quality control operating procedures for
TRANSCRIPT
APPROVED:
QUALITY CONTROL OPERATING PROCEDURES FOR
MULTIPLE QUALITY CHARACTERISTICS AND WEIGHTING FACTORS
by
Peter S. Hsing
. Thesis submitted to the Graduate Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment for the degree of
DOCTOR OF PHILOSOPHY
in
Industrial Engineering and Operations Research
Dr. Prabhakar M. Ghare, Chairman
Dr. Paul E. Torgersen Dr. G. Kemble Bennett
Dr. Kenneth E. Case
November 1973
Blacksburg, Virginia
Dr. B. Harshbarger
ACKNOWLEDGEMENTS
The author is grateful for this opportunity to express his appre-
ciation to the follo~,ing individuals for their help and encouragement
during the completion of this study and the pursuit of his doctorate:
Dr. P. M. Ghare, the author's major advisor, for providing the
initial impetus for research and his invaluable advice and
knowledge during all stages of the study.
The other members of his graduate committee, Dr. G. K. Bennett,
Dr. K. E. Case, Dr. B. Harshbarger and Dr. P. E. Torgersen for their
encouragement, assistance and constructive criticisms.
Mr. Dave Calhoun, Head of the Department of Biostatistics at G. D.
Searle Company for his professional advice and help.,
Mr. E. F. Chace, Mr. A. W. Zeiner and Mr. J. S. Lyday at IBM for
their editorial comment and valuable advice on the organization of the
writing.
The author's parents, Mr. and Mrs. H. T. Hsing, who inspired the
author to work toward the doctorate degree, the author's wife Mina who
sacrificed her own needs in order to help his pursuit of the doctorate
and who offered constant encouragement when it was greatly needed. To these people this dissertation is dedicated.
Mrs. Margie Strickler for her excellent typing of the final draft
of the manuscript.
ii
Chapter
1
2
3
TABLE OF CONTENTS
INTRODUCTION ...
1.1 Multivariate Quality Control Operational Procedure 2
l .2 Importance of Quality Characteristic and Variation in Weights Assigned to Different Characteristics 3
1. 3 Survey of the Literature . . 3
1.4 Overview of the Dissertation . 6
TESTING THE STABILITY OF THE MANUFACTURING PROCESS WITH RESPECT TO DISPERSION .......... .
2.1 Brief Review of the Univariate Case.
8
8
2.2 Notation and Symbols. . . . . 9
2.3 Statistical Test Procedure for Establishing Equivalence of Several Variance-Covariance Matrices. 11
2.4 Establishing the Stability of Past Operation with respect to Dispersion. . . . . . . . . . . . . . . . 12
2.5 Computational Procedure for Testing the Stability of the Manufacturing Process with respect to Dispersion 14
2.6 Computer Program for testing the Stability of Process Dispersion. . . . . . . . . . . . . . . . . 15
2.7 Example of Computations Involved in proving the Stability of the Manufacturing Process with respect to Dispersion. . . . . . . . . . . . . . . . . . . . 16
TESTING THE STABILITY OF THE MANUFACTURING PROCESS WITH RESPECT TO CENTRAL TENDENCY
3.1 Brief Review of Univariate Case.
3.2 Wilks Likelihood Ratio Test ....
3.3 The Critical Value for Decision.
24
24
25
26
3.4 Establishing the Stability of Past Operations with respect to Central Tendency. . . . . . . . . . . . . 27
i i i
iv
TABLE OF CONTENTS (continued)
Chapter Page
4
5
3.5 Computational Procedure for Testing the Stability of the Manufacturing Process with respect to Central Tendency. . . . . . . . . . . . . 30
3.6 Computer Program for Testing the Stability of Process Central Tendency . . . . . . . . . . . 32
3.7 Example of Computations Involved in proving the Stability of the Manufacturing Process with respect to Central Tendency. . . . . . . . . . . . . . . . . 32
PROCEDURE FOR MONITORING DISPERSION OF THE MANUFACTURING PROCESS ............. . 43
4.1 Brief Review of the Univariate Case. . . . . . 43
4.2 Theorems used in the Development of Dispersion Monitoring Procedure . . . . . . . . . . . . 44
4.3 Statistical Test for the Hypothesis that the Population Variance-Covariance Matrix Equals a Standard Matrix. . . . . . . . . . . . . . . . 45
4.4 Monitoring the Dispersion of the Manufacturing Process 47
4.5 Identification of Characteristics contributing to the Dispersion Control Problem . . . . 48
4.6 Computational Procedure for Monitoring the Dispersion of the Manufacturing Process 51
4.7 Computer Program for Monitoring the Process Dispersion................. 53
4.8 Examples of Computations involved in Monitoring Dispersion of the Current Manufacturing Process. 54
PROCEDURE FOR MONITORING THE CENTRAL TENDENCY OF THE MANUFACTURING PROCESS 59
5.1 Brief Review of the Univariate Case. . . . . . . 59
5.2 Theorems used in the Development of the Central Tendency Monitori no Procedure. . . . . . . . . 60
Chapter
V
TABLE OF CONTENTS (continued)
5.3 Development of the Central Tendency Monitoring Procedure. . . . . . . . . . . . . . . . . . .
5.4 Identification of Characteristics contributing to the Central Tendency Control Problem.
5.5 Computational Procedure for Monitoring the Central Tendency of the Manufacturing Process ....... .
5.6 Computer Program for Monitoring the Process Central Tendency . . . . . . . . . . . . . . . . . . . . . .
5.7 Example of Computations involved in Monitoring
Page
61 61
65
72
75
Central Tendency of the Current Manufacturing Process 76
6 SIMULATION ......... .
6.1 Random Number Generation .
6.2 Pre-analysis of Simulation
6.3 Simulation St11dy with Two Variables.
6.4 Simulation Study with Four Variables .
6.5 Computer Program for Simulation.
7 SUMMARY AND RECOMMENDATION.
7. 1 Summary. . . . . . . .
7.2 Areas for Further Study.
BIBLIOGRAPHY
VITA . . .
Appendix A MAIN PROGRAM FOR TESTING STABILITY OF PROCESS
Appendix B MAIN PROGRAM FOR MONITORING PROCESS ..
Appendix C TWO VARIABLES SIMULATION,MAIN PROGRAM
. . . .
80
80
82
83
86
88
91
91
94
96
99
100
l 06
110
vi
TABLE OF CONTENTS (continued)
Appendix D FOUR VARIABLES SIMULATION,MAIN PROGRAM ..
Page
115
121
130
131
Appendix E
Appendix F
SUBROUTINES .
TWO VARIABLES SIMULATION,RESULTS ..
(1) Correlation Coefficient= Standard Value
(2) Correlation Coefficient= Standard Value x 0.8 135
(3) Correlation Coefficient= Standard Value x 1.1 139
Appendix G FOUR VARIABLES SIMULATION RESULTS ... 143
(1) Correlation Coefficient= Standard Values and All Variances= Standard Values. . . . . . 144
(2) Correlation Coefficients= Standard Values and All Means= Standard Values. . . . . . . . 148
LIST OF TABLES
Table Page
I Sample Averages and Sample Variance-Covariance Matrices. 17
II
II I
IV
V
Log10 (Determinant of Si) ............. .
Transformation of W-Statistic to Provide Exact Upper Tail Tests Using F-Distribution.
I
Matrix of Xi Xi ..... . - I -
Matrix of n (!i-K) (!;-!)
V
21
28
33
38
Chapter l
INTRODUCTION
The manufacturing of a product by any successful industrial firm
involves, among many other aspects, an attempt to ensure that the
product meets certain specifications. Originally known as Quality
Control, these attempts were limited to inspection by various methods
and subsequent acceptance or rejection of the item. As the volume of
work grew, sampling techniques were evolved to extend the effectiveness
of the inspections, but, as was soon noted, an improvement in the approach
was required. The search for improvement took the form of attempts to
control processes and avoid the need for a retro-active system, one which
operated after the fact as did quality control. Thi's control effort
became known as Quality Assurance. The names, quality assurance and
quality control, are now used interchangeably in industry.
It is recognized that no system can, in and by itself, assure the
quality of an item. Variations and uncertainties inherent within the
environment are not yet that well understood, predictable, and control-
lable. But within the bounds of the state-of-the-art, quality control as
used herein is meant to convey the idea of in-process techniques that
may be used to control the quality of the finished product.
This research is devoted to the development of a procedure for
manufacturing applications to cases involving items having more than one
quality characteristic. These characteristics are assumed to be measured
on a continuous scale.
1
2
1.1 Multivariate Quality Control Operational Procedure
Because the preponderance of industrial processes are characterized
by hav1ng more than one parameter requiring control, and because tech-
niques of effectively controlling such as operation are at best under-
developed, the procedures explained in this research should have wide
application throughout industry.
Control charts have been the traditional means by which the need for
action in a given situation was identified. Such charts have not,
however, taken into account the interaction between variables when more
than one parameter is measured. A univariate control chart aids in
determinir,g the stability of the manufacturing process, and action is
taken when the process is not stable. Once the manufacturing process
capabilities have been determined and the process is stable,
action is taken only when the control chart indicates that the process
has gone out of control.
Just as in the case of the univariate quality control charts, the
multivariate situation requires the determination of process
stability with respect to both dispersion and central tendency. After
this stability has been established, the standard values for dispersion
and central tendency may be determined by examination of representative
past data or by consideration of future requirements as expressed by the
management. Stability of the process in the past prognosticates a
continuation of such stability until the occurrence of some assignable
cause that upsets the process. The lack of stability in past operations,
by the same token, would indicate the need for adjustment to the process.
3
There is no economic advantage, therefore, to be gained from estab-
lishing a monitoring procedure in conjunction with a system that is
not stable.
Once stability of the process has been established and standard
values for dispersion and central tendency calculated, action to main-
tain the process within the established bounds of dispersion and central
tendency becomes management's objective. Maintenance of the process
requires that guidelines for action be stated, e.g., action is required
when sample values indicate a lack of control with respect to dispersion,
central tendency, or both.
l .2 Importance of Quality Characteristic and Variation in Weights Assigned to Different Characteristics
In the multivariate situation, it is likely that the functional or
economic importance of the various characteristics to be controlled will
vary between characteristics. It is therefore desirable that the central
tendency monitoring procedure take into account such variations in
importance. This has been incorporated in the procedure described in
this dissertation and assigning different weights for the central tendency
value both above and below the standard value is presented and explained.
l .3 Survey of the Literature
In 1933, Dr. W. A. Shewhart (33) first proposed the statistical
quality control charts for the univariate case. Further technical and
theoretical developments were carried out by Barnard (2), Duncan (7,8),
Goel, Jain and Wu (10), Hartley (14), Noether (29), Ostle and Steck (30),
4
Page (31) and Weiler (36,37), etc. In addition to the application of
statistical quality control methods, many of these authors further
developed quality control procedures based on costs and other economic
factors. Because of the need for simultaneous control for related
variables, several authors developed procedures for the joint monitoring
of the central tendencies. Their approaches are briefly described below.
Jackson (15,16) proposed to use Hotelling T-square control chart
for central tendency monitoring. Basically, the test statistic is
where xis the sample average vector,
u is the standard mean vector,
Sis the sample variance-covariance matrix,
k is the number of quality characteristics, and
n is the sample size.
If the computed T-square value exceeds the appropriate upper fractile
of the T-square distribution, then the manufacturing process is said to
lack control.
Ghare and Torgersen (9) proposed to use a Chi-square control chart
for central tendency monitoring. Briefly, the test statistic is
where Vis the variance-covariance matrix. If the computed Q exceeds
the appropriate uoper fractile of the Chi-square distribution with k
degrees of freedom, the manufacturing process is said to lack control.
5
Montgomery and Klatt (24,25) proposed a method of determining the
optimal sample size, interval between samples, and critical region of the
parameters, based on the T-square control chart. Here the objective
function is the cost per unit of product for the test procedure, i.e.,
E(c) = E[c(l)] + E[c(2)] + E[c(3)]
where E[c(l)] is the expected cost per unit of sampling and carrying out
the test procedure, E[c(2)] is the expected cost per unit associated
wi~h investigating and correcting the process when the test procedure
indicates the process is out of control, and E[c(3)] is the expected
cost per unit associated with producing defective products. The optimal
sample size, interval between samples, and the critical region parameter
are derived by minimizing the expected total cost function.
In addition to the T-square control chart, Montgomery and Wadsworth
(26) present a method for monitoring the dispersion of the manufacturing
process. Basically, it is to convert the variance-covariance matrix
into a univariate random variable, the logarithm of the determinant of
the variance-covariance matrix. The distribution of this random
variable can be approximated by the normal distribution. The construction
of the control chart is based on the assumption that the manufacturing
process has already reached the stable state. Several samples are
taken from the process and the logarithm of the determinant of each
sample variance-covariance matrix is computed. Then, if the mean and the
standard deviation of those logarithms are represented by y and sy,
respectively, the control limits are given by
6
and
where Za/ 2 is the appropriate percentage point of the nonnal distribution.
The process is said to be out of control with respect to dispersion if
the logarithm of the determinant of the sample variance-covariance
matrix computed from the sample falls outside the control limits.
Otherwise, the process is said to be under control with respect to
dispersion.
1.4 Overview of the Dissertation
A procedure for testing the stability of the manufacturing process
with respect to the dispersion is established in Chapter 2. This
procedure is derived from the multivariate statistical technique for
identifying the common dispersion for multiple populations.
A procedure for testing the stability of the manufacturing process
with respect to the central tendency is developed in Chapter 3. This
procedure is based upon the Wilks Likelihood Ratio test which is
usually used for identifying the common mean for multiple populations.
A procedure for monitoring the uniformity of the products is
presented in Chapter 4. When the manufacturing process is found to be
out of control with respect to dispersion, then a set of follow-up
statistical tests is presented as an aid in identification of those
manufacturing phases which need adjustment.
A procedure for monitoring the central tendency is presented in
Chapter 5. This procedure allows assignment of different weights to
7
different quality characteristics and also for deviations above and below
the standard values. Here again, when the process is found to be out of
control with respect to the central tendency, a set of statistical tests
is presented to assist in locating the manufacturing phases which need
adjustment.
In Chapter 6, two simulation studies are presented. The purpose of
these simulations is to present alternate methods of examining the way
in which a proposed system will respond to the situation in which the
manufacturing process departs from the standard with respect to the
central tendency and/or the dispersion.
In Chapter 7, a summary of the entire work and some recommendations
for further study are presented.
Chapter 2
TESTING THE STABILITY OF THE MANUFACTURING PROCESS
WITH RESPECT TO DISPERSION
The purpose of this chapter is to develop and present a procedure
for testing the stability of the manufacturing process with respect
to dispersion. The major assumptions regarding the testing procedure
are stated. The mathematical symbols used in this work are defined, and
a step-by-step computational procedure is given for the user's con-
venience. As an illustration, an example with two quality character-
istics is discussed. It should be noted that the same example is used
in all chapters for the sake of consistency and simplicity.
2.1 Brief Review of the Univariate Case
Traditionally to prove the stability of the manufacturing process
in the univariate case, 20 to 40 samples, each containing 4 or 5 items
are taken from the manufacturing process at relatively equal time
intervals. This sample size and number of samples have been considered
satisfactory by many industrial applications. After the measurements
are taken, the central line and the trial limits are computed for one
of the following control charts.
Upper Lower Control Limit Control Limit Central Line
R-Chart D4 [ o3 R R
a-Chart s4 a B3 cr a
8
9
where o3, o4, s3, and B4 are constants which can be found in most quality
control texts. The R-Chart is easier to use, but is is appropriate only
when the sample size is small.
If all the statistics of the dispersion computed from the samples
fall within the trial limits, then the dispersion of the past operation
is concluded to be in control. If the process capability is satisfactory,
then maintenance of the statistics of the central line becomes manage-
ment's goal and the trial limits become the criteria for monitoring the
future manufacturing process.
2. 2 Notation and Symbols
The following definitions apply to all discussions and explanations
presented herein.
n is the sample size.
k is the number of quality characteristics.
mis the number of subgroups, lots or populations.
xitj is a single measruement where:
i indicates the sequence of time interval or the manufacturing
lot and i = 1, 2, ... , m.
t indicates the quality characteristic and t = 1, 2, ... , k.
j indicates the individual item taken from the ith time
interval and j = l, 2, ... , n.
{xilj xi 2j xikj) are measurements of all k-quality charac-
teristics made on the jth individual item which was taken
during the ith time interval.
x. ,
10
Xill xil 2 . . xiln
Xi 21 Xi22 x.2 , n = [x; tj] = . . . .
xi kl xik2 . xikn
are measurements of all n individual items which were taken
during the ith time interval.
Ki = (xil x; 2 ... xik) is the sample average vector for the
ith manufacturing lot. I
§_ = (x1 x2 ... xk) is the grand sample average vector.
s. ,
I
Sill Sil 2
Si 21 si22 = [sith] =
Si kl sik2
where sith fort,h=l,2, ... ,k
and i = l, 2, ... , mis the sample variance-covariance matrix
and elements of this matrix are computed from the observations
taken from the ith manufacturing lot.
_!:!_ = (u1 u2 ... uk) is the population mean vector.
v,, v, 2
V 21 v22 V = [vth] =
vkl vk2
11
is the population variance-covariance matrix.
Xis a general notation for a k-variates random variable, such
that l is normally distributed with mean vector equal !Land
variance-covariance matrix equal to V.
2.3 Statistical Test Procedure for Establishing Equivalence of Several Variance-Covariance Matrices
Let there be m populations and the random variable in each
population follows a k-variates normal distribution with a common mean
vector!:!_ and a variance-covariance matrix Vi where i = l, 2, ... , m.
It is assumed that all the Vi have the same value V. Therefore, the
problem may be stated so as to test the null hypothesis:
Ha: v1 = V 2 = = V = V m
against the alternative hypothesis
V. f V · l J
where if j and i, j = l, 2, ... , m.
This testing procedure is given by Kramer and Jensen (21 ), and
Chakravati, Loha and Kay (5). First, a sample of size ni is taken from
the ith population and the estimator of the variance-covariance matrix
for the ith population, Si, is calculated for i = 1, 2, ... , m.
The pooled estimator of the variance-covariance matrix is computed in
the following way:
(n1-l)s1 + (n2-l)s2 + ... + (nm-l)Sm ( n 1 - l ) + ( n 2 - l ) + .. . + ( nm -1 )
12
In practice, it is preferable that all samples be of equal size.
In the case of the equal size, the pooled estimator of the variance-
covariance matrix can be simplified as
s1 + s2 + ... + Sm s =-------p m
The statistic used in testing the null hypothesis is
l l -----]} n .-1 m 1 I (ni-1)
i = 1
m m x{[ 1_I __ 1(ni-l)] x LogjSpl - _I [(ni-l) LogjSij]} (2.1)
1 = l
If all the sample sizes are equal, then
2k2+3k-l m2-l R = 2.3026 [l - 6(k+l )(m-1) x m(n-1 )]
m x [m(n-1) LogjSpl - (n-1 \~1 LogjSi I] (2.2)
When dealing with large samples, the procedure is to reject the null
hypothesis if the test statistic R exceeds the appropriate upper fractile
of the Chi-square distribution with k(k+l)(m-1 )/2 degrees of freedom.
