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Locating stations on rapid transit lines
Computers & Operations Research 29 (2002) 741}759
Locating stations on rapid transit lines Gilbert Laporte *, Juan A. Mesa, Francisco A. Ortega E! cole des Hautes E! tudes Commerciales, 3000 Chemin de la CoL te-Sainte-Catherine, Montre& al, Canada H3T 2A7 Departamento de Matema& tica Aplicada II, Escuela Superior de Ingenieros, Universidad de Sevilla, Sevilla, Spain Departamento de Matema& tica Aplicada I, Escuela Te& cnica Superior de Arquitectura, Universidad de Sevilla, Sevilla, Spain
Abstract When the alignment of a new line or section of a line of a rapid transit system has already been designed, the problem of locating stations arises. One of the measures used to predict the future utilization of the line is the coverage provided by the stations. In this paper the problem of locating a pre"xed number of stations so that the weighted coverage will be maximized is considered. To evaluate the coverage of each station an estimation of the street network metric and the triangulations of the census tracts are used. Finally, a longest path type algorithm over an acyclic graph is devised in order to maximize the objective function, in which it was necessary to approximately determine the catchment area for each station. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Estimation of actual distances; Rapid transit lines; Coverage
1. Introduction One of the main issues when planning a rapid transit system is to determine the network which consists of the lines and the stations. For this purpose, in practice, the most widely used procedure is the so-called scenario analysis which leads to a set of corridors where the alignments are to be constructed. Though the location of some important stations (those coincident with railway stations, hospitals, university areas, etc.) should be chosen before deciding the alignments, there remains the problem of locating a number of stations along a corridor so that they attract the largest number of riders. A station can be considered as a point where changing from a mode of transportation to that provided by the rapid transit system occurs. When the corridor crosses a residential area the demand of changing from a private motorized mode can be prevalent, but when the alignment is located in a central area most of the users will access the station on foot. * Corresponding author. Tel.:#1-514-343-6143; fax: #1-514-343-7121. E-mail address: [email protected] (G. Laporte) 0305-0548/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 0 ) 0 0 0 1 3 - 7
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One of the measures used to estimate the future utilization of the line is the coverage provided by the stations. Despite the existence of criticism on the use of this measure because not only does ridership depend on the distance to the nearest station but also on the design of the network itself, it remains as a measure of high value especially when the alignment has already taken into account the patterns of mobility in the area under consideration. On the other hand, the model proposed in this paper can be adapted to the case in which the objective is to maximize the number of trips covered instead of the people covered [1]. The coverage provided by a station situated in a central area will depend on the pedestrian network around it and the limits of the coverage will be determined by the maximum distance (or time) that users are able or decide to walk to access the transit network. The percentage of captured users depends on the required time to reach the nearest station, thus several levels of attraction are stated. The structure of the street network determines walking travel distances which are often non-Euclidean, and therefore a prediction function to approximate the distances between points on the street network and nodes on the transit network must be obtained before resolving the problem of locating stations along a section to provide maximum coverage. The number of stations to be located in a section with "xed extremes is a function of the length and the interstation spaces. The question of deciding the optimum interstation spacing [2}5] is a very complex task because it depends on the technical characteristics of the chosen mode, on the overall network con"guration, on the total travel times of all the passengers from their origin to destination, as well as on the ratio between the number of passengers traveling on the train and those along the line wanting to board it. Hence, bounds on the minimum and maximum interstation spacing, s and s , have been taken as parameters in this paper.
The number of stations, N, to be located is N"U ¸/s V, where ¸ is the length of the section and s is the average interstation space. From now on, N will be "xed. Furthermore, the distance between two adjacent stations is assumed to belong to the interval [s , s ] which is called &constraint on the interstation spacing' (CIS).
The organization of the paper is as follows. Section 2 is devoted to the introduction of the model. In Section 3, the coverage of suburban stations is analyzed and the catchment areas are determined by using an estimation of the street distance from points to stations on the line. The line segment coverage is introduced in Section 4. The procedure for optimally locating N stations along the line to maximize total population coverage by the computation of the longest path in a graph is considered in Section 5. An example is analyzed in Section 6. The conclusion follows in Section 7.
2. Objective function Assuming the number of stations N is "xed, the line is assumed to be discretized and contains N disjoint intervals of potential location sites. Each interval E , i"1,2, N, will have some G candidate locations, represented by their corresponding abscissa xF3E , ∀h"1,2, n , where G G G
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n '0, ∀i"1,2, N. Let X", LG xF be the set of candidate stations along the alignG G F G ment. Several candidate stations corresponding to a situation encountered in Sevilla, Spain, have been represented in Fig. 1. The "lled point indicates one pre"xed station. The Guadalquivir river is situated between this station and the following candidate stations (empty points on the right). In the "gure, arcs (whose center is represented by the "lled point) generate the interval [s , s ]
which must contain the following station, extending the line forward on the right-hand side. Distances are scaled, with s "500 m and s "800 m, and the empty points correspond to
actual sites along RepuH blica Argentina street (a proposed corridor in the Intermodal Transit Planning of the Agglomeration of Sevilla). As Fig. 1 indicates, points numbered by 1, 2, 3 and 4 are the sites where the "rst station could be located. If node 1 is chosen, new arcs would be drawn in order to show the corresponding sites where the following station could be located (see Fig. 2). On the other hand, if point 3 had been chosen for the "rst station, other new arcs should be drawn, as Fig. 3 shows. Let C be the catchment area of this transit line, divided into census tracts c , i.e., H ( C" c , H H where each zone c is assumed to be a polygonal region with population density , ∀j"1,2, J. If H H data about jobs were available, could be reinterpreted as a (possibly weighted) sum of the H population and employment density. In Fig. 4, two areas called Los Remedios ("rst quadrant) and Triana (fourth quadrant) are shown and partitioned into their respective census tracts. The gray level is in accordance with the population density. Furthermore, one number j is associated with each census tract c , ∀j"1,2, 35, and is located on its center of gravity; therefore, when a number is outside the H corresponding census tract, the non-convexity property of this polygonal region is shown. Let d(x, y) be a distance measure in the plane with ) as associated norm: d(x, y)"x!y. Denote by B(x , r) the set of points in the plane whose distance to the station x is not greater than G G r (usually, called ball of radius r. For each station x , K di!erent levels of attraction will be G considered. Then, ∀i"1,2, N and ∀k"1,2, K v3B(x , r ),B if and only if v!x )r . G I GI G I These attraction levels correspond to concentric annuli around each location site.
Fig. 1. Candidate stations.
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Fig. 2. Candidate stations from node 1.
Fig. 3. Candidate stations from node 3.
Fig. 4. Typical census tract division.
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There are several studies about modeling the attraction by function. We will use that derived by the gravitation model [6]: a (z)" , z
z3[d , d ], 0(d (d .
