p-Median Cluster Analysis Based on General-Purpose Solvers (1)

p-Median Cluster Analysis Based

on General-Purpose Solvers

Boris Goldengorin, Dmitry Krushinsky

University of Groningen, The Netherlands

Joint work with

Bader F. Albdaiwi and Viktor Kuzmenko

22

Outline of the talk

• Two Main PMP formulations

• Pseudo-Boolean polynomial

• Mixed Boolean pseudo-Boolean Model

(MBpBM)

• Experimental results

• Concluding Remarks

• Directions for Future Research

33

The p-Median Problem (PMP)

I = {1,…,m} – a set of m facilities (location points),

J = {1,…,n} – a set of n users (clients, customers or demand points)

C = [cij] – a m×n matrix with distances (measures of similarities or

dissimilarities) travelled (costs incurred)

nmnjn

iniji

nj

ccc

ccc

ccc

.........

..................

.........

..................

.........

1

1

1111

clients

location p

oin

ts

- location point (cluster center)

- Client (cluster points)

Costs Matrix

44

The PMP: combinatorial formulation

The p-Median Problem (PMP) consists of determining p locations

(the median points) such that 1 ≤ p ≤ m and the sum of distances (or

transportation costs) over all clients is minimal.

- location point

- client

- opened facility

p = 3

1 mp

co

mp

lex

ity

)!(!

!

pmp

mC p

m

55

The PMP: combinatorial formulation

JjpI, |S|S

ijSi

C cSf min min)(

I – set of locations

J – set of clients

cij – costs for serving j-th client from i-th location

p – number of facilities to be opened

66

The PMP: Applications

• Facilty location

• Cluster analysis

• Quantitative psychology

• Telecommunications industry

• Sales force territories design

• Political and administrative districting

• Optimal diversity management

• Cell formation in group technology

• Vehicle routing

• Topological design of computer and communication networks

7

Similarities and dissimilarities

Proc Natl Acad Sci U S A. 1996 Jun

11;93(12):5854-9.

Similarities and dissimilarities of phage

genomes.

Blaisdell BE, Campbell AM, Karlin S.

Department of Mathematics, Stanford

University, CA 94305-2125, USA.

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Blaisdell%20BE%22%5BAuthor%5D&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVAbstract

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Campbell%20AM%22%5BAuthor%5D&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVAbstract

http://www.ncbi.nlm.nih.gov/pubmed?term=%22Karlin%20S%22%5BAuthor%5D&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVAbstract

8

A comparative study of similarity measures for manufacturing cell

formationS. Oliveira a, J.F.F. Ribeiro, S.C. Seok

Journal of Manufacturing Systems 27 (2008) 19--25

However, the similarity measure uses only

limited information between machines and parts:

either the number of parts producedby the pair

of machines or the number of machines

producing the pair of parts. Various similarity

measures (coefficients) have been introduced to

measure the similarities between machines and

parts for manufacturing cells problems.

9

The PMP: Applications• Facility location

- consumer (client)

- possible location of supplier (server)

- supplier (server), e.g. supermarket, bakery, laundry, etc.

10

The PMP: Applications• Facility location

- consumer (client)

- possible location of supplier (server)

- supplier (server), e.g. supermarket, bakery, laundry, etc.

11

The PMP: Applications• Cluster analysis

cluster

1

cluster

2

cluster

3cluster

4

―best‖ representatives – p-medians

Input:

- finite set of objects

- measure of similarity

Output

12


• Quantitative psychology

patients symptoms

(behavioural patterns) type 1

mentality

features

type 2

mentality

features

―leaders‖ or typical representatives

13


• Telecommunications industry

14


• Sales force territories design

possible

outlets for

some

product

...

...

3

2

1

...321

m

n

customers

(groups of customers)

entries of the costs

matrix account for

customers’ attitudes

and spatial distance

Goal: select p best outlets for promoting the product

15


• Political and administrative districting

districts,

cities,

regions...

...

3

2

1

...321

m

n

districts,

cities,

regions

degree of relationship:

political, cultural,

infrastructural

connectedness

16

The PMP: Applications• Optimal diversity management

– given a variety of products (each having some

demand, possibly zero)

– select p products such that:

• every product with a nonzero demand can be

replaced by one of the p selected products

• replacement overcosts are minimized

17

The PMP: Applications• Optimal diversity management

– Example: wiring designs, p=3configurations

with zero

demand

18

The PMP: Applications• Cell formation in group technology

functional layout cellular layout

- machines

- products routes

cell 1 cell 2

see also video at

http://www.youtube.com/watch?v=q_m0_bVAJbA

drilling

thermal processing

19


• Vehicle routing

depot- clients

/ storage

- vehicle

routes

20


• Topological design of computer and

communication networks

21




22




23

Publications, more than 500

Elloumi, 2010;

Brusco and K¨ohn, 2008;

Belenky, 2008;

Church, 2003; 2008;

Avella et al, 2007;

Beltran et al, 2006;

Reese, 2006 (Overview, NETWORKS)

ReVelle and Swain, 1970;

Senne et al, 2005.

2424

The PMP: Boolean Linear Programming

Formulation (ReVelle and Swain, 1970)

s.t.

