Gravity model

The Gravity model uses the following basic form to determine the trips between an origin zone $i$ and a destination zone $j$.

$$t_{ij} = K_g^i \frac{f(a_j)}{h(d_{ij})}T_i$$ (3)

where, $T_i$ is the total number of trips being produced by origin zone $i$; $a_j$ is a set of in situ attributes of the destination zone $j$ which attract trip makers; $d_{ij}$ is a set of factors which inhibit travel between two zones; $K_g^i$ is a calibration constant; and, $f(\cdot)$, and $h(\cdot)$ are positive monotonically increasing functions. Note that $\sum_{\forall i} t_{ij}$ is basically the trip attractions of Zone $j$.

The expression $K_g^i \frac{f(a_j)}{h(d_{ij})}$ may be thought of as a factor which distributes the total trips produced by a zone among all the possible destination zones. In this sense, the sum of the expression over all destinations should be equal to unity.

$$\sum_j K_g^i \frac{f(a_j)}{h(d_{ij})} = 1$$ (4)

The above equation implies that,

$$K_g^i = \left( \sum_j \frac{f(a_j)}{h(d_{ij})} \right)^{-1}$$ (5)

Substituting, the expression for $K_g^i$ in Equation , the following expression, often referred to as the origin constrained gravity model, is obtained:

$$t_{ij} = \frac{f(a_j) / h(d_{ij})}{\sum_j f(a_j) / h(d_{ij})} T_i$$ (6)

Sometimes, in the gravity model, the number of trips attracted to a zone (when such data is independently available), $T_j$, is used as a surrogate for $f(a_j)$. When such a substitution is done the gravity model is typically written as

$$t_{ij} = K_g^iK_g^j \frac{1}{h(d_{ij})}T_iT_j$$ (7)

Very often, in this case the following two constraints are imposed on the gravity model to obtain the two calibration constants:

$$\begin{eqnarray*} \sum_j t_{ij} = T_i, \;\;\;\;\;\;\;\; \forall j \\ \sum_i t_{ij} = T_j, \;\;\;\;\;\;\;\; \forall i \end{eqnarray*}$$

On imposition of the two constraints, the constants $K_g^i$ and $K_g^j$ are obtained as:

$$K_g^i = \left(\sum_{j}\frac{K_g^jT_j}{h(d_{ij})}\right)^{-1}$$ (8)

$$K_g^j = \left(\sum_{i}\frac{K_g^iT_i}{h(d_{ij})}\right)^{-1}$$ (9)

Obviously, using this model requires an iterative solution technique as $K_g^i$ depends on $K_g^j$ and vice versa.

Example

Consider the following six zone model of a town. Zones 1, 2, and 3 are fully residential areas and Zones 4, 5 and 6 are purely shopping areas. The shopping areas, shopping trips attracted (per day), shopping trips produced (per day) and travel distances are as shown in Table . The cells which have a ``-'' imply that those data are irrelevant to the problem. Determine the trip distribution between the zones for the following different scenarios:
(a) Use the origin constrained gravity model, assuming $f(a_j)$ to be a linear function of the shopping area (in square meters) with a slope of 0.01 and constant term of 10. Also assume $h(d_{ij})$ to be $d_{ij}^2$ where $d_{ij}$ is the distance in kms.
(b) Use the origin-destination constrained gravity model with relevant assumptions same as those in (a).

表: Data for the example on Gravity Model

Zone	Shop	Trips	Trips	Distance (km) to
	Area ($m^2$)	Prod.	Attr.	1	2	3	4	5	6
1	-	1000	-	-	-	-	4	2	7
2	-	1000	-	-	-	-	3	1	6
3	-	2000	-	-	-	-	5	2	6
4	1000	-	800	4	3	5	-	-	-
5	2000	-	2000	2	1	2	-	-	-
6	3000	-	1200	7	6	6	-	-	-

Solution

Part a
The trips of interest here are $t_{ij}$, where $i \in {1,2,3}$ and $j \in {4,5,6}$. Also note, as per the problem description $f(a_j) = 0.01 \times (\mbox{shopping area in m$^2$}) + 10$, and $h(d_{ij}) = d_{ij}^2$. The trips are given by Equation .

$$\begin{eqnarray*} t_{14} & = & \frac{f(a_4) / d_{14}^2}{(f(a_4) / d_{14}^2) + (f... ...^2) + (f(a_6) / d_{36}^2)} T_3 = \frac{1.11}{9.41}2000 = 236 \\ \end{eqnarray*}$$

Part b
In this case, the only difference from Part (a) is that the trips are given by Equation  and the constants of Equation  are given in Equations  and . The initial value of $K_g^i$ is assumed to be the square root of the corresponding values obtained in the previous part. These values of $K_g^i$ are used in Equation  to obtain a set of $K_g^j$ values which are in turn used in Equation  to obtain a new set of $K_g^i$ values. The process continues till all the $K_g^i$ and $K_g^j$ values converge. The final values of $K_g^i$ and $K_g^j$ obtained are as follows:

Final values of $K_g^i$
$K_g^1 = 0.359$, $K_g^2 = 0.136$, and $K_g^3 = 0.357$

Final values of $K_g^j$
$K_g^4 = 0.015$, $K_g^5 = 0.002$, and $K_g^6 = 0.032$

Using these values of $K_g^i$ and $K_g^j$ in Equation  the following $t_{ij}$ values are obtained: $t_{14} = 272$, $t_{15} = 444$, and $t_{16} = 284$
$t_{24} = 182$, $t_{25} = 672$, and $t_{26} = 146$
$t_{34} = 346$, $t_{35} = 884$, and $t_{36} = 770$
The reader must verify that for the above trips $\sum_j t_{ij} = T_i$ and $\sum_i t_{ij} = T_j$.

Discussion: It may be pointed out here that the expression $\frac{f(a_j) / h(d_{ij})}{\sum_j f(a_j) / h(d_{ij})}$ in Equation  may be viewed as $\pi_{ij}$, the probability that destination $j$ is chosen from origin $i$. Once such a view is taken, then the Maximum likelihood technique  can be used to estimate the parameters of the gravity model. If observations on the trip distribution between different origin-destination pairs are made and denoted as $n_{ij}$ then the likelihood function can be written as

$$\mathcal{L} = \prod_{ij}(\pi_{ij})^{n_{ij}}$$ (10)

Other constraints like $\sum_i \pi_{ij} = 1$ could be accounted for by adding the constraint to the likelihood function and constructing a Lagrangian.