Robust Line Planning under Unknown Incentives and Elasticity of Frequencies

Robust Line Planning under Unknown Incentives and Elasticity of Frequencies ? Spyros Kontogiannis1,2 and Christos Zaroliagis1,3 1 R.A. Computer Technology Institute, N. Kazantzaki Str., Patras University Campus, 26500 Patras, Greece 2 Computer Science Department, University of Ioannina, Ioannina, Greece 3 Department of Computer Engineering and Informatics, University of Patras, 26500 Patras, Greece Email: , Abstract. The problem of robust line planning requests for a set of origin-destination paths (lines) along with their traffic rates (frequencies) in an underlying railway network infrastructure, which are robust to fluctuations of real-time parameters of the solution. In this work, we investigate a variant of robust line planning stemming from recent regulations in the railway sector that introduce competition and free railway markets, and set up a new application scenario: there is a (potentially large) number of line operators that have their lines fixed and operate as competing entities struggling to exploit the underlying network infrastructure via frequency requests, while the management of the infrastructure itself remains the responsibility of a single (typically governmental) entity, the network operator. The line operators are typically unwilling to reveal their true incentives. Nevertheless, the network operator would like to ensure a fair (or, socially optimal) usage of the infrastructure, e.g., by maximizing the (unknown to him) aggregate incentives of the line operators. We show that this can be accomplished in certain situations via a (possibly anonymous) incentivecompatible pricing scheme for the usage of the shared resources, that is robust against the unknown incentives and the changes in the demands of the entities. This brings up a new notion of robustness, which we call incentive-compatible robustness, that considers as robustness of the system its tolerance to the entities’ unknown incentives and elasticity of demands, aiming at an eventual stabilization to an equilibrium point that is as close as possible to the social optimum. 1 Introduction An important phase in the strategic planning process of a railway (or any public transportation) company is to establish a suitable line plan, i.e., to determine the routes of trains that serve the customers. In the line planning problem, we ? This work was partially supported by the Future and Emerging Technologies Unit of EC (IST priority – 6th FP), under contract no. FP6-021235-2 (project ARRIVAL). ATMOS 2008 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems http://drops.dagstuhl.de/opus/volltexte/2008/1581 2 Spyros Kontogiannis and Christos Zaroliagis are given a network G = (V, L) (usually referred to as the public transportation network), where the node set V represents the set of stations (including important junctions of railway tracks) and the edge set L represents the direct connections or links (of railway tracks) between elements of V . A line is a path in G. Typically, a line pool is also provided, i.e., a set of potential lines among which the final set of lines will be decided. The frequency of a line l is a rational number indicating how often service to customers is provided along l within the planning period considered. For an edge ` ∈ L, the edge frequency f` is the sum of the frequencies of the lines containing ` and is upper bounded by the capacity c` of `, i.e., a maximum edge frequency established for safety reasons. The goal of the line planning problem is to provide the final set of lines offered by the public transportation company along with their frequencies (also known as the line concept). The line planning problem has mainly been studied under two main approaches (see e.g., [6, 7]). In the cost-oriented approach, the goal is to minimize the costs of the public transportation company, under the constraint that all customers can be transported. In the customer-oriented approach, the goal is to maximize the number of customers with direct connections (under a similar constraint), or at least minimize the traveling time of the customers. A recent approach aims at minimizing the travel times over all customers including penalties for the transfers needed [9, 11]. The aforementioned approaches do not take into account certain fluctuations of input parameters; for instance, due to disruptions to daily operations (e.g., delays), or due to fluctuating customer demands. This aspect introduces the so-called robust line planning problem: provide a set of lines along with their frequencies, which are robust to fluctuations of input parameters. Very recently, a game theoretic approach for robust line planning was presented in [10]. In that model, the lines act as players and the strategies of the players correspond to line frequencies. Each player aims to minimize the expected delay of her own lines. The delay depends on the traffic load and hence on the frequencies of all lines in the network. The objective is to provide lines that are robust against delays. This is pursued by distributing the traffic load evenly over the network (respecting edge capacities) such that the probability of delays in the system is as small as possible. In this work, we investigate a different perspective of robust line planning stemming from recent regulations in the railway sector (at least within Europe) that introduce competition and free railway markets, and set up a new application scenario: there is a (possibly large) number of line operators that should operate as commercial organizations, while the management of the network remains the responsibility of a single (typically governmental) entity; we shall refer to the latter as the network operator. Under this framework, line operators act as competing entities for the exploitation of shared goods and are (possibly) unwilling to reveal their actual level-of-satisfaction functions that determine their true incentives. Nevertheless, the network operator would like to ensure the maximum possible level of satisfaction of these competing entities, e.g., by maximizing the Robust Line Planning under Unknown Incentives & Elasticity of Frequencies 3 (unknown due to privacy) aggregate levels of satisfaction. This would establish a notion of a socially optimal solution, which could also be seen as a fair solution in the sense that the average level of satisfaction is maximized. Additionally, the network operator should ensure that the operational costs of the whole system are covered by a fair cost sharing scheme announced to the competing entities. This implies that a (possibly anonymous) pricing scheme for the usage of the shared resources should be adopted that is robust against changes in the demands of the entities (line operators). That is, we consider as robustness of the system its tolerance to the entities’ unknown incentives and elasticity of demand requests, and the eventual stabilization at a (...truncated)