- each client is served by exactly one facility

- p opened facilities

- prevents clients from being served

by closed facilities

min1 1

ij

m

i

n

j

ij xc

pym

i

i

1

Jjxm

i

ij , 11

JjIiyx iij ,

}1,0{, iij yx

xij = 1, if j-th client is served by i-th facility; xij = 0, otherwise

2525

The PMP: alternative formulation, Cornuejols et al. 1980

28134

33321

53212

43561

C

4 2 1 :client1 31

21

11 DDD

Let for each client j - sorted (distinct) distances (Kj – number

of distinct distances for j-th client)

jK

jj DD ,...,1

2626


28134

33321

53212

43561

C

5 4 3 2 :client5

8 3 :client4

5 3 2 1 :client3

6 3 2 1 :client2

4 2 1 :client1

45

35

25

15

24

14

43

33

23

13

42

32

22

12

31

21

11

DDDD

DD

DDDD

DDDD

DDD



jK

jj DD ,...,1

2727


28134

33321

53212

43561

C

5 4 3 2 :client5

8 3 :client4

5 3 2 1 :client3

6 3 2 1 :client2

4 2 1 :client1

45

35

25

15

24

14

43

33

23

13

42

32

22

12

31

21

11

DDDD

DD

DDDD

DDDD

DDD

closed are distance within sites all if ,1

opened is distance within site oneleast at if ,0kj

kjk

jD

Dz

Decision variables



jK

jj DD ,...,1

111121 )(...)(min jjj K

j

K

j

K

jjjjjijSi

zDDzDDDc

S - set of opened plants

2828for each client i - sorted distances

- iff all the sites within are

closed

s.t.


min)(),(

1

1

1

11n

j

K

k

ki

ki

kii

i

zDDDf yz

- p opened facilitiespy

m

i

i

1

,1 ,...,1,...,1

:

niKk

Ddj

jki

ikiij

yz

niz iKi ,...,1 ,0

niKk

ki

iz ,...,1

,...,1 ,0

mjy j ,...,1 },1,0{

- either at least one facility is open within

or

kiD

1kiz

- for every client it is an opened facility in some

neighbourhood

1kiz

kiD

iKii DD ,...,1

2929


Example, p=2

(Elloumi,2010)

28134

33321

53212

43561

C

Objective:

352515143323133222122111

352515

14

332313

322212

2111

52328

)45()34()23(2 :client5

)38(3 :client4

)35()23()12(1 :client3

)36()23()12(1 :client2

)24()12(1 :client1

zzzzzzzzzzzz

zzz

z

zzz

zzz

zz

5 4 3 2 :client5

8 3 :client4

5 3 2 1 :client3

6 3 2 1 :client2

4 2 1 :client1

45

35

25

15

24

14

43

33

23

13

42

32

22

12

31

21

11

DDDD

DD

DDDD

DDDD

DDD

+ +

+ +

13 coefficients

only distinct (in a

column)

distances are

meaningful

3030


Example

28134

33321

53212

43561

C

Objective:

352515143323

133222122111

52

328

zzzzzz

zzzzzz

Constraints:

jKkjjkz

zz

zzz

yyyyz

yyyyz

yyyyz

yyyz

yyyz

yyyz

yyyyz

yyz

yyyyz

yyz

yyz

yyyz

yz

yyyz

yz

yz

yyz

pyyyy

,...,1 ;5,...,1

4524

434231

432145

432143

432142

43135

43233

43232

432131

4325

432124

4223

3222

32121

415

32114

413

212

3111

4321

0

0,0

0,0,0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

4,...,1}1,0{ iiy

13 coefficients, 23 linear constr., 12 non-negativity constr., 4 Boolean

31

.eliminated becan variabledefinesthat

1 constraint and )1(by dsubstitute becan Variable solution. feasible

anyfor holds 1 then }{singleton a is if ,client any For :R1 Rule

1

11

11

j

ijij

ijij

z

yzyz

yzyVj

The p-Median Problem:a tighter formulation, Elloumi 2010

6 3 2 1 :2client 32

32

22

12 DDDD

28134

33321

53212

43561

C

)1(

open is

2facility

)0(

open is within

facility some

212

12

yz

D

212 1 yz

}:{ : within facilities ofset Let k

jijk

jk

jk

j DciVDV

Informally:

if for client j some neighbourhood

k contains only one facility i then

there is a simple relation between

corresponding variables ikj yz 1

32

.eliminated becan variabledefines

that constraint and by replaced becan Variable

solution. feasibleany for holds then any for If :R2 Rule1

k'j'

Di:c

k'-j'j

k'j'

kj

k'j'

k'j'

kj

k'i'

kj

z

zyzzz

zzVk', Vj',k,j,

k'j'ij'


28134

33321

53212

43561

C

{1,2,3,4} {1,2,3}

8 3 :4client

}4321{ }321{ }31{

4 2 1 :1client

24

14

24

14

31

21

11

31

21

11

VV

DD

,,,V,,V,V

DDD

14

21 zz

Informally:

if two clients have equal

neighbourhoods then the

corresponding z-variables are

equivalent and in the objective

function terms containing them

can be added.

33

. constraint eliminatecan

we,0 as Finally, .such that facilities of sets the toequal is

such that facilities ofset thecase, in this Further, .that

deduce toapplied becan R2 Rule then any for If :R3 Rule

1

11

11

j'K

j'ij'

j'j'

j'jj'

jj'j

j'j

Di:c

-Kj'i

K

j'

K

j'

K

j

K

j'ij'

K

jij

K

j'

K

j

K

j'

K

j

zyz

zzDci

Dcizz

Vj', Vj,


}432{1, }432{ {2,4} {4}

5 3 2 1 :3client

}432{1, }432{ }32{ }2{

6 3 2 1 :2client

33

33

23

13

33

33

23

13

32

32

22

12

32

32

22

12

,,V,,VVV

DDDD

,,V,,V,VV

DDDD

331

43 zyz

after applying Rule R2 becomes

redundant and can be eliminated

3434

The PMP: a tighter formulation, Elloumi 2010

A possible definition of variables : kjz

j

Dci

ikj Kknjyz

kjij

,...,1 ;,...,1 ,)1(

:

Or recursively:

j

Dci

ikj

kj

Dci

ij

,...,Kknjyzz

njyz

kjij

jij

2 ;,...,1 ,)1(

;,...,1 ,)1(

:

1

:

1

1

11221

3111

32121

3111 1 toequivalent is

1

1 e.g.

