Exam. Name. Use the indicated region of feasible solutions to find the maximum and minimum values of the given objective function.

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1 Exam Name Use the indicated region of feasible solutions to find the maximum and minimum values of the given objective function. 1) z = 12x - 22y y (0, 6) (1.2, 5) Solve the 3) The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2 man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $30 and on an SST ring is $40? (5, 0) Use graphical methods to solve the linear programming 2) Maximize z = 8x + 12y subject to: 40x + 80y x + 8y 72 x 0 y 0 y x x 4) Zach is planning to invest up to $50,000 in corporate and municipal bonds. The least he will invest in corporate bonds is $6000 and he does not want to invest more than $29,000 in corporate bonds. He also does not want to invest more than $28,174 in municipal bonds. The interest is 8.5% on corporate bonds and 6.8% on municipal bonds. This is simple interest for one year. What is the maximum value of his investment after one year? 5) An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide transportation for a minimum of 400 first class, 900 tourist class, and 1500 economy class passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accomodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and can accomodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost? Pivot once about the circled element in the simplex tableau, and read the solution from the result. 6) 1

2 7) 8) Write the solutions that can be read from the simplex tableau. 13) x1 x2 x3 s1 s2 z ) x1 x2 s1 s2 s3 z ) 15) x1 x2 s1 s2 s3 z Introduce slack variables as necessary, and write the initial simplex tableau for the 10) Find x1 0 and x2 0 such that 5x1 + 10x x1 + 15x2 136 and z = 2x1 + 5x2 is maximized. 11) Find x1 0 and x2 0 such that 2x1 + 5x2 7 3x1 + 3x2 18 and z = 4x1 + x2 is maximized. 12) Find x1 0 and x2 0 such that x1 + x2 64 3x1 + x2 171 and z = 2x1 + x2 is maximized. 16) x1 x2 x3 s1 s2 z ) x1 x2 x3 s1 s2 z A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and $15 for C. The production requirements and departmental capacities are as follows: Department Production requirement by product (hours) Assembling Painting Finishing A B C Departmental capa (Total hours) 30,000 38,000 28,000 18) What is the maximum profit if the capacity of the painting department changes to 40,000 hours? 19) What is the maximum profit if the profit margin on A changes to $7.00? 20) What is the constraint for the finishing department? 2

3 The initial tableau of a linear programming problem is given. Use the simplex method to solve the 21) x1 x2 x3 s1 s2 s3 z Use the simplex method to solve the linear programming 22) Maximize z = 2x1 + 5x2 + 3x3 subject to: 2x1 + x2 + 3x3 9 4x1 + 3x2 + 5x3 12 with, x3 0 23) Maximize z = x1 + 3x2 + x3 + 2x4 subject to: 2x1 + x2 + 5x3 + 6x4 25 5x1 + 3x2 + 4x3 + x4 60 with, x3 0, x4 0 24) Maximize z = x1 + 2x2 + 4x3 + 6x4 subject to: x1 + 2x2 + 3x3 + x x1 + x2 + 2x3 + x4 75 with, x3 0, x4 0 A toy making company has at least 300 squares of felt, 700 oz of stuffing, and 230 ft of trim to make dogs and dinosaurs. A dog uses 1 square of felt, 4 oz of stuffing, and 1 ft of trim. A dinosaur uses 2 squares of felt, 3 oz of stuffing, and 1 ft of trim. 25) It costs the company $1.65 to make each dog and $1.52 for each dinosaur. What is the company's minimum cost? 26) It costs the company $1.72 to make each dog and $1.88 for each dinosaur. The company wants to minimize its costs. What are the coefficients of the objective function? 27) It costs the company $1.62 to make each dog and $1.83 for each dinosaur. The company wants to minimize its costs. What are the coefficients of the constraint inequality for trim? Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill #1 contains 4 units of A and 3 of B. Pill #2 contains 1 unit of A, 2 of B, and 4 of C. Pill #3 contains 10 units of A, 1 of B, and 5 of C. 28) Pill #1 costs 10 cents, pill #2 costs 7 cents, and pill #3 costs 3 cents. Larry wants to minimize cost. What is the constraint inequality for vitamin A? 29) Pill #1 costs 9 cents, pill #2 costs 9 cents, and pill #3 costs 1 cent. How many of each pill must Larry take to minimize his cost? State the dual Use y1, y2, y3 and y4 as the variables. Given: y1 0, y2 0, y3 0, and y ) Maximize z = 5x1 + 7x2 subject to: 9x1 + 3x2 190 x1 + x2 18 2x1 + 6x ) Minimize w = 2x1 + 3x2 + x3 subject to: x1 + 3x2 + 2x3 31 2x1 + 4x2 + 3x3 67, x3 0 32) Minimize w = 6x1 + 2x2 + 3x3 + 2x4 subject to: 4x1 + 5x2 + 4x3 + 3x4 45 4x1 + 6x2 + 5x3 + 11x4 55, x3 0, x4 0 33) Maximize z = x1 + 2x2 + 3x3 subject to: 6x1 + 3x2 + x3 16 4x1 + 7x2 + 3x3 21, x3 0 34) Minimize w = 6x1 + 3x2 subject to: 3x1 + 2x2 28 2x1 + 5x2 48 3

