The Braess paradox, introduced by D. Braess in 1968, describes the situation in which the total time spent in a system for at least two vehicles travelling from one node to another within a network of several nodes may increase if an additional path is added. This effect is due to non-cooperative behavior of the subjects involved and the fact that time consumption on the paths depends on their respective congestion. In this Game Theoretical paradox, full information is given to all vehicles about the theoretically shortest path for one vehicle, measured in time units, disregarding any congestion, which leads to a routing decision of individual rationality. This paper analyzes the Braess paradox if several shipping units have to be transported by multiple vehicles and load weight influences traffic congestion. A mixed integer linear programming formulation is used to model the problem. We introduce the model in detail, give numerical examples and analyze the results. A brief literature review on previous publications dealing with the Braess paradox is given as well.
Table of Contents
1. Introduction
2. Literature review
3. Model formulation
3.1 Notation
3.2 MIP-formulation
4. Numerical results
4.1 Description of the case
4.2 Data for the example
4.3 Results
5. Discussion
6. Conclusion and research outlook
Objectives & Research Topics
This paper aims to investigate the occurrence of the Braess paradox within logistics networks, specifically examining how the assignment of shipping units to transportation carriers influences traffic latency. The research addresses whether load-dependent travel times per vehicle and different assignment strategies exacerbate or mitigate the phenomenon of increased total system travel time when additional paths are introduced.
- Mathematical modeling of the Braess paradox using Mixed Integer Linear Programming (MIP).
- Analysis of load-based latency functions in transportation networks.
- Impact assessment of varying vehicle counts and shipping unit distributions.
- Evaluation of different assignment strategies (unbalanced, balanced, Pareto-distributed) on network performance.
- Identification of critical network parameters and their influence on the Braess effect.
Excerpt from the Book
1. Introduction
Traffic flows can be modeled using a network consisting of several nodes which are connected via arcs. From a single start node to a single sink node, one or more paths can provide the opportunity to travel from origin to destination via different available directed arcs. Braess stated that adding a path to a network may increase its optimal total traffic flow time. The idea behind this is that each vehicle, obtaining all information about the theoretical time required from a source node to a sink node, takes the path that looks most preferable to it, neglecting the decisions of other vehicle drivers and therefore neglecting any congestion influences. The resulting total time spent for all vehicles in the whole system need not be minimal as the duration of travelling on a specific path depends on the congestion on the arcs that are used by the vehicles and that are part of the respective path.
Extensions of an existing network may cause a redistribution of flows that can result in longer individual running times [1], [2]. Therefore, the Braess paradox occurs in a graph if the traffic flow is not Pareto-optimal [3]. The linear mixed integer programming formulation (MIP) presented in this paper aims at minimizing the total latency occurring in the system, respectively the maximum time amongst all paths that connect start and sink nodes and that are used by at least one vehicle. We make the assumption that each vehicle takes an individual choice of its path, neglecting any path congestions that may arise due to the decisions of other vehicles. This can be modeled by forcing at least one vehicle to use the theoretically shortest path of all available paths, measured in time units. We use the original traffic network as provided in [1], but make an extension that shipping units have to be carried from origin to destination, which can be operated by several vehicles. However, the amount of load causes the vehicles to slow down speed based on a load-based latency function. Therefore it is analyzed how the assignment of shipping units to transportation carriers causes the Braess paradox and how the load-latency costs influence the time functions.
Summary of Chapters
1. Introduction: Introduces the Braess paradox and outlines the research objective of applying a MIP formulation to assess load-dependent latency in transportation networks.
2. Literature review: Provides a survey of existing research on the Braess effect in traffic, communication, and engineering networks, establishing the research gap.
3. Model formulation: Details the mathematical framework, including notation and the Mixed Integer Programming formulation used to model vehicle and unit assignments.
4. Numerical results: Describes the test cases, data sets, and experimental setup, presenting computational findings across different network configurations.
5. Discussion: Analyzes the experimental results, examining how load-latency parameters and distribution strategies impact the frequency of the Braess effect.
6. Conclusion and research outlook: Summarizes the findings regarding the link between load-based costs and the Braess paradox, while suggesting future research directions.
Keywords
Braess Paradox, Mixed Integer Linear Programming, Game Theory, Truck Assignment, Traffic Flows, Latency Functions, Load-dependent Travel, Network Equilibrium, Logistics Optimization, Pareto-optimality, Routing Decisions, Transportation Networks, System Latency, Vehicle Scheduling, Network Congestion.
Frequently Asked Questions
What is the fundamental subject of this research?
The research focuses on the Braess paradox, a phenomenon where adding infrastructure to a network can paradoxically increase total travel time, specifically within the context of cargo and truck-based logistics.
What are the central thematic areas?
The key themes include network flow optimization, game-theoretic decision making by vehicle drivers, load-dependent latency modeling, and the impact of shipping unit assignments on network efficiency.
What is the primary objective of this study?
The primary goal is to analyze how the distribution of shipping units onto vehicles affects the occurrence of the Braess paradox, supported by a novel Mixed Integer Linear Programming (MIP) model.
Which scientific method is employed?
The authors employ a Mixed Integer Linear Programming (MIP) formulation to simulate traffic flows and identify non-Pareto-optimal routing decisions under congestion.
What is covered in the main section of the paper?
The paper moves from a theoretical literature overview to the mathematical modeling of the network, followed by empirical numerical tests that vary vehicle counts, load weights, and network arcs to identify when the paradox emerges.
Which keywords characterize this paper?
The study is characterized by terms such as Braess Paradox, Mixed Integer Linear Programming, Game Theory, Truck Assignment, and load-dependent latency.
How does the load on a vehicle impact the Braess paradox according to this study?
The study finds that the amount of load a vehicle carries affects its speed via a load-based latency function; higher loads increase congestion, which alters the attractiveness of paths and influences the overall occurrence of the paradox.
Is the Braess effect always predictable when changing the number of vehicles?
No, the authors note that it is not easy to detect a simple, direct linear relationship between the number of trucks and the paradox; however, they observe that 4-truck configurations sometimes minimize the effect compared to others.
What role does the distribution of goods play in the findings?
The distribution is critical; the study shows that uniform distribution of goods across all trucks leads to the globally highest frequency of the Braess paradox (86%), whereas uneven distribution can help avoid the paradox.
- Quote paper
- Marc-André Dr. Weber (Author), Florian Koch (Author), 2014, Analysis of the Braess paradox under various assignments from shipping units to carriers, Munich, GRIN Verlag, https://www.grin.com/document/280983
