Mathematics, Operations Research and Statistics
MOSI group-VUB
On a Multiperiod School Location Problem Using
Interaction Between CPLEX and GIS
Eric Delmelle
December 10 2012
University of North Carolina, Charlotte, U.S.A.
Center for Operations Research and Econometrics (CORE), Belgium
Delmelle
School Location
1 / 28
Introduction
Motivation and background
Motivation
1
Rapidly expanding urban areas are dynamic environments
• Need to expand existing network of public facilities to meet
anticipated increase or decrease in demand
• schools, libraries, emergency services
2
Closing existing facilities in areas characterized by population
decline
• Decision making process → political decisions (bad press)
Delmelle
School Location
2 / 28
Introduction
Motivation and background
Motivation
1
Rapidly expanding urban areas are dynamic environments
• Need to expand existing network of public facilities to meet
anticipated increase or decrease in demand
• schools, libraries, emergency services
2
Closing existing facilities in areas characterized by population
decline
• Decision making process → political decisions (bad press)
Delmelle
School Location
2 / 28
Introduction
Motivation and background
Motivation
1
Rapidly expanding urban areas are dynamic environments
• Need to expand existing network of public facilities to meet
anticipated increase or decrease in demand
• schools, libraries, emergency services
2
Closing existing facilities in areas characterized by population
decline
• Decision making process → political decisions (bad press)
Delmelle
School Location
2 / 28
Introduction
Motivation and background
Background
1
Location models→ tools for regional and urban planners and
decision makers
• Focus is on multi-period school location planning
• Explicit temporal component
• Each student must be assigned to a school
• Budget determines number of schools (p-median)
2
Flexible capacity enabled by modular units
3
School as age investment: age constraint
Delmelle
School Location
3 / 28
Introduction
Motivation and background
Background
1
Location models→ tools for regional and urban planners and
decision makers
• Focus is on multi-period school location planning
• Explicit temporal component
• Each student must be assigned to a school
• Budget determines number of schools (p-median)
2
Flexible capacity enabled by modular units
3
School as age investment: age constraint
Delmelle
School Location
3 / 28
Introduction
Motivation and background
Background
1
Location models→ tools for regional and urban planners and
decision makers
• Focus is on multi-period school location planning
• Explicit temporal component
• Each student must be assigned to a school
• Budget determines number of schools (p-median)
2
Flexible capacity enabled by modular units
3
School as age investment: age constraint
Delmelle
School Location
3 / 28
Introduction
Motivation and background
Background
1
Location models→ tools for regional and urban planners and
decision makers
• Focus is on multi-period school location planning
• Explicit temporal component
• Each student must be assigned to a school
• Budget determines number of schools (p-median)
2
Flexible capacity enabled by modular units
3
School as age investment: age constraint
Delmelle
School Location
3 / 28
Introduction
Motivation and background
Existing objectives in the literature
• General objectives in public facility location models:
1
minimize total travel distance to the facility j (Hakimi 1964)
2
minimizing necessary infrastructure while keeping a certain level of
coverage (Toregas et al. 1971)
3
Models should address capacity constraints (CFLP)
• Mirchandani and Francis (1990) and Sridharan (1995).
Delmelle
School Location
4 / 28
Introduction
Motivation and background
Existing objectives in the literature
• General objectives in public facility location models:
1
minimize total travel distance to the facility j (Hakimi 1964)
2
minimizing necessary infrastructure while keeping a certain level of
coverage (Toregas et al. 1971)
3
Models should address capacity constraints (CFLP)
• Mirchandani and Francis (1990) and Sridharan (1995).
Delmelle
School Location
4 / 28
Introduction
Motivation and background
Existing objectives in the literature
• General objectives in public facility location models:
1
minimize total travel distance to the facility j (Hakimi 1964)
2
minimizing necessary infrastructure while keeping a certain level of
coverage (Toregas et al. 1971)
3
Models should address capacity constraints (CFLP)
• Mirchandani and Francis (1990) and Sridharan (1995).
Delmelle
School Location
4 / 28
Introduction
Motivation and background
Existing objectives in the literature
• General objectives in public facility location models:
1
minimize total travel distance to the facility j (Hakimi 1964)
2
minimizing necessary infrastructure while keeping a certain level of
coverage (Toregas et al. 1971)
3
Models should address capacity constraints (CFLP)
• Mirchandani and Francis (1990) and Sridharan (1995).
