Hierarchical Optimization Algorithm for Timetable Generation Problems

C" I''' i)( 111 © I F.. \ C l.a r )4l' Scall' S"lellls . lit- di ll . ( ;I> R. I ~IH~I

HIERARCHICAL OPTIMIZATION ALGORITHM FOR TIMETABLE GENERATION PROBLEMS W. Grega I 11.1 tit IItl'

of A lltolllatics.

of M in ing awl .\1f1a /lll1 g\'. 3{) -{)59 I\ rakull'. A/. M ickifwicU/ 30, Po /a ll lf

Awdf lllY

AbstrAct. The paper describes an appliCAtion of the d~a.position technIque to ti . .tabl. preparAtion of urbAn bus rout... It su. . .riz •• the different approaches that have baen proposed, describ.. the probleM context And proposes A solution .ethod. An opti.al ti.etAble generation strAt&9Y has been developed on the bAsis of A discrete bus route model which describes the evolution of headways for successive trips of vehicle.. The opti.izAtion prObl.. i . to deter.ine the optiMAl control sequence of lAyover ti . . . At the ter.inal point thAt minimizes the averAge delay of pA.senger. and the deviation of control variAbles fro. the no.inal condition •• Due to the high order of the Model and existence of bounds both on the state and control variable. A hierarchicAl technique has baen suggested to produce the opti.Ai tiMetable. The approach developed in this pAper i . ca.pAred with other methods and SOMe conclusions concerning the efficiency of the AlgorithM are drAwn. Keywords.

Transportation control, support syst ••••

INTRODOCTION

large

sCAle

.yste.s,

d~i.ion

.et of s.aller partiAl ti . .table. is a first step of this proc.... Partial ti . .tables Are generAted u.ing the AlgorithM developed in the next 5eCtion •• EAch ti.etAble ha. A specified ti . . horizon of planning (deter.ined by the nu.oer of bus trips) and hi. own objective function. A proper coordinate &Ch_ i . applied to 6lu. the partial ti . .tables together and to create the final ti . .table. The analytical MOdel for partiAl ti . .table prepAration i . presented in this paper. The MOdel i . based on discrete-ti.. .tate eqUAtions, which describe the evolution of headway. for successive trip. of vehicles. Th. layover ti . . . Allowed at the ter.inal point. are the control variabl... An adequate objective function consi.t. of two ter_. The first terM represents the total Averag. delay of passenger., the .econd penAlizes the d.viAtion. of control VAriable. frOM the nOMinal condition•• IneqUAlity constrAint. on the ad.i •• ible state and control are given. The prObl . . is to deter.ine the opti.al control .equence of the layover tiMeS (at ter.inus) thAt .ini.izes the objective function subject to the .tate equation. and constrAints.

This paper describ.s A . .thodology developed for the auta.Atic generAtion of optimal timetables for urban bus network. The urban public transportation sy.te. offers a wide variety of service. The .o.t obvious categorization is between regular and irregular services, Wren (1990). Re~tar services Are those which operate up And down one or .are rout.. .uch that buses turn round in a short t i . . at .ach ter.inus. Vehicles follON fixed routes and leave at fixed dep~rture ti . . . , deterMined by timetables. lrr.~tar services .AY include extra peak buses a. well a. diAl-a-bus syste.s. This study is concerned with the urban fixed route system_ It i. A feature of this type of service that planners forecast the transportAtion deaand., prepAre the ti . .table to . .et the de. .nds and offer the trAnsportation .ervice. according to ti.-table. Once the planner has decided the headWAY for each route And for each ti . . of the day (i.e. the level of service to be provided), the n.xt broad decision is to deter.ine the ti-.s at which the bus trip. should .tart. Timetabl. preparation is A difficult task And it requires .any triaL -and-error operations if it i. .ade by hu. .n plAnners. This process lends it •• lf well to cOMputer-Aided autOMAtion, Although there app.Ar prOble.s in ensuring the rRgularity of .ervice during . .ai-break. And other (planned) di.turbances on the route.

