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I wondered if someone could explain to me the intuition of what it would mean if the objective function in a Linear Programme is a multiple of one of the constraints?

I am thinking it means that the lines are parallel with the objective function being at a different contour. Does this imply anything about the uniqueness of an optimal solution? Would appreciate some intuition

InvestingScientist
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1 Answers1

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It means indeed that the optimal solution is not unique, if it exists. The corresponding constraint must be active. In a linear program with two variables it can be shown graphically. An example. Le´t suppose you have the following program:

$\textrm{max} \ Z=500x_{1}+300x_{2}$

s.t.

$15x_{1}+5x_{2}\leq 300$

$10x_{1}+6x_{2}\leq 240$

$8x_{1}+12x_{2}\leq 450$

$x_{1},x_{2}\geq 0$

We can solve the objective funktion for $x_2$: $ \, \, \, x_2=\frac{Z}{300}-\frac53x_1$. This function has a slope of $\color{red}{-\frac53}$. And we can solve the second constraint (equality) for $x_2$ as well:

$10x_{1}+6x_{2}=240\Rightarrow x_2=40\color{red}{-\frac53}x_2$

For more detailed information with a graph to this example see here.

callculus42
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