$($b$)$ An Origin$-$Destination pair $($OD$)$ with a demand of $1500ve\frac{h}{h}r$ is initially

connected by Link $1$ and Link $2$ only. Following complaints about the

time vehicles have to spend travelling between $O$ and $D,$ an additional

link $($Link $3)$ is built to connect the two zones.

Assume that the cost function of the links is of the BPR type, with

$k=p=1,$ and that the characteristics of the links are those reported in the

Table $6\mathrm{.}1$ below.

Table $6\mathrm{.}1$: Link characteristics

Calculate the User Equilibrium solution with and without Link $3,$ and

determine whether the introduction of the new link actually improves

the situation. Solve the problem with the two links by algebraic means,

and that with three links using the Frank$-$Wolfe algorithm.