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 study area with a base year and a target year, an inverse-square deterrence function and a fixed distance matrix.

Find. The four-cell trip matrix in the base year and in the target year, split into intra-zonal and inter-zonal totals, and the non-distance factors that affect trip distribution.

Approach. Apply the singly constrained (production-constrained) gravity model cell by cell, computing a separate denominator for each production zone so that every row sums exactly to its production.

1. Evaluate the friction factors. With \(F_{ij} = 1/d_{ij}^2\), \[F_{11} = F_{22} = \frac{1}{5^2} = 0.04,\qquad F_{12} = F_{21} = \frac{1}{10^2} = 0.01.\] Doubling the distance divides the friction factor by four, so an internal trip is four times as attractive per unit of attraction as an external one. That single ratio drives everything that follows.
2. Write the production-constrained gravity model. The share of zone \(i\)’s productions that goes to zone \(j\) is the attraction-weighted friction share, \[T_{ij} = P_i\,\frac{A_j F_{ij}}{\sum_k A_k F_{ik}}.\] The denominator must be computed separately for each origin zone; using one common denominator for both rows breaks the production constraint.
3. Distribute zone 1’s productions (base year). The denominator is \[\textstyle\sum_k A_k F_{1k} = 700(0.04) + 300(0.01) = 28 + 3 = 31,\] so \[T_{11} = 450 \times \frac{28}{31} = 406.45,\qquad T_{12} = 450 \times \frac{3}{31} = 43.55,\] and the row sums to 450 as required.
4. Distribute zone 2’s productions (base year). The denominator is \[\textstyle\sum_k A_k F_{2k} = 700(0.01) + 300(0.04) = 7 + 12 = 19,\] so \[T_{21} = 550 \times \frac{7}{19} = 202.63,\qquad T_{22} = 550 \times \frac{12}{19} = 347.37.\]
5. Collect the base-year intra-zonal and inter-zonal totals (part a). \[T_{\text{intra}} = T_{11} + T_{22} = 406.45 + 347.37 = \boxed{753.82\ \text{trips}}\] \[T_{\text{inter}} = T_{12} + T_{21} = 43.55 + 202.63 = 246.18\ \text{trips}\] The two sum to the 1,000 productions in the system, and 75.4 per cent of all travel stays inside its own zone.
6. Repeat for the target year, zone 1. With the new attractions, \[\textstyle\sum_k A_k F_{1k} = 950(0.04) + 450(0.01) = 38 + 4.5 = 42.5,\] \[T_{11} = 600 \times \frac{38}{42.5} = 536.47,\qquad T_{12} = 600 \times \frac{4.5}{42.5} = 63.53.\]
7. Repeat for the target year, zone 2. \[\textstyle\sum_k A_k F_{2k} = 950(0.01) + 450(0.04) = 9.5 + 18 = 27.5,\] \[T_{21} = 800 \times \frac{9.5}{27.5} = 276.36,\qquad T_{22} = 800 \times \frac{18}{27.5} = 523.64.\]
8. Collect the target-year totals (part b). \[T_{\text{intra}} = 536.47 + 523.64 = \boxed{1{,}060.11\ \text{trips}}\] \[T_{\text{inter}} = 63.53 + 276.36 = 339.89\ \text{trips}\] Total travel grows 40 per cent with the productions, but the growth is not uniform: intra-zonal travel grows 40.6 per cent and inter-zonal travel 38.1 per cent. The reason is that zone 2’s attractions grow proportionally faster than zone 1’s (450/300 = 1.50 against 950/700 = 1.357), which lifts zone 2’s internal share from \(12/19 = 63.2\) per cent to \(18/27.5 = 65.5\) per cent. That is enough to outweigh both the small fall in zone 1’s internal share (\(28/31 = 90.3\) to \(38/42.5 = 89.4\) per cent) and the shift of productions toward zone 2, which is the less internally-oriented of the two zones and whose productions grow the faster (800/550 = 1.455 against 600/450 = 1.333).

(c) Factors affecting trip distribution other than travel distance

Distance is only a proxy for the real deterrent, which is generalised cost. The factors a Canadian regional model would add fall into four groups.

Better measures of the impedance itself: travel time rather than distance, and congested rather than free-flow time, since a 10 km freeway trip and a 10 km arterial trip are not equally attractive; out-of-pocket cost including fuel, tolls, transit fare and above all parking charges; the composite generalised cost that combines time and money at a value of time; and, for transit trips, the number of transfers and the waiting and walking time, which travellers weight two to three times more heavily than in-vehicle time. Reliability — the variance of travel time, not just its mean — is increasingly included.

Attributes of the zones and their activities: the type and quality of the attraction, not merely its size, so that a regional shopping centre draws from a far wider catchment than an equal floor area of neighbourhood retail; the match between the socio-economic character of the origin and the job types at the destination, which is why practice segments the distribution by trip purpose and by income or car-ownership group; land-use mix and the presence of competing intervening opportunities, which is the mechanism the intervening-opportunities model makes explicit.

Attributes of the traveller and the trip: trip purpose above all — work trips are far less distance-sensitive than shopping trips, so each purpose needs its own deterrence parameter; household income and vehicle availability; and the time of day, since peak-period distribution is compressed relative to off-peak.

Network and geographic structure: physical and psychological barriers such as rivers, inlets, rail corridors and municipal or provincial boundaries; network connectivity and route circuity, which make the airline distance a poor impedance measure; and zone size and shape, which determine the intra-zonal distance and therefore control the diagonal of the matrix — the single largest cell in this problem. Modelling artefacts matter too: the choice of deterrence function (inverse power, exponential or gamma) and its calibrated parameter strongly influence how concentrated the resulting matrix is.

The sensitivity of the answer to the deterrence function alone is easy to demonstrate on this problem. Setting \(F_{ij} = 1\) for every cell removes distance from the model entirely and reduces it to a pure attraction share; the intra-zonal total then falls from 753.82 to 480 trips, or from 75.4 per cent of all travel to 48.0 per cent. The deterrence function is therefore responsible for essentially the entire concentration of travel inside the zones, which is why calibrating it against observed trip-length distributions is the central task in fitting a gravity model.