Optimal Location of School Facilities

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1

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t

OPTIt!.t\L LOCATIO::

OF

SCI-IOOL F/1CILITIES

j.

D. COELHO

Working -Paper nO .92

UNIVERSIDADE NOV A DE LISBOA

Faculdade

de

Economia

Travessa Estevao Pinto

1000

LISBOA

2

I

t t ! i ~ '1

OPTIlvlAL

LOCATIOI'~

OF SCHOOL FACILITIES

ADSTRACT

A project involving the development of a -relational da.lt\ base of educational information at a. national level for Portugal and"the optimization of school facilities is described. A few new theoretical developments in relation with facility location modelling are mentioned. Information regarding our experience is provided.

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Keywords: schools location, relational data base, discrete programming, multi-criteria. analysis.

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The location of educational facilities may be-formulated in many different ways dependina ho\v the main componen.ts .are considered and the nature of the decision process whose optimal solution is- pursued.

A most typical approach (see, for example, Balinski, 1961; Ha.kimi, 1964; Revelle and Swain, 1970; Leonardi, 1978; Laporte, 1987>" consists in assuming the study area divided into zones with the demand in each.z9ne concentrated in one of its central points and a number of potential facility"location sites that may coincide or note with the demand points. The optiJnization is accomplished

by

finding out which potential location sites offer greatest benefit according to the criteria assumed and the constraints placed to bind the feasible region. In this way, no consideration is given to existing facilities and an optimal set of new facilities

would be provided. _/

In real life the scenario above is, however, very seldom found in the educational facility location context. It is uncommon tha.t no facility of the type for which the optimization procedure is set up will exist in the study area. Th&-location of educational facilities is often confined to marginal improvements to an existing

s~tuation, that may consist in opening some new facilities, closing a few ot~ers, increasing or decreasing capacities by transforming a.vailable spaces or adapting the facilities to different uses.

Allother aspect that has started receiving attention (Laporte and Nobert, 1981; Branco and Coelho, 1986) is the joint location of facilities and routing of users (or goods, in different settings), The routing/location problem has a specia.l relevance in low density regions where the attraction area for a facility can be

quite extensive. Important savings may be obtained. by having fewer facilities and scho?l buses collecting facility users if a joint optimization is accomplished.

In the educational facility location modelling, we note that four main objectives are pursued:

1. Providing education services at full population coverage, this means ascribing a school to every student ha.ving in account some maximunl

distance or travel time constraints;

2. Minimization of overall costs - these include set up costs for new facilities.

location dependant operation an~ transp~rtation costs for· open faci~ities

and costs assigned to closing downs;

3.

Maximization of users benefit - a measure or-user benefit usually related

to a.ccessibility and therefore dependant on the location of facilities.

4. Political objectives - these are often ill-defined but. for exa.mple,

ma.Y

take the form of maximization of the voting population in t.he communities where new school are built or upgraded.

In addition. the constraints bindi.ng the optimization search are the balance 'between demand and supply for educational services, maximum distance or travel

time.imputed to users. budget and eventually some extra planning- constraints,

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2. MODELS

We carry on a brief review of optimization combinatorial models that fit to

the location of educational facilities and point out som.e adjustments that make them

more adequate to the peculiarities of the real 'World problems in the educational

field that have been mentioned previously. the review wi11 cover the following· models:

A. Simple Facility Location Mode~

B. P-Median

C. Capacitated Facility Location Model

D. Surplus fviaximization Capacitated Facility Location Model E. Covering Location Model

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.F. Maximal Covering Location Model

G. Hamiltonean P.-M~dian Model

H. Routing Location !rfodels

The notation adopted in the formulation of tho models is as follows:

I - set of indices of zones

l

J - set of indices of potential facility sites

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Jo - set of indices of facility location sites fixed' closed' J1 - set.of indices of facility location sites fixed lopen'

Jd - set of indices of facility location sites subject to the decision process of openlclose

p _{- total number of users in the study area.}

- number of users in zone i

Pi

Yj - boolean variable assigned to facility j

Yj - is equal to one if site j is used to locate a facility and zero otherwise

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y - vector for location variabb~~Yj. j e J

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Yo - vector of location vadablesYj. j e 10

Yl - vector of locatio~ variables Yj. j e

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Yd - vector of location variables Yj.

i

e Jd

lj . -minimum size of the facility at site j

Uj - maxinlum capacity o~ the facility at site j

tij - flow of users from zone i to facility j

Xij -proportion of users from zone i assigned to facility j

Cij -'generalized cost of assigning a user of zo~e i to a. facility j

fj - set up cost for facility j

fOj - cost of closing down facility j

flj - cost of opening or maintaining open facility j

A,B -matrices of coefficients of planning constraints

b - vecl!>r rhs of planning constraints .'

