An Analytical Procedure for the Location Problem and the Determination of a Transportation Line

AN ANALYTICAL PROCEDURE FOR THE LOCATION PROBLEM AND THE DETERMINATION OF A TRANSPORTATION LINE J.C DIVERREZ - M. STAROSWIECKI Universite des Sciences et Techniques de Lille 1 Centre d'Automatique B.P. n° 36 - 59650 Villeneuve d'Ascq - France

ABSTRACT

which depends on the distance between the points Xi

Based on a non linear model of user's behaviour, this paper presents an optimizing procedure applied to location, distribution and transportation problems. The objectives are both to locate points and lines in an urban area, the transportation needs of which are supposed known.

and y .. The basic hypothesis of our method is that each J user is not obligatorily served by only one source. Thus, we consider that everyone of the users is a fuzzy subset of the total set of the sources { x., i = 1, ... n } Let U be the set of all sourceslto be located, and V the set of the N '\, . points to be served. A fuzzy subset yjlS defined as the set of couples /3/.

INTRODUCTION The problems of location,distribution and transportation have been the subject of endless studies /1/, /2/, /6/. Solutions have been applied particularly to urban public services : firefighting, garbage collecting, ambulance service, taxis, etc .. In this paper, we are not proposing just another heuristic procedure among many others, but an analytical method where all the possible solutions are envisioned and dynamically balanced on successive adjustments.This method is clearly detailed on the first example, dealing with optimal location; its basis is the modeling of some distribution pattern /5/. Later on, it will be extended to various problems : location with equality constraints, determination of a transportation line and of its stations.Our objective is calculating neither the number of buses, nor their frequency /6/ but only establishing a " wished " line, taking into consideration origins and destinations of travellers, and a global criterion which varies according with the type of problem.

The problem under consideration may be stated either as researching the locations of source points, or sectorizing a given area. The overall demand of the area is given, being a deterministic data. The consumers are supposed to be concentrated in a finite number of points. The overall demand is then characterized by the set of couples ; = 1,

,11. J

(x )})

n

In order to have some easy interpretation of hhe results, we shall define the set M. as : M. J

{ 11 J'

(x i); 0 .:;; IJ J'

(xi)

.:!;

PJ"

nJ l:

i=l

IJ .

J

(x. ) =p.} l J

(2 )

Thus, the function IJ . (x.) represents the part of the total demand p. Jfro~ point y., serviced by the J J source x .. l

Cost formulation Within these conditions, the cost of service of point Yj may be written; n

J

Problem formulation

n

(1 )

C.

LOCATION OF POINTS OF SERVICE

j

(x

l:

i=l

11. J

(x.) d. l J

(x.)

(3)

l

The choice of d. (x.) would essentially depend on the problem beiJg c6nsidered. Applying this to the problem of transportation the cost is generally a function of a distance V. between the points y. Ji

J

and x .. l

d. J

(x. ) l



(4)

(V. ) Ji

••. N

The cost of the service of the total network is then: where Yj are the coordinates of a point, and Pj is the demand emanating from it. Thus, the problem is to locate n source points in order to meet the overall demand. The cost of service rendered by a source located in x. to a consumer in Yj is defined by a function d~ (xi)

C T

N

D

l:

l:

j=l

i=l

IJ.

J

(x. ) d. l J

(x.) l

(5)

Thus, the objective is to minimize C , entirely defined by the costs d. (x.) and the distribution J l

237

).1. J model ).1.

(x.)

J

The purpose of this model of the consumers, that is decisions ).1. (x.). From a of vue, thi~ moael should (x. )

is to define the behaviour to say their affectation well known classical point be :

(x.) = min d. 1. J

i f d.

Pj

1.

J

1=1, (x. )

).1j

=

if a < d. J

0

1.

(x.)

(9b)

1.

The denominator standardizes the functions ).1j (xi) according to formula (2). The figure 1 gives the characteristic shapes of the numerators.

1.

The distribution model

).1j

(x.)

...

numerators

(xl) (6)

n

a

q

o otherwise

1.

