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

Question 5 of 7: Singly Constrained Gravity Model for Two Zones

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

Question 5: Singly Constrained Gravity Model for Two Zones (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 study area with equal productions in each zone, unequal attractions, and a travel-time matrix that is symmetric with a shorter intra-zonal than inter-zonal time.

Find. The four-cell trip matrix, and hence the intra-zonal and inter-zonal totals, for both the base year and the target year.

Approach. Apply the singly constrained (production-constrained) gravity model, in which each origin zone distributes exactly its own production among the destinations in proportion to the product of attraction and friction factor.

Note first that productions and attractions already balance in both years — 500 against 500 in the base year and 700 against 700 in the target year — so no preliminary scaling of attractions is required and the model can be applied directly.

1. Compute the friction factors. With \(F_{ij}=1/t_{ij}\), $$ F_{11}=F_{22}=\frac{1}{5}=0.2000, \qquad F_{12}=F_{21}=\frac{1}{15}=0.06667 $$ The intra-zonal friction factor is three times the inter-zonal one, so at equal attraction a zone keeps three times as many trips at home as it sends away.
2. Write the singly constrained gravity formula. For each origin \(i\), $$ T_{ij} = P_i\,\frac{A_j F_{ij}}{\displaystyle\sum_{k} A_k F_{ik}} $$ which guarantees \(\sum_j T_{ij}=P_i\) row by row.
3. Evaluate the denominators for the base year. For origin zone 1, $$ \sum_k A_k F_{1k} = 200(0.2000) + 300(0.06667) = 40.00 + 20.00 = 60.00 $$ and for origin zone 2, $$ \sum_k A_k F_{2k} = 200(0.06667) + 300(0.2000) = 13.33 + 60.00 = 73.33 $$
4. Distribute each production — part (a). Applying the formula row by row, $$ T_{11}=250\frac{40.00}{60.00}=166.7, \qquad T_{12}=250\frac{20.00}{60.00}=83.3 $$ $$ T_{21}=250\frac{13.33}{73.33}=45.5, \qquad T_{22}=250\frac{60.00}{73.33}=204.5 $$ Each row sums back to 250 as required. Collecting the diagonal and the off-diagonal terms, $$ \boxed{\text{intra-zonal} = 371.2\ \text{trips},\qquad \text{inter-zonal} = 128.8\ \text{trips}} $$

The column totals of 212.1 and 287.9 do not exactly reproduce the target attractions of 200 and 300, which is the expected behaviour of a singly constrained model: only the production constraint is enforced. A doubly constrained model would iterate row and column balancing factors until both margins were satisfied.

5. Repeat the denominators for the target year — part (b). With \(A_1=300\) and \(A_2=400\), $$ \sum_k A_k F_{1k} = 300(0.2000)+400(0.06667) = 60.00+26.67 = 86.67 $$ $$ \sum_k A_k F_{2k} = 300(0.06667)+400(0.2000) = 20.00+80.00 = 100.00 $$
6. Distribute the increased productions. With \(P_1=P_2=350\), $$ T_{11}=350\frac{60.00}{86.67}=242.3, \qquad T_{12}=350\frac{26.67}{86.67}=107.7 $$ $$ T_{21}=350\frac{20.00}{100.00}=70.0, \qquad T_{22}=350\frac{80.00}{100.00}=280.0 $$ so that $$ \boxed{\text{intra-zonal} = 522.3\ \text{trips},\qquad \text{inter-zonal} = 177.7\ \text{trips}} $$

Total travel grows from 500 to 700 trips, a 40 per cent increase, but the split shifts slightly toward intra-zonal travel: the intra-zonal share rises from 74.2 to 74.6 per cent. The reason is that zone 2's attraction grew proportionally less than zone 1's (from 300 to 400, a factor of 1.33, against 200 to 300, a factor of 1.50), which raises the relative pull of the nearby zone-1 destinations on zone-1 residents while leaving zone 2's own strong self-attraction essentially unchanged.

(c) Factors affecting trip distribution other than travel time

Travel time is only one component of the generalised cost that actually drives destination choice. The additional factors fall into four groups.

Other components of generalised cost. Out-of-pocket travel cost (fuel, transit fare, tolls, parking charges), travel-time reliability, comfort and the number of transfers all enter the impedance term. A composite impedance built from a generalised cost, rather than from time alone, is standard practice in Canadian regional models.

Attributes of the destination beyond its size. The mix and quality of opportunities matter, not just the count: the type of employment relative to the skills of the origin zone's residents, retail rent and price levels, parking supply, school catchment boundaries, and the presence of complementary activities that allow trips to be chained.

Attributes of the traveller and the trip. Trip purpose is the largest single factor — work trips tolerate much greater impedance than shopping trips, which is why the friction-factor curve is calibrated separately by purpose. Household income, vehicle availability, age, and whether the trip is part of a multi-stop chain all shift the willingness to travel further.

Spatial and institutional structure. Physical and political barriers (a river with few crossings, a provincial or municipal boundary, a rail corridor), historical and social ties between neighbourhoods, language and cultural affinity, the arbitrary size and shape of the traffic zones themselves, and any intervening opportunities lying between the origin and a distant destination all bias the distribution in ways travel time alone does not capture. Competition among destinations — the basis of the intervening-opportunities and destination-choice model families — is the formal expression of that last effect.