Let X_1 and X_2 are optimal solutions of a LPP, then: - Vidyadip

MCQ

Let $X_1$ and $X_2$ are optimal solutions of a $\text{LPP}$, then:

A
$\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,\lambda\in$ R is also an optimal solution
✓
$\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,0\leq\lambda\leq1$ given an optimal solution
C
$\text{X}=\lambda\ \text{X}_1+(1+\lambda)\text{X}_2,0\leq\lambda\leq1$ given an optimal solution
D
$\text{X}=\lambda\ \text{X}_1+(1+\lambda)\text{X}_2,\lambda\in$ R given an optimal solution

✓

Answer

Correct option: B.

$\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,0\leq\lambda\leq1$ given an optimal solution

A set A is convex if, for any two points $X_1, X_2$$\in\text{A}$ and $\lambda\in0,1$ imply that $\lambda\times1+1-\lambda\times2\in\text{A}$.
Since, here $X_1$ and $X_2$ are optimal solution
Therefore, their convex combination will also be an optimal solution
Thus, $\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,0\leq\lambda\leq1$ gives an optimal solution.

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