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p-Median Cluster Analysis Based
  on General-Purpose Solvers


Boris Goldengorin, Dmitry Krushinsky
       University of G...
Outline of the talk
• Two Main PMP formulations
• Pseudo-Boolean polynomial
• Mixed Boolean pseudo-Boolean Model
  (MBpBM)...
The p-Median Problem (PMP)
                  I = {1,…,m} – a set of m facilities (location points),
                  J = ...
The PMP: combinatorial formulation
The p-Median Problem (PMP) consists of determining p locations
(the median points) such...
The PMP: combinatorial formulation


    f C (S )              min cij                    min
                           i...
The PMP: Applications
• Facilty location
• Cluster analysis
• Quantitative psychology
• Telecommunications industry
• Sale...
Similarities and dissimilarities
 Proc Natl Acad Sci U S A. 1996 Jun
 11;93(12):5854-9.
 Similarities and dissimilarities ...
A comparative study of similarity measures for manufacturing cell
                           formation
               S. O...
The PMP: Applications
• Facility location




      - consumer (client)
      - possible location of supplier (server)
   ...
The PMP: Applications
• Facility location




      - consumer (client)
      - possible location of supplier (server)
   ...
The PMP: Applications
• Cluster analysis                        Output
Input:
                            cluster   cluste...
The PMP: Applications
• Quantitative psychology

patients         symptoms
            (behavioural patterns)             ...
The PMP: Applications
• Telecommunications industry




                                13
The PMP: Applications
• Sales force territories design
                            customers
                       (group...
The PMP: Applications
• Political and administrative districting
                              districts,
                ...
The PMP: Applications
• Optimal diversity management
  – given a variety of products (each having some
    demand, possibl...
The PMP: Applications
• Optimal diversity management
  – Example: wiring designs, p=3
    configurations
      with zero
 ...
The PMP: Applications
• Cell formation in group technology
   functional layout                  cellular layout

        ...
The PMP: Applications
• Vehicle routing



  - clients
  / storage           depot

  - vehicle
  routes




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




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




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




                                  ...
Publications, more than 500
Elloumi, 2010;
Brusco and K¨ohn, 2008;
Belenky, 2008;
Church, 2003; 2008;
Avella et al, 2007;
...
The PMP: Boolean Linear Programming
      Formulation (ReVelle and Swain, 1970)
                                 m   n
   ...
The PMP:    alternative formulation, Cornuejols et al. 1980

                                           Kj
        Let for...
The PMP: alternative formulation, Cornuejols et al. 1980

                                       Kj
    Let for each clien...
The PMP: alternative formulation, Cornuejols et al. 1980

                                        Kj
     Let for each cli...
The PMP: alternative formulation, Cornuejols et al. 1980

                                   n               Ki 1
        ...
The PMP: alternative formulation, Cornuejols et al. 1980

Example, p=2
(Elloumi,2010)                               1
    ...
The PMP: alternative formulation, Cornuejols et al. 1980

       Example             1 6 5 3 4             Objective:
    ...
The p-Median Problem:
               a tighter formulation, Elloumi 2010
        Let V j k    set of facilities within D j...
The p-Median Problem:
                 a tighter formulation, Elloumi 2010
Rule R2 : If for any j, k, j', k', V j k Vi' k'...
The p-Median Problem:
                a tighter formulation, Elloumi 2010
                               Kj 1         K j'...
The PMP: a tighter formulation, Elloumi 2010

A possible definition of variables          :     zk
                       ...
The PMP: a tighter formulation, Elloumi 2010

                               n                   Kj 1
             f ( z, ...
PMP Example with p=2 borrowed from S. Elloumi,
       J Comb Optim 2010,19:69–83
                                         ...
The PMP: pseudo-Boolean formulation
                       (Historical remarks)

• Hammer, 1968 for the Simple Plant Locat...
The PMP and SPLP differ in the following details

• SPLP involves fixed cost for location a facility at the given site, wh...
The PMP: pseudo-Boolean formulation
              Numerical Example: m=5, n=4, p=2



