - EOQ and Newsvendor Examples (10 points)
Question 1 (5 points)
Give an example (not mentioned in class) of a real-world problem that fits the EOQ model. You can draw from your personal experience, previous job or internship, or an example you observed in the real world. For this example, explain the variable unit cost, the fixed ordering cost, the holding cost, and the demand. If you can, provide values for these parameters. If you cannot find precise parameter values, explain how you would obtain these costs.

Question 2 (5 points)

Give an example (not mentioned in class) of a real-world problem that fits the Newsvendor model. You can draw from your personal experience, previous job or internship, or an example you observed in the real world. For this example, explain the decision, the uncertainty, the underage cost, and the overage cost. If you can, provide specific values for underage and overage costs. If you cannot find specific values, explain what data and other information you need to compute underage and overage costs.

Part 2 - Ordering GT Laptop Cases (20 points)
Question 1 (8 points)

The demand for GT-branded laptop cases in the campus bookstore is fairly regular throughout the year, at 200 cases per year. The wholesale cost of each laptop case is $50. Assuming a fixed shipping cost of $20/order and a holding cost of $2 per case per year, determine the optimal order quantity, re-ordering interval, and the yearly average costs using the EOQ model.

Question 2 (5 points)

Assume a year consists of 52 weeks. Also, assume a fixed 4-week lead time for an order (it takes four weeks for the supplier to deliver the cases). What is your on-hand inventory level when you place the order? This value is called the reorder point.

Question 3 (7 points)

Your supplier offers you a 2% per-unit discount for all cases if you order 100 cases or more. Would your optimal order quantity change in such a case? If so, what is the new recommended order quantity and reorder interval?

Part 3 - Sensitivity of the EOQ Model (20 points)
After graduating from Georgia Tech, you accept a job offer to work at BuzzBread, an Atlantan bakery chain created by your uncle George P. Burdell. Your uncle, who is an expert in “leading” organizations but not so much in handling day-to-day business operations, makes you responsible for managing the inventory of various bread and pastry ingredients.

Your first task is to set-up the reorder policy for the flour supply of BuzzBead’s Atlanta locations. You estimate the cost of flour to be about $1 per kg and the demand to be stable at 50,000kg per year. After analyzing data, talking to the managers of several BuzzBread bakeries, and accounting for your uncle’s “business intuition,” you estimate that:

The fixed ordering cost, K, of flour is between $50 and $70 per order.
The holding cost, h, of flour is between $0.02 per kg per year and $0.05 per kg per year.
Question 1 (10 points)

You decide to set-up a flour reordering policy according to the EOQ model. Given the fixed and holding costs parameter ranges, what is the range of possible optimal ordering quantities?

Question 2 (10 points)

You now examine how much money could be “left on the table” by misestimating the cost parameters. Specifically, let OPT_COST(K, h) be the minimum average holding and fixed ordering costs (i.e., the EOQ cost excluding the variable cost of $50,000 per year of flour) for parameters (K, h). Also, let COST(Q, K, h) be the average holding and fixed ordering costs for order quantity Q and parameters (K, h).

a) For ordering quantities in the range in Question 1, and for the parameters (K, h) in the range from the problem statement, what is the maximum value of the ratio COST(Q, K, h)/OPT_COST(K, h)? Note that, for a given Q, the parameters (K, h) should be the same in the numerator and the denominator of the ratio for a proper comparison. Hint: you only need to consider combinations of the extreme values for h, K, and Q.

b) How does this maximum ratio compare to the range of ordering quantities in Question 1?

Part 4 - Back to the Bagel Store (15 points)
In this part of the assignment, we revisit the bagel store from Assignment 5. Assume that you open a new store and the daily demand for bagels follows a Uniform distribution with a range [0, 475]. The wholesale price for a bagel is $2.00, and the retail price is $3.00. Any leftover inventory at the end of the day is donated for free. The dynamics are the same as in Assignment 5.

Question 1 (3 points)

Calculate:

The underage and overage costs.
The service level (also called the critical ratio) and optimal ordering quantity.
The expected sales, lost sales, expected leftover inventory, and expected profit.
Question 2 (4 points)

A local organization offers to purchase any leftover bagels you have for $1.00 per bagel (they then resell them to factories that have a night shift).

How do your underage and overage costs change? What is your new ordering quantity?
How many bagels do you expect to sell to the local organization at the end of the day?
What is your expected profit in this case?
Question 3 (8 points)

You are considering purchasing a small oven for your store. The oven allows you to “bake fresh bagels to order.” However, in-store baking is more expensive than sourcing from suppliers, and you estimate that each in-store baked bagel costs $2.50 to make and bake (in-store baked bagels are still sold for $3.00). The oven allows you to capture all demand: if demand exceeds what you ordered from your supplier at the start of the day, you can just “bake-to-order.” Conversely, if there are any leftover bagels, you can still sell them to the local organization for $1.00 per bagel.

