Principal component analysis: A tool for assembly management

Computers and Industrial Engineering Vol. 25, Nos 1-4, pp. 77-80, 1993 Printed in Great Britain. All rights reserved

0360-835219356.00+0.00 Copyright © 1993 Pergamon Press Ltd

Principal Component Analysis: A Tool for Assembly Management Farhad Tadayon Beeing Commercial Airplanes Wichita, KS 67208 Ming C. Liu Department of Industrial Engineering Wichita State University Wichita, KS 67260-0035

Abstract

misfit. In this situation, the principal component analysis could be implemented to identify the problems and their

The dimensional variations of components or detail parts in a complex assembly could be within the

share of involvement in the overall assembly misfit.

acceptable specifications, but when put together, the

By implementing the principal component analysis

proper fit of the final assembly can not be obtained. It

technique in a job shop assembly environment, it is

becomes difficult to locate the root cause of the assembly

possible to identify the few factors or elements which

misfit. The principal component analysis, a multi-variant

account for the variability problem of the assembly.

statistical method, can define the position of the principal variables or the major parameters which signify the

Depending on the correlation strength of the original variables, it is conceivable to represent the variation

misfit condition. Depending on the correlation strength

behavior of 20 or 30 variables by only two or three

of the original variables, it is possible to present the

principal components.

variation characteristics of 20 or 30 variables by only

The principal component analysis can be a simple

two or three principal components. The method is simple

and powerful technique when there are not many original variables. In those situations where the number of

to use, and could save time and money by accelerating the identification of the assembly problems.

variables in defining the assembly is high and the overall correlation of these variables are not so strong, the

Introduction

cluster analysis can be implemented to construct zones

Variability control is the global corporate goal and

and groups with fewer variables. Then the principal

commitment of many industries. Identifying the real

component could be utilized for each cluster or zone to

cause of the variabilities, which are the root of the

further focus on the key variables.

problem, is the major milestone in reaching that goal. Experts agree that identifying the root causes of

The principal component technique can provide assistance and guidelines to the shop manager or the

variability problems consumes most of the variation and

quality analyst to diagnose the assembly misfit problems.

quality control activities. Individual variation of each part

Significant time and costs could be reduced by reducing

along with the combined variability of the components in an assembly could prevent an acceptable and adequate

the time and the manpower needed to look for the root cause of the assembly problems.

fit. The present and the traditional approaches to diagnose the root cause of assembly misfits are usually

Principal

Analysis

Principal component analysis is a simple multi-

time consuming, dependent on the discretion of different individuals and organizations, and are not standard. In

variant statistical approach. It considers a set of variables or characteristics Z1, Z2 ...... Zp and combines them to form indices X1, X2 ...... Xp that are uncorrelated. The

another word, the issue is treated as a "stand alone" nature. However, in an assembly product, the effect of

lack of correlation means that the indices arc measuring different dimensions in the data set. Certainly, it is

variability is transmitted throughout the assembled parts like a network system. There are many cases in which a combination or a string of characteristics would create the part or assembly

t a l e 25-1~4--G

Component

important to have significant correlation among the original variables Z1, Z2 . . . . . . Zp. The indices are

77

Proceedings of the 15th Annual Conference on Computers and Industrial Engineering

78

ordered so that X l displays the largest amount of

Figure la shows an airplane door assembly with

variation, X2 displays the second largest, and so on. The

five forward door fittings, where the door interfaces with

Xi's are the principal components,

the main body of the airplane. The key characteristic of this assembly is the contour of the door and the proper

Xi = ail Z1 + ai2 Z2 + ............ + aip Zp. Principal component analysis involves finding the eigenvalues and eigenvectors of the sample correlation

location of the fittings along that contour. Figure l b shows an access panel assembly which

matrix, with the variances of the principal components

is made of two parts. The key characteristic of this

being the eigenvalues of the matrix. The weight aij are

assembly is the flatness of the bottom plate.

the normalized eigenvectors corresponding to the i-th

From measurement data, a correlation matrix was estimated for each of the above assemblies. The

eigenvalue.

correlation coefficients are shown in Tables la and lb.

