Discussion Questions and Problems

Discussion Questions

  1. 10-1 Compare the similarities and differences of linear and goal programming.

  2. 10-2 A linear programming problem was developed, and the feasible region was found. If the additional restriction that all variables must be integers were added to the problem, how would the size of the feasible region change? How would the optimal value of the objective function change?

  3. 10-3 List the advantages and disadvantages of solving integer programming problems by (a) rounding off and (b) enumeration.

  4. 10-4 How do the three types of integer programming problems differ? Which do you think is most common, and why?

  5. 10-5 What is meant by satisficing, and why is the term often used in conjunction with goal programming?

  6. 10-6 What are deviational variables? How do they differ from decision variables in traditional LP problems?

  7. 10-7 If you were the president of the college you are attending and were employing goal programming to assist in decision making, what might your goals be? What kinds of constraints would you include in your model?

  8. 10-8 What does it mean to rank goals in goal programming? How does this affect the problem’s solution?

  9. 10-9 Which of the following are NLP problems, and why?

    1. Maximize profit=3X1+5X2+99X3subject to        X1≥10                           X2 ≤5                           X3≥18

    2. Maximize cost=25X1+30X2+8X1X2subject to    X1                        ≥8   X1+             X2≥120.0005X1−X2=11

    3. Maximize Z=P1d1−+P2d2++P3+subject toX1+X2+d1−−d1+=300          X2+d2−−d2+=200X1+           d3−−d3+=100

    4. Maximize profit=3X1+4X2subject toX12−5X2≥83X1+4X2≥12

    5. Maximize cost=18X1+5X2+X22subject to4X1−3X2≥8X1+  X2≥18

      Are any of these quadratic programming problems?

Problems

  1. 10-10 Elizabeth Bailey is the owner and general manager of Princess Brides, which provides a wedding planning service in southwestern Louisiana. She uses radio advertising to market her business. Two types of ads are available—those during prime-time hours and those at other times. Each prime-time ad costs $390 and reaches 8,200 people, while the off-peak ads each cost $240 and reach 5,100 people. Bailey has budgeted $1,800 per week for advertising. Based on comments from her customers, Bailey wants to have at least two prime-time ads and no more than six off-peak ads.

    1. Formulate this as a linear program.

    2. Find a good or optimal integer solution for part (a) by rounding off or making an educated guess at the answer.

    3. Solve this as an integer programming problem using a computer.

  2. 10-11 A group of college students is planning a camping trip during the upcoming break. The group must hike several miles through the woods to get to the campsite, and anything that is needed on this trip must be packed in a knapsack and carried to the campsite. One particular student, Tina Shawl, has identified eight items that she would like to take on the trip, but the combined weight is too great to take all of them. She has decided to rate the utility of each item on a scale of 1 to 100, with 100 being the most beneficial. Each item’s weight in pounds and utility value are given below.

    ITEM 1 2 3 4 5 6 7 8
    WEIGHT 8 1 7 6 3 12 5 14
    UTILITY 80 20 50 55 50 75 30 70

    Recognizing that the hike to the campsite is a long one, a limit of 35 pounds has been set as the maximum total weight of the items to be carried.

    1. Formulate this as a 0–1 programming problem to maximize the total utility of the items carried. Solve this knapsack problem using a computer.

    2. Suppose item 3 is an extra battery pack, which may be used with several of the other items. Tina has decided that she will take item 5, a CD player, only if she also takes item 3. On the other hand, if she takes item 3, she may or may not take item 5. Modify the problem to reflect this, and solve the new problem.

  3. 10-12 Student Enterprises sells two sizes of wall posters, a large 3- by 4-foot poster and a smaller 2- by 3-foot poster. The profit earned from the sale of each large poster is $3; each smaller poster earns $2. The firm, although profitable, is not large; it consists of one art student, Jan Meising, at the University of Kentucky. Because of her classroom schedule, Jan has the following weekly constraints: (1) up to three large posters can be sold, (2) up to five smaller posters can be sold, and (3) up to 10 hours can be spent on posters during the week, with each large poster requiring 2 hours of work and each smaller one taking 1 hour. With the semester almost over, Jan plans on taking a 3-month summer vacation to England and doesn’t want to leave any unfinished posters behind. Find the integer solution that will maximize her profit.

