Assessing the Efficiency of Rapid Transit Configurations

Sociedad de Estadistica e lnvestigaci6n Operativa Top (1997) Vol. 5, No. 1, pp. 95-104 Assessing the Efficiency of Rapid Transit Configurations Gilbert Laporte Centre de Recherche sur ies Transports Universitd de Montrdal, Canada Juan A. Mesa Departamento de Materndtica Aplicada 1I Escuela Superior de Ingenieros. Universidad de Sevilla, Spain Francisco A. Ortega Departamento de Materndtica Aplicada I Escuela Tgcnica Superior de Arquitectura. Universidad de Sevilla, Spain Abstract Eight basic transit network configurations are analyzed with respect to two measures: passenger/network ef[ectiveness and passenger/plane effectiveness. Assumptions are made with respect to trip distribution and competition with other transportation modes. Key Words: rapid transit configurations, network effectiveness 1. Introduction Over the last decades, several cities throughout the world have looked at the construction or extension of rapid transit systems as a partial answer to increased traffic congestion and urban sprawl. Such systems include traditional underground metros, surface rail and light rail transit networks, and monorails (see Jim@nez Solano (1993) for a taxonomy). In a recent article, Gendreau, Laporte and Mesa (1995) have examined the criteria considered by decision makers in the planning of rapid transit systems. The main considerations include purposes, cost, network characteristics, coverage and utilization, and external attributes. Network design lies at the heart of the problem: how to design a network configuration capable of improving the population's mobility by providing shorter travel times. While the operations research literature on network design is rich and still developing (see Ahuja et al. (1995) for a survey), standard optimization This research was in part supported by Canadian Natural Sciences and Engineering Research Council under grant OGP0039682 and by the Junta de Andalucia. This support is gratefully acknowledged. Received: March 1996; Accepted: July 1996 96 G. Laporte, J.A. Mesa and F. Ortega methods will rarely be applicable to large civil engineering projects such as the construction of highways, airports and mass transit systems (Magnanti and Wong (1984)). This is partly because of the size of the integer programming models involved, but also because of the fact that these models typically involve non-linearities, stochastic elements, as well as multiple and conflicting objectives. Traditionally, scenario analysis has been the favored planning tool. Several sensible network configurations are drawn up and assessed with respect to a number of criteria. From these, a "best" solution is selected, typically after a lengthy consultation process involving planners, engineers, politicians and citizen groups. While operations research tools can in no way be a substitute for a multi-player decision process, it can assist it in a number of ways. Two examples come to mind. One is the use of heuristics such as tabu search to locate an alignment maximizing population coverage, subject to station spacing constraints (Dufourd, Gendreau and Laporte (1996)). Here, the search space does not have to be an entire city; it can be restricted, for example, to one or several promising corridors targeted by planners. In such a context, optimization is used primarily to fine tune proposed alternatives. Another example is the analysis of proposed or prototype networks on the basis of their topological characteristics. This line of research is rooted in the work of Musso and Vuchic (1988) who analyzed a number of station travel paths, line overlapping, directness of travel, overall network connectivity, etc. This work was later extended by Laporte, Mesa and Ortega (1994) who proposed two efficiency measures. The first, "passenger/network effectiveness", compute in a idealized network the ratio ,~ of the sum of all O/D passenger travel times over the sum of all node to node travel times in the network. The second, "passenger/plane effectiveness", computes an index Qp to compare passenger travel time on the network to what it would be to travel in the plane in which is embedded, using an Ip norm (p = 1 corresponds to a Manhattan distance, p = 2 corresponds to a Euclidean distance). These two measures were computed for several typical networks such as stars, cartwheels, triangles, grids, etc. under the assumption of uniformity. In other words, it was assumed that all node pairs on the network were equally likely O/D pairs. This limits in some respect the degree of realism of this study as in practice, the most likely trips are from the periphery to the center of the network, and competition with other modes is a function of trip length. In