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Management Science 461
Lecture 3 – Covering ModelsSeptember 23, 2008
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Covering Models
We want to locate facilities within a certain distance of customers
Each facility has positive cost, so we need to cover with minimum # of facilities
Easy “upper bound” for these problems. What is it?
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Defining Coverage Geographic distance
Euclidean or rectilinear – distance metrics Time metric Network distance
Shortest Paths Coverage is usually binary: either node i is
covered by node j or it isn’t A potential midterm question would be to relax this
assumption…
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Network example14
A
E
D
C
B
10
13
12
1723
16
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Network example
If coverage distance is 15 km, a facility at node A covers which nodes?
14A
E
D
C
B
10
13
12
1723
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Example Network (cont.)
When D = 22km, what is the coverage set of node A?
14A
E
D
C
B
10
13
12
1723
16
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Algebraic formulation
Assume cost of locating is the same for each facility (again – possible HW / midterm relaxation)
The objective function becomes … (Set of facility locations – J; set of customers – I)
not if0
node candidateat locate weif1 jX j
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Example – D = 15
XXXXX EDCBA min
14A
E
DC
B
10
13
12
1723
16
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Example – D = 15
1s.t.min
XXXXXXXX
CBA
EDCBA
14A
E
DC
B
10
13
12
1723
16
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Example – D = 15
11s.t.
min
XXXXXX
XXXXX
EBA
CBA
EDCBA
14A
E
DC
B
10
13
12
1723
16
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Complete Model
1,0,,,,11111s.t.
min
XXXXXXX
XXXXX
XXXXXX
XXXXX
EDCBA
EB
DC
DCA
EBA
CBA
EDCBA
14A
E
DC
B
10
13
12
1723
16
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Algebraic formulation More generally, we can define
The value of aij does not change for a given model run.
We can include cost of opening a facility
not if 0
facility of units D within is node demand if 1 jiaij
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General Formulation
JjX
IiXa
Xc
j
Jjjij
Jjjj
1,0
1s.t.
min Cost of covering all nodesEach node covered
Integrality
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The Maximal Covering Problem
Locate P facilities to maximize total demand covered; full coverage not required
Extensions:Can we use less than P facilities?Each facility can have a fixed cost
Main decision variable remains whether to locate at node j or not
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The Maximal Covering Problem
14A
E
D
C
B
10
13
12
1723
16
100
200125
150
250
Demand
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Max Covering Solution for P=1
Locate at __ which covers nodes ___ for a total covered demand of ___ .
Distance coverage: 15 Km
14A
E
D
C
B
10
13
12
1723
16
100
200125
150
250
Demand
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Modeling Max Cover
If we use a similar model to set cover, we might double- and triple-count coverage.
To avoid this and still keep linearity, we need another set of binary variables
Zi = 1 if node i is covered, 0 if not Linking constraints needed to restrict the
model
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Max Cover Formulation (D=15)
1or 0 variablesall
s.t.
150125200250100Max
PXXXXX
ZXXZXXZXXXZXXXZXXX
ZZZZZ
EDCBA
EEB
DDC
CDCA
BEBA
ACBA
EDCBA
Total covered demand
Linkage constraints
Locate P sites
Integrality
14A
E
DC
B
1013
12
172316
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Max Covering Formulation
Ii,Z
Jj ,X
PX
IiXaZ
Zh
i
j
Jjj
Jjjiji
Iiii
10
10
s.t.
Max Covered demands
Node i not covered unless we locate at a node covering it
Locate P sites
Integrality
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Max Covering – Typical Results
Typical Max Covering Results
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
120.00%
0 5 10 15 20 25
Number of Facilities
% C
over
age
150 citiesDc= 250
Decreasing marginal coverage
Last few facilities cover relatively little demand~ 90% coverage with ~ 50% of facilities
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Problem Extensions
The Max Expected Covering ProblemFacility subject to congestion or being busyApplication: in locating ambulances, we need
to know that one of the nearby ambulances is available when we call for service
Scenario planningData shifts (over time, cycles, etc) force
multiple data sets – solve at once