zyz

yyz

yyyz

yyz

Thus:

3535for each client j - sorted distances

s.t.


min)(),(

1

1

1

11n

j

K

k

kj

kj

kjj

j

zDDDf yz

py

m

i

i

1

n 1,...,j ,1

:

1

kjij Dci

ij yz

njz jK

j ,...,1 ,0

nj

Kkkj

jz

,...,1

,...,1 ,0

miyi ,...,1 },1,0{jK

jj DD ,...,1

,,...,1

,...,21

:

nj

Kkkj

Dci

ikj

jkjij

zyz

,1,...,1

,...,1

:

nj

Kk

Dci

ikj

jkjij

yz

Cornuejols et al. 1980

3636

PMP Example with p=2 borrowed from S. Elloumi,

J Comb Optim 2010,19:69–83

28134

33321

53212

43561

C

Objective:

3525233222211142 57)1(2)1(8 zzzzzzzyy

Constraints:

22432

2322

214

11221

3111

4321

1

1

zyz

yyz

zy

zyz

yyz

pyyyy

4,...,1

352

25135

4325

4223

321

}1,0{

0

1

1

jj

ki

y

z

zy

zyz

yyz

yyz

zy

10 (13) coefficients

11 (23) linear constr.

7 (12) non-negativity

constr.

4 Boolean constr.

3737

The PMP: pseudo-Boolean formulation(Historical remarks)

• Hammer, 1968 for the Simple Plant Location Problem (SPLP) called

also Uncapacitated Faciltiy Location Problem. His formulation

contains both literals and their complements, but at the end of this

paper Hammer has considered an inversion of literals;

• Beresnev, 1971 for the SPLP applied to the so called standardi-

zation (unification) problem. He has changed the definition of

decision variables, namely for an opened site a Boolean variable is

equal to 0, and for a closed site a Boolean variable is equal to 1.

This is exactly what is done by Cornuejols et al. 1980 and later on

by Elloumi 2010 but as we will show by means of computational

experiments with a larger number of decision variables and

constraints. Beresnev’s formulation contains complements only for

linear terms and all nonlinear terms are without complements.

3838

The PMP and SPLP differ in the following details

• SPLP involves fixed cost for location a facility at the given site, while

the PMP does not;

• Unlike the PMP, SPLP does not have a constraint on the number of

opened facilities;

• Typical SPLP formulations separate the set of potential facilities

(sites location, cluster centers) from the set of demand points

(clients);

• In the PMP the sets of sites location and demand points are

identical, i.e. I=J;

• The SPLP with a constraint on the number of opened facilities is

called either Capacitated SPLP or Generalized PMP.

3939

28134

33321

53212

43561

C

The PMP: pseudo-Boolean formulationNumerical Example: m=5, n=4, p=2

28134

33321

53212

43561

C

If two locations are opened at sites 1 and 3, i.e S ={1,3}

1133311

}:min{)(

5

1j

ijC SicSf

5 clients

4 locations

2 facilities

4

3

2

1

54321

4040

PMP: pseudo-Boolean formulation

1C

4

1

2

1

5

4

3

2

3213111

min

min

min

min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

c

c

yyyyyyc

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

28134

33321

53212

43561

C

4

2

3

1

2

1

0

1

1C 1 1C

4141

5

4

3

2

3213111

min

min

min

min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

c

c

yyyyyyc

4

2

3

1

2

1

0

1


1C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

4

1

2

1

28134

33321

53212

43561

C

1C 1 1C

equal distances lead to

terms with zero coefficients

that can be dropped

i.e. only distinct distances

are meaningful (like in

Cornuejols’ and Elloumi’s

model)4

1

2

1

1C

4242

5

4

3

2

3213111

min

min

min

min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

c

c

yyyyyyc


1C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

4

1

2

1

28134

33321

53212

43561

C

4

2

3

1

2

1

0

1

1C 1 1C

4343

5

4

3

2

3213111

min

min

min

min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

c

c

yyyyyyc

2

1

0

1

4

2

3

1


1C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

4

1

2

1

1C 1 1C

28134

33321

53212

43561

C

4444

28134

33321

53212

43561

C


2C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

5

4

3

4323222

3213111

min

min

min

3111min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

c

yyyyyyc

yyyyyyc

3

2

1

6

2C 2 2C

1

4

3

2

3

1

1

1

4545

28134

33321

53212

43561

C


3C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

1

3

2

5

3C 3 3C

1

3

2

4

2

1

1

1

5

4

4324243

4323222

3213111

min

min

2111min

3111min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

yyyyyyc

yyyyyyc

yyyyyyc

4646

28134

33321

53212

43561

C


4C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

8

3

3

3

4C 4 4C

4

3

2

1

5

0

0

3

5

3212114

4324243

4323222

3213111

min

5003min

2111min

3111min

2101min

iSi

iSi

iSi

iSi

iSi

c

yyyyyyc

yyyyyyc

yyyyyyc

yyyyyyc

4747


5C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

2

3

5

4

5C 5 5C

28134

33321

53212

43561

C

2

1

3

4

1

1

1

2

4314345

3212114

4324243

4323222

3213111

1112min

5003min

2111min

3111min

2101min

yyyyyyc

yyyyyyc

yyyyyyc

yyyyyyc

yyyyyyc

iSi

iSi

iSi

iSi

iSi

4848


5

1

min)(j

ijSi

C cB y

BC(y) can be constructed

in polynomial time

BC(y) has polynomial size

(number of terms)

4314345

3212114

4324243

4323222

3213111

1112min

5003min

2111min

3111min

2101min

yyyyyyc

yyyyyyc

yyyyyyc

yyyyyyc

yyyyyyc

iSi

iSi

iSi

iSi

iSi+

+ +

+

4949

431434

321211

432424

432322

321311

1112

5003

2111

3111

2101

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyy

28134

33321

53212

43561

C


two possible

permutation

matrices24124

13342

32233

41421

1

1 2 43 5

but

a unique polynomial

=

+ +

+ +

+ +

+ +

)(yCB=

24124

12342

33231

41423

1

1 2 43 5

431434

321311

432424

432322

321313

1112

5003

2111

3111

2101

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyy

5050

PBP: combining similar terms

=

4324313214342323142 5171111218 yyyyyyyyyyyyyyyyyyy

20 terms

17 nonzero terms

10 terms

PBP formulation allows compact representation of the problem !

