On a network equilibrium problem with mixed demand

© Springer International Publishing Switzerland 2016.In the present paper, we formulate the network equilibrium problem with mixed demand containing the fixed and variable components. We present the equilibrium conditions and the conditions for existence of solution of this problem. In addition, we show that the network equilibrium problem with mixed demand generalizes the network equilibrium problems with fixed demand and elastic demand and establish the connection with the auction equilibrium problem. Preliminary computational experiments are also presented

The network equilibrium problem with mixed demand

© 2017, Pleiades Publishing, Ltd. We formulate the network equilibrium problem with mixed demand which generalizes the problems of network equilibrium with fixed and elastic demand. We prove the equilibrium conditions for this problem and propose some conditions of existence of a solution that are based on the coercivity property.We establish a connection between the problem of network equilibrium with mixed demand and the problem of auction equilibrium. The results of test calculations are presented for a model example

Quasi-variational inequality formulation of the mixed equilibrium in multiclass routing games

In the modeling of competition on networks it is usually assumed that users either behave following the Wardrop equilibrium or the Nash equilibrium concept. Nevertheless, in several equilibrium situations, for instance in urban traffic flows, intercity freight flows and telecommunication networks, a mixed behavior is observed. This paper presents a time-dependent network model shared by two types of users: group users (Nash players) and individual users (Wardrop players). A group user has a significant impact on the load of the network, whereas an individual user has a negligible impact. Both classes of users choose the paths to ship their jobs so as to minimize their costs, but they apply different optimization criteria. The source of interaction of users is represented by the travel demand, which is assumed to be elastic with respect to the equilibrium solution. Thus, the equilibrium distribution is proved to be equivalent to the solution of an appropriate time-dependent quasi variational inequality problem. A result on the existence of solutions is discussed as well as a numerical example.Nash equilibrium, Wardrop equilibrium, routing, quasi-variational inequality

Quasi-variational inequality formulation of the mixed equilibrium in multiclass routing games

In the modeling of competition on networks it is usually assumed that users either behave following the Wardrop equilibrium or the Nash equilibrium concept. Nevertheless, in several equilibrium situations, for instance in urban traffic flows, intercity freight flows and telecommunication networks, a mixed behavior is observed. This paper presents a time-dependent network model shared by two types of users: group users (Nash players) and individual users (Wardrop players). A group user has a significant impact on the load of the network, whereas an individual user has a negligible impact. Both classes of users choose the paths to ship their jobs so as to minimize their costs, but they apply different optimization criteria. The source of interaction of users is represented by the travel demand, which is assumed to be elastic with respect to the equilibrium solution. Thus, the equilibrium distribution is proved to be equivalent to the solution of an appropriate time-dependent quasi-variational inequality problem. A result on the existence of solutions is discussed as well as a numerical example

A Study of Problems Modelled as Network Equilibrium Flows

This thesis presents an investigation into selfish routing games from three main perspectives. These three areas are tied together by a common thread that runs through the main text of this thesis, namely selfish routing games and network equilibrium flows. First, it investigates methods and models for nonatomic selfish routing and then develops algorithms for solving atomic selfish routing games. A number of algorithms are introduced for the atomic selfish routing problem, including dynamic programming for a parallel network and a metaheuristic tabu search. A piece-wise mixed-integer linear programming problem is also presented which allows standard solvers to solve the atomic selfish routing problem. The connection between the atomic selfish routing problem, mixed-integer linear programming and the multicommodity flow problem is explored when constrained by unsplittable flows or flows that are restricted to a number of paths. Additionally, some novel probabilistic online learning algorithms are presented and compared with the equilibrium solution given by the potential function of the nonatomic selfish routing game. Second, it considers multi-criteria extensions of selfish routing and the inefficiency of the equilibrium solutions when compared with social cost. Models are presented that allow exploration of the Pareto set of solutions for a weighted sum model (akin to the social cost) and the equilibrium solution. A means by which these solutions can be measured based on the Price of Anarchy for selfish routing games is also presented. Third, it considers the importance and criticality of components of the network (edges, vertices or a collection of both) within a selfish routing game and the impact of their removal. Existing network science measures and demand-based measures are analysed to assess the change in total travel time and issues highlighted. A new measure which solves these issues is presented and the need for such a measure is evaluated. Most of the new findings have been disseminated through conference talks and journal articles, while others represent the subject of papers currently in preparation

Selective Bi-coordinate variations for network equilibrium problems with mixed demand

© Springer International Publishing AG, part of Springer Nature 2018. In the present paper, we propose a modification of the method of bi-coordinate variations for network equilibrium problems with mixed demand. This method is based on the equilibrium conditions of the problem under consideration. It uses a special tolerance control and thresholds for constructing descent directions and a variant of the Armijo-type line-search procedure as a rule of step choice. Some results of preliminary numerical calculations which confirm efficiency of the method are also presented