2.4 _Establishin~ the Stability of Past Operation with respect to Dispersion
The products fabricated during each time interval can be considered
as a population and, as there are m time intervals, so there are m
13
populations. It is assumed that the random variable in each of the
populations follows a k-variate normal distribution. Furthermore, the
central tendency is assumed to be under control during all the manufac-
turing periods. However, populations formed by the products fabricated
in different time intervals may have different variance-covariance
matrices if the manufacturing process is not stable with respect to the
dispersion. The statistical test just discussed in section 2.3 may be
applied to find out whether the past operations were actually in control
or not. Suppose a sample of size n was taken from each of them time
intervals. Now, m-sample variance-covariance matrices and the pooled
samples variance-covariance matrix are computed as Si where i = 1, 2,
... , m and Sp, respectively. The test statistic R for the equal sample
size can be computed according to equation (2.2). If the test statistic
R does not exceed the appropriate upper fractile of the Chi-square
distribution with k(k+l )(m-1 )/2 degrees of freedom, it may be concluded
that all the variance-covariance matrices are not significantly different
and the stability of dispersion for the manufacturing process has been
achieved. Furthermore, all the samples may be pooled to estimate
the variance-covariance matrix for the manufacturing process. If the
capability of the manufacturing process is satisfactory and a standard
value for dispersion of the process has been determined, maintenance
of the standard variance-covariance matrix becomes management's goal.
14
2.5 Computational Procedure for Testing the Stability of the Manufacturing Process with Respect to Dispersion
A sample of size n was taken from each of m manufacturing lots.
There are k measurements made on each of the individual items:
1) Record the measurements for the ith manufacturing lot, Xi,
in the matrix form defined in Section 2.2, where i = l, 2,
•.. ' m.
2) Compute the sample average vector, Ki' for the ith manu-
facturing lot, where i = l, 2, ... , m.
3) Compute the sample variance-covariance matrix, S;, for
the ith manufacturing lot, where i = 1, 2, ... , m.
4) Compute the determinant of the ith sample variance-covariance
matrix, IS; I, where i = 1, 2, ... , m.
5) Compute the pooled sample variance-covariance matrix s1 + s2 + ... + Sm
s = --------p m
6) Compute the determinant of the pooled variance-covariance
matrix, !Sp!•
7) Compute the R-statistic according to equation (2.2).
8) Compute the degrees of freedom for the Chi-square distribution;
d . f . = k ( k + 1 ) ( m-1 ) / 2 .
9) Find the table value of the upper fractile of the Chi-square
distribution with k(k+l )(m-1)/2 degrees of freedom and a
predetermined type one error.
10) Compare the value of the R-statistic in step 7) and the table
value of Chi-square distribution in step 9). The
15
manufacturing process is stable if the value in step 7) is
less than or equal to the value in step 9). The manufac-
turing process has not achieved the stable state with respect
to dispersion if the value in step 7) is greater than the
value in step 9).
2.6 Computer Program for Testing the Stability of Process Dispersion
1) Input Variables
IR= sample size
2)
JC= number of quality characteristics per inspected item
M = number of lots inspected
x(I,J,L) = measurement of the Lth quality characteristic on
the Jth item in the Ith inspected lot where
L = 1, 2,
J=l,2,
... ,
... ' JC;
IR;
I = 1, 2, ..• , M.
TABLEV = Chi-square table value with degree of freedom equal to
(M-1 )JC(JC+l )/2
Input Format
Variables
IR' JC, M
TABLEV
DO I = 1 ' M
DO J = 1 ' IR
(x(I,J,L), L = 1, JC)
Format
(315)
(Fl0.4)
(8Fl0.4)
16
3) FORTRAN Listing of Main Program
See Appendix A.
2.7 Example of Computations Involved in Proving the Stability of the Manufacturing Process with respect to Dispersion
ABC Company manufactures an antidiarrheal tablet preparation con-
taining two active drug ingredients, A and B. The major ingredient,
A, has the antidiarrheal effect exclusively. Component Bis present
in the tablet to prevent the side effects. It does not contribute
to the antidiarrheal activity, but produces an unpleasant physiological
effect when more than the recommended number of tablets are consumed.
The rest of the tablet contains a neutral ingredient. The Food and Drug
Administration has specified a range for each ingredient and a penalty
to be imposed for each detected violation of the specification.
Therefore, it ·is necessary to control both ingredients, A and B. This
example will be used throughout this whole research for illustrative
purposes.
In order to prove the stability of the manufacturing process, 30
samples of 20-tablets each are taken ar relatively equal time intervals.
Each tablet is assayed for both components A and B. The sample averages
and sample variance-covariance matrices are then calculated and
recorded in the manner shown in Table I.
17
Table I
Sample Averages and Sample Variance-Covariance Matrices
x, = (249.5301 2.5184) S = [ 16.0526 0.5403] l 0.5403 0.0287
x = (250.6852 2. 5084) S = [ 19.6316 0.6244] 2 0.6244 0.0275
x = (248.6164 2. 4587) [ 9. 9474 0. 3583} -3 s = 3 0. 3583 0. 0198
x = (250.4709 2.5004) [ 19. 3158 0.5654] s = 4 0.5654 0.0218
x = (250.1059 2.4902) [ l 0. 6842 0.2570] -'..:5 S5 = 0.2570 0.0163
x = (250.4703 2.5066) [ 8.8421 0.2763] s6 =
0.2763 0.0132
x = (251.5789 2.5817) [ 8.8947 0.3228] -7 S7 = 0.3228 0.0184
x = ( 248. 1619 2.4193) [10.4737 0. 3481] .'.'.8 s = 8 0.3481 0.0185
x = (250.9381 2.5403) [14.0526 0.4426] s = 9 0.4426 0.0193
18
X10 = (250.8936 2.5369) S = [12.5263 o. 5021] lO 0.5021 0.0280
K,, = (249.7131 2.4697) = [17 .8947 0.5718] s,, 0.5718 0.0268
K12 = (249. 3701 2.5057) S = [1 o. 9474 0.3366] 12 0.3366 0.0160
X13 = (251.7016 2.5592) S = [ 13. 7895 o. 5066] 13 0.5066 o. 0242
!14 = (250.0506 2.5048) S = [ 13.4737 0.4056] 14 0.4056 0.0209
!15 = (250.3838 2.5304) S = [13.6842 0.4751] 15 0.4751 0.0247
K16 = (247. 0457 2.4733) S = [14 .4211 0. 5313] 16 0.5313 0.0291
!17 = (249.6074 2.4751) S = [14.1053 o. 5136] 17 0.5313 o. 0291
K18 = (249.8517 2.5093) S = [13.5263 0.4001] 18 0. 4001 0.0208
x 9 = (250. 1998 2.5091) S = [10.2105 o. 2995] -1 19 0.2995 0.0154
19
K20 = (249.5176 2.4525) S = [13. 2105 0.3651] 20 0.3651 0.0172
K21 = (249.2402 2.4948) = [15.4737 0. 5222] s21
0.5222 0.0233
K22 = (249. 6689 2.4789) [ 9. 4737 0. 3045] s -22 - o. 3045 o. 0138
!23 = (251.7006 2.5601) S = [21. 7368 0.6605] 23 0.6605 0.0248
!24 = (251.3898 2.4929) 524 = [
8.3684 0.2876]
0.2876 0.0202
!25 = (250. 7766 2.4819) S = [ 10. 5789 0.2819] 25 0.2819 0.0178
K26 = (250. 1666 2.5082) S = [ 23. 8421 0.7177] 26 0.7177 0.0265
K27 = (250.3498 2.5125) S = [ 10.0526 0. 2911] 27 0.2911 0.0174
x28 = (250.4281 2.5148) [ 8.8421 0.2541] s -28 - 0. 2541 0.0139
K29 = (249.8111 2.4978) S =[13.8947 0.4147] 29 0.4147 0. 0189
!30 = (250.6680 2.4936) S = [15.8421 0.5249] 30 0.5249 0.0261
20
All the determinants and the logarithms of determinants are
calculated and recorded in the manner shown in Table II.
According to equation (2.2)
8 + 6 - l 899 R = 2.3026 X [l - (l 8)( 29 ) X ( 30)( 29 )]
X [(30)(19)(-0.9927) - (19)(-31.0448)]
= 2.3026 X 0.9743 X 34.01222 = 53.8690
i
l
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
21
Table II
Log10 (Determinant of Si)
Determinant of Si logl 0 (Determinant of Si)
0.1683 -0. 7739
0.1507 -0.8227
0.0690 -1.1611
0.1023 -0.9101
0. l 077 -0.9678
0. 0400 -1.3971
0.0599 -1. 2285
0.0726 -1 . 1391
0.0753 -1 . 1232
0.0982 -1. 0079
0.1527 -0.0162
0.0623 -1.2055
0.0777 -1 . 11 35
0. 1166 -0. 7799
o. 1117 -0.9516
0.1376 -0.8614
0. 1392 -0.8564
0.1218 -0.9143
0.0677 -1. 1694
0.0936 -1. 0207
0. 0885 -1 . 0531
22
22 0.0383 -1.4168
23 0. 1037 -0.9942
24 0.0860 -1.0655
25 0. 1089 -0.9598
26 o. 1172 -0.9311
27 0.0903 - 1. 0443
28 0.0582 -1.2351
29 o. 0901 -1.0453
30 0.1383 -0.8595
23
If the predetermined level of significance is one percent, then
x2 [ ( k ) ( k + 1 )( m- 1 ) / 2 , 0 . 9 9] = x2 ( 8 7 , 0 • 9 9 ) = 11 3 •
The test statistic R does not exceed the appropriate upper fractile
of the Chi-square distribution with k(k-l)(m-1)/2 degrees of freedom,
and the null hypothesis is therefore accepted. In other words, since
the products manufactured in different time intervals have the same
dispersion, it may be concluded that stability of the manufacturing
process has been achieved. Since the manufacturing p~ocess capability
is satisfactory, a decision to use
r 113.4596 ! , 0.4301 i
o. 43oil I 0.0213:
as the standard value of the variance-covariance matrix is made, and
maintaining
-ll3.4596
0. 4301
o. 43011 o. 0213j
as the process dispersion matrix becomes management's goal.
Chapter 3
TESTING THE STABILITY OF THE MANUFACTURING PROCESS
WITH RESPECT TO CENTRAL TENDENCY
The purpose of this chapter is to establish a procedure for testing
the stability of the manufacturing process with respect to the central
tendency. The major assumptions regarding the testing procedure are
stated and a step-by-step computational procedure is given for the
user's convenience. The example used in the last chapter is carried
over for illustrative purposes.
3.1 Brief Review of Univariate Case
Traditionally to prove the stability of the manufacturing process
in the univariate case, 20 to 40 samples, each containing 4 or 5 items are
taken from the manufacturing process at relatively equal time intervals,
as in the case of proving the stability of dispersion. The central line
is m
X = I xi i = l
and the trial limits are
x - A2R and x + A2R where A2 is a constant which can be found in most statistical quality
control texts. If all the sample means fall within the trial limits,
then the mean of the manufacturing process is concluded to be in control.
If management is satisfied with the capability of the manufacturing
process, then maintenance of the central line becomes management's goal
24
25
and the trial limits become the criteria for monitoring the future manu-
facturing processes.
3.2 Wilks Likelihood Ratio Test
The likelihood ratio test was originally developed by Wilks (38)
and discussed in detail by Anderson (1) and Kramer and Jensen (20).
Let there be m multivariate populations, such that the random variable
(xil xi 2 ... xik) from each population is normally distributed with mean
vector .!:!_i and a common variance-covariance matrix V, where i = 1, 2, ... ,
m. It is desired to test the null hypothesis
against the alternative hypothesis
H1: U. I U. -1 -J
where i; j, and i, j = l, 2, ... , m.
First of all, a sample of size n is randomly selected from each
population and k measurements are taken for each item. The measurements
of all the inspected items from the ith population can be written in I
the matrix form as Xi as defined in section 2.2, where i = 1, 2, ... ,
m. The sample average vector for the ith population and the grand
average vector can be computed as Ki and§_ respectively, where i = l,
2, ... , m. Now T, a (kxk) matrix, is defined as the 11 sum of square of
total , 11 and can be computed in the followinq wa.v:
m T = l
i=l
I x. x. l l
26
A, a (kxk) matrix, is defined as the "sum of square of among-group," and
can be computed in the following way:
I m A = n l
i=l (X.-G) (X.-G} -, - -, -
and E, a (kxk) matrix, is defined as the "sum of square of within-group,"
and can be computed in the following way:
E=T-A.
Finally, the Wilks Likelihood Ratio test is defined as
The distribution of the test statistic W depends upon the parameters of
k (the number of the measurements per inspected item), va (degrees of
freedom among-group= m - 1), and ve (degrees of freedom within-group=
m x n - m). The null hypothesis
is to be rejected for the small value of the computed W-statistic.
3.3 The Critical Value for Decision
The distribution of the test statistic W depends upon the parameters
of k, va, ve. Kramer and Jensen (20) have prepared a simple table for
type one error equal to 0.05 and 0.01 where k ranges from 1 to 5, where
va takes on values up to 6, and where ve assumes values between 2 and
1000 inclusive. Wall (35) has prepared a table for type one error
equal to 0.05 and 0.01 where k ranges from l to 8, where va takes on
several values between land 120, and where ve assumes values between
27
1 and 1000 inclusive. When values of k and va fall outside the ranges
covered in the available tables, Kramer and Jensen (20) presented some
alternatives. One of the alternatives requires a transformation of the
computed value of W-statistic and is shown in Table III.
When values of k and va are as in the first column, then the trans-
formation indicated in the second column generates a quantity having an
F-distribution with degrees of freedom as listed in the third column.
The null hypothesis
is rejected when the transformed value of W-statistic exceeds the upper
fractile of the F-distribution with degrees of freedom listed in the
corresponding third column and the predetermined type one error.
Other alternatives should be used when the values of k and va are other
than those in Table 1. An approximation for large samples can be used
as follows. Let
b = ve _ (k-va+l) 2
the hypothesis in question is rejected if-bx LnW exceeds the upper
fractile of Cni-square distribution with (k x va) degrees of freedom.
3.4 Establishing the Stability of Past Operations with respect to Central Tendency
The products fabricated during each time interval can be considered
as a population and, as there are m time intervals, so, there are m
populations. It is assumed that the random variable in each of the
populations follows a k-variate normal distribution. Furthermore, the
28
TABLE II I
Transformation of W-Statistic to Provide Exact Upper Tail Tests Using F-Distribution
Parameters (k,va) Statistic havinq F-dist. Deqrees of Freedom
va = 1, any k 1 - w ve + va - k W X k k, (ve+va-k)
va = 2, any k 1 - /W ve + va - k - 1 X k /W 2k, 2(ve+va-k-1)
k = 1, any va 1 - W ve W X va va, ve
k = 2, any va 1 - 1W ve -X /W va 1 2va, 2(ve-1)
29
process dispersion is assumed to be in control in all the manufacruring
periods. Populations formed by the products fabricated in different
time intervals may have different central tendency if the manufacturing
process is not stable with respect to the central tendency. Wilks
Likelihood Ratio test just discussed in section 3.3 may be applied to
find out whether the past operations were actually in control or not.
Suppose a sample of size n was taken from each of them time intervals.
Now, m-sample average vectors and the grand sample average vector can be
computed as Ki where i = 1, 2, ... , m and §_ respectively. The "sum of
square of tota 1 , 11
m T = l
i = 1
I
X. X. , the "sum of square of among-group, 11 l l
m I
A= I n(Ki-§_) (!-§.), the "sum of square within-group," i = 1
E = T - A, and Wilks Likelihood Ratio test statistic,
w = ffi can be calculated. If the test statistic Wis less than the corresponding
critical value with the parameters k, va and ve, then, the
conclusion is that the past operations were not stable. Otherwise,
the past operations were stable and one may expect the stability to
continue. At this time, the capability of the manufacturing process has
been proved satisfactory; a standard value for the central tendency has
been determined; and maintenance of this standard value becomes manage-
ment's new goal.
30
3.5 Computational Procedure for Testing the Stability of the Manufacturing Process with Respect to Central Tendency
A sample of size n was taken from each of m manufacturing lots.
There are k measurements made on each of the individual items. Note
that all the samples were the same as those used in the last chapter.
1) Record measurements for the ith manufacturing lot, X;,
in the matrix form defined in section 2.2, where i = 1,
2, ... , m.
2) Perform the matrix multiplication for the ith manufacturing
lot, X; X;, where i = 1, 2, ... , m.
3) Compute the "sum of square of total" matrix, I I I
T = x1 x1 + X2 x2 + ... + Xm Xm
4) Compute the determinant of the II sum of square of tota 1"
matrix, !Tl. 5) Compute the sample average vector,!;• for the manufacturing
lot, where i = 1, 2, ... , m.
6) Compute the grand sample average vector,§_.
7) Perform the matrix multiplication for the ith manufacturing I
lot, (!;-§.) CK;-§.), where i = 1, 2, ... , m.
8) Compute the "sum of square of among-group" matrix, I I I
A= n[(!1-§_) ([1-§_)+(~-§_) (~-§_)+ ... +(~-§_) (~-§_)]
9) Compute the determinant of the "sum of square of group"
matrix, !Al. 10) Compute the "sum of square of within-group" matrix,
E = T - A.
31
11) Compute the determinant of the "sum of square of within-
group" matrix, jEj.
12) Compute the test statistic,
w =
13) Compute values of parameters, va = m - l and ve = m(n-1).
14) Find the table value of W-statistic indexed by parameters,
k, va, and ve from Kramer and Jensen (20) or Wall (35). The
manufacturing process is said to be stable with respect to
central tendency if the W-statistic exceeds the table value.
Otherwise, the process is concluded not to be stable.
15) If the table value for the W-statistic indexed by param-
eters k, va, and ve is not available:
a) Convert the W-statistic into an F-statistic according
to the transformation given in Table I and compare it
with the appropriate upper fractile of F-distribution.
The process is said to be not stable if the converted
F-statistic exceeds the table value. Otherwise, the
process is concluded to be stable.
b) If Table I cannot be applied, then compute
Z = -[ve - (k-va+l)/2](LnW) .
The manufacturing process is said to be not stable if
the Z value exceeds the upper fractile of Chi-square
distribution indexed by k(va) degrees of freedom and a
predetermined type one error. Otherwise, the process
is concluded to be stable.
32
3.6 Computer Program for Testing the Stability of Process Central Tendency
This computer program is combined with the computer program for
Testing the Stability of Process Dispersion described in section 2.6;
therefore, there is no need to duplicate those variables already
inputed in the last computer program.
1) Input Variables
VA= Degrees of freedom among-group= m - 1
VE= Degrees of freedom within-group= m(n-1)
INDEX= 1 indicates table value is available,
= 2 indicates F-transformation,
= 3 indicates Chi-square transformation.,
TABLEM = Table value according to the given index number.
2) Input Format
Variables
VA, VE
INDEX, TA BLEM
Format
(2Fl0.4)
(15, Fl0.4)
3) Fortran Listing of Main Program
See Appendix A.
3.7 Example of Computations Involved in Proving the Stability of the Manufacturing Process with Respect to Central Tendency
The example in section 2.7 is carried over for discussing the test
of the stability of the manufacturing process with respect to central
tendency. From the same 30 samples of 20-tablet size used in the
33
I I I
last example, Xi Xi' n(Xi -X) (Xi-X), where i = 1, 2, ... , m, T
matrix, A matrix and E matrix are calculated and recorded as shown in
Table IV.
