Capacitated Assignment Research Articles

Schedule-based transit assignment with online bus arrival information

Most public transportation services deviate from their published schedule. To cope with the delay caused by the unreliable service, passengers use online information about the bus arrival time which affects their route choice behavior. Current schedule-based transit assignment models fail to capture the passengers’ adaptive response to unreliable service, resulting in an inaccurate estimation of passenger wait time and passenger loads on various transit routes. The current study proposes schedule-based transit assignment models that incorporate online bus arrival information when modeling the passenger route choice in a stochastic and time-dependent transit network. The authors propose that passengers employ strategies when traveling between different origin–destination pairs not only due to the limited capacity of vehicles but also to cope with the transit delay. The passenger routing problem is modeled as a Markov Decision Process, and efficient algorithms are developed to solve this problem. Depending on the vehicle capacity, two types of assignment models are presented, namely, uncapacitated and capacitated assignments. When penalties for arriving at the destination outside the desired arrival time window are not applied, the uncapacitated assignment problem is formulated as a linear program. On the other hand, the capacitated assignment is formulated as a variational inequality problem for which an efficient Method of Successive Averages-based heuristic solution algorithm is proposed. Computational experiments are presented for a small and a large schedule-based transit network. The results show that denied boarding in an unreliable network leads to higher expected costs to passengers compared to the reliable and uncongested network. Furthermore, the analysis shows that the strategies evaluated with reliable schedule assumption lead to unreliable paths in the network and produces more transferring flow than should happen in practice. The application of our method to a subnetwork of the Twin Cities transit network with artificial demand reveals that passengers traveling from a residential area to the University of Minnesota campus may prefer taking a path with transfer in the event of highly unreliable transit service on the direct routes.

Online Bipartite Matching with Amortized O (log 2 n ) Replacements

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O (log 2 n ) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n ) lower bound. The previous best strategy known achieved amortized O (√ n ) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O ( n ) bound was known except in special cases. Our analysis immediately implies the same upper bound of O (log 2 n ) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L , then the SAP protocol makes amortized O (min { L log 2 n , √ n log n }) reassignments. We also show that this is close to tight, because Ω (min { L , √ n }) reassignments can be necessary.

A Transit Bottleneck Model for Optimal Control Strategies and its Use in Traffic Assignment in Paris

Control strategies are of crucial importance in operations of saturated transit lines. At a given bottleneck, delay propagates due to an excess of demand over supply. Operators therefore use optimal strategies (e.g. holding, fast-running, train-canceling etc.) in order to recover the level of service. While toolbox for train operation control system has thrived in past years, few traffic assignment models for large-scale transit networks on the planning side have touched on the impact of control dynamics: static macroscopic models perform quite well in simulating platform/vehicle saturation, but less so with bottleneck delay; dynamic models deal with dynamic passenger behavior under congestion, yet single vehicle runs are not explicitly considered either. The objective of this paper is to (1) capture the dynamic control features in a bottlenecked rail system; (2) provide a computation-friendly simulation approach to optimize the control; and finally (3) fit the bottleneck model in a static capacitated assignment model to improve its performance. The paper first provides mathematical formulae to describe bottleneck phenomena in a high-demand high-frequency transit line. Formulae are then cast into a control problem of dynamic system, to which we apply dynamic programming to reduce the dimension of decision variables. A numerical application to the Line RER A in Paris is followed, in which control strategies are optimized at key stations in a rolling horizon; on-platform passenger activities are simulated accordingly. Results are compared with both static model and real-life data, showing the effectiveness of the control-based assignment.

Note on the applicability of the VCG mechanism to capacitated assignment problems and extensions

For the allocation of heterogeneous items, it is known that the buyers‐are‐substitutes condition is necessary and sufficient to ensure that a pricing equilibrium can yield the same allocation and payments as the VCG mechanism. Furthermore, concavity of the corresponding transferable utility TU‐game guarantees that this VCG outcome can also be achieved by an ascending price auction. We show that concavity, and hence the buyers‐are‐substitutes condition, holds for the TU‐game of the assignment problem with general capacities. Therefore, the VCG mechanism is supported by a pricing equilibrium which can also be achieved by an ascending auction. We also show that the buyers‐are‐substitutes condition, and hence concavity, does not hold anymore for very natural and straightforward extensions of this problem. This shows that the necessity of the substitutes property is a considerable restriction on the applicability of the VCG mechanism.

An augmented lagrangean dual algorithm for link capacity side constrained traffic assignment problems

As a means to obtain a more accurate description of traffic flows than that provided by the basic model of traffic assignment, there have been suggestions to impose upper bounds on the link flows. This can be done either by introducing explicit link capacities or by employing travel time functions with asymptotes at the upper bounds. Although the latter alternative has the disadvantage of inherent numerical ill-conditioning, the capacitated assignment model has been studied and applied to a limited extent, the main reason being that the solutions can not be characterized by the classical Wardrop equilibrium conditions; they may, however, be characterized as Wardrop equilibria in terms of a well-defined, natural generalized travel cost. The introduction of link capacity side constraints makes the problem computationally more demanding. The availability of efficient algorithms for the basic model of traffic assignment motivates the use of dualization approaches for handling the capacity constraints. We propose and evaluate an augmented Lagrangean dual method in which the uncapacitated traffic assignment subproblems are solved with the disaggregate simplicial decomposition algorithm. This algorithm fully exploits the subproblem's structure and has very favourable reoptimization capabilities; both these properties are necessary for achieving computational efficiency in iterative dualization schemes. The dual method exhibits a linear rate of convergence under a standard nondegeneracy assumption. The efficiency of the overall algorithm is demonstrated through experiments with capacitated versions of well-known test problems, with the conclusion that the introduction of link capacities increases the computing times with no more than a factor of four. The introduction of capacities and the algorithm suggested can be used to derive tolls for the reduction of flows on overloaded links. The solution strategy can be applied also to other types of traffic assignment models where side constraints have been added in order to refine a descriptive or prescriptive assignment model.