In the given example 50% reduction is achieved!

431434

321211

432424

432322

321311

1112

5003

2111

3111

2101

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyy+ +

+ +

This procedure is equivalent to application of Elloumi’s Rule R2

5151

PBP: combining similar terms

5252

Truncated polynomial BC,p=2 (y) (7 terms):

PBP: truncation

If p=2 each cubic term

contains at least one zero

variable

Observation:

The degree of the pseudo-Boolean

polynomial is at most m-p

Initial polynomial BC (y) (10 terms):

Truncation allows further

reduction of the problem size!

p = 2

4324313214342323142 5171111218 yyyyyyyyyyyyyyyyyyy

4342323142 1111218 yyyyyyyyyy

5353

PBP: truncation

p = 2

p = 3

p = 4

If p=m/2+1 then memory

needed to store the polynomial

is halved!

full polynomial

truncated

polynomialp = m/2+1

MEMORY

431434

321211

432424

432322

321311

1112

5003

2111

3111

2101

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyy+ +

+ +

5454

28134

33321

53212

43561

C


3C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

1

3

2

5

3C 3 3C

1

3

2

4

2

1

1

1

5

4

4324243

4323222

3213111

min

min

2111min

3111min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

yyyyyyc

yyyyyyc

yyyyyyc

5555

23121

33221

33211

33221

3C

28134

33321

53212

43561

C

Truncation and preprocessing

Initial matrix p-truncated matrix, p=3

Thus, truncation allows reduction of search space!

Corollary

Instances with p=p0>m/2 are easier to solve then those with

p=m-p0<m/2, even though the numbers of feasible solutions are the

same for both cases.

If i-th row contains all maximum

elements, then corresponding

location can be excluded from

consideration ( yi can be set to 0).

In truncated matrix

this is more likely

to happen

y3=1

4

3

2

1

56

Pseudo-Boolean formulation:

outcomes

• Compact but nonlinear problem

• Equivalent to a nonlinear knapsack (NP-hard)

• Goal: obtain a model suitable for general-purpose MILP solvers, e.g.:– CPLEX

– XpressMP

– MOSEK

– LPSOL

– CLP

5757

MBpBM: linearization

4342323142 28 y y y y y y y y y y

876542 28 z z z z y y

Example of the pseudo-Boolean

polynomial:

Linear function of new variables:

438427326315

4321

, , ,

, , , ,

yyzyyzyyzyyz

yyyy

28134

33321

53212

43561

C p = 2

Compare: in Elloumi’s model variables y2 and y4 were introduced into

objective via Rule R1.

5858

MBpBM: constraints

l

k

kk

l

k

k ylyzyz

11

}1,0{ ,1 Simple fact:

Example:

8...5

4...1

438438

427427

326326

315315

0

}10{

1

1

1

1

kk

kk

z

,y

yyzyyz

yyzyyz

yyzyyz

yyzyyz

nonnegativity is

sufficient !

5959

MBpBM: reduction

Lema:Let Ø be a pair of embedded sets of Boolean variables yi.

Then, the two following systems of inequalities are equivalent:

Obtained reduced constraints are similar to Elloumi’s constraints

derived from recursive definition of his z-variables.

60

MBpBM: reduction

• set covering problem

965431 yyyyyy

431 yyy

31 yy

54 yy 96 yy

531 yyy631 yyy

931 yyy

61

MBpBM: reduction

• set covering problem

965431 yyyyyy

431 yyy

31 yy

54 yy 96 yy

531 yyy631 yyy

931 yyy

NP-hard!

2965431965431 yyyyyyyyyyyy

6262

Example, p=2; S. Elloumi, J Comb Optim 2010,19:69–83

28134

33321

53212

43561

C

Objective:

876542 28 zzzzyy

Constraints:

438

427

326

315

4321

1

1

1

1

2

yyz

yyz

yyz

yyz

yyyy

4,...,1

8,...,5

}1,0{

0

ii

ii

y

z

7 coefficients.

5 linear constr.

4 non-negativity

constr.

4 Boolean constr.

In Elloumi’s model these figures are, correspondingly, 10 (13), 11 (23), 7(12)

and 4

63

Comparison of the models

876542 28 zzzzyy

438

427

326

315

4321

1

1

1

1

2

yyz

yyz

yyz

yyz

yyyy

4,...,1

8,...,5

}1,0{

0

ii

ii

y

z

3525233222211142 57)1(2)1(8 zzzzzzzyy

352

25135

4325

4223

321

22432

2322

214

11221

3111

4321

1

1

1

1

2

zy

zyz

yyz

yyz

zy

zyz

yyz

zy

zyz

yyz

yyyy

4,...,1

3,...,1 ; 5,...,1

}1,0{

0

ii

kjkj

y

z

our MBpBM Elloumi’s NF

64

MBpBM: preprocessing

• every term (product of variables) corresponds to a subspace of solutions with all these variables equal to 1

• like in Branch-and-Bound:

– compute an upper bound by some heuristic

– for each subspace define a procedure for computing a lower bound (over a subspace)

– if the constrained lower bound exceeds global upper bound then exclude the subspace from consideration