Here, (d(v, x ))"a/v!x , in which a is a constant to be calibrated. G G In order to discretize the objective function, the attraction in each annulus, determined by two consecutive levels, will be considered constant. Thus, the coverage provided by the station located at x is G ( ) a H Area ((B B ) c ). R(x )" GI GI\ H G (r #r /2) I H I I\ Fig. 5 shows the variety of regions obtained by intersections between census tract C and the attraction annuli. Finally, the objective function is the cover provided by the stations situated in the alignment l, which in the plane corresponds to the section of the line under consideration subject to the CIS. This yields a , ( ) H max R(l) " Area((B B ) c ). : GI GI\ H (r #r /2) X I G Z6 !'1 G H I I\ 3. Station-covering calculation Stations can be roughly classi"ed according to the zone of the city in which they are going to be constructed. Thus, following the methodology often used in trip modeling, the city is divided into two zones: the central business district and the residential areas. Stations in the residential areas
Fig. 5. Intersections.
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can be considered as points at which users change from a private motorized mode of transportation to the transit system; those in central areas mostly attract pedestrian users. Even though other passengers can reach stations by feeder buses (or other modes of transport), these contributions are not relevant now, because the stations (intermodal, near hospital, etc.) which provide a high number of these users had before been decided in order to anchor the alignment. Most of the recent planning projects use either a band around the proposed alignment or circles around station sites. However, Schabas [7], in this description of a project for Honolulu, addresses attention to the fact that actual walking distances are signi"cantly di!erent than that measured by the Euclidean norm. Thus, an estimation of the walking distances for users to reach the stations is required. Following some computational experiments [8] in which the weighted l norm resulted as the N most accurate, we will rely on an earlier study in which a function l , '0, p3[1, 2], has already N been determined in order to estimate actual walking distances in the street network (in Fig. 5, the borders were curves obtained by p"1.5, i.e. x !s N#x !s N"rN). I In order to determine the intersection areas (Area((B B ) c )) involved in the objective GI GI\ H function, a triangulation [9] of census tracts (see Fig. 6) can be used to get the intersections with the attraction annulus, given that census tracts are usually non-convex polygons and generate an excessive number of cases when intersecting with attraction levels. Alternatively, the use of a GIS [10] can be useful if analytical formulae for the contours of the levels of attraction can be incorporated and graphical representations of them are available.
Fig. 6. Decomposing census tracts.
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At this point, one could ask whether solving the problem in which demands are discrete and concentrated at the centroids of the triangles would yield a di!erent result from the problem in which demands are continuous. The following examples can help to clarify this question and show that the method proposed in the paper is more accurate to calculate coverage provided by stations. (A) A triangular census tract is considered "rst as a single block (Fig. 7(1)) and after as a decomposition into three census subtriangles (Fig. 7(2)). Coverage formulated by using continuous demand maintain the same value in both cases. Nevertheless, if we assume demands concentrated at centroids, two di!erent levels of coverage are possible in accordance with the relative position of these points. In the "rst case, coverage is nothing since the centroid is outside the attraction ball, while in the second scheme, the inner centroid generates a coverage di!erent from zero. (B) Two candidate stations S and S are adjacent along the alignment. In the upper band, several census triangles, whose densities are , and (gray level in accordance with these values), intersect the attraction areas built from the respective stations by using the distance generated by the l norm, for p"1.35. Taking into account that the centroid on the left hand is not N covered, we conclude that S provides more coverage in comparison with S if the objective were formulated in discrete terms. Nevertheless, a su$ciently high value of relative to could produce the opposite conclusion, i.e. S is better than S , when the demand is considered continuous. To establish this fact, let us assume that ( "k , ( "k "k k , for certain coe$cients k , k '1. Let A denote the area corresponding to one half-quadrant inside the attraction ball, and let ¹ be the area of each triangle in the "gure. Population covered by
Fig. 7. (1) Triangular census tract. (2) Decomposing tract. (3) Discrete demands versus continuous demand.
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stations S and S in accordance with discrete and continuous (denoted by PC (S ), PC (S ), PC (S ) and PC (S ), respectively), models can be easily calculated: PC (S )"3A #A "(3#k )A , PC (S )"2A #¹ #2(¹!A) "(2A#¹#2k k (¹!A)) , PC (S )"3¹ #¹ "(3#k )¹ , PC (S )"3¹ . Obviously, the relation PC (S )(PC (S )(PC (S ) is satis"ed and by imposing that k '[((1#k )A!¹)/2k (¹!A)], then the sequence can be extended, i.e. PC (S )(PC (S ) (PC (S )(PC (S ), producing converse decisions depending on the setting (continuous or discrete) considered. Taking into account the possible contradictions derived from the use of the discrete model, we adopt the continuous formulation. To compute the intersections between census triangles and balls (more exactly, half-balls), the relative position between tracts and half-balls has been analyzed and eight di!erent cases have been obtained. Figs. 8(1)}(8) illustrate the possible cases. The number of vertices that are inside/outside the half-ball and the number of triangle edges that intersect the contour of the half-ball permit us to establish the case and to elaborate a procedure to determine the intersected area. Following on, we show how to calculate the intersection area in the case in which two vertices are inside the half-ball (see Fig. 8(9), similar to Fig. 8(7)). In order to determine the intersection area, the following formula can be used: Area(Reg. ABCD)"Area(Sec. AsB) # Area(Trian. DsA) #Area(Trian. BsC)!Area(Trian. DsC), where the "rst additive term can be calculated by
Area(Sec. AsB),A( , )"
P dx y d!x( )y( )#x( )y( ) d P
when (x(), y()) represents the parametric curve depending on the angle (see Fig. 9). The total area A(0, 2) inside the curve x N#x N"rN can be obtained exactly by using the Euler beta function, i.e.,
2r 1 1 A(0, 2)" , . p p p If integration limits were integer divisors of then it would be possible to calculate A( , ) exactly by using parameter : r r x()" cos() N, y()" sin() N.
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Fig. 8. (1) Outside triangle. (2) Outside vertices and one edge intersects. (3) Outside vertices and two edges intersect. (4) Outside vertices and three edges intersect. (5) One inside vertex and one opposite edge intersects. (6) One inside vertex and one opposite edge does not intersect. (7) Two inside vertices. (8) Inside triangle. (9) A case of intersected area.
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Fig. 9. Sector area.
For instance, if "0 and "/2 then
1 r 1 1#p 2 r cos\N N() sin>N N() d" , A(0, /2)" p p p p
1 1 r (1/p) r " " , . 2p (2/p) 2p p p When the general case happens ( and are not integer divisors of ), it is necessary to employ some numerical method in order to approximate the exact value of A( , ). In the paper of Mesa and Ortega [11], approximating the calculation with l norms, p3[1, 2), by using -inclined block N norms was suggested. De5nition 1. A -inclined block norm ) is a norm whose unit ball B is the polytope with vertices: $(1, 0),$(cos , sin ) where 3(0, /2]. In Mesa and Ortega [11], the one-to-one correspondence between values of p3[1, 2) and the angles 3(0, /2] was established in the following way:
,(p)"2 arccos
(2
(2 N
3(0, /2].