The pAper is organized a. follows. Fir.t, the probleM context and soee prior ApprOAches are described. Then an anal yt i cal MOdel of a bus route i. presented iUld the objective function is introduced. Afterwards the decoeposi tion &Ch... based on cost Ate coordination . .thod i. applied to the prObl . . . Finally, so-. conclusion. concerning the efficiency of the algorithM are drawn_

The ti_table gen...-ation probleM i • •olved sequentially. PrObla. decOMposition into a -HI

44 2 PRIOR WORK

A good way of ~xalllining ~he na~ur. of ~he ~i . .~able genera~ing process is ~o consid.... ~he ...ark of ~he hUlllan opera~or who has been engaged wi~h ~hese ~a.ks for a long ~ime and has enough .xp .... ience in ~h.se ~asks. The basic ~i . .~able i. prepared as a se~ of ~rips, vehicle by v~hicle, und.r ~he boundary cons~rain~s (s~ar~ and end of ~he service) and service pa~tern chos.n pr.viously by ~h. operator. The nex~ step is the op~illliza~ion of ~h. ~illletabl •• In gen.ral, ~he try-and-error proce55 is considered as a search in ~he proble. space. If ~he problelll space becOllles large, ~he huaan exper~ us~s ~he .e~hod of heuristic. The ~echnique u •• d is ba.ed on ~h. cons~ruc~ion of p.rti.l ~illletables. Succes.ive s~ar~ ~i . . . of the journeys are .hifted under ~he cons~rain~ of .axilllUlII layover ~i . . a~ ~he ter.inu. in order ~o increase ~he regulari~y of ~he service. The hUlllan expert's knowledge was acquired and impl_lm~ed by lida (1988), ~o described ~he applica~ion of an ar~ificial in~elligence ~echnique ~o ti . .~able prepara~ion of a single-~rack railway. His algori ~h.. gen.ra~es • 1 arge nulllber of par~ial ~illletable. and, as ~he nex~ s~ep, selec~. ~he ~i . .~able which ha. ~he miniMUm loss waiting ~ime (as a local optimulll) • A good survey of ~he use of ca.puter in cons~ructing bus ~i . .tables was given by Wren (1980). He concentrates in his book upon irregular .ervice., but .ome of ~he lIIe~hods can be easily adopted when . . include extra peak buses toge~her wi~h odd journeys of infrequen~ service in ~hi. Nirregular" ca~egory. The approaches are classified by .olu~ion me~hods .s heuris~ic, lIIathema~ical and in~erac~ive

directly applicable to bus syste•• sys~e_ general, fixed route particular. TIMETABLE PREPARATION PROCESS

Let's us con.id .... to prepare ~he ti . .table covering three tillle periods, each of thRIll charac~erized by given number of vehicles, the nulllber of trips and the initial and final sta~es. The ~i . .table preparation problelll is solved sequentially and the par~ial tillletables are 6lu~d together crea~ing ~he final ti . .table. The first partial ti . . table in the sequence is generated with the given initial s~ate t xI(O) and de.ired final sta~e x(K ). The I state x(K ) a~tained after opti.iz.tion of the partial ti . .~able I is used as the initial state for optilllization of the next partial tillletable. The initial state xJ(O) and desired state. K (K I ), K (KII) are predeterlllined, on the basis of so. . heuristic technique, not discussed in this paper. PAaTIAL TIaa:TA.LE I

hecadvcr.y I x.' 0)

x x

I

f

•

• • • •

~

a discre~e tr.ffic lIIodel, state space forMUlation and optimal quadratic con~rol theory have be~n developed by Breusege., Campion and Bastin (1988), in ord.... ~o g~nerate ~he op~illlal sRquence of ~he depar~ure ~i . . s of . .~ro ~rains. The objective func~ion ~akes in~o account regularity with respect to the na.inal sch~dule and regularity of the ti.e intervals be~ween successiv~ ~rains. The physical con.traints (e.g. boundedness of the con~rol ac~ions) are not included in the .adel. In order to ~est the efficiency of the proposed algorithM und.... reali.tic op .....tion conditions, the authors have illlPl..anted a .iMUlation .oftware.

