**Problem Solving Assignment (Newsvendor Approach, Revenue Management, and Service Levels and Lead Times in a Supply Chain)**

1. The Supermarket Store is about to place an order for Halloween candy. One best- selling brand of candy can be purchased at $ 2.40 per box and usually is sold for $

4.25 per box before and up to Halloween. After Halloween, all the remaining candy can be sold for $ 1.00 per box. Demand for the candy at the regular price is a random variable with the following discrete probability distribution:

0.25

0.15

Demand ( boxes ) |
Probability |
||||

8 |
0.25 |
||||

9 |
0.15 |
||||

10 |
0.20 |
||||

11 |
|||||

12 |

You are required to:

a. Complete the following table to determine the optimal order quantity (Q*).

Each entry in the table represents the profit made by the store for a given combination of demand and stocking quantity.

Probability

80.25

90.15

100.20

110.25

120.15

Demand in boxes |
Q=8 |
Q=9 |
Q=10 |
Q=11 |
Q=12 |

Expected profit |

b. Determine the optimal order quantity using the critical ratio. Does this order quantity correspond to the answer in part a?

2. Inn at Penn has 200 rooms. For regular-fare customers, rooms are priced at $300 per night while the rooms are priced at $700 per night for the high-paying customers who generally arrive at the last minute. The demand for such high fare customers is distributed normally with mean 60 and standard deviation 50. Assume that there is ample demand for regular-fare customers.

a. What should the protection level for the high fare be to maximize expected profit?

b. Suppose that Inn at Penn operates with the protection level of 80 rooms for high fare customers. On average, how many high-fare customers are turned away because of lack of rooms?

**c. **Suppose again that the Inn at Penn operates with the protection level of 80 rooms for high fare customers. What is the probability that there are at least 5 rooms left unoccupied?

3. JetRed Airways flies several daily flights from Philadelphia to Chicago. Based on historical data, the flight on Wednesday evening before Thanksgiving is always sold- out. However, there are usually no-shows so the airline decides to improve revenues by overbooking. The no-shows are Poisson-distributed with mean 8 and the airline estimates that the cost of bumping a passenger is about 10 times more than ticket price.

a. How many seats should JetRed overbook?

b. JetRed management is dreading bad publicity around Thanksgiving so it decides instead that it does not want to bump passengers more than 5% of the time. What is the maximum number of seats that JetRed can overbook?

c. Suppose 6 seats are overbooked. How many seats can JetRed expect to have empty, on average?

4. Jtrix Inc. offers high end, specialized testing services to contract manufacturers (CM)

of semiconductor equipment. Its weekly testing capacity is 125 hours, and it is sold in

1-hour blocks for $140/hour. The internal operating cost of its testing equipment is

$100/hour. Because of demand uncertainty induced by the recession, in the recent past, CMs have often cancelled their orders for testing at the last moment. Jtrix estimates that the number of cancelled hours can be approximated by a Poisson distribution with a mean of 14 hours (see the table below). Jtrix does charge a cancelation penalty of $40 per canceled hour (that is, if a CM books an hour but then cancels, they still pay Jtrix $40). If Jtrix ends up with unused capacity due to cancellations, it is unable to sell that capacity to another customer, but it also doesn’t incur the operating cost of $100/hour (that is, the $100 is incurred only when operating). If its available capacity is insufficient to meet the booked demand, Jtrix

has the option to cannibalize testing time from its sister facility. However, doing so is expensive for Jtrix, and increases its operating cost to $150/hour.

a. Suppose Jtrix decides to overbook by 10 hours each week, i.e., it sells exactly

135 hours of testing capacity each week. On average, how many hours of unused testing capacity will it have in a given week?

b. Suppose Jtrix decides to overbook its capacity. How many hours should Jtrix sell to CMs?

5. Hotel manager Basil Fawlty and his resourceful assistant Manuel run a 26 room hotel in a quaint little town called Tourkey on the eastern sea coast. A combination of Mr. Fawlty’s genial attitude and the absence of a respectable inn in the nearby vicinity imply that Mr. Fawlty enjoys ample demand at his low fare of $159 per night.

Manuel notes that some customers will walk into the inn requesting a room for that evening and they are willing to pay a high fare of $325 per night. Manuel knows this demand is variable. (In reality, this demand is Poisson distributed with mean 7.5). He suggests some rooms should be kept unsold to the low fare customers, so that they

can serve the high fare customers.

a. To maximize profits with Manuel’s plan, what is the booking limit that should be set for low fare customers?

b. On average, how many empty rooms will the hotel have under Manuel’s plan, if the protection level is 9 rooms?

c. Basil scoffs at the idea: “Empty rooms! A bird in hand is better than two in the bush. This brilliant idea of yours, Manuel, might work in Barcelona but definitely not at Tourkey. Let me run the hotel my way. We will sell to everyone who reserves in advance and ignore the walk‐in demand”. If Basil

has his way, what will be the hotel’s expected revenue?

d. Manuel replies: “If you are going to forgo the opportunity to sell to last minute customers, let’s at least accept more than 26 reservations for the evening.” Checking the data, Manuel observes that the number of “no-shows” is Poisson distributed with mean 2.75. (Recall, a “no-show” is when a customer makes a reservation but doesn’t show up to use the room that evening.) Manuel also notes that a $100 non-refundable deposit is required with all reservations. However, if the hotel does not have a room for a reservation holder, then they need to book that person in a B&B in the nearest town. They decide that in those cases they would refund the customer’s deposit, and pay for the customer’s stay in the B&B, which is $450. The customer would not be happy, but they are getting a free night, so Manuel figures that there would be no loss of good will. Finally, if they have an empty room due to a no-show, they also figure that they would not be able to fill the room with a last minute customer. What is the critical ratio they should use to choose an overbooking quantity to maximize revenue?

