Problem $1$

$7-$ Consider the following urban transportation mode$-$choice model:

$p(A)=\frac{e^{U_A}}{e^{U_A}+e^{U_T}},p(T)=\frac{e^{U_T}}{e^{U_A}+e^{U_T}},$ and $U_m={}_m+{}_1{t}_m+{}_2\frac{x_m}{d}+{}_3\frac{c_m}{y}$

where

$p(A)=$ probability of an individual choosing auto mode $A$

$p(T)=$ probability of an individual choosing transit mode $T$

$t_m=$ in$-$vehicle time $($minutes$,$ one$-$way$)$

$x_m=$ out$-$of$-$vehicle time $($minutes$,$ one$-$way$)$

$d=$ distance $($miles$,$ one$-$way$)$

$c_m=$ out$-$of$-$pocket cost $($cents$,$ one$-$way$)$

$y=$ annual income $($$$/$year$)$

$m=$ automobile $(A)$ or transit $(T)$

Parameters ${}_1,{}_2,$ and ${}_3$ are the same for both modes A and $T,$ while ${}_m$ is specific to

each mode. Parameter values have been determined empirically and given below:

Given the following situation, determine the market share of transit and automobile, when

$($a$)y=5,000\frac{\text{\$}}{?}$ year and $($b$)y=10,000\frac{\text{\$}}{?}$ year.Problem

In the following network, derive and draw the minimum cost path

trees rooted from nodes $C$ and $F$ to all other nodes using the Dijkstra algorithm. Note that

the network is an undirected network and the numbers shown on each link represent the

cost of traversing that link.