Mode choice by the multinomial logit model: 16-Civ-A6 Highway Design, Construction, and Maintenance, May 2013, Q7

Question 7: Mode choice by the multinomial logit model (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. Calibrated utility coefficients and the level‑of‑service data for each mode.

Utility coefficients and level‑of‑service (part b)
Mode	$a$	$\beta_1$ (IVTT)	$\beta_2$ (OVTT)	$\beta_3$ (TC)	IVTT	OVTT	TC ($)
Automobile	0.3	−0.04	−0.20	−0.06	12	7	2.50
Bus	0.5	−0.06	−0.20	−0.06	20	12	0.75
Light rail	0	−0.06	−0.20	−0.06	18	10	1.20

Find. Interpretation of the coefficients (a); the three modal shares (b); and the percentage increase in the automobile share after the parking‑lot improvement (c).

Check / source reading: The coefficient table prints the OVTT and travel‑cost columns as “0.2” and “0.06” without a sign, while the IVTT column carries its minus sign. Taken literally, positive OVTT and cost coefficients would mean more waiting and higher fares increase a mode appeal — which is not physically sensible and would make part (c) reduce the auto share when accessibility is improved. The coefficients are therefore read as negative ($\beta_2=-0.20$, $\beta_3=-0.06$); this is the physically correct model and the reading under which part (c) is meaningful.

(a) Effect of the coefficients — do they make sense?

All three level‑of‑service coefficients are negative, so increasing in‑vehicle time, out‑of‑vehicle time, or travel cost lowers the utility — and hence the probability — of choosing that mode. This is intuitively correct: travellers dislike longer and costlier trips. The magnitude of the out‑of‑vehicle‑time coefficient ($-0.20$) is several times that of the in‑vehicle coefficient ($-0.04$ to $-0.06$), meaning a minute spent walking, waiting or transferring is penalized three to five times more heavily than a minute riding. That too matches observed behaviour: people find out‑of‑vehicle time more onerous than sitting in the vehicle, which is why service planners prioritise reducing wait and access times. The alternative‑specific constants ($a$: bus 0.5, auto 0.3, rail 0) capture all the unmodelled attributes (comfort, privacy, image) that make one mode preferred at equal time and cost.

Approach (b, c). Compute each mode utility $V$, then the multinomial‑logit share $P_i=e^{V_i}/\sum_j e^{V_j}$.

Utilities, base case (b).
$$V_\text{auto}=0.3-0.04(12)-0.20(7)-0.06(2.50)=-1.730,$$

$$V_\text{bus}=0.5-0.06(20)-0.20(12)-0.06(0.75)=-3.145,\qquad V_\text{rail}=0-0.06(18)-0.20(10)-0.06(1.20)=-3.152.$$
Shares, base case (b). With $e^{V}=0.1773,\ 0.0432,\ 0.0428$ (sum 0.2632):
$$\boxed{P_\text{auto}=67.4\%,\quad P_\text{bus}=16.4\%,\quad P_\text{rail}=16.3\%.}$$
Improved automobile utility (c). With OVTT 7→5 and TC 2.50→2.00:
$$V_\text{auto}'=0.3-0.04(12)-0.20(5)-0.06(2.00)=-1.300\quad(\text{bus, rail unchanged}).$$
New automobile share (c). Now $e^{V'_\text{auto}}=0.2725$ against the unchanged 0.0432 and 0.0428 (sum 0.3585):
$$P'_\text{auto}=\frac{0.2725}{0.3585}=76.0\%.$$
The automobile share rises from 67.4 % to 76.0 % — an increase of
$$\boxed{8.7\ \text{percentage points}\ (\approx 12.9\%\ \text{relative increase}).}$$

Question 7 — modal shares
Mode	(b) Base	(c) After parking lot
Automobile	67.4 %	76.0 %
Bus	16.4 %	13.5 %
Light rail	16.3 %	13.4 %

Topic: Mode choice — the multinomial logit model and its coefficient interpretation.

Key relations: systematic utility $V_i=a_i+\beta_1\text{IVTT}+\beta_2\text{OVTT}+\beta_3\text{TC}$; logit share $P_i=e^{V_i}/\sum_j e^{V_j}$.

Why this works: Under the Gumbel‑distributed random utility assumption the choice probabilities take the closed‑form logit expression. Because only utility differences matter, an equal shift applied to every mode leaves the shares unchanged; here only the automobile utility changes, so its exponential grows while the others hold, raising its share. The ratio of coefficients is a direct measure of trade‑offs — e.g. $\beta_2/\beta_3$ is the value the traveller places on out‑of‑vehicle time in dollars per minute.

Common pitfall: Exponentiating the level‑of‑service variables instead of the utilities, or mis‑reading the (sign‑stripped) OVTT/cost coefficients as positive — which makes an accessibility improvement perversely reduce the auto share.

Source: Ortuzar & Willumsen, Modelling Transport, ch. 7 (discrete choice, logit); Papacostas & Prevedouros, ch. 5 (mode choice).

Question 7 of 7: Mode choice by the multinomial logit model

Question 7: Mode choice by the multinomial logit model (20 marks)

(a) Effect of the coefficients — do they make sense?