This relation enables us to express the computation of l (x), ∀x31 (argument 3[0, 2)), as N a decomposition into Euclidean norm and p-bias factors. The second factor is approximated as the following relation shows: l (x)"l (x)r ( )"l (x)(cos N#sin N) N N N +l (x)
cos( /(/2)!/2) . cos(/2)
This formula can be extended to the general case 3[0, 2) taking into account the periodicity of the l function. Furthermore, the analytical study of the approximation error can be found in N Ortega [12].
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Proposition 1. For all , 3[0, /2], the sector area A( , ) can be obtained approximately by the relation:
r !/2 !tan !/2 A( , )+ cos(/2) tan /2 /2 4
,
N /2]. where r, and p3[1, 2) are given and ,(p)"2 arccos((2/(2)3(0, Proof. The curve l (x)"r can be expressed in polar coordinates in the following way: N
l (x)r ( )"rN"(r/ )(1/r ( )). Using the approximation formula r ( )+cos( /(/2)! N N N /2)/cos(/2), the earlier curve can be approximated by "(r/ )(cos(/2)/cos( /(/2)!/2)) and the calculation of the corresponding sector area
P 1 1 P r A( , )+ d " cos(/2) d cos( /(/2)!/2) 2 2 P P gives the required result
r !/2 !tan !/2 A( , )+ cos(/2) tan 4 /2 /2
.
䊐
In order to appreciate the approximation error when A( , ) is obtained by the previous relation, we consider the particular case corresponding to "0 and "/2, where r r A(0, /2)+ cos(/2)(tan(/2)#tan(/2))" sin(), 4 4 and we de"ne the functions S(p)"(r/2 ) (1/p, 1/p)/p and SH(p)"(r/2 )( sin((p))/2(p)), which represent the areas of the sector corresponding to the "rst quadrant for the unit balls associated to l ( ) ) norm and its approximation in the sense indicated by Proposition 1. The N proximity between S(p) (heavy print) and SH(p) (light print), for r/2 "1, can be noted in Fig. 10. In the extremes p"1 and p"2, and functions S(p) and SH(p) coincide. The maximum deviation between both functions occurs in a central value of p3(1, 2) which cannot be explicitly obtained by setting d/dp(S(p)!SH(p))"0. Summarizing this section, the problem involved has been formulated in terms of continuous demand. Coverage calculation presents a geometrical variety of cases which can be reduced if a triangulation of the census tracts is previously done. An approximation method has been proposed to estimate the sector intersected between each triangle and each attraction ring. So, the coverage of stations can be obtained by taking these points into account. Nevertheless, constraints in relation with the CIS must be included in the objective. For reaching it, coverage R determined GH by the pair of stations (S , S ) along the line segment is introduced in the next section by making an G H assumption over the common behavior of the users.
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Fig. 10. Proximity between curves.
4. Line segment-covering calculation In what follows, we will make the following assumption: Given two adjacent stations S and S and the mediatrix M (perpendicular to S S in the middle G H GH G H of that segment), people to the right of the mediatrix M (point X in Fig. 11) always prefer S to S . GH H G To obtain the coverage associated to segment (S , S ) in the line, the intersection areas between G H half-balls B> and B\ and the triangles arising from decomposition of census tracts must be GI HI determined. Two cases are possible: (1) The catchment areas of S and S can be disjoint arcs for attraction level r (see Fig. 12). H G I In general, triangles contained in the band of vertices S and (S #S )/2 must be intersected with G G H ball B , and triangles located in the band of vertices (S #S )/2 and S will be intersected with ball GI G H H B . In order to avoid those triangles which could intersect simultaneously with both half-balls HI B> and B\, a previous decomposition of each triangle into triangles inside the adequate band GI HI should be established. (2) The catchment areas of S and S intersect for attraction level r . H G I Despite half-balls B> and B\ having a common cut zone, the assumption that people always GI HI choose the closest station allows us to reduce this case to the earlier case, in which the key consists of calculating intersections between a portion of catchment area and triangles totally contained in the band determined by this portion (Fig. 13). There are six di!erent topological con"gurations to be considered depending on the number of vertices inside the band and the number of vertices that are on the right- and left-hand sides of the band. These con"gurations are shown in Figs. 14(1)}(6). In the "rst case the vertex distribution generates an empty intersection. The second con"guration gives rise to a new triangle. Two new triangles are originated from the initial one in third and fourth cases. Finally, in Fig. 14(5) we can note the decomposition of the intersected area into three new triangles. Denote by ¹\ the set of triangles that arise by decomposing "rst each of the census tracts GH involved and then by generating the new triangles inside the [S , M ] band from the earlier ones. G GH Similarly, denote by ¹> the set of triangles obtained by using the [M , S ] band. Thus, the GH GH H
G. Laporte et al. / Computers & Operations Research 29 (2002) 741}759
Fig. 11. Choosing the closest station.
Fig. 12. Disjoint catchment areas.
Fig. 13. Overlapped catchment areas.
coverage R provided by the line segment situated between stations S and S is GH G H ) a F Area((B B ) t ) R " GI GI\ F GH t \ ((r #r )/2) F Z2GH I I\ I a ) F # Area((B B ) t ). HI HI\ F ((r #r )/2) > t I\ I F Z2GH I
753
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Fig. 14. (1) Vertices in the (3, 0, 0) form. (2) Vertices in the (2, 1, 0) form. (3) Vertices in the (2, 0, 1) form. (4) Vertices in the (1,2,0) form. (5) Vertices in the (1, 1, 1) form. (6) Vertices in the (0, 3, 0) form.
Fig. 15. Acyclic graph generated.
