Applications of the PMP

   Cell Formation in Group
         Technology
Outline
•   Introduction
•   Brief overview of models
•   PMP approach to CF
•   Examples
Group technology
• A paradigm in industrial engineering
  suggesting structural decomposition of a
  manufacturing system ...
Group technology: advantages
• Smaller systems are easier to manage
  (e.g. scheduling)
• Better plant layout:
  – shorter...
Cell formation (CF)
• Grouping machines into manufacturing
  cells …
• … and parts into product families …



• … such tha...
Cell formation
• Cell Formation becomes possible by
  exploiting similarities in the manufacturing
  processes for differe...
Example
Drilling and cutting




Thermal processing
Example
Drilling and cutting




Thermal processing
Example
Drilling and cutting




Thermal processing
Example
Drilling and cutting




Thermal processing
Example
Drilling and cutting




Thermal processing
Example
Drilling and cutting




Thermal processing
Example
Drilling and cutting




Thermal processing
Example
         Drilling and cutting
Cell 1

Cell 2




         Thermal processing
                                     ...
Machine-part incidence matrix
                      parts



              1   .   1   .   .   1
              .   1   .  ...
Machine-part incidence matrix
     1 2 3 4 5 6                 1 3 2 4 5 6

1    1   .   1   .   .   1   1   1   1   .   ....
Performance measures
                           r parts

           1 1 1 . . .       . . . . . .        - exceptions
mach...
Performance measures

     voids, nv

                 exceptions, ne




                      number of cells
 1        ...
Performance measures
• ne
• ne + nv

         no ne              m r no nv
•                   (1   )
       no ne nv     ...
Problem size reduction
         machine-part                                   machine-machine
       incidence matrix    ...
Problem size reduction
        machine-part                                 machine-machine
      incidence matrix        ...
CF: existing approaches
• Clustering based on energy functions:
  BEA, ROC, MODROC, DCA, …
• Similarity based hierarchical...
Existing approaches: BEA
 BEA = bond energy analysis
 Goal: minimize the length of the border

1    .   1    .    .      1...
Existing approaches: hierarchical
             clustering
   SLC/CLC/ALC = single/complete/average linkage clustering

Alg...
Existing approaches: hierarchical
             clustering
   SLC/CLC/ALC = single/complete/average linkage clustering

Alg...
Existing approaches: hierarchical
             clustering
   SLC/CLC/ALC = single/complete/average linkage clustering

Alg...
Existing approaches: hierarchical
             clustering
   SLC/CLC/ALC = single/complete/average linkage clustering

Alg...
Existing approaches: hierarchical
             clustering
   SLC/CLC/ALC = single/complete/average linkage clustering

Alg...
P-Median approach
Goal:
• select p “central” machines – representatives of p
cells
• assign all other machines to cells......
P-Median approach
Goal:
• select p “central” machines – representatives of p
cells
• assign all other machines to cells......
Example 1: functional grouping
                                         Goal: group machines into clusters
             Ma...
Example 1: functional grouping
Cost matrix for the PMP                               r
is a machine-machine       d (i, j ...
Example 1: functional grouping
      12   24   24   17   29     BC (y )     12   5 y1    7 y1 y 4   0 y1 y 2 y 4   5 y1 y ...
Example 1: functional grouping
      12   24     24    17   29     BC (y )        12        5 y1     7 y1 y 4   0 y1 y 2 y...
Example 1: functional grouping
                               MBpBM formulation
68 5 y1 6 y2          6 y3 1y4     7 y5 18...
Example 1: functional grouping
                 1 3 2 4 5 6

     0
            1    1   1   .   .   .   1
     0
        ...
Example 2: workforce expenses
                                                     Goal: group machines into clusters
    ...
Example 2: workforce expenses
           Cost matrix for the PMP
           is a machine-machine      cij : d (i, j )
    ...
Example 2: workforce expenses
      0    0.80   0.83   1.00   0.80
    0.80     0    0.83   0.80   1.00
C   0.83   0.83   ...
Example 2: workforce expenses
                                  MBpBM formulation

 0.8 y1 0.8 y 2     0.83y3 0.8 y 4     ...
Example 3: from [BhaSad] (2010)*
                            105 parts
 46 machines




                                  ...
Future research directions
• Additional real-life constraints
  – capacities
  – workload
• Additional real-life factors
 ...
Conclusions
• An efficient model for CF:
  – low computing times
  – high quality solutions
Conclusions
• An efficient model for CF:
   – low computing times
   – high quality solutions


• BUT: all models in liter...
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Applications of the PMP. Cell Formation in Group Technology

