Consider the typical road network map on the next page. The streets are all one$-$way

with the arrows indicating the direction of traffic flow. The flow of traffic in and out of

the network is measured in vehicles per hour $($vph$).$ The figure given here is based on

midweek peak traffic hours, $7$ A$.$M$.$ to $9$ A$.$M$.$ and $4$ P$.$M$.$ to $6$ P$.$M$.$

Assume that the following traffic law applies.

All traffic entering an intersection must leave that intersection.

$($a$)$ Construct a mathematical model that can be used to analyze this network.

$($b$)$ Find the solution$($s$)$ of the mathematical model obtained in $($a$).$

$($c$)$ Suppose it becomes necessary to perform road work on the stretch DC of Monroe

Street. It is desirable to have as small a flow $x_3$ as possible along this stretch of

road. The flows can be controlled along various branches by means of traffic

lights. What is the minimum value of $x_3$ along $DC$ that would not lead to traffic

congestion? Use the preceding system of equations to answer this question.