          1 6 5 3 4                 ...
PMP: pseudo-Boolean formulation
                        BC (y)              min
                                         y...
PMP: pseudo-Boolean formulation
                        BC (y)              min
                                         y...
PMP: pseudo-Boolean formulation
                        BC (y)              min
                                         y...
PMP: pseudo-Boolean formulation
                        BC (y)              min
                                         y...
PMP: pseudo-Boolean formulation
                    BC (y)              min
                                     y, yi m p...
PMP: pseudo-Boolean formulation
                     BC (y)              min
                                      y, yi m...
PMP: pseudo-Boolean formulation
                         BC (y)              min
                                         ...
PMP: pseudo-Boolean formulation
                    BC (y)              min
                                     y, yi m p...
PMP: pseudo-Boolean formulation
               5
    BC (y )          min cij
                     i S
               j 1
...
PMP: pseudo-Boolean formulation
                                                   1 6 5 3 4
                             ...
PBP: combining similar terms
                        1 0 y1 1 y1 y3 2 y1 y 2 y3




                      +
              ...
PBP: combining similar terms




                               51
PBP: truncation
                                     p=2
 Initial polynomial BC (y) (10 terms):
 8 1y 2 2 y 4 1y1 y3 1y 2 ...
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 y...
PMP: pseudo-Boolean formulation
                     BC (y)              min
                                      y, yi m...
Truncation and preprocessing
        Initial matrix                  p-truncated matrix, p=3
        1   6   5   3    4   ...
Pseudo-Boolean formulation:
              outcomes
• Compact but nonlinear problem
• Equivalent to a nonlinear knapsack (N...
MBpBM: linearization
                                                     1 6 5 3 4
                                      ...
MBpBM: constraints
                       l                           l
Simple fact: z              yk              z     ...
MBpBM: reduction

Lema:
Let    Ø            be a pair of embedded sets of Boolean variables yi.
Then, the two following sy...
MBpBM: reduction
• set covering problem
         y y y       y1 y3 y5                    y1 y3 y 6
          1 3 4


     ...
MBpBM: reduction
• set covering problem
         y y y       y1 y3 y5                          y1 y3 y 6
              1 3...
Example, p=2; S. Elloumi, J Comb Optim 2010,19:69–83

                                             1 6 5 3 4
             ...
Comparison of the models
          our MBpBM                                                    Elloumi’s NF
8    y2    2 ...
MBpBM: preprocessing
• every term (product of variables)
  corresponds to a subspace of solutions
  with all these variabl...
PMP: pseudo-Boolean formulation implies a decomposition
  of the search space into at most n(m-p) subspaces
              ...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5               y1 y 3   Objective:
   2 1 2 3 5 z6               y 2 y3   8 ...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                         y1 y 3       Objective:            consider some te...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                         y1 y 3    Objective:            consider next term
...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                         y1 y 3       Objective:            and so on …
   2...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                         y1 y 3       Objective:           and so on …
   2 ...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                           y1 y 3       Objective:           and so on …
   ...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                           y1 y 3       Objective:           and so on …
   ...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                 y1 y 3       Objective:
   2 1 2 3 5 z6                 y 2...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5               y1 y 3       Objective:
   2 1 2 3 5 z6               y 2 y3 ...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5         y1 y 3    Objective:
   2 1 2 3 5 z6         y 2 y3                ...
Preprocessing from linear to
          nonlinear terms

• The preprocessing should be done
  starting from linear terms......
MBpBM: preprocessing (impact)




results from P. Avella and A. Sforza, Logical reduction tests
for the p-median problem, ...
Computational results
                                                  OR-library instances




[3]   Avella P., Sassano ...
Computational results, m=900
Results for different number of medians for two OR instances




                            ...
Computational results
             Results for different numbers of medians in BN1284




[3] Avella P., Sassano A., Vasil...
Computational results
Running times (sec.) for 15 largest OR-library instances




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




                                          82
Results for
   our
 complex
instances


              83
Concluding remarks
• a new Mixed Boolean Pseudo-Boolean
  linear programming Model (MBpBM) for
  the p-median problem (PMP...
Future research directions
• compact models for other location
  problems (e.g. SPLP or generalized PMP)
• revised data-co...
Next two lectures
• How many instances do we really solve
  when solving a PMP instance

• Why some data lead to more comp...
Literature
•   B. F. AlBdaiwi, B. Goldengorin, G. Sierksma. Equivalent instances of the simple
    plant location problem....
Literature (contd.)
•   Elloumi, S.: A tighter formulation of the p-median problem. Journal of
    Combinatorial Optimizat...
Thank you!




Questions?


             89
Application to Cell Formation
                                                                             parts
         ...
Application to Cell Formation
Example 1: functional grouping (contd.)

                                                   ...
Application to Cell Formation
        Example 1: functional grouping (contd.)
       0     1.00 0.50 1.00 0.33            ...
Application to Cell Formation
        Example 1: functional grouping (contd.)
MBpBM
0.33 y1 0.5 y3 0.33 y5 0.58 z 6 1.5 z ...
Application to Cell Formation
                                                                                workers
    ...
Application to Cell Formation
Example 2: workforce expences (contd.)

