Question

Let X₁ and X₂ are optimal solutions of a LPP, then
(a) X = λ X₁ + (1 − λ) X₂, λ ∈ R is also an optimal solution
(b) X = λ X₁ + (1 − λ) X₂, 0 ≤ λ ≤ 1 gives an optimal solution
(c) X = λ X₁ + (1 + λ) X₂, 0 ≤ λ ≤ 1 gives an optimal solution
(d) X = λ X₁ + (1 + λ) X₂, λ ∈ R gives an optimal solution

Answer 1

(b) X = λX₁ + (1 − λ)X₂, 0 ≤ λ ≤ 1 gives an optimal solution

A set A is convex if, for any two points, x₁, x₂ ∈ A, and $\mathrm{\lambda }\in \left[0,1\right]$ imply that
$\lambda x_1+\left(1-\lambda \right)x_2\in A$.

Since,here X₁ and X₂ are optimal solutions
Therefore, their convex combination will also be an optimal solution

Thus, X = λX₁ + (1 − λ)X₂, 0 ≤ λ ≤ 1 gives an optimal solution.