PROBLEM $4(25$ pts$)$

Figure $1$ presents a simple three$-$zone system, the link travel times in minutes $(w_{ij})$ for this system, and the observed zonal productions and attractions. Assuming a doubly constrained gravity model with friction factor $F_{ij}=w_{ij}^{-b}$ :

$($a$)$ Compute P$-$A flows $($production$-$attraction matrix$)$ for zone $1$ for values of $b$ of $1\mathrm{.}0.$

Table $1$: Observed P$-$A flows for the three$-$zone system.

$\backslash$table$[[$From$,$To$],[1,2,3],[1,130,50,15],[2,80,5,80],[3,30,,55]]$

$4$