# Problem #1: Graphically solve the following LP: Maximize 30X1 + 2...

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Problem #1: Graphically solve the following LP: Maximize 30X1 + 20X2 subject to: 8X1 + 4X2 <= 640 4X1 + 6X2 <= 540 X1 + X2 <= 100 X1, X2 >= 0 Problem #2 Graphically solve the following LP: Minimize 2X1 + 3X2 subject to: X2 >= 2 4X1 + 3X2 >= 12 X1 + 2X2 <= 8 X1, X2 >= 0 Problem #3 Do review problem 52 (paper recycling plant) on page 122 of the text. Formulate the LP only. A paper-recycling plant processes box board, tissue paper, newsprint, and book paper into pulp that can be used to produce three grades of recycled paper (grades 1, 2, and 3). The prices per ton and the pulp contents of the four inputs are shown in Table 82. Two methods, de-inking and asphalt dispersion, can be used to process the four inputs into pulp. It costs \$20 to de-ink a ton of any input. The process of de-inking removes 10% of the input’s pulp, leaving 90% of the original pulp. It costs \$15 to apply asphalt dispersion to a ton of material. The asphalt dispersion process removes 20% of the input’s pulp. At most, 3,000 tons of input can be run through the asphalt dispersion process or the de-inking process. Grade 1 paper can only be produced with newsprint or book paper pulp; grade 2 paper, only with book paper, tissue paper, or box board pulp; and grade 3 paper, only with newsprint, tissue paper, or box board pulp. To meet its current demands, the company needs 500 tons of pulp for grade 1 paper, 500 tons of pulp for grade 2 paper, and 600 tons of pulp for grade 3 paper. Formulate an LP to minimize the cost of meeting the demands for pulp. Problem #4 Do review problem 62 (Ole' Oil) on page 124 of the text. Formulate the LP only. Olé Oil produces three products: heating oil, gasoline, and jet fuel. The average octane levels must be at least 4.5 for heating oil, 8.5 for gas, and 7.0 for jet fuel. To produce these products Olé purchases two types of oil: crude 1 (at \$12 per barrel) and crude 2 (at \$10 per barrel). Each day, at most 10,000 barrels of each type of oil can be purchased. Before crude can be used to produce products for sale, it must be distilled. Each day, at most 15,000 barrels of oil can be distilled. It costs 10¢ to distill a barrel of oil. The re- sult of distillation is as follows: (1) Each barrel of crude 1 yields 0.6 barrel of naphtha, 0.3 barrel of distilled 1, and 0.1 barrel of distilled 2. (2) Each barrel of crude 2 yields 0.4 barrel of naphtha, 0.2 barrel of distilled 1, and 0.4 barrel of distilled 2. Distilled naphtha can be used only to produce gasoline or jet fuel. Distilled oil can be used to produce heating oil or it can be sent through the catalytic cracker (at a cost of 15¢ per barrel). Each day, at most 5,000 barrels of distilled oil can be sent through the cracker. Each barrel of distilled 1 sent through the cracker yields 0.8 barrel of cracked 1 and 0.2 barrel of cracked 2. Each barrel of dis- tilled 2 sent through the cracker yields 0.7 barrel of cracked 1 and 0.3 barrel of cracked 2. Cracked oil can be used to produce gasoline and jet fuel but not to produce heating oil. The octane level of each type of oil is as follows: naph- tha, 8; distilled 1, 4; distilled 2, 5; cracked 1, 9; cracked 2, 6. All heating oil produced can be sold at \$14 per barrel; all gasoline produced, \$18 per barrel; and all jet fuel pro- duced, \$16 per barrel. Marketing considerations dictate that at least 3,000 barrels of each product must be produced daily. Formulate an LP to maximize Olé’s daily profit. Problem #5 Do review problem 35 (greenhouse operator) on page 118 of the text. Formulate and solve the LP using LINDO. Formulate the following as a linear programming problem: A greenhouse operator plans to bid for the job of providing flowers for city parks. He will use tulips, daffodils, and flowering shrubs in three types of layouts. A Type 1 layout uses 30 tulips, 20 daffodils, and 4 flowering