Project 2 YOUR NAME____________________________________________ This project is given on page 149 in the textbook. Refer to the furniture manufacturing problem discussed in Ch3.1 pages 117-120 (Read the introduction to the problem page117-118) Maximize 80x +70y Subject to 6x + 3y <= 96 x + y <= 18 2x + 6y <= 72 x >=0, y >= 0 carpentry finishing upholstery constraints on labor hours in each manufacturing stage. (Now read the introduction in Shadow Prices page 149, stop reading before question1 ) Study the solution of this problem below .This is called base case. The graph of the feasible set is made here for the formulated linear programming problem above 6x + 3y <= 96 y <= 32 - 2x x 0 2 4 6 8 10 12 14 16 18 20 x + y <= 18 y <= 18 - x 32 28 24 20 16 12 8 4 0 -4 -8 2x + 6y <= 72 y <= 12 - 1/3x y >= 0 y >= 0 18 12 16 11.3333333333 14 10.6666666667 12 10 10 9.3333333333 8 8.6666666667 6 8 4 7.3333333333 2 6.6666666667 0 6 -2 5.3333333333 0 0 0 0 0 0 0 0 0 0 0 35 30 25 20 c y <= 32 - 2x 15 y <= 18 - x y <= 12 - 1/3x 10 y >= 0 b 5 d a 0 0 -5 5 10 15 20 25 20 c y <= 32 - 2x 15 y <= 18 - x y <= 12 - 1/3x 10 y >= 0 b 5 d a 0 0 5 10 15 20 25 -5 -10 Corner point a b c d point a point b point c point d x,y 80x + 70y 14,4 9,9 0,12 16,0 1400 optimal solution 1350 840 1280 32-2x=18-x 32-18=2x-x 14=x y=32-2*14 y=4 18-x=12-1/3x 18-12=x-1/3x 6=2/3x 9=x y=18-9 y=9 y=12-1/3*0 y=12 x=0 0=32-2x x=16 y=0 Now follow the direction answer the questions. If necessary insert more rows. Show your work here. Save when finished with your name in the name of the file and submit it. 1.Suppose that the finishing constraint now has 19 instead of 18 labor hours limit. Write he new finishing constrain. x+y<=19 2. Add the graph of the new finishing constraint on the graph for the base case below Label the two new corner points as A and B (see the picture in the project page 149) Find the coordinates of points A and B. 35 30 25 20 35 30 25 20 c y <= 32 - 2x y <= 18 - x y <= 12 1/3x y >= 0 x+y<=19 15 10 b 5 d a 0 0 5 10 15 20 25 -5 -10 3. Determine the optimal solution for the problem with the new finishing constraint (ignore points a and b when objective functi 4. Calculate the shadow price as the difference between the optimal profit with the new constraint and the optimal profit of the 5. Return to 18 labor-hour limit in finishing constraint and increase the labor-hour limit by 1 in carpentry constraint. Solve the problem with the new carpentry constraint. Determine the shadow price for the carpentry constraint. 6.Return to base case value in carpentry constraint and increase the right-hand side value of the upholstery constraint by 1. Solve the problem with the new upholstery constraint. Determine the shadow price for the upholstery constraint. 7. Fill in the blanks in the following sentence. The shadow price associated with a constraint can be interpreted as the change in the value of the ______________________ _______________per unit change of the constraint's right-hand side limit. urs in each manufacturing stage. y <= 32 - 2x y <= 18 - x y <= 12 - 1/3x y >= 0 y <= 32 - 2x y <= 18 - x y <= 12 - 1/3x y >= 0 new finishing constrain. y <= 32 - 2x y <= 18 - x y <= 12 1/3x y >= 0 x+y<=19 oints a and b when objective function is calculated). nstraint and the optimal profit of the base case. n carpentry constraint. rpentry constraint. of the upholstery constraint by 1. pholstery constraint