5. Line-covering calculation Repeating the process explained in Section 2 and in the Figs. 1}3, a connected acyclic graph G"(<, E) can be obtained where < collects the feasible nodes (
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Nodes i and j are connected by edge (i, j) if station j satis"es the constraint on the interstation spacing with respect to previous station i. We can observe that nodes 5, 6 and 13 are not feasible. As we have stated, this graph is deduced from an actual case in which the total length of the corridor does not permit more than two stations to satisfy the CIS. The procedure can be extended to deal with the case in which the number of stations to be located is higher. An optimal solution to the problem of locating N stations on a line to maximize total population coverage is obtained by computing a longest path between the given station (numbered in the graph by 0) and dummy last station (numbered by <#1), if no last station is given, or a real last station C< if a last station is given. This longest path is easy to compute by using the Dijkstra shortest path algorithm [13] over an acyclic graph with costs c "M!R , where R indicates the cost GH GH GH associated with edge (i, j) and M'max R . G H GH The following algorithm summarizes the total procedure for solving the problem of locating stations maximizing the coverage. Algorithm begin 1. compute estimation parameters , p. 2. compute decomposition of census tracts into T F and population density ; j"1,2, M. H 3. read attraction levels r ; k"1,2, K. I 4. read candidate sites S 3X; i"1,2, I. G 5. compute the distribution of the candidate stations S which satisfy the constraints into N blocks G of an acyclic graph G"(<, E), whereLband[M , S ]. GH GH H ) a F 6.4. compute R\" Area((B B ) t ). HI HI\ F GH t > (r #r /2) F Z2GH I I\ I 6.5 R QR>#R>. GH GH GH 7. compute M'max R . G H GH 8. compute longest path between station C0 and the last station using edge costs c "M!R . GH GH end 6. An example A previous study to determine the l function as the most accurate which estimates actual N walking distances in those areas considered in Sevilla was made and the results obtained were
"1.105 and p"1.2.
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Table 1 Stations
Numerical results Coverage 1
(4, 12)
Coverage 2
p"2
p"2 9.27994
p"1.2
p"1.2
p"2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
(10.29%) 12.3492
p"2 9.6167
p"1.2 3.33007
(12.00%) 15.41137
9.01152
5.63694
(8.75%) 12.56046
10.2495
3.33769
(10.06%) 15.75026
8.89257
5.16187
(15.65%) 11.61147
9.82641
3.66789
(7.14%) 16.26166
6.3181
5.92385
(9.34%) 12.47986
10.0053
5.29337
(6.96%) 16.29341
7.14278 p"2
p"1.2
(8.56%) 12.58685
p"2
p"1.2
6.25636
(2, 9)
p"1.2
9.6665
5.33708
(9.11%) 15.91657
7.24929 p"2
p"1.2
(2, 10)
p"2
p"1.2
6.62691
(3.24%) 13.31854
8.72325
5.33756 p"2
17.51375 p"1.2
6.6232
7.19332
(2, 11)
p"2
p"1.2
p"2
13.76588
9.41689
6.69534
(3, 9)
p"1.2
p"2
p"1.2
(4.27%) 16.76537
7.0527
8.09686
(3, 10)
p"2
p"1.2
p"2
(18.12%) 11.27097
8.577213
6.71318
(3, 11)
p"1.2
p"2
p"1.2
(21.54%) 13.74086
3.81704
8.19324
(4, 10)
p"2 4.46092
7.45393 (4, 11)
Total (dev. %)
(12.90%) 15.25364
p"1.2 8.49827
(14.07%) 11.82834
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Table 1 (Continued) Stations
Numerical results Coverage 1
(2, 8)
Coverage 2
p"2
p"2 4.30336
p"1.2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
(24.81%) 10.3492
p"2 8.54932
p"1.2 2.30184
(35.41%) 11.31137
8.01152
2.53763
(19.26%) 11.11379
8.7495
2.33768
(29.34%) 12.37413
8.63026
2.56187
(15.07%) 11.69073
9.5199
2.48353
(21.28%) 13.75026
9.18792
2.85423
(1, 7)
p"2
p"1.2
p"2
(19.39%) 11.09632
10.82641
2.50281
(1, 8)
p"1.2
p"2
p"1.2
(24.28%) 13.26106
7.83422
2.92385
(1, 9)
p"2 8.9577
3.2621 (1, 10)
Total (dev. %)
(36.69%) 11.08695
p"1.2 7.39787
(29.53%) 9.69971
The total length of alignment analyzed is 900 m (in earlier Figs. 4 and 6, the unit corresponds to 75 m) and the attraction annuli were [0, 200 m], (200 m, 400 m] and (400 m, 600 m], i.e., r "100 m"1.33333 u, r "300 m"4 u and r "500 m"6.66666 u. Let be the attraction factor 1/r. The coverage provided for each station s over each triangle I I T is calculated by (A !A )# (A !A )# A , where A is the intersection of triangle T and the l -ball centered in station s with radius r , taking into account the population density of N I the census tract in which triangle T belongs. Simplifying the earlier expression when the numerical values are used, we obtain the combination A # A #A , for measuring the coverage provided. All pairs of stations were evaluated as the algorithm proposed for ( "1, p"2) (Euclidean distance) and ( "1.105, p"1.2) (estimated actual distance), and summarized in Table 1. We can observe that the best locations for ( "1, p"2) correspond to nodes 4 and 10 and, on the other hand, the best choice for ( "1.105, p"1.2) is the pair (4, 11). Moreover, the second and third best allocations are the pairs (4, 11) and (3, 10) for the Euclidean distance case and the pairs (4, 10) and
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(3, 11) for the estimated distance case, respectively. The percentage of deviation from the optimal solution has been included in order to show the sensitivity involved. The size of the l -ball is smaller than the corresponding size of the Euclidean ball, hence point N 11 produces better results than node 10 owing to its proximity to the right-hand tracts with higher density. Node 11 remains the third best choice for ( "1.105, p"1.2), combined with node 3 as the "rst station, instead of point 10 as happens in the Euclidean case. Data for computation involved only the population distributions in addition to pedestrian accessibility. If data about employment had also been taken into account, the disparity between zones would have increased (actual data about population in the areas considered show similar density levels) and other di!erent solutions would probably have been generated.
7. Conclusion Locating stations on a prede"ned alignment is an important problem encountered in the design of a rapid transit systems. A standard objective is the maximization of the total potential ridership covered by the stations. We have presented a methodology for solving this problem, based on the triangulation of census tracts. Since such data is typically available, the proposed method could be implemented in most contexts. Although the mathematical developments are sometimes involved, the proposed procedure can easily be coded on a computer and combined with a geographic information system.
Acknowledgements This research was in part supported by Canadian Natural Sciences and Engineering Research Council under grand OGP0039682 and by Ministerio de EducacioH n y Ciencia under grant DGICYT PB-95-1237-C03-01. The authors are also grateful for the helpful comments and referee reports of two anonymous referees that have allowed us to improve upon the original version.
References [1] Mesa JA, Ortega FA. Suggestions for improving the coverage accuracy in urban transit systems. Working paper, University of Sevilla, 1998. [2] Vuchic VR, Newell GF. Rapid transit interstation spacings for minimum travel time. Transportation Science 1968;2:303}39. [3] Hurdle VF, Wirasinghe SC. Location of rail stations for many to one travel and several demand feeder modes. Journal of Advanced Transportation 1980;14:29}46. [4] Halder DK, Majunder A. A method for selecting optimum number of stations for a rapid transit system by network approach: an application in Calcutta tube rail. In: Jaieswal NK, editor. Scienti"c management of transport systems. New York: North-Holland, 1981. [5] Wirasinghe SC, Vandebona U. Some aspects of the location of subway stations and routes. Fourth International Symposium on Locational Decisions. Belgium: Namur, 1987. [6] OrtuH zar JD, Willumsen LG. Modelling transport. New York: Wiley, 1990.
