Consider the following linear program.

Min	8X	+	12Y
s.t.
	1X	+	3Y		9
	2X	+	2Y		14
	6X	+	2Y		18
	X,	Y	0

(a) Use the graphical solution procedure to find the optimal solution. (Graph the constraint lines, the feasible region, the objective function line, and the optimal solution.)

What is the value of the objective function at the optimal solution? ____ at (X, Y) = (____________)

(b) Assume that the objective function coefficient for X changes from 8 to 6. Use the graphical solution procedure to find the new optimal solution. (Graph the constraint lines, the feasible region, the objective function line, and the optimal solution.)

Does the optimal solution change? The extreme point (_____) remains optimal. optimal. The value of the objective function becomes______.

(c) Assume that the objective function coefficient for X remains 8, but the objective function coefficient for Y changes from 12 to 6. Use the graphical solution procedure to find the new optimal solution. (Graph the constraint lines, the feasible region, the objective function line, and the optimal solution.)

Does the optimal solution change? The extreme point (______) becomes optimal. The value of the objective function becomes _________.

(d) The computer solution for the linear program in part (a) provides the following objective coefficient range information.

Variable	Objective Coefficient	Allowable Increase	Allowable Decrease
X	8.00000	4.00000	4.00000
Y	12.00000	12.00000	4.00000

How would this objective coefficient range information help you answer parts (b) and (c) prior to resolving the problem?

The objective coefficient range for variable X is____ to ___ . Since the change in part (b) is within this range, we know the optimal solution will not change. The objective coefficient range for variable Y is_____ to_____ . Since the change in part (c) is outside this range, we know the optimal solution will change.