this paper, we analyze the passenger/network effectiveness and the Assessing Efficiency of R. T. C. 97 passenger/plane effectiveness of eight basic configurations by dropping the uniformity assumption and introducing mode competition. As we work on idealized networks and not on real data, a number of normative assumptions were made and some parameters were fixed to realistic values. Assumption had to be made between tradeoff functions between the use of public transit and private automobile, peak hour congestion was ignored, aversion for transfer between different lines (beyond added travel time) was not considered, etc. Sensitivity analyses were then conducted. We believe that in spite of the simplifications that were made, this type of analysis can help compare basic configurations and can easily be applied to real situations using proper parameter settings. The remainder of this paper is organized as follows. Our model is developed in Section 2, followed by computational results in Section 3 and by the conclusion in Section 4. 2. The Model We consider a circular city with a business core Z1 and an outer annulus Z2 corresponding to a residential area. In some cities with a natural barrier like a river, a semi-circular representation may be more appropiate: Here Z1 is a semi-circular inner centre and Z2 is an outer semi-annulus. Some typical circular and semi-circular configurations are illustred in figures 1 to 8 in the appendix. There are three classes of O/D trips, according to whether they are made inside Z1, inside Z2, or between Z1 and Z2. The transit network is represented by an undirected graph G -= (V, E, t), where V is a vertex set representing stations, E is an edge set representing direct transit links between adjacent stations, and t = (tij) is the travel time matrix on the edges. This network is embedded in a plane. Each vertex vi of V correspond to a point P/of that plane. The catchment area of vi is then = II ll<6 } where 6/is a constant that could represent the maximum acceptable travel time for using station vi. All catchment areas are assumed to be disjoint. 98 G. Laporte, J.A. Mesa and F. Ortega 2.1. Passenger/Network effectiveness To define passenger/network effectiveness, consider nij, the number of trips between vi and vj, and vii, the shortest travel time between vi and vj, and let Oij = nij vii(T) . Then the total cost incurred by passengers is i<j while the total network cost is T= Z tij (vi,vj)~E The required passenger/network effectiveness coefficient is therefore A = O/T 2.2. Passenger/Plane effectiveness The idea behind the passenger/plane effectiveness coefficient is to measure how well the network is embedded in the plane. Two networks with the same vertex set, but different edge sets will typically have a different coefficient. As mentionned in the introduction, the computation coefficient depends on the metric lp used for travels in the plane. Denote by lpij the travel time in the plane between vi and vj using an lp norm. Let also mpij -- nijlpij, 0 = (Oij) and Mp = (mpij). Then the passenger/plane effectiveness coefficient is defined as Qp _ IIO-Mpll IVI when II 9 II is the Frobenius norm, i.e., if H = (hij) is an m x n matrix, then m n i=1 j=l 2.3. Trip distribution To evaluate nij we consider the two zones to which vi and vj belong, the travel demand by any mode between points of Ai and Aj, and the Assessing Efficiency of R. T.C. 99 fraction of the demand corresponding to the use of the transit network. More specifically, nij = cij fij gij 9 We now explain the three factors used in the determination of nij. a) The coefficient cij takes one of three values cl, c2 or c3, with Cl + c2+c3 = 1, according to whether vi and vj are both in Z1, both in Z~ or in two different zones. (One could use more refined coefficients to reflect the fraction of Ai and Aj intersecting with each zone). b) The second factor, fij, is a "friction coefficient" representing the travel demand between Ai and Aj. This is a deterrence function that depends on the travel time x (called impedance) according to a relation of the form f(x) = x -zx ; > o (see Ortfizar and Willumsen (1990), p. 138). In our application, x is the travel time d(P, Q) between P and Q. Thus, fij =/PEAi fOeA~ d(P'o)~ e-zd(P'o) dPdo " To approximate this expression, we replace d(P, Q) by its mean value -dij over the integration domain, using the formula for the expected distance between two points belonging to disjoint circles of centres Pi and Pj and of radii pi and pj (Koshizuka and Kurita (1991)): pi 2 + pj2 -3ij = d(P. Pj) + Sd( , Pj) so that 2 2 2 -- a fij ,~ 7r Pi pj(dij) e -~dO c) Finally, we use a logit function for the modal split distribution. Assuming several modes of transportation, one of which being public transit, the proportion of trips between vi and vj using the transit network will be gij = (1 + e-~(~q-'"')) -1 , where rij and Ttij are the shortest travel time between vi and vj using public transit network and all the other transportation modes, respectively, and 7 is a parameter. 