Metropolis

## Team

Scientific advisers:

<table>
								<tr>
									<th>Name</th>
									<th>Status</th>
									<th>Affiliation</th>
								</tr>
								<tr>
									<th>Araldo, Andrea</th>
									<td>Associate Professor</td>
									<td>Télécom SudParis, Institut Polytechnique de Paris</td>
								</tr>
								<tr>
									<th>Coulombel, Nicolas</th>
									<td>Associate Professor</td>
									<td>LVMT, École des ponts ParisTech</td>
								</tr>
								<tr>
									<th>Geroliminis, Nikolas</th>
									<td>Professor</td>
									<td>LUTS, École Polytechnique Fédérale de Lausanne</td>
								</tr>
								<tr>
									<th>Kilani, Moez</th>
									<td>Professor</td>
									<td>Université du Littoral Côte d'Opale</td>
								</tr>
								<tr>
									<th>Lindsey, Robin</th>
									<td>Professor</td>
									<td>Sauder School of Business, University of British Columbia</td>
								</tr>
								<tr>
									<th>Picard, Nathalie</th>
									<td>Professor</td>
									<td>BETA, University of Strasbourg</td>
								</tr>
								<tr>
									<th>Seshadri, Ravi</th>
									<td>Assistant Professor</td>
									<td>Technical University of Denmark</td>
								</tr>
							</table>

## History of the Project

<ul>
								<li class="fragment" data-fragment-index="0"><b>1997-2009:</b> Simulator v1 [C++] (de Palma, Marchal, Nesterov)</li>
								<li class="fragment" data-fragment-index="1"><b>2001-2009:</b> Interface for the simulator v1 [Java] (Marchal)</li>
								<li class="fragment" data-fragment-index="2"><b>2018-2021:</b> Web interface for the simulator v1 [Python Django] (Javaudin et al.)</li>
								<li class="fragment" data-fragment-index="3"><b>2021-:</b> Simulator v2 [Rust] (de Palma, Javaudin)</li>
								<li class="fragment" data-fragment-index="4"><b>2021-:</b> Interface for the simulator v2 [Python + JavaScript] (Javaudin et al.)</li>
							</ul>

Basic Principle

Metropolis is an iterative model
At each iteration, four models are run successively (network skims computation, pre-day model, within-day model and day-to-day model)
The simulation stops when a convergence criteria is met or when the maximum number of iterations is reached

## Input of the Model

### Population

<ul>
								<li class="fragment"><b>Mode choice model</b> (e.g., Logit, deterministic)</li>
								<li class="fragment"><b>Schedule-utility model</b> (a function of the departure and arrival time; e.g., alpha-beta-gamma model)</li>
								<li class="fragment">Description of the <b>modes</b> available to the agent</li>
							</ul>

<div class="fragment">
								<hr>
								Currently, two different mode types are supported:
								<ul>
									<li>Mode with <b>time-independent utility</b> (e.g., walking, cycling): only the utility level needs to be known</li>
									<li><b>Road modes</b> (e.g., car, motorcycle, truck) with:
										<ul>
											<li>Origin and destination node</li>
											<li>Vehicle type</li>
											<li>Departure-time model (either fixed or a continuous Logit model)</li>
											<li>Travel-utility function (a function of the travel time)</li>
										</ul>
									</li>
								</ul>
							</div>

Network skims computation

Input: time-dependent travel-time function for each road of the road-network graph
Step 1: compute a time-dependent Hierarchy Overlay of the graph
Step 2: compute search spaces for each origin and destination node
Step 3: compute profile queries for each origin-destination pair
Output: time-dependent travel-time function for each origin-destination pair (with at least one trip)

Geisberger, R. and Sanders, P., 2010. Engineering time-dependent many-to-many shortest paths computation. In 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

Pre-Day Model

Input: time-dependent travel-time function for each origin-destination pair
Departure-time choice: choose the departure time of the agent, given the travel-time function, the schedule-utility and travel-utility functions and the mode (continuous Logit model)
Route choice: choose the route that minimizes the travel time of the agent for the departure time chosen
Mode choice: choose the mode of the agent, given the expected utility for each mode available (Logit model, deterministic model or any other choice model)
Output: mode, departure-time and route chosen by each agent

Note. For some modes, there is no departure-time and route choice.

Bottleneck Equation

\[ C = \alpha \cdot {tt} + \beta \cdot [t^* - t_a]_+ + \gamma \cdot [t_a - t^*]_+ \]

Day-to-Day Model

Input: expected and simulated time-dependent travel-time function for each road
Compute the expected travel-time functions for next iteration given the expected and simulated travel-time functions of current iteration, using a Markov process (e.g., exponential process with weight $\lambda$ for simulated values)
Stop the simulation if a stopping criteria is reached (e.g., maximum number of iterations, EXPECT threshold, departure-time shift threshold)
Output: expected time-dependent travel-time function for each road