The 11 sum of square of total 11
30 I
T = I x. x. = i =l l l
[37,558,460.0000
375,919.8000
375 '919 .8000·1
3,770.9130 .....J
and the determinant of T = 315,621,300.0000. The 11 sum of square among-
group 11
and the determinant of A= 0.9502.
r 431.1323
13. 2196
7 13. 2196 I !
0. 6410 !
Therefore, the 11 sum of square within-group 11 is
[37,558,460.0000 375,906.500071
E=T-A= 375,906.5000 3,770.2720 I
and the determinant of Eis 299,892,7000.0220.
Hence, the test statistic
W = 299,892,700.0220/315,621 ,300.2730 = 0.9501 .
Here, k = 2, va = 29, ve = 570 and F-transformation is applied to this
case. The degrees of freedom of converted F-statistic are (2va, 2ve-2)
= (58, 1138), and the converted F-statistic is calculated in the
fo 11 owing way:
F = l - 0.9747 570 - 1 0.9747 X 29 = 0.050l '
34
Suppose the type one error is one percent, then the 99 percent upper
fractile of. the F-distribution with degrees of freedom (58, 1138) is
1.48. Since 0.0501 is less than 1.48, the null hypothesis is accepted
and the past operations are said to be in control.
35
Table IV
I
Matri X of X; X;
i -
1 [ 1 , 245,609.0000 12,578.7900]
12,578.7900 127.3955
2 [ 1 ,257 ,233. 0000 12,588.2600]
12,588. 2600 126.365~
3 [1 , 236,390.0000 12,232.2500]
12,232.2500 121.2809
4 [1,255,079.0000 12,536.3700]
12,536.3700 125.4569
5 [ 1 ,251 ,261. 0000 12,461 . 0300]
12,461.0300 124.3288
6 [1,254,875.0000 12,561.7000]
12,561.7000 125.9085
7 [1,266,007.0000 1 2 , 996. 3500]
12,996.3500 133. 6581
8 [ 1 , 231 ,885. 0000 12 , 041 • 01 00]
12,014.0100 117 .4087
36
i
9· [ 1 ,259 ,665. 0000 12,757.4300]
12,757.4300 129.4265
10 [1,259,189.0000 12 , 719 . 11 00 ]
12,719.1100 128 ,8392
11 [ 1 ,247. 472. 0000 12,344.9400]
1 2 , 344 . 94 00 122.4927
12 [1,243,916.0000 1 2 • 503 . 5100 l 12,503.5100 125.8793_
13 [ 1 ,267. 334. 0000 1 2 , 892 . 9200 ]
12,892. 9200 131. 4551
14 [1,250,761.0000 1 2 , 534 . 0800]
12, 34. 0800 125.8737
15 [1,254,100.0000 12,680.5800]
12,680.5800 128.5299
16 [ 1,240,749.0000 12,329.5300]
12,329.5300 122. 9004
17 [1,246,344.0000 12 , 365. 7 300 ]
12,365.7300 123.0636
18 [1,248,774.0000 12 , 546. 6400 ]
12,546.6400 126.3273
37
i -
19 [1,252,192.0000 12,561.1000]
12,561.1000 126.2023
20 [1 , 245,431.0000 12,245.9700]
12,245.9700 120. 6255
21 [1,242,707.0000 12,445.9600]
12,445.9600 124.9233
22 [ 1 , 246,871 . 0000 12,384.0700]
12,384.0700 123.1657
23 [1,267,476.0000 1 2 , 900 . 1 000]
12,900.1000 l 31 . 5538
24 [1,264.095.0000 12,539.1000]
12,559.1000 124.6713
25 [ 1 ,257, 978. 0000 12,453.2300]
12,453.2300 123.5317
26 [1,252,119.0000 12,562.9900]
12,562.9900 126. 3254
27 [1,253,691.0000 12,585.5100]
12,585.5100 126.5823
28 [1,254,452.0000 12,600 .4400J
12,600.4400 126.7500
38
; -
29 [ 1 ,248. 376. 0000 1 2 , 48 7 . 51 00]
12,487.5100 125. 1400
30 [1,256,989.0000 12,511.2100]
l 2,511 . 21 00 124. 8558
39
Table V
- I
Matrix of n{!;-!) {!;-!)
; -
1 [ 8.5186 -0. 1735]
-0.1735 0.0035
2 [ 5.0489 0. 0326]
0.0328 0.0002
3 [ 49.0668 1.4550] 1. 4550 0. 0431
4 [ 1 ,6609 -0,0272]
-0.0272 0.0004
5 [ 0.1181 0.0230] 0.0230 0.0045
6 [ 1. 6542 0.0082] 0.0082 0.0001
7 [ 38.9872 2. 1388] 2. 1388 0. 1173
8 [81.6725 3.4707]
3.4707 0. 1475
9 [11.4117 0.5307]
0.5307 0.0247
40
; -
10 [ 10.1060 0.3941]
0.3941 0.0154
11 [ 4.4111 0.3334] 0.3334 0.0252
12 [13.2061 -0.0097]
-0.0097 0. 0001
13 [46. 1378 1 .6433] l . 6433 0.0585
14 [ 0. 3491 0.0010] 0.0010 0. 0001
15 [ 0.8087 0.1017] 0. l 017 0.0126
16 [25.8558 0.7235] 0.7235 0.0202
17 [ 6.6191 0. 3459] 0.3459 0.0181
18 [ 2.1907 -0.0275]
-0.0275 0.0003
19 [ 0.0058 0.0013]
o. 0013 0.0003
41
i -
20 [ 8.8480 0.6998] 0.6998 0.0553
21 [17.7652 0.1951] 0. 1951 0. 0021
22 [ 5.2791 0.26931 0.2693 0.0137
23 [46.0784 1. 6682] l. 6682 0.0604
24 [29.1429 -0.2963] -0. 2963 0.0030
25 [ 7.0530 -0.2764] -0.2764 0. 0108
26 [ 0. 0052 -0.0010] -0.0010 0.0002
27 [ o. 5583 0. 0245] 0.0245 0.0011
28 [ 1. 2045 0.0475] 0.0475 0.0019
42
i -
29 [ 2.7615 0.0545 ]
0.0545 0.0019
30 [ 4. 7095 -0.1122 J -0. 1122 0.0027
Chapter 4
PROCEDURE FOR MONITORING DISPERSION
OF THE MANUFACTURING PROCESS
The purpose of this chapter is to develop an operational procedure
for monitoring the current manufacturing process with respect to the
dispersion. When the manufacturing process is out of control, a set of
statistical tests can be used to locate the manufacturing phases which
need to be adjusted. A step-by-step computational procedure is included
for the user's convenience. The example discussed in previous chapters
is carried over for illustrative purposes.
4.1 Brief Review of the Univariate Case
In the univariate case, either the R-control or the a-chart is
usually employed for monitoring dispersion of the manufacturing process,
and control limits are as follows:
R-chart
a-chart
Upper Control Limit
o4R
B4;-
Lower Control Limit
If the statistic of the dispersion computed from the samples falls out-
side of the control limits, then it is said that the manufacturing
process lacks control and it becomes necessary to search for the
assignable cause. If, on the other hand, the statistic of the dis-
persion falls within the control limits, it is more economical to leave
the manufacturing process alone. However, there are exceptions,
43
44
sometimes, the manufacturing process is said to lack control even though
the computed statistic of dispersion falls within the control limits,
i.e., seven points in a row fall at one side of the central line. A
more detailed discussion is given by Duncan (6) and Grant (11).
4.2 Theorems Used in the Development of the Dispersion Monitoring Procedure
The following are some theorems used in the development of the
dispersion monitoring procedure. These theorems are either stated or
proved by Stein (34).
Theorem 4-1
If the matrix Bis (kxk), then the determinant .of Bis
where the symbol l denotes the summation of k! terms. (j )
Theorem 4-2
If matrices A (kxk) and B (kxk) are of order k, then the deter-
minant of AB equals (determinant of A) x (determinant of B).
Theorem 4-3
If the matrix A has an inverse, then the determinant of A-l equals
(determinant of A)-l.
45
Theorem 4-4
Suppose a matrix A (kxk) equals a matrix nB, where n is a constant
and Bis a (kxk) matrix. Then the determinant of A equals (nk x
determinant of B).
Theorem 4-5
If matrix Bis (kxk) then the trace of Bis
k tr(B) = l b ...
i =l 11
Theorem 4-6
Suppose that matrix A (kxk) equals matrix nB, where n is a constant
and B is a (kxk) matrix. Then
tr(A) = n tr(B)
4.3 Statistical Test for the Hypothesis that the Population Variance-Covariance Matrix Equals a Given Matrix
Suppose that a sample of size n with k measurements made on each
individual item is taken from a k-variates normal distribution with
mean vector !1._ and the sample variance-covariance matrix is E. The
likelihood ratio criterion for testing the null hypothesis
H0: E = V
against the alternative hypothesis
46
is e kn/2 1 n/2 -1/2 tr(nsv-l)
A = ( n) I nSV- I e
The above test procedure was given by Anderson (l ).
Expansion of the likelihood function by Theorem (4-2), Theorem
(4-4) and Theorem (4-6), results in
by Theorem (4-3). The above equation can be simplified as
Taking the logarithm of equation (4.3), results in
Ln >. = kn/2 + f LnlSI - f LnlVI - f tr(SV-l)
(4. 1)
(4. 2)
(4.3)
( 4 .4)
Multiplying (-2) by equation (4.4) generates the likelihood function
L = 2Ln >.=-kn - nLnlSI + nLnjVI + ntr(sv-l)
or
L = n{Ln ffi -k + tr(sv-1 )} (4. 5)
The test statistic Lis asymptotically Chi-square distributed with
degrees of freedom k(k+l )/2 when n becomes large. This test statistic
L was investigated by Box (4), Lawlty (23), Kullback (3) and Korin (17)
etc., and all the investigations show that the approximation appears to
be very good.
47
4.4 Monitoring the Dispersion of the Manufacturing Process
Suppose the manufacturing process is proved to be stable and the
products fabricated by this manufacturing process follow a k-variate
normal distribution with the standard mean vector U and the standard
variance-covariance matrix V. It is further assumed that the products
fabricated by the current manufacturing process follow the k-variate
normal distribution with the mean vector U and some variance-covariance
matrix E. If the variance-covariance matrix Eis not significantly
different from the standard variance-covariance matrix V, then it may be
concluded that the current manufacturing process is in control. In
order to determine whether the matrix E equals the matrix V, the
statistical technique may be employed to test the n~ll hypothesis
against the alternative hypothesis
The actual operating procedure is to take a sample of size n from the
current manufacturing process, then compute the estimator, S, for the
current process variance-covariance matrix, E. Because the standard
variance-covariance matrix Vis predetermined and the inverse of the
matrix V can be calculated long before the monitoring, therefore, the
statistic, L, is computed as follows:
V -1 L = n {Ln - 5- - k + tr(SV )}
The null hypothesis is rejected if the test statistic, L,exceeds
the appropriate fractile of Chi-square distribution with degrees of
48
freedom k(k+l)/2. In this case, the conclusion is that the current
manufacturing process lacks control and it becomes necessary to
determine the assignable cause. On the other hand, when the test
statistic, L, does not exceed the appropriate upper fractile of the
Chi-square distribution with degrees of freedom k(k+l)/2, then it is
more economical to leave the manufacturing process alone.
4.5 Identification of Characteristics Contributing to the Dispersion Control Problem
Since there are k quality characteristics to be monitored, and
since the sample results indicate that the current manufacturing process
lacks control, it is then helpful to have some test to identify which
of these k quality characteristics contributes to the lack of control
in dispersion. With this kind of information, efforts can be concen-
trated in those areas which were suspected from the statistical test
results. This enables detection of the trouble spots in relatively
short time. Essentially, the statistical test helps to reduce the
manufacturing process downtime.
For the univariate case, the statistical procedure for determining
the variance of a population, a2, from a given value, a02, is to test
the null hypothesis
against the alternative hypothesis
49
The procedure for this statistical test is to take a sample of size n
from the population and calculate the estimator of the population
variance, say s2, and the test statistic is computed as
The null hypothesis is rejected if the computed value of C exceeds the
appropriate upper fractile of the Chi-square distribution with degrees
of freedom (n-1 ). The above statistical test is given by Duncan (6).
Suppose there are k quality characteristics to be monitored, so k
populations, each with a variance ejj is assumed. Further assume a
desirable value for the variance of each population, say vjj' where
jj = 1, 2, ... , k. To determine whether the value df ejj is equal to
the value vjj is to test the null hypothesis
against the alternative hypothesis
e · • f V • • JJ JJ
where jj = 1, 2, ... , k.
This is accomplished by taking a sample of size n from the jth
population, computing the estimator of the variance, Sjj' and calculating
the test statistic, Cj, as follows:
(n-1 )Sjj C • = ---.-,,...---=-=--J V .•
JJ (4. 1)
50
where j = 1, 2, ... , k. The null hypothesis is rejected if the test
statistic falls in the reject region of the Chi-square distribution with
degrees of freedom (n-1 ). In this case, the conclusion is
that the manufacturing phase responsible for the jth quality charac-
teristic is out of control and some adjustment of this characteristic
may be necessary.
Second, the interaction between two quality characteristics is
investigated. The approach taken here is to partition the standard
variance-covariance matrix, the current process variance-covariance
matrix and the sample variance-covariance matrix as follows:
[V, V ij] V = 11 sub V .. V .. Jl JJ
Esub = [ ei; eij] e .. e .. Jl JJ
[5" •;j] 11 Ssub = s .. s .. Jl JJ
where i t j and i, j = 1 , 2, ... , k.
An interest in only two quality characteristics is assumed for each
test and the likelihood ratio approximation test discussed in Section 4.3
is used to test whether the (2x2) population variance-covariance matrix
Esub equals the predetermined variance-covariance matrix, Vsub' The
test statistic, Lsub' can be calculated as follows:
51
If the test statistic, Lsub' exceeds the appropriate upper fractile of
the Chi-square distribution with degrees of freedom 2(2+1 )/2 = 3, the
practical conclusion is that Esub does not equal Vsub· Now, suppose
that all the variances in the set of "sub" are proved not siginficatnly
different from their corresponding predetermined variances by previous
Chi-square tests. It is then logical to say that the interaction between
these quality characteristics in the set of "sub 11 is the cause disturbing
the manufacturing process. The total number of this kind of "sub"
statistical tests is
4.6 Computational Procedure for Monitoring Dispersion of the Manufacturing Process
The standard values for the central tendency,!!_, and the variance-
covariance matrix, V, are either computed from the past data or deter-
mined by management. The inverse and the determinant of V can be com-
puted before monitoring the process. Assume the tyoe one error is also predetermined.
A sample of size n is taken from the current manufacturing process
and there are k measurements made on each of then individual items.
1) Compute the sample variance-covariance matrix S based
on the sample measurements.
52
2) Perform the matrix multiplication, (sv-l ). -1 3) Compute the trace of SV .
4) Compute the determinant of the sample variance-covariance.
5) Compute the Ln(IVI/ISI),
6) Compute L = n[Ln(IVI/ISI) - k + tr(sv-l )].
7) Find the Chi-square table value indexed by k(k+l )/2 degrees
of freedom and the predetermined type one error.
8) The current manufacturing process is said to be out of
control with respect to the dispersion if the computed value
of Lin step 6) exceeds the Chi-square table value in step
7). Otherwise, leave the process alone.
If the current manufacturing process is out of.control
with respect to the dispersion, then the following statistical test can
be used to assist in locating the trouble spot.
9) Compute the test statistic for the jth population, Cj'
according to equation (4.1) for j = 1, 2, ... , k.
10) If the test statistic Cj exceeds the Chi-square table value
indexed by degrees of freedom (n-1) and the predetermined
type one error, then the manufacturing phase corresponding
to the jth quality characteristic is said to be out of control
and requires adjustment. Here, j = 1, 2, ... , k.
11) Partition the standard variance-covariance matrix, V, and
the sample variance-covariance matrix, S, into (2x2) sub-
matrices. There are k{k-1 )/2 ways to partition matrices
V and S.
53
12) Compute
Lsub = n {Ln(IVsubl/lSsubl) - 2 + tr(Ssubvsub-l)}
The total number of these sub-tests equals k(k-l)/2.
13) If the variances in the set of 11 sub 11 were proved not signi-
ficantly different from their corresponding standard variances
by Chi-square tests defined in step 9), but if the Lsub
exceeds the Chi-square table value indexed by degrees of
freedom 3 and the predetermined type one error, then it is
concluded that the interaction between the two quality
characteristics in the set of 11 sub 11 is the cause of troub 1 e.
4.7 Computer Program for Monitoring the Process Dispersion
l) Input Variables
IR= Sample size
JC= Number of quality characteristics per inspected item
x(J,L) = Measurement of the Lth quality characteristic on the
Jth sampling item
where L = 1, 2,
J = l, 2,
... ,
. . . ' JC;
IR •
CHIK = Chi-square upper fractile value with d.f. = JC(JC+l)/2
PVI(I,J) = Inverse of standard variance-covariance matrix
DPVL = Log of determinant of standard variance-covariance matrix.
2) Input Format
Variables
IR, JC
DO J = l, IR
(x{J,L), L = l, JC)
CHIK
DO I= l, JC
( PV I ( I , J) , J = l , JC)
DPVL
54
Format
(215)
(8Fl0.4)
{Fl0.4)
(8Fl0.4)
(Fl0.4)
3) FORTRAN Listing of Main Program
See Appendix B.
4.8 Examples of Computations Involved in Monitoring Dispersion of the Current Manufacturing Process
In Chapters 2 and 3, 30 samples of 20-tablet size were used to
prove that the manufacturing process was stable with respect to
dispersion as well as central tendency. It is expected that the
stability of the manufacturing process will continue until the occurrence
of some assignable cause that upsets the process. The standard value
for the central tendency is determined by management as
!!_' = (250.1689 2.5027)
and the standard value for the variance-covariance matrix is determined
by management as
V = [13.4596 0.4301
0. 4301]
0.0213
55
The inverse of the matrix Vis computed as
v-l = [ 0.2094 -4.2291
-4.2291] 132.3461
and the determinant of the matrix Vis calculated as 0.1017. The type
one error for the following examples is one percent.
Example A
The 31st sample of 20-tablet size is taken from the current manu-
facturing process and the sample variance-covariance matrix is computed
as
S = [14.0000 0.4000
0.4000] 0.0220
Assume that the central tendency is under control. Management is now
interested in whether the current manufacturing process is still under
control. Here, n = 20 and k = 2. The upper fractile of the Chi-
square distribution with 3 degrees of freedom and one percent of type
one error is 11.3000.
The determinant of the matrix Sis computed as 0.1480 and the trace
of (sv-l) is equal to 2.6033, and the test statistic
L = n {Ln(IVI/ISI) - k + tr(SV-l )}
= 20 [Ln(O. 1017/0.1480) - 2 + 2.6033]
= 4.5580
56
Since the test statistic, L, is less than the upper fractile of
Chi-square distribution, it is concluded that the process is in control
and it is more economical to leave the process alone.