• • • • • • • • •

•• • ••

(0)

• ,

(0)

2

hea.dvGY x

)(

2

<.0)

11

•

. .L•.L.

k-.

trip.

(0)

}~(

le)

~I•• z

(le)

I

PAaTIAL TIaa:TA.LE 11

• • • • • •• •

11

• , 2

lIIe~hods.

Curry, GoIIIide and Mendes (1980) have developed • hierarchical algori~hlll for de~erlllining a ~i . .table which r.~ionalizes ~he use of ~rains of Sao Paulo Underground Railway Sys~elll. A dynalllic, discre~e lIIOdel has been created to presen~ ~he .ys~_ behaviour: trains and pa.senger.' IIIDvemen~ s.

in in

...

• • • • ..L_.L.

k-.

l r i. p.

}II Z ( le) 11

~ )( t (

IC)

11

PAaTIAL

x

~ ; ;T'''T''JE I11

III( 0 )

'f-........................ . .L . .L. •

2

,

lr\.p.

k-l

I1I

Recen~ly,

On the ba.,tS of this revi_ i t . _ that only a f_ .y.t... utilizing the opti.ization technique can be proposed in the ti . .table preparation field. While con.iderable literature on creating ti . .tabl . . for train. i . .vailable, the .ethod. that have b . .n develooed are not

Fig.l. Generation of partial timetables Figure 1 shows the schellle that decomposes the timetable preparation problelll into lIIanageable subproblems along the ti.e axis. In the next section, linear 5tate space models for subproblems are forlllulated making possible the implementation of optimal control methods. Costate decomposition is carried out inside each subproblem. PROBLEM DEFINITION Basic Notations. We introduce here the basic notations for the lIIodel and SOMe related definition. "hi ch character-t ze the problem.

H ier,lrc hi ca l 0 pli m iza l iO Il .-\I go ril hm

A bus ro~t8 consists of tNO terminal points and intermediate stops over which service is provided. A day 01 operation is divided into periods. These need not be of equal duration. For each period a fixed nu~er of buses are put in operation, according to the travel demand and available resources.

Travel ti8eS and layover tiees eay vary with the period. Buses operate up and dOMn the route at different intervals, such that buses turn round at each tereinus point. A single two - way traversal of the route is called a trip. It consists of runs between terminal points and layovers at the terminal point.

A

ro~t8 timetable is defined to be a sequence of trips assigned to the buses.

{ ti. (k)} \. =j. •

••

N

k = 1 • • • 1C

where ti. (k) is a departure tiee fra. tereinus of i-bus during k-trip.

the

It is possible to describe the .otion of buses by treating headways between th . . as the state variables.

Objecti ve Functi on The objective function pen.liz_ irregularity ~ service and deviations ~ the control v..-iabl_ their d_ired v.lue. :l(x (k) ,u(k) ,k) .. N

E [ri. (k)x~ (k) + q(ui. (k) - Ui. (k»z] z i.~. or in the equivalent far •• !.

:l(x(k),u(k),k) ~("x(k)"z + lIu(k) ~(k).z),

z

a( le)

(4)

Q

The first tar. represents the .v.....Vdel.y for p •••eng ......... iting for • group ~ N bu_s. The aecond ter. corresponds to the devi.tion bet ...ean the actual control sequence .nd the reference control sequence u(k). The ....ight par . . .ter rl(k) describ_ the tr.vel d..and ob ..... ved .long t.he rout.e. The el___.t.. ri. (k) correspond t.o the .t.t.istic.l . .an v.lu_ ~ p.ssenger Ih whi ch ent.er the .top.. The .atrix Q is diagonal and positively definite. The t.otal objective t.he for •• :l
State Equations

443

IC-' E

function i .

:l(x (k) ,u(k) ,k) + :l(x

given in (I(»,

(:s)

k&o

A discrete .adel of the tiee table generation probl_ is given by the set of difference equations of the type. 1 x (k+l) = I x (k) + 8 u(k) + 8 u(k-ll, x(k),u(k) E RN

(1)

... here :l(x :l(x

(I(»

(I(» ..

is t.he ter.inal cost.