  4. 10-13 An airline owns an aging fleet of Boeing 737 jet airplanes. It is considering a major purchase of up to 17 new Boeing model 787 and 767 jets. The decision must take into account numerous cost and capability factors, including the following: (1) the airline can finance up to $1.6 billion in purchases; (2) each Boeing 787 will cost $80 million, and each Boeing 767 will cost $110 million; (3) at least one-third of the planes purchased should be the longer-range 787; (4) the annual maintenance budget is to be no more than $8 million; (5) the annual maintenance cost per 787 is estimated to be $800,000, and it is $500,000 for each 767; and (6) each 787 can carry 125,000 passengers per year, whereas each 767 can fly 81,000 passengers annually. Formulate this as an integer programming problem to maximize the annual passenger-carrying capability. What category of integer programming problem is this? Solve this problem.

  5. 10-14 Trapeze Investments is a venture capital firm that is currently evaluating six different investment opportunities. There is not sufficient capital to invest in all of these, but more than one will be selected. A 0–1 integer programming model is planned to help determine which of the six opportunities to choose. Variables X1, X2, X3, X4, X5, and X6 represent the six choices. For each of the following situations, write a constraint (or several constraints) that would be used.

    1. At least three of these choices are to be selected.

    2. Either investment 1 or investment 4 must be undertaken but not both.

    3. If investment 4 is selected, then investment 6 must also be selected. However, if investment 4 is not selected, it is still possible to select investment 6.

    4. Investment 5 cannot be selected unless investments 2 and 3 are both also selected.

    5. Investment 5 must be selected if investments 2 and 3 are both also selected.

  6. 10-15 Horizon Wireless, a cellular telephone company, is expanding into a new era. Relay towers are necessary to provide wireless telephone coverage to the different areas of the city. A grid is superimposed on a map of the city to help determine where the towers should be located. The grid consists of 8 areas labeled A through H. Six possible tower locations (numbered 1–6) have been identified, and each location could serve several areas. The following table indicates the areas served by each of the towers.

    TOWER LOCATION 1 2 3 4 5 6
    AREAS SERVED A, B, D B, C, G C, D, E, F E, F, H E, G, H A, D, F

    Formulate this as a 0–1 programming model to minimize the total number of towers required to cover all the areas. Solve this using a computer.

  7. 10-16 Innis Construction Company specializes in building moderately priced homes in Cincinnati, Ohio. Tom Innis has identified eight potential locations to construct new single-family dwellings, but he cannot put up homes on all of the sites because he has only $300,000 to invest in all projects. The following table shows the cost of constructing homes in each area and the expected profit to be made from the sale of each home. Note that the home-building costs differ considerably due to lot costs, site preparation, and differences in the models to be built. Note also that a fraction of a home cannot be built.

    LOCATION COST OF BUILDING AT THIS SITE ($) EXPECTED PROFIT ($)
    Clifton 60,000 5,000
    Mt. Auburn 50,000 6,000
    Mt. Adams 82,000 10,000
    Amberly 103,000 12,000
    Norwood 50,000 8,000
    Covington 41,000 3,000
    Roselawn 80,000 9,000
    Eden Park 69,000 10,000

    1. Formulate Innis’s problem using 0–1 integer programming.

    2. Solve with QM for Windows or Excel.

  8. 10-17 A real estate developer is considering three possible projects: a small apartment complex, a small shopping center, and a mini-warehouse. Each of these requires different funding over the next 2 years, and the net present values of the investments also vary. The following table provides the required investment amounts (in $1,000s) and the net present value (NPV) of each (also expressed in $1,000s):

    INVESTMENT
    NPV YEAR 1 YEAR 2
    Apartment complex 18 40 30
    Shopping center 15 30 20
    Mini-warehouse 14 20 20

    The company has $80,000 to invest in year 1 and $50,000 to invest in year 2.

