Two parallel highways connect two cities A and B and their performance functions areProblem $2.$

$($also see Problem $3\mathrm{.}3$ and $3\mathrm{.}4$ in the textbook$).$ Given the network shown below.

$($The link numbers are shown on the respective links$).$ The O$-$D trip rate is $4$ units of flow

between nodes $1$ and $3,$ and the link cost functions are given as follows:

$t_1=2+x_1^2$;$t_2=3+x_2$;$t_3=1+2x_3^2$;$t_4=2+4x_4.$

$($a$)$ Formulate the user equilibrium traffic assignment problem using the path$-$link variables and

solve it analytically $($i$.$e$.,$ using the first$-$order conditions$).$ Comment the uniqueness issue of the

$1$

solution $($e$.$g$.,$ link flows, link travel times, path flows, path travel times, O$-$D travel time, and

objective value. See the course slides for more information$).$

$($b$)$ Formulate the user equilibrium traffic assignment problem using the node$-$link variables and

solve it analytically. Comment the uniqueness issue of the solution $($e$.$g$.,$ link flows, link travel

times, O$-$D travel time, and objective value$).$

$1011$ t $=1+0.$ v and $20052$ t $=10+0.$ v respectively, where $1$t and $2$t are travel times $($in minutes$)$

and $1$ v and $2$ v are traffic flows $($in vehicles$).$ The total demand between two cities is $3900$

vehicles