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tii - apriori probabi.1ity that users from zone i select facility j

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f

_I

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f

1 1

t

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A - SIMPLE FACILITY LOCAtION MODEL

The simple (uncapacitated) facility location model was put forward by

Balinski (1961) and received many contributions until a very efficient dual based

solution method was proposed by Bilde and Kra{'up (977) and Erlenkotler (1978).

The sim.ple facility location model is formulated as foHows:

(A.I) _{lti~ ty Z}

_-2:

_..

cr

_{J J . J tr} +

L

f· y'_J

, lJ J

subject to

(A.2) _{~tij = Pi} ieI

J

(A,3) _{t·· IJ} ~ p' y' _{1 J} i&l,ieJ

(AA) too> _{1J ­} 0

(A.) _Yj=O,l j . j

In this formulation, constraint (A.2) indicates that aU users are assigned to a.

facility and constraint (A,3) prevents users to be allocated to a closed fa.cility. The objective is theminimizatlon of s~t up costs for ilopen facilities" and the overall

location dependant costs of serving users at facilities. An alternative formulation

involving variables Xij defined as

too

I'-'~~

1) - Pi

and cost parameters-elij = eij Pi consists of: (A.6)

~

_i~

_Z

_=-

_2:

_{.. eli' xi' J J •} +

L

f· _J y'_J

xy_{' I J J}

subject t.o

(A.7) _{~ Xij = 1 i.I}

J

(A.S) _{Xij :S Yj iaI,jaJ}

(A.10) _Yj:O,1 jeJ

Since no capacity constraint are considered in the formulations above it is

easy to conclude that variables Iij are 'at optima~ity either zero or one, and

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therefore users in the same zone are assigned to a single facility,

A step· forward is achieved consid(!ri'l1g the vector'y

=

(Yl,YO,Yd~ and

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introducing the· costs for closing down facilities. The model is than" formulated as follpws:

(A.1I) Min Z

=!

c'r xi' +

2:

ft, Y' +

2:

fO' (I-y')

(x,yd) ij ) .) j ) J j J )

subject to constraints (A.7) - (A.9) and

(A.12) _{Yj=O,1 j} &

Jd .

The first terr.o. in the objective function denotes the locatio!l dependant

opera.t~on and t~ansportation costs, the second term the 'Set. uP/costs for new

.\

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B - P-!\1EDIAN

The concept of p-medjan on 3. graph was introduced by Hakjmi (1964) and

. since then the p-median model for facility location has attracted considerable

attention. In this modol, the nUlnbcr of facilities is fh~ed to p and tota.~ overaH costs

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are minimized. ~ concise formulation is obt:"ined if it is considered

J

I and

assi~ned Xjj = 1 jf a facility is located at site j. The p-median model is formulated as

follows: (B.I) Mif Z:

L

_{. .} c',· _{IJ lJ} X" +

L

_{• J} f· X" _{JJ .} (x _{1J J '} (B.2) is! , 1 /.

J

(B.3) X_{IJ. JJ} ··Sx·· 16 ',J 6 (BA) ~Ijj :: p' J (B.5) _Xjj

_·OJ

_jeJ

As above, constraint (B.2) ensures that ali users cells ars allocated to a

facility and (B.3) prevents allocation to closed fa.c~lities, while (B.4) fixes the number of facilities to p. If it is assumed that facilities may also be shut down, then (B.1) is replaced by

(B.l)

A recent study of heuristic and exact methods to solve the p-median Ju·oblem

C - CAPACITATED'FACILITY LOCATION l\10DEL

A natural extension of the simple facility locatjon 'm~del is achiev"ed by ·t

c9nsidering constraints regarding the min!IDUm size and maximum capacity of

facilities. Adopting again the flow v3!iables. (tij) extensively used in th~

geographical and transportation fields, the model

is

written down

as:

(C.1) _{ty i~ Z}

₌

_2:

_{.. ci' t1'} +

l:

fl' Y' +

2:

fO· (l-y·)

