4. Consider the following resource-allocation problem and the accompanying optimal tableau (X5, X6, and x, are...

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4. Consider the following resource-allocation problem and the accompanying optimal tableau (X5, X6, and x, are the respective slack variables). maximize z = 15x + 8x + 10x + 12x, (Profit $) Subject to x + 2x x + x + x 2x + x4 S 20 + x4 54 + x3 + x 36 X X X3 X4 20 Z 1 9 0 0 1/2 1 0 X0 X60-3/2 0 0 x3 0 2 0 1 X 2 (Resource 1) (Resource 2) (Resource 3) X X RHS 440 0 10 4 1/2 1/2 0 0 10 1 -1 8 -1/2-1/2 1 0 0 1 36 (a) Write the dual problem and specify the optimal dual solution from the foregoing tableau. (b) What are the shadow prices of the resources? If you were to choose between increasing the amount of resource 1, 2, or 3, which would you choose to increase and why? (c) Suppose that the coefficient of x, in the objective function changes from 12 to 16. Use sensitivity analysis to find the new optimal solution. (d) Suppose that available amount of resource I changes from 20 to 40. Use sensitivity analysis to find the new optimal solution. (e) If the constraint x, 210 is added to the problem, use sensitivity analysis to find the new optimal solution. (f) Suppose that the constraint 3x + 2x + 2x + x 580 is added to the problem, use sensitivity analysis to find the new optimal solution. (g) Suppose that a new product is proposed with objective coefficient 16 and consumption vector (1 2 1). Use sensitivity analysis to find the new optimal solution. 4. Consider the following resource-allocation problem and the accompanying optimal tableau (X5, X6, and x, are the respective slack variables). maximize z = 15x + 8x + 10x + 12x, (Profit $) Subject to x + 2x x + x + x 2x + x4 S 20 + x4 54 + x3 + x 36 X X X3 X4 20 Z 1 9 0 0 1/2 1 0 X0 X60-3/2 0 0 x3 0 2 0 1 X 2 (Resource 1) (Resource 2) (Resource 3) X X RHS 440 0 10 4 1/2 1/2 0 0 10 1 -1 8 -1/2-1/2 1 0 0 1 36 (a) Write the dual problem and specify the optimal dual solution from the foregoing tableau. (b) What are the shadow prices of the resources? If you were to choose between increasing the amount of resource 1, 2, or 3, which would you choose to increase and why? (c) Suppose that the coefficient of x, in the objective function changes from 12 to 16. Use sensitivity analysis to find the new optimal solution. (d) Suppose that available amount of resource I changes from 20 to 40. Use sensitivity analysis to find the new optimal solution. (e) If the constraint x, 210 is added to the problem, use sensitivity analysis to find the new optimal solution. (f) Suppose that the constraint 3x + 2x + 2x + x 580 is added to the problem, use sensitivity analysis to find the new optimal solution. (g) Suppose that a new product is proposed with objective coefficient 16 and consumption vector (1 2 1). Use sensitivity analysis to find the new optimal solution.

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