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# p-Median Cluster Analysis Based on General-Purpose Solvers (2)

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AACIMP 2010 Summer School lecture by Dmitry Krushinsky. "Applied Mathematics" stream. "The p-Median Problem and Its Applications" course. Part 2.

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### p-Median Cluster Analysis Based on General-Purpose Solvers (2)

1. 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. 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. 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. 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. 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. 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. 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. 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. 9. The PMP: Applications • Facility location - consumer (client) - possible location of supplier (server) 9 - supplier (server), e.g. supermarket, bakery, laundry, etc.
10. 10. The PMP: Applications • Facility location - consumer (client) - possible location of supplier (server) 10 - supplier (server), e.g. supermarket, bakery, laundry, etc.
11. 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. 12. The PMP: Applications • Quantitative psychology patients symptoms (behavioural patterns) type 1 mentality features type 2 mentality features 12 “leaders” or typical representatives
13. 13. The PMP: Applications • Telecommunications industry 13
14. 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. 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. 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. 17. The PMP: Applications • Optimal diversity management – Example: wiring designs, p=3 configurations with zero demand 17
18. 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. 19. The PMP: Applications • Vehicle routing - clients / storage depot - vehicle routes 19
20. 20. The PMP: Applications • Topological design of computer and communication networks 20
21. 21. The PMP: Applications • Topological design of computer and communication networks 21
22. 22. The PMP: Applications • Topological design of computer and communication networks 22
23. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 42. PBP: combining similar terms 42
43. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 69. Computational results Results for different number of medians for two OR instances 69
70. 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. 71. Computational results Running times (sec.) for 15 largest OR-library instances 71
72. 72. Computational results Running times (sec.) for RW instances 72
73. 73. Results for our complex instances 73
74. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 88. Thank you! • Questions? 88
89. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 103. The p-Median Problem: a tighter formulation Elloumi 2009 103
104. 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. 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
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