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[7] Schabas M. Quantitative analysis of rapid transit alignment alternatives. Transportation Quarterly 1988;42:403}16. [8] Brimberg J, Love RF, Walker JH. The e!ect of axis rotation on distance estimation. European Journal of Operational Research 1995;80:357}64. [9] O'Rourke J. Computational geometry in C. Cambridge: Cambridge University Press, 1994. [10] Schweiger CL. Current use of geographic information systems in transit planning. Transportation Research Records 1992;1349:93}106. [11] Mesa JA, Ortega FA. Determining regular networks for each given l function (p3 [1, 2]). Studies in Locational N Analysis 1996;10:91}108. [12] Ortega FA. LocalizacioH n y trazado de redes. Ph.D. Thesis, University of Sevilla, Sevilla, Spain, 1997. [13] Dijkstra E. A note on two problems in connexion with graphs. Numeriche Mathematik 1959;1:269}71.
Locating stations on rapid transit lines Gilbert Laporte *, Juan A. Mesa, Francisco A. Ortega E! cole des Hautes E! tudes Commerciales, 3000 Chemin de la CoL te-Sainte-Catherine, Montre& al, Canada H3T 2A7 Departamento de Matema& tica Aplicada II, Escuela Superior de Ingenieros, Universidad de Sevilla, Sevilla, Spain Departamento de Matema& tica Aplicada I, Escuela Te& cnica Superior de Arquitectura, Universidad de Sevilla, Sevilla, Spain
Abstract When the alignment of a new line or section of a line of a rapid transit system has already been designed, the problem of locating stations arises. One of the measures used to predict the future utilization of the line is the coverage provided by the stations. In this paper the problem of locating a pre"xed number of stations so that the weighted coverage will be maximized is considered. To evaluate the coverage of each station an estimation of the street network metric and the triangulations of the census tracts are used. Finally, a longest path type algorithm over an acyclic graph is devised in order to maximize the objective function, in which it was necessary to approximately determine the catchment area for each station. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Estimation of actual distances; Rapid transit lines; Coverage
1. Introduction One of the main issues when planning a rapid transit system is to determine the network which consists of the lines and the stations. For this purpose, in practice, the most widely used procedure is the so-called scenario analysis which leads to a set of corridors where the alignments are to be constructed. Though the location of some important stations (those coincident with railway stations, hospitals, university areas, etc.) should be chosen before deciding the alignments, there remains the problem of locating a number of stations along a corridor so that they attract the largest number of riders. A station can be considered as a point where changing from a mode of transportation to that provided by the rapid transit system occurs. When the corridor crosses a residential area the demand of changing from a private motorized mode can be prevalent, but when the alignment is located in a central area most of the users will access the station on foot. * Corresponding author. Tel.:#1-514-343-6143; fax: #1-514-343-7121. E-mail address: [email protected] (G. Laporte) 0305-0548/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 0 ) 0 0 0 1 3 - 7
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One of the measures used to estimate the future utilization of the line is the coverage provided by the stations. Despite the existence of criticism on the use of this measure because not only does ridership depend on the distance to the nearest station but also on the design of the network itself, it remains as a measure of high value especially when the alignment has already taken into account the patterns of mobility in the area under consideration. On the other hand, the model proposed in this paper can be adapted to the case in which the objective is to maximize the number of trips covered instead of the people covered [1]. The coverage provided by a station situated in a central area will depend on the pedestrian network around it and the limits of the coverage will be determined by the maximum distance (or time) that users are able or decide to walk to access the transit network. The percentage of captured users depends on the required time to reach the nearest station, thus several levels of attraction are stated. The structure of the street network determines walking travel distances which are often non-Euclidean, and therefore a prediction function to approximate the distances between points on the street network and nodes on the transit network must be obtained before resolving the problem of locating stations along a section to provide maximum coverage. The number of stations to be located in a section with "xed extremes is a function of the length and the interstation spaces. The question of deciding the optimum interstation spacing [2}5] is a very complex task because it depends on the technical characteristics of the chosen mode, on the overall network con"guration, on the total travel times of all the passengers from their origin to destination, as well as on the ratio between the number of passengers traveling on the train and those along the line wanting to board it. Hence, bounds on the minimum and maximum interstation spacing, s and s , have been taken as parameters in this paper.
The number of stations, N, to be located is N"U ¸/s V, where ¸ is the length of the section and s is the average interstation space. From now on, N will be "xed. Furthermore, the distance between two adjacent stations is assumed to belong to the interval [s , s ] which is called &constraint on the interstation spacing' (CIS).
The organization of the paper is as follows. Section 2 is devoted to the introduction of the model. In Section 3, the coverage of suburban stations is analyzed and the catchment areas are determined by using an estimation of the street distance from points to stations on the line. The line segment coverage is introduced in Section 4. The procedure for optimally locating N stations along the line to maximize total population coverage by the computation of the longest path in a graph is considered in Section 5. An example is analyzed in Section 6. The conclusion follows in Section 7.
2. Objective function Assuming the number of stations N is "xed, the line is assumed to be discretized and contains N disjoint intervals of potential location sites. Each interval E , i"1,2, N, will have some G candidate locations, represented by their corresponding abscissa xF3E , ∀h"1,2, n , where G G G
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n '0, ∀i"1,2, N. Let X", LG xF be the set of candidate stations along the alignG G F G ment. Several candidate stations corresponding to a situation encountered in Sevilla, Spain, have been represented in Fig. 1. The "lled point indicates one pre"xed station. The Guadalquivir river is situated between this station and the following candidate stations (empty points on the right). In the "gure, arcs (whose center is represented by the "lled point) generate the interval [s , s ]
which must contain the following station, extending the line forward on the right-hand side. Distances are scaled, with s "500 m and s "800 m, and the empty points correspond to
actual sites along RepuH blica Argentina street (a proposed corridor in the Intermodal Transit Planning of the Agglomeration of Sevilla). As Fig. 1 indicates, points numbered by 1, 2, 3 and 4 are the sites where the "rst station could be located. If node 1 is chosen, new arcs would be drawn in order to show the corresponding sites where the following station could be located (see Fig. 2). On the other hand, if point 3 had been chosen for the "rst station, other new arcs should be drawn, as Fig. 3 shows. Let C be the catchment area of this transit line, divided into census tracts c , i.e., H ( C" c , H H where each zone c is assumed to be a polygonal region with population density , ∀j"1,2, J. If H H data about jobs were available, could be reinterpreted as a (possibly weighted) sum of the H population and employment density. In Fig. 4, two areas called Los Remedios ("rst quadrant) and Triana (fourth quadrant) are shown and partitioned into their respective census tracts. The gray level is in accordance with the population density. Furthermore, one number j is associated with each census tract c , ∀j"1,2, 35, and is located on its center of gravity; therefore, when a number is outside the H corresponding census tract, the non-convexity property of this polygonal region is shown. Let d(x, y) be a distance measure in the plane with ) as associated norm: d(x, y)"x!y. Denote by B(x , r) the set of points in the plane whose distance to the station x is not greater than G G r (usually, called ball of radius r. For each station x , K di!erent levels of attraction will be G considered. Then, ∀i"1,2, N and ∀k"1,2, K v3B(x , r ),B if and only if v!x )r . G I GI G I These attraction levels correspond to concentric annuli around each location site.
