Figure $1$: Braess paradox network topology

The user link cost functions are:

ca$($fa$)=10$fa$,$ cb$($fb$)=$ fb $+50,$

cc$($fc$)=$ fc $+50,$ cd$($fd$)=10$fd$,$

ce$($fe$)=$ fe $+10.$

The O$/$D pair w$1=(1,4)$ and the travel demand dw$1=20.$

$($a$).$ Compute the system$-$optimizing $($S$-$O$)$ path flow pattern and the link flows corresponding to

this path flow pattern.

$($b$).$ What is the U$-$O pattern? Support your answer fully.

$($c$).$ What do you observe in your solutions to parts $($a$)$ and $($b$)$ above?