Example B
Suppose the results of the 32nd sample of 20-tablet size gives the
variance-covariance matrix
S = [15.0000 0.5000
0.5000] 0.0450
Assume that the central tendency is under control. Management now is
interested in whether the current manufacturing process is still under
control. Again, n = 20 and k = 2. The upper fractile of the Chi-square
distribution with 3 degrees of freedom and one percent of type one error
is 11.3000.
The determinant of the sample variance matrix Sis 0.425, the trace
of (SV-l) is equal to 4.8674, and the test statistic
L = n {Ln(!VI/ISI) - k + tr(sv-l )}
= 20 [Ln(0.1017/4250) - 2 + 4.8674]
= 28.7480
Since the test statistic, L, exceeds the upper fractile of Chi-
square distribution, the process is said to lack control and further
investigation is needed to locate the cause. For ingredient A, the
sample variance, s2, is 15.00 and the null hypothesis is
57
H0: cr 2 = 13.4596
tested against the alternative hypothesis
2 a f 13.4596 .
The test statistic
c = (n-1)s2 2 a
= 19 X 15.00/13.4596
= 21 . 17 44
The upper fractile of Chi-square distribution with 19 degrees of freedom
and 99.5 percent type one error is 38.60, and the lower fractile of
Chi-square distribution with 19 degrees of freedom and 0.5 percent type
one error is 6.84.
Since the test statistic does not fall in the reject region, the
dispersion for ingredient A seems to be in control. For ingredient 8,
the sample variance, s2, is 0.0450 and the null hypothesis is
H0: cr2 = 0.0213
tested against the alternative hypothesis
2 a f 0.0213
The test statistic
C = (n-1 )S2 2
(J
= 19 X 0.0450/0.0213
= 40.1408
58
Since the test statistic exceeds the upper fractile of Chi-square
distribution with 19 degrees of freedom and 99.5 percent type one error,
the dispersion of ingredient Bis assumed to be the cause of the
disturbance. The process phase responsible for ingredient B requires
adjustment.
Chapter 5
PROCEDURE FOR MONITORING THE CENTRAL TENDENCY
OF THE MANUFACTURING PROCESS
The purpose of this chapter is to develop an operational procedure
for monitoring the current manufacturing process with respect to the
central tendency. When the manufacturing process is out of control with
respect to the central tendency, a set of statistical tests can be used
to locate the manufacturing phases which need to be adjusted. A step-
by-step computational procedure is presented for the user's convenience.
The example discussed in previous chapters is again carried over for
further discussion.
5. l Brief Review of the Univariate Case
In the univariate case, the x-chart is usually employed for
monitoring the central tendency of the manufacturing process. The
central line is x, the upper control limit is x + A2R and the lower
control limit is x - A2R, where A2 is a constant which can be found in
most statistical quality control texts. If the sample average taken
from the current process falls outside the control limits, the manu-
facturing process lacks control and it becomes worthwhile to determine
the assignable cause. If, on the other hand, the sample mean falls
within the control limits, it is more economical to leave the process
alone. There are, however, some exceptions; sometimes the manufacturing
process is said to lack control even when the sample mean falls within
the control limits, e.g., seven points in a row fall on one side of the
59
60
central line. A more detailed discussion is given by Duncan (6) and
Grant (11 ).
5.2 Theorems Used in the Development of the Central Tendency Monitoring Procedure
The following are some theorems which will be used in the development
of the monitoring procedure.
Theorem 5-1
The following theorem is given by Graybill (12):
If a vector y_ (lxk) is normally distributed with mean vector 0 I
and variance-covariance matrix W, then y_ By_ is distributed as the Chi-
square distribution with k degrees of freedom, if and only if that BW
is idempotent and Bis a (kxk) matrix.
Theorem 5-2
The followin9 theorem is given by Schmidt (32):
If y_ is a (lxk) vector and Tis a (kxk) symmetric matrix, then
I
Y. Ty_ =
Theorem 5-3
k I
i = l 2 t .. y. +
1 1 1
k-1 k I I i=l j=i+l
t .. y.y .. 1 J 1 J
The following theorem is given by Mood and Graybill (12):
If xi is normally distributed with mean ui and variance vii' then
(X;-Ui) 2
V •. 11
follows the Chi-square distribution with degrees of freedom n.
61
Theorem 5-4
The fo l1 owing theorem is given by Ha 1 d (13):
If xj is normally distributed with mean ui and variance vii' then
n (x .-x/ I ---=-J -. 1 V •• J= 11
follows the Chi-square distribution with degrees of freedom (n-1), where
the x is the average.
Theorem 5-5
The following theorem is given by Hald (13):
The Chi-square distribution with n degrees of freedom can be par-
titioned into two terms; the first term has the Chi-square distribution
with one degree of freedom and the second term has the Chi-square distri-
bution with (n~l) degrees of freedom.
5.3 Development of the Central Tendency Monitoring Procedure
Assume that the dispersion of the manufacturing process represented
by the variance-covariance matrix, V, is under control and the level
of the central tendency intended to be maintained is!!.= (u1 u2 ... uk).
Suppose a sample of size n is taken from the current manufacturing
process, and the unit weight assigned to the deviation of the quality
characteristic from the standard value.
62
-if g. Xi > u , -0 if W, = X, = u.
1 1 , -
l i if X, < U, 1 ,
where i = l, 2, ... , k. The utility function is therefore defined as
- )2 , 1 k w.n(x.-u. -n {X-!:!J v- (!-!:D - I 1 1 1
i=l V;; (5.1)
The term -n([-Q) 1 v-l (!-Q) in the exponential function represents the
total deviation amono all the major components and the sum of all the two
components interactions. - 2 k w-n(x.-u.)
The term - I 1 1 1 in the exponential . l V •• , = 11
function represents the sum of weighted deviation of all the major
components.
An analysis of the utility function just defined indicates that if
the manufacturing process is to maintain the desirable quality, then
the sample average vector,!, should be close to the desirable mean
vector, Q. Therefore, the value of the utility function would be
relatively large. On the other hand, if the current manufacturing
process is no longer producing desired quality, the sample average
vector, K, would deviate from the desirable mean vector, Q, and the
value of the utility function would be relatively small. Because the
monitoring procedure is operated in the probabilistical environment, it
is desirable to find a critical value, say c, which will have the
probability of acceptance as equal to {l-a) if the utility function
has value greater than or equal to the critical value c. Statistically,
the task is to find the critical value, c, such that
63
Substituting equation (5.1) into the above equation gives
k -n ([-!!_) • v-1 ([-!!_) - I
i = l P[e
- 2 w.n(x.-u.) l l l
V •• l l
> c] = l
By Theorem (5-1 ), the above equation can be written as
2 k 2 -x (k) - I wix (1)
P[e i=l ~c]=l-a
By Theorem (5-5), the following equation is obtained.
P[e
k -/(1) + i(k-1) - x2(1) L w. . l l 1= c] = l - a
Use of Theorem (5-5) recursively results in
2 k -x (l)(k + l wi)
P[e i=l ] c = l - a
Taking the logarithm of the above equation gives
k P[x2 (1 )(k + l wi) Lnc] = l - a
i =l
or
Therefore,
or
2 k P[x (1) -Lnc/(k + I wi)] = l - a
i =l
- a (5.2)
64
Substituting the c value back into equation (5.2) gives
or
2 k -x (1 ), (k + l w.)
-a i = l , ] = > e
I l k P[n ([-!!.) v- (K-U) + l
i =l
l - a
{- )2 w.n x.-u. 1 l 1
v .. 11
l - a
(5.3)
(5.4)
Now, suppose that a sample of size n is taken from the current
manufacturing process. It is assumed to have the specified variance-
covariance matrix, V, and it is desired to maintain a specified mean
vector,!!_, for this manufacturing process. The unit weight wi is assigned
to the deviation of the ith quality characteristic from the specified
mean value. The unit weights are given as
gi if x1 > Ui
Wi = l . if X, < u. 1 1 1
0 if X· 1 = u. 1
65
where i = l, 2, ... , k, should be predetermined according to the impor-
tance of each characteristic to the function of the product. The
decision is to leave the manufacturing process alone if
k A = n (K-!:D' v- l (K-U) + I
i =l
is less than or equal to
k 2 Q = (k + L wi) x (1 ), _
i = l a.
- 2 w.n(x.-u.) 1 1 1
Vii
5.4 Identification of Characteristics Contributinq to the Central Tendency Control Problem
(5.5)
(5.6)
When the manufacturing process is out of control, the ideal proce-
dure is to have all the dimensions restored to the standard values.
In some cases, however, it is desirable to adjust a minimum number
of characteristics in order to restore the manufacturing process to
normal operation. For example, it may be technically very difficult to
adjust all the dimensions in the short time interval available; or perhaps
management intends to schedule all the major maintenance work during the
evening shift. It is therefore desirable to have some procedure for
rapidly identifying the quality characteristics which need adjustment.
Examination of the inequality portion of equation (5.4) shows that
, l k w . n (x. -u . ) 2 n (!-!D v- CK-~) + L 1 1 1
i=l Vii
(5. 7)
66
Let y_ = (K-Q) and T = nv-1, then the first term of equation (5.7) I I
becomes Y TY. By Theorem (5-2), y_ T! can be written as
i 2 k-1 k I t .. y. + I I t .. y.y.
i=l ll l i=l j=i+l lJ l J
The second term of equation (5.7)
w.n(x.-u. )2 l l l k
I i=l V .• ,, can be written as
where d;
k 2 I d.y. i = l l l
w.n l
V .. l l
and Y; = (x. -u.). l l The right side of equation (5.7)
is essentially a constant, say h. Therefore, equation (5.7) can be
simplified as
k 2 k-1 k k 2 l t .. y. + l l ti .y.y. + I d.y. < h i=l 11 l i=l j=i+l J l J i=l l l -
With some algebraic manipulation, the above equation can be rewritten as
k 2 k-1 k L aiiyi + L i a;jY;Yj - h < 0
i=l i=l j=1+l
67
The above equation is recognized as a k-dimensional ellipsoid.
Furthermore, this ellipsoid is the control region for the k variates
which corresponds to the k quality characteristics of interest and
the boundary of the control region is represented by
k 2 k-1 k I a .. y. + I I a .. y.y. - h = 0
i=l 11 l i=l j=i+l lJ l J (5.8)
Therefore, the process is said to be in control if the sample average
vector falls within this ellipsoid. Otherwise, the process lacks
control. Now, it can be concluded that the manufacturing process is
out of control if either of the following two statements is true:
1) Sample averages of one or more variables fall outside
their ranges on the ellipsoid.
2) No sample average falls outside its respective range,
but the combination of the sample averages falls outside
the ellipsoid.
The range for all the quality characteristics on the ellipsoid
can be calculated. When a lack of control is indicated, the sample
average of each quality characteristic may be compared with its own
range. Any sample average falling outside its range represents a
cause for concern and indicates that the corresponding phase of the
manufacturing process should be adjusted. However, when the manu-
facturing process is out of control and the sample averages of all the
quality characteristics fall within their own ranges, it is logical
68
to say that the lack of control is attributed to interaction between
the characteristics.
Derivation of a procedure for finding the range for all the quality
characteristics can be illustrated using the two variables case.
Substitute k = 2 into equation (5.8) to obtain
The above equation can also be rewritten as
{5.9)
If the variable y1 in equation {5.9) is expressed as a dependent variable,
then the following equation can be obtained:
1/2 Yi = {-by2 [y22(b2-4ac) - 4ad] }/2a (5.10)
or
2 2 2 112 Yl = {-by2 + [b y2 - 4a(cy2 +d)] }/2a (5.11)
and
(5.12)
In order to find the extreme points for y1 on the ellipsoid, take
the first derivative of equation (5.10) with respect to y2 and set the
derivative equal to zero, i.e.,
dy l 1 2 2 - l / 2 2 dy2 = 2a {-b [y2 (b -4ac)-4ad] [y2(b -4ac)]} = o
Some algebraic manipulation results in
69
or
2 b2d Y2 = c(b2-4ac)
(5.13)
Therefore, the solutions for y2 are
and 2 l /2
y = -(b d/f) 22
where f = c(b2-4ac). Substitute y21 into equation (5.11) and equation
(5.12) and two solutions for the variable y1, say y11 and y12 , can be
found:
(5.14)
2 1/2 1/2 y '= -b(b d/f) - [b4d/f - 4a(b2cd/f + d)]
12 2a (5.15)
Substitute y22 into equation (5.11) and equation (5.12) and another two
solutions for y1, say y13 and y14 , can be computed:
1/2 1/2 y13 = +b(b2d/f) + [b4d/f + 4a(b2cd/f + d)]
2a (5.16)
(5.17)
Hence the maximum value of y1 and the minimum value of y1 can be found
by simple comparison, i.e.,
70
Maximum value of y1 = max(y11 y12 y13 Y14 ) and
Minimum value of Yi = min(y11 Y12 Y1 3 Y14).
The range for the variable y1, then, is
(maximum value of y1, minimum value of y1).
Similarly, the maximum value and the minimum value for the variable
y2 are found by rearranging equation (5.9) as
(5.18)
Express the variable y2 as the dependent variable of y1 and rewrite the
above equation as
{5.19)
Take the first derivative of the above equation with respect to the
variable y1 and set the derivative equal to zero. Algebraic manipulation
would yield the solutions for the variable y1:
1/2 y15 = +(b2d/g) (5.20)
and 2 1/2
y16 = -(b d/g) (5.21)
where g = a(b2-4ac). Substitute the values of y15 and y16 into equation
(5.19) to obtain four solutions for the variable y2, say Y23 , Y24 , Y25 ,
Y26:
(5.22)
71
2 1/2 2 1/2 = -b(b d/g) - [b4d/g - 4c(ab d/g + d)] (5.23)
2c
(5.24)
(5.25)
The maximum value and the minimum value for the variable y2 can now be
found by simple comparison, i.e.,
Maximum value of y2 = max(y23 y24 y25 Y26 ) and
Minimum value of y2 = min(y23 Y24 Y25 Y25).
The range for the variable y2, then, is
(maximum value of y2, minimum value of y2).
For three variables case, we substitute k = 3 into equation (5.8)
and the equation becomes
or
(5.26)
If the variable y1 is expressed as the dependent variable, then, the
above equation can be rewritten as:
72
Take the first derivative of the above equation with respect to y2 and
y3 respectively and set the resulted equations equal to zeros, i.e.,
3 Y1 0 ---
3 Y2 ( 5. 27)
and
~= 0 3 y3 (5.28)
Solve for y2 and y3 from the above equations and substitute the solutions
of y2 and y3 into equation (5.26). Then, the maximum value of y1 and the
minimum value of y1 can be obtained by comparison, and the range for
Yi is, therefore,
(maximum value of y1, minimum value of y1).
Similarly, the maximum values, minimum values and ranges for
variables y2 and y3 can be obtained. Clearly, the similar procedure can
be applied to the case of more than three variables.
In general it is necessary to adjust those dimensions
which have sample means fall outside their own boundaries. However, if
the lack of control is attributed to the interaction, then the (xi-ui) 2
of lxi-ui I can be arranged in the descending order. Then the process
engineer can adjust the dimensions which have larger deviations or choose
from those dimensions which are considered relatively easy to adjust.
5.5 Computational Procedure for Monitoring the Central Tendency of the Manufacturing Process
The standard values for the central tendency !Land the variance-
covariance matrix V are either computed from the past data or determined
73
by management. The inverse and the determinant of V can be computed
before monitoring the process. Assume the type one error is a.
A sample of size n is taken from the current manufacturing process
and k measurements are taken on each of then individual items.
1) Assign the unit weight to the imperfect quality characteristic -if g. X, > u.
l l l -0 if W• = Xi = u.
l l -l . if x. < u.
l l l
where i = l, 2, ... , k.
2) Perform matrices multiplication f. : ([-Q.) I v-1
3) Perform the vectors multiplication
f = I.([-Q.)
4) Compute - 2 w.n(x.-u.)
l l l
Vii
where i = l , 2, ... ' k.
5) Compute k
A= nf + 2 z. i = l l
6) Compute
Q = (k + k 2 l W;) x (1 ), _
i=l a
74
7) If the current manufacturing process with respect to the
central tendency is out of control, all the dimensions
should ideally be reset back to their standard values.
However, if for any economic or technical reason, it is
desirable not to adjust too many dimensions, then the
following procedure may provide some guidelines for
choosing the characteristics to adjust.
8) Compute the range for each characteristic using the
procedure illustrated in section 5.4.
9) If any sample mean falls outside its own range, then the
corresponding dimension must be adjusted in order to
restore the normal process.
10) If all sample means fall within their own ranges, the values
of (~i-ui) 2 or jxi-ui I may be arranged in descending order
and the characteristics with larger deviations may be
chosen for adjustment. Alternatively, those dimensions
which are considered relatively easy to adjust may be
chosen for action.
75
5.6 Computer Program for Monitoring the Process Central Tendency
This computer program is combined with the computer program for
monitoring the process dispersion described in Section 4.7; therefore,
there is no need to duplicate those variables already entered in the
computer program.
1) Input Variables
SMALL(L) = Weighting factor for the Lth quality characteristic
if the sample average is less than the standard mean.
GREAT(L) = Weighting factor for the Lth quality characteristic
if the sample average is greater than the standard
mean.
TRUEAV(L) = Standard mean vector.
FPV(I,J) = Standard variance-covariance matrix.
CHIONE = Chi-square upper fractile value with d.f. = 1.
2) Input Format
Variables
(SMALL(L), L=l,JC)
(GREAT(L), L=l ,JC)
(TRUEAV(L), L=l ,JC)
DO I=l, JC
(FPV(I,J), J=l, JC)
CJ IONE
Format
(8Fl0.4)
{8Fl0.4)
{8Fl0.4)
(8Fl0.4)
(Fl0.4)
76
3) FORTRAN Listing of the Main Computer Program
See Appendix B.
5.7 Example of Computations Involved in Monitoring Central Tendency of the Current Manufacturing Process
In Chapters 2 and 3, 30 samples of 20-tablet size were used to prove
the stability of the manufacturinq process. The standard value for
the central tendency was defined as
U = (250. 1689 2.5027)
and the variance-covariance matrix was defined as
V = [13.4596
0.4301 0.4301] 0.0213
The inverse of the matrix V was calculated as
v-l = [ 0.2094 -4.2291
4.2291] 132.3461
and the determinant of the V matrix was 0.1017.
For the following examples, it is assumed that the current manufac-
turing process has been proven to be in control with respect to
dispersion by previous tests. Management is now interested in knowing
whether the current manufacturing process is still under control with
respect to the central tendency. Furthermore, the unit weights for quality
characteristic deviations are assigned by management as:
77
10 if x1 > 250.1689
w, = 0 if x, = 250.1689
5 if x, < 250.1789
and
2 if x2 > 2.5027
W2 = 0 if x2 = 2.5027
6 if x2 < 2.5027
The type one error is assigned as one percent.
Example A
A sample of 20-tablet size is taken from the current manufacturing
process and the sample averages are calculated as I K = (250.0000 2.6000) .
Management wants to know whether the manufacturing process is in
control. Here, n = 20, k = 2, w1 = 5, w2 = 2 and the 99 percent fractile
of the Chi-square distribution with one degree of freedom is 6.6300.
