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I,ÍATHEII{ATICA], PROeRAl,llvf f Ne ¡,IODULS
r'OR TRAT'FIC NETYIORI( PROBLEI,IS
by
JOTÍN A. TO}ÍLIN B.Sc. (Hons.) (¡.¿er.)
ÍIhesis subnltted. for the Degree of Doctor of Philosopt\y
1n the University of Adelald.et
Department of l¡fathematlcet December, 1967. '
TABLE OF CONTE}I]IS
Summary Lv
Slgned. Statement vil
Aclcnov¡leilgements v1il
Chapter 1: Introd.uctlon 1
Chapter 2z Linear Programming Assignment 9 2.1 Practicabillty of tlre nod'el 9
2.2 Nod-e-Ârc Formulation 13
2"3 Arc-Chaln Formulation 18
2.1+ Comparlscn of the l,Íethod-s 21
chapter 3z Equillbrium Distributlon of Trafflc ?Lr
3.1 Statement of the Problem 2+
3.2 Equilibrium Distribution 26
3,3 Calibration of the l'{od'e1 3Z
3.4 Experlmental Results 3l+
3,5 Conclusions 36
Chapter l+: Cornbined- Distrlbution-'4'sslgnment \ '-'O
L+,1 Interd-epend.enee of Dlstrlbution and. Assignment ¿+o t+,,2 Linear Distribution-Assì-gnment \z
\.3 Eclullibriun Disirlbutlon-Assignment 5C
¿+.4 trfod-lflcation of the Deteffenee Functlon 53 t+.5 Further Extensions and- the General l'lod.el 57
Chapter 5: Applications to a }lod'el- City 62
5.1 Use of a lJlod-el CitY 62
5.2 Basic Data on U.K' Newtovrrn 63
5.3 All-or-Nothing and. I,lnear Programmlng Assigrunent
5.4 Equilibrlum Dlstribution ard. Dlstrlbutlon- ' AssJ-gnment
5.5 Conclusion to tlre Example Chapter 6: Discussion
Append.ix f : Computational Erçerfunents wlth the Arc-Chaln Algorithm
I.1 Relationship to Prevlous StuiLles
Í..2 Descrlption of the Test Prograrunes
T.3 The Test Problems
I.4 Experlmental Result s
I.5 Discussion
Append.ix II: Equivalence of the Multlcommod.ity Flow Problem wltTr the Sol1d. lransportation
ProbLen
II.1 The Single Commod.itY Case
ÍT.2 Corrstruction of an Equivalent Solld.
Transpor tatlon Problem
ÍT.3 Significance of the Result Blbllography
Supportlng l4aterial (tnsld.e Back Coven)
1l',J+
66
I¿
71
-11
/rJ
((
81
B2 B3 B7
B8
AD
96
sut{t[¿-RY
Thie thesis Ís concer::ecL wlth the use of rnathenat- lcal programmJ.ng nod.els for trafflc netv¡ork problens anil
in particular Êorne ctr the problems encountered. in trans- portation planning, The lntroductory chapter d.lscusses
previor:s r'.rork iir thls field. and. the use of prograrrunlng
mod.els ln transportation planning in relatlon to the sinulat- ion or rrsystemstt approach.
The use of mathematical and. particularly I1near prograrlming for the traffic asslgnment problen i e then
consid.erecL in the 11ght of reeent l:'ork on tÏre flotv/travet-
t1¡ne relationship and- the system of tor¡¡n plannlng advocatecl
1n the Buchanan Seport rrTraffic in lolrnsrr. the llnear pro-
granme is formulated. in both nod-e-arc ar¡l arc-chain forms lvhich are then ehol1ryn to be equivalent. Some renarks are
also mad.e on the computatlonal properties of these algo-
rlthms but experimental resul ts ."" å ef er¡ed. to an appencì.1x
for the sahe of continuitY.
A prerequislie of any assignnrent procedure is the
obtainlng of a traffic Ölstributlon or d-esj.re l1ne pattern.