100 G. Laporte, J.A. Mesa and F. Ortega 3. Computational Results In order to carry out the various computational tests, the following problem generation rules were used. The radii of Z1 and Z2 were taken as Pl = 2.5 and P2 : 9. The average distance between two adjacent vertices was set to approximately 1, and the radius 5i of each catchment area Ai was set equal 0.5. We used average surface speeds of vl = 20 and v2 = 40 for the zones Zt and Z2, respectively. The speed in the transit network is 60 so that it takes one time unit to cross each edge. We assumed a train stopping time of 0.4 at each station and a transfer time of 4 at connecting stations. In addition, we used cl = 1/3, c2 = 1/2 and c3 = 1/6. The values of the parameter a and /~ were obtained by an indirect method. The mean and variance of the trip length distribution are ~+1 Using the experimental values # = 3.551224 and a 2 = 3.23154 obtained by Blumenfeld, Shrager and Weiss (1975), we were able to compute { ~ = ~ = 1.09893 c~=#-~-1= 2.90257 so that the value of fij would be fij /-" 7I"2 (0.5) 4 (dij) 2'90257 6 -1"09893~i'/ The parameter 7 of the logit function is calculated using the fact that the maximal interstation distance is 16, which corresponds to two stations located at both ends of a diameter. Traveling between two extreme stations requires crossing 16 edges and 15 vertices. Thus, maxi<j 7-ij = 16 + 15 • 0.4 --- 22, while the same trip across the urban network has a length of Ttij -~ 11 • (60/40) + 5 • (60/20) = 31.5. Thus the maximum difference is 31.5 - 22 = 9.5. Using this information, we obtain a reasonable value of 7 = 0.309941. The two effectiveness indices A and Q2 were first computed using (~ = 2.90257,/~ = 1.09893 and 7 = 0.309941 for each of the eight basic network configurations shown in Figures 1 to 8 (see Appendix). The results are Assessing Efficiency of R. T.C. 101 presented in Table 1. In addition, we performed sensitivity analyses for a E [0.5, 4] and/3 E [0.5, 2]. Circular cities PARAMETERS CIRCUMF. A 0.0095 Qe 0.0219 CARTWH. 0.0124 0.0220 TRIANG. 0.0127 0.0207 Circular cities PARAM ETERS i GRID 0.0156 STAR U AND C. m m[f]O]!~ Q2 0.0228 0.0221 0.0281 Semi-circular cities PARAMETERS HALF-WHEEL A 0.0201 Q2 0.0317 HALF-RADIAL 0.0219 0.0303 Table 1: Values of the effectiveness indices for eight basic networks Computational results indicate that for circular cities, the circumferential configuration by far offers the best passenger/network effectiveness. The cartwheel configuration is the second best except when c~ E [2.25, 4] and /3 = 0.5, or c~ E [3, 4] and/3 = 0.75, in which case the triangle configuration is the second best. The best passenger/plane effectiveness is obtained for the triangle and circumferential configurations. For a given value of c~, the triangle configuration is best if/3 is low; as/3 becomes larger, the circumferential configuration is the best choice. For example, when a = 2, triangle is best for/3 _< 1; when a = 4, triangle is best for/3 _< 1.5. It should be noted that in the case of uniform trip distributions (Laporte, Mesa and Ortega (1994)), these three configurations (circumferential, triangle and cartwheel) also came out best. In that paper, the half-radial and half-wheel configurations always yield the best values of effectiveness for semicircular cities. Now, regarding to passenger/plane effectiveness, half-wheel is better than half-radial if c~ is low or/3 is high. Finally, the half-wheel always produces the best passenger/network effectiveness value. 102 G. Laporte, J.A. Mesa and F. Ortega 4. Conclusion We have analyzed a number of basic rapid transit network configurations with respect to two measures introduced in Laporte, Mesa and Ortega (1994), but under more realistic assumptions. Here, the hypothesis of uniform travel distribution between all station pairs is removed and replaced by a more realistic scenario: two concentric zones with different travel characteristics are used and, in addition, competition with an alternative travel mode is considered using simple modeling assumption and realistic parameter settings, we derive a comparative evaluation of several network designs. Sensitivity analyses point to the robustness of the results. We do not suggest that our modeling assumptions and choices of paramaters hold in all settings. We believe, however, that this type of analysis can help compare alternative network designs. 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