Delmelle
School Location
4 / 28
Introduction
Motivation and background
Existing objectives in the literature
1
Dynamic models have received a lot of attention:
• Fong and Srinivasan 1981, Van Roy and Erlenkotter 1982 and Alberada et al. 2009
• Drezner (1995b) suggests a progressive p-median (no relocation of facilities)
2
Significant body of literature in school location
• Dost 1968, Maxfield 1972, Viegas 1987, Greenleaf and Harrison 1987 as well as
Church and Murray 1994
• Antunes and Peters (2000, 2001) → modified capacitated median
• Handling opening of new facilities, expansion, reduction
• Age of facility, number of time periods
• Uncertainty during time periods
• Distance penalty beyond threshold
• Different growth scenarios
Delmelle
School Location
5 / 28
Introduction
Motivation and background
Existing objectives in the literature
1
Dynamic models have received a lot of attention:
• Fong and Srinivasan 1981, Van Roy and Erlenkotter 1982 and Alberada et al. 2009
• Drezner (1995b) suggests a progressive p-median (no relocation of facilities)
2
Significant body of literature in school location
• Dost 1968, Maxfield 1972, Viegas 1987, Greenleaf and Harrison 1987 as well as
Church and Murray 1994
• Antunes and Peters (2000, 2001) → modified capacitated median
• Handling opening of new facilities, expansion, reduction
• Age of facility, number of time periods
• Uncertainty during time periods
• Distance penalty beyond threshold
• Different growth scenarios
Delmelle
School Location
5 / 28
Introduction
Research Objectives
Research Objectives
1
Number of open facilities p is increased or decreased at each time
period or is controlled by cost function.
2
Each facility has a pair of minimum and maximum capacity constraints.
• Variation in the maximal capacity through modular equipments.
3
Closing, expansion and reduction of facilities and age of school are
controlled.
Delmelle
School Location
6 / 28
Introduction
Research Objectives
Research Objectives
1
Number of open facilities p is increased or decreased at each time
period or is controlled by cost function.
2
Each facility has a pair of minimum and maximum capacity constraints.
• Variation in the maximal capacity through modular equipments.
3
Closing, expansion and reduction of facilities and age of school are
controlled.
Delmelle
School Location
6 / 28
Introduction
Research Objectives
Research Objectives
1
Number of open facilities p is increased or decreased at each time
period or is controlled by cost function.
2
Each facility has a pair of minimum and maximum capacity constraints.
• Variation in the maximal capacity through modular equipments.
3
Closing, expansion and reduction of facilities and age of school are
controlled.
Delmelle
School Location
6 / 28
ViFCLP - Formulation
Notation: indices, sets and parameters
The Vintage Flexible Capacitated Location Problem
Indices and sets:
cfjm : cost of operating a school j at m
i, I: index and set of demand nodes
csm : marginal cost per student at m.
j, J: index and set of candidate school locations
cNjm : cost to open a new school j at m
m, M: index and set of periods
cEjm : cost to close an existing school j at m.
J N : set of new candidate locations, J N ⊂ J, and J E the set of
Capacities and additional parameters:
existing facilities, J E ⊂ J
z+
jm : max. capacity for facility j at m.
Travel costs:
Kjm : max. number of additional soft units at j during time m.
cijm : travel cost between an individual i and facility j in period
z++ : capacity of a soft unit (e.g. 30 students).
m.
aim : demand at location i at time m
dijm : travel distance between an individual i and facility j in
eE
jm : age of existing facility j at time m
period m.
Ē: minimal age for a facility to close
max
max
dim
: distance threshold; dim
> min dijm .
|{z}
j∈J
wm : discount factor reflecting the importance of period m with
P
m wm = 1.
