Assessing the Efficiency of Rapid Transit Configurations

Sociedad de Es adis ica e ln es igaci6n Ope a i a Top (1997) Vol. 5, No. 1, pp. 95-104 Assessing he E iciency o Rapid T ansi Con igu a ions Gilbe Lapo e Cen e de Reche che su ies T anspo s Uni e si d de Mon dal, Canada Juan A. Mesa Depa amen o de Ma e nd ica Aplicada 1I Escuela Supe io de Ingenie os. Uni e sidad de Se illa, Spain F ancisco A. O ega Depa amen o de Ma e nd ica Aplicada I Escuela Tgcnica Supe io de A qui ec u a. Uni e sidad de Se illa, Spain Abs ac Eigh basic ansi ne wo k con igu a ions a e analyzed wi h espec o wo mea- su es: passenge /ne wo k e [ec i eness and passenge /plane e ec i eness. Assump- ions a e made wi h espec o ip dis ibu ion and compe i ion wi h o he ans- po a ion modes. Key Wo ds: apid ansi con igu a ions, ne wo k e ec i eness 1. In oduc ion O e he las decades, se e al ci ies h oughou he wo ld ha e looked a he cons uc ion o ex ension o apid ansi sys ems as a pa ial answe o inc eased a ic conges ion and u ban sp awl. Such sys ems include a- di ional unde g ound me os, su ace ail and ligh ail ansi ne wo ks, and mono ails (see Jim@nez Solano (1993) o a axonomy). In a ecen a icle, Gend eau, Lapo e and Mesa (1995) ha e examined he c i e ia conside ed by decision make s in he planning o apid ansi sys ems. The main conside a ions include pu poses, cos , ne wo k cha ac e is ics, co e age and u iliza ion, and ex e nal a ibu es. Ne wo k design lies a he hea o he p oblem: how o design a ne wo k con igu a ion capable o imp o ing he popula ion's mobili y by p o iding sho e a el imes. While he ope a ions esea ch li e a u e on ne wo k design is ich and s ill de eloping (see Ahuja e al. (1995) o a su ey), s anda d op imiza ion This esea ch was in pa suppo ed by Canadian Na u al Sciences and Enginee ing Resea ch Council unde g an OGP0039682 and by he Jun a de Andalucia. This suppo is g a e ully acknowledged. Recei ed: Ma ch 1996; Accep ed: July 1996 96 G. Lapo e, J.A. Mesa and F. O ega me hods will a ely be applicable o la ge ci il enginee ing p ojec s such as he cons uc ion o highways, ai po s and mass ansi sys ems (Mag- nan i and Wong (1984)). This is pa ly because o he size o he in ege p og amming models in ol ed, bu also because o he ac ha hese mod- els ypically in ol e non-linea i ies, s ochas ic elemen s, as well as mul iple and con lic ing objec i es. T adi ionally, scena io analysis has been he a o ed planning ool. Se - e al sensible ne wo k con igu a ions a e d awn up and assessed wi h espec o a numbe o c i e ia. F om hese, a "bes " solu ion is selec ed, ypically a e a leng hy consul a ion p ocess in ol ing planne s, enginee s, poli i- cians and ci izen g oups. While ope a ions esea ch ools can in no way be a subs i u e o a mul i-playe decision p ocess, i can assis i in a num- be o ways. Two examples come o mind. One is he use o heu is ics such as abu sea ch o loca e an alignmen maximizing popula ion co e - age, subjec o s a ion spacing cons ain s (Du ou d, Gend eau and Lapo e (1996)). He e, he sea ch space does no ha e o be an en i e ci y; i can be es ic ed, o example, o one o se e al p omising co ido s a ge ed by planne s. In such a con ex , op imiza ion is used p ima ily o ine une p oposed al e na i es. Ano he example is he analysis o p oposed o p o- o ype ne wo ks on he basis o hei opological cha ac e is ics. This line o esea ch is oo ed in he wo k o Musso and Vuchic (1988) who analyzed a numbe o s a ion a el pa hs, line o e lapping, di ec ness o a el, o e all ne wo k connec i i y, e c. This wo k was la e ex ended by Lapo e, Mesa and O ega (1994) who p oposed wo e iciency