Top (2009) 17: 335–346DOI 10.1007/s11750-008-0061-4

O R I G I NA L PA P E R

Stop location design in public transportation networks:covering and accessibility objectives

Dwi Retnani Poetranto Groß ·Horst W. Hamacher · Simone Horn · Anita Schöbel

Received: 29 October 2006 / Accepted: 2 September 2008 / Published online: 24 September 2008© Sociedad de Estadística e Investigación Operativa 2008

Abstract We consider the location of new stops along the edges of an existing publictransportation network. Examples of StopLoc include the location of bus stops alongsome given bus routes or of railway stations along the tracks in a railway system.In order to evaluate the decision assume that potential customers in given demandfacilities are known. Two objectives are proposed. In the first one, we minimize thenumber of stations such that any of the demand facilities can reach a closest stationwithin a given distance of r . In the second objective, we fix the number of new stationsand minimize the sum of the distances between demand facilities and stations. Theresulting two problems CovStopLoc and AccessStopLoc are solved by a reduction toa classical set covering and a restricted location problem, respectively. We implementthe general ideas in two different environments, the plane, where demand facilitiesare represented by coordinates, and in networks, where they are nodes of a graph.

Keywords Location · Public transportation · Covering · Accessibility

Mathematics Subject Classification (2000) 90B80 · 90C10 · 90C35 · 90B20

D.R.P. Groß, H.W. Hamacher and S. Horn were partially supported by DeutscheForschungsgemeinschaft (DFG) grant HA 1737/7 “Algorithmik großer und komplexer Netzwerke,”by New Zealand’s Julius von Haast award, and by the Rheinland-Pfalz cluster of excellence“Dependable adaptive systems and mathematical modeling.”A. Schöbel was partially supported by the Future and Emerging Technologies Unit of EC (ISTpriority—6th FP), under contract no. FP6-021235-2 (project ARRIVAL).

D.R.P. Groß · H.W. Hamacher · S. HornFachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, Kaiserslautern, Germany

A. Schöbel (�)Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen,Göttingen, Germanye-mail: schoebel@math.uni-goettingen.de

336 D.R.P. Groß et al.

1 Introduction

The acceptance of public transportation depends on various components such as com-petitive fares, convenience, punctuality, reliability, etc. In this paper, we address thequestion of convenience. In particular, we investigate the problem of finding “good”locations for stops in order to allow potential customers to access the public trans-portation system within a reasonable amount of time. The public transportation sys-tem is defined by a Public Transportation Network, which is an undirected graphPTN = (V ,E) representing the railway tracks or the bus routes. The potential cus-tomers of the public transportation system are modeled as demand facilities, forwhich we assume that the distance to any point of the PTN is known. We will studythe following two different objective functions.

In our first objective, the goal is to minimize the number of stops such that eachdemand facility is within a tolerable distance from at least one stop. The maximal dis-tance that a customer is willing to accept is called covering radius, and we thereforecall the problem Covering Stop Location problem, or (CovStopLoc) for short. Forbus stops, a covering radius of 400 m is common. In rail transportation, the coveringradius is larger and is usually assumed to be 2 km. In the literature, CovStopLoc hasbeen introduced in Gleason (1975) and is considered in Murray et al. (1998), Murray(2001a, 2001b), Laporte et al. (2002), Wu and Murray (2005), see also referencestherein. In these papers, StopLoc is treated in a discrete setting, i.e., a finite set isconsidered as potential new stops. Schöbel et al. (2008) allow a continuous set ofpossible locations for the stops, for instance, all points on the current bus routes orrailway tracks. The problem is a variant of the planar maximal covering problem asintroduced in Church (1984). An application of continuous CovStopLoc is given inHamacher et al. (2001), where the authors report on a project with the largest Germanrail company (Deutsche Bahn) and consider the trade-off between the positive andnegative effects of additional stops. Based on this application, variants of continuousstop location problems have been treated in Kranakis et al. (2003), Schöbel (2005).The problem has been solved for the case of two intersecting lines, see Mammana etal. (2004). Algorithmic approaches for solving the underlying covering problem havebeen studied in Ruf and Schöbel, Mecke and Wagner (2004, 2004). Complexity andapproximation issues have been presented in Mecke et al. (2005). For an overview onvarious models, we also refer to Schöbel (2006).