Facility costs:
γ1,2 : importance given to travel and school operating costs,
c++ : leasing cost of a mobile classroom at m
respectively, with γ1 + γ2 = 1
Delmelle
School Location
7 / 28
ViFCLP - Formulation
Decision variables
The Vintage Flexible Capacitated Location Problem
Xijm

 1 if demand i is assigned to j at time m
=
 0 otherwise
Yjm

 1 if location j is opened at time m
=
 0 otherwise
Bjm = modular units necessary at j at time m
Delmelle
School Location
8 / 28
ViFCLP - Formulation
ViFCLP: objective function
The Vintage Flexible Capacitated Location Problem
Minimize F = (γ1 ∗ f1 ) + (γ2 ∗ f2 )
f1 =
f2 =
"
|
"
X X X
i∈I m∈M
j∈J
{z
#
}
transportation costs
X X
j∈J m∈M
|
X X
cNjm · (Yjm − Yj,m−1 ) +
cfjm · Yjm +
j∈J N m∈M
{z
}
fixed costs
X X
j∈J E m∈M
|
{z
|
opening cost
}
XX X
cEjm · (Yj,m−1 − Yjm +
csm · aim · Xijm +
i∈I j∈J m∈M
{z
closing cost
X X
c++ · Bjm
j∈J m∈M
|
wm · cijm · aim · Xijm
{z
leasing cost
Delmelle
#
}
|
{z
student cost
}
}
School Location
9 / 28
ViFCLP - Formulation
ViFCLP: Assignment, capacity, age, closing constraints
The Vintage Flexible Capacitated Location Problem
X
Xijm = 1
∀i ∈ I, ∀m ∈ M
j∈J
Xijm ≤ Yjm
X
∀i ∈ I, ∀m ∈ M, ∀j ∈ J
+
++
aim · Xijm ≤ (zjm · Yjm ) + (Bjm · z
)
∀j ∈ J, ∀m ∈ M
i∈I
Bjm ≤ A · Yjm
∀j ∈ J, ∀m ∈ M
Bjm ≤ Kjm
∀j ∈ J, ∀m ∈ M
E
E
Ē − ejm ≤ A · Yjm
∀j ∈ J , ∀m ∈ M
N
∀j ∈ J , ∀m ∈ M
Yj,m−1 ≤ Yjm
E
Yjm ≤ Yj,m−1
∀j ∈ J , ∀m ∈ M
Xijm , Yjm ∈ 0, 1
∀i ∈ I, ∀j ∈ J, ∀m ∈ M
+
∀j ∈ J, ∀m ∈ M
Bjm ∈ Z
γ1 , γ2 ∈ {0, 1}
Delmelle
School Location
10 / 28
GIS
Linking GIS to solvers
Generates locations of:
*demand nodes
*facilities
*city centers
Computes:
*distances
Visualization:
*capacities
*assignments (spiders)
ArcGIS
python IDLE
File exchange: calls
reads
LP, results
CPLEX
The semi-loosed coupled system. A python script is built in the IDLE environment, automatically calling Lingo. The exchange of
information is carried out through text files.
(Related literature: Ribeiro et al. 2002, Church 2002, Ghosh 2008, Alexandris and Giannikos 2010, Murray 2010).
Delmelle
School Location
11 / 28
Application: testing on a simulated city
Baseline situation t0
• We simulate a school system in a hypothetical urban area, shaped
as a square of size 100 × 100 (python programming).
1
81 demand nodes are positioned on a quasi-grid with a small
random perturbation (xi + ∆x, yi + ∆y).
2
In base year t0 , we simulate population ∼ U(5, 100)
• Nine potential schools on quasi-grid.
1
School system is optimally organized by solving a capacitated
p-median problem (four schools open).
2
School located the closest to the center of the study region is forced
to be open, specifically Y3,0 = Y4,0 = Y5,0 = Y8,0 = 1.
Delmelle
School Location
12 / 28
Application: testing on a simulated city
Baseline situation t0
• We simulate a school system in a hypothetical urban area, shaped
as a square of size 100 × 100 (python programming).
1
81 demand nodes are positioned on a quasi-grid with a small
random perturbation (xi + ∆x, yi + ∆y).
2
In base year t0 , we simulate population ∼ U(5, 100)
• Nine potential schools on quasi-grid.
1
School system is optimally organized by solving a capacitated
p-median problem (four schools open).
2
School located the closest to the center of the study region is forced
to be open, specifically Y3,0 = Y4,0 = Y5,0 = Y8,0 = 1.
Delmelle
School Location
12 / 28
Application: testing on a simulated city
Baseline situation t0
• We simulate a school system in a hypothetical urban area, shaped
as a square of size 100 × 100 (python programming).
1
81 demand nodes are positioned on a quasi-grid with a small
random perturbation (xi + ∆x, yi + ∆y).
2
In base year t0 , we simulate population ∼ U(5, 100)
• Nine potential schools on quasi-grid.
