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~ '- ~ ..............
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14.1 [ I*1
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