J J • J J • J J

n

, 1J J 1 subject to (C.2) _{~tij =Pi} iaI J (C.3) _{l j Yj :S} ~_{tij :$ Uj Yj} jaJ , ! 1 / / (CA) (C.S) iel,jeJ

where (C.2) defines the demand for the facilities, (C.3) provides minimum size an.d

maximum capacity bounds and (C.4) stands for any additional planning const.raints. This model was put forward by Balinski (1961) and received contributions,

among others, by Kuehn and Hamburger (1963), Feldman, Lehrer and Ray (1966),

Sa. (1969), Akinc and Khumawala (1977), Geoffrion and Macbride (1978) and Van

Roy (1986). This last author has proposed a very efficient algorithm based on a

cross decomposition technique.

At . this stage, we note that embedded in the simple facility location, p­

-media.n and capacitated facility location models is the linear transportation modeL

The behavioral assumption implicit there is that users select facility according t.o

the minimum cost criterion. This is adequate when decision are taken centra.llyas in the transportation of goods. However, when individual users take their own decision on the facility where _{. .} to _. go, some level of dispersion must. be considered, _

since users may have many different reasons for deviating from the minimum cost

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•

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•

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•• ?

•

LINEAR TAANS-P-.--,~..

MOD~L

)

,

,

O.K.

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Fig, I - Facility location models wUh embedded linear ./ transportation su b-models

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D - SURPLUS rv1AXIMIZATION CAPACITATED FACILITY LOCATION MODEL

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In this model proposed by Coelho (1980) the users' dispersion of preferences

is incorporate~ by a gravit.y type transportation. sub-mode1. The model is, as follows:

. 1

tr

. .

' .

(D.1) Ma~ _{S :: - -}

_L

_ti' (10g .~ -1) -

2:

ci' ti' -

L

.f

1.' y' -

L

fO' (1-Y' )

(t Y'J

p..

J fiiJ " J J • J J . J J ' I J IJ J J 'subject to '(D.2) _{~tij :: Pi} i.I J (D,3) . _{lj Yj :S} ~ _{tij :S Uj Yj} j E

J

1 " (D.4) At+By~b "'" / .\ I (D.) _{Yj =}

_{0.1 .}

_j

a

_J

It may be shown that tho first two terms in the objective function are a m£!8SUre of consumers' surplus associated to the gravity spatial demand model (W Hson et al (1981), Coelho (1983», The objective function (D.1) is therefore an aggregate

measure of Iocationa1 surplus. Numerical experience is provided by Erlenkotter and

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E - COVERING LOCATION MODEL

The aim in this model is to locate- facilities in such a way that every demand

zone is 'covered'. for exa.mple. at a maximum distance. time" or genera.tized transportation cost. The model is written as follows: ..

(E.1) ~i(l Z=

2:

fr y' +

2:

fO' (l-y')

YJ . J J • J J

~

J J

subjE!ci to

(E.2) "t"a··y·t _{.f' IJ J} 1 iaI

J

(E,3) _Yj =0 or 1 jeJ

whore

,/

.. _

{I

if zone i is 'covered' by facility j

/

a_{IJ - 0}

othetwis,e

The covering lpcation model was put forward by Toregas et al (1971), It is a

particular application of the set-covering problem, which has received very_{. . .} substantial attention in the literature (see, for example, Pierce (1968), Balas and

Padberg (1972), Christofides and Hey (1978), Almeida, Paixtlo and Coelho. (1982), A very efficient algorithm for large scale set-covering problems is given by Paix!o

P - MAXIlViAL COVERING LOCATION 1,10DEL

Assume that from budgetary constraints or o~et's, it is not possible to' have

more than p facilities which wHl not provide full population coverage. Then, the

location of facilities that ensures nlaxjm~l popu,ation covering is given by th~

maximal covering location model, proposed by Church and Revelle

t

1974), which is

fot'Jnulated, as follows: (F.l ) subjecllo (F.2) . _{~ai; Yj}

_c:

_zi iel J (F.3) _{~Yj -p} J /

(F.4) _{Yj = 0,1 and zi ~}

_O'f

_i.I,j-J

where zi is a boolean variable that is equal to one if zone i is 'covered' by at least

on~ facility. The objective function corresponds to the ma,ximization of the

p~pulation covered by the facility set. Constraints (F.2)-ensures that zi is equ,a.l to zero if no facility covers Zone i and constraints (F.3) fixes at p the number of

facilities.

