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1. The Otto Maddick Machine Tool Company produces two products, muffler bearings and torque amplifiers. One muffler bearing requires 0.125 hours of assembly labor, 0.25 hours in the stamping department and 9 square feet of sheet metal. Each torque amplifier requires 1/3 hours in assembly and 0.25 hours in stamping and uses 6 square feet of sheet metal. Current capacities are 400 hours of assembly labor and 375 hours of stamping capacity. Sheet steel costs $0.15 per square foot. Muffler bearings sell for $9 and torque amplifiers sell for $11.00. Use linear optimization to find the number of each product to make to maximize profit. Do not constrain the solution to integers. The available amount of sheet steel may be considered unlimited. The cost of Labor is not charged to the products. A new product called the furtengrinder uses 0.5 hours of assembly time and 10 square feet of sheet metal and sells for $32.10. Use shadow prices from Solvers sensitivity report to determine the maximum number of stamping hours it could use before it would become unprofitable to produce furtengrinders. Answer: cost impacting the profitability of the product is the steel sheet (sq feet). However, the total number of products that can be made is limited by the available capacities of labour and stamping department hours. Let M, T and F denote the number of Muffler bearings, Torque amplifiers and Furtengrinder Profitability of Muffler bearings = Price steel sheet*cost per sq ft = 9 (9*0.15) = $7.65 Torque amplifiers = 11 (6*0.15) = $10.1 Furtengrinder = 32.1 (10*0.15) = $30.6 Objective function Maximize 7.65*M + 10.1*T +...

2. A post office requires sufficient full- and part-time staff to satisfy the following daily needs for employee hours: Day M T W R F Sa Su Hours Needed 136 104 120 152 112 128 88

A full-time employee works 8 hours a day for five straight days. A part-time employee works 4 hours a day for five straight days. Full- or part-time employees can be hired to begin their five straight days on any day that the post office chooses. Full time employees earn $20/ hour while part time employees earn $15/hour. A union agreement limits the number of part-time employees to no more than 25% of the total number of employees. Determine how many full- and part-time workers should begin their work on each day of the week to satisfy the manpower requirements at minimum cost. Ignore lunch and other breaks. Produce a two-way Parameter Analysis to show the minimum cost as the part time wage varies from $10/hour to $20/ hour in increments of $5/hour while the full time wage varies from $15/hour to $25/hour in increments of $5/hour. 3. A political candidate wants to use mass mailing to counteract some negative remarks made recently by her opponent. Our candidate has identified five mailing lists that contain names and addresses of voters she might like to reach. Each list can be purchased for a price. Partial lists are not for sale. The number of names on each list in each of four professional categories are listed below along with the desired number of names to reach in each category and the price of each list. Find the subset of lists that will provide the desired coverage at minimum cost. List Law HealthCare Bus. Execs. Academe Price 1 28 4 7 2 $41,000 2 9 29 11 3 $52,000 3 6 3 34 18 $61,000 4 2 4 6 20 $32,000 5 8 9 12 14 43,000 Desired coverage 20 18 22 20

4. The Fleek Corporation has 5 warehouses designated A-E from which they supply 6 customers numbered 1-6. The table below gives the shipping cost in dollars per ton from each warehouse to each customer along with the monthly amounts in tons required by each customer, the capacity in tons for each warehouse and the fixed monthly cost of maintaining each warehouse. Management at Fleek suspects that the company may be over-warehoused and that the customers needs could be met more economically with fewer warehouses. Determine the shipping plan and which warehouses should remain open to satisfy the customers at minimum cost. Customers Fixed Cost $ Whse 1 2 3 4 5 6 Capacity Tons A 1675 400 685 1630 1160 2800 18 7650 B 1460 1940 970 100 495 1200 24 3500 C 1925 2400 1425 500 950 800 27 5000 D 380 1355 543 1045 665 2321 22 4100 E 922 1646 700 508 311 1797 31 2200 Demand 10 8 12 6 7 11

5. Design the steel framework shown below at minimum cost. The cost of each horizontal member in one direction is $20w and in the other direction its $30d. The cost of a vertical column is $50h. The length units are in meters. The frame must enclose a total volume of 600m3. (Textbook: Problem 7.5, 7.11, 8.8,Case 8-2) 1. A real-estate development firm, Peterson and Johnson, is considering five possible development projects. Using units of millions of dollars, the following table shows the estimated long run profit (net-present value) that each project would generate, as well as the amount of investment required to undertake the project. DEVELOPMENT PROJECT 1 2 3 4 5 Estimated Profit (Millions) 1 1.8 1.6 0.8 1.4 Capital Required (Millions) 6 12 10 4 8

