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Description
This AIMMS demo describes an example of a mixed complementarity problem (MCP). The problem described here is a so-called traffic equilibrium model. The goal is to find a traffic distribution over the roads, that follows from the behavior of the drivers.
In this example, we assume that every driver wants to minimize his own travel time in a noncooperative way, his noncooperative behavior means that he will change his route only when he can find a faster route for himself. There is always a faster route when somebody else with the same origin and destination has a faster route. This means that in the equilibrium situation, all the drivers with the same origin and destination will have the same travel time. This travel time is the minimum travel time for that origin and destination pair. This has been formulated in the complementarity variable 'Traffic'.
Beside the complementarity variable, it is also allowed to use normal variables and constraints.
This model has been extended with the possibility of using toll roads and maximize the income from the toll fees. This problem is formulated as a mathematical program with complementarity constraints (MPCC). This is a special case of mathematical program with Equilibrium Constraints (MPEC).
This example contains a small dataset because this formulation demands that every route between every origin destination pair is given. For larger dataset this is almost impossible. In that case a path generation method needs to be implemented, which is beyond the scope of this example.
In AIMMS currently the PATH and KNITRO solvers are available for solving mixed complementarity problems (MCP). The KNITRO solver also supports mathematical programs with complementarity constraints (MPCC).
Keywords
Braess's paradox, Non-cooperative Game, Stackelberg Game
Industries
Model Types
Mixed Complementarity Programming
Download AIMMS Example
You can download an AIMMS example dealing with this problem via the link below, and run it after installing the AIMMS software. If you don't have an AIMMS license yet, you can download a free license of AIMMS.
ftp://ftp.aimms.com/pub/Download/Examples/Traffic Equilibrium.aimmspack
Please make sure to save this file including the .aimmspack extension so that it can be opened by AIMMS.
This example application is a simplification of reality. Please do not hesitate to contact us to discuss how AIMMS enables you to build a complete optimization application that captures the full complexity of your problem.
Screenshot AIMMS Example
