AACIMP 2010 Summer School lecture by Dmitry Krushinsky. "Applied Mathematics" stream. "The p-Median Problem and Its Applications" course. Part 2. More info at http://summerschool.ssa.org.ua

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

2. 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
2

3. 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)
Costs Matrix
c11 ... c1 j ... ... c1n
location points
... ... ... ... ... ...
ci1 ... cij ... ... cin
... ... ... ... ... ...
c n1 ... c nj ... ... c nm
clients
- location point (cluster center)
- Client (cluster points)
3

4. 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.
4

5. A comparative study of similarity measures for manufacturing cell
formation
S. 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.
5

6. 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.
p m!
C m
p!(m p )!
complexity
1 m
p
- opened facility
- location point
- client
p=3 6

7. The PMP: combinatorial formulation
f C (S ) min cij min
i S S I, |S| p
j J
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
7

8. 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
8

9. The PMP: Applications
• Facility location
- consumer (client)
- possible location of supplier (server)
9
- supplier (server), e.g. supermarket, bakery, laundry, etc.

10. The PMP: Applications
• Facility location
- consumer (client)
- possible location of supplier (server)
10
- supplier (server), e.g. supermarket, bakery, laundry, etc.

11. The PMP: Applications
• Cluster analysis Output
Input:
cluster cluster cluster cluster
- finite set of objects 1 2 3 4
- measure of similarity
11
“best” representatives – p-medians

12. The PMP: Applications
• Quantitative psychology
patients symptoms
(behavioural patterns) type 1
mentality
features
type 2
mentality
features
12
“leaders” or typical representatives

13. The PMP: Applications
• Telecommunications industry
13

14. The PMP: Applications
• Sales force territories design
customers
(groups of customers)
1 2 3 ... n
entries of the costs
1 matrix account for
customers’ attitudes
possible 2 and spatial distance
outlets for
some 3 ...
product
...
m
Goal: select p best outlets for promoting the product 14

15. The PMP: Applications
• Political and administrative districting
districts,
cities,
regions
1 2 3 ... n
degree of relationship:
1 political, cultural,
infrastructural
districts, 2 connectedness
cities,
regions
3 ...
...
m
15

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
16

17. The PMP: Applications
• Optimal diversity management
– Example: wiring designs, p=3
configurations
with zero
demand
17

18. The PMP: Applications
• Cell formation in group technology
functional layout cellular layout
drilling
cell 1 cell 2
thermal processing
see also video at
http://www.youtube.com/watch?v=q_m0_bVAJbA
- machines
18
- products routes

19. The PMP: Applications
• Vehicle routing
- clients
/ storage depot
- vehicle
routes
19

20. The PMP: Applications
• Topological design of computer and
communication networks
20

21. The PMP: Applications
• Topological design of computer and
communication networks
21

22. The PMP: Applications
• Topological design of computer and
communication networks
22

23. Publications, more than 500
Elloumi, 2009;
Brusco and K¨ohn, 2008;
Belenky, 2008;
Church, 2003; 2008;
Mladenovic et al, 2007 (Overview, EJOR)
Avella et al, 2007;
Beltran et al, 2006;
Reese, 2006 (Overview, NETWORKS)
Senne et al, 2005.
23

24. The PMP: Boolean Linear Programming
Formulation
m n
cij xij min
i 1 j 1
m
s.t.
xij 1, j J - each client is served by exactly one facility
i 1
m
yi p - p opened facilities
i 1
xij yi i I, j J - prevents clients from being served
by closed facilities
xij , yi {0,1}
xij = 1, if j-th client is served by i-th facility; xij = 0, otherwise
24

25. The p-Median Problem:
a tighter formulation, Elloumi 2010
Let V j k set of facilities within D j k : V j k {i : cij Djk}
Rule R1 : For any client j, if V j1 is a singleton { yi } then z j1 1 yi holds for any
feasible solution. Variable z j1 can be substituted by (1 yi ) and constraint z j1 yi 1
that defines variable z j1 can be eliminated .
1 2 3 3
1 6 5 3 4 client 2 : D2 1 D2 2 D2 3 D2 6
2 1 2 3 5 some facility
C facility 2
1 2 3 3 3 1
within D2 is open is open
4 3 1 8 2
( z1
2 0) ( y 2 1)
Informally:
if for client j some neighbourhood
k contains only one facility i then
there is a simple relation between z1
2 1 y2
k
corresponding variables z j 1 yi 25

26. The p-Median Problem:
a tighter formulation, Elloumi 2010
Rule R2 : If for any j, k, j', k', Vj k Vi' k' then z j k z j' k' holds for any feasible solution.
Variable z j' k' can be replaced by z j k and constraint z j' k' k' y j z j' k'-1 that
i:cij' D j'
defines variable z j' k' can be eliminated .
1
1 6 5 3 4 client 1 : D1 1 D12 2 3
D1 4
2 1 2 3 5 V11 {1,3} V12 {1,2 ,3} V13 {1,2 ,3,4}
C
1 2 3 3 3
1 2
4 3 1 8 2 client 4 : D4 3 D4 8
1
V4 {1,2,3} V42 {1,2,3,4}
Informally:
if two clients have equal
neighbourhoods then the
corresponding z-variables are
2
equivalent and in the objective z1 z1
4
function terms containing them
26
can be added.