    1. Develop an integer programming model to maximize the NPV in this situation.

    2. Solve the problem in part (a) using computer software. Which of the three projects would be undertaken if NPV is maximized? How much money would be used each year?

  9. 10-18 Refer to the real estate investment situation in Problem 10.17.

    1. Suppose that the shopping center and the apartment complex would be on adjacent properties and that the shopping center would be considered only if the apartment complex were also built. Formulate the constraint that would stipulate this.

    2. Formulate a constraint that would force exactly two of the three projects to be undertaken.

  10. 10-19 Triangle Utilities provides electricity for three cities. The company has four electric generators that are used to provide electricity. The main generator operates 24 hours per day, with an occasional shutdown for routine maintenance. Three other generators (1, 2, and 3) are available to provide additional power when needed. A start-up cost is incurred each time one of these generators is started. The start-up costs are $6,000 for 1, $5,000 for 2, and $4,000 for 3. These generators are used in one of the following ways: a generator may be started at 6:00 a.m. and run for either 8 hours or 16 hours, or it may be started at 2:00 p.m. and run for 8 hours (until 10:00 p.m.). All generators except the main generator are shut down at 10:00 p.m. Forecasts indicate the need for 3,200 megawatts more than provided by the main generator before 2:00 p.m., and this need goes up to 5,700 megawatts between 2:00 and 10:00 p.m. Generator 1 may provide up to 2,400 megawatts, generator 2 may provide up to 2,100 megawatts, and generator 3 may provide up to 3,300 megawatts. The cost per megawatt used per 8-hour period is $8 for 1, $9 for 2, and $7 for 3.

    1. Formulate this problem as an integer programming problem to determine the least-cost way to meet the needs of the area.

    2. Solve using computer software.

  11. 10-20 The campaign manager for a politician who is running for reelection to a political office is planning the campaign. Four ways to advertise have been selected: TV ads, radio ads, billboards, and social media advertising buys. The costs of these are $900 for each TV ad, $500 for each radio ad, $600 for a billboard for 1 month, and $180 for each buy on social media (approximately 40,000 unique impressions). The audience reached by each type of advertising has been estimated to be 40,000 for each TV ad, 32,000 for each radio ad, 34,000 for each billboard, and 17,000 for each social media buy. The total monthly advertising budget is $16,000. The following goals have been established and ranked:

    1. The number of people reached should be at least 1,500,000.

    2. The total monthly advertising budget should not be exceeded.

    3. Together, the number of ads on either TV or radio should be at least 6.

    4. No more than 10 ads/buys of any one type should be used.

    1. Formulate this as a goal programming problem.

    2. Solve this using computer software.

    3. Which goals are exactly met and which are not?

  12. 10-21 Geraldine Shawhan is president of Shawhan File Works, a firm that manufactures two types of metal file cabinets. The demand for her two-drawer model is up to 600 cabinets per week; the demand for a three-drawer cabinet is limited to 400 per week. Shawhan File Works has a weekly operating capacity of 1,300 hours, with the two-drawer cabinet taking 1 hour to produce and the three-drawer cabinet requiring 2 hours. Each two-drawer model sold yields a $10 profit, and the profit for the large model is $15. Shawhan has listed the following goals in order of importance:

    1. Attain a profit as close to $11,000 as possible each week.

    2. Avoid underutilization of the firm’s production capacity.

    3. Sell as many two- and three-drawer cabinets as the demand indicates.

    Set this up as a goal programming problem.

  13. 10-22 Solve Problem 10-21. Are any goals unachieved in this solution? Explain.

  14. 10-23 Hilliard Electronics produces specially coded computer chips for laser surgery in 64MB, 256MB, and 512MB sizes. (1MB means that the chip holds 1 million bytes of information.) To produce a 64MB chip requires 8 hours of labor, a 256MB chip takes 13 hours, and a 512MB chip requires 16 hours. Hilliard’s monthly production capacity is 1,200 hours. Mr. Blank, the firm’s sales manager, estimates that the maximum monthly sales of the 64MB, 256MB, and 512MB chips are 40, 50, and 60, respectively. The company has the following goals (ranked in order from most important to least important):

    1. Fill an order from the best customer for thirty 64MB chips and thirty-five 256MB chips.

    2. Provide sufficient chips to at least equal the sales estimates set by Mr. Blank.

    3. Avoid underutilization of the production capacity.

    Formulate this problem using goal programming.

