The buses in the numerical example of a logit mode split model presented in Sec

$5\mathrm{.}5\mathrm{.}1($i$.$c$.,$ Eq$.[5\mathrm{.}23])$ are operated by the Red Bus Transit Co$.$ Assume that a

new operator, Blue Coach Lines, introduces a service that is identical to that of

the Red Bus Transit Co$.($i$.$e$.,15-$minute travel time, $$1\mathrm{.}50$ fares$),$ except that the

service is provided by blue buses rather than red. We now have four "modes": in

the choice set: auto, red bus, blue bus, and walk. Compute the probabilities of

each of these four modes being chosen using Eq$.[5\mathrm{.}23]($and assuming that the

blue bus utility function is the same as the red bus function$).$ Compare these

probabilities with those for the three$-$mode example in Sec. $5\mathrm{.}5\mathrm{.}1.$ Are these

results reasonable, especially given that the model. ignores frequency of service?

Why or why not?

The following is from the textbook Page $292.$

As an example of a multinomial logit model, consider a three$-$mode choice sit$-$

uation in which a worker must choose between auto, bus, and walking for the jour$-$

ney to work. The systematic utility functions associated with these alternatives

might take the form

$V_{\text{a}\text{u}\text{t}\text{o}}=1\mathrm{.}0-0\mathrm{.}1(T{T}_{\text{a}\text{u}\text{t}\text{o}})-0\mathrm{.}05(T{C}_{\text{c}\text{u}\text{t}\text{o}})$

$V_{\text{b}\text{u}\text{s}}=-0\mathrm{.}1(T{T}_{\text{b}\text{u}\text{s}})-0\mathrm{.}05(T{C}_{\text{b}\text{u}\text{s}})$

$V_{\text{w}\text{a}\text{l}\text{k}}=-0\mathrm{.}5-0\mathrm{.}1(T{T}_{\text{w}\text{a}\text{l}\text{k}})$

where

$T{T}_i=$ travel time $by$ mode $i,$ minutes

$T{C}_i=$ travel cost $by$ mode $i,$ dollars

Assume that a given individual is faced with travel times of $5,15,$ and $20$ min$-$

utes for the auto, bus, and walk modes, respectively. Similarly, out$-$of$-$pocket travel

costs by auto and bus are $$1\mathrm{.}60$ and $$1\mathrm{.}50,$ respectively. In this case, the values of

the systematic utility functions given by Eq$.[5\mathrm{.}24]$ are

$V_{\text{a}\text{u}\text{t}\text{o}}=0\mathrm{.}42,V_{\text{b}\text{u}\text{s}}=-1\mathrm{.}575,V_{\text{w}\text{a}\text{l}\text{k}}=-2\mathrm{.}5$

Substituting these values into Eq$.[5\mathrm{.}23],$ the probability of this worker choos$-$

ing the auto mode is

$P_{\text{a}\text{u}\text{t}\text{o}}=\frac{e^{0\mathrm{.}42}}{e^{0\mathrm{.}42}+e^{-1\mathrm{.}575}+e^{-2\mathrm{.}5}}=\frac{1\mathrm{.}522}{1\mathrm{.}811}=0\mathrm{.}841$

Similarly, $P_{\text{b}\text{u}\text{s}}=\frac{0\mathrm{.}207}{1\mathrm{.}811}=0\mathrm{.}114,$ and $P_{\text{w}\text{a}\text{l}\text{k}}=\frac{0\mathrm{.}082}{1\mathrm{.}811}=0\mathrm{.}045$