27. The p-Median Problem:
a tighter formulation, Elloumi 2010
Kj 1 K j' 1
Rule R3 : If for any j, j', V j V j' then Rule R2 can be applied to deduce
Kj 1 K 1 Kj
that z j z j' j' . Further,in this case, the set of facilities i such thatcij Dj
K Kj K
is equal to the sets of facilities i such thatcij' D j' j' . Finally, as z j z j' j' 0, we
K
can eliminate constraint z j' j' K yi z j' K j' -1 .
i:cij' D j' j'
1 2 3 3
client 2 : D2 1 D2 2 D2 3 D2 6
V21 {2} V22 {2,3} V23 {2 ,3,4} V23 {1,2,3,4}
1 2 3 3
client 3 : D3 1 D3 2 D3 3 D3 5
1
V3 {4} V32 {2,4} V33 {2,3,4} V33 {1,2 ,3,4}
4 3
z3 y1 z3
after applying Rule R2 becomes
redundant and can be eliminated 27

28. 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
standardization (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. Beresnev’s
formulation contains complements only for linear terms and all
nonlinear terms are without complements.
28

29. 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.
29

30. The PMP: pseudo-Boolean formulation
Numerical Example: m=5, n=4, p=2
1 6 5 3 4 5 clients
2 1 2 3 5
C 4 locations
1 2 3 3 3
2 facilities
4 3 1 8 2
If two locations are opened at sites 1 and 3, i.e S ={1,3}
1 6 5 3 4 1
5
2 1 2 3 5 2 f C (S ) min{cij : i S}
C
1 2 3 3 3 3 j 1
4 3 1 8 2 4 1 1 3 3 3 11
1 2 3 4 5
30

31. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C1
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2
i S
+
C1 1 C1 min ci 3
i S
+
1 1 1
min ci 4
2 3 0 i S
+
1 2 1 min ci 5
i S
4 4 2
31

32. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C1
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2 equal distances lead to
i S
+
C1 terms with zero coefficients
C1 C1 min ci 3 that can be dropped
1
i S 1
+
1 1 1
min ci 4 2 i.e. only distinct distances
2 3 0 i S are meaningful (like in
1
+
1 2 1 min ci 5 Cornuejols’ and Elloumi’s
i S
4
4 4 2 model)
32

33. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C1
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2
i S
+
C1 1 C1 min ci 3
i S
+
1 1 1
min ci 4
2 3 0 i S
+
1 2 1 min ci 5
i S
4 4 2
33

34. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C1
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2
i S
+
C1 1 C1 min ci 3
i S
+
1 1 1
min ci 4
2 3 0 i S
+
1 2 1 min ci 5
i S
4 4 2
34

35. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C2
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2 1 1 y 2 1 y 2 y3 3 y 2 y3 y 4
i S
+
C2 2 C2 min ci 3
i S
+
6 2 1
min ci 4
1 3 1 i S
+
2 4 1 min ci 5
i S
3 1 3
35

36. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C3
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2 1 1 y 2 1 y 2 y3 3 y 2 y3 y 4
i S
+
C3 3 C3 min ci 3 1 1y 4 1y 2 y 4 2 y 2 y3 y 4
i S
+
5 4 1
min ci 4
2 2 1 i S
+
3 3 1 min ci 5
i S
1 1 2
36

37. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C4
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2 1 1 y 2 1 y 2 y3 3 y 2 y3 y 4
i S
+
C4 4 C4 min ci 3 1 1y 4 1y 2 y 4 2 y 2 y3 y 4
i S
+
3 1 3
min ci 4 3 0 y1 0 y1 y 2 5 y1 y 2 y3
3 2 0 i S
+
3 3 0 min ci 5
i S
8 4 5
37

38. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C5
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2 1 1 y 2 1 y 2 y3 3 y 2 y3 y 4
i S
+
C5 5 C5 min ci 3 1 1y 4 1y 2 y 4 2 y 2 y3 y 4
i S
+
4 4 2
min ci 4 3 0 y1 0 y1 y 2 5 y1 y 2 y3
5 3 1 i S
+
3 1 1 min ci 5 2 1y 4 1y3 y 4 1y1 y3 y 4
i S
2 2 1
38

39. PMP: pseudo-Boolean formulation
5
BC (y ) min cij
i S
j 1
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3 BC(y) can be constructed
i S
+
min ci 2 1 1 y 2 1 y 2 y3 3 y 2 y3 y 4 in polynomial time
i S
+
min ci 3 1 1y 4 1y 2 y 4 2 y 2 y3 y 4
i S
BC(y) has polynomial size
+
min ci 4 3 0 y1 0 y1 y 2 5 y1 y 2 y3 (number of terms)
i S
+
min ci 5 2 1y 4 1y3 y 4 1y1 y3 y 4
i S
39

40. PMP: pseudo-Boolean formulation
1 6 5 3 4
2 1 2 3 5
C
1 2 3 3 3
1 2 3 4 5 4 3 1 8 2 1 2 3 4 5
1 2 4 1 4 3 2 4 1 4
3 3 2 2 3 two possible 1 3 2 3 3
1
2 4 3 3 1 permutation 1
2 4 3 2 1
4 2 1 4 2 matrices 4 2 1 4 2
but
1 0 y1 1 y1 y3 2 y1 y 2 y3 1 0 y3 1 y1 y3 2 y1 y 2 y3
a unique polynomial
+ + + +
+ + + +
1 1 y 2 1 y 2 y3 3 y 2 y3 y 4 1 1 y 2 1 y 2 y3 3 y 2 y3 y 4
1 1y 4 1y 2 y 4 2 y 2 y3 y 4
1 1y 4 1y 2 y 4
3 0 y1 0 y1 y 2
2 y 2 y3 y 4
5 y1 y 2 y3
= BC (y) = 3 0 y1 0 y1 y3 5 y1 y 2 y3
2 1 y 4 1 y3 y 4 1 y1 y3 y 4 2 1 y 4 1 y3 y 4 1 y1 y3 y 4
40

