MAT540 Homework Week 2 Page 1 of 4 MAT540 Week 2 Homework Chapter 12 1. A local real estate investor in Orlando is considering three alternative investments; a motel, a restaurant, or a theater. Profits from the motel or restaurant will be affected by the availability of gasoline and the number of tourists; profits from the theater will be relatively stable under any conditions. The following payoff table shows the profit or loss that could result from each investment: Weather Conditions Investment Shortage Stable Supply Surplus $-7,500 $12,000 $23,000 Restaurant 3000 7,000 6,500 Theater 5000 6,000 4,000 Motel Determine the best investment, using the following decision criteria. a. Maximax b. Maximin c. Minimax regret d. Hurwicz ( = 0.4) e. Equal likelihood 2. A concessions manager at the Tech versus A&M football game must decide whether to have the vendors sell sun visors or umbrellas. There is a 35% chance of rain, a 25% chance of overcast skies, and a 40% chance of sunshine, according to the weather forecast in college junction, where the game is to be held. The manager estimates that the following profits will result from each decision, given each set of weather conditions: MAT540 Homework Week 2 Page 2 of 4 Weather Conditions Decision Rain Overcast Sunshine 0.35 0.25 0.40 Sun visors $-400 $-200 $1,500 Umbrellas 2,100 0 -800 a. Compute the expected value for each decision and select the best one. b. Develop the opportunity loss table and compute the expected opportunity loss for each decision. 3. Place-Plus, a real estate development firm, is considering several alternative development projects. These include building and leasing an office park, purchasing a parcel of land and building an office building to rent, buying and leasing a warehouse, building a strip mall, and selling condominiums. The financial success of these projects depends on interest rate movement in the next 5 years. The various development projects and their 5- year financial return (in $1,000,000s) given that interest rates will decline, remain stable, or increase, are in the following payoff table. Place-Plus real estate development firm has hired an economist to assign a probability to each direction interest rates may take over the next 5 years. The economist has determined that there is a 0.45 probability that interest rates will decline, a 0.35 probability that rates will remain stable, and a 0.2 probability that rates will increase. a. Using expected value, determine the best project. b. Determine the expected value of perfect information. Interest Rate Project Decline Stable Increase Office park $0.4 $1.55 $3.5 Office building 2.5 1.8 2.75 Warehouse 1.7 1.45 1.5 Mall 0.8 2.3 3.5 Condominiums 3.2 1.5 0.5 MAT540 Homework Week 2 Page 3 of 4 4. The director of career advising at Orange Community College wants to use decision analysis to provide information to help students decide which 2-year degree program they should pursue. The director has set up the following payoff table for six of the most popular and successful degree programs at OCC that shows the estimated 5-Year gross income ($) from each degree for four future economic conditions: Economic Conditions Degree Program Recession Average Good Robust Graphic design 150,000 175,000 220,000 200,000 Nursing 160,000 180,000 205,000 215,000 Real estate 125,000 165,000 220,000 210,000 Medical technology 135,000 180,000 210,000 270,000 Culinary technology 110,000 145,000 235,000 205,000 Computer information 130,000 150,000 190,000 245,000 technology Determine the best degree program in terms of projected income, using the following decision criteria: a. Maximax b. Maximin c. Equal likelihood d. Hurwicz (=0.4) 5. Construct a decision tree for the following decision situation and indicate the best decision. Fenton and Farrah Friendly, husband-and-wife car dealers, are soon going to open a new dealership. They have three offers: from a foreign compact car company, from a U.S. producer of full-sized cars, and from a truck company. The success of each type of dealership will depend on how much gasoline is going to be available during the next few years. The profit from each type of dealership, given the availability of gas, is shown in the following payoff table: MAT540 Homework Week 2 Page 4 of 4 Gasoline Availability Dealership Shortage Surplus 0.7 0.3 Compact cars $25,000 $150,000 Full-sized cars -90,000 650,000 Trucks 125,000 170,000 Decision Tree diagram to complete: Shortage 0.7 $250,000 2 Compact cars 1 Full-sized cars Surplus 0.3 $150,00 0 Shortage 0.7 -$90,000 3 $650,000 Surplus 0.3 Trucks Shortage 0.7 $125,000 Surplus 0.3 $170,000 4 Investmen t Motel Restauran t Theater Shortag e Stable Surplus Maximu Supply m Profit Minimu m Profit -7500 3000 12000 7000 23000 6500 23000 7000 5000 6000 4000 Hurwicz Average Criterio n -7500 4700 9166.67 3000 4600 5500 6000 4000 4800 5000 a) Maximax The maximum profits for investment in Motel, Restaurant and Theatre are $23,000, $7,000, $6,000 respectively. The highest in all of 3 is $23,000. Hence, by Maximax criterion, the investor should invest in Motel. b) Maximin The minimum profits for investment in Motel, Restaurant and Theatre are $ -7,500, $3,000, $4,000 respectively. The maximum in all of 3 is $4,000. Hence, by Maximin criterion, the investor should invest in Theatre. c) Minimax regret Change the given payoff matrix into regret matrix by taking the difference between the maximum value and corresponding cell