16-Civ-A6 Highway Design, Construction, and Maintenance · December 2016

Question 5 of 7: Trip Distribution by the Gravity Model

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

Question 5: Trip Distribution by the 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. A two-zone system with equal productions, unequal attractions, and a reciprocal-time friction factor.

Find. The four-cell trip matrix and the intra-/inter-zonal split, for both the base and the target year, plus the non-time determinants of trip distribution.

Approach. Apply the singly constrained gravity model, in which each production is distributed among destinations in proportion to attraction times friction factor, so that productions are reproduced exactly and attractions emerge as an output.

1. Write the model. The singly constrained (production-constrained) gravity model is $$T_{ij}=P_i\,\frac{A_jF_{ij}}{\displaystyle\sum_{j'}A_{j'}F_{ij'}},\qquad F_{ij}=\frac{1}{t_{ij}}$$ The denominator normalises the row so that $\sum_j T_{ij}=P_i$ identically — a useful arithmetic check on every row.
2. Base year, trips produced by zone 1. The two destination weights are $$A_1F_{11}=\frac{50}{2}=25,\qquad A_2F_{12}=\frac{100}{5}=20,\qquad \sum=45$$ so $$T_{11}=75\left(\frac{25}{45}\right)=41.67,\qquad T_{12}=75\left(\frac{20}{45}\right)=33.33$$ Although zone 2 offers twice the attraction, the 2.5-times-longer trip more than offsets it, and the majority of zone 1's trips stay home.
3. Base year, trips produced by zone 2. Now the near zone is zone 2 itself: $$A_1F_{21}=\frac{50}{5}=10,\qquad A_2F_{22}=\frac{100}{2}=50,\qquad \sum=60$$ $$T_{21}=75\left(\frac{10}{60}\right)=12.50,\qquad T_{22}=75\left(\frac{50}{60}\right)=62.50$$ Here proximity and attraction pull the same way, so zone 2 retains five-sixths of its trips.
4. Base-year split. Collecting the diagonal and the off-diagonal, $$\boxed{\text{intra-zonal}=T_{11}+T_{22}=41.67+62.50=104.17\ \text{trips}}$$ $$\boxed{\text{inter-zonal}=T_{12}+T_{21}=33.33+12.50=45.83\ \text{trips}}$$ The two sum to 150, matching $P_1+P_2$, so the row constraints are satisfied.
5. Target year, both rows. The travel times are unchanged, so only the attraction weights move: $$\text{row 1: } \frac{75}{2}=37.5,\ \frac{175}{5}=35,\ \sum=72.5\;\Rightarrow\;T_{11}=125\left(\frac{37.5}{72.5}\right)=64.66,\ T_{12}=60.34$$ $$\text{row 2: } \frac{75}{5}=15,\ \frac{175}{2}=87.5,\ \sum=102.5\;\Rightarrow\;T_{21}=125\left(\frac{15}{102.5}\right)=18.29,\ T_{22}=106.71$$ $$\boxed{\text{intra-zonal}=171.36\ \text{trips},\qquad \text{inter-zonal}=78.64\ \text{trips}}$$ Total trips rise from 150 to 250, and the inter-zonal share rises from 30.6 % to 31.5 % — a modest increase, driven entirely by zone 2's attraction growing faster than zone 1's.
6. Check the attraction balance. A singly constrained model reproduces productions but not attractions. Summing the columns of the target-year matrix gives 82.95 trips arriving in zone 1 against the stated target of 75, and 167.05 arriving in zone 2 against 175 — errors of $+10.6\%$ and $-4.5\%$.

(c) Factors affecting trip distribution other than travel time

Travel time is only one component of the generalised cost that actually deters travel, and several classes of variable sit alongside it. Other cost components: out-of-pocket money cost (fuel, tolls, transit fare, parking charge), and for transit the access, waiting and transfer time, which travellers weight two to three times more heavily than in-vehicle time. Attractiveness of the destination beyond its size: the type and quality of the activity offered — retail floorspace and tenant mix, employment type against the traveller's skills, school reputation, hospital specialisation — so two zones with equal employment are not equally attractive for a given purpose. Competition and spatial structure: intervening opportunities (a nearer zone offering the same activity suppresses longer trips), agglomeration effects, and physical barriers such as rivers, rail corridors, mountains or an international border, which raise perceived separation far above the measured time. Traveller characteristics: income, car availability, age, household structure and employment status, all of which change the value of time and hence the shape of the deterrence function. Trip purpose and timing: work trips are long and time-insensitive, shopping and social trips are short and cost-sensitive, so distribution must be modelled purpose by purpose; peak-period and off-peak matrices differ substantially. Service quality and reliability: travel-time variability, comfort, crowding, safety and personal security, and habit or inertia, which keeps established travel patterns in place after the network has changed. In model form these enter either as extra terms in a generalised-cost friction factor, as separate purpose- and segment-specific deterrence functions, or as $K$-factors calibrated to correct residual bias between particular zone pairs.