1
School system is optimally organized by solving a capacitated
p-median problem (four schools open).
2
School located the closest to the center of the study region is forced
to be open, specifically Y3,0 = Y4,0 = Y5,0 = Y8,0 = 1.
Delmelle
School Location
12 / 28
Application: testing on a simulated city
a- location of facilities and age (t0)
EE
j=3
=39
j=3 e
=39
e3,0
3,0
N
N =0
j=6
j=6 e
=0
e6,0
6,0
N
N =0
j=9
j=9 e
=0
e9,0
9,0
ZZ3,0
=1500
=1500
3,0
ZZ6,0
=2000
=2000
6,0
ZZ9,0
=2000
=2000
9,0
++
N
++
E
E =34
j=8
e8,0
j=8 e
=34
8,0
=1500
ZZ5,0
=1500
5,0
=1500
ZZ8,0
=1500
8,0
N
++
E
++
N
N =0
j=1
e1,0
j=1 e
=0
1,0
E =26
j=4
e4,0
j=4 e
=26
4,0
N =0
j=7
e7,0
j=7 e
=0
7,0
=2000
ZZ1,0
=2000
1,0
=1500
ZZ4,0
=1500
4,0
=2000
ZZ7,0
=2000
7,0
++
schools
j=index of school
ej,t =
age of existing
school j at time t
+
Zj,t =
maximum capacity of
school j at time t
E
E
E =26
j=5
e5,0
j=5 e
=26
5,0
=2000
ZZ2,0
=2000
2,0
++
Delmelle
b- capacitated p-median solution (t0)
++
N =0
j=2
e2,0
j=2 e
=0
2,0
++
Solution to baseline situation t0
++
capacity usage
allocation:
1
0.1
existing (E)
candidate (N)
not selected
demand (population #)
1
100
0
School Location
aggregated
demand
25
5
52
90
90
130
200
3878
50
km
13 / 28
Application: testing on a simulated city
Growth Scenarios
• Four distinct scenarios of evolution of the spatial distribution of
school demand are taken to represent four possible growth
patterns of the simulated urban area.
1
Direct and inverse monocentric growth.
2
Direct and inverse polycentric growth.
• School demand is determined following a distance decay/increase
function from the city center.
• The time periods are indexed by M = 1 . . . 3 with a five-year
increment.
min(t0 )
max(t0 )
min(t1 )
max(t1 )
min(t2 )
max(t2 )
min(t3 )
max(t3 )
direct monocentric
5
99
10
321
21
635
21
3878
inverse monocentric
5
99
7
243
13
329
17
582
direct polycentric
5
99
7
203
10
430
11
925
inverse polycentric
5
99
12
222
15
400
19
1661
Case
School Location
Table: Demand characteristics
for the four growth scenarios.
Delmelle
14 / 28
Application: testing on a simulated city
Growth Scenarios
• Four distinct scenarios of evolution of the spatial distribution of
school demand are taken to represent four possible growth
patterns of the simulated urban area.
1
Direct and inverse monocentric growth.
2
Direct and inverse polycentric growth.
• School demand is determined following a distance decay/increase
function from the city center.
• The time periods are indexed by M = 1 . . . 3 with a five-year
increment.
min(t0 )
max(t0 )
min(t1 )
max(t1 )
min(t2 )
max(t2 )
min(t3 )
max(t3 )
direct monocentric
5
99
10
321
21
635
21
3878
inverse monocentric
5
99
7
243
13
329
17
582
direct polycentric
5
99
7
203
10
430
11
925
inverse polycentric
5
99
12
222
15
400
19
1661
Case
School Location
Table: Demand characteristics
for the four growth scenarios.
Delmelle
14 / 28
Application: testing on a simulated city
time 1
time 2
time 3
polycentric inverse
polycentric direct
monocentric increasing
monocentric decreasing
time 0
Spatial variation demand based on growth scenarios
demand node
city center
Delmelle
proximity buffer
(10,20,30km)
demand (population #)
5
52
90
School Location
0
25
50
km
130 200 3878
15 / 28
Application: testing on a simulated city
Spatial variation demand based on growth scenarios
Calibration of cost parameters
Fixed annual cost to operate a school cfjm :
Capacity of a modular unit z++ : 25
$710,000.
students per unit.
Marginal annual cost of a student csm :
Cost of closing an existing school cEjm : $1
$7,200.
