Consider the following linear programming problem P with three resources and three activities:

Maximize Z = 4x₁ + 3x₂ + 6x₃ (0)

subject to 2x₁ + 5x₂ + 9x₃ 500 (1) 4x₁ + 5x₃ 350 (2) 2x₂ + 3x₃ 150 (3)

and x₁, x₂, x₃ 0

Let x₄, x₅, and x₆ denote the slack variable of functional constraint (1), (2), and (3), respectively. When we apply the simplex method to problem P, it yields the following final simplex tableau:

Basic Variable	Eq. No	Coefficient of:							Right Side
		Z	x1	x2	x3	x4	x5	x6
Z	(0)	1	0	0	2.9	0.6	0.7	0	545
x1	(1)	0	1	0	1.25	0	0.25	0	87.5
x6	(2)	0	0	0	0.4	-0.4	0.2	1	20
x2	(3)	0	0	1	1.3	0.2	-0.1	0	65

Answer the following independent questions:

(a) (15 points) Use algebraic analysis to find the allowable range for resource #1 (i.e., b₁) so that the current basis remains optimal.

(b) (15 points) The allowable change for unit profit of activity #1 and #2 (i.e., 1 and 2) is given as follows: -2.32 1 and -2.23 2 7. Using this information and the result obtained in part (a), answer the following independent questions (use 100 percent rule, no need to re-optimize):

If c₁ decreased by 1 and c₂ increased by 5, can you find the optimal profit without changing the basis (tableau)? Yes or no, and why?

If b₁ becomes 400, what would the optimal profit Z* be?

(c) (20 points) Suppose the coefficients of resource #3 have been changed to a₃₁ = 1, a₃₂ = 2, a₃₃ = 2, and b₃ = 100. Will the optimal solution remain unchanged? If not, re-optimize to find the new optimal solution.

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