Substitute all these values into equations (5.5) and (5.6) to obtain
A= 27.9600 + 0.2177 + 17.8403 = 46.0120
and
Q = (2+5+2)(6.63) = 59.6700.
Since 46.0120 is less than 59.6700, the manufacturing process is
concluded in control and it is more economical to leave the process
alone than to adjust it.
78
Example B
A sample of 20-tablet size is taken from the current manufacturing
process and the sample mean vector is computed as
x = (254.oooo 2.5000) .
Management wants to know whether the manufacturing process is in
control. Here, n = 20, k = 2, w1 = 10, w2 = 6 and the 99 percent
fractile of the Chi-square distribution with one degree of freedom is
6.6300. Substitute all these values into equations (5.5) and (5.6) to
obtain
A= 61.4460 + 218.0941 + 0.0411 = 279.5812
and
Q = (2+10+6)(6.63) = 119.3400.
Since 279,.5812 is greater than 119.3400, the manufacturing process
is out of control. A further investigation is needed for
identifying which characteristics lack control.
Let Yi = x1 - u1, y2 = x2 - u2 and substitute the variables of y1 and y 2 into equation (5.8). This gives
[ 0.2094 -4.2291] I
20(y, Y2) 132.3461
(yl Y2) -4.2291
+ 10(20)(y,2) 6(20)(y/)
- (2+10+6)(6.63) 0 + 0.0213 = 13.4596
Some algebraic manipulation results in
19.0472y12 - 169.1640y1y2 + (8280.7248y22 - 119.3400) = 0 (5.30)
79
By equation (5.13), the solutions for the variable y2 are computed as
y21 = +0.0260 and y22 = -0.0260.
Substitute the value of y21 into equations (5.14) and (5.15)
resulting in y11 = 2.3306 and y12 = -2.5616.
Substitute the value of y22 into equations (5.16) and (5.17)
resulting in y13 = 2.5616 and y14 = -2.3306.
By inspection, the maximum value of y1 is 2.5616 and the minimum
value of y1 is -2.5616 or the maximum value of the x1 is 252.7300 and
the minimum value of the xi is 247.6073. Since the sample average of x1,
254.0000, exceeds the maximum value of x1, the phase of the manufacturing
process responsible for ingredient A is identified as the characteristic
which requires some adjustment.
Rewrite equation (5.31) in the following form:
8280.7248y22 - 169. 1640y1y2 + {19.0472y12 - 119.3400) = 0 (5. 31)
The use of equations (5.20) and (5.21) gives
y15 = +0.5455 and y16 = -0.5455.
Substitute the values of y15 and y16 into equation (5.31) to obtain
the maximum value of y2, 0.0056, and the minimum value of y2, -0.0056.
The maximum value of x2 is 2.5073 and the minimum value of x2 is 2.4971.
Since the sample average of the variable x2 is 2.5000 and it falls within
its own range, ingredient Bis concluded to be in control.
Chapter 6
SIMULATION
The purpose of this chapter is to present two simulation studies.
Through the simulation results, the reader may see how the proposed
system responds to the situations when the manufacturing process actually
departs from the standard.
6.1 Random Number Generation
The development of the random number generating procedure is given
by Newman and Odell (28). If the random vector, I, follows a k-variate
normal distribution with mean vector G and a variance-covariance matrix
H, and where~ is a constant vector and A is a matrix of rank k, then
! =AI+~ is a random vector following a k-variate normal distribution I
with mean vector AG+ Mand variance-covariance matrix AHA. This theorem
provides a convenient means of generating a random vector! with
specified mean vector!!_ and variance-covariance V,provided two basic
requirements can be met:
1) A means of generating a random vector Z with mean vector
0 and variance-covariance matrix I is available, and
2) There is a convenient means of factoring matrix Vin the I
form of V = AA where A is a (kxk) matrix.
If these two requirements are met, then!= AZ+ U will follow normal I
distribution with mean U and variance-covariance matrix AIA = V.
80
81
Concerning the first requirement, it is easily seen that if x1 x2 ... xk are one-dimensional random variables, independent, and all
being normally distributed with mean O and variance l, then the random I
vector f = (x1 x2 ... xk) follows a k-variates normal distribution
with mean vector O and variance-covariance matrix I. The second require-
ment can be met because the variance-covariance matrix, V, is a positive,
definite, real, symmetric matrix. For any positive definite, real,
symmetric matrix, there exists a lower triangular matrix, A, with I
positive elements on the main diagonal such that V =AA. The elements
of A can be computed recursively in the order of 11, 21, ... , kl; 22,
23, ... , kk. Since A is lower triangular aij = 0 for j > 1.
Hence,
V •• lJ
For i = j, v11 = a11 2, so that a11 = v11 112 The remaining elements in
the first column of A are then given by
for i = 1, 2, ... , k.
Once the first j-1 columns of A are computed,
j-1 2 1 /2 a .. = (v .. - I a. ) JJ JJ m=l Jm
Now if j = k, the task is completed. Otherwise,
j-1 a .. = (v .. - I a. a. )/a .. lJ lJ m=l ,m Jm JJ
for i = j+l, j+2, ... , k.
82
6.2 Pre-analysis of Simulation
In quality control procedures, deviation of the parameters, the
central tendency and dispersion from the standard is measured in terms
of the standard deviation of the distribution of the parameters. For
example, in controlling the central tendency in the univariate case,
management would like to know the probability of detecting the deteri-
oration if the process average deviates one, two or three standard
deviations from the standard central tendency. In fact, the process
average can be either above or below the standard central tendency.
The actual process average can, therefore, take values as:
the standard central tendency,
the standard central tendency::!:_ (l)(standard deviation of
the mean),
the standard central tendency::!:_ (2)(standard deviation of
the mean), and
the standard central tendency+ (3)(standard deviation of
the mean).
The same principle can be applied to the control of the dispersion and
the actual process dispersion can also take seven values. Therefore,
it becomes necessary to consider (7)(7) - l = 48 ways in the univariate
case.
For the two variables case, the ways of the process deviating
from the standard can be laid out as:
83
Central Tendency Di seers ion
(x) (y) (x) (y)
0 0 0 0
+l +l +l +l
+2 +2 +2 +2
+3 +3 +3 +3
The total ways of the process deviating from the standard process
is (7) (7) (7) (7) - l - 2,401 . However, in the two variables case, if
the correlation is present, then the combinations wi 11 be greatly
increased and the upper bound for the combinations is, theoretically,
infinite.
For three variables case, the total combinations for deviating
from the standard process are {7)(7)(7)(7)(7)(7) - l = 117,648.
Again, if the partial correlation is present, the number of combinations
will be greatly increased. The number of the combinations increases
with the number of quality characteristics of interest. In order to
obtain a meaningful interpretation, it seems reasonable to select some
key combinations to see how the system responds to the deteriorations
when many quality characteristics are being observed.
6.3 Simulation Study with Two Variables
In order to give a complete example, the drug manufacturing problem
used throughout this research is again used for this simulation. The
standard values for the central tendency and dispersion are determined
as
and
I
U = (250.1689 2.5027)
V = [ 13.4596 0.4301
0.4301] 0.0213
84
respectively. The correlation coefficient is computed as 0.8033. The
unit weights for the deviations of quality characteristics from the
standard values are assigned by management as
10 if x, > 250.1689
w. = 0 if X1 = 250.1689 l
5 if X1 < 250.1689
2 if x2 > 2.5027
w2 = 0 if x2 = 2.5027
6 if x2 < 2.5027
To generate random variates for the two dimensional normal
distribution with mean vector U and varfance-covariance matrix V,
it is necessary to find a matrix A such that the variance-I
covariance matrix V can be represented by AA Based on the discussion
in Section 6.1, the elements of matrix A can be computed in the
fo 11 owing way:
= V 1/2 11
85
and
a12 = 0.
For every combination of the simulated system departing from the
standard, 100 samples are to be generated. Each sample contains 20
individual items and each item has two measurements generated for it.
Because it is not very economical to exhaust all the combinations of
the simulated system departing from the standard, only a selected set
were simulated. Three such runs are presented here.
1) In the first run of the simulated system, the correlation
coefficient between the ingredients A and B has the standard
value. For the central tendency, the simulated process
takes values as standard process, two standard deviations
above and below the standard process. For the dispersion,
the simulated process takes values as the standard process,
two standard deviations above and below the standard process.
2) In the second run of the simulated system, the correlation
coefficient between ingredients A and B has 80 percent of the
standard value. For the central tendency and dispersion,
the simulated process has the same combination as in step 1).
3) In the third run of the simulated system, the correlation
coefficient between ingredients A and B has 110 percent of
the standard value. For the central tendency and dispersion,
the simulation process has the same combination as in
run l ) .
86
All the simulated results are presented in Appendix F. The
simulated results seem to follow what is expected and the proposed
system appears to function reasonably well.
6.4 Simulation Study with Four Variables
The simulation work for two variables was discussed in Section 6.4.
Now, going another step further indicates how the system will respond
to the more complicated case of more than two variables. The results
of the investigation indicate that the simulation would be very
expensive. The computer program developed here can handle as many
variables as the practical situation requires with some limited
modifications. The simulation attempted in this section is a
case with four variables, say A, B, C and D. The standard values for
the centra 1 tendency and the dispersion are given as
u = (20.00 30.00 50.00 60.00)
and
1.00 1.00 1.00 1.00
1.00 2.00 1.00 1.00 V =
1.00 1.00 3.00 1.00
1.00 1.00 1.00 4.00
The partial correlation coefficients between variables were calculated as
Variables Partial Correlation Coefficient
( 1 , 2) = 0.7071
( 1 , 3) = 0.5774
( 1 ,4) = 0.5000
87
(2.3) = 0.4082
(2,4) = 0.3536
(3,4) = 0.2887
The unit weight for the imperfect quality characteristic with respect
to the central tendency is assigned as
Wl ={ 1.00 3.00
W2 = { 3.00
2.00
W3 = { 5.00
1.50
, { 3.00 w -4 - 0.50
if x(A) > 20.00
if x(A) < 20.00
if x(B) > 30.00
if x(B) < 30.00
if x(C) > 50.00
if x(C) < 50.00
if x(D) > 60.00
if x(D) < 60.00
Originally, it was planned to simulate all the combinations with
three levels for each variance and mean. Using a sample size of 20 and
a simulated sample number of 100, this required CPU time on the IBM
360/50 of approximately 80 hours, and was not economically feasible.
Therefore, the plan for this study was revised as follows:
l) Hold all means as the standard values and take dispersion
values as standard dispersion, two standard deviations
above and below the standard dispersion for all four
variables.
88
2) Hold all variances as the standard values and take central
tendency values as standard central tendency, two
standard deviations above and below the standard central
tendency for all four variables.
For each of these runs, the CPU time on an IBM 360/50 was approximately
80 minutes. The simulated results are presented in Appendix G. The
simulated results seem to match what is expected and the proposed system
appears to function reasonably well.
6.5 Computer Program for Simulation
l) Input Variables
IX= Random number initiator; it has to be a five digit number
with last digit an odd number
R0CF = Multiple of correlation coefficient away from the
standard correlation coefficient
CHI0NE = Chi-square upper fractile value with d.f. = l
CHIK = Chi-square upper fractile value with d.f. = JC(JC+l)/2
IR= Sample size
JC= Number of quality characteristics
AVE(L) = Standard mean vector
PV(I,J) = Standard variance-covariance matrix
PVI{IMJ) = Inverse of standard variance-covariance matrix
SPV{L) = Standard deviation for the variance of the Lth variable
MM= Number of sample for every combination of departure from
the standard process
89
GREAT(L) = Weighting factor for the Lth quality characteristic if
the sample average is greater than the standard mean
SMALL(L) = Weighting factor for the Lth quality characteristic if
the sample average is less than the standard mean
NM(L) = Maximum number of S.D. to be varied for the mean of Lth
quality characteristic
NV(L) = Maximum number of S.D. to be varied for the variance of
the Lth quality characteristic
ZM(J,L) = Actual number of S.D. departed from the standard mean
of the Lth quality characteristic
ZV(J,L) = Actual number of S.D. departed from the standard mean
of the Lth quality characteristic.
2) Input Format
Vari.ables
IX
ROCF
CHIONE, CHIK
IR, JC
(AVE(L), L = l, JC)
DO I= l, JC
( PV ( I , J) , J = 1 , JC)
DO I= l, JC
(PVI(I,J), J = 1, JC)
(SPV(L), L = l, JC)
MM
(!5)
(FlO.4)
(2Fl0.4)
(215)
(8Fl0.4)
(8Fl0. 4)
(8FlO.4)
(8FlO.4)
(15)
(GREAT ( L ) , L = l , JC )
(SMALL(L), L = l, JC)
(NM(L), L = l, JC)
(NV(L), L = l, JC)
DO J = l, JC
(ZM(J,L), L = l, NM(J))
( ZV ( J , K) , L = 1 , NV ( J ) )
90
(8Fl0.4)
(8Fl0.4)
(1015)
(1015)
(8Fl0.4)
(8Fl0.4)
3) FORTRAN Listing of Main Program for Two Variables Case
See Appendix C.
4) FORTRAN Listing of Main Program for Four Variables Case
See Appendix 0.
Chapter 7
SUMMARY AND RECOMMENDATIONS
Previous chapters of this research were devoted to the development
of the quality control procedure for the multivariate situation. In
this chapter, a summary of the entire work is presented and recom-
mendations are made concerning areas for further study.
7. l Summary
The primary purpose of this dissertation is to develop an appli-
cation system which can be used for the quality control areas. Because
of the nature of the application, the computational procedures and
numerical examples are particularly emphasized. An antidiarrheal
tablet manufacturing process was used as an illustration. The tablet
preparation contained two active drug ingredients, A and B. First of
all, it was desired to prove the stability of the manufacturing
process with respect to the dispersion. For illustrative purpose, 30
samples of 20-tablet size were taken at relatively equal time intervals
during the past operation. Here, the number of samples, m, is 30, the
sample size, n, is 20, and the number of quality characteristics, k, is
2. All the measurements taken in the ith time interval were designated
by
=
xi,2,1
. xi, l ,20]
X. 2 2 ' ' ' X. 2 20 ,, ' ,, '
91
92
The sample variance-covariance matrix for the ith time interval,
Si' was computed based on Xi. The pooled sample variance-covariance
matrix, SP, was computed from the values of all Si. The R-statistic
is then calculated according to the formula:
2k2 + 3k - 1 m2 - 1 R = 2.3026 x {l - [6(k+l )(m-1) x m(n-1)]
k x [m(n-1) Log I Sp! - (n-1) i~l LogjSi !]}
The R-statistic was found to be 53.8690 and the corresponding Chi-square
table value was found to be 113.00. Since the R-statistic was less
than the critical value, it was concluded that the manufacturing process
was reasonably stable with respect to dispersion. Next, it was desired
to prove the stability of the manufacturing process with respect to the
central tendency. The following were computed:
m I
T = l X X i = 1
= [37,558,460.0000 375,919.8000
375,919.8000] 3,770.9130
m A = I
i = l n (x. -x) (x. -x) =
-. - [431.1323 -l - -l - 13.2191
13.2196] 0.6410
[37,558,460.0000
E=T-A= 375,906.5000
W = ffi = 0.9501
375,906.5000]
3,770.2720
93
and va = 29, ve = 570, and k = 2. The W-statistic was converted to an
F-statistic. The value of the converted F-statistic was 0.0501 and its
corresponding F-table value was 1.48. Since the value of 0.0501 was
less than 1.48, it was concluded that the manufacturinq process was
reasonably stable with respect to central tendency, too. At this time,
management indicated satisfaction with the manufacturing process.
The standard values for the dispersion matrix and the central tendency
were determined as
and
[13.4596
V = 0. 4301
I
0.4301] 0.0213
U = (250. 1689 2.5027)
The determinant of the standard dispersion matrix was computed as 0.1017.
The 31st sample of 20-tablet size was taken; the sample mean and
the sample dispersion matrix were calculated as
and
X = (250.0000 2.6000)
s = [14.0000
0.4000 0. 4000] 0.0220
respectively. The L-statistic was calculated by the following formula:
L = n[Ln(jVj/jSj) - k + tr(SV- 1)]
and the L-statistic was found to be 4.5580 and the corresponding Chi-
square table value was found to be 11.3000. Because 4.5580 was less
94
than 11.3000, it was concluded that the current manufacturing process
was under control with respect to the dispersion. To investigate
whether the central tendency of the manufacturing process was still under
control, management assigned the weight factor w1 = 5 to ingredient A
and the weight factor w2 = 2 to ingredient B.
and
- 2 w.n(x.-u.) l l l
V •• 11
= 46.0120
Since A was less than Q, it was concluded that the current manu-
facturing process was also under control with respect to central
tendency. Therefore, it was more economical to leave the process alone.
Operational procedures and examples for the cases lacking control were
also discussed. Simulations were developed to show how the system
responded to the situations when the manufacturing process actually
departed from the standard. The simulation results showed that the
proposed system was adequate to perform this function.
7.2 Areas for Further Study
1) Terminal Operation: Because of the large amount of data
manipulation, an on-line computer support seems ioost
appropriate for this case. Through the terminal, the
data can be entered to permit a quick decision based on
the calculated results.
95
2) Trend Analysis: According to the quality control chart
for the univariate case, the manufacturing process is
said to be out of control when seven consecutive points
fall at one side of the central line but with no point
falling outside the limits. For the multivariate
situation, it would be helpful to develop some kind of
trend analysis for detecting the deterioration when the
manufacturing process degrades slowly.
3) Warning Zones: In the quality control chart
for the univariate case, when sample points fall in the
warning zone, the quality special attention should be paid
to preventing the process from departing from the
standard. For the multivariate situation, the warning
zone should be a good means of getting the engineer's
attention before the process goes too far out of control.
4) Sample Size: The optimal sample size in the multivariate
case, intuitively, should be some function of the number
of quality characteristics to be controlled and of the
various costs such as the cost of inspection, cost of
failing to detect the lack of control and cost of false
alarms, etc.
5) Time Interval Between Sampling: The optimal time interval
between sampling should be derived from some cost function.
The above areas are recommended for further study.
BIBLIOGRAPHY
1. Anderson, T. W., An Introduction to Multivariate Statistical Analysis, John Wiley & Sons, Inc., New York, 1966.
2. Barnard, A. G., "Control Charts and Stochastic Process," Journal of the Royal Statistical Society, Series B, Volume 21, No. 2, 1959.
3. Bartlett, M. S., "Test of Significance in Factor Analysis," British Journal of Psychology Statistics, Sec. 3, 1950.
4. Box, G. E., "A General Distribution Theory for a Class of Likelihood Criterion," Biometrika, Vol. 36, 1949.
5. Chakravati, I. M., Laha, R. G., and Ray, J., Handbook of Methods of Applied Statistics, John Wiley & Sons, Inc., New York, 1967.
6. Duncan, A. J., Quality Control and Industrial Statistics, Richard D. Irwin, Inc., Homewood, Illinois, 1965.
7. Duncan, A. J., "The Economic Design of x-Chart Used to Maintain Current Control of Process," Journal of the American Statistical Association, Vol. 51, 1956.
8. Duncan, A. J., "The Economic Design of x-Charts When There is A Multiplicity of Assignable Causes," Journal of American Statistical Association, Vol. 66, 1971.