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AACIMP 2010 Summer School lecture by Dmitry Krushinsky. "Applied Mathematics" stream. "The p-Median Problem and Its Applications" course. Part 5.
More info at http://summerschool.ssa.org.ua

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Applications of the PMP. Cell Formation in Group Technology

  1. 1. Applications of the PMP Cell Formation in Group Technology
  2. 2. Outline • Introduction • Brief overview of models • PMP approach to CF • Examples
  3. 3. Group technology • A paradigm in industrial engineering suggesting structural decomposition of a manufacturing system into smaller subsystems.
  4. 4. Group technology: advantages • Smaller systems are easier to manage (e.g. scheduling) • Better plant layout: – shorter in travelling distances (up to 95%) – less intersecting routes
  5. 5. Cell formation (CF) • Grouping machines into manufacturing cells … • … and parts into product families … • … such that each family is produced (mainly) within one cell
  6. 6. Cell formation • Cell Formation becomes possible by exploiting similarities in the manufacturing processes for different parts and increases the throughput of the manufacturing system without sacrificing the products quality.
  7. 7. Example Drilling and cutting Thermal processing
  8. 8. Example Drilling and cutting Thermal processing
  9. 9. Example Drilling and cutting Thermal processing
  10. 10. Example Drilling and cutting Thermal processing
  11. 11. Example Drilling and cutting Thermal processing
  12. 12. Example Drilling and cutting Thermal processing
  13. 13. Example Drilling and cutting Thermal processing
  14. 14. Example Drilling and cutting Cell 1 Cell 2 Thermal processing Cell 1 Cell 2
  15. 15. Machine-part incidence matrix parts 1 . 1 . . 1 . 1 . 1 . 1 machines . . . 1 1 1 1 . 1 . . . . 1 . . 1 .
  16. 16. Machine-part incidence matrix 1 2 3 4 5 6 1 3 2 4 5 6 1 1 . 1 . . 1 1 1 1 . . . 1 2 . 1 . 1 . 1 4 1 1 . . . . 3 . . . 1 1 1 2 . . 1 1 . 1 4 1 . 1 . . . 3 . . . 1 1 1 5 . 1 . . 1 . 5 . . 1 . 1 .
  17. 17. Performance measures r parts 1 1 1 . . . . . . . . . - exceptions machines 1 1 1 . . . . . . 1 . . - voids 1 1 1 . . . . . . . . . ne – number . . . 1 . 1 1 1 . . 1 . of exceptions m . . . 1 1 1 . 1 . . . . nv – number of voids . . . 1 1 1 1 1 . . . . no – total . . . . . . . . 1 1 . 1 number of 1s . . . . 1 . . . 1 1 1 1
  18. 18. Performance measures voids, nv exceptions, ne number of cells 1 m
  19. 19. Performance measures • ne • ne + nv no ne m r no nv • (1 ) no ne nv m r no nv ne [0;1] weighting factor • many others
  20. 20. Problem size reduction machine-part machine-machine incidence matrix machine-machine similarity matrix similarity measure mxr mxm Similarity measures: r parts Each machine is characterized by a M1 1 . 1 . . 1 vector in r-dimensional space M2 . 1 . 1 . 1 M3 . . . 1 1 1 similarity  any computable metrics M4 1 . 1 . . . s(i, j ) M5 . 1 . . 1 . Dissimilarity: d (i, j ) s(i, j ) const
  21. 21. Problem size reduction machine-part machine-machine incidence matrix machine-machine incidence matrix similarity measure mxr mxm Wei and Kern similarity measure: r s(i, j ) (aik , a jk ) r – number of parts, k 1 m – number of machines r 1, if aik a jk 1 (aik , a jk ) 1, if aik a jk 0 0, if aik a jk