41. PBP: combining similar terms
1 0 y1 1 y1 y3 2 y1 y 2 y3
+
1 1 y 2 1 y 2 y3 3 y 2 y3 y 4
20 terms
+
1 1 y 4 1 y 2 y 4 2 y 2 y3 y 4
17 nonzero terms
+
3 0 y1 0 y1 y 2 5 y1 y 2 y3
+
2 1y 4 1y3 y 4 1y1 y3 y 4
=
8 1y2 2 y4 1y1 y3 1y2 y3 1y2 y4 1y3 y4 7 y1 y2 y3 1y1 y3 y4 5 y2 y3 y4
10 terms
This procedure is equivalent to application of Elloumi’s Rule R2
PBP formulation allows compact representation of the problem !
In the given example 50% reduction is achieved!
41

42. PBP: combining similar terms
42

43. PBP: truncation
p=2
Initial polynomial BC (y) (10 terms):
8 1y2 2 y4 1y1 y3 1y2 y3 1y2 y4 1y3 y4 7 y1 y2 y3 1y1 y3 y4 5 y2 y3 y4
If p=2 each cubic term
Observation:
contains at least one zero
The degree of the pseudo-Boolean variable
polynomial is at most m-p
Truncated polynomial BC,p=2 (y) (7 terms):
8 1y2 2 y4 1y1 y3 1y2 y3 1y2 y4 1y3 y4
Truncation allows further
reduction of the problem size!
43

44. PBP: truncation
1 0 y1 1 y1 y3 2 y1 y 2 y3
If p=m/2+1 then memory
+
1 1 y 2 1 y 2 y3 3 y 2 y3 y 4 needed to store the polynomial
+
1 1 y 4 1 y 2 y 4 2 y 2 y3 y 4 is halved!
+
3 0 y1 0 y1 y 2 5 y1 y 2 y3
full polynomial
+
2 1y 4 1y3 y 4 1y1 y3 y 4
p=2
MEMORY
p=3
p=4 truncated
polynomial
p = m/2+1
44

45. PMP: pseudo-Boolean formulation
BC (y) min
y, yi m p
i I
C3
1 6 5 3 4 5
2 1 2 3 5 BC (y ) min cij
i S
C j 1
1 2 3 3 3
min ci1 1 0 y1 1y1 y3 2 y1 y 2 y3
4 3 1 8 2 i S
+
min ci 2 1 1 y 2 1 y 2 y3 3 y 2 y3 y 4
i S
+
C3 3 C3 min ci 3 1 1y 4 1y 2 y 4 2 y 2 y3 y 4
i S
+
5 4 1
min ci 4
2 2 1 i S
+
3 3 1 min ci 5
i S
1 1 2
45

46. Truncation and preprocessing
Initial matrix p-truncated matrix, p=3
1 6 5 3 4 1 1 2 2 3 3
2 1 2 3 5 2 1 1 2 3 3
C C3
1 2 3 3 3 3 1 2 2 3 3 y3=1
4 3 1 8 2 4 1 2 1 3 2
If i-th row contains all maximum
elements, then corresponding In truncated matrix
location can be excluded from this is more likely
consideration ( yi can be set to 0). to happen
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.
46

47. 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
47

48. MBpBM: linearization
1 6 5 3 4
2 1 2 3 5
C p=2
1 2 3 3 3
Example of the pseudo-Boolean
4 3 1 8 2
polynomial:
8 y2 2 y4 y1 y3 y2 y3 y2 y4 y3 y4
8 y2 2 y4 z5 z6 z7 z8
Linear function of new variables:
y1 , y 2 , y3 , y 4 ,
z5 y1 y3 , z 6 y 2 y3 , z 7 y 2 y 4 , z8 y3 y 4
Compare: in Elloumi’s model variables y2 and y4 were introduced into
objective via Rule R1. 48

49. MBpBM: constraints
l l
Simple fact: z yk z yk l 1, yk {0,1}
k 1 k 1
Example:
z5 y1 y3 z5 1 y1 y3
z6 y 2 y3 z6 1 y2 y3
z7 y2 y4 z7 1 y2 y4
z8 y3 y 4 z8 1 y3 y4
yk {0 ,1}k 1...4
nonnegativity is
zk 0k 5...8 sufficient !
49

50. 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.
50

51. MBpBM: reduction
• set covering problem
y1 y3 y4 y1 y3 y5 y1 y3 y6
y1 y3 y4 y5 y6 y9
y1 y3 y1 y3 y9
y4 y5 y6 y9
51

52. MBpBM: reduction
• set covering problem
y1 y3 y4 y1 y3 y5 y1 y3 y6
NP-hard! y1 y3 y4 y5 y6 y9
y1 y3 y1 y3 y9
y4 y5 y6 y9
y1 y3 y4 y5 y6 y9 y1 y3 y4 y5 y6 y9 2 52