If you purchase the oven, how many bagels would you order from your supplier at the start of the day?
If you purchase the oven, how many bagels do you expect to bake in-store each day? How many leftover bagels do you expect to sell to the local organization at the end of each day?
How much are you willing to pay for the oven per day?

Part 5 - Vaccines (20 points)
After graduating from Georgia Tech, Georgina Burdell started a company that manufactures and sells flu vaccines at a discount to underdeveloped countries' governments. Georgina decided to use her Operations Management skills to optimize the manufacturing process and reduce production costs. The optimized process will enable her to offer vaccines at a lower price than her competitors.

After some serious process optimization, Georgina reduces the vaccine production costs to $6 per dose (a "dose" is a vaccine for one person). However, because of capacity constraints, her company must manufacture the vaccine before the flu season. During the flu season, Georgina sells doses to governments for $20 per dose. The selling price is significantly lower than other suppliers in the market but is sufficient to cover her fixed costs such as salaries etc.

While the manufacturing process is optimized, Georgina realizes that she can do nothing about the world's uncertainty, where sometimes the flu season is mild and not all doses that are produced are sold – if a dose is not sold during the season then it is worthless and must be thrown out.

For the forthcoming flu season, Georgina estimates that her vaccine's demand follows the Normal distribution with a mean of 50 million doses and a standard deviation of 15 million (thus = 50 million and = 15 million).

Question 1 (7 points)

Find the profit-maximizing number of doses that Georgina should manufacture (i.e. find her newsvendor ordering quantity). What is her expected profit in this case?

Question 2 (7 points)

New regulations may come out soon. It stipulates that, for sanitary and environmental reasons, vaccine manufacturers must hire a specialized agency to assist with the disposal of unsold vaccines. The disposal fee is $1 per unsold vaccine. How would this change the optimal order quantity? Calculate the optimal order quantity and expected profits in this case.

Question 3 (6 points)

In 2012, there was a huge shortage of flu vaccines in the world. To improve public health, the World Health Organization (WHO) is considering incentives for Georgina’s company to encourage higher levels of vaccine production.

In particular, WHO would like Georgina to make enough vaccines so that the probability of not meeting the demand is 20%. One possible incentive is to purchase all unused doses of vaccine (a ’buy-back’ program). For example, if Georgina manufactures 50 million doses and sells 40 million, WHO will buy back each of the remaining 10 million doses for some price. There are no disposal fees in this case.

What price per unit must WHO offer for each unused dose to best incentivize Georgina to meet the 20% goal?

Part 6 - BuzzAir (15 points)
A group of recent Georgia Tech grads decides to open a new airline company called BuzzAir. They buy one airplane that has n seats. They estimate that two types of travelers will purchase tickets for a certain flight on a certain date:

Leisure travelers, who are willing to pay only the discounted fare $d ;
Business travelers, who are willing to pay the full fare $f (where f > d ).
After quite a bit of market research, they conclude that the number of leisure travelers requesting tickets for this flight will be greater than n for sure, while the number of business travelers requesting tickets is random. Please assume that leisure travelers always purchase their tickets before business travelers do. (In practice, this is roughly true, which is why airfares increase as the flight date gets closer).

BuzzAir wishes to sell as many seats as possible to business travelers since they are willing to pay more. However, since the number of such travelers is random and these customers arrive near the flight's departure date, a sensible strategy is for BuzzAir to allocate a certain number of seats Q for full fares and the remainder, n - Q for discount fares.

The discount fares are sold first: The first n-Q customers requesting tickets will be charged $d per ticket and the remaining (at most Q) customers will be offered the full price $f. Since leisure travelers are only willing to pay $d, they will decline to buy a full-fare ticket. Thus, if there are less than Q ticket requests from business travelers, some seats will not be sold (and BuzzAir regrets not selling them to leisure travelers). Conversely, it is possible that some of the seats sold to leisure travelers for $d could have been sold to business travelers who would have been willing to pay $f.

Question 1 (7 points)

Show that the problem of finding the optimal number of full-fare seats, Q, is equivalent to a newsvendor problem. Clearly define the underage cost, overage cost, and uncertainty. What is the critical ratio?

Question 2 (3 points)

Suppose that demand for full-fare seats is normally distributed with a mean of 40 and a standard deviation of 18 (thus = 40 and = 18). There are n=100 seats on the flight and the fares are d = $175 and f=$400. What is the optimal number of full-fare seats? (Fractional solutions are OK)

Question 3 (5 points)

For each situation below, explain how the underage cost and the overage cost will change. How will this affect the optimal quantity reserved for full-fare customers? (No need to recalculate the optimal quantity - a qualitative answer is sufficient)

Situation 1: The full-fare tickets are fully refundable and, with some probability, each business traveler will cancel his or her ticket at the last minute, too late for BuzzAir to re-sell the newly vacant seat.
Situation 2: Any unsold seats may be sold at the very last minute for a steeply discounted price, $s (s < d). These tickets are made available after the business travelers have requested tickets.
Part 7 - Bonus 20 points
Derive the lost sales formula for the normal distribution and the uniform distribution shown in the lecture.