!.b (I() Z

-

;e(K) MZ

a( X)

,

and.

x = (x(0)T,x(1)T, ••• x(K- 1)T) u

(u(0)T,u(1)T, ••• u(K-1)T)

where: x (k) is a head ...ay defined as: xi. (k)

ti. (k)

ti.-1(k),

x1(k)

t1(k)

tN(k-1)

i .. 2 •• N (2)

I is N x N identity .atrix and u(k) is the layover ti8e at the ter.inus. The variable u(k) will be seen as a control variable. 8, 8 1 are N x N .atrix in th* for.:

0 0 0... 0 01

1

-~ .. ~ .. ~ .. ~:::~ ... ~

8

1 o

0

0

0 •• -1

1

0

0

0

0 ••• 0

-11

1

0

0

0 •• 0

0

~ .. ~ .. ~ .. ~:::~ ... ~

o

CS)

CS' )

interconnections bet~ adjacent buses in the string. For k - 0 .... get the initial conditions:

reprl~senting

Xi. (0)

ti.(O)

X1(O)

ti.-1(O),

i " 2 ••• N

tN(-ll ,

The departure ti . . tN ... ill be fulf LL the condition:

chosen

N

E

xi.

(0)

~

T

\.=&

where T is the route travel ti . . .

to

The st..t.e and control v.riable. are con.t.r.ined by upper and IDMRr bounds. o ~ x(k) ~ x (k) ..... x unri.n(k) ~ u(k) ~ u(k)_x

(6)

The probl.. of deter.ining the opt.i . .l ti . .t.ble can be posed as the opt.i.iz.t.ion proble •• .in :l(x,u), subject to (1) and (6), (7) x. u

It can be e •• ily seen th.t. the order ~ the probl_ grDMS proportionally to the nu~ar of buses under consider.tion. The _in require.ent of an opti.iz.tion . .thod for this proble. is the ability to handle syst.. of high di..nsion.lity, with inequality constr.ints on .t.tes and controls. It is i~ortant to ..ntion that without (6) the probl_ beca.es .n unconstrained LQ-Probl_ for which the opti . . l control sequences .re knDNn.The another i~ortant observation is th.t the discrete syste. (1)-(3) consi.t. of interconnected subsyst . . . ~ lower order. As the additivity require..nt on the objective function is fulfilled, e.ch of the subsyste•• could be opti.izad f.irly eaSily, even when the over.ll .yst_ i . not a.en.ble to solution because of the ca.putation.l difficulties. Traditionally/in such probl . .s the ap.ce is au~ted by introd\lcing adi1::ianal variabl_ for the del.y tar_ and by converting the origin.l hi Qh-ord....

444

W. Grega

diffRr.nce equation to a ••t of coupled fir.t order equ.tion.. HotMtVRr, this incr •• ses the di . .n.ion.lity and consequently .ake. the ca.put.tion.1 burd.n even .ore seVRre. To ov.rca.. this difficulty a hier.rchic.1 .ppro.ch due to La.don (1970), TalllUr. (197~) and Singh (1975) Mas proposed in the previous paper, GrRlila (1999). The . .thod is based on the Observ.tion that the dual of the original prOble. can be deco.posed (by the index k) into K independent . .all .ini.ization problems which are solved by the standard . .thod. The limit.tion of the . .thod is that N x K, the nu-oer of vehicle. ti . . . number of stages, 8USt not b. too large fro. the ca.putational point of vieM. The optimization of even a 2O-variabl. sy.t. . conSUmRS much of the c.pacity of IB"/XT COMputRr. ConsidRring a typical ti . .table designing problem, Mith N 20 and the nulllber of trip. K - 10 then 200 iterations are necess.ry to Obt.in the convergence of the TalllUra .ethod. Th. ca.putation ti . . MOuld be roughly 20 min. by I~/XT cOMputer. Since the nu-oRr of off-di.gonal interconnections in the .atrix B i . . . .11, it .akes sense to exploit the d.centralized structure of the prObl . . . SOLUTI ON I1ETHOD