53. Example, p=2; S. Elloumi, J Comb Optim 2010,19:69–83
1 6 5 3 4
2 1 2 3 5
C
1 2 3 3 3
4 3 1 8 2
Objective:
8 y2 2 y4 z5 z6 z7 z8
Constraints:
y1 y2 y3 y4 2
z5 1 y1 y3 7 coefficients.
z6 1 y2 y3 5 linear constr.
z7 1 y2 y4 zi 0i 5,...,
8 4 non-negativity
constr.
z8 1 y3 y4 yi {0,1}i 1,...,
4
4 Boolean constr.
In Elloumi’s model these figures are, correspondingly, 10 (13), 11 (23), 7(12) 53
and 4

54. Comparison of the models
our MBpBM Elloumi’s NF
8 y2 2 y4 z5 z6 z7 z8 8 (1 y2 ) 2(1 y4 ) z11 7z21 z22 5z32 z23 z25 z35
y1 y2 y3 y4 2 y1 y2 y3 y4 2
z11 y1 y3 1
z5 1 y1 y3
z 21 y2 z11
z6 1 y2 y3 y4 z 21
z7 1 y2 y4 z 22 y3 1 y2
z 32 y4 z 22
z8 1 y3 y4
y1 z 32
zi 0i 5,...,
8 z 23 y2 1 y4
z 25 y3 1 y4
yi {0,1}i 1,...,
4
z 35 y1 z 25 z kj 0j 1,..., ; k 1,...,
5 3
y2 z 35 yi {0,1}i 1,...,
4
54

55. 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
55

56. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective:
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4
p 2
Constraints:
UB
f 9 (by greedy heuristic)
y1 y2 y3 y4 2
z5 1 y1 y3
z6 1 y2 y3
z7 1 y2 y4
z8 1 y3 y4
zj 0j 5,...,
8
yj {0,1} j 1,...,
4
56

57. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective: consider some term
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 2 y4 z5 z6 z7 z8
p 2
Constraints:
UB
f 9 (by greedy heuristic)
y1 y2 y3 y4 2
f (y 34 ) 11 f UB z8 y3 y 4 0 z5 1 y1 y3
thus, z8 can be deleted z6 1 y2 y3
from the model
z7 1 y2 y4
z8 1 y3 y4
zj 0j 5,...,
8
yj {0,1} j 1,...,
4
Tr
def 57
yi 1 iff yi Tr

58. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective: consider next term
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 2 y4 z5 z6 z7 z8
p 2
Constraints:
UB
f 9 (by greedy heuristic) y1 y2 y3 y4 2
f (y 34 ) 11 f UB z8 y2 y3 0 z5 1 y1 y3
f (y 24 ) 12 f UB z7 y2 y4 0 z6 1 y2 y3
thus, z7 can be deleted z7 1 y2 y4
from the model
z8 1 y3 y4
zj 0j 5,...,
7
yj {0,1} j 1,...,
4
Tr
def 58
yi 1 iff yi Tr

59. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective: and so on …
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 2 y4 z5 z6 z7 z8
p 2
Constraints:
UB
f 9 (by greedy heuristic) y1 y2 y3 y4 2
f (y 34 ) 11 f UB z8 y3 y 4 0 z5 1 y1 y3
f (y 24 ) 12 f UB z7 y2 y4 0 z6 1 y2 y3
f (y 23 ) 10 f UB z6 y 2 y3 0 z7 1 y2 y4
z8 1 y3 y4
zj 0j 5,...,
6
yj {0,1} j 1,...,
4
Tr
def 59
yi 1 iff yi Tr

60. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective: and so on …
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 2 y4 z5 z6 z7 z8
p 2
Constraints:
UB
f 9 (by greedy heuristic)
y1 y2 y3 y4 2
f (y 34 ) 11 f UB z8 y3 y 4 0 z5 1 y1 y3
f (y 24 ) 12 f UB z7 y2 y4 0 z6 1 y2 y3
f (y 23 ) 10 f UB z6 y 2 y3 0 z7 1 y2 y4
f (y13 ) 9 z8 1 y3 y4
z5 0
yj {0,1} j 1,...,
4
Tr
def 60
yi 1 iff yi Tr

61. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective: and so on …
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 2 y4 z5 z6 z7 z8
p 2
Constraints:
UB
f 9 (by greedy heuristic)
y1 y2 y3 y4 2
34 UB
f (y ) 11 f z8 y3 y 4 0
z5 1 y1 y3
f (y 24 ) 12 f UB z7 y2 y4 0 z6 1 y2 y3
f (y 23 ) 10 f UB z6 y 2 y3 0 z7 1 y2 y4
f (y13 ) 9 z8 1 y3 y4
f (y 4 ) 10 f UB y4 0 z5 0
yj {0,1} j 1,...,
4
Tr
def 61
yi 1 iff yi Tr

62. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective: and so on …
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 2 y4 z5 z6 z7 z8
p 2
Constraints:
UB
f 9 (by greedy heuristic)
y1 y2 y3 y4 2
34 UB
f (y ) 11 f z8 y3 y 4 0
z5 1 y1 y3
24 UB
f (y ) 12 f z7 y2 y4 0
z6 1 y2 y3
f (y 23 ) 10 f UB z6 y 2 y3 0 z7 1 y2 y4
f (y13 ) 9 z8 1 y3 y4
f (y 4 ) 10 f UB y4 0 z5 0
f (y 2 ) 9 yj {0,1} j 1,...,
4
Tr
def 62
yi 1 iff yi Tr

63. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective:
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 2 y4 z5 z6 z7 z8
p 2
Constraints:
UB
f 9 (by greedy heuristic)
y1 y2 y3 y4 2
f (y 34 ) 11 f UB z8 y3 y 4 0
z5 1 y1 y3
24 UB
f (y ) 12 f z7 y2 y4 0 z6 1 y2 y3
f (y 23 ) 10 f UB z6 y 2 y3 0 z7 1 y2 y4 unnecessary
13
f (y ) 9 z8 1 y3 y4 restrictions !
f (y 4 ) 10 f UB y4 0 z5 0
f (y 2 ) 9 yj {0,1} j 1,...,
4
63

64. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective:
2 1 2 3 5 z6 y 2 y3 8 y2 2 y4 z5 z6 z7 z8
C
1 2 3 3 3 z7 y2 y4
4 3 1 8 2 z8 y3 y 4 8 y2 z5
p 2
Constraints:
UB
f 9 (by greedy heuristic)
y1 y2 y3 0 2
f (y 34 ) 11 f UB z8 y3 y 4 0 0 1 y1 y3
f (y 24 ) 12 f UB z7 y2 y4 0 0 1 y2 y3
f (y 23 ) 10 f UB z6 y 2 y3 0
f (y13 ) 9
f (y 4 ) 10 f UB y4 0 z5 0
f (y 2 ) 9 yj {0,1} j 1,...,
4
64

65. MBpBM: preprocessing (example)
1 6 5 3 4 z5 y1 y3 Objective:
2 1 2 3 5 z6 y 2 y3 8 y2 z5
C
1 2 3 3 3 z7 y2 y4 Constraints:
4 3 1 8 2 z8 y3 y 4 y1 y2 y3 2
p 2
1 y1 y3
1 y2 y3
3 (10) coefficients
z5 0
3 (11) linear constr.
yj {0,1} j 1,...,
4
1 (7) non-negativity constr.
4 Boolean (1 fixed to 0)
y4 0
Note: number of Boolean variables is 4 in all considered models.
65

66. 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
66

67. MBpBM: preprocessing (impact)
results from P. Avella and A. Sforza, Logical reduction tests
for the p-median problem, Ann. Oper. Res. 86, 1999, pp. 105–115.
our results 67

68. Computational results
OR-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)
68

69. Computational results
Results for different number of medians for two OR instances
69

70. Computational results
Results 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) 70

71. Computational results
Running times (sec.) for 15 largest OR-library instances
71

72. Computational results
Running times (sec.) for RW instances
72

73. Results for
our
complex
instances
73

74. 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
74

75. 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
75

76. 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 detail
76

77. Literature
• 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)
• Boros, E., Hammer, P.L.: Pseudo-Boolean optimization. Discrete Applied
Mathematics, 123, 155-225 (2002)
• 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)
• 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., AlBdaiwi B.F.: Complexity evaluation of
benchmark instances for the p-median problem (submitted to Mathematical and
Computer Modelling )
77

78. Literature (contd.)
• Hammer, P.L.: Plant location -- a pseudo-Boolean approach. Israel Journal
of Technology, 6, 330-332 (1968)
• Mladenovic, N., Brimberg, J., Hansen, P., Moreno-Perez, J.A.: The p-
median problem: A survey of metaheuristic approaches. European Journal
of Operational Research, 179, 927-939 (2007)
• 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)
78

79. Application to Cell Formation
parts
1 2 3 4 5
Example 1: 0 1 0 1 1
1
Machine-part
machines
functional 1 0 1 0 0
2
incidence matrix
grouping 0 1 1 1 0
3
4
1 0 1 0 0
5
0 1 0 0 1
The task is to group machines into clusters (manufacturing cells)
such that to to minimize intercell communication.
Dissimilarity measure for machines
number of parts that need both machines i and j
d (i, j )
number of parts that need either of the machines
79

80. Application to Cell Formation
Example 1: functional grouping (contd.)
machines
Cost matrix for the PMP 0 1.00 0.50 1.00 0.33
machines
is a machine-machine 1.00 0 0.75 0 1.00
dissimilarity matrix: 0.50 0.75 0 0.75 0.75
1.00 0 0.75 0 1.00
c[i, j] : d (i, j)
0.33 1.00 0.75 1.00 0
parts
2 4 5 1 3 intercell communication is
1 1 1 0 0 caused by only part # 3
1
In case of
machines
1 1 0 0 1 that is processed in both
3
two cells cells
1 0 1 0 0
5
the solution
4
0 0 0 1 1
is:
2
0 0 0 1 1 80

81. Application to Cell Formation
Example 1: functional grouping (contd.)
0 1.00 0.50 1.00 0.33 BC (y ) 0.33 y1 0.16 y1 y5 0.25 y 2 y3 y 4
1.00 0 0.75 0 1.00 0 y2 0.75 y 2 y 4 0.25 y 2 y3 y 4
C 0.50 0.75 0 0.75 0.75 0.5 y3 0.25 y1 y3 0 y1 y 2 y3
1.00 0 0.75 0 1.00 0 y2 0.75 y 2 y 4 0.25 y 2 y3 y 4
0.33 1.00 0.75 1.00 0 0.33 y5 0.42 y1 y5 0.25 y1 y3 y5
BC , p 2 (y ) 0.33 y1 0.5 y3 0.33 y5 0.58 y1 y5 1.5 y 2 y 4 0.25 y1 y3 0.75 y1 y3 y5 0.5 y 2 y3 y4
Linearization:
f (y, z) 0.33y1 0.5 y3 0.33y5 0.58z6 1.5z7 0.25z8 0.75z9 0.5z10
where:
z6 y1 y5 z9 y1 y3 y5
z7 y2 y4 z10 y2 y3 y4
z8 y1 y3
81

