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

1. Applications of the PMP
Cell Formation in Group
Technology

2. Outline
• Introduction
• Brief overview of models
• PMP approach to CF
• Examples

3. Group technology
• A paradigm in industrial engineering
suggesting structural decomposition of a
manufacturing system into smaller
subsystems.

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. Cell formation (CF)
• Grouping machines into manufacturing
cells …
• … and parts into product families …
• … such that each family is produced
(mainly) within one cell

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. Example
Drilling and cutting
Thermal processing

8. Example
Drilling and cutting
Thermal processing

9. Example
Drilling and cutting
Thermal processing

10. Example
Drilling and cutting
Thermal processing

11. Example
Drilling and cutting
Thermal processing

12. Example
Drilling and cutting
Thermal processing

13. Example
Drilling and cutting
Thermal processing

14. Example
Drilling and cutting
Cell 1
Cell 2
Thermal processing
Cell 1
Cell 2

15. Machine-part incidence matrix
parts
1 . 1 . . 1
. 1 . 1 . 1
machines
. . . 1 1 1
1 . 1 . . .
. 1 . . 1 .

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. 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. Performance measures
voids, nv
exceptions, ne
number of cells
1 m

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Future research directions
• Additional real-life constraints
– capacities
– workload
• Additional real-life factors
– operational sequences
– processing and setup times

43. Conclusions
• An efficient model for CF:
– low computing times
– high quality solutions

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