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

If we increase the objective function coefficient of x by 2, i.e., 4 becomes 6, the new optimal solution includes point C.

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

If we increase the objective function coefficient of x by 2, i.e., 4 becomes 6, the new optimal solution includes point C.

BlackTom AI · Accepted Answer

Here is the problem restated: We have an optimization problem with Maximize profit = 4X + 4Y subject to the constraints 3X + 2Y ≤ 150, X − 2Y ≤ 10, 2X + 3Y ≤ 150, and X, Y ≥ 0. The statement claims that if we increase the objective function coefficient of x by 2 (from 4 to 6), the new optimal solution includes point C. The answer options are: True or False.

Option 1: True. This option asserts that increasing the coefficient of X in the objective function changes the optimal solution to include the point labeled C on the graph. To evaluate this, we consider how changing the objective coefficients affects the optimal vertex of the feasible region. When the profit coefficients change, the set of supporting hyperplanes (or iso-profit lines) rotates. A vertex becomes optimal only if its objective value is at least as large as that of all other feasible vertices under the new coefficients. The point C lies at the intersection of certain constraint lines, and whether it becomes optimal depends on how the rotated iso-profit line with slope determined by the new coefficients intersects the feasible region. If, under the new objective, the highest profit line touches or passes through C before other vertices, then C becomes part of the optimal solution. The claim is specific: the new optimum includes C. If the rotated iso-profit line does not pass through C or if another vertex yields greater profit with coefficients 6 and 4, then C would not be optimal. Therefore, this statement requires checking which vertex yields the maximum 6X + 4Y over all feasible points; it is not automatically guaranteed by simply increasing X’s coefficient. Consequently, the truth of this option depends on the geometry and is not inherently guaranteed by the change alone.

Option 2: False. This option states that increasing the coefficient of X from 4 to 6 does not lead to the new optimal solution including C. To assess this, we analyze the effect of raising the X coefficient on the slope of iso-profit lines, which is -a2/a1 in standard form a1X + a2Y = b for objective a1X + a2Y. Increasing a1 (the coefficient of X) while keeping a2 the same rotates the iso-profit line about the feasible region, potentially changing which vertex is optimal. If, after rotation, another vertex yields higher profit than C, then C would not be optimal. In many linear programming cases, adjusting coefficients can shift the optimum to a different vertex; whether that vertex is C depends on the relative position of C within the feasible region and the other vertices. If the graph shows that after increasing X’s coefficient to 6, the maximum occurs at a vertex other than C, then the statement would be false. This explanation aligns with the principle that changing objective coefficients can alter the optimal corner, but it does not guarantee inclusion of C without calculation or geometric inspection.

Summary of reasoning for both options: The central idea is that changing the objective coefficients can rotate the objective lines and change which corner is optimal, but this change does not guarantee that C becomes optimal unless the new iso-profit line touches C with a higher profit than all other feasible vertices. The correctness hinges on comparing profits of all candidate vertices under the new coefficients.

Note: Without performing explicit calculations or referencing the plotted figure’s exact vertex coordinates and their profits under the updated objective, we evaluate the logical possibilities: True would require C to indeed maximize 6X + 4Y, while False would indicate some other vertex has the larger value. Both outcomes are plausible depending on the geometry; thus the determination requires analysis beyond the given statement alone.

BU.520.601.T2.FA25 Final Exam

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