In the literature ("ah.aud, 1977a, Papageorgiou, 1982) a nulllbRr of hierarchical control methods have been suggested for optimization of discrete ti.e dynamical syst •••• Inherent in all these aethods i . the central idea of the decoaposition-coordination procedure. Th. methods that have been proposed by "ahmoud involve the formulation of the Lagrangian dual to the problem and a.sociated Hamiltonian. To develop a IllUltil.vel control structure, the set of variables involved is divided into tMO distinct subsets. The first subset, the dependent variables, are closely related to the local prOble.. The second sub.et, the independent variable., are .anipulat.d on the coordination level. GenRrally speaking, there is no restriction on the choice of either subset of variable. so long as the level structures can be implemented practically. The methods that have been proposed fall essentially into folloMing categoriesl a/ Feasible methods, with additional variables playing a role of independent variables in the .ubprOble•• , b/ Nonfeasible methods, Mith Lagrange IllUltipliers modified on the coordination level, c/ Costate coordination . . thod., Mith costate variables playing • role of interconnection variables. In this p.perJthe last . .thod h •• been adapted to produce the opti.al bus ti . .tables. The ca.putation.l effort is shared between tMO levels, .nd the opt i.i z at i on prOb 1_ is conVRrted into successive iterations on the set of 10Mer-order subprobl . .s . . thod. The b •• i . of this algorithm is to forlllUl.te the Lagrangian dual to the prObl . . (7)1

•• xilllize • Od Mi th respect to A, MhRrel • (A) - .in l!.Cx,u,A) x. u

(8)

.ubject to (1) .nd (6). Equation (1) in the decOlllPosed for. can be forlllUl.ted by adjoing loc.l variables x, (k), u, (k) .nd separ.ble interconnection variabl_

1h

(k), such that:

[uT(k), ••••••• u T (k)l

uT(k)

S

NS

X T (k)

MhRre Ns is the nw.bRr of subprOblemsl x . (k+l) '" I . x. (k) + B. u . (k) + B. n .(k) + \.

\.

\.

\.

,

+

B

S

\.

\.

\.

,

u . (k-ll

(9)

Ns

,

n.

(k)

F.sD'j

Uj(k)

, "j

MhRre D'j are the ••trices of sUbsyste.s> i nterconn ect ions. In this forlllUlation control-delay term u, (k-l) is al May. considered as the local variable. The Lagrangian functional the overall prOblelll can be for.ulated

Ns

\.-.

E {

l!.(x,u, n, A)

Ji.

(x, (K»

for a.:

+

(k) ,u ,, (k), n ,, (k»

+

Ns D .. u . (k»]}

E

'J

j =s , .. j

( 11)

J

Interchanging the sUmlllation in the last ter. of the equation (11) results in the folloMing subprOblellls:

Ns l!.(x,u,n,A) -

E

l!., (x"u"n,'\)

i. =s

IC-S

+ f=~Ji. (x, (k), u, (k),

n, (k»

+

Ns

E

XT(k) D.. u, (k»l

j:tJ

\.Jl

, .. j

In this forlllUlation only local state and control variables app,ar in the equations of subprOble.s, X appears a. a coordination vari.ble. The local subproble.s are in the form: .in

l!..

)( ,u., , R ,.

\.

(x . \.

,u , ,n. ,X , ) \.

\.

\.

(12)

subject to (9), and (6) The subproblem (12) has which can be found .ethod.

a

solution

by

The .ain re.ponsibility of the higher level is to upd.te the value of the intRrconnection IllUltipliers recursively, ~sing any gradient-type algorith •• Knowing that the effect of X, (k) i . tending to incr . . . . the Lagrangi.n ("-hlllOUd 1977b),

Hiera rchi ca l Optimiza ti oll :\Igu rithlll the iter~tive be taken as:

Nt XL+! (k) = xL(~)+"'x~(1T.(k)-E D.. u .( k»} (13) L

L

\.

j:,tJ

\.

