Question

The origin-destination (O-D) trips for a three-zone system were predicted using a trip distribution model as shown below. O-D Trip Table (Person trips) Destination zone Origin zone 2. 3 2120 233 1831 2 263 4825 3227 3 229 358 1293 1 Assume that person trips are made by two modes – automobile and transit. Interzonal and intrazonal travel times by these two modes are shown below: Auto Travel Time (min) Destination zone Origin zone 2 3 2.4 11.0 8.2 2 11.0 2.1 7.4 3 8.2 7.4 3.4 1 Origin zone Transit Travel Time (min) Destination zone 2 3 8.1 22.8 15.0 22.8 8.2 13.4 15.0 13.4 7.5 1 1 2 3 The utility functions for trips from zone i to zone j were calibrated as follows: V. = 1.2 - 0.06 * (tija Vi = -0.06 * (tij) where Va, Vi = observable utilities for auto and transit, respectively; (tij)a, (tij)= the travel times from zone i to zone j for auto and transit (minutes), respectively.

Answer 1

We are given the utility functions for auto and transit. So we can calculate the utility function values for each O-D pair for each mode as follows

O-D table (person Trips) 1 2 3 1 233 2 2120 263 229 4285 358 1831 3227 1293 3 Auto Travel time (min) 1 2 3 1 3 15.0 1 8.2 2.4 11.0 1 Transit travel time (min) 2 8.1 22.8 2 22.8 8.2 3 15.0 13.4 11.0 2.1 7.4 2 7.4 13.4 7.5 3 8.2 3.4

We are given

V_a = 1.2 - (0.06*t_ij)_a

V_t = - (0.06*t_ij)_t

So we can calculate the utility function values for each O-D pair for each mode as follows

For example V_{auto trips} for Zone 1-3 = 1.2 - 0.06* Auto Travel time for Zone (1,3) = 1.2 - 0.06*8.2 = 1.2 -.492 = .708

V_{auto transit trips} for Zone 1-3 = - 0.06* Transit Travel time for Zone (1,3) = -0.06*15 = -0.9

1 V (Utility Fn Value) for Auto Trips 1 2 3 1 1.056 0.540 0.708 2 0.540 1.074 0.756 3 0.708 0.756 0.996 V (Utility Fn Value) for Transit Trips 2 3 1 -0.486 -1.368 -0.900 2 -1.368 -0.492 -0.804 3 -0.900 -0.804 -0.450

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Now when you have two modes of travel namely Auto (a) and Transit (t), with utility function values V_a and V_t, then

Probability of taking mode a = P_a = (e^V_a)/ ((e^V_a + e^V_b))

Probability of taking mode t = P_t = (e^V_t)/ ((e^V_a + e^V_b))

The denominator is common for all O-D pairs and is given by

3 Denominator 1 1 3.490 2 1.971 2.436 2.436 2 1.971 3.538 2.577 2.577 3.345 3

So calculate the probability of taking auto and transit for each O-D Pair. We have the following

For example for O-D Pair (2,3) Probability of choosing Auto = e^{^0.756}/(e^0.756 + e^-0.804) = 2.1297/2.577 = 0.8264

For example for O-D Pair (2,3) Probability of choosing Transit = e^{^-0.804}/(e^0.756 + e^-0.804) = .4475/3.490 = 0.174

3 Probability of choosiing Auto 1 2. 0.824 0.871 2 0.871 0.827 3 0.833 0.826 1 0.833 Probability of choosiing Transit 1 2 1 0.176 0.129 2 0.129 0.173 3 0.167 0.174 3 0.167 0.174 0.191 0.826 0.809

So the number of trips taking auto and transit can be calculated by multiplying the resective probabilities by the trips from the O-D matrix.

For example # of person trips by auto for OD (1,1) = 2120 * 0.824 = 1746

For example # of person trips by transit for OD (1,1) = 2120 * 0.176 = 374

These are shown for all O-D pairs below

1 3 # of person trips by auto 1 2 1 1746 203 2 229 3545 3 1525 # of person trips by transit 2 1 374 30 34 740 3 38 62 306 2667 560 N 3 191 296 1047 246

Vehicle occupancy is 1.29. That means 1.29 people travel in one vehicle. So # of auto trips = # of person auto trips/1.29

So for O-D pair (1,1) # of auto trips = 1746/1.29 = 1354

Like this we can calculate for all O-D pairs.

1 # of vehicle trips by auto 2 1354 157 2 178 2748 3 148 229 1 3 1183 2067 811

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