                                                    ...
Application to Cell Formation
     Example 2: workforce expences (contd.)
     0   0.80 0.83 1.00 0.80     BC (y )   0.80 ...
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...
Application to Cell Formation
                   Example 3: from Yang,Yang (2008)*
                                 105 pa...
Thank you!
• Questions?




                            99
The PMP:    alternative formulation, Cornuejols et al. 1980

                                           Kj
        Let for...
The PMP: alternative formulation, Cornuejols et al. 1980

                                       Kj
    Let for each clien...
The PMP: alternative formulation, Cornuejols et al. 1980

                                        Kj
     Let for each cli...
The PMP: alternative formulation, Cornuejols et al. 1980

                                   n               Ki 1
        ...
The PMP: alternative formulation, Cornuejols et al. 1980

Example
(Elloumi,2009)                               1
         ...
The PMP: alternative formulation, Cornuejols et al. 1980


Example          1 6 5 3 4




                               1...
The PMP: alternative formulation, Cornuejols et al. 1980

Example
(Elloumi,2009)                               1
         ...
The PMP: alternative formulation, Cornuejols et al. 1980

       Example             1 6 5 3 4             Objective:
    ...
The PMP: a tighter formulation, Elloumi 2009


                                          k
A possible definition of variab...
The PMP: a tighter formulation, Elloumi 2009

                               n                   Kj 1
             f ( z, ...
The PMP: a tighter formulation, Elloumi 2009

        Let V j k    set of facilities within D j k : V j k          {i : ci...
The PMP: a tighter formulation, Elloumi 2009

Rule R2 : If for any j, k, j', k', V j k Vi' k' then z j k z j' k' holds for...
p-Median Cluster Analysis Based on General-Purpose Solvers (1)
p-Median Cluster Analysis Based on General-Purpose Solvers (1)
p-Median Cluster Analysis Based on General-Purpose Solvers (1)
p-Median Cluster Analysis Based on General-Purpose Solvers (1)
p-Median Cluster Analysis Based on General-Purpose Solvers (1)
p-Median Cluster Analysis Based on General-Purpose Solvers (1)
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p-Median Cluster Analysis Based on General-Purpose Solvers (1)

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AACIMP 2010 Summer School lecture by Boris Goldengorin. "Applied Mathematics" stream. "The p-Median Problem and Its Applications" course. Part 1.
More info at http://summerschool.ssa.org.ua

