Module 2: Transportation planning

Lecture 8 Trip distribution

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Gravity model

This model originally generated from an analogy with Newton's gravitational law. Newton's gravitational law says, $F = G M_1 M_2/d_2$ Analogous to this, $T_{ij} = C O_i D_j/c_{{ij}^n}$ Introducing some balancing factors, $T_{ij} = A_i O_i B_j D_j f(c_{ij})$ where $A_i$ and $B_j$ are the balancing factors, $f(c_{ij})$ is the generalized function of the travel cost. This function is called deterrence function because it represents the disincentive to travel as distance (time) or cost increases. Some of the versions of this function are:

$$f(c_{ij}) = e^{-\beta{c_{ij}}}$$ (1)

$$f(c_{OJ}) = c_{ij}^{-n}$$ (2)

$$f(c_{ij}) = c_{ij}^{-n} \times e^{-\beta{c_{ij}}}$$ (3)

The first equation is called the exponential function, second one is called power function where as the third one is a combination of exponential and power function. The general form of these functions for different values of their parameters is as shown in figure.

As in the growth factor model, here also we have singly and doubly constrained models. The expression $T_{ij} = A_i O_i B_j D_j f(c_{ij})$ is the classical version of the doubly constrained model. Singly constrained versions can be produced by making one set of balancing factors $A_i$ or $B_j$ equal to one. Therefore we can treat singly constrained model as a special case which can be derived from doubly constrained models. Hence we will limit our discussion to doubly constrained models.

As seen earlier, the model has the functional form, $T_{ij} = A_i O_i B_j D_j f(c_{ij})$

$$\Sigma_{i}{T_{ij}} = \Sigma_{i}{A_i O_i B_j D_j f(c_{ij})}$$ (4)

But

$$\Sigma_{i}{T_{ij}} = D_j$$ (5)

Therefore,

$$D_j = B_j D_j\Sigma_{i}{A_i O_i f(c_{ij})}$$ (6)

From this we can find the balancing factor $B_j$ as

$$B_j = 1 / {\Sigma_{i}{A_i O_i f(c_{ij})}}$$ (7)

$B_j$ depends on $A_i$ which can be found out by the following equation:

$$A_i = 1/{\Sigma_{j}{B_j D_j f(c_{ij})}}$$ (8)

We can see that both $A_i$ and $B_j$ are interdependent. Therefore, through some iteration procedure similar to that of Furness method, the problem can be solved. The procedure is discussed below:

1. Set $B_j$ = 1, find $A_i$ using equation 8
2. Find $B_j$ using equation 7
3. Compute the error as $E= \Sigma{\vert O_i - O_i^1\vert}+\Sigma{\vert D_j - D_j^1\vert}$ where $O_i$ corresponds to the actual productions from zone $i$ and $O_i^1$ is the calculated productions from that zone. Similarly $D_j$ are the actual attractions from the zone $j$ and $D_j^1$ are the calculated attractions from that zone.
4. Again set $B_j$ = 1 and find $A_i$, also find $B_j$. Repeat these steps until the convergence is achieved.