9. Ghare, P. M. and Torgersen, P. E., "The Multicharacteristic Control Chart," Journal of Industrial Enqineerinq, June 1968.
10. Goel, A. L., Jain, S. C., and Wu, S. M., "An Algorithm for the Determination of the Economic Design of x-Charts Based on Duncan's Model," Journal of the American Statistical Association, Vo 1. 6 3 , 1 968 .
11. Grant, E. L., Statistical Quality Control, 3rd Edition, McGraw-Hill Book Company, New York, 1964.
12. Graybill, F. A., An Introduction to Linear Statistical Models, McGraw-Hill Book Company, New York, 1961.
13. Hald, A., Statistical Theory With Engineering Applications, John Wiley & Sons, Inc., New York, 1952.
14. Hartley, H. 0., ''The Range in Random Samples," Biometrika, Vol. 32, 1942.
96
97
15. Jackson, J. E., 11 Quality Control Methods for Two Related Variables, 11
Industrial Quality Control, January 1956.
16. Jackson, J. E., "Quality Control Methods for Several Related Variables, 11 Technometric, Vol. 1, No. 4, November 1959.
17. Korin, B. P., 11 0n the Distribution of a Statistic Used for Testing a Covariance Matrix, 11 Biometrika, Vol. 56, 1969.
IH. Kramer, C. Y. and Jensen, D. R., "Fundamentals of Multivariate Analysis--Part I. Inferences About Means," Journal of Quality Technology, Vol. l, No. 3, July 1969.
19. Kramer, C. Y. and Jensen, D. R., "Fundamentals of Multivariate Analysis--Part II. Inference About Two Treatments, 11 Journal of Quality Technology, Vol. 2, No. l, January 1969.
20. Kramer, C. Y. and Jensen, D. R., 11 Fundamentals of Multivariate Analysis--Part III. Analysis of Variance for One-Way Classi-fication,11 Journal of Quality Technology, Vol. 1, No. 4, October 1969.
21. Kramer, C. Y. and Jensen, D. R., 11 Fundamentals of Multivariate Analysis--Part IV. Analysis of Variance for Balanced Experiments, 11 Journal of Quality Technology, Vol. 2, No. 1, January 1970.
22. Kullback, S., Information Theory and Statistics, John Wiley & Sons, Inc., New York, 1959.
23. Lawlty, D. N., 11 The Estimation of Factor Loadings by the Method of Maximum Likelihood," ProLeeding of the Royal Society uf Edinburgh, 1939-1940.
24. Montgomery, D. C. and Klatt, P. J., "Economic Design of T2 Control Chart to Maintain Current Control of a Process, 11 Manaqement Science, Vol. 19, No. 1, September 1972.
25. Montgomery, D. C. and Klatt, P. J., "Minimum Cost Multivariate Quality Control Tests, 11 AIIE Transactions, June 1972.
26. Montgomery, D. C. and Wadsworth, H. M., 11 Some Techniques for Multi-variate Quality Control Applications, 11 Transactions of the ASQC, Washington, D. C., 1g72,
27. Morrison, D. F., Multivariate Statistical Methods, McGraw-Hill Book Company, New York, 1967.
28. Newman, T. G. and Odell, P. L., The Generation of Random Variation, Hafner Publishin9 Company, New York, 1971.
98
29. Noether, G. E., "Use of the Range Instead of the Standard Deviation," Journal of the American Statistical Association, Vol. 50, 1955.
30. Ostle, B. and Steck, G. P., "Correlation Between Sample Means and Sample Ranges, 11 Journal of the American Statistical Association, Vo 1 . 54, 1959.
31. Pages,!:. S., 11 A Modified Control Chart with Warning Limits, 11
Biometrika, Vol. 49, 1962.
32. Schmidt, J. W., "Mathematical Operations Research--Class Handout," Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University, Blacksburg.
33. Shewhart, W. D., Statistical Method from the Viewpoint of Quality Control, Department of Agriculture, Washington, D. C., 1939.
34. Stein, F. M., Introduction to Matrices and Determinants, Wadsworth Publishing Co., Belmont, California, 1967.
35. Wall, F. J., "The Generalized Variance Ratio or U-Statistic, 11
Technical Report, The Dikewood Corporation, Albuquerque, New Mexico, 87106.
36. Weiler, G. H., "New Type of Control Chart Limits for Means, Ranges and Sequential Runs," Journal of the American Statistical Association, Vol. 49, 1954.
37. Weiler, G.' H., 11 0n the Most Economical Sample Size for Controlling the Mean of a Population," The Annals of Mathematical Statistics, Vol. 23, 1952.
38. Wilks, S.S., "Certain Generalizations in the Analysis of Variance," Biometrika, Vol. 24, 1932.
The vita has been removed from the scanned document
Appendix A
MAIN PROGRAM FOR TESTING STABILITY OF PROCESS
100
C C MAIN PROGRAM FOR TESTING THE STABILITY OF PROCESS C
1000 1010 1100 1101 1110 1200
1250
1308 1350 1400 1450
C
COMMON /BLOKA/ AVE(5),XBAR(5),SV(5,5),PV(5,5),PVJ(5,S),DEV(5), X C RE AT ( 5 ) , S MAL L ( 5 ) , X ( 5 0, 5 ) , T RU EA V ( 5 ) , F P V ( 5 , 5 ) 'COMMON /BLCKil/ DPV,NOOC,IX,DOCC,~CCC,IR,R,JC,C,DPVL
COMMON DET COMMO~ /BLCKH/ SS(l0,10),TOT(lO) DIMENSION U(40,20,5),DS(40),SP(5,5),A(S,5),AA(5,5),XBAL(40,5) DIMENSION TT(40,20),T(5,5),G(5),Y(40,5),E(5,5) FORMAT (3 I 5) FCRMAT (Fl0.4) FORMAT (8Fl0.4) FORMAT (4Fl2.2) FORMAT (15,Fl0.4) FORMAT (/, 1 PROCESS IS NOT STABLE WITH RESPECT TO DISPERSION
X ', 2F 12. 4) FORMAT(/,' PROCESS IS STABLE WITH RESPECT TO DISPERSION
X ',2Fl2.4) FORMAT ( 1 PROCESS IS STABLE WITH RESPECT TO CENTRAL TENDENCY') FORMAT (' F-DISTR IBUTION IS NOT APPROPRIATE 1 )
FORMAT(' INDEX IS INVALID 1 ) FORMAT ( 1 PROCESS IS NOT STABLE WITH RESPECT TO CENTRAL TENDENCY
X I )
C TEST STABILITY OF PRCCESS WITH RESPECT TO DISPERSION C
READ (5,lCOC) IR,JC,M READ (5,1010) TABLEV R=IR AN=IR AK=JC
0
AM=M S DS=O .O DO 1500 I=l,JC G(Il=C.O DO 1500 J-=1,JC A(I,J)=O.O TCI,Jl=O.O
1500 SP(I,J)=O.O DO 20CO I=l,M DO 2500 J=l, IR READ (5,1100) (X(J,Ll,L=l,JC)
2500 CONTINUE CALL MANDV DO 2440 LA=l,JC G(LA)=G(LA)+XBAR(LA) XBAL(I,LA)=XBAR(LA) DO 2430 LB=l,JC SP(LA,LB)=SP(LA,LB)+SV(LA,LB) T(LA,LB)-=T(LA,LR)+SS(LA,LB) PV(LA,LB)=SV(LA,LB)
2430 CONTINUE 2.'.t40 CUNTINUE
CALL DETER OS( I)=.'\LOG!O(OET) SDS=SDS+DS(I)
2000 CCJNTINUE DO 27CO 1-=1,JC DO 2700 J=l,JC PV(I,J)=SP(l,J)/AM
2700 C !JNTI NUE CALL DETER OSP=ALOGlO(CJET)
.... 0 N
AZ=2.0*AK*AK+3.0*AK-l.O oB= AM*AM-1.0 CC=6.C*(AK+l.O)*(AM-l.O) DD=AM*( AN-1.0) R=2.3026*(1.0-(AZ*B8/CC*DD))~(DD*DSP-(AN-l.O)*SDS) IF (R .LE. TABLEV) GO TO 5000 WRITE (6,1200) R,TABLEV GO TO 6001
5000 hRITE (6,1250) R,TABLEV 6001 CONTINUE C C TEST STABILITY OF PROCESS WITH RESPECT TO CENTRAL TENDENCY C
READ (5,1100) VA,VE READ (5,1110) INDEX,TABLEM DO 4550 1=1,JC DO 4600 J=l,JC PV( I,J)=T(I,J)
4600 CONTI~UE 4550 CONT I NUE
CALL DETER DfTT=DET DO 4640 J=l ,JC
4640 G(J)=G{J)/AM DO 50 0 5 I= 1 , M DO 5500 l= 1,JC Y(I,L)=XBAL(J,L)-G(LJ
5500 CONTINUE 5005 CONTINUE
DO 59<;9 L=l ,M DO 6000 I=l,JC DO 65CO J=l,JC
__, 0 w
6500 6000 59<;9
7000
8500 8000
A(I,J)=Y{I,I)*Y(I,J)+A(I,J) CONTINUE CONTINUE CONTINUE DO 7000 I=l,JC DO 7000 J=l,JC A(I,J)=AN*A(I,J) P V { I, J) =A ( I, J ) C>JNTINUE CALL DETER DETA=OET DO 8000 I=l,JC 00 8500 J=l ,JC EC I,J)=T( I,JJ-A(I,J) PV ( I, J )= E (I, J) CGrHI NUE CONTINUE CALL DETER DETE=DET W=DETE/DETT ~RITE (6,1100) ~,TABLEM IF (INDEX .EQ. 1) GG TO 510 IF (INDEX .EQ. 2) GO TO 520 IF { l~DEX .E:Q. ~) GC TO 530 WR.I TF ( 6, 1400) INDEX GO TO 2450
510 IF (W .LT. TABLE~) GO TO 2410 515 WRITE (6,1450)
GO TO 2450 5 2 0 I F ( V A • N f. l • 0 ) GO TO 2 412
F=(l.O-W)*(VE+VA-AK)/(W*AK) WRITE (6,1100) F,TABLEM
_,
IF (F .GT. TABLEM) GO TO 515 GO TO 2410
2 412 I F ( V A • N E. 2 • 0 ) GC TO 2 414 F=(l.C-SQRT(Wll*(V[+VA-AK-1.0}/(SQRT(W)*AK) v,RITE (6,1100) F,TABLEM _ IF (F .er. TA8LEM) GO TO 515 GO TO 2410
2414 IF (AK .NE. 1.0) GO TO 2416 F={l.0-W)•VE/(W*VA) wRITE (6,1100) F,TABLEM IF (F .GT. TABLEM) GO TO 515 GO TO 2410
2416 IF (AK .r~E. 2.0) GO TO 2418 F=(l.O-SQRT(W)*(VE-1.))/(SQRT(W)*VA) WRITE (6,1100) F,TABLEM IF (F .GT. TABLEM) GO TO 515 GO TO 2410
2418 WRITE (6,1350) GO TO 2450
530 B=VE-(AK-VA+l.0)/2.0 B=-8*ALOG(W) WRITE (6,1100) B,TABLEM IF (8 .GT. TABLEM) GO TO 515
2410 WRITE (6,1308) 2450 STOP
END
...... 0 u,
Appendix B
MAIN PROGRAM FOR MONITORING PROCESS
106
C C MAIN PROGRAM FOR MONITORING THE MANUFACTURING PROCESS C
C C C 1130 1140 1000 1100 1164
2000
2100
2200
CO~MUN /3LOKA/ AVE(5),XBAR(5),SV(5,~),PV(S,5),PVI(5,5),DEV(5), XGREAT(5),SMALL(5),X(S0,5),T~UEAV(5),FPV{5,5)
COMMON /glCKB/ DPV,NOOC,IX,COCC,~OOC,IP,R,JC,C,DPVL CO~MON /BLOKD/ CHIONE,CHIK,DSV CUMMON /BLOKG/ DET DIMENSION F(5J,ZC5),TRS(5,5),WEIG(5)
MONITORING OF THE DISPERSION
FORMAT ( 1 DISPEPSION IS CUT CF CGNTROL') FORMAT ( 1 MEAN IS OUT OF CONTROL ') FORMAT (215) FORMAT (8Fl0.4) FORMAT ( 1 END OF TEST 1 )
REAO (5,1000) IR,JC R=IR DO 2000 l=l,IR READ (5,llOC) (XC I,J) ,J=l,JC) CCN TINUE READ (5,1100) CHIONE,CHIK READ (5,ll0C) CS~ALL(J) ,J=l,JC) READ (5,1100) (GREAT(J},J=l,JC) READ (5,1100) CTPUfAV(J),J=l,JC) DO 2100 I=l,JC READ (5,1100) (FPVCI,J),J=l,JC) CONTINUE DO 2200 I=l ,JC READ (5,1100) (PVI(I,J),J=l,JC) CCNTINUE
...... 0 '-I
READ (5,1100) DPVL TR=O. FIR=O. S EC=O. Tvl=O. CALL MANDY OD 1155 I=l,JC DO 1155 J=l ,JC PV( I,Jl=SV( 1,J)
1155 CONTINUE CALL DETER DSV=DET DO 200 1=1,JC DO 180 J=l,JC TRS(I,J)=O. DO 170 K=l,JC TRS{I,J)=TRS(I,J)+SV(l,K)*PVI(K,J)
170 CONTINUE 180 CONTINUE 200 CONTINUE
DO 210 J=l,JC TR=TR+TRS(J,J)
210 CON Tl NUE OSVL=ALCG(DSV) STAT=R*(DPVL-DSVL-C+TR) IF (STAT .GE. CHIK) WRITE (6,1130)
C C MGNITORING OF THE CENTRAL TENDENCY C 260 DO 370 J=l,JC
Z(J)=XBAR(J)-TRUEAV(J) IF (XBAR(JJ .GE. TRUEAV(J)) GO TO 350
__,
W E I G ( J ) =SM A LL ( J ) GO TO 360
350 ~EIG(J)=GREAT(J) 360 TW=TW+WEIG(J) 370 CON TI NUE 390 CRIT=CHIONE*(C+TW)
00 "•10 1=1, JC F(l)=O. DO 400 J=l,JC
400 F(l)=Z(J)*PVl(J,I)+F(I) 410 CONTINUE
DO 420 J=l,JC FIR=FIR+F(J)*Z(J) SEC=SEC+R*WEIG(J)*Z(J)**2/FPV(J,J)
420 CON TI NUE FIR=FIR*R ST=FIR+SEC IF (ST .GT. CRIT) WRITE {6,1140) hRITE (6,1164) STOP END
..... 0 1.0
Appendix C
TWO VARIABLES SIMULATION MAIN PROGRAM
110
C C MAIN PRCGRAM FOR SIMULATION ( TWO VARIABLES) C
110 120 230 250 350
351
352 360 373
805
COMMON /BLOKA/ AVE(5),XBAR(~),SV(5,5),PV(5,5),PV1(5,5),DEV(5), XGREAT(5),SMALL(5),X(50,5),TRUEAV(5),FPV(5,5)
COMMON /BLC~B/ DPV,NOOC, IX,DOOC,MOOC,IR,R,JC,C,DPVL cu~~ON /BLOKC/ A(5,5) COMMON /BLOKD/ CHIONE,CHIK,OSV CO~MON /BLGKG/ DET OIMENSICN SPV(5,5),DVE(5) DIMENSION NM(lO),NV(lO),ZM(5,10),ZV(5,10) ,R0(5,5),FPV(5,5) OIMENSICN ID(20) FORMAT ( 8Fl0.4) FORMAT '1015) FORMAT (4Fl 0.6) FORMAT (lHl,///) FORMAT (I,' S.O. AWAY FROM STANDARD PROCESS
XMEAN TOTAL . ) FORMAT (/,' VARIANCE
XERCENT PERCENT 1 ) FORMAT (X,4Il0,2I9,2X,I9,/) FORMAT (/) FORMAT (/, 1
X. 0 • C • 0 .o. C. ' ) FORMAT (lOF5.l) REAU (5,120) IX READ (5,110) ROCF
( B)
READ (5,230t CHIONE,CHJK READ (5,120} IR,JC READ (5,110) (AVE(J),J=l,JC) DO 270 1=1,JC
CENTRAL TENDENCY
(A) (8) (A)
DISPERSION
PERCENT
o.o.c.
p
0
..... ..... .....
READ (5,110) (PV(I,JJ,J=l,JC) 270 CONTil\:UE
DO 2 8 0 I = 1 , JC PEAD (5,110) ( P VI ( I, J) , J= 1, JC)
280 CON TI NUE DO 290 I=l,JC PEAD (5,110) SPV(I,I)
290 CONTINUE READ (5,120) MM
300 R !:AD ( 5,110) (GREAT(J),J=l,JC) READ (5,110) (SMALL(J),J=l,JC) READ ( 5, 12 0 ) (NM( I) ,1=1,JC) READ (5,120) (NV(IJ,I=l,JC) DO 308 1=1,JC NMDUM=NM( I) NVDUM=NV( I) READ (5,805) ( ( I, L ) , L = 1, NMDUM) ...... READ (5,8C5) ( ZV( I, L) ,l=l ,NVDUM) ......
N 308 CONTINUE
R=IR C=JC DPV=PV(l,l)*PV(2,2)-PV(l,2}**2 CPVL=ALCG{DPV) ~ 1Rl TE (6,250) w RITE (6,350) \<.RITE (6,351) wRI TE (6,373) WRITE (6,360) DO 301 l=l,JC TRUEAV( I )=AVE( I) DVE(l)=SQRT(PV(I,I))/SQRT(R) DO 302 J=l,JC
FPV(l,J)=PV(I,J) 302 CONTINUE 301 CONTINUE
00 3777 I=l,JC 3777 RO(I,I)=l.O
JCA=JC-1 DO 317 l=l,JCA l=I+l DO 317 J=L,JC RO (I, J) = PV ( I, J ) /SQRT ( PV ( I, I ) *PV ( J, J) ) RO(J,l)=RO(I,J)
317 CONTINUE NV2=NV(2) ~Vl=NV(l) DO 3200 L2=1,NV2 PV(2,2)=FPV(2,2)+ZV(2,L2)*SPV(2,2) DO 3300 Ll=l,NVl PV(l,l)=FPV(l,l)+ZV(l,ll)*SPV(l,1) PV(l,2)=RC(l,2)*SCRT(PV(l,ll*PV{2,2)) PV( 2, ll=PV( 1,2) CALL CONVE2 r-. M2=N~ (2) I\' i-11 = N M ( 1 ) DO 4200 M2=1,NM2 AVE(2)=TRLE~V(2)+ZM(2,M2)*DVE(2) DO 4300 Ml=l,NMl AVE(l)=TRUEAV(lJ+ZM(l,Ml)*DVE(l) OOOC=O. MOOC=O NOOC=O DO 1500 KK=l,MM CALL RANGE2
...... ...... w
1500
4300 4200 3300 3200
CALL MANDV CALL TEST CONT lNUE AA=NOOC AMOOC=MOOC BB=MM PER=AA*lOO. 0/BB APER=lOC.O*AMOOC/88 DPER=lOO.O*COCC/BB ID ( 1) = Z V ( 2, l2) ID( 2)=ZV( 1,Ll) ID(3)=ZM(2,M2) ID(4J=ZM(l,,-,l) l0(5)=DPER l0(6)=APER 10(7)=PER \-,RITE ( f,352) ( ID(l ),1=1,7) PUNCH 352,IC(l),10(2),10(3),10(4),ID(S),I0(6),ID(7) CONTINUE CONTINUE CONTINUE CONTINUE STOP END
--.i:,.