  15. 10-24 An Oklahoma manufacturer makes two products: speaker telephones (X1) and pushbutton telephones (X2). The following goal programming model has been formulated to find the number of each to produce each day to meet the firm’s goals:

    Minimize   P1d1−+P2d2−+P3d3++P4d1+subject to              2X1+4X2+d1−+d1+=808X1+10X2+d2−−d2+=3208X1+6X2+d3−−d3+=240All Xi,di≥0

    Find the optimal solution using a computer.

  16. 10-25 Major Bill Bligh, director of the Army War College’s new 6-month attaché training program, is concerned about how the 20 officers taking the course spend their precious time while in his charge. Major Bligh recognizes that there are 168 hours per week and thinks that his students have been using them rather inefficiently. Bligh lets

    X1=number of hours of sleep needed per weekX2=number of personal hours (eating, personalhygiene, handling laundary, and so on)X3=number of hours of class and studyingX4=number of hours of social time off base(dating, sports, family visits, and so on)

    He thinks that students should study 30 hours a week to have time to absorb material. This is his most important goal. Bligh feels that students need at most 7 hours sleep per night on average and that this goal is number 2. He believes that goal number 3 is to provide at least 20 hours per week of social time.

    1. Formulate this as a goal programming problem.

    2. Solve the problem using computer software.

  17. 10-26 Mick Garcia, a certified financial planner (CFP), has been asked by a client to invest $250,000. This money may be placed in stocks, bonds, or a mutual fund in real estate. The expected return on investment is 13% for stocks, 8% for bonds, and 10% for real estate. While the client would like a very high expected return, she would be satisfied with a 10% expected return on her money. Due to risk considerations, several goals have been established to keep the risk at an acceptable level. One goal is to put at least 30% of the money in bonds. Another goal is that the amount of money in real estate should not exceed 50% of the money invested in stocks and bonds combined. In addition to these goals, there is one absolute restriction. Under no circumstances should more than $150,000 be invested in any one area.

    1. Formulate this as a goal programming problem. Assume that all of the goals are equally important.

    2. Use any available software to solve this problem. How much money should be put in each of the investment options? What is the total return? Which of the goals are not met?

  18. 10-27 Hinkel Rotary Engine, Ltd., produces four- and six-cylinder models of automobile engines. The firm’s profit for each four-cylinder engine sold during its quarterly production cycle is $1,800−$50X1, where X1 is the number sold. Hinkel makes $2,400−$70X2 for each of the larger engines sold, with X2 equal to the number of six-cylinder engines sold. There are 5,000 hours of production time available during each production cycle. A four-cylinder engine requires 100 hours of production time, whereas six-cylinder engines take 130 hours to manufacture. Formulate this production planning problem for Hinkel.

  19. 10-28 Motorcross of Wisconsin produces two models of snowmobiles, the XJ6 and the XJ8. In any given production-planning week, Motorcross has 40 hours available in its final testing bay. Each XJ6 requires 1 hour to test and each XJ8 takes 2 hours. The revenue (in $1,000s) for the firm is nonlinear and is stated as (Number of XJ6s)(4 − 0.1 number of XJ6s) + (Number of XJ8s)(5 − 0.2 number of XJ8s).

    1. Formulate this problem.

    2. Solve using Excel.

  20. 10-29 During the busiest season of the year, Green-Gro Fertilizer produces two types of fertilizers. The standard type (X) is just fertilizer, and the other type (Y) is a special fertilizer and weed-killer combination. The following model has been developed to determine how much of each type should be produced to maximize profit subject to a labor constraint:

    Maximize profit=12X+0.04X2+15Y−0.06Y2subject to  2X+4Y≤160hours             X,Y≥0

    Find the optimal solution to this problem.