This problem 1s approached. þere from a nel'¡ cllrectiont
making use of the equillbrium propertles of traffic and
d.raving an analogy from thernod.ynamlc processes rattrer
iv
than ttre gravitatlonal analogy so often used.. Thls nelv
approaeh al-l-olys a d.lrect formulatlon of the d.lstribution problen as a matbrema.tlcal- pr.ograrilme. Some su-ggestlons
for the cal-ibration of tlre rnod.el are macle and. illustraied.
by applicati-on to some clata collected. by the ì,fetro_ooIltan Ad.elaid.e Transpor tation Stud.y group.
It is customary to treat the assignment and d.istri-
butlon problems sepanatelyralthough tIæy are c1-earl-y j.nter.-
related.. thls inter-relatlonsh1p 1s taken into account 1n
the for¡nulation of a combined. d.lstribution-assigrunent mocle1.
Thls iclea is initlally cÌenonstrated. lvith a llurely Ilnear
mod.el- uslng' the Dantzlg-\Yolfe d.ecomposition principle to
interpret the process. îhis formulation and its interpret- ation are then extend.ed. to a comblned- linear progrannning -
equilibrlum d.lstrlbution assignnent mode1. Further exten- sion and. generallzation of the mod.el 1s also conmented. upon.
These mathematlcal- programming mod.eLs aue then
lllustrated. by application to a simpie inod.el city. The
comblned. d-istribution-assignment algorithm in partieular 1s
d.emonstrated. ex¡gl1cltly through geveraL steps of the
d.ecomposed. fornr of tÌre mod.eL.
;\ chapter of d-iscussion preced.es the tuto a3¡rend.ices
supplementing the main body of the thesls. lþoend-lx I
d.eals rzith the lmportant topic of choosing an efflclent
v
forn of the llnear progranming asslgnment algorlthn.
Some eomputatlonal experimente to thie end. are deecrlbed.
and. the results and concluslons prese-nted.. Append.lx II
1s a eontrlbution to the general field. of multicornnod.lty fJ.ow theory of v¡hich linear nrogramning asslgnment forme
a part. The ecluivalence of the mininlun cost multlconmod.lty
problen and. the rfgolld.rr or nul-t1-ind.ex tranoportatlon problen is proved., and. the elgnlflcance of thls result
d.iscussed..
vl
SIGNED STAIEI,.{E}TT
Thls tiresls cotrtalns no natenlaL whlch hae been accepted. for the award. of ar\y other d.egree or d.lp1ona
ln any Unlvereity. To ttre þest of my lcrowledge ancl
beL1ef, the thesls contains no materlal prevlously publlshed. or wrltten by any other personr except where due reference is mad.e in the text of the thesigr
vi1
AC Ifl lIOl'¡tED GE L/Í El{ T S
The author acknowredges the generous support of the
Australian Road. Research Board. and. General Motors-Hold.ens
Pty. r,td.., vrhich has enabled thls research to be car.rled.
out. The author is indebted. to h1s supervisors,
Professor R.B. Potts and. Dr. R.G; Keats, for tJreir \
encouragenent and. assi.stance. He has also greatly þ
benefited. from d.iscussion wlür professor 17.R. Bl-und.en
of the unlverslty of New south \fales, who also suggested.
the use of the nod.el eity ln chaptet 5. The d.ata for the
experiments of chapter J rzere supplied. by the i,ietropolitan
Adelaid.e Transportation Study group, and. the author
panticularly acknowledges the helpful comments of LIr. R. Payze.
The auttror ls also ind.ebted for their eneouragement
to lilr. D.F. Glyrur, Director of the Australian Road. Research Board., Dr. D.J. Buckley (rrow at the University of Nev¡ South
iïales) and. Dr. R¡L. Pretty, also of the Boarcl. Marty thanko
are also extenclect to the ty¡rist, l.{lss D.J. Potter.
vi11
