Discussion
Consider the LPP :
Max \(Z = 3x + 5y\)
Subject to \(x + 5y \le 10\);
\(2x + 2y \le 5\);
\(x \ge 0,\ y \ge 0\).
Which of the following statements is true ?
\(S_{1}\) : There exists a unique optimal solution to the LPP.
\(S_{2}\) : There exists a unique optimal solution to the dual problem.
Answer: C
Both \(S_{1}\) and \(S_{2}\) are true.
The feasible region has corners (0, 0), (2.5, 0), (0, 2) and the intersection of \(x + 5y = 10\) with \(x + y = 2.5\):
\[ \begin{aligned} 4y &= 7.5,\ y = 1.875,\ x = 0.625 \\ Z &= 3(0.625) + 5(1.875) = 11.25 \end{aligned} \]
The other corners give 0, 7.5 and 10, so the optimum 11.25 is attained only at (0.625, 1.875). The objective (3, 5) is not parallel to either constraint, (1, 5) or (2, 2), so no edge is optimal and \(S_{1}\) holds.
The optimal primal solution is non-degenerate (both variables positive, both constraints tight), so complementary slackness fixes the dual solution uniquely: \(u_{1} + 2u_{2} = 3\), \(5u_{1} + 2u_{2} = 5\), giving \(u_{1} = 0.5\), \(u_{2} = 1.25\) with value \(10(0.5) + 5(1.25) = 11.25\). So \(S_{2}\) holds too.
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