4 35) Maximize z = 3x1 + 2x2 subject to: x1 + x2 28 2x1 + x2 22 Use the simplex method to solve the linear programming 36) Minimize w = y1 + 4y2 subject to: 4y1 + 3y2 75 3y1 + 5y2 93 y1 0, y2 0 37) Minimize w = 5y1 + 3y2 subject to: 2y1 + 3y2 9 2y1 + y2 11 y1 0, y2 0 4

5 Answer Key Testname: 1324-SIMP-PT 1) Maximum of 60; minimum of ) Maximum of 100 when x = 8 and y = 3 3) 0 VIP and 24 SST 4) $53,893 5) 14 P1 planes and 7 P2 planes 6) x2 = 2, s2 = 4, s3 = 2, z = 12; x1, x3, s1 = 0 7) x3 = 5, x2 = -7, z = -10; x1, s1, s2 = 0 8) x3 = 5, s2 = 1, s3 = -27, z = 15; x1, x2, s1 = 0 9) x3 = 14, s2 = -6, s3 = 4, z = 42; x1, x2, s1 = 0 10) x1 x2 s1 s2 z ) x1 x2 s1 s2 z ) x1 x2 s1 s2 z ) x1 = 4, s1 = 3, z = 2; x2, x3, s2 = 0 14) s1 = 9, s3 = 16, x2 = 20, z = 21; x1, s2 = 0 15) s2 = 16, s1 = 18, x1 = 17, z = 10; x2, s3 = 0 16) s2 = 10, x2 = 8, z = 16; x1, x3, s1 = 0 17) x1, x2, s1 = 0, x3 = 21, s2 = 14, z = 20 18) $225,000 19) $225,000 20) 2A + 3B + C 28,000 21) Maximum at 15 for x2 = 3, s1 = 2, s3 = 8 22) Maximum is 20 when x1 = 0, x2 = 4, x3 = 0 23) Maximum is 60 when x1 = 0, x2 = 20, x3 = 0, x4 = 0 24) Maximum is 450 when x1 = 0, x2 = 0, x3 = 0, x4 = 75 25) $317 26) 1.72, ) 1, 1 28) 4P1 + P2 + 10P ) P1 = 0, P2 = 0, P3 = 12 30) Minimize w = 190y1 + 18y2 + 85y3 subject to: 9y1 + y2 + 2y3 5 3y1 + y2 + 6y3 7 31) Maximize z = 31y1 + 67y2 subject to: y1 + 2y2 2 3y1 + 4y2 3 2y1 + 3y2 1 32) Maximize z = 45y1 + 55y2 subject to: 4y1 + 4y2 6 5y1 + 6y2 2 4y1 + 5y2 3 3y1 + 11y2 2 33) Minimize w = 16y1 + 21y2 subject to: 6y1 + 4y2 1 3y1 + 7y2 2 y1 + 3y2 3 34) Maximize z = 28y1 + 48y2 subject to: 3y1 + 2y2 6 2y1 + 5y2 3 35) Minimize w = 28y1 + 22y2 subject to: y1 + 2y2 3 y1 + y2 2 36) 31 when y1 = 31 and y2 = 0 37) 27.5 when y1 = 5.5 and y2 = 0 5