Improving Customer Proximity to Railway Stations

We consider problems of (new) station placement along (existing) railway tracks, so as to increase the number of users. We prove that, in spite of the NP-hardness for the general version, some interesting cases can be solved exactly by a suitable dynamic programming approach. For variants in which we also take into account existing connections between cities and railway tracks (streets, buses, etc.) we instead show some hardness results. 1 Models and Problems There are many instances when public or private sector bodies are faced with making decisions on how to allocate facilities optimally. Such problems with mathematically quantifiable optimization constraints have been studied extensively in the scientific literature (e.g., see the book [3]). Recently the European Union has been encouraging the privatization of railway assets in various EU countries in order to improve system efficiency as well as customer satisfaction. In this paper we approach one such problem by studying how customer proximity can affect the railway station location. More specifically, given a set of settlements and an existing track, one wishes to build a set of new stations such that (some of) the settlements can easily access those stations and, thus, use the railway. This gives a gain in terms of (potentially) new users, but it also turns into a cost for the old ones (for instance, a new station results into a delay for those trains travelling on the track). Let us consider the following problem: Work partially supported by the Swiss Federal Office for Education and Science under the Human Potential Programme of the European Union under contract no. HPRN-CT-1999-00104 (AMORE). Research of E. Kranakis was supported in part by NSERC (Natural Sciences and Engineering Research Council of Canada) and MITACS (Mathematics of Information Technology and Complex Systems) grants. R. Petreschi et al. (Eds.): CIAC 2003, LNCS 2653, pp. 264–276, 2003. c © Springer-Verlag Berlin Heidelberg 2003 Improving Customer Proximity to Railway Stations 265 Input: A set of P = {p1, . . . , pn} of settlements (i.e. points) on the Euclidean plane, each of them with an associated demand di, and an existing railway, that is, a set of straight-line segments forming a connected polygonal and whose endpoints represent existing stations. Solution: A set of new stations along the track. Given a solution to this problem, we have a gain and a cost function due to the new stations. The cost of building a new station, in general, depends on the position we are placing it. In the sequel, we describe some possible definitions for the gain function. All such definitions are distance-based, that is, the gain due to the new stations depends on how far a settlement is from its closest new station. We will first assume that the distance is the Euclidean one (although, some of the results can be extended to other metrics). Single radius. We first consider the following (simplified) scenario. A certain settlement pi is far away from every existing station. So, for the people living there it is not worth to use the railway. If we build a new station which is close enough to pi (let us say at distance less than R) then the railway transportation becomes “competitive” with respect to other transportations and all the people in pi (let their number be di) will use this new station. We then have the following model: a settlement pi uses a (newly built) station if and only if (a) this station is at distance less than or equal to some radius R and (b) no existing station was at distance less than R. Notice that we can assume w.l.o.g. that no settlement in P is currently “covered” by the existing stations. Hence, the gain of a set S of new stations is the sum of the demands di of those pi that are covered by the radius of some s ∈ S. Formally, n ∑ i=1 di · cover(S, pi), where cover(S, pi) equals 1 if there exists an s ∈ S at distance less than or equal to R from pi, and it equals 0 otherwise. Distance based costs. Notice that the single radius model is, in some cases, too unrealistic since it assumes that a station at distance R = 500m, for instance, is accessible, while a station at distance R′ = 550m is not. A more realistic model should take into account the fact that the closer a station is the more (potential) customers from a settlement are expected. For instance, we could say that the expected number of users from pi is di/(δ + 1), where δ is the distance of pi to the closest station. More generally, given a monotone (decreasing) function α(·), the gain of a set of new stations can be expressed as ∑n i=1 di ·α(δ(pi, S)), where δ(pi, S) is the distance between pi and the closest station in S. Multiple radii. This setting is somewhat in between the two previous ones. Indeed, it can be used to approximate any distance based cost function with a fixed set of radii. Roughly speaking, these radii result from a “discretization” of an arbitrary function α(·). For instance, the function α(δ) = 1/(δ + 1) can be 266 E. Kranakis et al. approximated by two (or more) radii. Clearly, the more radii we consider, the better we approximate α(·). Two optimization problems for single radius. We now focus on the single radius model and we assume the cost of building a new station to be constant. For this version, one can envision the following two optimization problems: – Min Number of Stations (Min Station): minimize the number of stations needed to cover all the settlements. – Budget Constrained Max Gain (Max Gain): given an integer k, with 1 ≤ k ≤ n, find the placement of k stations that maximizes the gain. We first observe that the second problem can be easily used to solve the first one: one just has to try all the k from 1 up to the smallest one for which the gain is the biggest possible, that is, we cover all the settlements. On the other hand, the other way round does not necessarily work. The limitation on the number of new stations seems to complicate things: with only k new stations at hand, we may not be able to cover all the settlements. In this case, our task is to find the best subset of settlements that can be covered with k stations only.