Management
Science 461
Lecture 3 – Covering Models
September 23, 2008
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?

2
Defining Coverage

Geographic distance
 Euclidean


Time metric
Network distance
 Shortest

or rectilinear – distance metrics
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…
3
Network example
14
A
B
13
10
C
E
17
23
16
12
D
4
Network example

If coverage distance is 15 km, a facility at
node A covers which nodes?
A
14
B
13
10
C
E
17
23
16
12
D
5
Example Network (cont.)

When D = 22km, what is the coverage set
of node A?
A
14
B
13
10
C
E
17
23
16
12
D
6
Algebraic formulation

Assume cost of locating is the same for each facility
(again – possible HW / midterm relaxation)
1 if we locate at candidate node j
Xj 
0 if not


The objective function becomes …
(Set of facility locations – J; set of customers – I)
7
Example – D = 15
14
A
10
B
23
16
12
XA
XB
XC 
XD
E
17
C
min
13
D
XE
8
Example – D = 15
14
A
10
B
23
16
12
s.t.
XA
XA
XB
XB
XC 
XC
XD
E
17
C
min
13
D
XE
1
9
Example – D = 15
14
A
10
B
23
16
12
s.t.
XA
XA
XB
XB
XA 
XB
XC 
XC
XD
E
17
C
min
13
D
XE
1
XE 1
10
14
A
Complete Model
10
B
23
16
12
s.t.
XA
XA
XB
XB
XA 
XB
XA
X A,
XC 
XC,
D
XE
1
XC
XC 
XC 
XB
X B,
XD
E
17
C
min
13
XD
XE 1
1
XD
1
X D,
XE 1
X E  0,1
11
Algebraic formulation

More generally, we can define
1 if demand node i is within D units of facility j
aij  
0 if not


The value of aij does not change for a given
model run.
We can include cost of opening a facility
12
General Formulation
min
c X
a X
jJ
s.t.
jJ
j
ij
Cost of covering
all nodes
j
j
 1 i  I
X j  0, 1
j  J
Each node
covered
Integrality
13
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
14
The Maximal Covering Problem
250
100
A
14
B
13
150
10
C
200
E
17
23
16
12
Demand
D
125
15
Max Covering Solution for P=1
250
100
A
14
B
13
150
10
23
200
C
E
17
16
12
Demand
D
Locate at __
which covers
nodes ___ for
a total covered
demand of
___ .
Distance
coverage: 15
Km
125
16
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

17
Max Cover Formulation
(D=15)
A
14
10
B
23
16
12
100 Z A  250 Z B  200 Z C  125 Z D  150 Z E
s.t.
XA
 XB
XA
 XB
 XE
 XC
 XC
XA
 ZA
 XC
 XD
 XD
 XB
XA
XB
 XC
all variables 0 or 1
XD
E
17
C
Max
13
D
Total covered demand
Linkage constraints
 ZB
 ZC
 ZD
 XE
 ZE
XE
P
Locate P sites
Integrality
18
Max Covering Formulation
Max
h Z
iI
s.t.
i
Covered demands
i
Z i   aij X j i  I
Node i not covered
unless we locate at
a node covering it
jJ
X
jJ
j
P
X j  0 ,1
Z i  0 ,1
Locate P sites
j  J
i  I
Integrality
19
Max Covering – Typical Results
150 cities
Dc= 250
Typical Max Covering Results
120.00%
% Coverage
100.00%
Decreasing
marginal
coverage
80.00%
60.00%
40.00%
20.00%
0.00%
0
5
10
15
20
Number of Facilities
~ 90% coverage with ~ 50% of facilities
25
Last few
facilities cover
relatively little
demand
20
Problem Extensions

The Max Expected Covering Problem
 Facility
subject to congestion or being busy
 Application: in locating ambulances, we need
to know that one of the nearby ambulances is
available when we call for service

Scenario planning
 Data
shifts (over time, cycles, etc) force
multiple data sets – solve at once
21