Discussion

As can be seen, the variables for the door assemblies are

In this study, different types of assembled

and

strongly correlated. However, the coefficients for the

detailed parts were examined and analyzed. Every point

panel assembly variables, in general, are not so strongly

or location of an assembly which was measured by a

significant.

Coordinate Measuring Machine (CMM), was considered

Based on the obtained correlation matrix, the

as a variable. Depending on the nature and the

principal component analysis was performed for both

complexity of the assembly, the number of variables or

assemblies. Since this method depends directly on the

locations could vary significantly. One important

strength of the correlation of the variables, the results of

reminder is that the analytical procedure should revolve

the principal components were more pronounced for the

around the primary idea of reducing the number of

door assembly. Tables 2a and 2b illustrate the results in

variables and focus on those locations or variables which

the form of eigenvalues and the associated eigenvector

contribute the most to the variability problem of an

coefficients. It is true that the principal component analysis is

assembly. According to the information survey, the principal component analysis approach has been tested for the

one of the simplest multivariate statistical methods to describe the variability, by the fact that the factor

variation study of automobile door assembly in an

measurement for the original variables could be directly

automated manufacturing environment [1]. It was

obtained from the principal components. Wu and Hu [1,

decided to initiate the analytical procedure by varifying

2, 3] have shown the application of this method in

this technique for assemblies made in job

shop

automobile body assembly. Their effort was focused on

manufacturing area, where more incidents of variations

the evaluation of the variability of the body assemblies

due

to many different sources could occur.

based on the measurements of a few identified variables. Table la. Door Assembly Con'elation Coefficients

3

4

Variable

t

1

1,000

2

2

3

4

5

0.779 1.000

0.832

0.818

0.630

0.958

0.972

0,810

3

1.000 0.966 0.713

4

Figure la. Airplane Door

i.000

0.867

5

1,000

Table l b . Access Panel Conelation Coefficients Vmiable 1 2 3

Figure lb. Access Panel

4

1 1.000

2

3

4

-0.089

0.468

0.341

1.000

0.030

0.261

1.000

0.111 1.000

TADAYON and LIU: Principal Component Analysis Table 2a. Door Assembly, F.,igenvaluusand Eigenvectors C.omp. ~ "

~

eisem'ecmrCoemcimm Z2 23 7.,4 Zl

79

assembly by just relying on the principal components. Therefore, number of variables should be reduced to a

7_,5

manageable quantity.

1

4~t$

87.10

0.416

0.466

0.461

0.476

0.412

2

0.39

7.91

.0.615

0.047

-0.7.49

0.092

0.741

3

0,21

4.38

-0.395

-0.436

-0.151

0,444

variables into subgroups which differ based on the

4.

0.03

0.61

.0.121

,.0.7¢30

0.4.73

0.365

0.1064

similarities and the attitudes of the variables in each

S

0.00

0.02

0.0~

0.0~

-0.5~/

0.7110

-02~

cluster. This technique has been popular in the social and

0.6,58

Cluster analysis can be used to divide a set o f

behavior science studies, however, not many studies and

Table 2b. Panel Assembly, Eigenvalues and Eigenvectors

1

applications of this method in the assembly and

Eigenvector Coefficients ZI Z2 Z3 7_.4

Comp. Eigenvalue 2.44

51.17

0.552

-0.592

0.543

production management have been reported. The key element in investigating the network

-0.221

2

1.25

H.28

0.331

-0.325

-0.362

0.808

3

0.31

7.55

-0.619

-0.184

0.613

0.454

4

0.00

0.00

0.449

0.444

0.302

variability is to identify and categorize the behavior and the similarities of the entities o f the assembly. This way the target areas are distinguished better, and the number

0.714

of variables would be reduced to those which have greater contributions to the network variability.