measu es. The i s , "pas- senge /ne wo k e ec i eness", compu e in a idealized ne wo k he a io ,~ o he sum o all O/D passenge a el imes o e he sum o all node o node a el imes in he ne wo k. The second, "passenge /plane e ec i eness", compu es an index Qp o compa e passenge a el ime on he ne wo k o wha i would be o a el in he plane in which is embedded, using an Ip no m (p = 1 co esponds o a Manha an dis ance, p = 2 co esponds o a Euclidean dis ance). These wo measu es we e compu ed o se e al ypical ne wo ks such as s a s, ca wheels, iangles, g ids, e c. unde he assump ion o uni o mi y. In o he wo ds, i was assumed ha all node pai s on he ne wo k we e equally likely O/D pai s. This limi s in some espec he deg ee o ealism o his s udy as in p ac ice, he mos likely ips a e om he pe iphe y o he cen e o he ne wo k, and compe i ion wi h o he modes is a unc ion o ip leng h. In his pape , we analyze he passenge /ne wo k e ec i eness and he Assessing E iciency o R. T. C. 97 passenge /plane e ec i eness o eigh basic con igu a ions by d opping he uni o mi y assump ion and in oducing mode compe i ion. As we wo k on idealized ne wo ks and no on eal da a, a numbe o no ma i e as- sump ions we e made and some pa ame e s we e ixed o ealis ic alues. Assump ion had o be made be ween adeo unc ions be ween he use o public ansi and p i a e au omobile, peak hou conges ion was igno ed, a e sion o ans e be ween di e en lines (beyond added a el ime) was no conside ed, e c. Sensi i i y analyses we e hen conduc ed. We belie e ha in spi e o he simpli ica ions ha we e made, his ype o analysis can help compa e basic con igu a ions and can easily be applied o eal si ua ions using p ope pa ame e se ings. The emainde o his pape is o ganized as ollows. Ou model is de eloped in Sec ion 2, ollowed by compu a ional esul s in Sec ion 3 and by he conclusion in Sec ion 4. 2. The Model We conside a ci cula ci y wi h a business co e Z1 and an ou e annulus Z2 co esponding o a esiden ial a ea. In some ci ies wi h a na u al ba ie like a i e , a semi-ci cula ep esen a ion may be mo e app opia e: He e Z1 is a semi-ci cula inne cen e and Z2 is an ou e semi-annulus. Some ypical ci cula and semi-ci cula con igu a ions a e illus ed in igu es 1 o 8 in he appendix. The e a e h ee classes o O/D ips, acco ding o whe he hey a e made inside Z1, inside Z2, o be ween Z1 and Z2. The ansi ne wo k is ep esen ed by an undi ec ed g aph G -= (V, E, ), whe e V is a e ex se ep esen ing s a ions, E is an edge se ep esen ing di ec ansi links be ween adjacen s a ions, and = ( ij) is he a el ime ma ix on he edges. This ne wo k is embedded in a plane. Each e ex i o V co espond o a poin P/o ha plane. The ca chmen a ea o i is hen = II ll<6 } whe e 6/is a cons an ha could ep esen he maximum accep able a el ime o using s a ion i. All ca chmen a eas a e assumed o be disjoin . 98 G. Lapo e, J.A. Mesa and F. O ega 2.1. Passenge /Ne wo k e ec i eness To de ine passenge /ne wo k e ec i eness, conside nij, he numbe o ips be ween i and j, and ii, he sho es a el ime be ween i and j, and le Oij = nij ii(T) . Then he o al cos incu ed by passenge s is i<j while he o al ne wo k cos is T= Z ij ( i, j)~E The equi ed passenge /ne wo k e ec i eness coe icien is he e o e A = O/T 2.2. Passenge /Plane e ec i eness The idea behind he passenge /plane e ec i eness coe icien is o measu e how well he ne wo k is embedded in he plane. Two ne wo ks wi h he same e ex se , bu di e en edge se s will ypically ha e a di e en co- e icien . As men ionned in he in oduc ion, he compu a ion coe icien depends on he me ic lp used o a els in he plane. Deno e by lpij