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

  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. 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 4
  5. 5. 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 5
  6. 6. 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 6
  7. 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. 7
  8. 8. 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. 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, 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. 23
  24. 24. The PMP: Boolean Linear Programming Formulation (ReVelle and Swain, 1970) 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 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 2 3 1 6 5 3 4 client1 : D1 1 D1 2 D1 4 2 1 2 3 5 C 1 2 3 3 3 4 3 1 8 2 25
  26. 26. 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 26
  27. 27. 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 27
  28. 28. 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 - either at least one facility is open within Dik zik yj 1, i 1,..., n k 1,..., K i or zi 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 Di1 ,..., DiK i - sorted distances 28
  29. 29. The PMP: alternative formulation, Cornuejols et al. 1980 Example, p=2 (Elloumi,2010) 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 1 2 3 3 3 client3 : D3 1 2 3 4 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 (in a + client3 : 1 ( 2 1) z13 (3 2) z 23 (5 3) z 33 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 29
  30. 30. 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 3z32 z13 C 1 2 3 3 3 z 23 2 z33 5 z14 z15 z 25 z35 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 z 31 y1 y2 y3 y4 1 z 31 0, z 42 0, z 43 0 z15 y4 1 z 32 y2 y3 y4 1 z 24 0, z 45 0 z 21 y1 y2 y3 1 z 33 y2 y3 y4 1 z jk 0 j 1,..., 5; k 1,..., K j yi {0,1}i 1,..., 4 13 coefficients, 23 linear constr., 12 non-negativity constr., 4 Boolean 30
  31. 31. 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) ( y2 1) Informally: if for client j some neighbourhood k contains only one facility i then there is a simple relation between k z1 2 1 y2 corresponding variables z j 1 yi 31
  32. 32. The p-Median Problem: a tighter formulation, Elloumi 2010 Rule R2 : If for any j, k, j', k', V j 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 32 can be added.
  33. 33. 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 that cij Dj K Kj K is equal to the sets of facilities i such that cij' 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 33
  34. 34. The PMP: a tighter formulation, Elloumi 2010 A possible definition of variables : zk 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 D1 j zk j zk 1 j (1 yi ), j 1,..., n; k 2,...,K j 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 34
  35. 35. The PMP: a tighter formulation, Elloumi 2010 n Kj 1 f ( z, y ) D1 j (Dk 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,..., K j 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,..., K j j k 1,..., K j i:cij D k j yi {0,1}, i 1,..., m Cornuejols et al. 1980 Kj for each client j D1 ,..., D j j - sorted distances 35
  36. 36. PMP Example with p=2 borrowed from 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 (1 y2 ) 2(1 y4 ) z11 7 z 21 z 22 5 z32 z 23 z 25 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. z 32 y4 z 22 z ki 0 yj {0,1} j 1,..., 4 36
  37. 37. 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. 37
  38. 38. 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. 38
  39. 39. 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 39
  40. 40. 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 40
  41. 41. 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 2 min ci 4 i.e. only distinct distances 2 3 0 i S 1 are meaningful (like in + 1 2 1 min ci 5 4 Cornuejols’ and Elloumi’s i S model) 4 4 2 41
  42. 42. 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 42
  43. 43. 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 43
  44. 44. 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 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. 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 46
  47. 47. 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 47
  48. 48. 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 48
  49. 49. 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 1y4 1y2 y4 2 y 2 y3 y 4 1 1y4 1y2 y4 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 49
  50. 50. 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 1 y 4 1 y3 y 4 1 y1 y3 y 4 = 8 1y 2 2 y 4 1y1 y3 1y 2 y3 1y 2 y 4 1y3 y 4 7 y1 y 2 y3 1y1 y3 y 4 5 y 2 y3 y 4 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! 50
  51. 51. PBP: combining similar terms 51
  52. 52. PBP: truncation p=2 Initial polynomial BC (y) (10 terms): 8 1y 2 2 y 4 1y1 y3 1y 2 y3 1y 2 y 4 1y3 y 4 7 y1 y 2 y3 1y1 y3 y 4 5 y 2 y3 y 4 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 1y 2 2 y 4 1y1 y3 1y 2 y3 1y 2 y 4 1y3 y 4 Truncation allows further reduction of the problem size! 52
  53. 53. 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 1 y 4 1 y3 y 4 1 y1 y3 y 4 p=2 MEMORY p=3 p=4 truncated polynomial p = m/2+1 53
  54. 54. 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 54
  55. 55. 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. 55
  56. 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 56
  57. 57. 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 y 2 y3 y2 y4 y3 y 4 8 y2 2 y4 z5 z6 z7 z8 Linear function of new variables: y1 , y2 , y3 , y4 , z5 y1 y3 , z6 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. 57
  58. 58. 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 ! 58
  59. 59. 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. 59
  60. 60. MBpBM: reduction • set covering problem y y y y1 y3 y5 y1 y3 y 6 1 3 4 y1 y3 y 4 y5 y6 y9 y1 y 3 y1 y3 y9 y 4 y5 y 6 y9 60
  61. 61. MBpBM: reduction • set covering problem y y y y1 y3 y5 y1 y3 y 6 1 3 4 NP-hard! y1 y3 y 4 y5 y6 y9 y1 y 3 y1 y3 y9 y 4 y5 y 6 y9 y1 y3 y 4 y5 y6 y9 y1 y3 y 4 y5 y 6 y9 2 61
  62. 62. 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) 62 and 4