6. JetAirways flight from Philadelphia to Boston has 350 seats. The high fare on the flight is $780 and the restricted/low fare is $500. There is ample demand for the low fare class but high fare demand is uncertain. Demand for the high fare is normally distributed with mean 150 and standard deviation of 45. Further, the customers buy low fare tickets well in advance of high fare customers.

a. What is the optimal protection level for the high fare tickets?

b. What is the expected revenue (in thousands) from high fare passengers when a booking limit of 200 is selected for the low fare tickets?

c. The JetAirways Customers’ Bill of rights states that “Customers who are involuntarily denied boarding shall receive $1,200 in addition to a ticket refund.” The RM department notices that the number of no-shows is Poisson distributed with mean of 7.5. What is the maximum number of reservations in excess of plane capacity that the airline should accept?

7. Jim is a CFO of a mid-sized construction company. One of his key tasks is to ensure that the company has sufficient cash to pay its daily and hourly workers who are hired whenever need arises. The company operates all 365 days a year and Jim estimates that on any day his payroll is Normally distributed with mean $25,000 and standard deviation $7,000. Jim manages his cash reserves using the order-up-to model: each morning he places an order for cash with the bank and the armored vehicle arrives 2 days later with the money. It is company’s policy to pay all workers on the same day, but in rare cases when the company runs out of cash it pays workers as soon as possible while increasing their paycheck by 1% for each day of delay. For each

dollar carried the company incurs 15% cost annually which includes cost of capital and insurance.

a. Suppose Jim is interested in ensuring that the company immediately satisfies all employee salary requirements with a 95% probability. What is the corresponding order-up-to level?

b. Suppose Jim uses the order-up-to level of $150,000. One of these days the armored car is robbed. What is the probability that more than $30,000 is stolen?

8. Supreme Cola is a supplier of fountain equipment to restaurants, bars and cafeterias.

The fountain equipment is manufactured at their York PA plant site. A national distribution center (DC) for the fountain equipment is also maintained adjacent to the plant. Supreme has one common platform design to which they add various features and accessories to create 10 different product options. The lead time for manufacturing and delivering a batch of products to the distribution center is 2 weeks. They review inventory and order weekly. For product ACola, Supreme uses a Normal distribution with mean 25 and standard deviation 20 to model weekly

demand. Demands across weeks are independent. ACola sells for $15,000 and they enjoy a 50% gross margin. The annual holding cost for inventory in the DC is 25% of the product’s cost.

In the event that a customer order cannot be filled from the warehouse due to an out-of-stock situation, Supreme expedites the manufacture and

delivery of the item. It estimates that such expediting increases their cost by $770 per unit.

a. What order upto level should Supreme choose to minimize their inventory for

ACola while achieving at least a 99.25% in-stock probability?

b. Supreme uses an order up-to policy with a base stock level equal to 250 for ACola. What is the probability that Supreme will have more than 40 units on order of that product at the start of any given week?

9. An industrial company requires argon gas cylinders for its work. Weekly demand is normally distributed with a mean of 35 and a standard deviation of 15. Demands are independent across weeks. Orders are placed weekly and the lead time to receive an order is 1 week. They want to hold enough cylinders to ensure a 99.75% in-stock probability. If they use an order up-to model, what base stock level should they implement? (Do not round to an integer value, i.e., leave your response in decimal form.)

10. Radio Shack sells a 32GB flash drive. Weekly demand for the 32GB flash drive in one of their stores is Poisson distributed with a mean of 1.25. The store places orders weekly and there is a one week lead time to receive orders.

a. On average, how many units will the store have on order?

b. Suppose they operate with a base stock level of 4. What in-stock probability would they achieve?

11. Radio Shack sells the same 32GB flash drive on their e-commerce site which has a single distribution center. Daily demand at their e-commerce DC is forecasted to be normally distributed with a mean of 150 and a standard deviation of 75. The lead time to receive a replenishment at the distribution center from their supplier is 2 days.

They review their inventory and place orders every two days, on the same day that they receive deliveries of new inventory. They operate 7 days a week. If they were to implement an order upto model, what base stock level should they choose for the DC if they want to achieve a 99.3% in-stock probability? (Leave your answer in decimal form, i.e., no need to round to an integer value.)

12. A large toy company Müttel currently allows toy retailers to place orders with delivery in 2 weeks. The Gigantic Pocket Monster (Gipokmon) is a new toy that Müttel has introduced. Müttel charges a wholesale price of $10 for the Gipokmon. The manager of a small boutique toy retail company, TOYS-are-MINE, plans to sell the toy for $20 and incurs a holding cost of $0.1 per toy per week. At this price, demand per week for the toy at one of their stores is estimated to be Poisson distributed with a mean of 1.5 units. Assume that the backorder cost is equal to the product’s retail margin. Assume TOYS-are-MINE uses the order-up-to model to plan orders and deliveries to this store.

a. Suppose TOYS-are-MINE uses an order-up-to level of 10. After receiving their delivery for this week, they have 2 units on-hand. Last week’s order was for 5 units. How many units will they order this week?

b. Again, suppose they use an order-up-to level of 10 for this store. On average, how many units will this store have on-order?

c. Suppose an order-up-to level of 5 is established. What is the resulting in- stock probability?

d. Suppose an order-up-to level of 3 is established. What would be the expected end-of-period on-hand inventory of Gipokmons?