(7)

Such a model ·corresponds to a total affectation of customers, defining a partition of the set {y.; j = 1, ... N } into n equivalence classes, each J

of these being constituted of the set of points which are served by the same source. Nevertheless, the user's behaviour obeys rarely such a rational law, one of the reasons for this being that generally, the servicing cost is not the only criterion of interest, even when i t may be considered as the preponderant one. Thus, according to the hypothesis that every user is a fuzzy subset of the total set of the sources, we shall adopt a distribution model based on some fuzzy, subjective proposition, taking into account the preponderance of · the servicing cost when making some decision. So we shall say : the nearer a point of consummation is to a given source, the more it will be served by that source. A lot of functions can be used to describe subjective proximity /4/; we can use, for example :

Figure 1: Distribution functions given by formulas (7) , (8),

Remark

We have here as an example the appurtenance function (7) and the quadratic cost function :

).1 . J

(x. )

Pj

1.

(x. )

d. J

n

E

e

l.

-T

(7)

d. J

j

(xi)

N L j=l

C T

).1j

).1. J

if

(x. )

Pj

1.

d . J

l+k (d. J

(x.) 1.

~

(xl)

(8)

~l.

)q

(d. J

a

n E i=l

Pj

j l.

(xl'·· . x n ) d. J

(x. )

(11 )

l.

(9a)

(xl)

(xl'

...

e x ) n

-T d.

J

(x. ) l.

n -T d. e L J 1=1

(12) (xl)

The necessary condition for an extremum gives

1.

n L 1

(10)

(Yj - Xi )

where

)q

(x. ) - a )q

(d. J

T

1.

k > 1 n E 1

(x.)

'"

(x.)

Pj

1.

0

(d. J

= (Yj - Xi)

where the superscript T denotes transposition . The total cost is then :

(xl)

1

1 + k

It is easy to see that the distribution model defined by (7) gives as a limit, the total affectation of model (6) when T becomes infinite.

Conditions for an extremum

d e-T

(9).

- a )q

q even

238

a

being the minimization of the cost of the service or, in other words , the maximization of its quality. But the problem rises now of the choice of n, for it is evident that the greater is n, the better is the services quality. We s hall now take into account the cost of implementing the sources, and we shall s uppos e that it can be expressed with the same unit as the cost of the service, and that the sum of these two costs is of some interest. The figure 2 s hows a characteristic shape of these costs.

N j 2 x. 1: Pj ~ ( xl' ... x n ~ j=l

c1'

a x.

~

N

- 2 1: n + 1: k=l

j

Pj

j=l

xl' ... x n

~

a j k

N 1: j=l

Yj

(13)

xl' ..• x n (x - y ) k j

x.

T

(x - Y )= 0 k j

Costs

~

o •

Service cost Impl ementation Cos t Global Cost

1, ... n

i

,_ 0

The partial derivatives are eas ily calculated fr om (12). We obta in :

x

n

2

a

(14)

T

x.

~

(x .

-

~

Y.) J

where 6 is the Kronecker symbo l. ik The s t eepes t descent algorithm leads to

-...

a CT x.

(t+l)

xi (t) - P a x.

~

(t)

1,

i

"

... n (15)

~

...

For a large scale problem, the choice of the iteration coefficient, P , i s very delicate, also, the convergence i s s low. However, the n ecess ary conditions can be written :

i

N 1: ]J. j=l J

(x . )

N 1: ]J j j=l

x .

~

Pj

(x. ) ~

y.+T J

(d. J

~

- Yj

(x . )

]J. J

vmb ....

Servi ce cost , implementation cost, globa l cost as flUlctions of the number of service points .

1) Minimization of the global cost - Such a policy l eads to a number of sources equal to n .It may occur that nand n + 1 give the same °global cost due to the d~s creteOnature of the curves. Then, the' second example will apply and lead to the choice of n + 1. o

~

(16)

i

1,

...

n

his form can serve as the ba s i s of an iterative rocedure and accelerate the convergence. If, in he course of iterations, the quantity :

=1

Figure 2

N

or s ta. teo ns

Le t us give two examples of policies which can be used :

c.

...1.)

-r---·----x----~

O~-1--1_.:------"..L.------"r\..L-------~

2) Maximisation of the quality for a given cost - Let Cl be an admissible cost. Then one seeks to maximize, for a given expense, the quality of the service. So the point n has to be chos e n. ( fig.2 ) 1

x . ) becomes null, the source x . ~

~

ill not service any point of consummation. Only f the initial condition is poorly chosen Mill his phenomenon interfere, b ecause the problem is ~rried out to the positionnement of ( n - 1 ) ~ints of service. It is therefore convenient to hoose another initial condition since the total ~st reduces as the number of sources increases. ?timal number of sources s far, we have been dealing with optimal location f n sources in a given area, the criterion

Optimal location under equality constraints Supposing that a transportation line has already been d e termined, we must locate a certain number of stations along it. This number may be a priori fixed or adjusted after different simulations, in order to insure a guaranteed quality of service. The cost functions d. (x.) represent, in this case, the walking distan6es ~from the generating points y. to the stations x . on the line. The global criteriort defines the qualit9 of the service provided by the line to the area under consideration. Now, the service points, the stations, should satisfy the defining constraints of the line. We assume a line given by the function :

239

x.

