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 = ...
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. ...
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 industr...
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, 2009;
Brusco and K¨ohn, 2008;
Belenky, 2008;
Church, 2003; 2008;
Mladenovic et al, 20...
The PMP: Boolean Linear Programming
                  Formulation
                               m   n
                   ...
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', Vj k Vi' k' ...
The p-Median Problem:
                a tighter formulation, Elloumi 2010
                              Kj 1         K j' ...
The PMP: pseudo-Boolean formulation
                         (Historical remarks)

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

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




                               42
PBP: truncation
                                 p=2
 Initial polynomial BC (y) (10 terms):
 8 1y2 2 y4 1y1 y3 1y2 y3 1y2 ...
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...
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
         y1 y3 y4         y1 y3 y5            y1 y3 y6

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

 NP-hard!            ...
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 y4...
MBpBM: preprocessing
• every term (product of variables)
  corresponds to a subspace of solutions
  with all these variabl...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5              y1 y3    Objective:
   2 1 2 3 5 z6               y 2 y3   8 ...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5                        y1 y3        Objective:           consider some ter...
MBpBM: preprocessing (example)
   1 6 5 3 4 z5                         y1 y3     Objective:           consider next term
 ...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5                        y1 y3        Objective:           and so on …
   2 ...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5                        y1 y3        Objective:           and so on …
   2 ...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5                          y1 y3        Objective:           and so on …
   ...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5                          y1 y3        Objective:           and so on …
   ...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5                     y1 y3        Objective:
   2 1 2 3 5 z6               ...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5               y1 y3        Objective:
   2 1 2 3 5 z6                y 2 y...
MBpBM: preprocessing (example)
    1 6 5 3 4 z5          y1 y3    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
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




                                          72
Results for
   our
 complex
instances


              73
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
•   Avella, P., Sforza, A.: Logical reduction tests for the p-median problem. Annals
    of Operations Research...
Literature (contd.)
•   Hammer, P.L.: Plant location -- a pseudo-Boolean approach. Israel Journal
    of Technology, 6, 33...
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 0.58z 6 1.5 z 7 0...
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 ) ...
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?




                            88
The PMP:    alternative formulation, Cornuejols et al. 1980

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

                                      Kj
    Let for each client...
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             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, y )  ...
The PMP: a tighter formulation, Elloumi 2009

        Let V j k    set of facilities within D j k : V j k         {i : cij...
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 ...
The PMP: a tighter formulation, Elloumi 2009
                              Kj 1         K j' 1
Rule R3 : If for any j, j',...
Example (from Elloumi, 2009)

                                    1 6 5 3 4
                                    2 1 2 3 5
...
The PMP: a tighter formulation, Elloumi 2009

                               n               Ki 1
             f ( z, y ) ...
The p-Median Problem:
a tighter formulation Elloumi 2009




                                     103
MBpBM: preprocessing
f UB       some (global) upper bound
                     m
if for some y :            yi   m    p ho...
MBpBM: preprocessing
Claim:
The inequality f (y Tr )        f UB must be strict.
Counter-example (p=2):
We can show that i...
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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.
More info at http://summerschool.ssa.org.ua

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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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