Consider the following optimization problem and the constraint boundary lines given below.

Maximize profit = 4X + 4Y
Constraints
3X + 2Y ≤ 150
X - 2Y ≤ 10
2X + 3Y ≤150
X, Y ≥ 0

When the constraint coefficient of x in the blue constraint changes from 1 to 3, the optimal solution changes.

Question

Consider the following optimization problem and the constraint boundary lines given below.

Maximize profit = 4X + 4Y 
Constraints
3X + 2Y ≤ 150
X - 2Y ≤ 10
2X + 3Y ≤150
X, Y ≥ 0

When the constraint coefficient of x in the blue constraint changes from 1 to 3, the optimal solution changes.

BlackTom AI · Accepted Answer

To analyze whether altering the blue constraint from 1x - 2y ≤ 10 to 3x - 2y ≤ 10 can change the optimal solution, we first restate the problem and the options in our own words. The goal is to maximize profit 4X + 4Y, which is equivalent to maximizing X + Y, subject to the three constraints 3X + 2Y ≤ 150, X - 2Y ≤ 10, 2X + 3Y ≤ 150, and X, Y ≥ 0. The blue constraint line originally has the form X - 2Y = 10 (slope 1/2 when solved for Y), and would become 3X - 2Y = 10 (slope 3/2 when solved for Y) if the coefficient of X changes from 1 to 3.

Option 1 (true): The claim is that changing the coefficient of X in the blue constraint changes the optimal solution. This is plausible because the feasible region is defined by the intersection of all constraint half-spaces, and the blue constraint is one of them. Changing its slope alters where this line intersects the other constraint boundaries, thus altering the set of feasible points and potentially which vertex yields the maximum X + Y. In linear programming, the optimum occurs at a vertex (or along a segment if multiple optima exist), and moving a constraint line typically shifts which vertex is the binding one. Therefore, it is reasonable that the optimal solution could change when the blue constraint's slope changes.

Option 2 (false): The claim would be that the optimum never changes regardless of the blue constraint’s coefficient. This would imply that the same corner or same binding set of constraints always yields the maximum X + Y, no matter how the blue line is tilted. However, altering a constraint that directly involves X and Y can reorder which constraints are binding at the optimum, especially since two other lines (the black and red lines in the diagram) define the existing feasible region. Since the intersection points between the blue line and the other constraint boundaries move as the slope changes, the vertex achieving the maximum can shift, contradicting the notion that the optimum is invariant.

Additional nuance to consider: the objective is to maximize X + Y, so the optimal point tends to lie at a vertex where X and Y are both relatively large but still feasible. If the blue constraint becomes steeper (slope 3/2), it may become more or less restrictive in regions where X and Y are large, potentially selecting a different vertex as optimal. Conversely, making the line less steep (slope 1/2) could favor a different corner as feasible and optimal. In short, the optimal vertex depends on the relative positioning of all constraint lines, and changing the blue constraint’s slope can indeed change which vertex is optimal.

Concisely, the feasibility region is a polygon determined by all constraints. A change in one constraint’s slope redefines that polygon’s shape and the corner that yields the maximum X + Y. Since both the feasible region and the objective interact to determine the optimum, it is quite plausible for the optimal solution to change when the blue constraint coefficient changes from 1 to 3. Therefore, the statement is true.

BU.520.601.T2.FA25 Final Exam

Consider the following optimization problem and the constraint boundary lines given below. Maximize profit = 4X + 4Y Constraints 3X + 2Y ≤ 150 X - 2Y ≤ 10 2X + 3Y ≤150 X, Y ≥ 0 When the constraint coefficient of x in the blue constraint changes from 1 to 3, the optimal solution changes.

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