2

9 ( x.1.

1.

1

off at x . If m > 1 the trip is called" go ", if m < 1 th~n it is called " return " The cost of the transportation from y. to Yk via ( Xl' xm ) is a function of three factors.

(17)

2

where x. and x . are the components of the location veEtor x .. 1. The global co§t and the appurtenance functions are the same as exhibited in (12) The introduction of the constraints by means of Lagrange multipliers leads to the extremum condition :

a a

x.

dg ( x. )

a

eT

1

a

x.

1.

1

d x.

1.

2) the trip in bus

:

V

( 18)

0

1.

a

the num-

ber of persons going from Yj to Yk by this itinera-

and x.

: v m to Yk mk

Yj to Yk via ( Xl' xm ) and qjk ( Xl' xm

eT

where the partial derivatives

a

jl

Let Tjk ( Xl' xm ) be the transportation cos t from

2

1.

v

lm

3) the walking distance from x

1

eT

1 ) the walking distance from y j to Xl

ry. The transportation cost of all passengers going from Yj to Y is then: k

1

1.

are obtained from formula (13).

a

x.

2

n

1.

In practice, the line is most generally described by a succession of linear sections. The symbol dgx. 1)/ d x. 1.

1.

1

should not be taken in the sense of a

dg ( x.

1.

dx.

g

( x.

1

1.

1.

1

+

_g(x.

£

1.

2

1

n E

1=1

m=l

x

(20)

m

The model

derivative( which possibly does not exist on several points along the line ) but as a quantity which is obtained on each point of the line by an interpolation formula such as : 1

E

The fraction of the total migrants from Y to Y k j choosing the ( Xl' xm ) itinerary may be taken as a decreasing function of this itinerary's cost, as in the first part of this paper. A good model is provided by formula (7) which gives, in this case :

_ £ ) (19)

£

Remark

When formulating the problem of service in a given area, one should take into account the origin-destination matrix. This is not the case here, because the line is supposed to be given, and the different points y. are only considered as demand generators.On theJother hand, constraints such as the maximum walking distance can be introduced; they influence the graphing of the line and the minimal number of stations to be placed.

e-T Tjk (xl'x ) m qjk (xl'xm)

qjk

n

E p =l

E 0=1 0

SIMULTANEOUS DETERMINATION OF A LINE AND ITS STATIONS

(21)

n

F

-T Tjk (x ,x- ) p 0 e p

The cost Tjk (xl'x ) will be, as before, quadratic. m Its expression is :

Problem formulation (22)

A given area is defined by the set { (Y . ' p. ), j = 1, ... N } and by the knowledge of th~ origin-destination matrix, which has been established for a typical time interval ( a day, or a specific period of the day, for example). The problem is to implement a transportation line taking into account not only the walking time ( or distance ) of the users to and from the line, but also some cost linked to the time spent in the bus by the passengers. Thus, we are concerned with the total cost of the service, from the origin to the destination. The formulation may be as follows the line is formed by n stations: Xl x ' .•. x which are en2 countered in this order along the l~ne. Let q'k be the population of y . travelling to Yk· Each 6f these users may then ch60se from among several itineraries : take the bus in Xl and get

+ a

where a is the balancing coefficient between walking cost and riding cost ( penibility coefficient) . The transportation cost of the whole population starting from Y is then: j

240

C.

J

N

n

n

l:

l:

l:

k=l

1=1

m=l

k;ofj

qjk (Xl,Xm)T jk (xl'x m)

(23)

P

jk i

n

i

l:

l:

a=i+1 ' p=l

qjk (x p x a ) + qjk (xa x p )

(30)

m;ofl

and the total cost on the network is Qjk

N

:L

C = l: T j=l

C. J

n

- a

l:

qjk

p=l

(24)

(x. x )

(31 )

P

:L

p;ofi

The problem becomes one of calculating and anihilating the quantities oCT

...

i = 1,

n

N

N

l:

l:

jk --= ox.

j=l k=l

ox.

