Sample Questions for In-Class Portion of Final Exam
QMETH 501 — Professor Hillier
These questions are representative of the type of question that might appear on the in-class portion of the final exam. However, the actual exam will not be this long (actual exam will likely include 3 or 4 questions). The actual exam may include questions on topics not covered by these questions (e.g., integer programming including the application of binary variables, etc.)—anything covered in class or the required readings that does not require the computer is fair game for inclusion in the in-class portion of the final exam.
1. Running the Farm (35 points)
A farm family owns 125 acres of land and has $40,000 in funds available for investment. Its members can produce a total of 7500 person-hours worth of labor. Any hours not spent farming can be spent in town working part time at the cannery earning $6/hour.
Cash income may be obtained from three crops and two types of livestock: dairy cows and laying hens. No investment funds are needed for the crops. However, each cow will require an investment outlay of $1200 and each hen will cost $9.
Each cow will require 1.5 acres of land, and 100 person-hours of work. Each cow will produce a net annual cash income of $1000 for the family. The corresponding figures for each hen are: no acreage, 0.9 person-hours of work, and an annual net cash income of $5. The chicken house can accommodate a maximum of 3000 hens, and the size of the barn limits the herd to a maximum of 32 cows.
Estimated person hours and income per acre planted in each of the three crops are
Soybeans / Corn / OatsPerson-hours of labor / 70 / 105 / 50
Net annual cash income ($) / 600 / 900 / 450
The family wishes to determine how much acreage should be planted in each of the crops, how many cows and hens should be kept, and how long to work in town (if at all), so as to maximize its net cash income. Formulate (but do not solve) the algebraic linear programming model for this problem.
2. Supersport Footballs, Inc. (35 points)
Supersport Footballs, Inc. has the problem of determining the best number of Pro (P), College (C), and High-School (H) models of footballs to produce in order to maximize profits. Constraints include production capacity limitations in each of three departments (cutting and dyeing, sewing, and inspection and packaging) as well as a contractual obligation with the N.F.L. that requires production of at least 1000 Pro footballs. The linear programming model of Supersport’s problem is shown below:
The above linear program was solved using the Excel Solver. The resulting spreadsheet (already solved), along with the Sensitivity Report is shown below:
For each of the following, answer the question as specifically and completely as is possible without re-solving the problem with the Solver. All problems are independent (i.e., any change made in one problem does not affect other problems).
a. What is the optimal solution and its profit.
b. Suppose the profit per college football decreases to $4. Will this change the optimal solution (P, C, and H)? What can be said about the change in profit?
c. Suppose the profit per pro football increases to $5. Will this change the optimal solution (P, C, and H)? What can be said about the change in profit?
d. Suppose an additional sewing machine can be leased for $500. This would add 1000 hours of available minutes in the sewing department. Should they lease the sewing machine? Explain.
e. Suppose the contract that Supersport has with the NFL changes so that now only 900 footballs need to be produced per year. Will this change the optimal solution (P, C, and H)? What can be said about the change in profit?
f. While producing this exam, Pepsi was accidentally spilled on the “Allowable Decrease” for the Cutting constraint. What value should be there. Explain why.
g*. Suppose some of the sewing machines break down, so that they lose 4000 of the available minutes in the sewing department. If they don’t replace the sewing machines, what can be said about the new optimal solution (P, C, and H) and its profit? If they can replace all of the sewing machines for $1000, should they?
3. Airline Ticket Purchasing. (25 points)
In the course of doing business, your company incurs significant costs from airline travel to meet with business associates in other cities. Due to highly unpredictable schedules on the part of these business associates, scheduled meetings often get canceled or moved to a new date and time. As a result, your company has had a policy of purchasing full-fare coach tickets. These tickets do not have any advance-purchase requirements, and are fully refundable if they are not used.
Due to recent increases in the cost of full-fare tickets, your company has decided to reconsider the possibility of using 21-day advance-purchase tickets for travel. These tickets must be purchased at least 3 weeks before the initial travel date, and are non-refundable. As a result, if a meeting gets rescheduled after such a ticket is purchased, the company may or may not be able to use the ticket. In some cases, the meeting will be rescheduled in such a way that the ticket can be re-issued (with new travel dates) for a moderate fee. In other cases, the new meeting date does not allow for the ticket to be re-issued, and the ticket is wasted. At this point the company would simply purchase a full-fare coach ticket for the new meeting date.