6565

28134

33321

53212

43561

C

PMP: pseudo-Boolean formulation implies a decomposition

of the search space into at most n(m-p) subspaces

3C

+ +

+ +

5

1

min)(j

ijSi

C cB y

Iii pmy,

CBy

y min )(

1

3

2

5

3C 3 3C

1

3

2

4

2

1

1

1

5

4

4324243

4323222

3213111

min

min

2111min

3111min

2101min

iSi

iSi

iSi

iSi

iSi

c

c

yyyyyyc

yyyyyyc

yyyyyyc

6666

MBpBM: preprocessing (example)

heuristic)greedy (by 9

2

UBf

p

Objective:

876542 28 zzzzyy

Constraints:

4,...,1

8,...,5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jj

jj

y

z

yyz

yyz

yyz

yyz

yyyy

28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz

6767


28134

33321

53212

43561

C

011)(


2

43834 yyzff

f

p

UB

UB

y

Objective:

876542 28 zzzzyy

Constraints:

4,...,1

8,...,5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jj

jj

y

z

yyz

yyz

yyz

yyz

yyyy

876542 28 zzzzyy438

427

326

315

yyz

yyz

yyz

yyz

thus, z8 can be deleted

from the model

consider some term

ri

defT

i Tyy r iff 1

6868


012)(

011)(


2

42724

32834

yyzff

yyzff

f

p

UB

UB

UB

y

y

Objective:

876542 28 zzzzyy

Constraints:

4,...,1

7,...,5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jj

jj

y

z

yyz

yyz

yyz

yyz

yyyy

876542 28 zzzzyy28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz consider next term

thus, z7 can be deleted

from the model

ri

defT

i Tyy r iff 1

6969


010)(

012)(

011)(


2

32623

42724

43834

yyzff

yyzff

yyzff

f

p

UB

UB

UB

UB

y

y

y

Objective:

876542 28 zzzzyy

Constraints:

4,...,1

6,...,5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jj

jj

y

z

yyz

yyz

yyz

yyz

yyyy

876542 28 zzzzyy28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz and so on …

ri

defT

i Tyy r iff 1

7070

9)(

010)(

012)(

011)(


2

13

32623

42724

43834

y

y

y

y

f

yyzff

yyzff

yyzff

f

p

UB

UB

UB

UB

MBpBM: preprocessing (example)Objective:

876542 28 zzzzyy

Constraints:

4,...,1

5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jjy

z

yyz

yyz

yyz

yyz

yyyy

876542 28 zzzzyy28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz and so on …

ri

defT

i Tyy r iff 1

7171

010)(

9)(

010)(

012)(

011)(


2

44

13

32623

42724

43834

yff

f

yyzff

yyzff

yyzff

f

p

UB

UB

UB

UB

UB

y

y

y

y

y


876542 28 zzzzyy

Constraints:

4,...,1

5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jjy

z

yyz

yyz

yyz

yyz

yyyy

876542 28 zzzzyy28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz and so on …

ri

defT

i Tyy r iff 1

7272

9)(

010)(

9)(

010)(

012)(

011)(


2

2

44

13

32623

42724

43834

y

y

y

y

y

y

f

yff

f

yyzff

yyzff

yyzff

f

p

UB

UB

UB

UB

UB


876542 28 zzzzyy

Constraints:

4,...,1

5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jjy

z

yyz

yyz

yyz

yyz

yyyy

876542 28 zzzzyy28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz and so on …

ri

defT

i Tyy r iff 1

7373


876542 28 zzzzyy

Constraints:

4,...,1

5

438

427

326

315

4321

}1,0{

0

1

1

1

1

2

jjy

z

yyz

yyz

yyz

yyz

yyyy

876542 28 zzzzyy

unnecessary

restrictions !

28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz

9)(

010)(

9)(

010)(

012)(

011)(


2

2

44

13

32623

42724

43834

y

y

y

y

y

y

f

yff

f

yyzff

yyzff

yyzff

f

p

UB

UB

UB

UB

UB

7474


876542 28 zzzzyy

Constraints:

4,...,1

5

32

31

321

}1,0{

0

10

10

20

jjy

z

yy

yy

yyy

528 zy28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz

9)(

010)(

9)(

010)(

012)(

011)(


2

2

44

13

32623

42724

43834

y

y

y

y

y

y

f

yff

f

yyzff

yyzff

yyzff

f

p

UB

UB

UB

UB

UB

7575


2p

Objective:

Constraints:

0

}1,0{

0

1

1

2

4

4,...,1

5

32

31

321

y

y

z

yy

yy

yyy

jj

528 zy

3 (10) coefficients


1 (7) non-negativity constr.

3 Boolean (1 fixed to 0)

28134

33321

53212

43561

C

438

427

326

315

yyz

yyz

yyz

yyz

Note: the number of Boolean variables was 4 in all considered

models and in MBpBM it is 3.

7676

Preprocessing from linear to

nonlinear terms

• The preprocessing should be done

starting from linear terms...

• ... as cutting some term T cuts also all

terms for which T was embedded

7777

MBpBM: preprocessing (impact)

our results

results from P. Avella and A. Sforza, Logical reduction tests

for the p-median problem, Ann. Oper. Res. 86, 1999, pp. 105–115.

