- Email: [email protected]

A MODEL

0

0191-2615/84 1984 Pergamon

$3.00 + .Xl Press Ltd.

FOR ALLOCATING CAR PARKING SPACES IN UNIVERSITIES S. K.

Department

of Quantitative

Methods,

GOYAL Concordia University,

Montreal,

Quebec,

Canada

and L. F. A. M. GOMES Department

of Industrial

Engineering,

Pontifical

Catholic

University

of Rio de Janeiro,

Brazil

(Received 15 February 1982; in revised form 18 August 1983) Abstract-In this paper a linear programming model has been presented for determining the optimum allocation of existing car parking facilities for different classes of users within a closed community, of which a university is a typical example. NOTATION probability that a user will bring his car on a day denotes the type of user having permit type i denotes the destination of car park user denotes the car park it is the total number of types of permits i = 1,2, ,, I total number of destinations for car park users j = 1,2, ., m parking tariff for i type users in the kth car park total number of car park users number of users having permit type i number of i type users having destinations j total number of parking spaces available number of parking places available in kth car park after reserving space for handicapped number of parking permits issued for the kth car park distance between the jth destination and the k th car park number of permits issued to type i users in the kth car park number of persons having permit type i destination j and permit for the kth car park.

users

INTRODUCTION

During the past fifteen years a number of mathematical models have been presented (see Ellis et al. 1972; Dirickx and Jennergren 1975; Woodie and May 1969; Narragon and Dessouky 1974; Maher and Birchall 1975; and Goyal 1978) to tackle the problem of providing and allocating car parking spaces. In the models presented in the published literature the problem of allocating car parking spaces is formulated as a linear programme. The suggested models can be categorised based on the number of user classes (single or multi-class) and the types of communities (open or closed) served by the car parks. An educational institution such as a university is a typical example of a closed community. There are generally many classes of users of car parks, i.e. administrators, lecturers, full-time and part-time students, visitors, etc. However, all the models suggested for closed communities suffer from a common drawback of assuming a single class of user which is far from reality among university car park users. The objective of this paper is to fill this gap in the theoretical models for optimising the allocation of car parking permits in universities. The linear programming formulation of the car parking problem given assumes multi-class users. Closed communities having single class of users of car parks are a particular case of the formulation given in this paper. THE

LINEAR

PROGRAMMING

MODEL

The following assumptions are being made in deriving the model: (1) The shortest possible walking distance between car park and a building car park user is accommodated can be obtained; 261

where a

268

S. K. GOYAL and L. F. A. M. GOMES

(2) Every user goes to his work place after parking his car and comes to the car park from his work place by the shortest route; (3) The probability of a car park user bringing his car is assumed equal for each type of user. (This assumption is made to simplify the formulation of the car parking problem); (4) The parking tariff for each type of user using a particular car park is known; (5) The probability of finding a parking space is assumed equal for each type of user. The total number of car park users is given by B = 2 B, = t i=l

The total

number

of available

,=l

parking N=

i

B,.

i=l

spaces is given by i Nk. k=l

Three cases may arise as given below: Case I: B > N Case II: N > B Case III: N = B. Case I: Number of car park users is greater than the number of available spaces for parking cars Mean=p.A, Standard

deviation

In order to have equal probability for everyone condition must be satisfied for the k th car park: Nk-P

1 - p)A,).

= J(p(

finding

a parking

.A,

qGTFwk)=

space, the following

(1)

YJ.

If the value of Y is known then the total number of permits issued for the kth car park can be determined. Goyal (1978) suggested determining Y from the following quadratic equation: nY2-2Y

i

=O

J(Nk)-2

k=l

and then determining

A, from the following: A k = (2Nk + y’) - 2’yJ(N,) 2P

Having obtained the number of permits to be issued for each car park, we embark upon the linear programming formulation of the allocation problem of parking permits to each type of user and for each destination. If XVkis the number of i type users having destination j and an allocation of a parking place in the kth park then the expected distance walked per day will be given by z’ = i -f 2&t&& k=lj=l If the objective

is to minimise

the total distance

walked

by car park users, then Z’ will

A model

for allocating

car parking

be the objective function classes of users, paying minimise:

of the linear programming different parking tariffs,

As p is assumed

so minimisation

constant

k=lj=l

subject

model. However, as there are different so a more appropriate objective is to

of Z’ is equivalent

z=i 2

$,

269

spaces in universities

to minimising

CikDjkxgk

to XVk= B,

i

i = 1,2,.

for

. ., I

and

(4)

k=l

j=1,2,...,m; X,,k = A,

k = 1,2, . . ., n.

for

Case II and Case III: Number of car park users is less than or equal to the number of available spaces for parking cars. Under these conditions every car park user can be guaranteed a parking space. The allocation problem can be formulated as a linear programming model given below: Z = Minimise

f

t

i

CikDjkxlljk

k=lj=li=l

subject

to

i: [email protected] = Bi,

for

i = 1,2,.

. ., 1

and

k=l

j= j$$+k

for

(6)

1,2 ,...,m;

k=1,2

,...,

n.

It may be pointed out that this model is far easier to solve than the model given by (4) as we do not need to compute A,. These linear programming formulations of the car parking problem can be solved by adopting standard methods for solving transportation problems. For a detailed description of solving transportation problems the readers are asked to refer to any standard textbook on operational research techniques. However, for solving practical problems use of computer packages is inevitable. Acknowledgements-The useful comments.

authors

are grateful

to the editor,

the associate

editor

and the referees

for their most

REFERE,NCES Dirickx M. I. and Jennergren L. P. (1975) An analysis of the situation in the downtown area of West Berlin, Transpn Res. 9, l-l 1. Ellis R. H., Rassam P. R. and Bennett J. C. (1972) Development and implementation of a parking allocation model. Highway Res. Rec. No. 395, 5-20. Goyal S. K. (1978) Two models for allocating car parking spaces. Traffic Engng and Confrol 19(2), 83-85. Maher M. J. and Birchall M. C. (1975) A stochastic parking problem. Trafic Engng and Control 16(5), 22&223. Narragon E. A., Dessouky M. I. and De Vor R. E. (1974) A probabilistic model for analyzing campus parking policies. 9per. Res. 22, 1025-1039. Woodie L. W. and May A. D. (1969) A computer program for parking distribution estimation. Mimeographed Paper, Institute of Transportation and Traffic Engineering, University of California, Berkeley.