  21. 10-30 Pat McCormack, a financial advisor for Investors R Us, is evaluating two stocks in a particular industry. He wants to minimize the variance of a portfolio consisting of these two stocks, but he wants to have an expected return of at least 9%. After obtaining historical data on the variance and returns, he develops the following nonlinear program:

    Minimize portfolio variance=0.16X2+0.2XY+0.09Y2subject to                         X+Y=1   (all funds must be invested)                   0.11X+0.08Y         ≥0.09   (return on the investment)                                                      X,Y≥0

    where

    X=proportion of money invested in stock 1Y=proportion of money invested in stock 2

    Solve this using Excel and determine how much to invest in each of the two stocks. What is the return for this portfolio? What is the variance of this portfolio?

  22. 10-31 Summertime Tees sells two very popular styles of embroidered shirts in southern Florida: a tank top and a regular T-shirt. The cost of the tank top is $6, and the cost of the T-shirt is $8. The demand for these is sensitive to the price, and historical data indicate that the weekly demands are given by

    X1=500−12P1X2=400−15P2

    where

    X1=demand for tank topP1=price for tank topX2=demand for regular T-shirtP2=price for regular T-shirt

    1. Develop an equation for the total profit.

    2. Use Excel to find the optimal solution to the following nonlinear program. Use the profit function developed in part (a).

    Maximize profitsubject toX1=500−12P1X2=400−15P2P1≤20P2≤25X1,P1,X2,P2≥0

  23. 10-32 The integer programming problem below has been developed to help First National Bank decide where, out of 10 possible sites, to locate four new branch offices: Xi represents Winter Park, Maitland, Osceola, Downtown, South Orlando, Airport, Winter Garden, Apopka, Lake Mary, and Cocoa Beach, for i equals 1 to 10, respectively.

    1. Where should the four new sites be located, and what will be the expected return?

    2. If at least one new branch must be opened in Maitland or Osceola, will this change the answers? Add the new constraint and rerun.

    3. The expected return at Apopka was overestimated. The correct value is $160,000 per year (i.e., 160). Using the original assumptions—that is, ignoring part (b)—does your answer to part (a) change?

  24. 10-33 In Solved Problem 10.3, nonlinear programming was used to find the best value for the smoothing constant, α, in an exponential smoothing forecasting problem. To see how much the MAD can vary due to the selection of the smoothing constant, use Excel and the data in Program 10.13A to find the value of the smoothing constant that would maximize the MAD. Compare this MAD to the minimum MAD found in the solved problem.

  25. 10-34 Using the data in Solved Problem 10-3, develop a spreadsheet for a two-period weighted moving average forecast with weights of 0.6 (w1) for the most recent period and 0.4 (w2) for the other period. Note these weights sum to 1, so the forecast is simply

         Forecast for next period = w1(Value incurrent period) + w2(Value in last period)

    Find the weights for this two-period weighted moving average that would minimize the MAD. (Hint: The weights must sum to 1.)

Case Study Schank Marketing Research

Schank Marketing Research has just signed contracts to conduct studies for four clients. At present, three project managers are free for assignment to the tasks. Although all are capable of handling each assignment, the times and cost to complete the studies depend on the experience and knowledge of each manager. Using his judgment, John Schank, the president, has been able to establish a cost for each possible assignment. These costs, which are really the salaries each manager would draw on each task, are summarized in the table on the next page.

Schank is very hesitant about neglecting NASA, which has been an important customer in the past. (NASA has employed the firm to study the public’s attitude toward the International Space Station.) In addition, Schank has promised to try to provide Ruth a salary of at least $3,000 on his next assignment. From previous contracts, Schank also knows that Gardener does not get along well with the management at CBT Television, so he hopes to avoid assigning her to CBT. Finally, as Hines Corporation is also an old and valued client, Schank feels that it is twice as important to assign a project manager immediately to Hines’s task as it is to provide one to General Foundry, a brand-new client. Schank wants to minimize the total costs of all projects while considering each of these goals. He feels that all of these goals are important, but if he had to rank them, he would put his concern about NASA first, his worry about Gardener second, his need to keep Hines Corporation happy third, his promise to Ruth fourth, and his concern about minimizing all costs last.