You have decided to evaluate the choice between a full-fare ticket and a 21-day advance-purchase ticket for a typical meeting. The cost of a full-fare coach ticket is $900, while the cost of a 21-day advance-purchase ticket is $500. The fee for re-issuing an advance-purchase ticket is $75. When travel plans are made 3 weeks ahead of a scheduled meeting, the probability that the meeting will be changed is estimated to be 0.4. If this occurs, the probability that your company can get the ticket re-issued to accommodate the new date is 0.5.
a. Construct a decision tree to help the company decide which type of ticket to purchase for this client. Solve the tree assuming that the company wants to minimize the expected cost of attending the meeting.
b. Suppose that, in addition to the two types of tickets considered in part a), the company would also like to consider a 14-day advance-purchase ticket. Suppose that when travel plans are made 2 weeks ahead of a scheduled meeting, the probability that the meeting will be changed is estimated to be 0.2. If this occurs, the probability that your company can get the ticket re-issued to accommodate the new date is 0.5. (The fee for re-issuing this type of ticket is also $75.)
What is the highest price that you would be willing to pay for such a ticket? (Perform and show whatever analysis is necessary to support your answer.)
4. New Watch Introduction. (40 points)
The Hip-Time Watch Company has decided to introduce a new inexpensive, fashionable watch targeting the teenage market. Hip-Time must now decide how much production capacity to plan for.
One option is to choose a low production capacity (20,000 units per year). Doing so would require an $80,000 investment in production equipment. At the end of the first year, the company has the option of expanding production capacity to a new total of 40,000 units per year. This would require an additional $80,000 investment. Alternately, the company could simply choose a high production capacity (40,000 units per year) at the beginning of year one. Doing so would require a $120,000 investment in production equipment.
Due to the highly volatile nature of watch fashions in this particular market, Hip-Time believes that it will only be able to sell this particular watch for the next two years. (Assume zero salvage value for the equipment at the end of two years.) In each of these two years, the company believes that demand for the new watch will be either low (20,000 units) or high (40,000 units). The probability that demand will be high in the first year is estimated to be 0.6. In the second year, the probability of high demand is estimated to be 0.8 if demand was high in the first year, and the probability of high demand is estimated to be 0.2 if demand was low in the first year.
In either year, the number of units of this watch that Hip-Time will sell is equal to the smaller of demand and production capacity. E.g., if production capacity is 20,000 units and demand is 40,000 units, the company can only sell 20,000 units. Each unit produced and sold earns Hip-Time a contribution margin of $10.
a. Construct a decision tree to help Hip-Time make the necessary decisions around production capacity for this new watch. Solve the tree to find the course of action that maximizes the expected two-year contribution to profit for the company. (Do not bother discounting future cash flows to reflect the time value of money—simply maximize the total expected contribution to profit over the next two years.)
b. Assuming that you are the Hip-Time manager responsible for the production-capacity decision, construct a utility function for this decision using the “equivalent lottery” method discussed in class. (Note: You will be graded on the process you use to construct the utility function, not the specific utilities you come up with.) Graph the utility function and use it to determine whether you are risk averse, risk indifferent, risk seeking, or none of these. Briefly justify your answer.
c. Use the utility function constructed in part b) to re-solve the problem facing Hip-Time, this time maximizing expected utility. (To save time and space, do this using your original tree in part a).)
5. Renting vs. Selling. (35 points)
Jerry Young is thinking about opening a bicycle shop in his hometown. He can open a small shop, a large shop, or no shop at all. Jerry feels that a large bicycle shop will earn $80,000 in the coming year if the market is favorable, but will lose $30,000 if the market is unfavorable. A small shop will return a $50,000 profit in the coming year in a favorable market and a $10,000 loss in an unfavorable market. At the present time he believes that there is a 50-50 chance that the market will be favorable in the coming year. If Jerry does not open a bicycle shop, he can take a job doing repairs in bike shop in a nearby town and earn $15,000 per year.
a. Construct a decision tree to help Jerry decide what to do. Solve the tree to determine the best course of action for Jerry, assuming he wishes to maximize his expected payoff for the coming year.
b. Jerry is considering hiring his old marketing professor to provide marketing research to support his decision. The professor has offered to predict the type of market that Jerry can anticipate (either favorable or unfavorable). Using 20 previous occasions when she has provided similar market research, the professor has compiled the following data regarding predictions and outcomes. She feels that this information is fairly representative of market situations like the one Jerry is facing.
Number of instancesPredict favorable / Predict unfavorable / Total
Market favorable / 8 / 2 / 10
Market unfavorable / 3 / 7 / 10
Total / 11 / 9 / 20
Using this information, compute the maximum amount that Jerry should be willing to pay his marketing professor to conduct the marketing research.
c. Compute the maximum that Jerry would be willing to pay to obtain a 100% accurate prediction of the market for the coming year.