82. Application to Cell Formation
Example 1: functional grouping (contd.)
MBpBM
0.33 y1 0.5 y3 0.33 y5 0.58z 6 1.5 z 7 0.25z8 0.75z9 0.5 z10 min
s.t.
y1 y 2 y3 y4 y5 5 2 MBpBM with reduction based on bounds
z 6 1 y1 y5 0.33 y1 0.5 y3 0.33 y5 min
z7 1 y2 y4 s.t
z8 1 y1 y3 y1 y 2 y3 y4 y5 5 2
z9 2 y1 y3 y5 0 1 y1 y5 0
z10 2 y2 y3 y4 0 1 y2 y4 1
0 1 y1 y3 y* 1
yi {0,1}i 1..5
0 2 y1 y3 y5 0
zi 0i 6..10
0 2 y2 y3 y4 1
yi {0,1}i 1..5
82

83. Application to Cell Formation
workers
1 2 3 4 5 6 7 8
Example 2: 1 0 0 0 1 0 1 0
1
machines
1 1 0 0 0 1 0 0
2
workforce Machine-worker
3
incidence matrix 0 1 1 0 1 0 0 1
expences
4
0 0 1 1 0 1 0 0
5
0 0 0 1 0 0 1 1
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.
Dissimilarity measure for machines
number of workers that can operate both machines i and j
d (i, j )
number of workers that can operate either of the machines
83

84. Application to Cell Formation
Example 2: workforce expences (contd.)
machines
Cost matrix for the PMP 0 0.80 0.83 1.00 0.80
machines
is a machine-machine 0.80 0 0.83 0.80 1.00
dissimilarity matrix: 0.83 0.83 0 0.83 0.83
1.00 0.80 0.83 0 0.80
c[i, j] : d (i, j)
0.80 1.00 0.83 0.80 0
workers
2 3 5 8 1 4 6 7 1 worker needs
1 1 1 1 1 0 0 0
3
additional training
In case of
machines
0 1 0 0 0 1 1 0
4
three cells 7 non-clustered
2
1 0 0 0 1 0 1 0 elements that
the solution represent the skills that
5
0 0 0 1 0 1 0 1
is: are not used (potential
1
0 0 1 0 1 0 0 1 overpayment) 84

85. Application to Cell Formation
Example 2: workforce expences (contd.)
0 0.80 0.83 1.00 0.80 BC (y ) 0.80 y1 0 y1 y2 0.03 y1 y2 y5 0.17 y1 y2 y3 y5
0.80 0 0.83 0.80 1.00 0.80 y2 0 y1 y2 0.03 y1 y2 y4 0.17 y1 y2 y3 y4
C 0.83 0.83 0 0.83 0.83 0.83 y3 0 y1 y3 0 y1 y2 y3 0 y1 y2 y3 y4
1.00 0.80 0.83 0 0.80 0.80 y4 0 y2 y4 0.03 y2 y4 y5 0.17 y2 y3 y4 y5
0.80 1.00 0.83 0.80 0 0.80 y5 0 y1 y5 0.03 y1 y4 y5 0.17 y1 y3 y4 y5
BC , p 3 (y ) 0.8 y1 0.8 y2 0.83 y3 0.8 y 4 0.8 y5
The objective is already a linear function !
85

86. Application to Cell Formation
Example 2: workforce expences (contd.)
MBpBM
0.8 y1 0.8 y 2 0.83 y3 0.8 y 4 0.8 y5 min
s.t.
y1 y2 y3 y4 y5 5 3
yi {0,1}i 1..5
1
1
y* 0
0
0
86

87. Application to Cell Formation
Example 3: from Yang,Yang (2008)*
105 parts
45 machines
(uncapacitated)
functional grouping
105 parts
grouping efficiency:
45 machines
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
87
ART1 method. EJOR, vol. 188, pp. 140-152

88. Thank you!
• Questions?
88

89. The PMP: alternative formulation, Cornuejols et al. 1980
Kj
Let for each client j D1 ,...,D j
j - sorted (distinct) distances (Kj – number
of distinct distances for j-th client)
1
1 6 5 3 4 client1 : D1 1 D12 2 3
D1 4
2 1 2 3 5
C
1 2 3 3 3
4 3 1 8 2
89

90. The PMP: alternative formulation, Cornuejols et al. 1980
Kj
Let for each client j D1 ,...,D j
j - sorted (distinct) distances (Kj – number
of distinct distances for j-th client)
1
1 6 5 3 4 client1 : D1 1 D12 2 3
D1 4
1 2 3 4
2 1 2 3 5 client2 : D2 1 D2 2 D2 3 D2 6
C
1 2 3 3 3 1 2 3 4
client3 : D3 1 D3 2 D3 3 D3 5
4 3 1 8 2 1 2
client4 : D4 3 D4 8
1 2 3 4
client5 : D5 2 D5 3 D5 4 D5 5
90

