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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 (a) Define the decision variables and write the linear programming model for the problem in the space provided below (b) Sketch the feasible region. Give your answer in the worksheet named P1-Graph (i) You can use QM for Windows to get the graph, copy and paste it onto the worksheet named P1-Graph OR (ii) You can sketch the feasible region on paper, scan and save it in your computer. Copy the image (i.e. graph) and paste it onto the worksheet named P1-Graph (c') Write all the vertices of the feasible region and identify the optimal vertex What is the optimal objective function value? Vertex 1 x1 x2 x1 x2 optimal vertex 2 3 optimal objective fuction value (a) Define the decision variables and write the linear programming model for the problem in the space provided below (b) Sketch the feasible region. Give your answer in the worksheet named P2-Graph (i) You can use QM for Windows to get the graph, copy and paste it onto the worksheet named P2-Graph OR (ii) You can sketch the feasible region on paper, scan and save it in your computer. Copy the image (Graph) and paste it onto the worksheet named P2-Graph Write all the vertices of the feasible region and identify the optimal vertex What is the optimal objective function value? Vertex 1 x1 x2 x1 x2 optimal vertex 2 3 optimal objective fuction value 4 5 (c') How much labor and wood will be unused if the optimal number of chairs and tables are used? Provide your answer in the space below (a) Define the decision variables and write the linear programming model for the problem in the space provided below (b) Sketch the feasible region. Give your answer in the worksheet named P4-Graph (i) You can use QM for Windows to get the graph, copy and paste it onto the worksheet named P4-Graph OR (ii) You can sketch the feasible region on paper, scan and save it in your computer. Copy the image (Graph) and paste it onto the worksheet named P4-Graph (c') Write all the vertices of the feasible region and identify the optimal vertex What is the optimal objective function value? Vertex 1 x1 x2 x1 x2 optimal vertex 2 3 optimal objective fuction value (a) Sketch the feasible region. Give your answer in the worksheet named P5-Graph (i) You can use QM for Windows to get the graph, copy and paste it onto the worksheet named P5-Graph OR (ii) You can sketch the feasible region on paper, scan and save it in your computer. Copy the image (Graph) and paste it onto the worksheet named P5-Graph (b) Write all the vertices of the feasible region and identify the optimal vertex What is the optimal objective function value? Vertex 1 x1 x2 x1 x2 optimal vertex 2 3 optimal objective fuction value MAT540 Homework Week 7 Page 1 of 3 MAT540 Week 7 Homework Chapter 3 1. Southern Sporting Good Company makes basketballs and footballs. Each product is produced from two resources rubber and leather. Each basketball produced results in a profit of $11 and each football earns $15 in profit. The resource requirements for each product and the total resources available are as follows: Product Resource Requirements per Unit Rubber (lb.) Leather (ft2) Basketball 2.8 3.7 Football 1.5 5.2 Total resources available 600 900 a. Find the optimal solution. b. What would be the effect on the optimal solution if the profit for the basketball changed from $11 to $12? c. What would be the effect on optimal solution if 400 additional pounds of rubber could be obtained? What would be the effect if 600 additional square feet of leather could be obtained? 2. A company produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows: Product Resource Requirements per Unit Line 1 Line 2 A 11 5 B 6 9 Total Hours 65 40 a. Formulate a linear programming model to determine the optimal product mix that will maximize profit. MAT540 Homework Week 7 Page 2 of 3 b. What are the sensitivity ranges for the objective function coefficients? c. Determine the shadow prices for additional hours of production time on line 1 and line 2 and indicate whether the company would prefer additional line 1 or line 2 hours. 3. Formulate and solve the model for the following problem: Irwin Textile Mills produces two types of cotton cloth denim and corduroy. Corduroy is a heavier grade of cotton cloth and, as such, requires 8 pounds of raw cotton per yard, whereas denim requires 6 pounds of raw cotton per yard. A yard of corduroy requires 4 hours of processing time; a yard od denim requires 3.0 hours. Although the demand for denim is practically unlimited, the maximum demand for corduroy is 510 yards per month. The manufacturer has 6,500 pounds of cotton and 3,000 hours of processing time available each month. The manufacturer makes a profit of $2.5 per yards of denim and $3.25 per yard of corduroy. The manufacturer wants to know how many yards of each type of cloth to produce to maximize profit. Formulate the model and put it into standard form. Solve it a. How much extra cotton and processing time are left over at the optimal solution? Is the demand for corduroy met? b. If Irwin Mills can obtain additional cotton or processing time, but not both, which should it select? How much? Explain your answer. 4. The Bradley family owns 410 acres of farmland in North Carolina on which they grow corn and tobacco. Each acre of corn costs $105 to plant, cultivate, and harvest; each acre of tobacco costs $210. The Bradleys' have a budget of $52,500 for next year. The government limits the number of acres of tobacco that can be planted to 100. The profit from each acre of corn is $300; the profit from each acre of tobacco is $520. The Bradleys' want to know how many acres of each crop to plant in order to maximize their profit. a. Formulate the linear programming model for the problem and solve. b. How many acres of farmland will not be cultivated at the optimal solution? Do the Bradleys use the entire 100-acre tobacco allotment? c. The Bradleys' have an opportunity to lease some extra land from a neighbor. The neighbor is offering the land to them for $110 per acre. Should the Bradleys' lease the land at that price? What is the maximum price the Bradleys' should pay their neighbor for the land, and how much land should they lease at that price? MAT540 Homework Week 7 Page 3 of 3 d. The Bradleys' are considering taking out a loan to increase their budget. For each dollar they borrow, how much additional profit would they make? If they borrowed an additional $1,000, would the number of acres of corn and tobacco they plant change? Sporting Goods Resource Requirements per Unit (a) Product Basketball Football Rubber (lb.) 2.8 1.5 Resource Requirements per Unit (c-1) Leather (f2) 3.7 5.2 Product Basketball Football Rubber (lb.) 2.8 1.5 Leather (f2) 3.7 5.2 1000 900 Basketball 11 Football 15 input Constraints Total resources available Profits ($) 600 Basketball 11 Total resources available 900 Football 15 Profits ($) Basketball Football Decision variables Basketball Football Maximized profits Objective function Maximized profits (b) Product Basketball Football Resource Requirements per Unit Rubber (lb.) Leather (f2) 2.8 3.7 1.5 5.2 (C-2) Product Basketball Football Resource Requirements per Unit Rubber (lb.) Leather (f2) 2.8 3.7 1.5 5.2 input Constraints Total resources available Profits ($) 600 Basketball 12 Total resources available 900 Football 15 Profits ($) Basketball Football Decision variables Basketball Football Maximized profits Objective function Maximized profits Please use computer method to solve the problem Please enter your solution in Yellow cells 600 1500 Basketball 11 Football 15 A & B products Hours/ Unit Line 1 Line2 11 5 6 9 (a) Product A B Constraints Total Hours Profits ($) Product A Product B Maximized profits 65 40 A 11 B 15 decision variables Objective function (b) Sensitivity Range for the coefficient of # of units of product A in the objectivefunction Sensitivity Range for the coefficient of # of units of product B in the objectivefunction (c') Shadow Price for additional hours of production time on Line 1 Shadow Price for additional hours of production time on Line 2 Which one of the two the company would prefer (Line 1 or Line 2?) Irwin textile mills (a) Profits for each product Corduroy 3.25 denim 2.5 Resources Cotton Labor Corduroy denim 8 4 6 3