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Let $X_1$ and $X_2$ are the optimum solutions of LPP, then

(a) $X = λX_1+(1- λ)X_2$, $λ \in \Bbb R$ is also an optimal solution

(b) $X = λX_1+(1-λ)X_2$, $0 \leq λ \leq 1$ gives an optimal solution

(c) $X = λX_1+(1+λ)X_2$, $0 \leq λ \leq 1$ gives an optimal solution

(d) $X = λX_1+(1+λ)X_2$, $λ \in \Bbb R$ gives an optimal solution

Please explain (in detail) how to solve it...

Thanks in advance.

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user124004
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1 Answers1

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A set $K$ is called convex if $X_1, X_2 \in K$ and for $0 \leq λ \leq 1 :$ $λ X_1+(1- λ )X_2 \in K$, that means that each convex combination of two points of $K$ is also a point of $K$. In other words if $X_1,X_2$ are the optimal solutions of LPP, then (b) $X = λX_1+(1-λ)X_2$, $0 \leq λ \leq 1$ gives an optimal solution. A convex combination of $x_1,x_2$ is each point $x$ of the form:$$x=\Sigma_{i=1}^{k}{λ_ix_i}$$ with $λ_i \geq 0$ and $\Sigma_{i=1}^{k}λ_i=1$

Mary Star
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