Logit for routes

Another major problem of our approach is that all travellers with the same situation will be put on the same route, that is, there is no ``spread'' of solutions.

A typical way to obtain some spread of solutions is to use a logit approach. Remember, a logit means that the probability of picking a solution $i$ is set to

$$p_i = {e^{\beta \, U_i} \over \sum_je^{\beta \, U_j}} \ ,$$

where $U_i$ is the utility of solution $i$. When the utility of a solution is high, then it will be selected with a high probability.

For routes, utility is negative, and it becomes more negative the longer the driving time. For example, one could set $U_j = - T_j$, where $T_j$ is the driving time for route choice $j$.

A major problem with this is that it is not easy to generate routing alternatives. Two approaches, and their drawbacks, are:

• It is possible to compute $k$-shortest paths.

  Then, it is problematic to use logit on routes (e.g. (25)). This is actually easy to see: In Fig. 19.1, there are three paths from A to B. Assume they have all the same travel time. The plausible solution then is that path 1 is used with probability $0.5$, and paths 2 and 3 are used with probability $0.25$ each.

  The logit solution will however be that all three paths are used with equal probabilities $1/3$.

  The example can be made arbitrarily pathologic by adding more ``short'' alternatives.

  It is however possible to use more sophisticated models than the logit models (25).

• Another method is to only generate routes which are ``real'' alternatives (94). This is however not an easy problem in itself.

  And the problem with the logit still applies, although to a weaker extent.

Figure 19.1: Correlations between paths