16-Civ-A6 Highway Design, Construction, and Maintenance · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: 98-Civ-A6 Transportation Planning & Engineering, National Examination May 2016. Seven questions, all of equal value (20 marks); any five constitute a complete examination. Closed book, one two-sided aid sheet permitted. Three hours. All seven questions are solved below as a study resource.
Reference texts. Mannering, Washburn & Kilareski, Principles of Highway Engineering and Traffic Analysis (Wiley) — queueing, shock waves and traffic-stream models; Papacostas & Prevedouros, Transportation Engineering and Planning (Prentice Hall) — the four-step demand model; Ortuzar & Willumsen, Modelling Transport (Wiley) — trip generation, distribution, mode choice and assignment; Roess, Prassas & McShane, Traffic Engineering (Pearson) — signalised-intersection delay.
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Given. Three modes with linear-in-parameters utility functions and the level-of-service table below. Travel costs are in dollars and remain unchanged throughout.
| Mode | IVTT (min) | OVTT (min) | Travel cost ($) | Alternative-specific constant |
|---|---|---|---|---|
| Automobile | 10 | 5 | 3.50 | +0.1 |
| Bus (base) | 20 | 10 | 1.50 | +0.2 |
| Light rail | 12 | 15 | 2.00 | 0 |
| Bus (improved, part b) | 15 | 8 | 1.50 | +0.2 |
Find. The choice probabilities for the three modes before and after the bus service improvement, and an assessment of the prediction in the light of the IIA property.
Approach. Evaluate each observable utility from the given coefficients, exponentiate, and normalise by the sum — the standard multinomial logit (MNL) formula. Then compare the ratio of any two probabilities before and after the change to expose the IIA restriction.
| Quantity | (a) base case | (b) improved bus | Change |
|---|---|---|---|
| \(V_a\) (auto) | −0.955 | −0.955 | 0 |
| \(V_b\) (bus) | −1.945 | −1.495 | +0.450 |
| \(V_r\) (light rail) | −2.670 | −2.670 | 0 |
| \(P_a\) | 0.645 | 0.567 | −0.077 |
| \(P_b\) | 0.239 | 0.331 | +0.091 |
| \(P_r\) | 0.116 | 0.102 | −0.014 |
| Odds ratio \(P_a/P_r\) | 5.557 | 5.557 | unchanged (IIA) |
| Share of new bus riders drawn from auto | — | 84.7 % | — |
The direction of the result is entirely sensible: a bus service that cuts in-vehicle time by five minutes and out-of-vehicle time by two minutes becomes more attractive, so its share rises from 23.9 to 33.1 per cent, and the other two modes lose riders. The allocation of that gain, however, is not credible. The model draws 84.7 per cent of the new bus riders out of automobiles and only 15.3 per cent from light rail, purely because automobile had the larger initial share. Behaviourally one would expect the opposite emphasis: bus and light rail are both scheduled public transit, sharing unobserved attributes such as the need to walk to a stop, exposure to weather, the absence of a private vehicle at the destination end, and a broadly similar rider population. A traveller already willing to accept those attributes on light rail is a far easier convert to an improved bus than a committed motorist. A realistic model would take a disproportionate share of the new bus patronage from light rail.
The cause is the IIA property, demonstrated numerically in step 6: because the ratio \(P_a/P_r\) depends only on \(V_a - V_r\), any change to the bus alternative must leave that ratio untouched, and the only way to satisfy that constraint is to reduce both competing modes by the same proportion — here 12.0 per cent each. IIA follows directly from the MNL derivation, which assumes the random components of utility are independently and identically Gumbel-distributed across alternatives. That assumption is false whenever two alternatives share unobserved attributes, which is exactly the situation here. The classical illustration is the red-bus / blue-bus problem: introducing a bus identical to an existing one except for its colour should split the bus market in half and leave the car share alone, but MNL instead predicts that the car share falls by a third.
Four remedies are available, in increasing order of generality. The nested logit model is the standard answer here: place bus and light rail together in a "transit" nest under a "motorised travel" root, so that the correlation of the unobserved terms within the transit nest is captured by a logsum (inclusive-value) variable and a nest scale parameter \(\lambda\) between 0 and 1. A bus improvement then competes primarily within its own nest, drawing most of its gain from light rail, and only the residual effect passes up to the automobile. Cross-nested or paired-combinatorial logit generalises this to alternatives that belong partly to more than one nest. The multinomial probit model abandons the Gumbel assumption altogether in favour of a multivariate normal error with a full covariance matrix, permitting any correlation structure at the cost of simulation-based estimation. The mixed (random-parameters) logit model achieves the same generality more tractably by letting coefficients vary randomly across the population, and can approximate any random-utility model arbitrarily closely. A cheaper partial fix is to add explanatory variables — a transit alternative-specific constant, reliability, service frequency, or a transfer penalty — that move the shared attributes from the unobserved error term into the observed utility, which reduces but does not eliminate the correlation. For this three-mode problem the nested logit is the appropriate and defensible choice.