The second objective function we consider leads to the Access Stop Location Prob-lem (AccessStopLoc). Here we minimize the sum of the distances between the de-mand facilities and their closest stations. Nearly nothing has been reported in theliterature about this problem so far. AccessStopLoc was first introduced in the thesisof Poetranto (2004), where it is treated as a restricted planar median location prob-lem. In Murray and Wu (2003), the accessibility distance is also used as objectivefunction, and a discrete version of this problem has been formulated as an integerprogram.

In our paper, we will consider (CovStopLoc) and (AccessStopLoc) both in a pla-nar and in a network environment. In the former case, the demand is concentratedin points pm = (pm1,pm2), m = 1, . . . ,M , of the plane R

2. In the latter case, thedemand originates in nodes of an extended network. For (CovStopLoc), we present a

Stop location design in public transportation networks: covering 337

unifying framework handling the planar and the network case and new results moti-vated by the thesis of Horn (2004). In (AccessStopLoc), we present new results bothfor the planar and for the network case.

The remainder of our paper is structured as follows. In the next section, (Cov-StopLoc) and (AccessStopLoc) are formally introduced. In Sect. 3, we show thatCovStopLoc can be reduced to a discrete set covering problem by identifying a finitedominating set (FDS). We show that such an FDS can be found efficiently. Finitedominating sets for AccessStopLoc are presented in Sect. 4. We finish the paper bysketching ideas on how to extend the presented theory.

2 Covering and access problems

In both problems discussed in this paper, the new stops are to be selected amongthe points of a given public transportation network (PTN). A PTN consists of anundirected connected graph G = (V ,E) where the edges e = (vi, vj ) ∈ E betweennodes vi and vj are part of public transportation lines. The nodes V = Vstop ∪ Vbreakcan be existing stations or breakpoints. The latter ones are needed to guarantee aplanar embedding of the graph in which the edges are represented as straight lines.Each edge e has an associated length le, which is a real number representing itsphysical length or travel time. A point s ∈ G of the graph G = (V ,E) is either anode or strictly contained in an edge e = (vi, vj ). We use the notation s = (e, t) with0 ≤ t ≤ 1, where the distance between vi and s is t le and between s and vj is (1− t)le .

Let M = {1, . . . ,M}, and let P = {pi : i ∈ M} be the demand facilities, i.e., thepotential users of the public transportation system. For each demand facility p ∈ Pand each point s ∈ G, we assume that a distance d(p, s) ∈ R is known. If S ⊆ G is acompact subset of points, let d(p, S) := mins∈S d(p, s) be the closest distance fromp ∈ P to any point in S . We consider two versions of stop location problems.

In the Covering Stop Location Problem, a real number r is given as radius. Wewant to find a set of locations for stations such that every customer can reach at leastone of the new stations with a distance of at most r :

minimize |S|subject to d(p, S ∪ Vstop) ≤ r ∀p ∈ P ,

S ⊆ G.

(CovStopLoc)

In the Accessibility Stop Location Problem, a positive integer k is given which is themaximal number of stations which can be built. We want to find a set S ⊆ G = (V ,E)

with |S| ≤ k such that the sum of distances to the demand facilities is minimized:

minimize f (S) := ∑p∈P d(p, S ∪ Vstop)

subject to S ⊆ G,

|S| ≤ k.

(AccessStopLoc)

Note that in the covering problem, we can assume w.l.o.g. that Vstop = ∅ by simplynot investigating demand facilities that are already covered by existing stops. In (Ac-cessStopLoc), we have to consider both cases, Vstop = ∅ and Vstop �= ∅. Both versions

338 D.R.P. Groß et al.

of the stop location problem include a distance d(p, s) between demand facilities andpoints in the PTN. We treat two different environments to define distances.

In the planar environment, we use the given straight line planar embedding of thePTN G = (V ,E) and interpret any point s of the graph as a point s = (s1, s2) ∈ R

2.Furthermore, let the demand facilities pm = (pm1,pm2) ∈ R

2, m = 1, . . . ,M . Thend(p, s) can be any distance function in R

2. In this paper, we will only consider thecase of rectangular distances d(p, s) := l1(p, s) := |p1 − s1| + |p2 − s2|. But ourresults can easily be extended to gauge distances as was done, for instance, in Schöbelet al. (2008).

In the network environment, the PTN is part of a larger network, the street andpublic transportation network (SPTN) G = (V , E ), where V = V ∪ P and E ⊆ E .The edges in E \E represent streets which can be used by the passengers to reach thestations, see Fig. 1.