However, in many assembly environments, there could part

Figure 2 illustrates the final cluster formation of the

measurements for an assembly. In this situation, this

variables of the entire airplane door frame assembly.

technique can not solely reduce the number o f variables

Since the variables inside every cluster are strongly

be

a

significant

number

of

variables

or

to identify the original key variables, and the size of the

correlated,

principal components could become extremely large and

components or factors for each cell would be obtained.

better

and

more

predictive

principal

difficult to analyze. Remember that the objective is to

Remember that the strength of the principal component

identify the key variables or locations of the key

technique depends on the significance o f the correlation

variabilities o f the assembly in the shortest period of time

of the variables. The principal component method was

to keep the cost low. Consider the frame assembly of an

executed for each of the popular cells at the 20 cluster

airplane door. If it is decided to investigate the variation

level for the door frame assembly. The results are

of different locations for the contour, there could be

tabulated in the following tables (4a - 4c):

some where between 30 to 200 locations or variables which need to be measured. Table 3 illustrates some of the results o f the principal component analysis for a door assembly with only 30 variables. This table shows that most of the variability could be described by the three following equations representing principal components Z1, Z2, and

Z3:

V

Z1 = 0.195 X1 + 0.188 X2 + 0.194 X3 + .... + 0.198 X30

Figure 2. Ouster Formation

Z2 = -0.233 X1 - 0.257 X2 - 0.234 X3 + .... + 0.096 X30 Z3 = -0.031 X1 - 0.064 X2 - 0.097 X3 + .... - 0.026 X30 Table 3. Eigenvalues and Eigenvectors for 30 Locations n lmcorn, nme 9t Zl I

2:2

16-$4 $.$.16 0.19~J 0 . I n 0.194 0.197 0.187 $.34 17.82 .0.233 41.257 -0.234 -0.2(B -0.132

3

4,715 15.95 .0.031 -0.064 -0.097 -0.124 -0.157

0-150 0.096

4

1.87

-0.009..0.256 j I

i

•

0.029

6.7.5 0.159 0-063 0.038-0.112 -0.299 . . . . i

•

i

.

.

.

.

.

.

.

.

BlSmComp. v ~

0.224 0-19(t

2

•

Table 4a. Cluster I, Eigenvalues and Eigenvectors

P.,ilpavect~Coalkiem 23 Z4 Z S . . . Z29 Z30

.

.

0.096

The previous three equations reveal that it is

]

Zl

i

8.61

2

o.28

3.12

3

0.13

1.43

4

om

Z2

O,338 ~

5

0.~

o/d

6

om

0.18

7

0101

0.14

extremely difficult to make any conclusions about the

8

~

o~

location or the behavior of the variability problem of the

9

o.eo

o,ol

7"4

Z4

0,34Z ~

0.411 4.12g 4.117 ~

Z5

7.6

0..~tl 0.DI

~

~

O&17.0.U2 ~

0,ON .0..,~ ~

0.1~

,o,ep3

,o.lk5

~

,0.,172

0.g84

oJot

0.1M -03~| 0.'F~ ~

0.121 O.e01 4.474 0,0'11 O.ITZ ~

Z7 ~

~

2:8

Z9

0.341

.0.15|

0,.339 ,,0.~3 4laat

Iklm

0.183

..0,131 ~O.4gO ,0.100

~

OJIm 4.3O4

Proceedings of the 15th Annual Conference on Computers and Industrial Engineering

80

T a b l e 4b. Cluster II, Eigenvalues and Eigenvectors

[Comp. Eiguevalue

~

1

4.31

86.25

m~Cmffieitnu Zl

Z2

7_.3

Za

Z5

0.464

0.463

0.417

0.433

0.4.55

. 0 . 3 1 3 41.307

is. In this case, although the weight factors are almost the same, it could be mentioned that variable 65 is more significant than variable 96.

