Discussion
The following Linear Programming problem has :
\(\text{Max}\quad Z = x_{1} + x_{2}\)
Subject to \(x_{1} - x_{2} \ge 0\)
\(3x_{1} - x_{2} \le -3\)
and \(x_{1}, x_{2} \ge 0\)
Answer: B
The problem has no feasible solution. The first constraint gives \(x_{1} \ge x_{2}\). The second gives \(x_{2} \ge 3x_{1} + 3\). Combining them, \(x_{2} \ge 3x_{1} + 3 \ge 3x_{2} + 3\), so \(-2x_{2} \ge 3\), that is \(x_{2} \le -1.5\). This contradicts \(x_{2} \ge 0\).
With the feasible region empty, there is no feasible point, no unbounded direction and no single optimal point. So options A, C and D are all ruled out.
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