Fig. 1. Candidate stations.
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Fig. 2. Candidate stations from node 1.
Fig. 3. Candidate stations from node 3.
Fig. 4. Typical census tract division.
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There are several studies about modeling the attraction by function. We will use that derived by the gravitation model [6]: a (z)" , z
z3[d , d ], 0(d (d .
Here, (d(v, x ))"a/v!x , in which a is a constant to be calibrated. G G In order to discretize the objective function, the attraction in each annulus, determined by two consecutive levels, will be considered constant. Thus, the coverage provided by the station located at x is G ( ) a H Area ((B B ) c ). R(x )" GI GI\ H G (r #r /2) I H I I\ Fig. 5 shows the variety of regions obtained by intersections between census tract C and the attraction annuli. Finally, the objective function is the cover provided by the stations situated in the alignment l, which in the plane corresponds to the section of the line under consideration subject to the CIS. This yields a , ( ) H max R(l) " Area((B B ) c ). : GI GI\ H (r #r /2) X I G Z6 !'1 G H I I\ 3. Station-covering calculation Stations can be roughly classi"ed according to the zone of the city in which they are going to be constructed. Thus, following the methodology often used in trip modeling, the city is divided into two zones: the central business district and the residential areas. Stations in the residential areas
Fig. 5. Intersections.
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can be considered as points at which users change from a private motorized mode of transportation to the transit system; those in central areas mostly attract pedestrian users. Even though other passengers can reach stations by feeder buses (or other modes of transport), these contributions are not relevant now, because the stations (intermodal, near hospital, etc.) which provide a high number of these users had before been decided in order to anchor the alignment. Most of the recent planning projects use either a band around the proposed alignment or circles around station sites. However, Schabas [7], in this description of a project for Honolulu, addresses attention to the fact that actual walking distances are signi"cantly di!erent than that measured by the Euclidean norm. Thus, an estimation of the walking distances for users to reach the stations is required. Following some computational experiments [8] in which the weighted l norm resulted as the N most accurate, we will rely on an earlier study in which a function l , '0, p3[1, 2], has already N been determined in order to estimate actual walking distances in the street network (in Fig. 5, the borders were curves obtained by p"1.5, i.e. x !s N#x !s N"rN). I In order to determine the intersection areas (Area((B B ) c )) involved in the objective GI GI\ H function, a triangulation [9] of census tracts (see Fig. 6) can be used to get the intersections with the attraction annulus, given that census tracts are usually non-convex polygons and generate an excessive number of cases when intersecting with attraction levels. Alternatively, the use of a GIS [10] can be useful if analytical formulae for the contours of the levels of attraction can be incorporated and graphical representations of them are available.
Fig. 6. Decomposing census tracts.
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At this point, one could ask whether solving the problem in which demands are discrete and concentrated at the centroids of the triangles would yield a di!erent result from the problem in which demands are continuous. The following examples can help to clarify this question and show that the method proposed in the paper is more accurate to calculate coverage provided by stations. (A) A triangular census tract is considered "rst as a single block (Fig. 7(1)) and after as a decomposition into three census subtriangles (Fig. 7(2)). Coverage formulated by using continuous demand maintain the same value in both cases. Nevertheless, if we assume demands concentrated at centroids, two di!erent levels of coverage are possible in accordance with the relative position of these points. In the "rst case, coverage is nothing since the centroid is outside the attraction ball, while in the second scheme, the inner centroid generates a coverage di!erent from zero. (B) Two candidate stations S and S are adjacent along the alignment. In the upper band, several census triangles, whose densities are , and (gray level in accordance with these values), intersect the attraction areas built from the respective stations by using the distance generated by the l norm, for p"1.35. Taking into account that the centroid on the left hand is not N covered, we conclude that S provides more coverage in comparison with S if the objective were formulated in discrete terms. Nevertheless, a su$ciently high value of relative to could produce the opposite conclusion, i.e. S is better than S , when the demand is considered continuous. To establish this fact, let us assume that ( "k , ( "k "k k , for certain coe$cients k , k '1. Let A denote the area corresponding to one half-quadrant inside the attraction ball, and let ¹ be the area of each triangle in the "gure. Population covered by
Fig. 7. (1) Triangular census tract. (2) Decomposing tract. (3) Discrete demands versus continuous demand.
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stations S and S in accordance with discrete and continuous (denoted by PC (S ), PC (S ), PC (S ) and PC (S ), respectively), models can be easily calculated: PC (S )"3A #A "(3#k )A , PC (S )"2A #¹ #2(¹!A) "(2A#¹#2k k (¹!A)) , PC (S )"3¹ #¹ "(3#k )¹ , PC (S )"3¹ . Obviously, the relation PC (S )(PC (S )(PC (S ) is satis"ed and by imposing that k '[((1#k )A!¹)/2k (¹!A)], then the sequence can be extended, i.e. PC (S )(PC (S ) (PC (S )(PC (S ), producing converse decisions depending on the setting (continuous or discrete) considered. Taking into account the possible contradictions derived from the use of the discrete model, we adopt the continuous formulation. To compute the intersections between census triangles and balls (more exactly, half-balls), the relative position between tracts and half-balls has been analyzed and eight di!erent cases have been obtained. Figs. 8(1)}(8) illustrate the possible cases. The number of vertices that are inside/outside the half-ball and the number of triangle edges that intersect the contour of the half-ball permit us to establish the case and to elaborate a procedure to determine the intersected area. Following on, we show how to calculate the intersection area in the case in which two vertices are inside the half-ball (see Fig. 8(9), similar to Fig. 8(7)). In order to determine the intersection area, the following formula can be used: Area(Reg. ABCD)"Area(Sec. AsB) # Area(Trian. DsA) #Area(Trian. BsC)!Area(Trian. DsC), where the "rst additive term can be calculated by
Area(Sec. AsB),A( , )"
P dx y d!x( )y( )#x( )y( ) d P
when (x(), y()) represents the parametric curve depending on the angle (see Fig. 9). The total area A(0, 2) inside the curve x N#x N"rN can be obtained exactly by using the Euler beta function, i.e.,
2r 1 1 A(0, 2)" , . p p p If integration limits were integer divisors of then it would be possible to calculate A( , ) exactly by using parameter : r r x()" cos() N, y()" sin() N.
