$2.$ Consider the simple network in the below figure where there are $100$ vehicles per hour travelling from $\backslash ($ A $\backslash )$ to $\backslash ($ X $\backslash )$ and $500$ from $\backslash ($ B $\backslash )$ to $\backslash ($ X $\backslash ).$ The travel time versus flow relationships $($volumedelay functions$)$ are depicted in the figure in minutes and the flow v in vehicles per hour.

a$.$ Express the objective function of the mathematical program corresponding to Wardrop's user equilibrium in terms of the flows and travel time$-$flow relationships in the figure. Hint: refer to the Traffic Assignment lecture slides.

b$.$ Calculate the equilibrium flows on each link and the travel time for each group of travelers. Calculate the value of the objective function above under equilibrium conditions $-$ i$.$e$.,$ the total expenditure in travel time on the network in veh$-$hr$.$ You should start with an incremental assignment. You can assume the following loading: $50\backslash \%$ A$->$X$,50\backslash \%$ B$->$X$,50\backslash \%$ B$->$X$,50\backslash \%$ A$->$X$.$ You should then use an all$-$or$-$nothing assignment as your second volume in the Frank$-$Wolfe $($FW$)$ algorithm. Perform only one FW iteration. You can use the same FW approximation as in the class example $-$ i$.$e$.,$ evaluate $\backslash (\backslash$lambda$=0,1/3,2/3\backslash ),$ and $1.$

c$.$ Local traffic engineers have decided to install speed restrictions on link C$->$D so that the new travel time versus flow function is:

$\backslash [$

t$=5\mathrm{.}2+0\mathrm{.}002$ v

$\backslash ]$

The total expenditure in travel time $($veh$-$hr$)$ in the system is less than in $($b$)$ under these conditions. Why would this be the case? The following two tables show the final FrankWolfe flow results for the relationships in part $($b$)$ and $($c$),$ respectively. Please use these tables in your response.