I

I

f

!

I

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_,

{

i

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G - HAMILTONEAN P-fv1EDIAN MODEL

The Hamiltonean p-Median problem (HPMP) is a mixed location and routing approach' pronosed by Branco and Coelho (1934 and 1986) which embeds a joint

routin g and location optimization.

It is clear that in school facilit.y location it is essential to take inw account simultaneously the 10cational aspects of the potential ,facility sites and the school

buses routing.

Let N denote a set of potential facility sites and demand points. The

Hamiltonean p-median problem consists in finding out p-Hamiltonean' circuits

serving all users in such a 'Way that every user is assigned to a. facilit.r in. the saIne

circuit. This model is therefore an .extension of' the p-median and travelling salesman problems.

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. \ Several formulations for the HPMP may be considered' (see Branco and

Coelho (1984» which may be.explored for different solution methods. In particular.

the.HPMP can be formulated as follows:

(G.1) _{tii~ Z}

_{-2: 2:}

cr

_xfk I,y k ij J J subje~t to (G.2) _LYik

₌

₁ k . (G.3)

_2:

_{Yik ~ 1} i (6.4)

+

_{Xijk =Yjk} 1 (G.5) _{~Xijk -Yik} J (G.6)

_2:

_{Xijk:S ISI-1} i,jeS where

t

t

I

i

iel

teK

ieI,keK i.l,keK V SC R k : lSI C: 1 teK

I if demand' point i bel~ngs to circuit k

Yik;.{

0

_otherwise

I if demand point' i precedes j in circuit k

xiJ'k ={ 0 _otherwise

the cardinality of set K is p and Rk ;. (i: Yik ;.'1) is the setofve'rtices in circuit

k ..

Constraints (6.2) ensures that each demand point is assigned to a circuit,

. (G.3) defines p circuits, (GA) and (G.~) together wit.h· (G.2) establishes that every

demand points belongs to just one ~jrcuit and (G.6) prevents· the existence of

'subcircuits in Rk.

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H - ROUTING - LOCATION

~10DELS

A more general setting regarding joint..routing and location modelling than

the one con'sidcred in the HamHtonean p-me9iah problem, would "include multiple

routes serving the facilities. The optinljzation prQcess must obviously ~ave into

account., in t.his context, the location costs 8S$igne,d to facilities, and the transport.

cost associated with the routes that are designed to collect,users. This fram~work is

. depicted in figure 2 diagrammatically.

*

demand point

o

facility site _ arc in a route

Fig. 2 - Routing-Location Diagram

A recent survey of routing location problems is given by Laporte (1987), A particular routing location model for just one facility and m routes is studied by

Laporte and Norbert (1981). It corresponds 00 the following formulation:

(H.l) _{i~ z ..}

_l:

_{cj' uj' +}

_l:

_{f· y'} .. J J . J J

rx

xy_{' 1 < J J} subject. to (H.2) . _~Yj

₌

₁ J

18

I

(H.3) _~ur +

2:

u'k = 2+ 2(m-l) y' jeJ •• J k' J J 1<J . >J I

t

f

(RAJ _~.Uij:5ISI-1 + (m-l)'.LYj,

vs

C I: ISlt2

t

lq JeS

i,jeS _J

. t

.

~here Uij =0,1,2 is the number of times a~ arc (undirected) is used in a route.

Constraint (R.2) fixes the number of faciliHes to op.e, (R.3) states that the degree of

a mode is 2m if it corresponds to a open facility and 2 otherwise, while (HA)

prevents illegal subtours.

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The implementation of the modelling approach depends on the nature of the

educationa.l facilities being located, availa.bility of school transport, whether

schools shut downs are considered or not, type. of constraints to be taken into consideration and objectives assumed to drive the search of optimal policies.

A common strategy may however be identified in several applic.ations consisting of:

1. Definition of the study area - this may be at a national, regipnai or local scale depending on the facilities being located, attraction areas a.nd degree .of interdependence between altern~tive facilities;

2. Retrieving from database the information required to run the m.odel or _, models according to the constraints a.nd objectives set up; /'

3. Optimization phase consisting in exploring several algorithms and . optimization procedures in relation with the criteria and constraints 'considered;

.f. Analysis by a planning team of the results generated in the optimization phase to ensure adequacy to reality and add considerations that the optimization procedure has been unable to embrace;

5. Prepa.ring and documenting consistent proposals to submit to political and financial approval;

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The data collecting in our particular experience at the pla.nning bureau of the M:inistry o! Education is based upon a relational data base .management system which allows storing retrieving, querying and protecting data in a users friendly

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_. environment.