he a el ime in he plane be ween i and j using an lp no m. Le also mpij -- nijlpij, 0 = (Oij) and Mp = (mpij). Then he passenge /plane e ec i eness coe icien is de ined as Qp _ IIO-Mpll IVI when II 9 II is he F obenius no m, i.e., i H = (hij) is an m x n ma ix, hen m n i=1 j=l 2.3. T ip dis ibu ion To e alua e nij we conside he wo zones o which i and j belong, he a el demand by any mode be ween poin s o Ai and Aj, and he Assessing E iciency o R. T.C. 99 ac ion o he demand co esponding o he use o he ansi ne wo k. Mo e speci ically, nij = cij ij gij 9 We now explain he h ee ac o s used in he de e mina ion o nij. a) The coe icien cij akes one o h ee alues cl, c2 o c3, wi h Cl + c2+c3 = 1, acco ding o whe he i and j a e bo h in Z1, bo h in Z~ o in wo di e en zones. (One could use mo e e ined coe icien s o e lec he ac ion o Ai and Aj in e sec ing wi h each zone). b) The second ac o , ij, is a " ic ion coe icien " ep esen ing he a el demand be ween Ai and Aj. This is a de e ence unc ion ha depends on he a el ime x (called impedance) acco ding o a ela ion o he o m (x) = x -zx ; > o (see O iza and Willumsen (1990), p. 138). In ou applica ion, x is he a el ime d(P, Q) be ween P and Q. Thus, ij =/PEAi OeA~ d(P'o)~ e-zd(P'o) dPdo " To app oxima e his exp ession, we eplace d(P, Q) by i s mean alue -dij o e he in eg a ion domain, using he o mula o he expec ed dis ance be ween wo poin s belonging o disjoin ci cles o cen es Pi and Pj and o adii pi and pj (Koshizuka and Ku i a (1991)): pi 2 + pj2 -3ij = d(P. Pj) + Sd( , Pj) so ha 2 2 2 -- a ij ,~ 7 Pi pj(dij) e -~dO c) Finally, we use a logi unc ion o he modal spli dis ibu ion. As- suming se e al modes o anspo a ion, one o which being public ansi , he p opo ion o ips be ween i and j using he ansi ne wo k will be gij = (1 + e-~(~q-'"')) -1 , whe e ij and T ij a e he sho es a el ime be ween i and j using public ansi ne wo k and all he o he anspo a ion modes, espec i ely, and 7 is a pa ame e . 100 G. Lapo e, J.A. Mesa and F. O ega 3. Compu a ional Resul s In o de o ca y ou he a ious compu a ional es s, he ollowing p oblem gene a ion ules we e used. The adii o Z1 and Z2 we e aken as Pl = 2.5 and P2 : 9. The a e age dis ance be ween wo adjacen e ices was se o app oxima ely 1, and he adius 5i o each ca chmen a ea Ai was se equal 0.5. We used a e age su ace speeds o l = 20 and 2 = 40 o he zones Z and Z2, espec i ely. The speed in he ansi ne wo k is 60 so ha i akes one ime uni o c oss each edge. We assumed a ain s opping ime o 0.4 a each s a ion and a ans e ime o 4 a connec ing s a ions. In addi ion, we used cl = 1/3, c2 = 1/2 and c3 = 1/6. The alues o he pa ame e a and /~ we e ob ained by an indi ec me hod. The mean and a iance o he ip leng h dis ibu ion a e ~+1 Using he expe imen al alues # = 3.551224 and a 2 = 3.23154 ob ained by Blumen eld, Sh age and Weiss (1975), we we e able o compu e { ~ = ~ = 1.09893 c~=#-~-1= 2.90257 so ha he alue o ij would be ij /-" 7I"2 (0.5) 4 (dij) 2'90257 6 -1"09893~i'/ The pa ame e 7 o he logi unc ion is calcula ed using he ac ha he maximal in e s a ion dis ance is 16, which co esponds o wo s a ions loca ed a bo h ends o a diame e . T a eling be ween wo ex eme s a ions equi es c ossing 16 edges and 15 e ices. Thus, maxi<j 7-ij = 16 + 15 • 0.4 --- 22, while he same ip ac oss he u ban ne wo k has a leng h o T ij -~ 11 • (60/40) + 5 • (60/20) = 31.5. Thus he maximum di e ence is 31.5 - 22 = 9.5. Using his in o ma ion, we ob ain a easonable alue o 7 = 0.309941. The wo e ec i eness indices A and Q2 we e i s compu ed using (~ = 2.90257,/~ = 1.09893 and 7 = 0.309941 o each o he eigh basic ne wo k con igu a ions shown in Figu es 1 o 8 (see Appendix). The esul s a e Assessing E iciency o R. T.C. 101 p esen ed in Table 1. In addi ion, we pe o med sensi i i y analyses o a E [0.5, 4] and/3 E [0.5, 2]. Ci cula ci ies PARAMETERS CIRCUMF. A 0.0095 Qe 0.0219 CARTWH. 