An individual chooses a transport mode for a particular trip based on three attributes i.e., cost of journey (X), In-vehicle travel time to reach destination (Y), and Out-of-vehicle time taken to access mode at respective stops (Z). The values for these attributes for three modes Rail, Bus and Para-transit are given in the table. If the general utility (U) equation is \( U = - 0.5 \times X - 0.3 \times Y - 0.4 \times Z \), using the Logit model, the estimated probability of choosing Bus is _________ (rounded off to two decimal places).
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The Logit model is used to calculate the probability of choosing a particular mode of transportation based on the utility derived from different attributes. The utility function typically incorporates costs, travel times, and other factors.
The utility equation is:
\[
U = - 0.5 \times X - 0.3 \times Y - 0.4 \times Z
\]
Step 1: Calculate the utility for each mode:
For Rail:
\[
U_{{Rail}} = -0.5 \times 20 - 0.3 \times 20 - 0.4 \times 10 = -10 - 6 - 4 = -20
\]
For Bus:
\[
U_{{Bus}} = -0.5 \times 10 - 0.3 \times 40 - 0.4 \times 7.5 = -5 - 12 - 3 = -20
\]
For Para-transit:
\[
U_{{Para-transit}} = -0.5 \times 15 - 0.3 \times 35 - 0.4 \times 5 = -7.5 - 10.5 - 2 = -20
\]
Step 2: The Logit model is used to calculate the probability of choosing each mode. The probability of choosing Bus (denoted as \( P_{{Bus}} \)) is given by:
\[
P_{{Bus}} = \frac{e^{U_{{Bus}}}}{e^{U_{{Rail}}} + e^{U_{{Bus}}} + e^{U_{{Para-transit}}}}
\]
Substituting the values of \( U_{{Rail}} \), \( U_{{Bus}} \), and \( U_{{Para-transit}} \):
\[
P_{{Bus}} = \frac{e^{-20}}{e^{-20} + e^{-20} + e^{-20}} = \frac{1}{3}
\]
Step 3: The estimated probability of choosing Bus is:
\[
P_{{Bus}} = \frac{1}{3} = 0.33
\]
Thus, the estimated probability of choosing Bus is 0.33.