The owners of the firm, Dave Peterson and Ron Johnson, have raised $20 million of investment capital for these projects. Dave and Ron now want to select the combination of projects that will maximize their total estimated long run profit (net present value) without investing more than $20 million. a. Formulate a BIP model in algebraic form for this problem. b. Formulate and solve thismodel on a spreadsheet c. Perform What-if-analysis on the amount of investment capital made available for the development projects by generating a parameter analysis report with Analytic Solver to solve the model with the following amounts of investment capital (in millions of dollars): 16, 18, 20, 22, 24, 26, 28, and 30. include both the changing cells and the objective cell as output Cells in the parameter analysis report. 2. 2. An increasing number of Americans are moving to a warmer climate when they retire. To take advantage of this trend, Sunny Skies Unlimited is undertaking a major real-estate development project. The project is to develop a completely new retirement community (to be called Pilgrim Haven) that will cover several square-miles. One of the decisions to be made is where to locate the two paramedic stations that have been allocated to the community to respond to medical emergencies. For planning purposes, Pilgrim Haven has been divided into five tracts, with no more than one paramedic station to be located in any given tract. Each station is to respond to all the medical

(Textbook: Problem 7.5, 7.11, 8.8,Case 8-2) emergencies that occur in the tract in which it is located as well as in the other tracts that are assigned to this station. Thus, the decisions to be made consist of (1) the tracts to receive a paramedic station and (2) the assignment of each of the other tracts to one of the paramedic stations. The objective is to minimize the overall average of the response times to medical emergencies. The following table gives the average response time to a medical emergency in each tract (the rows) if that tract is served by a station in a given tract (the columns). The last column gives the forecasted average number of medical emergencies that will occur in each of the tracts per day. Paramedic Station in Tract 1 2 3 4 5 Average Frequency of Medical Emergencies per day Response 1 5 20 15 25 10 2 Time (mins) 2 12 4 20 15 25 1 To a medical 3 30 15 6 25 15 3 Emergency 4 20 10 15 4 12 1 In tract 5 15 25 12 10 5 3

3. Reconsider the portfolio selection example given in Section 8.2. A fourth stock (Stock 4) now has been found that gives a good balance between expected return and risk. Using the same units as in Table 8.2, its expected return is 17% and its risk is 18%. Its joint risk per stock with Stocks 1, 2, and 3 is -0.015, -0.025, and 0.003, respectively. a. Still using a minimum acceptable expected return of 18%, formulate the revised quadratic programming model in algebraic form for this problem b. Display and solve this model on a spreadsheet. c. Develop the revision of the parametric analysis report in figure 8.14 for this revised problem (use Analytic Solver) 4.Savvy Stock Selection Ever since the day she took her first economics class in high school, Lydia wondered about the financial practices of her parents. They worked very hard to earn enough money to live a comfortable middle-class life, but they never made their money work for them. They simply deposited their hard-earned paychecks in savings accounts earning a nominal amount of interest. (Fortunately, there always was enough money available when it came time to pay her college bills.) She promised herself that when she became an adult, she would not follow the same financially conservative practices as her parents. And Lydia kept this promise. She took every available finance course in her business program at college. Having landed a coveted job on Wall Street upon graduation, she now begins every morning by watching the CNN financial reports. She plays investment games on the internet, finding portfolios that maximize her return while minimizing her risk. And she reads The Wall Street Journal and Financial Times. Lydia also reads the investment advice columns of the financial magazines. She decides to follow the current advice given by her two favorite columnists. In his monthly column, editor Jonathan Taylor recommends three stocks that he believes will rise far above market average. In addition, the well known mutual fund guru Donna Carter advocates the purchase of three additional stocks that she thinks will outperform the market over the next year. Bigbell (ticker symbol on the stock exchange: BB), one of the nation's largest telecommunications companies, trades at a price-earnings ratio well below market average. Huge investments over the last eight months have depressed earnings considerably. However, with its new cutting-edge technology, the company is expected to significantly raise its profit margins. Taylor predicts that the stock will rise from its current price of $60 per share to $72 per share within the next year. Lotsofplace (LOP) is one of the leading hard drive manufacturers in the world. The industry recently underwent major consolidation, as fierce price wars over the last few years were followed by many competitors going bankrupt or being bought by Lotsofplace and its competitors. Due to reduced competition in the hard drive market, revenues and earnings are expected to rise considerably over the next year. Taylor predicts a one-year increase of 42 percent in the stock of Lotsofplace from the current price of $127 per share. Internetlife (ILI) has survived the many ups and downs of Internet companies. With the next Internet frenzy just around the corner, Taylor expects a doubling of this company's stock price from $4 to $8 within a year. Healthtomorrow (HEAL) is a leading biotechnology company that is about to get approval for several new drugs from the Food and Drug Administration, which will help earnings to grow 20 percent over the next few years. In particular, a new drug to significantly reduce the risk of heart attacks is supposed to reap huge profits. Also, due to several new great-tasting medications for children, the company has been able to build an excellent image in the media. This public relations coup will surely have a positive effect on the sale of its over-the-counter medications. Carter is convinced that the stock will rise from $50 to $75 per share within a year. Quicky (QUI) is a fast-food chain that has been vastly expanding its network of restaurants all over the United States. Carter has followed this company closely since it went public some 15 years ago when it had only a few dozen restaurants on the West Coast of the United States. Since then the company has expanded, and it now has restaurants in every state. Due to its emphasis on healthy foods, it is capturing a growing market share. Carter believes that the stock will continue to perform well above market average for an increase of 46 percent in one year from its current stock price of $150. Automobile Alliance (AUA) is a leading car manufacturer from the Detroit area that just recently introduced two new models. These models show very strong initial sales, and therefore the company's stock is predicted to rise from $20 to $26 over the next year. Using the internet, Lydia found data about the risk involved in the stocks of these companies. The historical variances of return of the six stocks and their covariances are shown in the following tables. Company BB LOP ILI HEAL QUI AUA Variance 0.032 0.1 0.333 0.125 0.065 0.08 Covariance LOP ILI HEAL QUI AUA BB 0.005 0.03 -0.031 -0.027 0.01 LOP 0.085 -0.07 -0.05 0.02 ILI -0.11 -0.02 0.042 HEAL 0.05 -0.06 QUI -0.02 a a. At first, Lydia wants to ignore the risk of all the investments. Given this strategy, what is her optimal investment portfolio; that is, what fraction of her money should she invest in each of the six different stocks? What is the total risk of her portfolio? b. Lydia decides that she doesn't want to invest more than 40 percent in any individual stock. While still ignoring risk, what is her new optimal investment portfolio? What is the total risk of her new portfolio? c. Now Lydia wants to take into account the risk of her investment opportunities. For use in the following parts, formulate a quadratic programming model that will minimize her risk (measured by the variance of the return from her portfolio) while ensuring that her expected return is at least as large as her choice of a minimum acceptable value.

