Entropy model

The Entropy model of trip distribution, unlike the previous models discussed here, is not a behavioral model. That is, the entropy model does not strive to predict the trip distribution by modeling the human behavioral aspects related to choosing a destination. This model, on the other hand, attempts to determine a distribution of trips which is most likely to occur assuming that each trip occurs independently of another.

For a total number of $Q$ trips in a network, it is assumed that there are a total of $Q$ independent decisions. Obviously, a given trip distribution matrix can be obtained through different combination of the decisions. Specifically, the number of ways a trip distribution matrix [$t_{ij}$], can be obtained is:

$$\frac{Q!}{\prod_{ij}t_{ij}!}$$ (22)

In the entropy models it is assumed that the likelihood of a particular trip distribution matrix occurring is proportional to the number of ways the matrix can be obtained. That is, the likelihood of a matrix occurring is proportional to the expression given in Equation . The entropy models determine the estimates of $t_{ij}$ by determining that set of $t_{ij}s$ which maximizes the likelihood of a trip distribution matrix occurring. Mathematically, the problem therefore is to solve the following

$$\max_{t_{ij}} \frac{Q!}{\prod_{ij}t_{ij}!}$$ (23)

Since $Q$ is a constant, this is equivalent to solving

$$\max_{t_{ij}} - \prod_{ij}t_{ij}!$$ (24)

or

$$\max_{t_{ij}} - \sum_{ij} \ln t_{ij}!$$ (25)

Using the approximation $\ln x! \approx x \ln x - x$, the above is equivalent to solving

$$\max_{t_{ij}} - \left( \sum_{ij} t_{ij} \ln t_{ij} - \sum_{ij} t_{ij} \right)$$ (26)

Since $\sum_{ij} t_{ij} = Q$, this is equivalent to solving

$$\max_{t_{ij}} - \left( \sum_{ij} t_{ij} \ln t_{ij} \right)$$ (27)

The unconstrained optimization problem in Equation  is the standard maximum entropy model of trip distribution. However, one can easily incorporate various constraints in this model. Some of the constraints which are often included in the maximum entropy model are $\sum_j t_{ij} = T_i$, $\sum_i t_{ij} = T_j$, $\frac{\sum_{ij} t_{ij}d_{ij}}{\sum_{ij} t_{ij}} =$ observed average trip length, etc.

Discussion: The constraints can easily be incorporated in the function to be maximized by constructing a Lagrangian. The Lagrangian can then be differentiated with respect to the $t_{ij}$'s and and the Lagrange multipliers to obtain a set of expressions which can be equated to zero. These equations when solved will give the estimate of the trip distribution.

Also note that this method does not use any calibration constants and hence the issue of calibrating the model does not arise.