0.0124 0.0220 TRIANG. 0.0127 0.0207 Ci cula ci ies PARAM ETERS i GRID 0.0156 STAR U AND C. m m[ ]O]!~ Q2 0.0228 0.0221 0.0281 Semi-ci cula ci ies PARAMETERS HALF-WHEEL A 0.0201 Q2 0.0317 HALF-RADIAL 0.0219 0.0303 Table 1: Values o he e ec i eness indices o eigh basic ne wo ks Compu a ional esul s indica e ha o ci cula ci ies, he ci cum e - en ial con igu a ion by a o e s he bes passenge /ne wo k e ec i eness. The ca wheel con igu a ion is he second bes excep when c~ E [2.25, 4] and /3 = 0.5, o c~ E [3, 4] and/3 = 0.75, in which case he iangle con igu a ion is he second bes . The bes passenge /plane e ec i eness is ob ained o he iangle and ci cum e en ial con igu a ions. Fo a gi en alue o c~, he iangle con igu a ion is bes i /3 is low; as/3 becomes la ge , he ci cum e - en ial con igu a ion is he bes choice. Fo example, when a = 2, iangle is bes o /3 _< 1; when a = 4, iangle is bes o /3 _< 1.5. I should be no ed ha in he case o uni o m ip dis ibu ions (Lapo e, Mesa and O ega (1994)), hese h ee con igu a ions (ci cum e en ial, iangle and ca wheel) also came ou bes . In ha pape , he hal - adial and hal -wheel con igu- a ions always yield he bes alues o e ec i eness o semici cula ci ies. Now, ega ding o passenge /plane e ec i eness, hal -wheel is be e han hal - adial i c~ is low o /3 is high. Finally, he hal -wheel always p oduces he bes passenge /ne wo k e ec i eness alue. 102 G. Lapo e, J.A. Mesa and F. O ega 4. Conclusion We ha e analyzed a numbe o basic apid ansi ne wo k con igu a ions wi h espec o wo measu es in oduced in Lapo e, Mesa and O ega (1994), bu unde mo e ealis ic assump ions. He e, he hypo hesis o uni- o m a el dis ibu ion be ween all s a ion pai s is emo ed and eplaced by a mo e ealis ic scena io: wo concen ic zones wi h di e en a el cha - ac e is ics a e used and, in addi ion, compe i ion wi h an al e na i e a el mode is conside ed using simple modeling assump ion and ealis ic pa ame- e se ings, we de i e a compa a i e e alua ion o se e al ne wo k designs. Sensi i i y analyses poin o he obus ness o he esul s. We do no sug- ges ha ou modeling assump ions and choices o pa ama e s hold in all se ings. We belie e, howe e , ha his ype o analysis can help compa e al e na i e ne wo k designs. Appendix Illus a ion o eigh basic ne wo k con igu a ions ...... ::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::: :::::' ......... Figu e 1: Ca whell ~ :~::::i~:::::::::!:::::::::!:i:::~ ' ::.:.:.:.. I!!!! Figu e 2: S a Assessing E iciency o R. T. C. 103 IJ ....... ~ I I .,.::7::": ........................ :~::~:i::.:.... I ~ .:.- 9 :!::?~::i~i:i!ii ========================== ............ ::::::::::::::::::::::::::: I I =============================================================== IIIII "i ' ~ TT'~k i i I ] I [ I I I Figu e 3: U and C oss I I I i .................... , i::~,:;~ i l~1 I I@1 , '%::[ : .... 14.1114.1 ........ 14. I I 14'1 ::!iii~;i; I~l I 14.1 .... i~i:! 14.1 ] I~1 .FI ! _ I :~:~ - :1~ '- ~ .............. -----I--: 14.1114.1 I 14.1 [ I*1 I ::![ i --:i;ii~ : I 14.11 14.1 ..... Figu e 5: G id _l I I , . %~ .............. I / i :::il 1% II i~::: i J Figu e 7: Hal -Radial Figu e 4: Ci cum e en ial / I I ~~' I ..; =================================================== Figu e 6: T iangle ~ :!::~,,:':i ~':~':: ::%:~ .... !iii::. I .... :!i~!!~:i~i~i~ i.,, ..... ' % Figu e 8: Hal -Wheel