Clusters of information regarding educational facilities, students, teachers . and other technical administrative staff and educational statistics are considered,

The cluster of educational facilities includes information .for every school, such· as name, address, postal code, telephone, school type and level, availability of special facilities and a unique index found on the geogra.phical location of the schoo1. Detailed information on buildings and grounds is also stored, Th.e cluster on students includes information on registration, by age groups, ...,classes and

succes~/fai1ure p~rformances from kinderga.rden tb high schoo1./The information

is registered by school unit and since this is referred to a geographical based index. a very fine spatial location grid'of students by age groups is produced. The cluster on teachers and other staff includes detailed. information' on academic ... degrees, previous school related experience and other relevant information for imputation of ·costs. Finally, the cluster on educational statistics contains information on rates of approval and failure by grade, rates of premature school leaving, average size of classe$, average number of students by teacher, etc.,. and many demographic and social indicators needed for school population forecasting.

Loading information into the data.b~e is a huge task demanding careful planning a.nd long term persistance. A project involving the survey of schools, buildings, spaces and equipments assigned to educational activities at a national scale has been set up. Routine surveys of school population have been improved to

feed the database. Data from the National Institute of Statitistics was transfered and substantial in house information previously dispersed was gathered and copied or inputed into the database.

A menu driven query system has been implemented, as well as procedures for validating and listing data and (tefining ,users protections. This overall data ­ management system intends to support planning in the Ministry of Education and provide data consistency to the modelHng efforts previously described.

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FACl).JTIES DATA SCHOOL POPULATION RELATIONAL DATA BASE (RDBNMS) .\

Fig. 3 - Relational Data Base Structure

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OVEPCQ~JII\!G

IITR/\PS'·'

~he size of the tasks underpinn'ing the optimization of school facilities at a

n~tiona.1 sc~.lo, and the l~r[',e number of bureaucrats that aU hie orc;anizations tend to attract, for which the Portuguese :Ministry of Education is no exception, creates an environment propitious to resistance to changes and growing fears of transfer· of power .

.In order to overcome this difficulty it is essential providing careful planning, conveying a substantial amount of effort explaining the usefulness of the approach and to show results fast.

A set of recommendations derived from our experience may be put forward: 1. Try to keep as many tasks as possible with those previously acquainted

~ ,;'

with them; /

2. Whenever possible. _. to decentralize do not hesitate in doing it; _,

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3. Engage in making database users feel part of the project;

... Stimulate users from other departments.to be~efit from the stock of information that the database has made available;

5. Provide facilities for querying the da.tabase in a users frierJ.dly environment:

6. Keep the computer system dedicated exclusively to database management in order to ensure short response time;

7. Sea.rch optimality but do not become slave of it - the uncertainty in data items such as costs and demographic forecasts and the existence of qualitative and political components that the models are unable to absorb. advocates precaution in proclaiming optimality;

8. Ma.ke all optimal model solutions pass through the sieve. of common sense

and the expertise of planners of different disciplines such as architects, urban planners, geographers and teachers;

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9. Stand the project as a glob'll integrated one and not as a project of the small team that has eventually had the opoortunity of starting it. o . . .

6.

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ALMEIDA, M.T., PAIXAO, J.P. and COELHO,

J.D.

(1982), "Ap1ic~Oes dos Problemas de

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BALINSKI, M.L. (1961), "Fixed Cost Transportation Problems", Naval Research

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.

_/

.\ BILDE,·O. and KRARUP,

J.

(1977). "Sharp Lower Bounds and Efficient Algorithms for

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BRANCO, I.M. and COELHO, J.D. (1984.), . "Formul~Oes Matem.lUicas . da p-Media1).& .

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Nota

nl

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25

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.

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.

/

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'\

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I

LEONARDI, G. (1978), "Optimum FacHity Location by Accessibility Maximizing", 1

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,

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. .

PAIXAO, J.P. (1983), "Algorithms for La,rge Scale Set Covering Problems", Ph.D.

.

. '

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lB.

(1968), ttApplication of Combinatorial Programming to

a

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