  22. 22. CF: existing approaches • Clustering based on energy functions: BEA, ROC, MODROC, DCA, … • Similarity based hierarchical clustering: SLC, CLC, ALC, LCC, … • Fuzzy logic methods • Genetic algorithms and simulated annealing • Neural networks: backpropagation learning, competitive learning, adaptive resonance theory (ART1), self-organizing maps, … • Graph partitioning • Integer Linear Programming
  23. 23. Existing approaches: BEA BEA = bond energy analysis Goal: minimize the length of the border 1 . 1 . . 1 1 1 1 . . . . 1 . 1 . 1 1 1 . . . . . . . 1 1 1 . . 1 1 1 . 1 . 1 . . . . . 1 1 . 1 . 1 . . 1 . . . . . 1 1 • equivalent to the Quadratic Cost Assignment Problem • only partial solution
  24. 24. Existing approaches: hierarchical clustering SLC/CLC/ALC = single/complete/average linkage clustering Algorithm: • start with each cluster containing one machine • at each step connect two most similar clusters
  25. 25. Existing approaches: hierarchical clustering SLC/CLC/ALC = single/complete/average linkage clustering Algorithm: • start with each cluster containing one machine • at each step connect two most similar clusters
  26. 26. Existing approaches: hierarchical clustering SLC/CLC/ALC = single/complete/average linkage clustering Algorithm: • start with each cluster containing one machine • at each step connect two most similar clusters
  27. 27. Existing approaches: hierarchical clustering SLC/CLC/ALC = single/complete/average linkage clustering Algorithm: • start with each cluster containing one machine • at each step connect two most similar clusters
  28. 28. Existing approaches: hierarchical clustering SLC/CLC/ALC = single/complete/average linkage clustering Algorithm: • start with each cluster containing one machine • at each step connect two most similar clusters Equivalent to the minimum spanning tree problem (MST)
  29. 29. P-Median approach Goal: • select p “central” machines – representatives of p cells • assign all other machines to cells... • ... such that the sum of dissimilarities is minimum
  30. 30. P-Median approach Goal: • select p “central” machines – representatives of p cells • assign all other machines to cells... • ... such that the sum of dissimilarities is minimum p=2
  31. 31. Example 1: functional grouping Goal: group machines into clusters Machine-part (manufacturing cells) such as to minimize incidence matrix intercell communication. parts 1 2 3 4 5 6 r 1 . 1 . . 1 1 d (i, j ) r (r 1) (aik , a jk ) machines . 1 . 1 . 1 2 k 1 . . . 1 1 1 3 1 . 1 . . . 4 Wei and Kern’s “commonality score” . 1 . . 1 . 5 r 1, if aik a jk 1 m = 4, r = 5 (aik , a jk ) 1, if aik a jk 0 0, if aik a jk r – number of parts, m – number of machines
  32. 32. Example 1: functional grouping Cost matrix for the PMP r is a machine-machine d (i, j ) r (r 1) (aik , a jk ) dissimilarity matrix: k 1 cij : d (i, j ) 1 . 1 . . 1 12 24 24 17 29 . 1 . 1 . 1 24 12 18 29 23 . . . 1 1 1 24 18 12 29 23 1 . 1 . . . 17 29 29 16 28 . 1 . . 1 . 29 23 23 28 16
  33. 33. Example 1: functional grouping 12 24 24 17 29 BC (y ) 12 5 y1 7 y1 y 4 0 y1 y 2 y 4 5 y1 y 2 y 3 y 4 24 12 18 29 23 12 6 y2 5 y 2 y3 1y 2 y3 y5 5 y1 y 2 y 3 y 5 12 6 y3 5 y 2 y3 1y 2 y3 y5 5 y1 y 2 y 3 y 5 C 24 18 12 29 23 16 1y 4 11 y1 y 4 1y1 y 4 y 5 0 y1 y 2 y 4 y 5 17 29 29 16 28 16 7 y5 0 y 2 y5 5 y 2 y3 y5 1y 2 y3 y 4 y5 29 23 23 28 16 BC , p 2 (y) 68 5 y1 6 y2 6 y3 1y 4 7 y5 18 y1 y 4 10 y 2 y3 1y1 y 4 y5 7 y 2 y3 y5