91. The PMP: alternative formulation, Cornuejols et al. 1980
Kj
Let for each client j D1 ,...,D j
j - sorted (distinct) distances (Kj – number
of distinct distances for j-th client)
1
1 6 5 3 4 client1 : D1 1 D12 2 3
D1 4
1 2 3 4
2 1 2 3 5 client2 : D2 1 D2 2 D2 3 D2 6
C
1 2 3 3 3 1 2 3 4
client3 : D3 1 D3 2 D3 3 D3 5
4 3 1 8 2 1 2
client4 : D4 3 D4 8
1 2 3 4
client5 : D5 2 D5 3 D5 4 D5 5
Decision variables
0, if at least one site within distance D k is opened
j
zk
j
1, if all sites within distance D k are closed
j
Kj Kj 1 Kj 1
min cij D1
j (D 2
j D1 ) z1
j j ... (D j D j )z j
i S
S - set of opened plants 91

92. The PMP: alternative formulation, Cornuejols et al. 1980
n Ki 1
f ( z, y ) Di1 ( Dik 1
Dik ) zik min
j 1 k 1
m s.t.
yi p - p opened facilities
i 1
k
D
- either at least one facility is open within i
zik yj 1, i 1,...,
n
k 1,..., i
K or z i
k
1
j:d ij Dik
- for every client it is an opened facility in some
ziKi 0, i 1,...,n neighbourhood
z ik 0, i 1,..., n - zi
k
1 iff all the sites within Dik are
k 1,..., K i closed
yj {0,1}, j 1,...,m
for each client i Di ,...,DiKi
1
- sorted distances 92

93. The PMP: alternative formulation, Cornuejols et al. 1980
Example
(Elloumi,2009) 1
1 6 5 3 4 client1 : D1 1 D12 2 3
D1 4
1 2 3 4
2 1 2 3 5 client2 : D2 1 D2 2 D2 3 D2 6
C 1 2 3 4
1 2 3 3 3 client3 : D3 1 D3 2 D3 3 D3 5
4 3 1 8 2 client4 : D1 3 2
D4 8
4
1 2 3 4
client5 : D5 2 D5 3 D5 4 D5 5
Objective:
client1 : 1 (2 1) z11 (4 2) z 21
+
client2 : 1 (2 1) z12 (3 2) z 22 (6 3) z 32 only distinct
+
client3 : 1 (2 1) z13 (3 2) z 23 (5 3) z 33 (in a column)
distances are
+
client4 : 3 (8 3) z14 meaningful
+
client5 : 2 (3 2) z15 (4 3) z 25 (5 4) z 35
8 z11 2 z 21 z12 z 22 3 z 32 z13 z 23 2 z 33 5 z14 z15 z 25 z 35
13 coefficients 93

94. The PMP: alternative formulation, Cornuejols et al. 1980
Example 1 6 5 3 4
1
2 1 2 3 5
plants
2 3
1
C client1 : D1 1 D12 2 3
D1 4
1 2 3 3 3
4 3 1 8 2
4
Constraints:
client1 :
z11 y1 y3 1 if plants 1 and 3 are closed ( y1 0, y3 0)
z 21 y1 y3 y2 1
then all plants within distance D11=1 are closed
z31 y1 y3 y2 y4 1
and z11 1
94

95. The PMP: alternative formulation, Cornuejols et al. 1980
Example 1 6 5 3 4 Objective:
2 1 2 3 5 8 z11 2 z 21 z12 z 22 3 z 32 z13
C
1 2 3 3 3 z 23 2 z 33 5 z14 z15 z 25 z 35
4 3 1 8 2
Constraints:
y1 y 2 y3 y 4 p z 22 y2 y3 1 z 35 y1 y3 y4 1
z11 y1 y3 1 z 23 y2 y4 1 z 42 y1 y2 y3 y4 1
z12 y2 1 z 24 y1 y2 y3 y4 1 z 43 y1 y2 y3 y4 1
z13 y4 1 z 25 y3 y4 1 z 45 y1 y2 y3 y4 1
z14 y1 y2 y3 1 z31 y1 y2 y3 y4 1 z 31 0, z 42 0, z 43 0
z15 y4 1 z32 y2 y3 y4 1 z 24 0, z 45 0
z 21 y1 y2 y3 1 z33 y2 y3 y4 1 z jk 0 j 1,...,5; k 1,..., j
K
yi {0,1}i 1,...,4
13 coefficients, 23 linear constr., 12 non-negativity constr., 4 Boolean 95

96. The PMP: a tighter formulation, Elloumi 2009
k
A possible definition of variables z j :
zk
j (1 yi ), j 1,...,n; k 1,...,K j
i:cij D k
j
Or recursively:
z1
j (1 yi ), j 1,...,n;
i:cij D1j
zk
j zk 1
j (1 yi ), j 1,...,n; k 2 ,...,Kj
i:cij D k
j
Thus:
z11 y1 y3 1 z11 y1 y3 1
e.g. is equivalent to
z 21 y1 y2 y3 1 z 21 y2 z11
96

97. The PMP: a tighter formulation, Elloumi 2009
n Kj 1
f (z, y ) D1
j (D k
j
1
Dk )z k
j j min
j 1 k 1
m s.t.
yi p
i 1
j 1,...,
n
z1
j yi 1, j 1,...,n zk
j yi z k 1,
j k 2,..., j
K
i:cij D k
j i:cij D k
j
Kj
zj 0, j 1,...,n
j 1,...,
n
j 1,...,
n zk yi 1,
zk
j 0, k 1,..., j
K
j k 1,..., j
K
i:cij D k
j
yi {0,1}, i 1,...,m Cornuejols et al. 1980
1 Kj
for each client j D j ,...,D j - sorted distances 97