(this cost can be negative, that is there is
Cost to build a new school cNjm :
a benefit to close a school).
$3,000,000 for a school of 60 acres.
Maximum number of modular units per
Leasing cost (per unit) c++ : $10,000 per
site Kjm : upper-limit of 100 units.
year, which includes rent, heating,
Minimum age for a school to close Ē: 32
maintenance.
years.
Delmelle
School Location
16 / 28
Application: testing on a simulated city
Spatial variation demand based on growth scenarios
• In the ViFCLP, we resort to inflating travel costs by incorporating a
distance-based penalty to limit the number of such instances,
which we refer to misassignments hereafter.
• Controlled by two parameters:
1
Maximum travel distance dmax which determines the assignment
distance above which an extra travel penalty will occur.
2
β, which regulates the magnitude of this penalty.
• When students are assigned to schools beyond a certain
threshold travel distance, travel costs increase exponentially with
distance.
Delmelle
School Location
17 / 28
Application: testing on a simulated city
Spatial variation demand based on growth scenarios
• In the ViFCLP, we resort to inflating travel costs by incorporating a
distance-based penalty to limit the number of such instances,
which we refer to misassignments hereafter.
• Controlled by two parameters:
1
Maximum travel distance dmax which determines the assignment
distance above which an extra travel penalty will occur.
2
β, which regulates the magnitude of this penalty.
• When students are assigned to schools beyond a certain
threshold travel distance, travel costs increase exponentially with
distance.
Delmelle
School Location
17 / 28
Application: testing on a simulated city
Spatial variation demand based on growth scenarios
• In the ViFCLP, we resort to inflating travel costs by incorporating a
distance-based penalty to limit the number of such instances,
which we refer to misassignments hereafter.
• Controlled by two parameters:
1
Maximum travel distance dmax which determines the assignment
distance above which an extra travel penalty will occur.
2
β, which regulates the magnitude of this penalty.
• When students are assigned to schools beyond a certain
threshold travel distance, travel costs increase exponentially with
distance.
Delmelle
School Location
17 / 28
Application: testing on a simulated city
Estimating travel costs
max
Maximal traveled distance dim
cijm : travel cost between an individual i and facility j in period m; cijm =
(a)
i
cij
{
diB
diA
max
if dijm ≤ dim
max
if dijm > dim
(b)
!
(B
!
(A

 αd
ijm
 αdmax + (d − dmax )β
ijm
im
im
diB - dimax
d imax
!
(B
max
ci
!
(A
i
distance from
diA travel
demand to facility
i demand
d iA d i
!
(A
d iBmax
d ij
facility
Once an individual i is assigned to a facility beyond a travel distance dimax , the impendence increases exponentially with rate β.
Delmelle
School Location
18 / 28
Application: testing on a simulated city
Results
Solution for different growth scenarios
time 1
time 2
time 3
inverse polycentric
direct polycentric
inverse monocentric
direct monocentric
time 0
Delmelle
School Location
19 / 28
Application: testing on a simulated city
Age constraint
Minimal age for school closing
time 2
time 3
60 years age limit
32 years age limit
time 1
Delmelle
School Location
20 / 28
Real-world application
Baseline situation t0
(a)
I-77
21
20
22
1
2
13
18
I-485
11
I-85
14
17
16
3
23
9
I-77
6
4
School
(id-name)
Age
1-Hopewell
2-NorthMeck
3-Harding
4-MyersPark
5-SouthMeck
6-EastMeck
7-Providence
8-Butler
9-Independence
10-Olympic
11-Vance
12-ArdreyKell
13-MallardCreek
14-WestMeck
15-Waddell
16-Garinger
17-WestCharlotte
2001
1951
1992
1951
1959
1950
1989
1997
1967
1966
1997
2001
2007
1951
2001
1959
1938
Remain
18-HucksRoad
19-Palisades
20-Stumpton
21-Bailey
22-Rama
23-RockyRiver
proposed
proposed
proposed
proposed
proposed
proposed
school (utilization)
> 3500 students
< 1500 students
(b)
x
x
x
x
<= 100% physical capacity
> 100% physical capacity
demand
TAZ centroid
student flow
100
50
10
modular units
< 30
31 - 40
41 - 50
I-485
15
10
5
7
school: opened
8
school: proposed
19
interstates
I-77
Mecklenburg
12
0
5
10
km
Delmelle
Schoolhigh
Location
Figure: Location
of existing and potential
schools in the CMS system in 2008 21
in/ 28
Real-world application
Baseline situation t0
t2 -2018
scenario B: ³1 = 0.5, dmax=5km
scenario A: ³1 = 0.1, dmax=5km
t1 -2013
modular units
< 30
31 - 40
41 - 50
Average
travel distance:
7.32km
Average
travel distance:
8.01km
Closest
assignments:
20,698(50.5%)
Closest
assignments:
28,835(61.9%)
Closest
assignments:
28,270(52.1%)
Average
travel distance:
6.68km
Average
travel distance:
6.97km
Average
travel distance:
7.49km
Closest
assignments:
26,405(64.4%)
Closest
assignments:
30,534(65.6%)
Closest
assignments:
30,711(56.5%)
demand
school (utilization)
TAZ centroid
student flow
100
50
t3 - 2023
Average
travel distance:
8.39km
10
> 3500 students
< 1500 students
0
5
10
km
Interstates
<=100% physical capacity
>100% physical capacity
Mecklenburg
note:
A larger proportional symbol for a school denotes a higher
number of students assigned to it. Schools color-coded in dark
orange require additonal soft capacity (modular units), while
schools color-coded in green operate within their physical capacity.
