Question

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

A. X = λ X₁ + (1 – λ)X₂, λ ϵ 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 give an optimal solution

D. X = λ X₁ + (1 + λ)X₂, λ ϵ R gives an optimal solution

Answer 1

Correct answer is B.

Given, X₁ and X₂ are optimal solutions of a Linear programming problem(LPP).

This means that, {X₁, X₂} C (a convex Set) as the optimal solution of a LPP is convex.

Now by using the definition of a Convex set,

A set of points C is called convex if, for all λ in the interval 0 ≤ λ ≤ 1, λy + (1 − λ)z is contained in C whenever y and z are contained in C.

By using this property of Convex set,

If {X₁ ,X₂} C (a convex set of optimal solutions), then

X = λX₁ + (1 − λ) X₂ where 0 ≤ λ ≤ 1, is also contained in C (the optimal solution set).

This proves that, also X ∈ C.