:L

n

R jk

oC

- a

i 0

~ =1 qjk ( xp Xi )

(32)

(25)

p ;ofi

:L

k;ofj Th e o verall problem reduces then to

which may be written

OC

jk

jk

V.

:L

ox.

+ M.

jk

:L

x. + :L

N

jk i

x _ i 1

:L

+ p.

jk

x i + 1 + Qi

:L

jk

Yj + R.:L

jk

Yk

where

V.

jk

:L

x x

oqjk

i

n

l:

l:

n=l

m=i

m T

ox.

jk

(x x n m

:L

oqjk (xmx n ) + ox.

:L

M.

jk

n (l+ a )

:L

l:

p=l

) + qjk (x.x qjk (xP x.) :L :L P

(28)

Y.

:L

i-1 + 2

l:

p=l

a=i+1

qjk (x p x a ) + qjk (x a x p ) :L

:L

jk

i-1

-

N

j=l

k=l kh

N

N

l:

l:

j=l

k=l kh

n

l:

N.

N.

N f

l:

Q i

jk

N.

:L

Yj + R.:L

jk

jk

Yk

P.

:L

N

N

L

l:

j=l k=l k;ofj

P

jk i

n

l:

l:

p=l

a=i

qjk (x p x a )+ qjk (x a x p )

(29) Simplifications In many practical problems encountered, the origindestination matrix is symetrical, at least over a long period of time. We will not deal here with the calculation of the frequency of the runs, as these are certain to depend on normal and peak hours. They must be calculated later on. The symetry of the

241

~~trix { q'k } leads to numerous simplifications. The calculations are reduced to a one way run, adopting as a convention : the cost from Yj to Yk via ( xl'X ) is

2) The observation of the users' behaviour shows that the station to be taken or left is chosen from among the two or three closest stations. The problem can be reduced to only these stations with a very good approximation. Another characteristic behaviour i s that people do not take into consideration the stations for which the walking distance is greater than a given number ( typical - 400 meters ), so that a reali s tic model is provided by a choice from among the Stations satisfying such a condition.

m

m

if

1 <

if

m < 1

(34)

The total cost is then expressed by

C T

N

N

n-1 n

E

E-

E

E

j=l k=l 1=1 m=l+l kh

qjk (xlxm) Tjk (xl'xm)

(35)

Discussion and remarks 1) When the number of points y. and of stations becomes large, the computing tim~ and the memory requirements become rapidly prohibitive. We can imagine a simplification based on the following postulate each user goes to the station nearest to his origin and gets off at the station closest to his destination. In reality this postulate is not always true because it does not take the passengers' habits into account, the transportation jams which often appear in certain sections, the user' s psychological perception as to his notions of distance or nearness of a station, and time s pent in the bus. Nevertheless, this method may provide some results concerning the POsition of the line, the number and the location of the stations. This approximate result may then serve as an initial solution to the mo s t general and precise process. Let q'k (W.,W )= q·k'so,w. is the nearest station J J k J J from y. and W the nearest from Yk. The walking di stan~e is t~en a preponderant factor in the minimization of the global cost. One will take qjk if Wj precedes W on the run, and qkj within the k inverse hypothesi s. The cost from a point Yj to a point Yk is then :

a (y.-w.)T(y.-W.) + J J J J

3) The walking distances are generally much less than the total length of the trip. However, by choosing a very large coefficient a, we can neglect the distance on the line. We solve then, a problem similar to the one presented in the second paragrah minimizing, now, the terminal costs : walking time from the origin to the near es t station; walking time from the getting off station to the destination, and taking into account the o rigin-destination matrix. 4) The motives of the decision mak e r are primordial when choosing a line. In fact, if a is small, the operator wants to minimize its trip, to the prejudiCe of the quality of service. Inversely, if a is large, the return o f the bus line will be mediocre. It is thus s uitable to find a balance between a service which attracts a maximum of user s by a convenient graph, and one which gives a faster network service, without affecting too much the riding time, or the productiveness. CONCLUSION Throughout this study, we have not considered the costs linked to the waiting time at the s tations. This problem naturally depend s on the number of buses and on the total lengh of the line, determining the frequency of the buses. It depends also on the affluence of passengers at the different s tations , on the capacity of the buses, etc ... This corresponds to the determination of a time schedule for the line, problem which is almost always considered after the line and the stations have been chosen. This paper does not pretend to provide a complete study of the establishment of line s and schedules, but rather to give a helping t oo l for decision problems involving optimal location of points or of transportation lines, taking into account the quality of the service and the productiveness. EXAMPLES Optimal location of n service points

E

i=i

o

This example i s an illustration of the location problem. The '~ rea under con s ideration consists of twenty points, the coordinates of which are given in table I. The demand emanating from each point is supposed to be the same for all points, and equal to 100. The number of sources to be located is four, and the repartition model is given by formula (7). The results consist of table 11, giving the optimal location for the four sources, and table III which shows the cor~esponding repartition of the overall demand. Figure 3 shows the geographical location

(36)

where W. J

x.

