Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Find the general flow pattern of the network. Assuming that the flows are all nonnegative, what is the smallest possible value for $x_4$?

Points of Intersection: Flow In = Flow Out

A: $x_1+x_4 = x_2$

B: $x_2 = x_3 + 100$

C: $x_3 + 80 = x_4$

The total flow in equals the total flow out, so you can set up other equations...

How do I turn this into a solvable matrix to get the answer?

share|improve this question
    
This question is entirely unintelligible to me -- you've explained almost none of the things you introduced. –  joriki Sep 12 '12 at 8:25

1 Answer 1

Put it each column the corresponding variable, last column is used for constants. So rewrite each equation in the form $...=0$. Then, for $A$ we get $x_1-x_2+x_4=0$, $B$: $x_2-x_3-100=0$, $C$: $x_3-x_4+80=0$. Put that in a matrix: $ \begin{pmatrix} 1 & -1 & 0 & 1 & 0\\ 0 & 1 & -1 & 0 & -100\\ 0 & 0 & 1 & -1 & 80\\ \end{pmatrix} $

Solving that matrix results into $ \begin{pmatrix} 1 & 0 & 0 & 0 & 20\\ 0 & 1 & 0 & -1 & -20\\ 0 & 0 & 1 & -1 & 80\\ \end{pmatrix} $

So, in other words, $x_1=20, x_2=-x_4-20, x_3=-x_4+80$. Since the flows are non-negative, it must hold that $x_4\le -20$ (for $x_2\ge 0$) and $x_4 \le 80$ for $x_3 \ge 0$. So, $x_4 \le -20$. But that is not possible, since $x_4 \ge 0$ so there are no solutions.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.