# Linear System - Network Flow Matrices

• 1LastTry
In summary, the minimum flow along AC when ED is closed due to roadwork is 20 cars per minute, assuming all roads have non-negative flow.
1LastTry

## Homework Statement

Consider the network of streets with intersections a,b,c,d and e below. The arrows indicate the direction of traffic flow along the oneway streets, and the numbers refer to the exact number of cars observed to enter or leave a,b,c,d and e during one minute. Each xi denotes the unknown number of cars which passed along the indicated streets during the same period.
See Attached Image
Linear System:
x1+x6-x5=55
x5-x4=35
x6+x3-x4=60
x3-x2=40
x1-x2=70

Reduced row-echelon form
1 0 0 0 -1 1 | 55
0 1 0 0 -1 1 |-15
0 0 1 0 -1 1 |25
0 0 0 1 -1 0 |-35
0 0 0 0 0 0 |0
s t
General Solution:
x1=55+s-t
x2=-15+s-t
x3=25+s-t
x4=-35-s
x5=s
x6=t

The question is: if ED were closed due to roadwork, find the minimum flow along AC, using your results in the general solution.
Note: x2 = ED therefor x2=0, and AC is x6

## The Attempt at a Solution

I know x2=0 and no idea what to do with S and T...
what I did is set x2=0 so -15+s-t=0 so S=t+15
And sub it into the general solution you get:
x1=70
x2=0
x3=40
x4=-20+t
x5=15+t
x6=t

so x4≥0 when t≥20
x6≥0 when t≥0
So minflow is 0?

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1LastTry said:

## Homework Statement

Consider the network of streets with intersections a,b,c,d and e below. The arrows indicate the direction of traffic flow along the oneway streets, and the numbers refer to the exact number of cars observed to enter or leave a,b,c,d and e during one minute. Each xi denotes the unknown number of cars which passed along the indicated streets during the same period.
See Attached Image
Linear System:
x1+x6-x5=55
x5-x4=35
x6+x3-x4=60
x3-x2=40
x1-x2=70

Reduced row-echelon form
1 0 0 0 -1 1 | 55
0 1 0 0 -1 1 |-15
0 0 1 0 -1 1 |25
0 0 0 1 -1 0 |-35
0 0 0 0 0 0 |0
s t
General Solution:
x1=55+s-t
x2=-15+s-t
x3=25+s-t
x4=-35-s
x5=s
x6=t

The question is: if ED were closed due to roadwork, find the minimum flow along AC, using your results in the general solution.
Note: x2 = ED therefor x2=0, and AC is x6

## The Attempt at a Solution

I know x2=0 and no idea what to do with S and T...
what I did is set x2=0 so -15+s-t=0 so S=t+15
And sub it into the general solution you get:
x1=70
x2=0
x3=40
x4=-20+t
x5=15+t
x6=t

so x4≥0 when t≥20
x6≥0 when t≥0
So minflow is 0?

No, you are not thinking it through! You need ALL xi >= 0, so you need x4 >= 0 and so you need t >= 20. Just having t >= 0 is not good enough.

RGV

I am trying to find the MINIMUM flow so...

## 1. What is a linear system?

A linear system is a mathematical representation of a set of linear equations or inequalities. It involves a set of variables and coefficients that are related through linear relationships.

## 2. What is a network flow matrix?

A network flow matrix is a type of linear system used to model the flow of resources or information in a network. It consists of nodes, which represent points in the network, and arcs, which represent the paths connecting the nodes.

## 3. How is a network flow matrix used?

A network flow matrix is used to analyze and optimize the flow of resources or information in a network. It can help identify the most efficient routes for transportation, the optimal distribution of goods, or the best way to allocate resources in a supply chain.

## 4. What are some real-world applications of network flow matrices?

Network flow matrices have many real-world applications, including transportation and logistics, telecommunications, energy distribution, and supply chain management. They are also used in computer network routing algorithms and in analyzing social networks.

## 5. What are some methods for solving network flow matrices?

There are several methods for solving network flow matrices, including the simplex method, the primal-dual method, and the network simplex method. These methods involve finding the optimal solution by iteratively adjusting the flow through each arc until the desired objective is achieved.

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