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Fig. 8. (1) Outside triangle. (2) Outside vertices and one edge intersects. (3) Outside vertices and two edges intersect. (4) Outside vertices and three edges intersect. (5) One inside vertex and one opposite edge intersects. (6) One inside vertex and one opposite edge does not intersect. (7) Two inside vertices. (8) Inside triangle. (9) A case of intersected area.
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Fig. 9. Sector area.
For instance, if "0 and "/2 then
1 r 1 1#p 2 r cos\N N() sin>N N() d" , A(0, /2)" p p p p
1 1 r (1/p) r " " , . 2p (2/p) 2p p p When the general case happens ( and are not integer divisors of ), it is necessary to employ some numerical method in order to approximate the exact value of A( , ). In the paper of Mesa and Ortega [11], approximating the calculation with l norms, p3[1, 2), by using -inclined block N norms was suggested. De5nition 1. A -inclined block norm ) is a norm whose unit ball B is the polytope with vertices: $(1, 0),$(cos , sin ) where 3(0, /2]. In Mesa and Ortega [11], the one-to-one correspondence between values of p3[1, 2) and the angles 3(0, /2] was established in the following way:
,(p)"2 arccos
(2
(2 N
3(0, /2].
This relation enables us to express the computation of l (x), ∀x31 (argument 3[0, 2)), as N a decomposition into Euclidean norm and p-bias factors. The second factor is approximated as the following relation shows: l (x)"l (x)r ( )"l (x)(cos N#sin N) N N N +l (x)
cos( /(/2)!/2) . cos(/2)
This formula can be extended to the general case 3[0, 2) taking into account the periodicity of the l function. Furthermore, the analytical study of the approximation error can be found in N Ortega [12].
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Proposition 1. For all , 3[0, /2], the sector area A( , ) can be obtained approximately by the relation:
r !/2 !tan !/2 A( , )+ cos(/2) tan /2 /2 4
,
N /2]. where r, and p3[1, 2) are given and ,(p)"2 arccos((2/(2)3(0, Proof. The curve l (x)"r can be expressed in polar coordinates in the following way: N
l (x)r ( )"rN"(r/ )(1/r ( )). Using the approximation formula r ( )+cos( /(/2)! N N N /2)/cos(/2), the earlier curve can be approximated by "(r/ )(cos(/2)/cos( /(/2)!/2)) and the calculation of the corresponding sector area
P 1 1 P r A( , )+ d " cos(/2) d cos( /(/2)!/2) 2 2 P P gives the required result
r !/2 !tan !/2 A( , )+ cos(/2) tan 4 /2 /2
.
䊐
In order to appreciate the approximation error when A( , ) is obtained by the previous relation, we consider the particular case corresponding to "0 and "/2, where r r A(0, /2)+ cos(/2)(tan(/2)#tan(/2))" sin(), 4 4 and we de"ne the functions S(p)"(r/2 ) (1/p, 1/p)/p and SH(p)"(r/2 )( sin((p))/2(p)), which represent the areas of the sector corresponding to the "rst quadrant for the unit balls associated to l ( ) ) norm and its approximation in the sense indicated by Proposition 1. The N proximity between S(p) (heavy print) and SH(p) (light print), for r/2 "1, can be noted in Fig. 10. In the extremes p"1 and p"2, and functions S(p) and SH(p) coincide. The maximum deviation between both functions occurs in a central value of p3(1, 2) which cannot be explicitly obtained by setting d/dp(S(p)!SH(p))"0. Summarizing this section, the problem involved has been formulated in terms of continuous demand. Coverage calculation presents a geometrical variety of cases which can be reduced if a triangulation of the census tracts is previously done. An approximation method has been proposed to estimate the sector intersected between each triangle and each attraction ring. So, the coverage of stations can be obtained by taking these points into account. Nevertheless, constraints in relation with the CIS must be included in the objective. For reaching it, coverage R determined GH by the pair of stations (S , S ) along the line segment is introduced in the next section by making an G H assumption over the common behavior of the users.
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Fig. 10. Proximity between curves.
4. Line segment-covering calculation In what follows, we will make the following assumption: Given two adjacent stations S and S and the mediatrix M (perpendicular to S S in the middle G H GH G H of that segment), people to the right of the mediatrix M (point X in Fig. 11) always prefer S to S . GH H G To obtain the coverage associated to segment (S , S ) in the line, the intersection areas between G H half-balls B> and B\ and the triangles arising from decomposition of census tracts must be GI HI determined. Two cases are possible: (1) The catchment areas of S and S can be disjoint arcs for attraction level r (see Fig. 12). H G I In general, triangles contained in the band of vertices S and (S #S )/2 must be intersected with G G H ball B , and triangles located in the band of vertices (S #S )/2 and S will be intersected with ball GI G H H B . In order to avoid those triangles which could intersect simultaneously with both half-balls HI B> and B\, a previous decomposition of each triangle into triangles inside the adequate band GI HI should be established. (2) The catchment areas of S and S intersect for attraction level r . H G I Despite half-balls B> and B\ having a common cut zone, the assumption that people always GI HI choose the closest station allows us to reduce this case to the earlier case, in which the key consists of calculating intersections between a portion of catchment area and triangles totally contained in the band determined by this portion (Fig. 13). There are six di!erent topological con"gurations to be considered depending on the number of vertices inside the band and the number of vertices that are on the right- and left-hand sides of the band. These con"gurations are shown in Figs. 14(1)}(6). In the "rst case the vertex distribution generates an empty intersection. The second con"guration gives rise to a new triangle. Two new triangles are originated from the initial one in third and fourth cases. Finally, in Fig. 14(5) we can note the decomposition of the intersected area into three new triangles. Denote by ¹\ the set of triangles that arise by decomposing "rst each of the census tracts GH involved and then by generating the new triangles inside the [S , M ] band from the earlier ones. G GH Similarly, denote by ¹> the set of triangles obtained by using the [M , S ] band. Thus, the GH GH H
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Fig. 11. Choosing the closest station.
Fig. 12. Disjoint catchment areas.
Fig. 13. Overlapped catchment areas.
coverage R provided by the line segment situated between stations S and S is GH G H ) a F Area((B B ) t ) R " GI GI\ F GH t \ ((r #r )/2) F Z2GH I I\ I a ) F # Area((B B ) t ). HI HI\ F ((r #r )/2) > t I\ I F Z2GH I
753
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Fig. 14. (1) Vertices in the (3, 0, 0) form. (2) Vertices in the (2, 1, 0) form. (3) Vertices in the (2, 0, 1) form. (4) Vertices in the (1,2,0) form. (5) Vertices in the (1, 1, 1) form. (6) Vertices in the (0, 3, 0) form.