7878

Computational resultsOR-library instances

[3] Avella P., Sassano A., Vasil’ev I.: Computational study of large-scale p-median problems. Math. Prog., Ser. A, 109, 89-114 (2007)

[12] Church R.L.: BEAMR: An exact and approximate model for the p-median problem. Comp. & Oper. Res., 35, 417-426 (2008)

[15] Elloumi S.: A tighter formulation of the p-median problem. J. Comb. Optim., 19, 69–83 (2010)

7979

Computational results, m=900

Results for different number of medians for two OR instances

8080

Computational resultsResults for different numbers of medians in BN1284

[3] Avella P., Sassano A., Vasil’ev I.: Computational study of large-scale p-median

problems. Math. Prog., Ser. A, 109, 89-114 (2007)

81

Computational resultsRunning times (sec.) for 15 largest OR-library instances

82

Computational resultsRunning times (sec.) for RW instances

83

Results for

our

complex

instances

8484

Concluding remarks

• a new Mixed Boolean Pseudo-Boolean

linear programming Model (MBpBM) for

the p-median problem (PMP):

instance specific

optimal within the class of mixed

Boolean LP models

allows solving previously unsolved

instances with general purpose software

8585

Future research directions

• compact models for other location

problems (e.g. SPLP or generalized PMP)

• revised data-correcting approach

• implementation and computational

experiments with preprocessed MBpBM

based on lower and upper bounds

86

Next two lectures

• How many instances do we really solve

when solving a PMP instance

• Why some data lead to more complex

problems than other

• Two applications in details

87

Literature• B. F. AlBdaiwi, B. Goldengorin, G. Sierksma. Equivalent instances of the simple

plant location problem. Computers and Mathematics with Applications, 57 812—820 (2009).

• B. F. AlBdaiwi, D. Ghosh, B. Goldengorin. Data Aggregation for p-Median Problems. Journal of Combinatorial Optimization 2010 (open access, in press) DOI: 10.1007/s10878-009-9251-8.

• Avella, P., Sforza, A.: Logical reduction tests for the p-median problem. Annals of Operations Research, 86, 105-115 (1999).

• Avella, P., Sassano, A., Vasil'ev, I.: Computational study of large-scale

p-median problems. Mathematical Programming, Ser. A, 109, 89-114

(2007).

• Beresnev, V.L. On a Problem of Mathematical Standardization Theory,

Upravliajemyje Sistemy, 11, 43–54 (1973), (in Russian).

• Church, R.L.: BEAMR: An exact and approximate model for the p-median problem. Computers & Operations Research, 35, 417-426 (2008).

• Cornuejols, G., Nemhauser, G., Wolsey, L.A.: A canonical representation of simple plant location problems and its applications. SIAM Journal on Matrix Analysis and Applications (SIMAX), 1(3), 261-272 (1980).

88

• Elloumi, S.: A tighter formulation of the p-median problem. Journal of Combinatorial Optimization, 19, 69-83 (2010).

• Goldengorin, B., Krushinsky, D.: Towards an optimal mixed-Boolean LP model for the p-median problem (submitted to Annals of Operations Research).

• Goldengorin, B., Krushinsky, D.: Complexity evaluation of benchmark instances for the p-median problem (submitted to Mathematical and Computer Modelling ).

• Hammer, P.L.: Plant location -- a pseudo-Boolean approach. Israel

Journal of Technology, 6, 330-332 (1968).

• Reese, J.: Solution Methods for the p-Median Problem: An Annotated

Bibliography. Networks 48, 125-142 (2006)

• ReVelle, C.S., Swain, R.: Central facilities location. Geographical Analysis,

2, 30-42 (1970)

Literature (contd.)

8989

Thank you!

Questions?

90

Application to Cell Formation

Machine-part

incidence matrix

Dissimilarity measure for machines

machines theofeither need that parts ofnumber

and machinesboth need that parts ofnumber ),(

jijid

Example 1:

functional

grouping

The task is to group machines into clusters (manufacturing cells)

such that to to minimize intercell communication.

10010

00101

01110

00101

11010

parts

machin

es

1 2 3 4 5

5

4

3

2

1

91


Cost matrix for the PMP

is a machine-machine

dissimilarity matrix:

000.175.000.133.0

00.1075.0000.1

75.075.0075.050.0

00.1075.0000.1

33.000.150.000.10

machines

machin

es

),(:],[ jidjic

Example 1: functional grouping (contd.)

In case of

two cells

the solution

is:11000

11000

00101

10011

00111

parts

machin

es

2 4 5 1 3

2

4

5

3

1

intercell communication is

caused by only part # 3

that is processed in both

cells

92


000.175.000.133.0

00.1075.0000.1

75.075.0075.050.0

00.1075.0000.1

33.000.150.000.10

C

Example 1: functional grouping (contd.)

531515

432422

321313

432422

432511

25.042.033.0

25.075.00

025.05.0

25.075.00

25.016.033.0)(

yyyyyy

yyyyyy

yyyyyy

yyyyyy

yyyyyyBC y

4325313142515312, 5.075.025.05.158.033.05.033.0)( yyyyyyyyyyyyyyyB pC y

Linearization:

109876531 5.075.025.05.158.033.05.033.0),( zzzzzyyyf zy

where:

318

43210427

5319516

yyz

yyyzyyz

yyyzyyz

93

Application to Cell FormationExample 1: functional grouping (contd.)

10..6

5..1

43210

5319

318

427

516

54321

109876531

0

}1,0{

2

2

1

1

1

25

..

min5.075.025.05.158.033.05.033.0

ii

ii

z

y

yyyz

yyyz

yyz

yyz

yyz

yyyyy

ts

zzzzzyyy

MBpBM

5..1

432

531

31

42

51

54321

531

}1,0{

20

20

10

10

10

25

.

min33.05.033.0

iiy

yyy

yyy

yy

yy

yy

yyyyy

ts

yyy

1

0

1

1

0

*y

MBpBM with reduction based on bounds

94


Machine-worker

incidence matrix

Dissimilarity measure for machines

machines theofeither operatecan that workersofnumber

and machinesboth operatecan that workersofnumber ),(

jijid

Example 2:

workforce

expences

The task is to group machines into clusters (manufacturing cells) such that:

1) every worker is able to operate every machine in his cell and cost of additional

cross-training is minimized;

2) if a worker can operate a machine that is not in his cell then he can ask for

additional payment for his skills; we would like to minimize such overpayment.