shrubs. A Type 2 layout uses 10 tulips, 40 daffodils, and 3 flowering shrubs. A Type 3 layout uses 20 tulips, 50 daffodils, and 2 flowering shrubs. The net profit is \$50 for each Type 1 layout, \$30 for each Type 2 layout, and \$60 for each Type 3 layout. He has 1,000 tulips, 800 daffodils, and 100 flowering shrubs. How many layouts of each type should be used to yield maximum profit? Problem #6 Do review problem 50 (city waste) on page 121 of the text. Formulate and solve the LP using LINDO. City 1 produces 500 tons of waste per day, and city 2 produces 400 tons of waste per day. Waste must be incinerated at incinerator 1 or 2, and each incinerator can process up to 500 tons of waste per day. The cost to incinerate waste is \$40/ton at incinerator 1 and \$30/ton at 2. Problem #7 Solve the following LP using the simplex method: Maximize 300X1 + 400X2 subject to: 2X1 + X2 <= 11 X1 + 8X2 <= 40 7X1 + 8X2 <= 52 X1 <= 5 X1, X2 >= 0 Problem #8 Solve the following LP using the simplex method: Minimize -4X1 + X2 subject to: 3X1 + X2 <= 6 -X1 + 2X2 <= 0 X1, X2 >= 0 Problem #9 Solve the following LP using the simplex method: Maximize 30X1 + 20X2 subject to: 8X1 + 4X2 <= 640 4X1 + 6X2 <= 540 X1 + X2 <= 100 X2 >= 50 X1, X2 >= 0 Problem #10 Solve the following LP using any method you choose: Maximize X1 + X2 subject to: 2X1 + X2 >= 3 3X1 + X2 <= 3.5 X1 + X2 <= 1 X1, X2 >= 0 Problem #11 Do review problem 8 (Wivco) on page 257 of the text. Wivco produces two products: 1 and 2. The relevant data are shown in Table 12. Each week, up to 400 units of raw material can be purchased at a cost of \$1.50 per unit. The company employs four workers, who work 40 hours per week. (Their salaries are considered a fixed cost.) Workers are paid \$6 per hour to work overtime. Each week, 320 hours of machine time are available. Product 1 Product 2 Selling Price (\$) 15 8 Labor Required (hours) 0.75 0.50 Machine time required (hours 1.5 0.80 Raw material required (units) 2 1 Problem #12 Do review problem 12 (Machinco) on page 259 of the text. Machinco produces four products, requiring time on two machines and two types (skilled and unskilled) of labor. The amount of machine time and labor (in hours) used by each product and the sales prices are given in Table 13. Each month, 700 hours are available on machine 1 and 500 hours on machine 2. Each month, Machinco can purchase up to 600 hours of skilled labor at \$8 per hour and up to 650 hours of unskilled labor at \$6 per hour. Formulate an LP that will enable Machinco to maximize its monthly profit. Solve this LP and use the output to answer the following questions: (1) will become Using the definition of shadow price, answer parts (a) and (b). 10 Use LINDO to solve the Sailco problem of Section 3.10, then use the output to answer the following questions: a If month 1 demand decreased to 35 sailboats, what would be the total cost of satisfying the demands during 􏰂2x6J 􏰂 x9J 􏰂 80,000􏰂 the next four months? b If the cost of producing a sailboat with regular-time labor during month 1 were \$420, what would be the new optimal solution to the Sailco problem? c Suppose a new customer is willing to pay \$425 for a sailboat. If his demand must be met during month 1, should Sailco fill the order? How about if his demand must be met during month 4? a By how much does the price of product 3 have to in- crease before it becomes optimal to produce it? b If product 1 sold for \$290, then what would be the new optimal solution to the problem? c What is the most Machinco would be willing to pay for an extra hour of time on each machine? d What is the most Machinco would be willing to pay for an extra hour of each type of labor? e If up to 700 hours of skilled labor could be pur- chased each month, then what would be Machinco’s monthly profits? Product Machine 1 Machine 2 Skilled Unskilled Sales (\$) 1 11 4 8 7 300 2 7 6 5 8 260 3 6 5 4 7 220 4 5 4 6 4 180

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