Dynamic User Assignment

# Introduction#

For a given set of vehicles with of origin-destination relations (trips), the simulation must determine routes through the network (list of edges) that are used to reach the destination from the origin edge. The simplest method to find these routes is by computing shortest or fastest routes through the network using a routing algorithm such as Dijkstra or A*. These algorithms require assumptions regarding the travel time for each network edge which is commonly not known before running the simulation due to the fact that travel times depend on the number of vehicles in the network.

Caution

A frequent problem with naive user assignment is that all vehicles take the fastest path under the assumption that they are alone in the network and are then jammed at bottlenecks due to the sheer amount of traffic.

The problem of determining suitable routes that take into account travel times in a traffic-loaded network is called user assignment. SUMO provides different tools to solve this problem and they are described below.

# Iterative Assignment (Dynamic User Equilibrium)#

The tool duaIterate.py can be used to compute the (approximate) dynamic user equilibrium.

Caution

This script will require copious amounts of disk space

python tools/assign/duaIterate.py -n <network-file> -t <trip-file> -l <nr-of-iterations>


duaIterate.py supports many of the same options as sumo. Any options not listed when calling duaIterate.py --help can be passed to sumo by adding sumo--long-option-name arg after the regular options (i.e. sumo--step-length 0.5). The same is true for duarouter options using duarouter--long-option-name arg. Be aware that those options have to come after the regular options.

This script tries to calculate a user equilibrium, that is, it tries to find a route for each vehicle (each trip from the trip-file above) such that each vehicle cannot reduce its travel cost (usually the travel time) by using a different route. It does so iteratively (hence the name) by

1. calling duarouter to route the vehicles in a network with the last known edge costs (starting with empty-network travel times)
2. calling sumo to simulate "real" travel times result from the calculated routes. The result edge costs are used in the net routing step.

The number of iterations may be set to a fixed number of determined dynamically depending on the used options. In order to ensure convergence there are different methods employed to calculate the route choice probability from the route cost (so the vehicle does not always choose the "cheapest" route). In general, new routes will be added by the router to the route set of each vehicle in each iteration (at least if none of the present routes is the "cheapest") and may be chosen according to the route choice mechanisms described below.

Between successive calls of duarouter, the .rou.alt.xml format is used to record not only the current best route but also previously computed alternative routes. These routes are collected within a route distribution and used when deciding the actual route to drive in the next simulation step. This isn't always the one with the currently lowest cost but is rather sampled from the distribution of alternative routes by a configurable algorithm described below.

## Route-Choice algorithm#

The two methods which are implemented are called Gawron and Logit in the following. The input for each of the methods is a weight or cost function (w) on the edges of the net, coming from the simulation or default costs (in the first step or for edges which have not been traveled yet), and a set of routes $R$ where each route $r$ has an old cost $c_r$ and an old probability $p_r$ (from the last iteration) and needs a new cost $c_r'$ and a new probability $p_r'$.

### Gawron (default)#

The Gawron algorithm computes probabilities for choosing from a set of alternative routes for each driver. The following values are considered to compute these probabilities:

• the travel time along the used route in the previous simulation step
• the sum of edge travel times for a set of alternative routes
• the previous probability of choosing a route

#### Number of Routes in each traveller's route set#

The maximum number of routes can be defined by users, where 5 is the default value. In each iteration, the route usage probability is calculated for each route. When the number of routes is larger than the defined amount, routes with smallest probabilities are removed.

The update rule is explained with the following example. Driver d chooses Route r in Iteration i. The travel time Tau_d(r, i+1) is calculated according to the aggregated and averaged link travel times per defined interval (default: 900 s) in Iteration i. The travel time for Driver d's Route r in Iteration i+1 equals to Tau_d(r, i) as indicated in Formula (1). The travel times of the other routes in Driver d's route set are then updated with Formula (2) respectively, where Tau_d(s, i) is the travel time needed to travel on Route s in Iteration i and calculated with the same way used for calculating Tau_d(r, i) an T_d(s, i-1). The parameter beta is to prevent travellers from strongly "remembering" the latest trave time of each route in their route sets. The current default value for beta is 0.3.

T_d(r, i+1) = Tau_d(r, i) ------------------------------------(1)

T_d(s, i+1) = beta * Tau_d(s, i) + (1 - beta) * T_d(s, i-1) ---(2)

, where s is one of the routes, which are not selected to use in Iteration i, in Driver d's route set.

The aforementioned update rules also apply when other travel cost units are used. The way to use simulated link costs for calcuating route costs may result in cost underestimation especially when significant congestion only on one of traffic movenents (e.g. left-turn or right-turn) exists. The existing ticket #2566 deals with this issue. In Formula (1), it is also possible to use Driver d's actual travel cost in Iteration i as Tau_d(r, i).

### Logit#

The Logit mechanism applies a fixed formula to each route to calculate the new probability. It ignores old costs and old probabilities and takes the route cost directly as the sum of the edge costs from the last simulation.

$c_r' = \sum_{e\in r}w(e)$

The probabilities are calculated from an exponential function with parameter $\theta$ scaled by the sum over all route values:

$p_r' = \frac{\exp(\theta c_r')}{\sum_{s\in R}\exp(\theta c_s')}$

Caution

It is recommended to set option --convergence-steps (i.e. to the same number as -last-step) to ensure convergence. Otherwise Logit route choice may keep oscillating, especially with higher values of --logittheta.

## Termination#

DuaIterate convergence is hard to predict and results may continue to vary even after 1000 iterations. There are several strategies in this regard:

### Default#

By default, a fixed number of iterations, configured via --first-step and --last-step (default 50) is performed.

### Deviation in Average Travel times#

The option --max-convergence-deviation may be used to detect convergence and abort iterations automatically. In each iteration, the average travel time of all trips is computed. From the sequence of these values (one per iteration), the relative standard deviation is computed. Onece a minimum number of iterations has been computed (--convergence-iterations, default 10) and this deviation falls below the max-convergence deviation threshold, iterations are aborted

### Forced convergence#

Option --convergence-steps may used to force convergence by iteratively reducing the fraction of vehicles that may alter their route.

• If a positive value x is used, the fraction of vehicles that keep their old route is set to max(0, min(step / x, 1) which prevents changes in assignment after step x.
• If a negative value x is used, the fraction of vehicles that keep their old route is set to 1 - 1.0 / (step - |x|) for steps after |x| which asymptotically reduces assignment after |x| steps.

## Usage Examples#

By default, vehicle types are taken from the input trip file and are then propagated through duarouter iterations (always as part of the written route file).

In order to use vehicle type definitions from an additional-file, further options must be set

duaIterate.py -n ... -t ... -l ...
duarouter--vtype-output dummy.xml


Options preceded by the string duarouter-- are passed directly to duarouter and the option vtype-output dummy.xml must be used to prevent duplicate definition of vehicle types in the generated output files.

# oneShot-assignment#

An alternative to the iterative user assignment above is incremental assignment. This happens automatically when using <trip> input directly in sumo instead of <vehicle>s with pre-defined routes. In this case each vehicle will compute a fastest-path computation at the time of departure which prevents all vehicles from driving blindly into the same jam and works pretty well empirically (for larger scenarios).

The routes for this incremental assignment are computed using the Automatic Routing / Routing Device mechanism. Since this device allows for various configuration options, the script Tools/Assign#one-shot.py may be used to automatically try different parameter settings.

# marouter#

The marouter application computes a classic macroscopic assignment. It employs mathematical functions (resistive functions) that approximate travel time increases when increasing flow. This allows to compute an iterative assignment without the need for time-consuming microscopic simulation.