Problem $4$

Peak$-$hour traffic demand between an origin and destination pair is $6000$ vehicles. The performance functions for the two routes are

$t_1=4+5\frac{x_1}{c_1}$

$t_2=3+7\frac{x_2}{c_2}$

Where t$\text{'}$s are in minutes and flows $($x$\text{'}$s$)$ and capacities are in thousands. The capacities of routes $1$ and $2$ are $4400ve\frac{h}{h}$ and $2200ve\frac{h}{h}$ respectively.

The State Department of Transportation decides to regulate traffic on routes $1$ and $2$ to achieve system$-$optimum conditions. Calculate the SO flows.

min $s(n)=2xn+n$