Each project manager can handle, at most, one new client.

Discussion Questions

  1. If Schank were not concerned about noncost goals, how would he formulate this problem so that it could be solved quantitatively?

  2. Develop a formulation that will incorporate all five objectives.

Source: Trevor S. Hale.

CLIENT
PROJECT MANAGER HINES CORP. NASA GENERAL FOUNDRY CBT TELEVISION
Gardener $3,200 $3,000 $2,800 $2,900
Ruth 2,700 3,200 3,000 3,100
Hardgraves 1,900 2,100 3,300 2,100

Case Study Oakton River Bridge

The Oakton River had long been considered an impediment to the development of a certain medium-sized metropolitan area in the Southeast. Lying to the east of the city, the river made it difficult for people living on its eastern bank to commute to jobs in and around the city and to take advantage of the shopping and cultural attractions that the city had to offer. Similarly, the river inhibited the access of those on its western bank to the ocean resorts lying 1 hour to the east. The bridge over the Oakton River had been built prior to World War II and was grossly inadequate to handle the existing traffic, much less the increased traffic that would accompany the forecasted growth in the area. A congressional delegation from the state prevailed upon the federal government to fund a major portion of a new toll bridge over the Oakton River, and the state legislature appropriated the rest of the needed monies for the project.

Progress in construction of the bridge has been in accordance with what was anticipated at the start of construction. The state highway commission, which will have operational jurisdiction over the bridge, has concluded that the opening of the bridge for traffic is likely to take place at the beginning of the next summer, as scheduled. A personnel task force has been established to recruit, train, and schedule the workers needed to operate the toll facility.

The members of the personnel task force are well aware of the budgetary problems facing the state. They have taken as part of their mandate the requirement that personnel costs be kept as low as possible. One particular area of concern is the number of toll collectors that will be needed. The bridge is scheduling three shifts of collectors: shift A from midnight to 8 a.m., shift B from 8 a.m. to 4 p.m., and shift C from 4 p.m. to midnight. Recently, the state employees union negotiated a contract with the state that requires that all toll collectors be permanent, full-time employees. In addition, all collectors must work a five-on, two-off schedule on the same shift. Thus, for example, a worker could be assigned to work Tuesday, Wednesday, Thursday, Friday, and Saturday on shift A, followed by Sunday and Monday off. An employee could not be scheduled to work, say, Tuesday on shift A followed by Wednesday, Thursday, Friday, and Saturday on shift B or on any other mixture of shifts during a 5-day block. The employees would choose their assignments in order of their seniority.

The task force has received projections of traffic flow on the bridge by day and hour. These projections are based on extrapolations of existing traffic patterns—the pattern of commuting, shopping, and beach traffic currently experienced with growth projections factored in. Standards data from other state-operated toll facilities have allowed the task force to convert these traffic flows into toll collector requirements—that is, the minimum number of collectors required per shift, per day, to handle the anticipated traffic load. These toll collector requirements are summarized in the following table:

Minimum Number of Toll Collectors Required per Shift
SHIFT SUN. MON. TUE. WED. THU. FRI. SAT.
A 8 13 12 12 13 13 15
B 10 10 10 10 10 13 15
C 15 13 13 12 12 13 8

The numbers in the table include one or two extra collectors per shift to fill in for collectors who call in sick and to provide relief for collectors on their scheduled breaks. Note that each of the eight collectors needed for shift A on Sunday, for example, could have come from any of the A shifts scheduled to begin on Wednesday, Thursday, Friday, Saturday, or Sunday.

Discussion Questions

  1. Determine the minimum number of toll collectors that would have to be hired to meet the requirements expressed in the table.

  2. The union had indicated that it might lift its opposition to the mixing of shifts in a 5-day block in exchange for additional compensation and benefits. By how much could the number of toll collectors required be reduced if this is done?

Source: Based on B. Render, R. M. Stair, and I. Greenberg, Cases and Readings in Management Science, 2nd ed., © 1990. Reprinted and electronically reproduced by permission of Pearson Education, Inc., New York, NY.

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