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1. An zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Optimal Desi Multivariate Control Charts
in the Presence 0 ultiple Assignable Causes
Joel K. Jolayemi .
Department zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
ofStatistics
Unitxmity of Zbadan,hdan, Nigeria
Julio N. Berrettoni
Case Western Reserve Uniuemity
Chekznd, Ohio 44106
Transmitted by John Casti
ABSTRACT
In quality surveillance, it is often necessary to measure the quality of a product on
more than one characteristic. Also, p-es subjected to only one assignable cause
are rare in practice. Hence there is need for a good technique for designing a control
chart that takes account of all the relevant characteristics and different assignable
causes. We present an economic model for designing a matched single-cause model
from the original multiple-cause model. The results of the final computations show
that, under the model, the multivariate control chart is generally more efficient than
the %
hart.
1. INTRODUCTION
In process quality control, two of the problems that face quality control
engineers are:
(i) that instead of a single quality characteristic, the quality of an indus-
trial process may depend on more than one variable characteristic, and
(ii) that instead of a single assignable cause, there may be multiple
assignable causes for an outaf-control situation. Chiu and Wetherill [2] noted
that processes with only one single assignable cause are not common. zyxwvutsrqponmlk
APPLZED MATHEMATZCSAND COMPUTATZON32: 17-33 (1989)
0 Elsevier Science Publishing
Co., Inc., 1989
655 Avenue of the Americas, New York,NY 10010
17
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2. 18 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
JOEL K. JOMYEMI AND JULIO N. BERRETTONI zyxwvutsrqp
In (i) above the quality characteristics zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
usua lly come as related variables.
Under this-condition, it could be both inefficient and ineffective to operate a
separate X-chart for each variable. Consequent to this, the multivariate
quality-controltechniques has been introduced. The application of this tech-
nique in process control involves the design of multivariate control charts
(MCC).
Duncan [3] developed an economic model for the design of x-charts when
the condition in (ii) prevails. So far, no similar economic model has been
developed for MCCs. However, a few authors, such as Heikes et al. [5],
Jolayemi and Berrettoni [I, Montgomery and Klatt [8, 91, and Yeung [ll],
have developed economic models for designing MCCs when there is a single
assignable cause.
In this paper, we shall develop a fully economic model for designing
MCCs in the presence of multiple assignable causes. The development of the
model will be based on [3], [7J,and 121.Using the model, the sample size, the
sampling interval, and the Type 1 error that minimize the total cost of
operating a MCC when condition (ii) prevails can be obtained.
A classical optimization procedure and algorithm developed by Goel and
Klu [4] and modified by Jolayemi and Berrettoni will be applied for deteimin-
ing the optimal design parameters that minimize the total operating costs.
Our model will be capable of producing an MCC involving any number of
variable characteristics. We shall also compare the performance of MCC and
&hart under the model.
2. ASSUMPTIONS AND OPERATING CONDITIONS
We have based the information of our model on the following as~ump-
tions:
(i) The process is characterized by o quality characteristics XT=
(X,, X,,..., X,) which are multivariate normal.
(ii) The covariance matrix 2 and the mean vector p, of the quality
characteristics are known. We assume they can be either derived from a large
number of past data or selected by management to attain a certain objective.
(iii) The sample elements of each quality characteristic are not serially
correlated.
(iv) The process is characterized by an in-control state p,, and when an
assignable cause A, OL’CU~S,
it takes the form of a shift from ~~ to cl0+ si
where