The distance d(p, s) is the shortest path distance in G if both p and s are nodes ofG , i.e., for p ∈ P and s ∈ V . We denote this distance with dN (p, s). If s = (e, t) isa point of the PTN inside the edge e = (vi, vj ) ∈ E (i.e., with 0 < t < 1) and p ∈ P ,then

dN (p, s) := min{dN (p, vi) + t le, dN (p, vj ) + (1 − t)le

}.

Note that dN (p, s) = dN (p, (e, t)) is continuous, concave, and piecewise linear in t

if we fix p ∈ P and move s = (e, t) along an edge e, see Fig. 2.

Fig. 1 Street and public transportation network

Fig. 2 Network distance

Stop location design in public transportation networks: covering 339

Having defined two objective functions and two environments, we obtain fourtypes of stop location problems which will all be tackled using the following proper-ties.

1. The finite dominating set (FDS) property identifies a finite candidate set Scand

which is sufficient to choose locations for new stations.2. The polynomiality property furthermore yields such a set Scand which is polyno-

mial in the size of the problem.

If both properties are satisfied, the following algorithmic approach is straightfor-ward.

Algorithm Generic algorithm for StopLocx

Input: PTN = (V ,E), demand facilities P = {p1, . . . , pM} ⊂ R2.

Output: S ∗ optimal stops.

Step 1: Determine the set of candidates Scand.Step 2: Let S ∗ = argmin{f (S) : S ∈ Scand}.

(Note that the algorithm is polynomial for single facility StopLoc problems, butnot in the multifacility case.)

3 Covering

In this section, we present a unifying framework for (CovStopLoc) both for networksand for a planar environment.

Definition 3.1 Let r > 0 be a given radius. A demand facility p ∈ P is covered by apoint s in the PTN if d(s,p) ≤ r . For a set S of points ,we define

cover(S) = {p ∈ P : d(s,p) ≤ r for some s ∈ S

}.

If only a finite set S ⊆ G of points is allowed for new stops, the problem reducesto the well-known Location Covering Problem (or set covering problem) introducedin Toregas et al. (1971):

min∑

s∈Scand

xs

s.t. Ax ≥ 1, ∀p ∈ P ,

xs ∈ {0,1}, ∀s ∈ S

with the binary matrix A = (aps)p ∈ Ps ∈ S

, where

aps ={

1 if d(p, s) ≤ r,

0 else,and xs =

{1 if s ∈ S ,0 else.

340 D.R.P. Groß et al.

We will now identify such finite set S satisfying the FDS and polynomiality propertyboth for the planar and the network case. To this end, for m ∈ M, let

Bm := {s : d(pm, s) ≤ r

}

be diamond-shaped balls of radius r , centered at the demand points. Consider theplanar arrangement induced by the set of boundaries of the Bm, and let Z be the cellsof the arrangement. Each point in a given cell Z ∈ Z (0-, 1- or 2-dimensional) coversthe same set of demand points. Since the sets Bm are closed, we obtain:

Lemma 3.2 Let s be a point in the interior of some cell Z ∈ Z . Then cover({s}) ⊆cover( ˜{s}) for any point s̃ on the (relative) boundary ∂Z of Z.

This yields the main result of this section.

Theorem 3.3 Let CovStopLoc be feasible. Then there exists an optimal solution S ⊆Scand := {s ∈ PTN : s ∈ δZ for some Z ∈ Z} if this set is nonempty. Otherwise, anypoint of PTN is optimal.

Proof Let Scand = ∅. Then PTN ⊆ Bm for all m ∈ M, hence any point of PTN canbe used to cover all demand points.

Otherwise, let S be optimal and S � Scand. Take any s /∈ Scand and let e be theedge of s and Z ∈ Z the cell where s belongs to. Due to Lemma 3.2 there existss̃ ∈ δZ ∩ e (i.e., s̃ ∈ Scand) with cover({s}) ⊆ cover({s̃}). Hence, S̄ = S \ {s} ∪ {s̃}covers all demand points and satisfies |S̄| ≤ |S|. We iterate until all stops in S \ Scandare replaced by stops in Scand. �

Scand contains at most O(M|E|) points, hence the polynomiality property holds,see Schöbel et al. (2008) for details and a generalization to gauges. For the networkenvironment, we even get:

Theorem 3.4 The candidate set Scand for the network case satisfies |Scand| ≤ 2M|E|.

