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Linear Programming A) An investor is considering four investments in order to create a stock portfolio. For each investment, she knows the expected rate of return and a risk factor (between 0 and 10, with 0 being no risk and 10 being very risky.) The data is below. The investor wants to invest $10,000. She also wants the average risk to be 7 or less and she wants to invest at least 25% of her money in investment B. How much money should she invest in each and what is her expected return? Investment Rate of Risk Return Factor A 15% 9 B 10% 6 C 6% 2 D 4% Homework Assignment 4 Problems: 1. Solve the following linear programming problem graphically. a. Maximize Z- x 10Y 1. Subject to: 4X + a. 2X + 4Y #<40 i. Y#3 2. Ed Silver Dog Food Company wishes to introduce a new brand of dog biscuits composed of chicken- and liver-flavored biscuits that meet certain nutritional requirements. The liver-flavored biscuits contain unit of nutrient A and 2 units of nutrient B; the chicken-flavored biscuits contain 1 unit of nutrient A and 4 units of nutrient B. According to federal requirements, there must be at least 40 units of nutrient A and 60 units of nutrient B in a package of the new mix. In addition, the company has decided that there can be no more than 15 liver-flavored biscuits in a package. If it costs le to make 1 liver-flavored biscuit and 2e to make 1 chicken- flavored, what is the optimal product mix for a package of the biscuits to minimize the firm's cost? a. Formulate this as a linear programming problem. b. Solve this problem graphically, giving the optimal values of all variables. c. What is the total cost of a package of dog biscuits using the optimal mix? 3. The Lauren Shur Tub Company manufactures two lines of bathtubs, called model A and model B. Every tub requires blending a certain amount of steel and zinc; the company has available a total of 25,000 lb. of steel and 6,000 lb. of zinc. Each model A bathtub requires a mixture of 125 lb. of steel and 20 lb. of zinc, and each yields a profit of $90. Each model B tub requires 100 lb. of steel and 30 lb. of zinc and can be sold for a profit of $70. a. Find by graphical linear programming the best production mix of bathtubs. 4. MSA Computer Corporation manufactures two models of minicomputers, the Alpha 4 and Beta 5. The firm employs 5 technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (that is, all 160 hours of time) be maintained for each worker during next month's operations. It requires 20 labor-hours to assemble each Alpha 4 computer and 25 labor-hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate a $1,200 profit per unit, and Betas yield $1,800 each. a. Determine the most profitable number of each model of minicomputer to produce during the coming month. 5. The Arden County, Maryland, superintendent of education is responsible for assigning students to the three high schools in his county. He recognizes the need to bus a certain number of students, for several sectors of the county are beyond walking distance to a school. The superintendent partitions to the county into five geographic sectors as he attempts to establish a plan that will minimize the total number of student miles traveled by bus. He also recognizes that if a student happens to live in a certain sector and is assigned to the high school in that sector, there is no need to bus him because he can walk to school. The three schools are located in sectors B, C, and E. a. The accompanying table reflects the number of high school age students living in each sector and the distance in miles from each sector to each school. Sector School in Sector School in Sector School in Sector E Number of B C Students A 5 8 9 700 B 4 12 500 C 4 7 100 D 7 2 5 800 E 12 7 400 b. Set up the objective function and constraints of this problem using linear programming so that the total number of student miles traveled by bus is minimized. c. Solve the problem. 6. Boston's famous Limoges Restaurant is open 24 hours a day. Servers report for duty at 3 a.m., 7 a.m., 11 a.m., 3 p.m., 7 p.m., or 11 p.m., and each works an 8- hour shift. The following table shows the minimum number of workers needed during the 6 periods into which the day is divided. Period Time Number of Servers Required 1 3 a.m.-7a.m. 3 2 7a.m. 11 a.m. 12 3 11 a.m.-3 p.m. 16 4 3 p.m.-7p.m. 9 5 7 p.m.- ll p.m. 11 6 11 p.m.-3a.m. 4 a. Owner Michelle Limoge's scheduling problem is to determine how many servers should report for work at the start of each period in order to minimize the total staff required for one day's operation. (Hint: Let Xi equal the number of servers beginning work in time period i, where i= 1, 2,3,4,5,6.)

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