For visualization purposes, only allocation flows with demand over
2 students are displayed. Black dots represent the geographic
location where demand originates.
Delmelle of the ViFCLP for theSchool
Figure: Results
CMSLocation
network of high schools with varying
22 / 28
Real-world application
t2 -2018
scenario D: ³1 = 0.999, dmax=0.1km
scenario C: ³1 = 0.9, dmax=5km
t1 -2013
modular units
< 30
31 - 40
41 - 50
Average
travel distance:
6.61km
Average
travel distance:
6.92km
Closest
assignments:
28,917(70.5%)
Closest
assignments:
33,333(71.6%)
Closest
assignments:
37,520(69.1%)
Average
travel distance:
5.91km
Average
travel distance:
6.04km
Average
travel distance:
6.25km
Closest
assignments:
34,589(84.3%)
Closest
assignments:
38,391(82.4%)
Closest
assignments:
42,327(77.9%)
demand
school (utilization)
TAZ centroid
student flow
100
50
t3 - 2023
Average
travel distance:
6.5km
10
0
> 3500 students
< 1500 students
5
10
km
Interstates
<=100% physical capacity
>100% physical capacity
Mecklenburg
note:
A larger proportional symbol for a school denotes a higher
number of students assigned to it. Schools color-coded in dark
orange require additonal soft capacity (modular units), while
schools color-coded in green operate within their physical capacity.
For visualization purposes, only allocation flows with demand over
2 students are displayed. Black dots represent the geographic
location where demand originates.
Figure: Results of the ViFCLP for the CMS network of high schools with varying
values of γ1 . β is kept constant (β = 2.5) throughout.
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Real-world application
Impact of dmax
(a)
85
(b)
8.5
t1
t2
t3
80
t1
t2
t3
average
average
8
average travel distance (km)
% of closest assignments
75
70
65
7.5
7
6.5
60
6
55
50
0
0.2
0.4
0.6
0.8
1
5.5
0
γ − weight of travel costs
0.2
0.4
0.6
0.8
1
γ − weight of travel costs
1
1
Figure: Proportion of cases of closest assignments with increasing γ1 -values in (a),
and average student travel distance (km) as a function of γ1 in (b).
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Discussion
Discussion
• This paper presents a multi-period location problem
• Capacity, age constraints and modular equipment.
• Model is flexible (time-varying weights for uncertainty)
1 Depending on growth scenario, increase in the school age population
in the center of the study region can either be met with the addition of
soft modular units, or require students to travel a longer distance.
2 Student assignments can be severely impacted by the minimum age
closing constraint.
3 Sharp increase in population in the periphery may require to close
schools in center of the city, generating longer commute for students
living in the center of the city.
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Discussion
Future research
1
Euclidean, Manhattan distances.
2
Population growth (decline), and uncertainty of the demand.
3
Fractional demand (split).
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Discussion
Future research
1
Euclidean, Manhattan distances.
2
Population growth (decline), and uncertainty of the demand.
3
Fractional demand (split).
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Discussion
questions . . . comments . . . concerns . . . suggestions
[email protected]
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