\.

i. .. j This ~uation describ.s a si~le steepest asc.nt whose rate of convergence d.p.nd. on the choice of a Xi. • Th. conjugat. gradient ~ethod for this prOble. has been proposed by Singh (1975). In ord.r to solve the overall prObl . . (b). it is possible to use the follONing two-Iev.l algorith.u St~ 1. Gue.s the sequence X(k) • Set the iteration index L z 1. Step 2. Solve the BubprOble.. (12) for i=l •• Nt using the method of Ta.ura. Specify the solutions xL(k), uL(k), 1T L (k)

,

,

,

Step 3. Update the gradient of .(X). I f 119 4>( X) 11 ~ & for sOllle prescr i bed accuracy & go to Step 4, else stop the procedure. Step 4. Update the current values of multipliers XL+t by equation (13), using one-di.ensional search technique to maximize dual function .(X).Go to step 2. EFFICIENCY OF THE ALGORITHM To estimate the computational efficiency of the dual decentralized .ethod J the relation between the order of the optimization problem and the computational time must be studied. The convergence of the dual algorithm for the overall problem (characterized by the same structure as the subproblems) can be achieved after NxK iterations, ~s reported by Ta.ura (1975). The problem (b) was solved for _f,3, •• 8 vehicles with an accuracy of 10 . The results indicate the follONing formula relating the order of the proble. with the computational ti.e needed for its solutionatc = N· / z . Thus, in view of the results of Papageorgiou (1983) it becomes apparent that only a slight ca.putational time reduction can be expected by application of the decomposition .ethod on a single computer. Decomposed opti.al control for two subprOblems was obtained for 4,5,b vehicles and for band 7 vehicles when three subprOblems were generated. Decomposition of the prOble. into subproblems was achieved by introducing interconnection variables for equation (1), defined as: (u (k) ,u (k»T, for Nt 2, N ~ t 2, N b 1T (k) "' (u (k) ,u (k»T; for Nt t Z (u • (k) , u (k»T, 1T (k) Ut (k), 1T (k) Z for Nt = 3, N .. b. 1T

(k)~

Z

•

,

•

1T (k) 2

Ut (k), 1T (k)

•

CONCLUSIONS

.illtply

procedure

,

.,

-

REFERENCES Breuseg. . , V.V., G.Ca~ion, and G.Ba.tin (1988). SiMUlation and control of the traffic of _tro line.. In P.Geril (Ed.) Proceedings of 2nd European Simulatlon HUltlconference, 001.2, 169 173. C.der, A., and N.H.". Wilson (1986). Bu. network design. Transportation Research,20B, 331 - 343. Curry, J.K., F.A.C. Go.id., and ".J.~de. (1980).A .ethodology for generation of of opti_l sch.dules for an underground railway syste.s. IEEE Transaction. on Automatic Control, At 15, 217 222. 6r~a, <1989). bIiC~OSl bon _thod for generation of opti.al ti_table. for urban bus network. Proceedings of Control, C~utRr., Coe.unlcation in Transportatlon 5Y~OSlU., Parl.
w.

or.-

(u .(k), u (k»T; for Nt = 3, N

A hierarchical cOGputational algorith. ha. been appli.d to solve large seale ti . .-table generation prObI... Typically, the di.ension of the .tat. vector x(k) is 20 - 40 and therefore the cOlllPutational burden of deter.ining the aanag--.nt policy for the bus line will be .xc.s.ive when using conventional .ethod. • The presented algorith. deal. with .i~le routines on each l.vel (the difficulties required to solve two-point boundary value proble~are by-pass.d ) and enable. to deco~ose the dyna.ic syst_ into subsyste.s, each representing a group of vehicle•• This re.ults in!di.tribution of the c~utational effor~. A distinct c~utation t i _ reduction can be expected when a mul ti processor sy.t •• is u.ed.

7.

It wa. found that the nu~er of coordination operations is independent of the nw.ber NI. of llUbprobl_.. Hence, a di.tinct c~utation ti . . reduction can be achieved for a sufficiently high order of the overlay probl_ if a .ultiproc_sor syst_ is applied.