Summary

2

0.46

931

0.638

0.481

-0.410

3

0.16

3.27

0.259

-0.272

-0.5911

0.699

.0.10~

The principal component analysis is a simple

4

0.04

0.75

.0.329

.0.~211

0.036

0,064

0.778

statistical method which can be used to identify the

5

0,~

0.~

0.714

-0.5111

0.243

.0.294

.0.079

location of the key elements that cause major variation in

T a b l e 4c. Cluster IlL Eigenvalues and Eigenvectors EigenEomp. value

qt

the assembly products. The present approaches to diagnose the root cause of assembly misfits are time

Ei~eav~tor Ceeffieients

Z5 0,378

consuming and dependent on the discretion of the

0.511

with multivariate statistical techniques can provide a

0.0~

1.15 -0.139 0.097 -0.002 -0.347 .0.042

useful tool to establish standard procedures to attack

4

O.O3

0.40 -0.361 -0.409 0.822 0.101 -0.004

assembly problems. By implementing these procedural

5

0.00

0.09

steps, the investigator can approach and solve the misfit

Zl 7?7 Z3 Z4 96.80 0.375 0.381 0.378 0.382

I

6.77

2

0.11

1_56 -0.583 -0.280 -0.343

3

0.520 -0.658 -0.175

0.113

0.260

0.263

investigator. Principal component analysis in conjunction

problems in a shorter time period, and consequently save The tables presented above, indicate that indeed for every cluster, there is one principal component which

time and money in the overall production cycle of the product.

describes the corresponding variability. The eigenvalue distribution for each cell also indicates that since there is one highly significant component for each cell, the variability behavior of the measured locations is uni-

References 1. Hu, S. J., Wu, S. M. "In-Process 100% Measurement and Statistical Analysis for an

directional. In situations where two, three, or more

Automotive Body Assembly Process", Transactions

eigenvalues are significant, consequently, two, three, or

of NAMRI/SME, May 1990.

more principal components would correspond to the

2. Wu, S. M., Hu, S. J. "Impact o f 100% In-Process

variation, which in turn reflects that the variability is

Measurement on Statistical Process Control in

two, three, or multi-directional. The following principal

Automobile Body assembly", Monitoring and

components represent the variability o f the entities in

Control in Manufacturing, Ed. S. Liang, T. C.

each popular cell (cluster): PC I = 0.328 zl + 0.343 Z2 + 0.342 Z3 + 0.302 z4 + 0.341 z5

Tsao, ASME Winter Annual Meeting, November

+ 0.331 Z6 + 0335 Z7 +0.341 7_.8 + 0334 Z9 PCII = 0.464 Z1 + 0.463 Z2 + 0.417 Z3 + 0.433 7-,4 + 0.455 Z5 PC m = 0.375 Z1 + 0.381 Z2 + 0.378 Z3 + 0.382 7-4 + 0.378 Z5

Analysis and Variation Reduction Case Studies in Automobile Assembly", Transactions of NAMRI/SME, May 1991. 4. Manly, B.EJ. "Multivariant Statistical Methods, A

To illustrate the above principal components, lets consider the first cluster. The fact that the factor measurement for the original variables could be directly obtained from the principal components, the above component equation could

1990. 3. Hu, S. J., Wu, S. K, Wu, S. M. "Multivariate

be

translated into the

following: PC I = 0328 (var62) + 0.343 (vat64) + 0.342 (vat65) + 0.302 (vat96) + 0.341 (varl00) + 0.331 (varl01) + 0335 (varl20) + 0.341 (var121) + 0.334 (var122) It indicates that there are nine variables that have the major role in defining the variability of that cluster. Each of the nine variables has a weight factor. The higher the weight value, the more significant that variable

Primer," Chapman and Hall, 1986, pp 59 - 71.