Applications of the PMP. Cell Formation in Group Technology

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 into smaller

subsystems.

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

Cell formation (CF)

• Grouping machines into manufacturing

cells …

• … and parts into product families …

• … such that each family is produced

(mainly) within one cell

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.

ExampleDrilling and cutting

Thermal processing

ExampleDrilling and cutting

Thermal processing

Cell 1

Cell 2

Cell 1

Cell 2

Machine-part incidence matrix

machin

es

parts

.

1

.

.

1

.

1

.

.

1

1

.

.

1

.

.

.

1

1

.

1

.

1

.

.

.

.

1

1

1

Machine-part incidence matrix

.1.1..

111...

1.11..

....11

1...11

.1..1.

...1.1

111...

1.1.1.

1..1.1

5

4

3

2

1

5

3

2

4

1

654321 654231

Performance measures

1111...1....

1.11........

....11111...

....1.111...

.1..111.1...

.........111

..1......111

.........111 - exceptions

- voids

ne – number

of exceptions

nv – number

of voids

no – total

number of 1s

r

m

machin

es

parts


number of cellsm1

exceptions, ne

voids, nv


• ne

• ne + nv

•

• many others

nenvnorm

nvnorm

nvneno

neno)1(

factor g weightin ]1;0[

Problem size reduction

machine-part

incidence matrix

mxr

machine-machine

similarity matrix

mxm

machine-machine

similarity measure

Similarity measures:

.1..1.

...1.1

111...

1.1.1.

1..1.1

M5

M4

M3

M2

M1Each machine is characterized by a

vector in r-dimensional space

similarity any computable metrics

r parts

Dissimilarity: constjisjid ),(),(

),( jis

Problem size reduction

machine-part

incidence matrix

mxr

machine-machine

incidence matrix

mxm

machine-machine

similarity measure

Wei and Kern similarity measure:

r

k

jkik aajis1

),(),(

jkik

jkik

jkik

jkik

aa

aa

aar

aa

if,0

0 if,1

1 if,1

),(

r – number of parts,

m – number of machines

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

Existing approaches: BEA

.1..1.

...1.1

111...

1.1.1.

1..1.1

BEA = bond energy analysis

Goal: minimize the length of the border

11....

1.11..

.111..

....11

...111

• equivalent to the Quadratic Cost Assignment Problem

• only partial solution

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

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)

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-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

Example 1: functional grouping

parts

ma

ch

ine

s

1 2 3 4 5 6

5

4

3

2

1

Machine-part

incidence matrix

Goal: group machines into clusters

(manufacturing cells) such as to minimize

intercell communication.

r

k

jkik aarrjid

1

),()1(),(

Wei and Kern’s “commonality score”

jkik

jkik

jkik

jkik

aa

aa

aar

aa

if,0

0 if,1

1 if,1

),(

r – number of parts, m – number of machines

m = 4, r = 5

.1..1.

...1.1

111...

1.1.1.

1..1.1


Cost matrix for the PMP

is a machine-machine

dissimilarity matrix:

),(: jidcij

.1..1.

...1.1

111...

1.1.1.

1..1.1

1628232329

2816292917

2329121824

2329181224

2917242412

r

k

jkik aarrjid

1

),()1(),(


1628232329

2816292917

2329121824

2329181224

2917242412

C

5432532525

5421541414

5321532323

5321532322

4321421411

150716

0111116

515612

515612

507512)(

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyyBC y

5325413241543212, 7110187166568)( yyyyyyyyyyyyyyyB pC y


1628232329

2816292917

2329121824

2329181224

2917242412

C

5432532525

5421541414

5321532323

5321532322

4321421411

150716

0111116

515612

515612

507512)(

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyyBC y

5325413241543212, 7110187166568)( yyyyyyyyyyyyyyyB pC y

New variables:

5329

5418

327

416

yyyz

yyyz

yyz

yyz

Additional constraints:

1or 2

1or 2

1

1

5795329

5685418

327

416

yzzyyyz

yzzyyyz

yyz

yyz


min7110187166568 987654321 zzzzyyyyy

s.t.

5...1 },1,0{

9...6 ,0

1

1

1

1

25

579

568

327

416

54321

iy

iz

yzz

yzz

yyz

yyz

yyyyy

i

i1

1

1

0

0

*y

MBpBM formulation


1

1

1

0

0

*y

.1.1..

111...

1.11..

....11

1...11

654231

5

3

2

4

1

Example 2: workforce expenses

Machine-worker

incidence matrix

11001000

00101100

10010110

00100011

01010001

workers

ma

ch

ine

s

1 2 3 4 5 6 7 8

5

4

3

2

1

Goal: 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

machines theofeither operatecan that workersofnumber

and machinesboth operatecan that workersofnumber ),(

jijid

Example 2: workforce expensesCost matrix for the PMP

is a machine-machine

dissimilarity matrix:

080.083.000.180.0

80.0083.080.000.1

83.083.0083.083.0

00.180.083.0080.0

80.000.183.080.00

machines

ma

ch

ine

s

),(: jidcij

11001000

00101100

10010110

00100011

01010001

workers

ma

ch

ine

s

5

4

3

2

1

87654321

Example 2: workforce expenses

080.083.000.180.0

80.0083.080.000.1

83.083.0083.083.0

00.180.083.0080.0

80.000.183.080.00

C

5431541515

5432542424

4321321313

4321421212

5321521211

17.003.0080.0

17.003.0080.0

00083.0

17.003.0080.0

17.003.0080.0)(

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyy

yyyyyyyyyyBC y

543213, 8.08.083.08.08.0)( yyyyyB pC y

The objective is already a linear function !

Example 2: workforce expensesMBpBM formulation

0

0

0

1

1

*y

5,...,1 },1,0{

2

s.t.

min8.08.083.08.08.0

54321

54321

iy

yyyyy

yyyyy

i

10010100

10101000

01010001

01100010

00011111

workers

ma

ch

ine

s

1

5

2

4

3

76418532 1 addtional training

7 redundant skills

Example 3: from [BhaSad] (2010)*

* 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.

105 parts

46 m

achin

es

(uncapacitated)

functional grouping

105 parts

46 m

achin

es

grouping efficiency:

[BhaSad] 90.98%

our result 95.20%

(solved within 1 sec.)

Future research directions

• Additional real-life constraints

– capacities

– workload

• Additional real-life factors

– operational sequences

– processing and setup times

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 literature (including our)

are heuristics from the CF perspective

• exact model – MINpCUT

Applications of the PMP. Cell Formation in Group Technology

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