Fig. 15. Acyclic graph generated.
5. Line-covering calculation Repeating the process explained in Section 2 and in the Figs. 1}3, a connected acyclic graph G"(<, E) can be obtained where < collects the feasible nodes (
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Nodes i and j are connected by edge (i, j) if station j satis"es the constraint on the interstation spacing with respect to previous station i. We can observe that nodes 5, 6 and 13 are not feasible. As we have stated, this graph is deduced from an actual case in which the total length of the corridor does not permit more than two stations to satisfy the CIS. The procedure can be extended to deal with the case in which the number of stations to be located is higher. An optimal solution to the problem of locating N stations on a line to maximize total population coverage is obtained by computing a longest path between the given station (numbered in the graph by 0) and dummy last station (numbered by <#1), if no last station is given, or a real last station C< if a last station is given. This longest path is easy to compute by using the Dijkstra shortest path algorithm [13] over an acyclic graph with costs c "M!R , where R indicates the cost GH GH GH associated with edge (i, j) and M'max R . G H GH The following algorithm summarizes the total procedure for solving the problem of locating stations maximizing the coverage. Algorithm begin 1. compute estimation parameters , p. 2. compute decomposition of census tracts into T F and population density ; j"1,2, M. H 3. read attraction levels r ; k"1,2, K. I 4. read candidate sites S 3X; i"1,2, I. G 5. compute the distribution of the candidate stations S which satisfy the constraints into N blocks G of an acyclic graph G"(<, E), where
"1.105 and p"1.2.
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Table 1 Stations
Numerical results Coverage 1
(4, 12)
Coverage 2
p"2
p"2 9.27994
p"1.2
p"1.2
p"2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
(10.29%) 12.3492
p"2 9.6167
p"1.2 3.33007
(12.00%) 15.41137
9.01152
5.63694
(8.75%) 12.56046
10.2495
3.33769
(10.06%) 15.75026
8.89257
5.16187
(15.65%) 11.61147
9.82641
3.66789
(7.14%) 16.26166
6.3181
5.92385
(9.34%) 12.47986
10.0053
5.29337
(6.96%) 16.29341
7.14278 p"2
p"1.2
(8.56%) 12.58685
p"2
p"1.2
6.25636
(2, 9)
p"1.2
9.6665
5.33708
(9.11%) 15.91657
7.24929 p"2
p"1.2
(2, 10)
p"2
p"1.2
6.62691
(3.24%) 13.31854
8.72325
5.33756 p"2
17.51375 p"1.2
6.6232
7.19332
(2, 11)
p"2
p"1.2
p"2
13.76588
9.41689
6.69534
(3, 9)
p"1.2
p"2
p"1.2
(4.27%) 16.76537
7.0527
8.09686
(3, 10)
p"2
p"1.2
p"2
(18.12%) 11.27097
8.577213
6.71318
(3, 11)
p"1.2
p"2
p"1.2
(21.54%) 13.74086
3.81704
8.19324
(4, 10)
p"2 4.46092
7.45393 (4, 11)
Total (dev. %)
(12.90%) 15.25364
p"1.2 8.49827
(14.07%) 11.82834
G. Laporte et al. / Computers & Operations Research 29 (2002) 741}759
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Table 1 (Continued) Stations
Numerical results Coverage 1
(2, 8)
Coverage 2
p"2
p"2 4.30336
p"1.2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
p"2
p"1.2
(24.81%) 10.3492
p"2 8.54932
p"1.2 2.30184
(35.41%) 11.31137
8.01152
2.53763
(19.26%) 11.11379
8.7495
2.33768
(29.34%) 12.37413
8.63026
2.56187
(15.07%) 11.69073
9.5199
2.48353
(21.28%) 13.75026
9.18792
2.85423
(1, 7)
p"2
p"1.2
p"2
(19.39%) 11.09632
10.82641
2.50281
(1, 8)
p"1.2
p"2
p"1.2
(24.28%) 13.26106
7.83422
2.92385
(1, 9)
p"2 8.9577
3.2621 (1, 10)
Total (dev. %)
(36.69%) 11.08695
p"1.2 7.39787
(29.53%) 9.69971
The total length of alignment analyzed is 900 m (in earlier Figs. 4 and 6, the unit corresponds to 75 m) and the attraction annuli were [0, 200 m], (200 m, 400 m] and (400 m, 600 m], i.e., r "100 m"1.33333 u, r "300 m"4 u and r "500 m"6.66666 u. Let be the attraction factor 1/r. The coverage provided for each station s over each triangle I I T is calculated by (A !A )# (A !A )# A , where A is the intersection of triangle T and the l -ball centered in station s with radius r , taking into account the population density of N I the census tract in which triangle T belongs. Simplifying the earlier expression when the numerical values are used, we obtain the combination A # A #A , for measuring the coverage provided. All pairs of stations were evaluated as the algorithm proposed for ( "1, p"2) (Euclidean distance) and ( "1.105, p"1.2) (estimated actual distance), and summarized in Table 1. We can observe that the best locations for ( "1, p"2) correspond to nodes 4 and 10 and, on the other hand, the best choice for ( "1.105, p"1.2) is the pair (4, 11). Moreover, the second and third best allocations are the pairs (4, 11) and (3, 10) for the Euclidean distance case and the pairs (4, 10) and
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G. Laporte et al. / Computers & Operations Research 29 (2002) 741}759
(3, 11) for the estimated distance case, respectively. The percentage of deviation from the optimal solution has been included in order to show the sensitivity involved. The size of the l -ball is smaller than the corresponding size of the Euclidean ball, hence point N 11 produces better results than node 10 owing to its proximity to the right-hand tracts with higher density. Node 11 remains the third best choice for ( "1.105, p"1.2), combined with node 3 as the "rst station, instead of point 10 as happens in the Euclidean case. Data for computation involved only the population distributions in addition to pedestrian accessibility. If data about employment had also been taken into account, the disparity between zones would have increased (actual data about population in the areas considered show similar density levels) and other di!erent solutions would probably have been generated.
7. Conclusion Locating stations on a prede"ned alignment is an important problem encountered in the design of a rapid transit systems. A standard objective is the maximization of the total potential ridership covered by the stations. We have presented a methodology for solving this problem, based on the triangulation of census tracts. Since such data is typically available, the proposed method could be implemented in most contexts. Although the mathematical developments are sometimes involved, the proposed procedure can easily be coded on a computer and combined with a geographic information system.
Acknowledgements This research was in part supported by Canadian Natural Sciences and Engineering Research Council under grand OGP0039682 and by Ministerio de EducacioH n y Ciencia under grant DGICYT PB-95-1237-C03-01. The authors are also grateful for the helpful comments and referee reports of two anonymous referees that have allowed us to improve upon the original version.
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