values from each row. The obtained regret matrix is as follows: First choose the maximum from each investment row. Investmen t Motel Restauran t Theater Shortage Stable Supply Surplus Maximum -7500 3000 12000 7000 23000 6500 23000 7000 5000 6000 4000 6000 Subtract each value of row from the corresponding maximum. The obtained regret matrix is as follows: Investmen t Motel Restauran t Theater Shortage Stable Surplus Maximum Supply 30500 11000 0 30500 4000 0 500 1000 0 2000 4000 2000 From the matrix choose the maximum regret value from each row and choose the minimum amount the maximum. Since $2,000 is the minimum among the maximum losses. The deicision is to invest in Theatre. d) Hurwicz Calculate Hurwicz value for each decision by multiplying the maximum value with 0.4 and minumum value with 1-0.4=0.6. Investment Shortage Stable Surplus Hurwicz Supply value Motel -7500 12000 23000 4700 Restaurant 3000 7000 6500 4600 Theater 5000 6000 4000 4800 From the Hurwicz value the maximum value is $4,800, the corresponding dicision is to invest in Theatre. e) Equal likelihood Give to each states of nature and calculate the expected payoff for each decision, choose 1 3 the maximum expected payoff. Investment Shortage Stable Surplus Average Supply Value Motel -7500 12000 23000 9166.667 Restaurant 3000 7000 6500 5500 Theater 5000 6000 4000 5000 Since the expected pay-off is high for Motel compared to other investments, the best decision is to invest in Motel. MAT540 Homework Week 6 Page 1 of 2 MAT540 Week 6 Homework Chapter 2 1. A Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 45 milligrams of vitamin A and 13 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 10 milligrams of vitamin A and 2 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 3 milligrams of B. An ounce of oats costs $0.06, and an ounce of rice costs $0.03. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. 2. A Furniture Company produces chairs and tables from two resources- labor and wood. The company has 125 hours of labor and 45 board-ft. of wood available each day. Demand for chairs is limited to 5 per day. Each chair requires 7 hours of labor and 3.5 board-ft. of wood, whereas a table requires 14 hours of labor and 7 board-ft. of wood. The profit derived from each chair is $325 and from each table, $120. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. Formulate a linear programming model for this problem. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. (Do not round the answers) c. How much labor and wood will be unused if the optimal numbers of chairs and tables are produced? 3. Kroeger supermarket sells its own brand of canned peas as well as several national brands. The store makes a profit of $0.28 per can for its own peas and a profit of $0.19 for any of the national brands. The store has 6 square feet of shelf space available for canned peas, and each can of peas takes up 9 square inches of that space. Point-of-sale records show that each week the store never sales more than half as many cans of its own brand as it does of the national brands. The store wants to know how many cans of its own brand of peas of peas and how many cans of the national brands to stock each week on the allocated shelf space in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. MAT540 Homework Week 6 Page 2 of 2 4. Solve the following linear programming model graphically: Minimize Z=8X1 + 6X2 Subject to 4X1 + 2X2 -6X1 + 4X2 X1 + X2 X1 , X2 20 MAT540 Homework Week 6 Page 1 of 2 MAT540 Week 6 Homework Chapter 2 1. A Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 45 milligrams of vitamin A and 13 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 10 milligrams of vitamin A and 2 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 3 milligrams of B. An ounce of oats costs $0.06, and an ounce of rice costs $0.03. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. 2. A Furniture Company produces chairs and tables from two resources- labor and wood. The company has 125 hours of labor and 45 board-ft. of wood available each day. Demand for chairs is limited to 5 per day. Each chair requires 7 hours of labor and 3.5 board-ft. of wood, whereas a table requires 14 hours of labor and 7 board-ft. of wood. The profit derived from each chair is $325 and from each table, $120. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. Formulate a linear programming model for this problem. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. (Do not round the answers) c. How much labor and wood will be unused if the optimal numbers of chairs and tables are produced? 3. Kroeger supermarket sells its own brand of canned peas as well as several national brands. The store makes a profit of $0.28 per can for its own peas and a profit of $0.19 for any of the national brands. The store has 6 square feet of shelf space available for canned peas, and each can of peas takes up 9 square inches of that space. Point-of-sale records show that each week the store never sales more than half as many cans of its own brand as it does of the national brands. The store wants to know how many cans of its own brand of peas of peas and how many cans of the national brands to stock each week on the allocated shelf space in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. MAT540 Homework Week 6 Page 2 of 2 4. Solve the following linear programming model graphically: Minimize Z=8X1 + 6X2 Subject to 4X1 + 2X2 -6X1 + 4X2 X1 + X2 X1 , X2 20