Discussion
Consider the following linear programming problem :
Maximize Z = \(2x_1 + 3x_2\)
Subject to : \(x_1 + 3x_2 \le 12\)
\(3x_1 + 2x_2 \le 12\)
\(x_1, x_2 \ge 0\)
At which points of graphical solution space the infeasible basic optimum solutions are represented ?
Answer: A
The infeasible corner points are (x1 = 0, x2 = 6) and (x1 = 12, x2 = 0).
The basic solutions are the intersections of the constraint lines with the axes and with each other. The line \(x_1 + 3x_2 = 12\) meets the axes at (12, 0) and (0, 4); the line \(3x_1 + 2x_2 = 12\) meets them at (4, 0) and (0, 6).
\[ \begin{aligned} (0, 6)&: 0 + 3(6) = 18 \gt 12 \\ (12, 0)&: 3(12) + 0 = 36 \gt 12 \end{aligned} \]
Each violates the other constraint, so these two basic solutions are infeasible. The points in B and C are not intersections of these lines at all, and D is wrong because two corner points are infeasible.
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