  34. 34. Example 1: functional grouping 12 24 24 17 29 BC (y ) 12 5 y1 7 y1 y 4 0 y1 y 2 y 4 5 y1 y 2 y 3 y 4 24 12 18 29 23 12 6 y2 5 y 2 y3 1y 2 y3 y5 5 y1 y 2 y 3 y 5 12 6 y3 5 y 2 y3 1y 2 y3 y5 5 y1 y 2 y 3 y 5 C 24 18 12 29 23 16 1y 4 11 y1 y 4 1y1 y 4 y 5 0 y1 y 2 y 4 y 5 17 29 29 16 28 16 7 y5 0 y 2 y5 5 y 2 y3 y5 1y 2 y3 y 4 y5 29 23 23 28 16 BC , p 2 (y) 68 5 y1 6 y2 6 y3 1y 4 7 y5 18 y1 y 4 10 y 2 y3 1y1 y 4 y5 7 y 2 y3 y5 New variables: Additional constraints: z6 y1 y 4 z6 y1 y4 1 z7 y 2 y3 z7 y2 y3 1 z8 y1 y 4 y5 z8 y1 y4 y5 2 or z 8 z6 y5 1 z9 y 2 y3 y5 z9 y2 y3 y5 2 or z 9 z7 y5 1
  35. 35. Example 1: functional grouping MBpBM formulation 68 5 y1 6 y2 6 y3 1y4 7 y5 18z6 10z7 1z8 7 z9 min s.t. y1 y2 y3 y4 y5 5 2 z6 y1 y4 1 0 z7 y2 y3 1 0 z8 z6 y5 1 y* 1 z9 z7 y5 1 1 zi 0, i 6...9 1 yi {0,1}, i 1...5
  36. 36. Example 1: functional grouping 1 3 2 4 5 6 0 1 1 1 . . . 1 0 4 1 1 . . . . y* 1 1 2 . . 1 1 . 1 1 3 . . . 1 1 1 5 . . 1 . 1 .
  37. 37. Example 2: workforce expenses Goal: group machines into clusters Machine-worker (manufacturing cells) such that: incidence matrix 1) every worker is able to operate every workers machine in his cell and cost of additional cross- 1 2 3 4 5 6 7 8 training is minimized; 1 0 0 0 1 0 1 0 1 machines 1 1 0 0 0 1 0 0 2) if a worker can operate a machine that is not 2 0 1 1 0 1 0 0 1 in his cell then he can ask for additional 3 0 0 1 1 0 1 0 0 payment for his skills; we would like to minimize 4 0 0 0 1 0 0 1 1 such overpayment. 5 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
  38. 38. Example 2: workforce expenses Cost matrix for the PMP is a machine-machine cij : d (i, j ) dissimilarity matrix: workers 1 2 3 4 5 6 7 8 1 1 0 0 0 1 0 1 0 machines 2 1 1 0 0 0 1 0 0 3 0 1 1 0 1 0 0 1 4 0 0 1 1 0 1 0 0 machines 0 0.80 0.83 1.00 0.80 5 0 0 0 1 0 0 1 1 machines 0.80 0 0.83 0.80 1.00 0.83 0.83 0 0.83 0.83 1.00 0.80 0.83 0 0.80 0.80 1.00 0.83 0.80 0
  39. 39. Example 2: workforce expenses 0 0.80 0.83 1.00 0.80 0.80 0 0.83 0.80 1.00 C 0.83 0.83 0 0.83 0.83 1.00 0.80 0.83 0 0.80 0.80 1.00 0.83 0.80 0 BC (y ) 0.80 y1 0 y1 y2 0.03 y1 y2 y5 0.17 y1 y 2 y3 y5 0.80 y2 0 y1 y2 0.03 y1 y2 y4 0.17 y1 y 2 y3 y 4 0.83 y3 0 y1 y3 0 y1 y2 y3 0 y1 y 2 y3 y 4 0.80 y4 0 y2 y4 0.03 y2 y4 y5 0.17 y 2 y3 y 4 y5 0.80 y5 0 y1 y5 0.03 y1 y4 y5 0.17 y1 y3 y 4 y5 BC , p 3 (y ) 0.8 y1 0.8 y 2 0.83 y3 0.8 y 4 0.8 y5 The objective is already a linear function !
  40. 40. Example 2: workforce expenses MBpBM formulation 0.8 y1 0.8 y 2 0.83y3 0.8 y 4 0.8 y5 min s.t. y1 y 2 y3 y4 y5 2 yi {0,1}, i 1,...,5 workers 2 3 5 8 1 4 6 7 1 addtional training 1 3 1 1 1 1 1 0 0 0 7 redundant skills 1 4 machines 0 1 0 0 0 1 1 0 y* 0 2 1 0 0 0 1 0 1 0 0 5 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 1
  41. 41. Example 3: from [BhaSad] (2010)* 105 parts 46 machines (uncapacitated) functional grouping 105 parts 46 machines grouping efficiency: [BhaSad] 90.98% our result 95.20% (solved within 1 sec.) * R. Bhatnagar, V. Saddikuti, Models for cellular manufacturing systems design: matching processing requirements and operator capabilities, Journal of the Operational Research Society, 61, 2010, pp. 827-839.
  42. 42. Future research directions • Additional real-life constraints – capacities – workload • Additional real-life factors – operational sequences – processing and setup times
  43. 43. Conclusions • An efficient model for CF: – low computing times – high quality solutions
  44. 44. Conclusions • An efficient model for CF: – low computing times – high quality solutions • BUT: all models in literature (including our) are heuristics from the CF perspective • exact model – MINpCUT

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