  63. 63. Comparison of the models our MBpBM Elloumi’s NF 8 y2 2 y4 z5 z6 z7 z8 8 (1 y 2 ) 2(1 y4 ) z11 7 z 21 z 22 5 z32 z 23 z 25 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,..., 5 ; k 1,..., 3 y2 z 35 yi {0,1}i 1,..., 4 63
  64. 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 64
  65. 65. PMP: pseudo-Boolean formulation implies a decomposition of the search space into at most n(m-p) subspaces 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 65
  66. 66. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 66
  67. 67. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 67 yi 1 iff yi Tr
  68. 68. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 68 yi 1 iff yi Tr
  69. 69. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 69 yi 1 iff yi Tr
  70. 70. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 70 yi 1 iff yi Tr
  71. 71. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 71 yi 1 iff yi Tr
  72. 72. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 72 yi 1 iff yi Tr
  73. 73. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 73
  74. 74. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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 74
  75. 75. MBpBM: preprocessing (example) 1 6 5 3 4 z5 y1 y 3 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. 3 Boolean (1 fixed to 0) y4 0 Note: the number of Boolean variables was 4 in all considered models and in MBpBM it is 3. 75
  76. 76. 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 76
  77. 77. 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 77
  78. 78. 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) 78
  79. 79. Computational results, m=900 Results for different number of medians for two OR instances 79
  80. 80. 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) 80
  81. 81. Computational results Running times (sec.) for 15 largest OR-library instances 81
  82. 82. Computational results Running times (sec.) for RW instances 82
  83. 83. Results for our complex instances 83
  84. 84. 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 84
  85. 85. 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 85
  86. 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 86
  87. 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). 87
  88. 88. Literature (contd.) • 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) 88
  89. 89. Thank you! Questions? 89
  90. 90. 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 90
  91. 91. 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 is: 0 0 0 1 1 2 0 0 0 1 1 91
  92. 92. 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 y2 y4 0.25 y1 y3 0.75 y1 y3 y5 0.5 y2 y3 y4 Linearization: f (y, z ) 0.33 y1 0.5 y3 0.33 y5 0.58 z 6 1.5 z 7 0.25 z8 0.75 z9 0.5 z10 where: z6 y1 y5 z9 y1 y3 y5 z7 y2 y4 z10 y 2 y3 y 4 z8 y1 y3 92
  93. 93. Application to Cell Formation Example 1: functional grouping (contd.) MBpBM 0.33 y1 0.5 y3 0.33 y5 0.58 z 6 1.5 z 7 0.25 z8 0.75 z9 0.5 z10 min s.t. y1 y2 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 y2 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 yi {0,1}i 0 1 y1 y3 y* 1 1..5 zi 0i 0 2 y1 y3 y5 0 6..10 0 2 y2 y3 y4 1 yi {0,1}i 1..5 93
  94. 94. 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 94
  95. 95. 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 7 non-clustered three cells 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) 95
  96. 96. 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 y4 0.8 y5 The objective is already a linear function ! 96
  97. 97. 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 y 2 y3 y4 y5 5 3 yi {0,1}i 1..5 1 1 y* 0 0 0 97
  98. 98. 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 98 ART1 method. EJOR, vol. 188, pp. 140-152
  99. 99. Thank you! • Questions? 99
  100. 100. 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 2 3 1 6 5 3 4 client1 : D1 1 D1 2 D1 4 2 1 2 3 5 C 1 2 3 3 3 4 3 1 8 2 100
  101. 101. 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 5 101
  102. 102. 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 102
  103. 103. 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,..., K i or zi 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 Di1 ,..., DiK i - sorted distances 103
  104. 104. 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 1 2 3 3 3 client3 : D3 1 2 3 4 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 104
  105. 105. The PMP: alternative formulation, Cornuejols et al. 1980 Example 1 6 5 3 4 1 2 1 2 3 5 plants 2 3 1 2 3 C client1 : D1 1 D1 2 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 z 31 y1 y3 y2 y4 1 and z11 1 105
  106. 106. 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 1 2 3 3 3 client3 : D3 1 2 3 4 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 106
  107. 107. 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 3z32 z13 C 1 2 3 3 3 z 23 2 z33 5 z14 z15 z 25 z35 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 z 31 y1 y2 y3 y4 1 z 31 0, z 42 0, z 43 0 z15 y4 1 z 32 y2 y3 y4 1 z 24 0, z 45 0 z 21 y1 y2 y3 1 z 33 y2 y3 y4 1 z jk 0 j 1,..., 5; k 1,..., K j yi {0,1}i 1,..., 4 13 coefficients, 23 linear constr., 12 non-negativity constr., 4 Boolean 107
  108. 108. 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 D1 j zk j zk 1 j (1 yi ), j 1,..., n; k 2,...,K j 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 108
  109. 109. The PMP: a tighter formulation, Elloumi 2009 n Kj 1 f ( z, y ) D1 j (Dk 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,..., K j 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,..., K j j k 1,..., K j i:cij D k j yi {0,1}, i 1,..., m Cornuejols et al. 1980 Kj for each client j D1 ,..., D j j - sorted distances 109
  110. 110. 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) ( y2 1) Informally: if for client j some neighbourhood k contains only one facility i then there is a simple relation between k z1 2 1 y2 corresponding variables z j 1 yi 110
  111. 111. The PMP: a tighter formulation, Elloumi 2009 Rule R2 : If for any j, k, j', k', V j 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 111 can be added.
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