Appendix D
FOUR VARIABLES SIMULATION MAIN PROGRAM
115
C C MAIN PRCGRA~ FOR SIMULATION (FOUR VARIABLES ) C
110 120 230 250 350
351
352 360 373
COMMON /BLCKA/ AVE(5),XBAR(5J,SV(5,5J,PV(5,5),PVl(5,5),DEV(5), XGREAT(5),SMALL(5),X(50,5),TRUEAV(S),FPV(5,5)
COMMON /BLOK8/ DPV,NOOC,IX,DOCC,MCCC,IR,R,JC,C,DPVL COMMON /BLOKC/ A(5,5)
CHIO~E,CHIK,DSV COMMON /BLCKG/ DET DIMENSION SPV(5,5),CVE(5) DIMENSION NM(lO),NV(lO),ZM(5,10),ZV(5,lO),R0{5,5),FPV(5,5) DIMENSION 10(20) FORMAT (8Fl0.4) FORMAT (1015) FORMAT (4Fl0.6) FORMAT (lHl,///) FORMAT(/,' S.D. AWAY FRCM STANDARD PROCESS DISPERSION
XMEAN TOTAL 1 )
FORMAT(/,' VARIANCE CENTRAL TENDENCY PERCENT XERCENT PERCENT ')
FORMAT (8I5,219,2X,19,/) FORMAT (/) FORMAT (/,' (0) (C) (8) (A) (D) (C) (B) (A) o.o.c.
x.o.c. o.o.c. 1 )
805 FORMAT (lOFS.l) READ ( 5,120) IX READ (5,110) ROCF READ (5,230) CHIONE,CHIK READ (5,120) IR,JC READ (5,110) (AVE(J),J=l,JC) DO 210 I=l,JC READ (5,110) (PV(I,J),J=l,JC)
p
0
..... ..... °'
270 CONTINUE DO 2 8 0 I = 1 , JC R EAO (5,110) (PVI(I,J),J:1,JC)
280 CONTINUE DO 290 l=l,JC READ ( 5, 110) SP V ( I, I)
290 CONTl"UE READ (5,120) MM
300 READ ( 5, 110 ) (GREAT(J),J=l,JC) READ (5,110) (SMALL<J),J=l,JC) PEAD (5,120) (NM( I) ,I=l ,JC) READ (5,120) (NV ( I ) , I= l , JC J DO 3 0 e I= l , JC M1DUM=N M ( I ) NVDUM=NV(I) READ (5,8C5) ( Z M ( I , L) , L = 1 , WW UM ) READ (5,805) ( Z V ( I, L) , L = 1, N VD UM) ......
308 CONTI f\UE ...... ...., P=IR C=JC CALL CETER 0 PV=DET DPVL=ALOG( OPV) wRITE (6,250) WRITE (6,350) WRITE (6,351) hRITE (6,373) WRITE (6,360) 00 301 I= l, JC TRUEAV(I)=AVE(I) DVE(l}=SQRT(PV(l,I))/SQRT(R) DO 302 J::::1,JC
FPV(I ,J)=PV(I,J) 302 CONTINUE 301 CONTINUE
DO 3777 1=1,JC 3777 RO(l,I)=l.O
JCA=JC-1 DO 317 I=l,JCA L=l+l DO 317 J=L,JC f<.O( I, J )= PV ( I, J ) / S QP. T ( PV ( I , I ) ,:,py ( J, J) ) RO(J,l)=RC(l,J)
317 CONTINUE NV4=NV(4) f\V3 =NV ( 3) NV2=NV ( 2) NVl=NV ( l) DO 3000 L4=1,NV4 PV(4,4)=FPV(4,4)+ZV(4,L4l*SPV(4 1 4) DO 3100 L3=1,NV3 PV(3,3)=FPV(3,3)+ZV(3,L3)*SPV(3,3J PV(3,4)=R0(3,4J*SQRT(PV(3,3)*PV(4,4J) PV(4,3)=PV(3,4) DO 3200 L2=1,NV2 PV(2,2)=fPV(2,2)+ZV(2,L2)*SPV{2,2) PV(2,3)=RC(2,31*SCRT(PV(2,2)*PV(3,3)) PV(2,4)=RC(2,4)*SORT(PV(2,2)*PV(4,4)) PV(3,2)=PV(2,3) PV(4,2)=PV(2,4) DO 3300 Ll=l,NVl PV(l,l)=FPV(l,l)+ZV(l,Ll)*SPV(l,1) PV( 1,2)=RC(l,2)*SCRT(PV(l,l)*PV(2,2J) PV(l,3)=RO(l,3)*SQRT(PV(l,l)*PV(3,3))
.... .... CX)
PV(l,4)=RCC1,4)*SQRT(PV(l,l)*PV(4 1 4JJ PY(2,U=PV(l,2) PV( 3, 1) =PV( 1,3) PV(4, U=PV( 1,4) CALL CCf\VER4 l\:M4=NM(4) NM3=NM'3) l':M2=N~( 2) Nt'il=NM(l) CO 4000 f"4=1,NM4 AVE(4)=TRUEAV(4)+ZM(4,M4)*DVE(4) DO 4100 M3=1,NM3 AVE(3)=TRUEAV(3)+Zr(3,M3)*0Vf(3) DO 4200 M2=1,NM2 AVE(2l=TRUEAV(2)+ZM(2,~2)*DVE(2) DO 4300 Ml=l,NMl AVE(l)=TRUEAV(l)+ZM(l,Ml)*DVE(l) OOOC=O. MOOC=O NOOC=O DO 1500 KK=l,MM CALL RANGEN4 CALL MANDV CALL TEST
1500 CON TI NUE AA=NOOC AMOOC=MCOC BB=MM PER=AA*lOO.C/88 APE~=lOO.O*AMOOC/BB OPER=lOO.O*COOC/8B ID( U=ZV(4,L4)
_, _, \0
4300 4200 4100 4000 3300 3200 3100 3300
ID(2)=ZV(3,L3) ID(3)=ZV(2,L2) ID( 4)=ZV( l,LU I D ( 5 ) = Z ( 4 , ~4 ) ID(6)=ZM(3,M3) IO( 7)=Z,..<2,M2) ID(8)=Z~<l,MU ID(9)=OPER ID( 10 )=APER ID(ll)=PER hRITE (6,352) (I0(I),1=1,11) PUNCH 3 5 2 , I D ( 1 ) , I D ( 2 ) , ID ( 3 ) , ID ( 4 ) , ID { 5 ) , I D ( 6) , ID ( 7 ) , I D ( 8 ) , l D ( 9) ,
XID(l0),10(11) CONTINUE CJNT INUE CONTINUE CONTINUE CONTINUE CONTINUE CONTINUE CUNTit\UE STOP END
__, N 0
Appendix E
SUBROUTINES
121
C
C
C
C
C
SUBqOUTINE CCNVER2
C O t-1 :-1 Q N / BL OK M AV E ( 5 ) , X B AR. C 5 ) , S V ( 5 , 5 ) , P V ( 5 , 5 ) , P V I ( 5 , 5 ) , D E V ( 5 ) , XGREAT(5),SMALL(5),X(50,5),T~UEAV(S),FPV(5,5)
COMMON /BLCKB/ DPV,NOOC,IX,DtCC,MOOC,IR,R,JC,C,DPVL COM~ON /BLCKC/ A(5,5)
A(l,l)=SQkT(PV(l,1)) A(2,l)=PV(2,1)/A(l,1) A(l,2)=0. A(2,2)=SQRT(PV(2,2)-A(2,1)**2) RETURN ENO
SUBROUTINE RANGEN2
COM ~ON / 8 LOK A/ AVE ( 5 ) , X B r.q ( 5 ) , S V ( 5, 5) , PV ( 5 , 5) , P VI ( 5, 5) , DEV ( 5) , X GREAT ( 5) , S ~ALL ( 5 ) , X ( 5 O, 5) , TRUE AV ( 5) , F PV ( 5, S)
COMMON /BLCKB/ DPV,f'..:OOC,IX,DOCC,r1ccc,1R,R,JC,C,DPVL CJ~MON /5LOKC/ A(5,5) OP1ENSICN WC5,5) DO 10 I=l,IR DO 350 J=l,2 CALL GAUSS (IX,1.0,0.0,V) W(l,J)=V
350 CONTINUE 10 CONTINUE
DO 300 1=1, IR X ( I , 1 ) = A ( 1 , 1 ) ~,w ( I , 1 ) + A ( 1, 2) * \·! ( I, 2 ) +AVE ( 1) X(I,2)=A(2,l)*W(l,l)+A(2,2)*W(I,2)+AVEC2)
300 CONTINUE
_, N N
C
C
C
C
RETURN END
SUBROUTINE CONV~R4
COMXON /BLCKA/ AVE(5),XBAR(~),SV(5,5),PV(5,5),PVl(S,5),DEV(5), XGREAT(5),SMALL(5),X(50,5),TRUEAV(5),FPV{5,5)
CO~MQN /BLCK6/ DPV,NOOC,IX,DOCC,PCOC,IR,R,JC,C,DPVL COMMON /BLOKC/ A(5,5) A(l,l)=SQKT(PV(l,1)) A(2,l)=PV(2,l)/A{l,1) A(3,l)=PV(3,1)/A(l,1) J(4,l)=PV(4,1)/A(l,1) A(2,2J=SQRT(PV(2,2)-A(2,1)**2) A(3,2)=(PV(3,2)-A(3,l)*A(2,1))/A(2,2) A{3,3J=SQRT(PV(3,3)-A(3,1)**2-A(3,2)**2) A{4,2)=(PV(4,2)-A(4,l)*A(2,1))/A(2,2) A(4,3)=(PV(4,3)-A(4,l)*A(3,1)-A(4,2)*A(3,2))/A(3,3) AC4,4)=SQRT(PV(4,4)-A(4,1)**2-A(4,2)**2-AC4,3)**2) A(l,2)=0.0 A(l,3)=0.0 A(l,4)=0.0 A(2,3)=0.0 A{2,4)=0.0 A(3,4)=0.0 RETURN E~
SU~RGUTINE RANGEN4
COM~ON /BLOKA/ AVE(5),XBAR(S),SV(5,5),PV(5,5),PVI(5,5),DEV(5), XCREAT(5),SMALL(5),X(50,5),TRUEAV(S),FPVC5,5)
..... N w
350
200
205 10
C
C
COMMON /BLOKB/ DPV,NOOC,IX,DCOC,~OOC,IR,R,JC,C,DPVL COMMO~ /BLCKC/ A(5,5) 0 !:''IE r~ SI ON h ( 5 0) DO 10 1=1, IR DO 350 J=l,JC CALL GAUSS (IX,1.0,0.0,V) W(J)=V CONTlf\UE DO 205 L=l,JC X (I, L )= 0. 0 DO 200 K=l,JC Xll,L)=X(l,L)+A(L,K)*W(K) CONTINUE X(I ,L)=X(I,U+AVE(L) CONTINUE CONTlf\UE RETURN END
SUBROUTINE MANDY
COMMON /BLCKA/ AVE(5),XBA~(5},SV(5,5),PVC5,SJ,PVl(5,5),DEV(5), XGREAT(5),SMALL(5),X(50,5),TRUEAV(5),FPV(5,5)
COMMON /BLCK8/ DPV,NOOC,IX,DCGC,MCCC,IR,R,JC,C,DPVL COMMON /BLOKC/ CHIONE,CHIK,DSV COMMON /BLCKH/ SS(l0,10),TOT(lO) DO 55 1=1,JC TOT ( I )=O.
55 SS(l,l)=O. DO 60 I=l,JC l=l+l DO 60 J=L,JC
N .i:.
60
70
85 83 80
90
110 100
122
C
SS(l,J)=O. DO 70 J= 1,JC 00 70 1=1,IR TOT ( J) = X ( I , J) + TOT ( J) SS(J,J)=SS(J,J)+X(I,J)**2 JCA=JC-1 DO 80 J=l,JCA L=J+l DO 83 K=L,JC DO 85 I=l, IR SS(J,K)=SS(J,K)+X(I,J>*X(l,K) C 01\ITI NUE CONTINUE DD 90 J=l,JC XBAR(J)=TOT(J)/R SV(J,J)=(SS(J,J)-TGT(J)**2/R)/(R-l.O) DO 100 J=l,JCA L=J+l DO 110 K=L,JC SV(J,K)=(SS(J,K)-TOT(J)*TGT(K)/R)/(R-1.0) CONTINUE CONTINUE DO 122 J=l,JCA l=J +l DO 122 K=L,JC SS(K,J)=SS(J,K) SV(K,J)=SV(J,K) RETUR~ END
SUB ROUT I NE CETER
__, N> <.Tl
C CO~MON /BLOKA/ AVE(5),XBAR(5),SV(5,5),PV(5,5),PVI(5,5),DEV(5),
XGREAT(5),SMALL(5),X(50,5),TPUEAV(5),FPV(5,5) CU~MON /OLOKB/ DPV,NOOC,IX,DCCC,MCOC,IP,R,JC,C,OPVL COM~O~ /eLOKG/ DET K=JC JCA=JC-1 DO 2001 1=1,JCA L=I +l DO 2001 J=L,JC
2001 PV(J,I)=PV(l,J) DO 7 M=2,K DO 7 I=M,K ~=PV(I,M-1)/PV(M-I,M-l) DO 7 J=M,K
7 PV(I,J)=PV(I,J)-PV(M-1,J)*W DET=l.O DO 8 I=l,K
8 CET=DET*PV(I,I) RETURN END
C SUBROUTINE TEST
C
C C C
COMMON /BLGKA/ AVE(5),XBAR(5),SV(5,5),PV(5,5),PVl(5,5),DEV(5), XGREAT(5),S~ALL(5),X(50,5),TRUEAV(5),FPV(5,5)
COMMON /BLOKS/ DPV,NOOC,IX,DOCC,~CCC,IR,R,JC,C,DPVL COMMON /BLOKD/ CHICNE,CHIK,DSV DIMENSION F(5),Z(5),TRS(5,5),WEIG(5)
MGNITORl~G OF THE DISPERSION
...... N O'I
170 180 200
?10
C
TR-=O. FIR=0. S EC=0. TW=0. CALL CETER CSV=DET DO 20C I=l,JC DO 18 0 J = l , JC TRS(I,J)=O. DO 170 K=l,JC TRS{I ,J)=TRS( I,J) +SV( I ,K)*PVI (K,J) CONTINUE CON TI NL: E CONTINUE DO 210 J=l,JC TR=TR+TRS(J,J) CONTINUE DSVL= ALOG ( DSV) STAT=R*{DPVL-OSVL-C+TR) IF (STAT .GE. CHIK) DOOC=DOOC+l.
C MCNITCRING Of lHE CENTRAL TE~DE~CY C 260 DO 370 J=l,JC
Z(J)=XBAR(J}-T~UEAV(J) IF (XBAP(J) .GE. TRUEAV(J)) GC TO 350 WEIG(J)=S~ALL(J} GO TO 360
350 WEIG(J)=GREAT(J) 360 TW=TW+WEIG(J) 370 CONTINUE 390 CRIT=CHIONE*(C+TW)
...... N ......,
DO 410 I= 1, JC F(l)=O. DO 40C J=l,JC
400 F(l)=Z(J)*PVI<J,I)+F(I) 410 CONTINUE
DO 420 J=l,JC FIR=FIR+F{J)*Z(J) SEC=SEC+R*WEIG(J)*Z(Jl**2/FPV(J,J)
420 CON TI NUE FIR=FIR*R S T=FI R+S EC IF (ST .GT. CRIT) tl.OCC=MOOC+l
C C TOTAL ~UM3ER OUT CF CO~TROL C
440
C
C
IF ((ST .GT. CRIT) .OR. (STAT .GT. CHI3)) NOOC=NOOC+l RETURN ENO
SUBROUTI~E GAUSS(IX,S,AM,V)
A=O.O DO 50 1=1,12 CALL RANDU(IX,IY,Y) IX=IY
50 A=A+Y
C
C
V=( A-6.C)*S+AM RETURN END
SUBROUTINE RANDU(IX,IY,YFL)
..... N CP
IY= IX*65539 IF (IY) 5,6,6
5 IY=IY+21474e3647+1 6 YFL=IY
YFL=YFL*0.4656613E-9 P.ETURN END
.... N
'°
Appendix F
TWO VARIABLES SIMULATION RESULTS
130
131
(1) Correlation Coefficient= Standard Value
S.D. AwAY FROM STANDARD PROCESS DISPERSION MEAN TOTAL
VARIANCE CENTRAL TENDENCY PERCENT PERCENT PERCENT
( 6) ( A ) ( R) (t.) iJ.o.c. n.o.c. o.o.c. 0 0 0 0 0 l l 0 0 0 2 2 22 23 0 0 0 -2 4 6 9 0 C 2 0 l 4 5 0 0 2 2 l 24 24 0 0 2 -2 l 93 93 0 0 -2 0 2 22 24 0 0 -2 2 1 60 60 0 0 -2 -2 l 31 31 .....
w 0 2 0 0 ~5 6 29 N
0 2 0 2 20 36 47 0 2 0 -2 24 19 39 0 2 2 0 18 6 24 0 2 2 2 38 20 51 0 2 2 -2 28 84 36 0 2 -2 0 zc; 19 43 () 2 -2 2 29 53 67 0 2 -2 -2 ll: 23 35 0 -2 0 C 52 0 52 0 -2 0 2 54 13 62 0 -2 0 -2 52 6 57 0 -2 2 0 50 3 51 0 -2 2 2 47 17 56 0 -2 2 -2 52 85 92 0 -2 -2 0 50 22 63 0 -2 -2 2 47 40 69
0 -2 -2 -2 57 21 70 2 G 0 0 16 4 19 2 0 0 2 11 18 28 2 C 0 -2 16 9 23 2 0 2 0 15 8 20 2 0 ? 2 q 33 40 .... 2 0 2 -2 lP 78 82 2 0 -2 0 12 24 32 2 0 -2 2 q 58 63 2 0 -2 -2 20 36 51 2 2 0 0 33 6 34 2 2 0 2 19 20 36 2 2 0 -2 25 10 34 2 2 2 0 14 9 22 2 2 2 2 ")";)
L..., 36 54 2 2 2 -2 20 86 90 ....
w 2 2 -? 0 7 .... 17 35 w -~ 2 2 -2 2 23 60 67 2 2 -2 -2 ?8 26 46 2 -2 0 0 8 <; 5 90 2 -2 0 2 31 16 82 2 -2 0 -2 3f 9 37 2 -2 2 0 91 3 91 2 -2 2 2 35 18 90 2 -2 2 -2 -~ C,~ 73 95 2 -? -2 0 87 20 88 2 -£ -2 2 81 48 90 2 -2 -2 -2 82 25 86
-2 C 0 0 22 l 23 -2 0 0 2 35 28 54 -2 0 0 -2 26 7 31 -2 0 2 0 22 5 27
-2 0 2 2 21 27 43 -2 0 2 -2 17 81 35 -2 0 -2 0 29 13 36 -2 0 -2 2 34 38 53 -2 0 -2 -2 23 27 43 -2 2 0 0 O] 3 91 -2 2 0 2 87 29 90 -2 2 0 -2 88 13 38 -2 2 2 0 90 8 90 -2 2 2 2 94 26 96 -2 2 2 -2 ::.7 73 94 -2 2 -2 0 so 12 90 -2 2 -2 2 86 53 93 -2 2 -2 -2 95 37 9€ -2 -2 0 0 45 0 45 -w -2 -2 0 2 50 9 54 -2 -2 0 -2 37 3 40 -2 -2 2 0 41 l 42 -2 -2 2 2 45 12 54 -2 -2 2 -2 47 85 94 -? ... -2 0 43 7 47 -L
-2 -2 -2 2 60 41 75 -2 -2 -2 -2 53 23 66
135
(2) Correlation Coefficient= Standard Value x 0.8
S.D. AWAY FROM STANDARD PROCESS DISPERSION MEAN TOTAL
VAR IANCF CENTRAL TENDENCY PE:RC ENT PER CENT PERCENT ( B) (A) ( f:'., ) ( A} C.l;.C. r1.-J.C. o.r.c.