d. Lydia wants to ensure that she receives an expected return of at least 35 percent. She wants to reach this goal at minimum risk. What investment portfolio allows her to do that? e. What is the minimum risk Lydia can achieve if she wants an expected return of at least 25 percent? Of at least 40 percent? f. Do you see any problems or disadvantages with Lydia's approach to her investment strategy?

1. The Otto Maddick Machine Tool Company produces two products, muffler bearings and torque amplifiers. One muffler bearing requires 0.125 hours of assembly labor, 0.25 hours in the stamping department and 9 square feet of sheet metal. Each torque amplifier requires 1/3 hours in assembly and 0.25 hours in stamping and uses 6 square feet of sheet metal. Current capacities are 400 hours of assembly labor and 375 hours of stamping capacity. Sheet steel costs $0.15 per square foot. Muffler bearings sell for $9 and torque amplifiers sell for $11.00. Use linear optimization to find the number of each product to make to maximize profit. Do not constrain the solution to integers. The available amount of sheet steel may be considered unlimited. The cost of Labor is not charged to the products. A new product called the furtengrinder uses 0.5 hours of assembly time and 10 square feet of sheet metal and sells for $32.10. Use shadow prices from Solvers sensitivity report to determine the maximum number of stamping hours it could use before it would become unprofitable to produce furtengrinders. Answer: cost impacting the profitability of the product is the steel sheet (sq feet). However, the total number of products that can be made is limited by the available capacities of labour and stamping department hours. Let M, T and F denote the number of Muffler bearings, Torque amplifiers and Furtengrinder Profitability of Muffler bearings = Price steel sheet*cost per sq ft = 9 (9*0.15) = $7.65 Torque amplifiers = 11 (6*0.15) = $10.1 Furtengrinder = 32.1 (10*0.15) = $30.6 Objective function Maximize 7.65*M + 10.1*T +...

2. A post office requires sufficient full- and part-time staff to satisfy the following daily needs for employee hours: Day M T W R F Sa Su Hours Needed 136 104 120 152 112 128 88

A full-time employee works 8 hours a day for five straight days. A part-time employee works 4 hours a day for five straight days. Full- or part-time employees can be hired to begin their five straight days on any day that the post office chooses. Full time employees earn $20/ hour while part time employees earn $15/hour. A union agreement limits the number of part-time employees to no more than 25% of the total number of employees. Determine how many full- and part-time workers should begin their work on each day of the week to satisfy the manpower requirements at minimum cost. Ignore lunch and other breaks. Produce a two-way Parameter Analysis to show the minimum cost as the part time wage varies from $10/hour to $20/ hour in increments of $5/hour while the full time wage varies from $15/hour to $25/hour in increments of $5/hour. 3. A political candidate wants to use mass mailing to counteract some negative remarks made recently by her opponent. Our candidate has identified five mailing lists that contain names and addresses of voters she might like to reach. Each list can be purchased for a price. Partial lists are not for sale. The number of names on each list in each of four professional categories are listed below along with the desired number of names to reach in each category and the price of each list. Find the subset of lists that will provide the desired coverage at minimum cost. List Law HealthCare Bus. Execs. Academe Price 1 28 4 7 2 $41,000 2 9 29 11 3 $52,000 3 6 3 34 18 $61,000 4 2 4 6 20 $32,000 5 8 9 12 14 43,000 Desired coverage 20 18 22 20