Proof Along an edge, g(s) = dN (p, s) is continuous, concave, and piecewise linearwith at most two linear pieces with slopes le and −le, respectively (Fig. 2). Hencethere exist at most two points on edge e with distance exactly r from p. �

Note that we can reformulate Scand = {s ∈ G : ∃pm ∈ P with dN (pm, s) = r} incase of the network environment.

4 Accessibility

(AccessStopLoc) is a restricted k-median problem, where the (at most) k new loca-tions are required to be points of the PTN. We consider two problems, namely com-pletely redesigning all stops (Vstop = ∅) and opening additional stops (Vstop �= ∅).

Stop location design in public transportation networks: covering 341

4.1 Accessibility problem in planar environment

We begin with the special case of establishing only a single new stop and there is noexisting stop in the PTN.

Let the construction lines L be given as the set of 2M vertical and horizontal linespassing through the M demand points, and let Z be the set of cells defined by thisplanar arrangement. Note that the objective function f is linear in each of these cells(see, e.g., Hamacher 1995).

Furthermore, let C be the set of intersection points between construction lines andthe PTN.

Theorem 4.1 Scand := Vbreak ∪ C is an FDS for (AccessStopLoc) in the case of asingle stop (see Fig. 3), i.e., there exists an optimal solution s∗ ∈ Scand.

Proof Let Z ∈ Z and e ∈ E. Then, e ∩ Z is either empty or a line segment, and f

is linear on e ∩ Z. Hence, to find an optimal solution of f in e ∩ Z it is sufficient toconsider its two endpoints. An endpoint is either a breakpoint or an intersection pointof the PTN and a construction line. �

Intersections l ∩ e in which a construction line coincides with an edge of the PTNneed not be considered due to the linearity of f over the cells of the arrangement.

There are at most |V | + 2M|E| candidates. The effort to compute the objectivevalue for each candidate is O(M), thus the overall complexity of a simple enumera-tion algorithm is O(M2|E|). It can be improved to O(M|E|) by handling a facilitylocation problem on each of the |E| edges in time O(M): On each edge e, the ob-jective function is the sum of M piecewise linear convex functions. Each of these M

functions has at most two breakpoints. Hence, the minimum of the objective on e canbe found in O(M) time by the algorithm in Zemel (1984).

Now we remove the condition that all demand facilities have to be served by thenew stop, i.e., we consider a subset Vstop �= ∅. We denote this problem by (AccessSto-pLoc’). For each demand facility pm,m ∈ M, let dm denote the (rectangular) distance

Fig. 3 Candidates for (AccessStopLoc) in planar environment

342 D.R.P. Groß et al.

of pm to the closest existing stop dm := minv∈Vstop l1(v,pm). Let Dm the (diamond-shaped) ball of radius dm about pm, i.e.,

Dm := {s ∈ R

2 : l1(pm, s) ≤ dm

}, m ∈ M.

Customers in demand facility pm will prefer a new stop in s to the previous stops ifand only if s lies inside Dm. Therefore, it is sufficient to consider points in the interiorof

D :=⋃

m∈MDm. (4.1)

Lemma 4.2 Let d := ∑m∈M dm. Then f (s) < d if and only if s ∈ int(D). Otherwise,

f (s) = d .

We add the boundaries δDm,m ∈ M, to the planar arrangement given by L fromabove. Let Z ′ be the cells of the new arrangement which are contained in D (seeFig. 4). On each cell Z ∈ Z ′, the objective function is given as

f (s) =∑

m∈M̃

wmd(pm, s) +∑

m∈M\M̃

dm

for some M̃ ⊆ M and is hence linear.

Fig. 4 Candidates for (AccessStopLoc’) in planar environment

Stop location design in public transportation networks: covering 343

Theorem 4.3 An optimal solution for (AccessStopLoc’) is contained in S ′cand :=

Scand ∩ int(D).

Proof f is linear in each cell Z ∈ Z ′, hence on e ∩ Z for each edge e. If e ∩ Z �= ∅,then f attains its minimum s∗ at an endpoint of e ∩ Z, i.e., s∗ ∈ Vbreak, or s∗ ∈ C , ors∗ = e ∩ ∂Dm, where ∂Dm is the boundary of Dm. It remains to show that there is noneed to consider the third case: To this end, let s̃ ∈ G ∩ ∂Dm̃ \ Scand for at least onem̃ ∈ M. Consider the contribution

gm(s) := min{dm, l1(pm, s)

