Question

How to write a linear equation that describes this image of flow of traffic along the various branches. How to use these equations to determine the minimum flow possible along x1. What are the other flows at that time ?

Answer 1

See below.

Explanation:

Begining from left and supposing that the flow is a continuous flow,

Applying Kirchoff's law to each knot,

`{ (x_5 = 110+x_4), (80 = x_5-x_6), (75+x_6 = x_7), (120=x_7-x_8), (155+x_8 = x_1), (100=x_1-x_2), (90+x_2=x_3), (130=x_3-x_4) :}`

Calling `X = (x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)` and

`A = ((0, 0, 0, -1, 1, 0, 0, 0),(0, 0, 0, 0, -1, 1, 0, 0),(0, 0, 0, 0, 0, 1, -1, 0),(0, 0, 0, 0, 0, 0, -1, 1),(-1, 0, 0, 0, 0, 0, 0, 1),(-1, 1, 0, 0, 0, 0, 0, 0),(0, 1, -1, 0, 0, 0, 0, 0),(0, 0, -1, 1, 0, 0, 0, 0))`

and

`b = (-110, 80, 75, 120, 155, 100, 90, 130)`

Note that by making `A cdot X + b = 0` we are reproducing the same set of linear constraints. The first two lines of `A cdot X + b = 0` read

`{(-x_4+x_5-110=0),(-x_5+x_6+80=0):}`

etc.

The minimization problem can be stated as a linear programming problem:

https://en.wikipedia.org/wiki/Linear_programming

`min x_1`

subject to

`A cdot X + b = 0`
`X ge 0`

The solution follows

`X = (155, 55, 145, 15, 125, 45, 120, 0)` with

`min x_1 = 155`