16-Civ-A6 Highway Design, Construction, and Maintenance · December 2014

Question 5 of 7: Singly-Constrained Gravity Model

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Examination — 98-Civ-A6 Transportation Planning & Engineering, December 2014. Closed book (one two-sided aid sheet), 3 hours. Seven questions of equal value (20 marks); any five constitute a complete paper. All seven are solved here as a study resource.

Reference texts (subject):

Papacostas, C.S. & Prevedouros, P.D., Transportation Engineering and Planning, 3rd ed. (Prentice Hall).
Roess, R.P., Prassas, E.S. & McShane, W.R., Traffic Engineering, 4th ed. (Pearson).
Mannering, F.L. & Washburn, S.S., Principles of Highway Engineering and Traffic Analysis (Wiley).
Ortúzar, J. de D. & Willumsen, L.G., Modelling Transport, 4th ed. (Wiley).
Sheffi, Y., Urban Transportation Networks (Prentice Hall).
Transportation Association of Canada (TAC), Geometric Design Guide for Canadian Roads (Canadian context).

Question 5: Singly-Constrained Gravity 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. Origin production $P_1$, destination attractions $A_j$, interzonal times $t_{1j}$, and friction $F_{1j}=1/t_{1j}$ (inverse proportionality).

Given data
Destination zone $j$	2	3	4	5
Travel time $t_{1j}$ (min)	25	50	75	100
Attraction $A_j$ — case (a)	75	300	115	675
Attraction $A_j$ — case (b)	150	340	150	900

Find. The distributed trips $T_{1j}$ for the base case (a), the future case (b), and the zone of largest increase (c). Base production $P_1=1500$; future $P_1=1875$.

Approach. Singly-constrained gravity model: weight each destination by $A_j/t_{1j}$, normalize so the shares reproduce the origin production, and multiply by $P_1$.

Model form. With $F_{1j}=1/t_{1j}$, $$T_{1j}=P_1\,\frac{A_j/t_{1j}}{\displaystyle\sum_{k}A_k/t_{1k}}.$$
(a) Weights and denominator. $A_j/t_{1j}=\{75/25,\;300/50,\;115/75,\;675/100\}=\{3.000,\;6.000,\;1.533,\;6.750\}$, summing to $17.283$.
(a) Distribute $P_1=1500$. $T_{1j}=1500\times(\text{weight})/17.283$ gives $$T_{12}=260.4,\quad T_{13}=520.7,\quad T_{14}=133.1,\quad T_{15}=585.8,$$ which sum to $\boxed{1500}$ (production conserved).
(b) Future weights. $A_j/t_{1j}=\{150/25,\;340/50,\;150/75,\;900/100\}=\{6.000,\;6.800,\;2.000,\;9.000\}$, summing to $23.800$.
(b) Distribute $P_1=1875$. $T_{1j}=1875\times(\text{weight})/23.800$ gives $$T_{12}=472.7,\quad T_{13}=535.7,\quad T_{14}=157.6,\quad T_{15}=709.0,$$ summing to $\boxed{1875}$.
(c) Largest increase. The changes are $+212.3$ (zone 2), $+15.0$ (zone 3), $+24.5$ (zone 4) and $+123.2$ (zone 5). The largest increase is to zone 2: although zone 5 gains the most attraction in absolute terms, zone 2's attraction doubled (75→150, the largest relative jump) while its short 25-minute travel time gives it the strongest friction weight, so the extra production is funnelled disproportionately toward it.

Question 5 — trips from zone 1
Destination	(a) Base	(b) Future	Increase
Zone 2	260.4	472.7	+212.3
Zone 3	520.7	535.7	+15.0
Zone 4	133.1	157.6	+24.5
Zone 5	585.8	709.0	+123.2
Total	1500	1875	+375

Topic: Production-constrained gravity model of trip distribution.

Key relations: $T_{ij}=P_i\,A_jF_{ij}/\sum_k A_kF_{ik}$ with $F_{ij}=1/t_{ij}$; the normalization guarantees $\sum_j T_{ij}=P_i$.

Why this works: Each destination competes for the origin's trips in proportion to its attractiveness discounted by travel impedance. Normalizing by the sum of weights is exactly the singly-constrained balancing that reproduces the known production, so the distribution shifts toward zones whose attraction grows fastest and whose impedance is lowest.

Common pitfall: Forgetting to re-normalize after the attractions change, or judging "largest increase" by absolute attraction rather than by the combination of relative attraction growth and low travel time.

Source: Ortúzar & Willumsen, Modelling Transport, Ch. 5; Papacostas & Prevedouros, Ch. 6.

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