Method of path selection with or without path search

You can choose from different procedures for path selection (Path search and path selection):

• In the parameters of dynamic assignment, in the Choice tab, select the procedure Use old volumes (no path search) to select a path without carrying out a path search. In this case, the probability of a path being used corresponds to its attribute value Volume (old) share in the total of attribute values Volume (old), of all paths of the same OD pair. These attribute values stem from an ANM import or the path file of a previous simulation run.
• If in the parameters of dynamic assignment, on the Choice tab, you selected the path choice model Stochastic assignment (Kirchhoff) or Equilibrium assignment, your path search is followed by path selection. The vehicles are then distributed across the paths depending on the distribution model and based on the distribution formula according to Kirchhoff or for equilibrium assignment.

The following descriptions require that the destination parking lot and potential routes to it are already known. Path search finds only the best possible path in each interval for each OD pair, but all found paths can be used in all intervals (Path search finds only the best possible path in each interval). For the dynamic assignment, the drivers select the route at the time they depart from the origin parking lot.

Calculating utility

One of the basic assumptions in path selection according to Kirchhoff is that not all drivers use the best path, but that all known paths are used and have different costs. However, a large percentage of the traffic should be distributed across the better paths. The quality of paths is evaluated using the generalized costs. Generalized costs are contrary to the "benefit" involved in the theory of discrete decisions. Thus the benefit is defined as the reciprocal of the generalized costs:

Where

Uj = the benefit of path j

Cj = the generalized costs of path j

Calculating the decision behavior using the Logit function

The most frequently used and thus also the most theoretically analyzed function for mapping the decision behavior is the Logit function:

Where

Uj = the benefit of path j

p(Rj) = the probability that path j is selected

μ = the sensitivity parameter of the model (>0), Logit scaling factor for destination parking lot selection

The sensitivity parameter determines how strongly the distribution responds to benefit differences. A low value would result in a quite similar distribution without any major influence of the benefit, and a high value would result in virtually every driver selecting the best path.

Distribution according to Kirchhoff

If the logit function is applied with the cost function defined above, this leads the model to attach the same importance to the difference between 5 and 10 minutes of travel time as the difference between 105 and 110 minutes of travel time because the logit function is translationally invariant and thus considers only the absolute difference of benefits. Obviously, this modeling is not particularly appropriate, because in reality two paths which have a travel time of 105 and 110 minutes are basically considered equally good, whereas paths of 5 and 10 minutes are perceived as significantly different. To approximate the real assessment, the distribution formula according to Kirchhoff is used in Vissim:

Where

Uj = the benefit of path j

p(Rj) = the probability that path j is selected

k = the sensitivity parameter of the model

The sensitivity parameter also determines here how sensitively the model responds to differences in the benefits. For Kirchhoff, the ratio of benefits determines the distribution and not the absolute difference of benefits, thus only slight variations arise in the paths with 105 and 110 minutes of travel time, whereas the path with 5 minutes of travel time receives much more traffic than the path with 10 minutes of travel time.

In fact, the Kirchhoff function is also a logit model. It arises from the logit function described above if the logarithmic benefit is used as a utility function:

Cj are the generalized costs of path j in this case.

Distribution with the equilibrium assignment

The equilibrium assignment redistributes demand across paths proportionally to costs, from expensive to inexpensive paths, for each parking lot OD pair.

The volume of paths that are more expensive than the average is reduced. The volume of these less expensive paths is also reduced and part of it assigned to cheaper paths. All paths that are cheaper than the average path costs are assigned additional volume. The cheaper the path, the more volume it is assigned (Equilibrium assignment – Example).

Assign normalized probability for path selection

As with the procedure according to Kirchhoff, each path j is a assigned a normalized probability .

where:

s: simulation run

n: time interval

v: vehicle class

The probabilities are calculated before each time interval n from the generalized costs . In equilibrium assignment, the target volume is determined, which is different from the method according to Kirchhoff .

Where: = the Target volume (relative) attribute and is the total volume of the OD parking lot relation. is iteratively calculated, so that is a function γ with the following variables:

• the generalized cost of the corresponding path
• the average generalized cost
• the relative target volumes of the previous simulation run

Redistributing volumes proportionally to costs

In the following

: is the number of paths at the beginning of a new time interval for each OD parking lot relation, including newly found paths and excluding previously deleted paths.

This includes:

: the costs of path j

 

: the average path costs, with the number of paths with the OD relation .

The demand is shifted towards the vector

with

Due to the definition of the following applies:

The volume is thus redistributed an no additional volume generated.

In iteration s the proportion of the total demand for a parking lot relation is redistributed:

Where is the content of CurIterIdx (Current iteration index attribute: index of the current iteration of an equilibrium assignment). The CurIterIdx index is incremented at the end of a simulation run, under the following conditions:

• A dynamic assignment has been carried out and matrices or trip chain files have been referenced, and
• the distribution model Equilibrium assignment has been selected.

CurIterIdx is saved to the path file *.weg.

CurIterId is restored when a simulation run is started without a path file.

To redistribute only the desired share of the total volume, the vector has yet to be scaled. For this purpose the scaled direction vector is calculated.

Thus the following conditions are met:

This means that, just as much volume is taken from paths that are more expensive than the average as is added to paths that are less expensive than the average.

Demand is shifted towards so that no negative demand is created on any of the paths:

If the algorithm implies that volume is taken from paths which have a volume of 0 already. To carry out the redistribution, these paths are temporarily taken from the set of paths, the OD pair. Volume balancing is restarted and only the temporarily reduced path set is taken into account.

If the following is set:

 

The new target volume is then given by:

Thus, a proportion of in the total demand for the parking lot relation is shifted.

If the remaining share of must be shifted. The remaining share is redistributed iteratively. For this purpose, paths with a relative target volume are temporarily removed from the path set . The volume balancing is restarted, however with instead of and with instead of .

The iterative procedure is stopped when is reached.

 

The new final volume is then saved and used for the new time interval, down to vehicle class level, in the new attribute Target volume (relative) (Attributes of paths). Target volume (relative) is saved to the path file *.weg.

If during assignment a path file is read in that does not contain the Target volume (relative) and Current iteration index, the following values are set:

• Current iteration index: 1
• Target volume (relative): empty for all vehicle classes and time intervals