User Guide

Traffic Networks

Motivation

Minimum cost flows coincide with user equilibrium and system optimum in traffic network if the edge cost are transformed by:

\[\begin{split}\begin{align} \text{User equilibrium:} \quad F_e &= \int_0^{x_e} l_e(s) ds \\ \text{System optimum:} \quad F_e &= x_e \cdot l_e \\ \end{align}\end{split}\]

The link travel time is defined as: show/hide

\[l_e(x_e) = \text{fft}_e \cdot \left( 1 + B_e \cdot \left(\frac{x}{\text{cap}_e}\right) ^ {p_e} \right).\]

Braess Paradox: Adding capacity to a network might decrease network efficiency due to selfish participants.

Adding capacity to a network might decrease network efficiency due to selfish participants.

Go to user guide

NetworkFW

Pigou’s Example: An upper bound for the Price of Anarchy

Simple network with a price of anarchy of \(\text{PoA} = \frac{4}{3}\).

Go to user guide

NetworkFW

SiouxFalls: calculating and plotting a fixed user equilibrium

We compute a user equilibrium and plot it using NetworkX.

Go to user guide

NetworkFW

SiouxFalls: parametric Price of Anarchy

We compute a parametric price of anarchy for SiouxFalls and a single commodity.

Go to user guide

MCA

Barcelona: Price of Anarchy

Go to user guide

We calculate user equilibrium and system optimum for Barcelona with SymbolicCost due to non-integer power factors.

NetworkFW SymbolicCost

Natural Gas

GasLib: Gas24

We compute the parametric edge flows for an affine demand function with MCA.

Go to user guide

MCA AffineDemandFunction

Network Setup

Motivation

You want to set up a custom graph with paminco? Learn how to specify graphs, how to allow for negative flows (we treat all graphs as directed), and how to specify cost and demand functions.

Graph Setup: directed graphs (with negative flows)

We set up directed graphs, learn about graph representations and how to set lower and upper boundaries for the edge flow in order to allow for negative edge flows (thus creating undirected graphs).

Go to user guide

Edges Nodes

Demand Function: linear and affine demand functions

We illustrate how to specify the demand function for a network. Typically this is a linear demand funtion that scales the node rates (encoded with a demand vector \(\mathbf{b}\)) by a demand multiplier \(\lambda\): \(\mathbf{h}_\text{linear}(\lambda) = \lambda \mathbf{b}\) or an affine demand function that adds a base demand \(\mathbf{h}_\text{affine}(\lambda) = \mathbf{b}_0 + \lambda \mathbf{b}\).

Go to user guide

DemandFunction (abstract) LinearDemandFunction AffineDemandFunction

Read-in from XML: How to read in network from XML

Go to user guide

Network

Plotting: How to specify node positions and plot with NetworkX

We specify node positions and used them to plot a network with NetworkX.

Go to user guide

Network Nodes