11001000

00101100

10010110

00100011

01010001

workers

machin

es

1 2 3 4 5 6 7 8

5

4

3

2

1

95


Cost matrix for the PMP

is a machine-machine

dissimilarity matrix:

080.083.000.180.0

80.0083.080.000.1

83.083.0083.083.0

00.180.083.0080.0

80.000.183.080.00

machines

machin

es

),(:],[ jidjic

Example 2: workforce expences (contd.)

In case of

three cells

the solution

is:10010100

10101000

01010001

01100010

00011111

workers

ma

chin

es

2 3 5 8 1 4 6 7

1

5

2

4

3

1 worker needs

additional training

7 non-clustered

elements that

represent the skills that

are not used (potential

overpayment)

96

Application to Cell FormationExample 2: workforce expences (contd.)

080.083.000.180.0

80.0083.080.000.1

83.083.0083.083.0

00.180.083.0080.0

80.000.183.080.00

C

5431541515

5432542424

4321321313

4321421212

5321521211

17.003.0080.0

17.003.0080.0

00083.0

17.003.0080.0

17.003.0080.0)(

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyyBC y

543213, 8.08.083.08.08.0)( yyyyyB pC y

The objective is already a linear function !

97

Application to Cell FormationExample 2: workforce expences (contd.)

MBpBM

0

0

0

1

1

*y

5..1

54321

54321

}1,0{

35

..

min8.08.083.08.08.0

iiy

yyyyy

ts

yyyyy

98

Application to Cell FormationExample 3: from Yang,Yang (2008)*

105 parts

45 m

achin

es

(uncapacitated)

functional grouping

105 parts

45 m

achin

esgrouping efficiency:

Yang, Yang* 87.54%

our result 87.57%

(solved within 1 sec.)

* Yang M-S., Yang J-H. (2008) Machine-part cell formation in group technology using a modified

ART1 method. EJOR, vol. 188, pp. 140-152

9999

Thank you!

• Questions?

100100


28134

33321

53212

43561

C

4 2 1 :client1 31

21

11 DDD



jK

jj DD ,...,1

101101


28134

33321

53212

43561

C

5 4 3 2 :client5

8 3 :client4

5 3 2 1 :client3

6 3 2 1 :client2

4 2 1 :client1

45

35

25

15

24

14

43

33

23

13

42

32

22

12

31

21

11

DDDD

DD

DDDD

DDDD

DDD



jK

jj DD ,...,1

102102


28134

33321

53212

43561

C

5 4 3 2 :client5

8 3 :client4

5 3 2 1 :client3

6 3 2 1 :client2

4 2 1 :client1

45

35

25

15

24

14

43

33

23

13

42

32

22

12

31

21

11

DDDD

DD

DDDD

DDDD

DDD

closed are distance within sites all if ,1

opened is distance within site oneleast at if ,0kj

kjk

jD

Dz

Decision variables



jK

jj DD ,...,1

111121 )(...)(min jjj K

j

K

j

K

jjjjjijSi

zDDzDDDc

S - set of opened plants


- iff all the sites within are

closed

s.t.


min)(),(

1

1

1

11n

j

K

k

ki

ki

kii

i

zDDDf yz

- p opened facilitiespy

m

i

i

1

,1 ,...,1,...,1

:

niKk

Ddj

jki

ikiij

yz

niz iKi ,...,1 ,0

niKk

ki

iz ,...,1

,...,1 ,0

mjy j ,...,1 },1,0{

- either at least one facility is open within

or

kiD

1kiz

- for every client it is an opened facility in some

neighbourhood

1kiz

kiD

iKii DD ,...,1

104104


Example

(Elloumi,2009)

28134

33321

53212

43561

C

Objective:

352515143323133222122111

352515

14

332313

322212

2111

52328

)45()34()23(2 :client5

)38(3 :client4

)35()23()12(1 :client3

)36()23()12(1 :client2

)24()12(1 :client1

zzzzzzzzzzzz

zzz

z

zzz

zzz

zz

5 4 3 2 :client5

8 3 :client4

5 3 2 1 :client3

6 3 2 1 :client2

4 2 1 :client1

45

35

25

15

24

14

43

33

23

13

42

32

22

12

31

21

11

DDDD

DD

DDDD

DDDD

DDD

+ +

+ +

13 coefficients

only distinct

(in a column)

distances are

meaningful

105105


Example

28134

33321

53212

43561

C

Constraints:

1

1

1

:client1

423131

23121

3111

yyyyz

yyyz

yyz

4 2 1 :client1 31

21

11 DDD

if plants 1 and 3 are closed

then all plants within distance D11=1 are closed

and

)0,0( 31 yy

111zpla

nts

1 2

3 4

106106


Example

(Elloumi,2009)

28134

33321

53212

43561

C

Objective:

352515143323133222122111

352515

14

332313

322212

2111

52328

)45()34()23(2 :client5

)38(3 :client4

)35()23()12(1 :client3

)36()23()12(1 :client2

)24()12(1 :client1

zzzzzzzzzzzz

zzz

z

zzz

zzz

zz

5 4 3 2 :client5

8 3 :client4

5 3 2 1 :client3

6 3 2 1 :client2

4 2 1 :client1

45

35

25

15

24

14

43

33

23

13

42

32

22

12

31

21

11

DDDD

DD

DDDD

DDDD

DDD

+ +

+ +

13 coefficients

only distinct

(in a column)

distances are

meaningful

107107


Example

28134

33321

53212

43561

C

Objective:

352515143323

133222122111

52

328

zzzzzz

zzzzzz

Constraints:

jKkjjkz

zz

zzz

yyyyz

yyyyz

yyyyz

yyyz

yyyz

yyyz

yyyyz

yyz

yyyyz

yyz

yyz

yyyz

yz

yyyz

yz

yz

yyz

pyyyy

,...,1 ;5,...,1

4524

434231

432145

432143

432142

43135

43233

43232

432131

4325

432124

4223

3222

32121

415

32114

413

212

3111

4321

0

0,0

0,0,0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

4,...,1}1,0{ iiy

13 coefficients, 23 linear constr., 12 non-negativity constr., 4 Boolean

108108


A possible definition of variables : kjz

j

Dci

ikj Kknjyz

kjij

,...,1 ;,...,1 ,)1(

:

Or recursively:

j

Dci

ikj

kj

Dci

ij

,...,Kknjyzz

njyz

kjij

jij

2 ;,...,1 ,)1(

;,...,1 ,)1(

:

1

:

1

1

11221

3111

32121

3111 1 toequivalent is

1

1 e.g.

zyz

yyz

yyyz

yyz

Thus:

109109for each client j - sorted distances

s.t.


min)(),(

1

1

1

11n

j

K

k

kj

kj

kjj

j

zDDDf yz

py

m

i

i

1

n 1,...,j ,1

:

1

kjij Dci

ij yz

njz jK

j ,...,1 ,0

nj

Kkkj

jz

,...,1

,...,1 ,0

miyi ,...,1 },1,0{jK

jj DD ,...,1

,,...,1

,...,21

:

nj

Kkkj

Dci

ikj

jkjij

zyz

,1,...,1

,...,1

:

nj

Kk

Dci

ikj

jkjij

yz

Cornuejols et al. 1980

110110

.eliminated becan variabledefinesthat

1 constraint and )1(by dsubstitute becan Variable solution. feasible

anyfor holds 1 then }{singleton a is if ,client any For :R1 Rule

1

11

11

j

ijij

ijij

z

yzyz

yzyVj


6 3 2 1 :2client 32

32

22

12 DDDD

28134

33321

53212

43561

C

)1(

open is

2facility

)0(

open is within

facility some

212

12

yz

D

212 1 yz

}:{ : within facilities ofset Let k

jijk

jk

jk

j DciVDV

Informally:

if for client j some neighbourhood

k contains only one facility i then

there is a simple relation between

corresponding variables ikj yz 1

111111

.eliminated becan variabledefines

that constraint and by replaced becan Variable

solution. feasibleany for holds then any for If :R2 Rule1

k'j'

Di:c

k'-j'j

k'j'

kj

k'j'

k'j'

kj

k'i'

kj

z

zyzzz

zzVk', Vj',k,j,

k'j'ij'


28134

33321

53212

43561

C

{1,2,3,4} {1,2,3}

8 3 :4client

}4321{ }321{ }31{

4 2 1 :1client

24

14

24

14

31

21

11

31

21

11

VV

DD

,,,V,,V,V

DDD

14

21 zz

Informally:

if two clients have equal

neighbourhoods then the

corresponding z-variables are

equivalent and in the objective

function terms containing them

can be added.

112112

. constraint eliminatecan

we,0 as Finally, .such that facilities of sets the toequal is

such that facilities ofset thecase, in this Further, .that

deduce toapplied becan R2 Rule then any for If :R3 Rule

1

11

11

j'K

j'ij'

j'j'

j'jj'

jj'j

j'j

Di:c

-Kj'i

K

j'

K

j'

K

j

K

j'ij'

K

jij

K

j'

K

j

K

j'

K

j

zyz

zzDci

Dcizz

Vj', Vj,


}432{1, }432{ {2,4} {4}

5 3 2 1 :3client

}432{1, }432{ }32{ }2{

6 3 2 1 :2client

33

33

23

13

33

33

23

13

32

32

22

12

32

32

22

12

,,V,,VVV

DDDD

,,V,,V,VV

DDDD

331

43 zyz

after applying Rule R2 becomes

redundant and can be eliminated

113113

Example (from Elloumi, 2009)

28134

33321

53212

43561

C

Objective:

3525233222211142 57)1(2)1(8 zzzzzzzyy

Constraints:

22432

2322

214

11221

3111

4321

1

1

zyz

yyz

zy

zyz

yyz

pyyyy

4,...,1

352

25135

4325

4223

321

}1,0{

0

1

1

jj

ki

y

z

zy

zyz

yyz

yyz

zy

10 (13) coefficients


7 (12) non-negativity

constr.

4 Boolean constr.


s.t.


min)(),(

1

1

1

11n

j

K

k

ki

ki

kii

i

zDDDf yz

py

m

i

i

1

,1 ,...,1,...,1

:

niKk

Ddj

jki

ikiij

yz

niz iKi ,...,1 ,0

niKk

ki

iz ,...,1

,...,1 ,0

mjy j ,...,1 },1,0{

iKii DD ,...,1

,1 ,...,1,...,2

:

niKk

Ddj

jki

ikiij

yz

additional constraints

+ reduction rules

(next slide)

115115

The p-Median Problem:a tighter formulation Elloumi 2009

116116

MBpBM: preprocessing

0 constraint a add and

objective thefrom excludecan weand

0 solution optimalevery for then

)( holds monomial somefor if I.e.

solution. optimalan not is

11 satisfying every then

holds : somefor if

boundupper (global) some

1

ri

rT

ri

Tyi

r

r

UB

Tyir

ii

UBm

ii

UB

y

T

T

ffyT

y'y'

f)f(pmy

f

y

y

yy

ri

defT

i Tyy r iff 1

117117

6)(

4)(

0)( 1)(

1

6 ,4 ,Let

3

2

1

11

432211

T

T

UBT

T

UB

f

f

yfff

f

yTyTyT

y

y

yy

MBpBM: preprocessingClaim:

strict. bemust )( inequality The UBTff ry

Counter-example (p=2):

: violatedisassertion previous the)( if that showcan We UBTff ry

045

992

801

660

334

213

142

421 4241214212, 231641)( yyyyyyyyyB pC y

cost

matrix

permuta-

tion

But in the unique optimal solution y1=1 !

suppose

0

1

0

1

opty

Education

p-Median Cluster Analysis Based on General-Purpose Solvers (1)