~O

and W k

x.

~1

242

of the servic e points in the area under consideration . The dashed lines give the main servi ce area, for each of the service points . The convergence o f the a l gori thm (16) i s fairly fast, as can be seen on figure 4, giving the total cost as a function of the number of itera tion s .

-10.

I (

2,.

4. 3· 0

i. Table I:Coordinates of the 20 points of consumat i on 1 Yi

1

2

3

4

1 2 3 4 5 6 7 8 9 10

0 0 0 0 1 1 6 0 0 0

100 96 96 83 98 95 94 78 19 5

0 0 0 0 0 0 0 2 7 2

0 4 4 17 1 4 0 20 74 93

11 12 13 14 15 16 17 18 19 20

0 0 88 98 6 70 5 10 1 1

3 1 2 0 2 0 0 0 0 0

16 4 10 2 74 30 86 88 92 96

81 95 0 0 18 0 9 2 7 3

Yj

5 5 7 7 8 9 9 10 13 13 16 15 18 19 19 21 21 23 23 25

8 11 10 12 8 9 7 11 13 15 14 16 6 4 11 7 11 10 12 12

f).

-'

• :& ,-

3

19 . 8

7.2

23 .

2 x.

5.2

9 .4

11.1 15 . 28

~

~

19

:,15 _

.20/

i'if

_ • • - • ':" ::- '7 _-::

,

o

_

.- "-

·1.6

o ~----------------------------------.------~~~ 1.

Figure 3

Geographical locations in the cons idered area

Cost

•

• o

4

1 x.

. 17.

.6

Table 11: Opt ima l l oca tio n of the four source s

2

3

o

-100

1

'1.1

\

·f

Tab le III:Repartition of the overa ll demand .

~

2 Yi

,

0

\

number of i teratioIl!

3

4

5

6

14. 2

Figure 4

The convergence of a l gorithm (16)

Optimal determination o f a bus line and its sta tions The area under consideration consists of twenty towns , or centroids, the coordinates of which are given in tab le IV. Table V gives the origin-destination matrix which will be the hypothesis for the determination of the transportation line. This one must link up to six s tations. Due to the symetric natur e of the origin destination matrix,

we will deal only with one direction of trave l . The global operation cost, and the repartition model are given by (23,24) and (21). Two lines have been plotted on figure 5, corresponding to two different choices of the walking Coeffici en t cx.

243

TABLE IV

6

3

9

2.53.5

4

Coordinates of the 20 centroYds

5.5

10

11

12

3

4.5

4

7 1. 5

13

14

15

16

6

17

18

7.5

8.5

19

20

REFERENCES 1 -

E. VALENSI - J.CASTAGNE " Probleme d'affectation - Localisation d 'entrepots " RI RO V-3 19 70

2 -

VO - KHAC - KHOAN " La regulari s ation dans les p r oblemes combinatoi r es et son application au p r obleme des tournees de li vrai s on " RIRO 1969 V-I

3 -

LA . ZADEH " Fuzzy sets Information and control Vo l. S 1965

9

3.5

TABLE V : The ori gi n des t i na t ion ma tr i x

10

11

12

13

14

15

16

17

18

19

20

11

10 10

4 -

A. KAUFMANN " Introduction a la theorie des sous ensembles flous " MASSON ET CIE 1973

5 -

M. STAROSWIECKI - J.C DIVERREZ " Implantation optima l e de centres de pro duction dans un e space economique " 2 ND Polish National Conference on Systems science Sept. 75

6 -

F. BYRNE and VUCHIC " Public transportation line positions and Headways for minimum cost " Fifth international symposium on the theory of traffic flow and transportation p.347 - 360 Tra ffic f l ow and transportation Ed . Gordo n F. Newell Juin 1971

1(1

HI

10

10 10

.14

·1 .13

·20

~

__

~

__________________________________-7?X

Figure 5

Two transportation lines corresponding to et 5 et

1

244