0 0 0 0 2 l 3 0 0 0 2 l 27 28 0 0 0 -2 3 12 14 0 0 2 0 0 6 6 0 0 2 2 " 18 23 _, 0 0 2 -2 2 89 89 0 0 -2 0 1 23 23 0 0 -2 2 4 44 46 0 0 -2 -2 2 28 30 .....
w 0 2 0 0 1~ 5 23 °' 0 2 0 2 29 27 52 0 2 0 -2 JG 19 43 0 2 2 0 26 6 30 0 2 2 -; zc, 36 61 L..
0 2 2 -? 25 37 92 0 ? -2 0 24 17 37 0 2 -? 2 35 52 67 0 2 -2 -2 18 36 50 0 -2 0 0 4f l 49 0 -2 0 2 51 8 55 0 -2 0 -2 54 0 54 0 -2 2 0 55 2 57 0 -2 2 2 55 24 68 0 -2 2 -2 50 86 92 0 -2 -?. 0 50 22 62 0 -2 -2 2 55 41 75
0 -2 -2 -2 5e 22 67 2 0 0 0 11 4 14 2 0 0 2 lt 24 37 2 0 0 -2 7 12 17 2 0 2 0 11 3 14 2 0 2 2 11 26 34 2 0 2 -2 10 86 86 2 0 -2 0 9 23 31 2 0 -2 2 E 53 59 2 C -2 -2 7 30 35 2 2 0 0 22 3 23 2 2 0 2 22 30 46 2 2 0 -2 27 21 40 2 2 2 0 24 5 29 2 2 2 2 18 33 43 2 2 2 -2 18 88 <JO
_, w
2 2 -2 0 lS 25 36 -..J
2 2 -2 2 ? r, ,_ 1,; 61 6<; 2 2 -2 -2 26 31 50 2 -2 0 0 C"=> _,_, 1 93 2 -2 0 2 86 17 88 2 -2 0 -2 79 7 81 2 -2 2 0 89 5 91 2 -2 2 2 C/2 28 94 2 -2 2 -2 gc J. 78 97 2 -2 -2 0 89 19 91 2 -2 -2 2 86 47 95 2 -2 -2 -2 85 32 90
-2 0 0 0 32 0 32 _, 0 0 2 21 24 40 '-
-2 0 0 -2 25 8 31 -2 0 2 0 23 1 24
-2 (' 2 2 29 25 50 -2 (' 2 -2 21 87 88 -2 0 -2 0 29 13 31 -2 0 -2 2 20 44 56 -2 r, -2 -2 30 22 47 I.,
-2 2 0 0 93 4 93 -2 2 0 2 95 26 98 -2 2 0 -2 g4 8 88 -2 2 2 0 g<; l 90 -2 2 2 2 oo 25 92 -2 2 2 -2 q~ .L 73 97 -2 2 -2 0 so 18 90 -2 2 -2 2 93 50 99 -2 2 -2 -2 £5 25 89 -2 -2 0 0 37 0 37 .....
w -2 -2 0 2 41 11 49 CX)
-2 -2 0 -2 35 2 36 -z -2 2 0 53 0 53 -2 -2 2 2 42 18 56 -2 -2 2 -2 44- 91 97 -2 -2 -2 C, 47 11 54 -2 -2 -2 2 47 24 60 -2 -2 -2 -2 4C 21 52
139
(3) Correlation Coefficient= Standard Value x l .1
S.D. AAAY FROM STA~OARD P~OCESS QISPEPSYDN MEAN TOT /\L
VAR I A":cr CENTRAL Tf~.;CENCY P rncu:r P [KC ENT PF.l?CHH
( 5) ( A J ( fl ) { /'. ) {' • r; • C • ri.ci.c. o.o.c. 0 0 0 0 0 4 4 0 0 0 2 2 15 17 0 0 0 -2 5 9 14 0 0 2 0 l 3 4 J 0 2 2 4 35 37 0 0 2 -2 0 87 87 0 C -2 0 5 8 13 0 0 -2 2 l 56 57 0 0 -2 -2 3 29 31 .....
.i:. 0 2 0 0 29 4 31 0
0 2 0 2 30 29 52 0 2 0 -2 25 19 37 a 2 2 0 30 5 32 0 2 2 2 27 31 45 0 ? 2 -2 27 34 87 0 2 -2 0 26 22 42 0 2 -2 2 21 59 69 0 2 -2 -2 29 29 49 0 -2 0 0 50 l 50 0 -2 0 2 64 9 67 0 -2 0 -2 ?3 4 56 0 -2 2 0 53 4 55 0 -2 2 2 53 24 66 0 -2 2 -2 49 85 92 0 -2 -2 0 57 15 63 0 ..., -2 2 48 41 69 -L
0 -2 -2 -2 55 31 71 2 0 0 0 13 4 17 2 (; 0 2 10 24 33 2 0 0 -2 12 10 21 2 0 2 0 7 9 16 2 0 2 2 14 23 35 2 0 2 -2 10 87 88 2 0 -2 0 14 21 35 2 C -2 2 12 63 68 2 0 -2 -2 22 23 39 2 2 0 0 2 fl 4 30 2 2 0 2 23 30 40 2 2 0 -2 lo 14 26 2 2 2 0 19 8 24 2 2 2 2 18 25 39 2 2 2 -2 33 38 92
__,
2 2 -2 0 25 19 39 __,
2 2 -2 2 25 56 68 2 2 -2 -2 27 38 58 2 -2 0 C 9(• l 91 2 -2 0 2 83 13 86 2 -2 0 -2 84 8 87 2 -? 2 0 32 7 84 <-
2 -2 2 2 81 24 S6 2 -2 2 -2 32 69 94 2 -2 -2 0 39 26 93 2 -2 -2 2 87 46 94 2 -2 -2 -2 88 27 92
-2 0 0 0 26 C 26 -2 0 a 2 30 27 48 -2 f", 0 -2 33 7 39 V
-2 0 2 0 19 2 21
-2 0 2 2 28 27 48 -2 0 2 -2 35 81 84 -2 0 -2 0 JS 11 41 -2 0 -2 2 33 47 67 -2 0 -2 -2 26 26 1t3 -2 2 0 0 10 3 90 -2 2 0 2 92 22 93 -2 2 0 -2 93 16 94 -2 2 2 0 93 5 93 -2 2 2 2 86 31 91 -2 2 2 -2 93 71 99 -2 2 -2 0 Gl 12 92 -2 2 -2 2 87 43 92 -2 2 -2 -2 89 25 91 -2 - ? 0 0 49 0 49
_, ... -2 -2 0 2 51 9 55 I'\)
-2 -2 0 -2 45 0 45 -2 -2 2 0 3c; 2 40 -2 -2 2 2 51 18 62 -2 -2 2 -2 50 i35 q 1 -2 -2 -2 C 43 12 51 -2 -z -2 2 47 35 67 -2 -2 -7 -2 4B 20 59
Appendix G
FOUR VARIABLES SIMULATION RESULTS
143
144
(1) Correlation Coefficients= Standard Values
and All Variances= Standard Values
S.D. AWAY FROM STANDARD PROCESS DISPERSION MEAN TOTAL VARIANCE CENTRAL TENDE~CY PERCENT PER CENT PERCENT
( D) (C) ( B ) (A) ( D) ( C) ( B ) (A) o.o.c. o.o.c. o.o •. c. 0 u 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 2 1 1 2 0 0 0 0 0 0 0 -2 3 1 4 0 0 0 0 0 0 2 0 3 1 4 0 0 0 0 0 0 2 2 2 1 3 0 0 0 0 0 0 2 -2 2 18 20 a 0 0 0 0 0 -2 0 2 0 2 0 0 0 0 0 0 -2 2 2 12 14 0 0 0 0 0 0 -2 -2 1 9 10 0 0 0 0 0 2 0 0 1 3 4 0 0 0 0 0 2 0 2 3 9 12 _, 0 0 0 0 0 2 0 -2 0 15 15 .,1::1,
U"I 0 0 0 0 0 2 2 0 4 13 16 0 0 0 0 0 2 2 2 3 11 14 0 0 0 0 0 2 2 -2 2 36 37 0 0 0 0 0 2 -2 0 2 11 13 0 0 0 0 0 2 -2 2 0 24 24 0 0 0 0 0 2 -2 -2 4 24 27 0 0 0 0 0 -2 0 0 2 1 3 0 0 0 0 0 -2 0 2 5 6 11 0 0 0 0 0 -2 0 -2 3 10 13 0 0 0 0 0 -2 2 0 4 4 7 0 0 0 0 0 -2 2 2 2 20 22 0 0 0 0 0 -2 2 -2 7 34 40 0 0 0 0 0 -2 -2 0 2 5 7 0 0 0 0 0 -2 -2 2 0 60 60 0 0 0 0 0 ...a.2 -2 -2 1 13 14
0 0 0 0 2 0 0 0 4 2 6 0 0 0 0 2 0 0 2 5 4 9 0 0 0 0 2 0 0 -2 4 7 10 0 0 0 0 2 0 2 0 4 6 10 0 0 0 0 2 0 2 2 3 11 14 . 0 0 0 0 2 0 2 -2 2 21 22 C a 0 0 2 G -2 C 6 3 8 0 0 0 0 2 0 -2 2 5 25 29 :) 0 0 0 2 0 -2 -2 2 10 11 0 0 0 0 2 2 0 0 3 1 10 0 0 0 0 2 2 0 2 0 14 14 0 0 0 0 2 2 0 -2 0 29 29 0 0 0 0 2 2 2 0 l 24 25 0 0 0 0 2 2 2 2 1 30 31 a 0 0 0 2 2 2 -2 2 62 62 0 0 0 0 2 2 -2 0 1 15 16 0 0 0 0 2 2 -2 2 1 37 38 ..... 0 0 0 0 2 2 -2 -2 3 32 34 .:::a
O'I
0 0 0 0 2 -2 0 0 5 5 9 0 0 0 0 2 -2 0 2 5 16 21 0 0 0 0 2 -2 0 -2 2 14 16 0 0 0 0 2 -2 2 0 4 20 23 0 0 0 0 2 -2 2 2 2 17 19 0 0 0 0 2 -2 2 -2 2 44 45 0 0 0 0 2 -2 -2 0 3 16 19 0 0 0 0 2 -2 -2 2 1 65 65 0 0 0 0 2 -2 -2 -2 3 26 28 0 0 0 0 -2 0 0 0 3 0 3 0 0 0 0 -2 0 0 2 l 4 5 0 0 0 0 -2 0 0 -2 2 7 9 0 0 0 0 -2 0 2 0 2 2 4 0 0 0 0 -2 0 2 2 3 5 8
0 0 0 0 -2 0 2 -2 5 21 24 C 0 0 0 -2 0 -2 0 3 3 6 0 0 0 0 -2 0 -2 2 3 36 38 G 0 0 0 -2 0 -2 -2 1 14 15 0 0 0 0 -2 2 0 0 1 8 8 0 0 0 0 -2 2 0 2 2 14 16 0 0 0 0 -2 2 0 -2 3 19 20 0 0 0 0 -2 2 2 0 1 22 23 C 0 0 0 -2 2 2 2 4 22 24 0 0 0 0 -2 2 2 -2 5 50 54 0 0 0 0 -2 2 -2 0 0 17 17 0 0 0 0 -2 2 -2 2 4 41 44 0 0 0 0 -2 2 -2 -2 1 23 24 ,.. 0 0 0 -2 -2 0 0 4 2 6 I.,
0 0 0 0 -2 -2 0 2 6 34 39 0 0 0 0 -2 -2 0 -2 0 14 14 .... 0 0 0 0 -2 -2 2 0 5 16 21 " 0 0 0 0 -2 -2 2 2 3 42 45 0 0 0 0 -2 -2 2 -2 4 35 37 0 0 0 0 -2 -2 -2 0 0 10 10 0 0 0 0 -2 -2 -2 2 1 88 88 a 0 0 0 .-2 -2. -2 -;2 3 16 18
148
(2) Correlation Coefficients= Standard Values
and All Means= Standard Values
S.D. AWAY FROM STANDARD PROCESS DI SP ER SION MEAN TOTAL
VARIANCE CENTPAL TENDENCY PERCENT PERCENT PERCENT
( D ) (C) (B) ( A ) (0) (C) ( B) ( A ) 0 .o .c. o.o.c. o.o.c. 0 0 0 0 0 0 0 0 1 0 1 0 0 0 2 0 0 0 0 8 1 9 0 0 0 -2 0 0 0 0 8 0 8 0 0 2 0 0 0 0 0 10 0 10 0 0 2 2 0 0 0 0 13 0 13 0 0 2 -2 0 0 0 0 26 0 26 0 0 -2 0 0 0 0 0 14 0 14 0 0 -2 2 0 0 0 0 37 0 31 0 0 -2 -2 0 0 0 0 21 0 21 0 2 0 0 0 0 0 0 3 1 4 ..... 0 2 0 2 0 0 0 0 13 0 13 \0 0 2 0 -2 0 0 0 0 19 1 20 0 2 2 0 0 0 0 0 17 0 17 0 2 2 2 0 0 0 0 19 2 21 0 2 2 -2 0 0 0 0 37 0 37 r) 2 -2 0 0 0 0 0 19 0 19 0 2 -2 2 0 0 0 0 30 1 31 0 2 -2 -2 0 0 0 0 36 0 36 0 -2 0 0 0 0 0 0 7 1 8 0 -2 0 2 0 0 0 0 18 0 18 0 -2 0 -2 0 0 0 0 11 0 11 0 -2 2 0 0 0 0 0 13 0 13 0 -2 2 2 0 0 0 0 21 0 21 0 -2 2 -2 0 0 0 0 31 0 31 0 -2 -2 0 0 0 0 0 30 0 30 0 -2 -2 2 0 0 0 0 46 0 46
0 -2 -2 -2 0 0 0 0 23 0 23 2 0 0 0 0 0 0 0 9 0 9 2 0 0 2 0 0 0 0 8 0 8 2 0 0 -2 0 0 0 0 21 l 22 2 0 2 0 0 0 0 0 17 0 17 2 0 2 2 0 0 0 0 22 1 23 2 0 2 -2 0 0 0 0 56 0 56 2 0 -2 ,.. 0 0 0 0 32 0 32 V
2 0 -2 2 0 0 0 0 45 0 45 2 a -2 -2 0 C 0 0 30 0 30 2 2 0 0 0 0 0 0 9 0 9 2 2 0 2 0 0 0 0 13 1 14 2 2 0 -2 0 0 0 0 34 0 34 2 2 2 0 0 0 C 0 21 0 21 2 2 2 2 0 0 0 0 25 0 25 2 2 2 -2 0 0 0 0 46 1 46 2 2 -2 0 0 0 0 0 35 0 35 ...... 2 2 -2 2 0 0 0 0 44 0 44 u,
0 2 2 -2 -2 0 0 0 0 50 l 51 2 -2 0 0 0 0 0 0 10 0 10 2 -2 0 2 0 0 0 0 16 0 16 2 -2 0 -2 0 0 0 0 26 0 26 2 -2 2 C 0 0 0 0 22 l 23 2 -2 2 2 0 0 0 0 21 0 21 2 -2 2 -2 0 0 0 0 49 0 49 '2. -2 -2 0 0 0 0 0 30 0 30 2 -2 -2 2 0 0 0 0 53 0 53 2 -2 -2 -2 0 0 0 0 37 0 37
-2 0 0 0 0 0 0 0 6 0 6 -2 0 0 2 0 0 0 0 26 0 26 -2 0 0 -2 0 0 0 0 21 0 21 -2 0 2 0 0 0 0 0 18 0 18
-2 0 2 2 0 0 0 0 27 0 21 -2 0 2 -2 0 0 0 0 44 0 44 -2 0 -2 0 0 0 0 0 33 0 33 -2 0 -2 2 0 0 0 0 60 0 60 -2 0 -2 -2 0 0 0 0 36 0 36 -2 2 0 0 0 0 0 0 19 0 19 -2 2 0 2 0 0 0 0 38 0 38 -2 2 a -2 0 0 0 0 30 0 30 -2 2 2 0 0 0 0 0 23 0 23 -2 2 2 2 0 0 0 0 33 0 33 -2 2 2 -2 0 0 0 0 51 0 51 -2 2 -2 0 0 0 0 0 38 0 38 -2 2 -2 2 0 0 0 0 56 0 56 -2 2 -2 -2 0 0 0 0 43 0 43 -2 -2 0 0 0 0 0 0 19 0 19 -2 -2 0 2 0 0 0 0 41 0 41 .... -2 -2 0 -2 0 0 0 0 25 0 25 U'I .... -2 -2 2 0 0 0 0 0 38 0 38 -2 -2 2 2 0 0 0 0 44 1 44 -2 -2 2 -2 0 0 0 0 50 0 50 -2 -2 -2 0 0 0 0 0 49 0 49 -2 -2 -2 2 0 0 0 0 65 0 65 -2 -2 -2 -2 0 0 ,0 0 38 0 38
QUALITY CONTROL OPERATING PROCEDURES FOR
MULTIPLE QUALITY CHARACTERISTICS AND WEIGHTING FACTORS
by
Peter S. Hsing
(ABSTRACT)
This research is devoted to the development of a procedure for
manufacturing use that applies to cases involving a manufactured item
having more than one quality characteristic and continuous measurements
of these characteristics. There are two phases in this procedure.
The first phase consists of testing the stability of the manufacturing
process with respect to dispersion and central tendency. Once the
stability of the process has been established, maintain the process
within the established bounds of dispersion and central tendency becomes
important. The second phase consists of monitoring dispersion and
central tendency of the current manufacturing process. During the
monitoring central tendency activity, the user is allowed to assign
different weights to the sample mean value both above and below the
standard value. For both dispersion and central tendency monitoring
activities, a method is provided to identify quality characteristics or
the interaction between quality characteristics as the cause of the
trouble when the sampling indicated a lack of control.
An example of the antidiarrheal tablet manufacturing process was
presented to illustrate how to test the stability of the manufacturing
process and how to monitor the current manufacturing process. Step-by-step
computational procedures and Fortran computer programs were presented.
The cases for dispersion and central tendency deviated from the standard
values were simulated and the results were presented.