Traffic Engineering

Engineers often analyze traffic flow patterns in urban areas to determine settings for traffic

control devices such as traffic lights. The figure below shows the flow of downtown traffic in a

certain city during the rush hours on a typical day.

The arrows indicate the direction of traffic flow on

each one$-$way road, and the average number of

vehicles per hour entering and leaving each

intersection appears beside each road.

$5$ th Avenue and $6$th Avenue can each handle up

to $2000$ vehicles per hour without causing

congestion, whereas the maximum capacity of

both $4$th Street and $5$th Street is $1000$ vehicles

per hour. The flow of traffic is controlled by traffic

lights installed at each of the four intersections.

a$)$ Arrange this information into a system of equations describing the traffic flow in terms of $x_1,$

$x_2,x_3,x_4.$ You should have $4$ equations $($one for each intersection$),$ and not all $x_i$ terms will

appear in each equation $($in other words, the coefficients will be zero for some $x$ terms in

some equations$).$ The intersection at $4^{th}St$&$5^{th}$ Ave has been done for you as an example.

vehicles $in=$ vehicles out $300+1200=x_1+x_{4}1x_1+0x_2+0x_3+1x_4=1500$

b$)$ Write MATLAB statements to define the arrays for this system of equations:

c$)$ Try to solve the system of equations in MATLAB using $x=$ Alb. What is the result?

d$)$ Use the rank and length functions to demonstrate that the system is underdetermined $($i$.$e$.,$

has an infinite number of solutions$).$ Show commands and results.