Heuristic Method of Solving Traffic Network Under Variable Parameter Condition

Copyright @ lFAC Transportation Systems.

Tiujin. PRC. 1994

HEURISTIC METHOD OF SOLVING TRAFFIC NETWORK UNDER VARIABLE PARAMETER CONDITION HE XIANCI and HUANG ZHONG XIANG Changsha Ccnnmw,ications U7Iiversity. Systnns Engineering J7Istitute. 410076. PRC

Abstract. A heuristic method is advanced to solve the two problems ·trip distribution· and ·traffic assignment" in a traffic network. The model depends on the overall optimum theory : The optimum trip distribution rel&tes not only to the trip volumn of node. but also to the path on which the resistance is minimum .

Key words . Transpon Network, Heuristic programming, Traffic Control

1. INTRODUCTION

method, we solve the two problems: trip dis-

The two major problems in traffic network

tribution and traffic assignment at the same

and trip forecasting are "trip distribution" and

time, trip fares (resistances) are drawn into

If

the model, if it is a variable.

traffic assignment If. There are interrelated

and interdependent problems either in theory or in practice, because in the problem of trip

2.HTV MODEL

distribution, trip fares (resistances on net-

" Heuristic method of solving traffic network

work links) are needed as parameters in order

under variable parameter condition" is called

11

to solve the 0 - D (origin - destination) trip

HTV model" for short.

matrix, and in the problem of traffic assign-

Given a traffic network G

men t, the resistances on network links (trip

of nodes, the number of nodes is m. A is the

fares) are also functions of trip flow. The

set of arcs, the number of arcs is L. The sub-

problem of traffic resistance is linked with and

set Z is the set of real nodes, the number of

used as a variable in the above two problems.

real nodes is n (n
(2, A), 2 is

the set

These problems have been studied for hundred

node Zi (i = 1", n) is: the output volume r;

years. In the past, it was presumed that the

and input volume s; are positive (r;>O, s;>

resistance on network link is not a variable but

0). The residual nodes Z.~I' "', Z.... ;, "', Zm

a constant, obviously the reliability in the so-

are so called "virtual nodes". r .+; and

lution would no longer be effective.

output and input volume of virtual node Z.+;,

After

s.~ ; ,

the

(i=l,"',n-m) are zero. In traffic network,

1976, the combined - trip distributions and

the points of intersection are virtual nodes. So

traffic assignment models (CDA) were arisen,

the HTV model is described as follows:

which depended on the principle of "Informa-

~tp;j p

tion Min." or "Entropy Max." to establish a

t;j=

non linear programming model. In this model

tp;j=k;j •

~r;

0) •

~Sj

• f;j-oi

(2)

the factor of traffic resistance had been omitted (Evans, 1976; Frank, 1978).

(3)

In our 949

step 3: Calculate parameters a, (kl and k;;'k)with v. = L: L: L: ~P;j • tpij . . p ~P;j=

iteration,

(4)

l,if arc a is contained in Pij . { 0, otherwise

.(i ,j= 1, •..

,n).

step 4: Calculate the OD matrix T (kl k;/kl t;j = M2 • r; • Sj • (f;;'k»-.)

(5)

tpij~O

(k)

,

(i.j=l,···,n)

i,j=I.···.n; aEL

(I3)

k " t;j (k) = " L..J t;j (k) , ( 1. >J. = 1 , ... ,n ) (14 ) k-l step 5: Calculate the flow volume of network

where: t;j- -an element of O-D trip matrix T. p;j - - the minimum resistance path from

v., a E L

to

k

n

n

2: (~p;;'k) • tp;;'k». a E L

v, (k) = L: ~

J.

11-1 i-I

v, - - the flow volumn on arc a, a E L.

j-l

(IS)

c, - - the capacity of arc a, a E L

If k=M, then V,=V.'M), aEL

f, (.::...) - - the resistance (or fare) of arc a. c, that is the function of congestion level v le, . In

step 6: Calculate the total fare F F(k)=

this paper. we consider the function is mono-

k

n

n

k=1

i=l

, -= 1

2: L: 2: (t;;'kl. L,'k»,

(16)

tonic and ascensive.

If k=M. then F=F(M),

f;j - - the resistance of p;j.

If k=M, then T=T(M)=[t;j(M)J •• c end the calculation. If k < M, k: = k

kij - - the parameter which satisfies the

+1, go to step 2.

eguilibrium of 0- D matrix T: (8)

L: L: tp;j=r;. ;E,

p

4. Algorithm Analysis Theornn 1. The theorem

(9)

L: L: tpij=Sjt

(~

-

existence

The optimum solution with real value must

(i,j=l,"',n) a; -

0/ solution

i Ea

p

a parameter which satisfies a statistic

exist, if graph G is connected.