98. The PMP: a tighter formulation, Elloumi 2009
Let V j k set of facilities within D j k : V j k {i : cij Djk}
Rule R1 : For any client j, if V j1 is a singleton { yi } then z j1 1 yi holds for any
feasible solution. Variable z j1 can be substituted by (1 yi ) and constraint z j1 yi 1
that defines variable z j1 can be eliminated .
1 2 3 3
1 6 5 3 4 client 2 : D2 1 D2 2 D2 3 D2 6
2 1 2 3 5 some facility
C facility 2
1 2 3 3 3 1
within D2 is open is open
4 3 1 8 2
( z1
2 0) ( y 2 1)
Informally:
if for client j some neighbourhood
k contains only one facility i then
there is a simple relation between z1
2 1 y2
k
corresponding variables z j 1 yi 98

99. The PMP: a tighter formulation, Elloumi 2009
Rule R2 : If for any j, k, j', k', Vj k Vi' k' then z j k z j' k' holds for any feasible solution.
Variable z j' k' can be replaced by z j k and constraint z j' k' k' y j z j' k'-1 that
i:cij' D j'
defines variable z j' k' can be eliminated .
1
1 6 5 3 4 client 1 : D1 1 D12 2 3
D1 4
2 1 2 3 5 V11 {1,3} V12 {1,2 ,3} V13 {1,2 ,3,4}
C
1 2 3 3 3
1 2
4 3 1 8 2 client 4 : D4 3 D4 8
1
V4 {1,2,3} V42 {1,2,3,4}
Informally:
if two clients have equal
neighbourhoods then the
corresponding z-variables are
2
equivalent and in the objective z1 z1
4
function terms containing them
99
can be added.

100. The PMP: a tighter formulation, Elloumi 2009
Kj 1 K j' 1
Rule R3 : If for any j, j', V j V j' then Rule R2 can be applied to deduce
Kj 1 K 1 Kj
that z j z j' j' . Further,in this case, the set of facilities i such thatcij Dj
K Kj K
is equal to the sets of facilities i such thatcij' D j' j' . Finally, as z j z j' j' 0, we
K
can eliminate constraint z j' j' K yi z j' K j' -1 .
i:cij' D j' j'
1 2 3 3
client 2 : D2 1 D2 2 D2 3 D2 6
V21 {2} V22 {2,3} V23 {2 ,3,4} V23 {1,2,3,4}
1 2 3 3
client 3 : D3 1 D3 2 D3 3 D3 5
1
V3 {4} V32 {2,4} V33 {2,3,4} V33 {1,2 ,3,4}
4 3
z3 y1 z3
after applying Rule R2 becomes
redundant and can be eliminated 100

101. Example (from Elloumi, 2009)
1 6 5 3 4
2 1 2 3 5
C
1 2 3 3 3
4 3 1 8 2
Objective:
8 (1 y2 ) 2(1 y4 ) z11 7 z21 z22 5z32 z23 z25 z35
Constraints:
y1 y2 y3 y4 p y1 z 32
z11 y1 y3 1 z 23 y2 1 y4 10 (13) coefficients
z 21 y2 z11 z 25 y3 1 y4 11 (23) linear constr.
y4 z 21 z 35 y1 z 25 7 (12) non-negativity
constr.
z 22 y3 1 y2 y2 z 35
4 Boolean constr.
z32 y4 z 22 z ki 0
yj {0,1} j 1,...,
4 101

102. The PMP: a tighter formulation, Elloumi 2009
n Ki 1
f ( z, y ) Di1 ( Dik 1
Dik ) zik min
j 1 k 1
m s.t.
yi p
i 1
additional constraints
zik yj 1, i 1,...,
n
k 1,..., i
K zik yj 1, i 1,...,
n
k 2,..., i
K
j:d ij Dik j:d ij Dik
ziKi 0, i 1,...,n + reduction rules
(next slide)
z ik 0, i 1,..., n
k 1,..., K i
yj {0,1}, j 1,...,m
for each client i Di ,...,DiKi
1
- sorted distances 102

103. The p-Median Problem:
a tighter formulation Elloumi 2009
103

104. MBpBM: preprocessing
f UB some (global) upper bound
m
if for some y : yi m p holds f(y) f UB
i 1
then every y' satisfying yi' 1 yi 1
is not an optimal solution.
I.e. if for some monomial Tr yi holds f (y Tr ) f UB
yi Tr
then for every optimal solution Tr 0
and we can exclude Tr from the objective
and add a constraint yi 0
yi Tr
def
Tr
yi 1 iff yi Tr
104

105. MBpBM: preprocessing
Claim:
The inequality f (y Tr ) f UB must be strict.
Counter-example (p=2):
We can show that if f (y Tr ) f UB the previous assertion is violated :
0 6 6 1 2 4 BC , p 2 (y ) 1y1 4 y 2 6 y 4 1y1 y 2 3 y1 y 4 2 y2 y4
1 0 8 2 4 1 Let T1 y1 , T2 4 y 2 , T3 6 y4
2 9 9 3 1 2 f UB 1
f (y T1 ) f UB y1 0
f (y T1 ) 1
5 4 0 4 3 3
cost permuta- f (y T2 ) 4
suppose 1
matrix tion T3
f (y ) 6 0
y opt
1
0
But in the unique optimal solution y1=1 !
105