Question 6 of 7: Multinomial Logit Mode Choice

Question 6: Multinomial Logit Mode Choice (20 marks)

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. Four modes, a linear-in-parameters utility function, and the attribute values of each mode.

Find. The modal shares before and after the bike-path programme cuts cycling time to 45 minutes, and an appraisal of how the four attributes drive the split.

Approach. Evaluate the deterministic utility of each mode, exponentiate, and normalise by the sum to obtain logit probabilities; then repeat with the single changed attribute and interpret the coefficients as implied values of time.

1. Evaluate the utility of each mode. Substituting each row of the attribute table into $$V_i = -0.075\,AT_i - 0.05\,WT_i - 0.04\,RT_i - 0.002\,TC_i$$ gives, for the automobile, $$V_{\text{auto}} = -0.075(6) - 0.05(1) - 0.04(25) - 0.002(300) = -0.45 - 0.05 - 1.00 - 0.60 = -2.100$$ and repeating for the other three modes, $V_{\text{bus}} = -3.220$, $V_{\text{rail}} = -2.375$, $V_{\text{bike}} = -2.475$. The automobile has the highest utility despite costing 300 cents, because it has by far the shortest riding time and almost no waiting.
2. Exponentiate and sum. The logit denominator is the sum of the exponentiated utilities, $$\sum_j e^{V_j} = 0.12246 + 0.03996 + 0.09301 + 0.08416 = 0.33959$$
3. Normalise to obtain the shares. Each probability is $P_i = e^{V_i}/\sum_j e^{V_j}$, for example $$P_{\text{auto}} = \frac{0.12246}{0.33959} = 0.3606$$ so the modal split before the bike programme is $$\boxed{P_{\text{auto}} = 36.06\%, \quad P_{\text{bus}} = 11.77\%, \quad P_{\text{rail}} = 27.39\%, \quad P_{\text{bike}} = 24.78\%}$$ The four shares sum to 100 per cent, which is the arithmetic check that the denominator was formed correctly.
4. Recompute the utility of cycling with the new bike paths. Only $RT_{\text{bike}}$ changes, from 60 to 45 minutes: $$V_{\text{bike}}^{\,\prime} = -0.075(1) - 0.05(0) - 0.04(45) - 0.002(0) = -0.075 - 1.800 = -1.875$$ an improvement of exactly $0.04 \times 15 = 0.60$ utility units, and $e^{-1.875} = 0.15335$ against the previous 0.08416.
5. Re-normalise. The other three exponentials are unchanged, so the new denominator is $$\sum_j e^{V_j'} = 0.12246 + 0.03996 + 0.09301 + 0.15335 = 0.40878$$ and the revised shares are $$\boxed{P_{\text{auto}} = 29.96\%, \quad P_{\text{bus}} = 9.77\%, \quad P_{\text{rail}} = 22.75\%, \quad P_{\text{bike}} = 37.52\%}$$
6. Interpret the shift. Cycling gains 12.74 percentage points and becomes the leading mode. The losses are drawn from the other three in strict proportion to their previous shares: the automobile loses 6.10 points, rail 4.64 and bus 1.99, and the ratio of the automobile share to the rail share is 1.3165 both before and after the change. That invariance is the independence of irrelevant alternatives at work, and it is worth flagging explicitly here because Question 1(b) asks why it can be unrealistic — a cycling improvement that in reality would draw disproportionately from short bus trips is modelled as drawing evenly from everything.

(c) Effects of the four attributes, and their realism

All four coefficients are negative, so an increase in any attribute reduces the utility of the mode that carries it and hence its share, while raising the shares of every competing mode. The relative magnitudes are what carry the behavioural content. Access time is penalised most heavily, at 0.075 per minute; waiting time next, at 0.05 per minute; riding time least among the time components, at 0.04 per minute. Access time is therefore weighted 1.875 times as heavily as in-vehicle time and waiting time 1.25 times as heavily. Converting each to money through the cost coefficient of 0.002 per cent gives implied values of time of 20 cents per minute for in-vehicle time, 25 cents per minute for waiting and 37.5 cents per minute for access — that is, about CAD 12.00, CAD 15.00 and CAD 22.50 per hour respectively.

These effects are broadly realistic and match the empirical mode-choice literature in both sign and ordering. Travellers consistently value out-of-vehicle time — walking to a stop, waiting on an exposed platform — at roughly one and a half to three times in-vehicle time, because it is uncomfortable, exposed, unproductive and, in the case of waiting, uncertain. The implied in-vehicle value of about CAD 12 per hour is a plausible order of magnitude for personal travel, sitting near half of average gross earnings, which is the usual empirical benchmark. The ordering also explains why the automobile does well here despite being the most expensive mode by a factor of four: it carries almost no access or waiting penalty at all.

Several features are nonetheless unrealistic, and a complete answer should say so. First, the utility functions contain no mode-specific constants, so the model attributes the entire choice to the four measured attributes and ignores comfort, privacy, reliability, safety, weather protection, cargo capacity and habit. This matters most for cycling, whose real share is limited by weather, terrain, fitness, safety perception and trip purpose rather than by riding time — a 37.5 per cent cycling share for a 45-minute ride is not credible in a Canadian city with winter conditions. Second, the specification is linear in each attribute, so the first minute of waiting is penalised exactly as heavily as the thirtieth, whereas real disutility grows more than proportionally with long waits, and a linear cost term implies that a fixed dollar amount matters equally to every traveller regardless of income. Third, cost enters only as an out-of-pocket amount, which for the automobile excludes depreciation, insurance and the real cost of parking, so the automobile's true generalised cost is understated. Fourth, there are no capacity or availability constraints: a mode not accessible to a household — no bicycle, no driving licence, no rail station within reach — still receives a positive predicted share. Finally, as part (b) has just shown numerically, the IIA property forces every alternative to lose share in fixed proportion, which misrepresents competition between the two transit modes and the bicycle. The practical remedies are the ones set out in Question 1(b): add mode-specific constants and richer attributes, segment the market by income and trip purpose, and adopt a nested or mixed logit structure that groups the similar alternatives.