~

Proof:

is a reflection of the characteristic of a giv-

If graph G is connected, from step 2. a shortest path L/ k> between node pail (i ,j)must (1).

en network):

~ k;j • Sj • Lj- ai = 1-~. jE,

(IO)

be found,(i,j=l,···,n ; k=1,"',M) and the flow volumn

(i=l,"',n)

~

t;j can be assigned to that

path. After M times, the OD flow tij have already been assigned.

3. Algorithm

step 1: Calculate the resistance of adjacent

tion.

arcs f.(v,(k)/c.). aEL. k=l"",M (M

IS

a

given

number

As all are shortest

paths, v, (a EL) must be an optimum solu(2).

of

In this paper, we have supposed that f,

(.::...) (a EL) is a monotonic and ascensive c. function, therefore Lj(k), k = 1 , ••• , M are real,

calculation). Initial value k = 1, v, (J)

=0. step 2: Calculate the shortest path p;;'k) and

the value of t;;'kl, which calculated by step 4

the resistance fij (k) , (i, j = 1 , ••• , n; k =

must have t;;'kl~O, (k=1,···,M;i,j=1.···,

1,"',M):

n), and from step 5, we have

f;;'k'= L:~Pi;'k'f,(k)(v.(k)/c.),

i

Theornn 2. The theornn

=

k 1 , if arc a is contained in Pi/ ) ,

aEL. So

the optimum solution is real.

(ID

,EL

~p;;'k)

v.~O,

0/ unique solution

If the number M is great enough. ~ t;;'k) ap(I2)

0, otherwise

proach to "a driver" (see theorem 5), then 950

Tii (Ml , (i, j = 1 •••. ,n) is the unique solution of

o__=

OD matrix, v,(Ml(aEL) is the unique solution

001

{ 1, if a on the path qii' 0, otherwise

(22)

of flow volumn on arc a.

proof:

Proof:

If M is great enough in order that the assigned Assume p/kl,(i,j=l,···,n) that we

folw volumn correspond to a driver, obviously

find from step I is unique shortest path, owing

fi/kl, calculated from step 2, is the minimun fare on the shortest path Pi/ kl for a driver, so

0).

to that f, (~) is monotonic and ascensive funcc

fi/kl satisfies the formula (8). The other !;i

tion, thus fa( ~), fi/kl and fi/kl, (k=l , •.• ,M; c, i , j = 1 , ••• , n; a EL) are all uniqueness, and

flow, therefore it satisfies the formula (9).

therefore v, (kl (k = 1, ... ,M, a EL) is unique

So in accordance with our algorithm, the solu-

too.

tion for Matrix T satisfies the first eguilibrium

which is not on the shortest path will have no

principle or wardrop.

If p/kl,(i,j=l,···,M) are not unique,

(2).

assume between node pair (i, j) there are d 5. Conclusion

shortest paths, then we assign _1_ t __ (k) flow dm'l volumn to those d paths. In accordance with

In this papre the heuristic method has been advanced to solve the two problems" trip dis-

our algorithm, the solution would be unique.

tribution" and " traffic assignment" at the

Theorem 3. Solutions of parameters k;j and a; (i , j

=

same time. The software of this method has

1 , ••• , n) are unique, there are surely

been held and used in some articles. If n<70,

convergence with iteration algorithm.

the calculation can be completed in personal

proof: (omitted)

computer .

Them -em 4. The solution of our algorithm is

In view of the universality of the principles of

the solution which accords with the first equi-

this method, we wish to popularize it to some

librium principle of wardrop.

other networks.

we can describe the first principle of wardrop in mathematics:

6. REFERENCES

If and only if the flow tii =

2:: toii

Evans, (7)

S. P. (976). Derivation and Analy-

sis of some Models for Combining Trip

o

Distribution and Assignment. Transpn while foii> 0 foii =

Res. , 10,37-57.

(8)

Frank, C. (1978), A study of Alternative

2:: Ooii • f, ( v,c, ) ~fii'

Approaches to Combined Trip Distribu-

,EL

while foii= 0

(9)

tion - Assignment Modeling. phD the-

toii~O,

(20)

sis, Dept. of Regional Science, Univer-

i,j=l,···,n

sity

where: Li=min foii

PA.

(21)

o

951

of

Pennsylvania,

Philadelphia,