Question

Let X₁ and X₂ are optimal solutions of a LPP, then

Options

• X = λ X₁ + (1 − λ) X₂, λ ∈ R is also an optimal solution

• X = λ X₁ + (1 − λ) X₂, 0 ≤ λ ≤ 1 gives an optimal solution

• X = λ X₁ + (1 + λ) X₂, 0 ≤ λ ≤ 1 gives an optimal solution

• X = λ X₁ + (1 + λ) X₂, λ ∈ R gives an optimal solution

   

Answer 1

X = λ X₁ + (1 − λ)X₂, 0 ≤ λ ≤ 1 gives an optimal solution

A set A is convex if, for any two points, x₁, x₂ ∈ A, and \[\lambda \in \left[ 0, 1 \right]\] imply that \[ \lambda  \text{ x } _1 + \left( 1 - \lambda \right) x_2 \in A\] .
Since, here  X₁ and X₂ are optimal  solutions 
Therefore, their convex combination will also be an optimal solution

Thus, X = λ X₁ + (1 − λ) X₂, 0 ≤ λ ≤ 1 gives an optimal solution.