}

of the demand points to the objective function. Let U := U(s̃) ∩ G be a (sufficientlysmall) neighborhood about s̃. Since s̃ /∈ Scand, we have that gm is linear on U forall m such that s /∈ ∂Dm. However, gm(s̃) attains its maximal value dm if s̃ ∈ ∂Dm.Consequently, f (s) cannot have a minimum at s̃. �

Note that S ′cand ⊆ Scand. Evaluating the objective function for each of these candi-

dates results in an O(M2|E|) algorithm which can be improved to O(M|E| logM)

as follows: Consider an edge e. Each function gm(s) is piecewise linear with at mostfour breakpoints on e. Therefore, f (s) has at most 4M breakpoints on e. Sort this setof breakpoints on e in O(M logM) time. Following this ordering and starting withthe first breakpoint, it now takes only a constant time to evaluate the objective at eachnew adjacent point in the sequence of the ordered breakpoints.

Now we analyze (AccessStopLoc) where we are allowed to establish k stops. Thefirst result is straightforward.

Theorem 4.4 For (AccessStopLoc), there exists an optimal solution S ∗ ⊆ Scand,|S ∗| ≤ k.

Proof Let S̃ be a solution with |S̃| ≤ k. Then, each s̃ in a solution S̃ is optimal fora single stop location problem with the demand facilities that are served by s̃, hencethe result follows. �

Looking at the complexity, we obtain:

Theorem 4.5 (AccessStopLoc) is N P -hard.

Proof Consider the (Rectangular k-median problem), known to be N P -hard (seeMegiddo and Supowit 1984): Given a set P = {p1, . . . , pM} ⊂ R

2 of points in theplane and a positive integer number k, is there a set S ⊂ R

2 of k points such that ifl1(pm,S) is the length of shortest l1 travel path from pm to the closest point in S,then

∑Mm=1 l1(pm,S) ≤ k?

To reduce rectangular k-median to (AccessStopLoc), we leave P and k as they areand define G such that it contains sufficiently large parts of the construction lines,where V consists of the endpoints of the segments and of the intersection pointsbetween pairs of the segments (see Poetranto 2005 for details). �

344 D.R.P. Groß et al.

4.2 Accessibility problem in street network environment

In the network environment, the following candidate sets can easily be obtained for(AccessStopLoc), i.e., if Vstop = ∅ and for (AccessStopLoc’) for Vstop �= ∅.

Theorem 4.6 An optimal solution of (AccessStopLoc) is contained in Scand := V .

Proof Note that d(pm, s) is concave over any edge e, hence f also is. Consequently,f attains its minimum at a node. �

For the case in which Vstop �= ∅, we follow the same approach as in the planar caseand determine D as in (4.1) but measuring the radius by the network distance. Hence,we only have to consider candidates in the interior of D.

Theorem 4.7 An optimal solution of (AccessStopLoc’) is contained in S ′cand :=

Vbreak ∩ int(D).

We remark that (AccessStopLoc) on a network can be considered as a restricted k-median problem in SPTN, where new facilities must be located in PTN ⊆ SPTN. Ourobservation hence generalizes the well-known node optimality property (see Hakimi1965) for k-median problems. Moreover, it has been shown in Hakimi and Kariv(1979) that the k-median problem is NP-hard. Thus, so are (AccessStopLoc) and(AccessStopLoc’).

5 Conclusion

In this paper, we have discussed solution approaches for four classes of stop locationproblems, the covering and the accessibility problem in planar and network environ-ments. In all cases, we were able to identify a finite (polynomial) dominating set.Heuristic methods based on a node partitioning scheme and some numerical resultsare presented in Poetranto (2005). Several other variations of the problems are possi-ble, three of them are mentioned below.

First, a generalization to polyhedral gauges is possible for (AccessStopLoc) bytaking the fundamental directions as construction lines leading to cells on which theobjective is linear. A generalization to arbitrary gauges is straightforward for (Cov-StopLoc), see Schöbel, Schöbel et al. (2006, 2008).

Second, an interesting variant is the Center Stop Location Problem with respect toa given positive integer k. It is defined by

minimize{

maxp∈P

d(p, S) : S ⊆ G, |S| = k}

and can be tackled using results either of (CovStopLoc) by a binary search strategyor of (AccessStopLoc) by looking at constrained center location problems.

Stop location design in public transportation networks: covering 345

Another extension is to take the number of potential passengers in each demandfacility into account by using weights for each p ∈ P . This leads to weighted objec-tive functions in the set covering problem and in the restricted location problem inSects. 3 and 4.

Acknowledgement The authors would like to thank an anonymous referee for very helpful comments.

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