Management Science 461

Lecture 3 – Covering ModelsSeptember 23, 2008

2

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?

3

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…

4

Network example14

A

E

D

C

B

10

13

12

1723

16

5

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

16

6

Example Network (cont.)

When D = 22km, what is the coverage set of node A?

14A

E

D

C

B

10

13

12

1723

16

7

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

8

Example – D = 15

XXXXX EDCBA min

14A

E

DC

B

10

13

12

1723

16

9

Example – D = 15

1s.t.min

XXXXXXXX

CBA

EDCBA

14A

E

DC

B

10

13

12

1723

16

10

Example – D = 15

11s.t.

min

XXXXXX

XXXXX

EBA

CBA

EDCBA

14A

E

DC

B

10

13

12

1723

16

11

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

12

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

13

General Formulation

JjX

IiXa

Xc

j

Jjjij

Jjjj

1,0

1s.t.

min Cost of covering all nodesEach node covered

Integrality

14

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

15

The Maximal Covering Problem

14A

E

D

C

B

10

13

12

1723

16

100

200125

150

250

Demand

16

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

17

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

18

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

19

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

20

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

21

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