Question

For a given urban area, a logit model for mode split was estimated and the following utility equations were obtained: ????? = 1.0−0.1(??????)−0.05(??????)
???? = −0.1(?????)−0.05(?????)
????? = −0.5−0.1(??????)
where TTi is the travel time by mode i in minutes and TCi is the travel cost by mode i, in dollars.  

Assume an individual facing travel times of 5,15, and 20 minutes for the auto, bus, and walk modes, respectively, as well as the out-of-pocket travel cost of $1.60 and $1.50 by auto and bus, respectively. If the buses in the above mode split model are operated by the Red Bus Transit Co, and a new operator, Blue Coach Lines, introduces a service that is identical to that of the Red Bus Transit Co. (i.e.15-min travel time and $1.50 fare), except that the service is provided by blue buses rather than red, making the choice to be four: auto, red bus, blue bus, and walk.  
(a) Compute the probabilities each of these four modes while assuming that the blue bus utility function is the same as the red bus function.
(b) Compare the probabilities computed in (a) above with probabilities computed using the original three mode choices (i.e., before introducing blue bus option).
(c) Are the results reasonable? Why or why not? Can you recommend a solution?

Answer 1

Solution:

Let's first compute the probabilities and number of trips for each utility model

For auto

v (auto) = Va = 1 - 0.1*5 - 0.05* 1.6 = 0.42

v(bus) = Vb = -0.1* 15 - 0.05*1.5 = - 1.575 for both blue and red

v(walking) = Vw = -0.5 - 0.1*20 = -2.5

probability of auto usage = Pa = e^ -0.42 / ( e^-0.42 + e^2.5 + 2*e^1.575)

= 0.029

probability of bus usage = red/ blue = Pb = e^1.575 / ( e^-0.42 + e^2.5 + 2*e^1.575)

= 0.215

probability of walking = Pw = e^2.5 / ( e^-0.42 + e^2.5 + 2*e^1.575)

= 0.541

b. Now using the original 3 modes:

auto usage = Pa = e^-0.42 /( e^-0.42 + e^2.5 + e^1.575)

= 0.037

bus usage = Pb = e^1.575 / ( e^-0.42 + e^2.5 + e^1.575)

= 0.273

walking = Pw = e^2.5 /  ( e^-0.42 + e^2.5 + e^1.575)

= 0.689

c. No , the results are not reasonable as the results are too much skewed towards walking. This means that the population is not liking any of the transportation services, which is not possible as the transport service is run after detailed studies. Auto usage is extremely low , whereas it is seen that in most situations autos are introduced because of its cost effectiveness and comfort and better travel time compared to other public modes.

the auto usage utility function seems to be a complimentary function, if it can be changed to a normal function then the probabilities might turn out to be comparable.