4. The Fleek Corporation has 5 warehouses designated A-E from which they supply 6 customers numbered 1-6. The table below gives the shipping cost in dollars per ton from each warehouse to each customer along with the monthly amounts in tons required by each customer, the capacity in tons for each warehouse and the fixed monthly cost of maintaining each warehouse. Management at Fleek suspects that the company may be over-warehoused and that the customers needs could be met more economically with fewer warehouses. Determine the shipping plan and which warehouses should remain open to satisfy the customers at minimum cost. Customers Fixed Cost $ Whse 1 2 3 4 5 6 Capacity Tons A 1675 400 685 1630 1160 2800 18 7650 B 1460 1940 970 100 495 1200 24 3500 C 1925 2400 1425 500 950 800 27 5000 D 380 1355 543 1045 665 2321 22 4100 E 922 1646 700 508 311 1797 31 2200 Demand 10 8 12 6 7 11

5. Design the steel framework shown below at minimum cost. The cost of each horizontal member in one direction is $20w and in the other direction its $30d. The cost of a vertical column is $50h. The length units are in meters. The frame must enclose a total volume of 600m3. (Textbook: Problem 7.5, 7.11, 8.8,Case 8-2) 1. A real-estate development firm, Peterson and Johnson, is considering five possible development projects. Using units of millions of dollars, the following table shows the estimated long run profit (net-present value) that each project would generate, as well as the amount of investment required to undertake the project. DEVELOPMENT PROJECT 1 2 3 4 5 Estimated Profit (Millions) 1 1.8 1.6 0.8 1.4 Capital Required (Millions) 6 12 10 4 8

The owners of the firm, Dave Peterson and Ron Johnson, have raised $20 million of investment capital for these projects. Dave and Ron now want to select the combination of projects that will maximize their total estimated long run profit (net present value) without investing more than $20 million. a. Formulate a BIP model in algebraic form for this problem. b. Formulate and solve thismodel on a spreadsheet c. Perform What-if-analysis on the amount of investment capital made available for the development projects by generating a parameter analysis report with Analytic Solver to solve the model with the following amounts of investment capital (in millions of dollars): 16, 18, 20, 22, 24, 26, 28, and 30. include both the changing cells and the objective cell as output Cells in the parameter analysis report. 2. 2. An increasing number of Americans are moving to a warmer climate when they retire. To take advantage of this trend, Sunny Skies Unlimited is undertaking a major real-estate development project. The project is to develop a completely new retirement community (to be called Pilgrim Haven) that will cover several square-miles. One of the decisions to be made is where to locate the two paramedic stations that have been allocated to the community to respond to medical emergencies. For planning purposes, Pilgrim Haven has been divided into five tracts, with no more than one paramedic station to be located in any given tract. Each station is to respond to all the medical

(Textbook: Problem 7.5, 7.11, 8.8,Case 8-2) emergencies that occur in the tract in which it is located as well as in the other tracts that are assigned to this station. Thus, the decisions to be made consist of (1) the tracts to receive a paramedic station and (2) the assignment of each of the other tracts to one of the paramedic stations. The objective is to minimize the overall average of the response times to medical emergencies. The following table gives the average response time to a medical emergency in each tract (the rows) if that tract is served by a station in a given tract (the columns). The last column gives the forecasted average number of medical emergencies that will occur in each of the tracts per day. Paramedic Station in Tract 1 2 3 4 5 Average Frequency of Medical Emergencies per day Response 1 5 20 15 25 10 2 Time (mins) 2 12 4 20 15 25 1 To a medical 3 30 15 6 25 15 3 Emergency 4 20 10 15 4 12 1 In tract 5 15 25 12 10 5 3

3.Reconsider the portfolio selection example given in Section 8.2. A fourth stock (Stock 4) now has been found that gives a good balance between expected return and risk. Using the same units as in Table 8.2, its expected return is 17% and its risk is 18%. Its joint risk per stock with Stocks 1, 2, and 3 is -0.015, -0.025, and 0.003, respectively. a. Still using a minimum acceptable expected return of 18%, formulate the revised quadratic programming model in algebraic form for this problem b. Display and solve this model on a spreadsheet. c. Develop the revision of the parametric analysis report in figure 8.14 for this revised problem (use Analytic slover)