Question 5 of 7: Two-Zone Gravity Model of Trip Distribution

Question 5: Two-Zone Gravity Model of Trip Distribution (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 from each zone, unequal attractions, and a friction factor that is the reciprocal of travel time.

Total productions equal total attractions in both years (150 and 250 respectively), so the trip table is balanced on its margins before distribution begins.

Find. The four cells of the trip matrix in each year, reported as intra-zonal and inter-zonal totals, and a list of the factors other than travel time that influence trip distribution.

Approach. Apply the singly constrained (production-constrained) gravity model: each production zone distributes its own trips in proportion to the attraction of each destination weighted by the friction factor, which requires a separate denominator for each production zone.

1. State the gravity model. In production-constrained form, $$T_{ij}=P_i\,\frac{A_jF_{ij}}{\displaystyle\sum_{x}A_xF_{ix}}$$ The denominator is computed separately for each origin zone \(i\); using one common denominator for both rows breaks the production constraint and is the commonest error on this question. With \(F=1/t\), the friction factors are \(F_{11}=F_{22}=1/2=0.50\) and \(F_{12}=F_{21}=1/5=0.20\).
2. (a) Build the denominators for the base year. For zone 1, $$\sum_x A_xF_{1x}=A_1F_{11}+A_2F_{12}=50(0.50)+100(0.20)=25+20=45$$ and for zone 2, $$\sum_x A_xF_{2x}=A_1F_{21}+A_2F_{22}=50(0.20)+100(0.50)=10+50=60$$ The two differ because zone 2 has the larger nearby attraction pool.
3. Distribute the base-year productions. Applying the model row by row, $$T_{11}=75\left(\frac{25}{45}\right)=41.67,\qquad T_{12}=75\left(\frac{20}{45}\right)=33.33$$ $$T_{21}=75\left(\frac{10}{60}\right)=12.50,\qquad T_{22}=75\left(\frac{50}{60}\right)=62.50$$ Each row sums to its production (41.67 + 33.33 = 75 and 12.50 + 62.50 = 75), confirming the constraint is satisfied. Collecting by type, $$\boxed{\text{intra-zonal} = 41.67+62.50 = 104.17 \text{ trips};\quad \text{inter-zonal} = 33.33+12.50 = 45.83 \text{ trips}}$$ Intra-zonal travel is therefore 69.4 per cent of all 150 trips — the short intra-zonal travel time dominates the distribution even though zone 2 attracts twice as many trips as zone 1.
4. Check the attraction margin. Summing down the columns, the model sends \(41.67+12.50=54.17\) trips to zone 1 against a target of 50, and \(33.33+62.50=95.83\) trips to zone 2 against a target of 100 — errors of \(+8.3\) and \(-4.2\) per cent. This is expected: a singly constrained model reproduces productions exactly but only approximates attractions. Matching both margins requires iterative proportional fitting (the Furness / Fratar procedure) with attraction balancing factors, which the question does not supply, so the single pass is the intended answer — but stating the discrepancy is part of a complete solution.
5. (b) Repeat for the target year. The travel times and hence the friction factors are unchanged, so only the margins move. The new denominators are $$\sum_x A_xF_{1x}=75(0.50)+175(0.20)=37.5+35.0=72.5$$ $$\sum_x A_xF_{2x}=75(0.20)+175(0.50)=15.0+87.5=102.5$$
6. Distribute the target-year productions. $$T_{11}=125\left(\frac{37.5}{72.5}\right)=64.66,\qquad T_{12}=125\left(\frac{35.0}{72.5}\right)=60.34$$ $$T_{21}=125\left(\frac{15.0}{102.5}\right)=18.29,\qquad T_{22}=125\left(\frac{87.5}{102.5}\right)=106.71$$ $$\boxed{\text{intra-zonal} = 64.66+106.71 = 171.36 \text{ trips};\quad \text{inter-zonal} = 60.34+18.29 = 78.64 \text{ trips}}$$ Row sums are 125 each, as required. The intra-zonal share slips slightly from 69.4 to 68.5 per cent, because attractions grew faster in zone 2 (+75 per cent) than in zone 1 (+50 per cent), pulling a little more travel across the boundary from zone 1.
7. Interpret the growth. Total travel rises from 150 to 250 trips (+66.7 per cent), but the two components grow unequally: intra-zonal travel rises 64.5 per cent while inter-zonal travel rises 71.6 per cent. To see what the friction factor is actually doing, re-run part (a) with \(F_{ij}=1\) for all pairs — a zero-deterrence model. The distribution collapses to a pure attraction share and the intra-zonal total falls from 104.17 to exactly 75.0 trips, i.e. from 69.4 per cent to 50 per cent of travel. The deterrence function, not the attraction pattern, is responsible for the entire concentration of travel within zones.

(c) Factors affecting trip distribution other than travel time

The friction factor is a proxy for generalised cost, and the attraction term is a proxy for destination opportunity; both are far richer than a single time value. The main additional influences are:

• Money cost of travel — fares, tolls, fuel and parking charges, which belong in the generalised cost alongside time.
• Trip purpose — work trips are far less deterred by distance than shopping or social trips, so each purpose needs its own friction function and its own calibrated exponent.
• Size and type of the attraction — employment by sector, retail floor space, school enrolment; a regional hospital or shopping centre draws from far beyond its own zone.
• Socio-economic characteristics of the traveller — income, vehicle ownership, household size and life-cycle stage all change the willingness to trade time for opportunity.
• Socio-economic and demographic matching between zones — occupational and skill matching between residents and jobs, language and cultural communities, and school catchment boundaries, all of which produce systematic preferences a pure time model cannot capture. These are handled with \(K\)-factors, which are an admission that the model is missing a variable.
• Physical and institutional barriers — rivers, rail corridors, municipal or provincial boundaries and, in the Canadian context, the international border, which suppress travel well beyond their travel-time cost.
• Modal availability and level of service — frequency, reliability, comfort and the number of transfers, none of which appear in an in-vehicle time.
• Competing and intervening opportunities — the number of acceptable destinations passed on the way, which is the basis of the intervening-opportunities model and of destination-choice logit specifications.
• Time-of-day and congestion effects — travel time is itself a function of the assignment, so distribution and assignment should strictly be solved together.