3. Multioariate
ControlCharts zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE
19
is the shift in the process corresponding to the assignable cause j, and ai is a
known standard deviation corresponding to the ith variable.
(v) The process is monitored by a multivariate control chart. Samples of
size rt are taken every s hours, and the quantity @(jr- - l@-r(r - po)
plotted. If a plotted point falls within the lower control limit zero and the
Upper control limit J&, the process is in control and its operation is allowed
to continue. Otherwise the process is shut down and a search for trouble is
made.
(vi) Once an assignable cause Ai occurs, it continues to affect the process
ttntil it is detected by the control chart, and during this period it is
ttn&sturbed by the occurrence of other assignable causes.
(vii) During the search for an assignable cause, the process is stopped
fwm operating, and after an adjustment the process starts afresh from a state
of control.
(viii) The delay in takin g the samples and plotting the points is propor-
tional to the sample size.
3. MODEL FORMULATION
3.2, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Notation and Symbols
The following notation is used in formulating the economic models:
Aj A the assignable causes j,
Bj -the average time the process operates after the occurrence of the
assignable cause A i, given by
S
*i
=--
‘i
7j+en+~j,
where
l-(1+ AjS)e-xjs S XjS2
W-P
“” Xj(l -e-xis) + 2 12
I
plus terms of order X’S4 or higher,
b *the fixed cost of sampling and plotting,
C *the Trariable cost per item of sampling, testing, and plotting,
Di *the average time taken to find an assignable cause Aj after it has
caused the process to go out of control,

4. 20 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
JOEL K. JOLAYEMIAND JULIO N. BERRE‘ITONI
e =the rate of increase of the time between taking a sample and plotting a
point on the MCC, with reSpect to the sample size,
Mj =the net loss per hour owing to a greater percentage of unacceptable
items brought about by cause Aj,
Wj =the average cost of finding the assignable cause Aj when it has
occurred, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
T =the average cost of looking for an assignable cause when none exists,
Xj =the average rate of occurrence of assignable cause j per unit of
operating time,
S = sampling interval in hours.
3.2. The Model
If a is the probability of Type 1 error when the process is out of control,
then in our multivariate case, aris given by
1
a=p(X~~~~,“)=Z”/pri~)/m~(v’2)-b-xdx,
XC”
(3.2.1)
where x’f;is a &i-square random variable with u degrees of freedom and x f, v
is the upper control limit of an MCC.
When an assignable cause Aj occurs, the process mean shifts in accor-
dance with the fourth assumption. Let the probability that the shift is
detected be Pi* In our multivariate case, Pj follows a noncentral chi-square
distribution given by
Pj=exp 4
/ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ
0/2+i-1
0/2+i-1
i=. i!2(v’2+2)‘I(u/2+ i) x2,,, ’
dx. (3.2.2)
Its noncentrality parameter hj is given by Aj = zyxwvutsrqponmlkjihgfedcbaZYXWV
nl$X- ‘8j.
Using (3.2.1) and (3.2.2) above together with the expressions and notation
defined in Section 3.1, and following Duncan [3], the total expected loss cost
per hour of operation is given by
c=
CAjBj + AAT +cXjWj b cn zyxwvutsrqponmlkjihgfedcbaZ
(l+CAj~j) + S + ~~
(3.2.3)

5. Mu&variate Control Chmts
where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
21
The only method by which the minimum value of C, as given in (3.2.3)
above, can be detErmined is by direct search. However, in optimizing a
similar model for X-charts, Chiu and Wetherill [2] realized that it is very
tedious to obtain the minimum value of C using the direct search method. A
way out of this is to obtain a matched single-cause model from the original
model. It has been adequately demonstrated by Chiu and Wetherill [2] and
Duncan [3] that the matched singlecause model approximates the original
model very well. In view of their results, we shall develop and optimize a
matched singlecause model that well approximates (3.2.3).
4. THE MATCHING AND SOLUTION PROCEDURES
This section will be divided into two. The first subsection presents the
matching procedure for our multivariate case, and the second develops the
solution.
4J. The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Matching Procedure
To simplify the matching process, the parameter to be matched can be
laid out as shown in Table 1. Based on this layout the matching of the
parameters is as follows:
(i) We have zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
TABLE 1
Assignable
Causes Xl Xg l . l X, l ** X, Ai Mi Wi Di

6. 22 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
JOEL K. JOLAYEMIAND JULIO N. BERRE’ITONI zyxwvutsrqpo
For example, the weighted shift in the first qu&ty characteristic due to the
assignable cause A I, A,, . . . , A, is given by
(ii) Following (i), we obtain weighted shifts for r = 2,3,. . . , o to obtain the
weighted mean vector
(iii) Using (ii), the matched noncentrality parameter is given by
A, = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON
nS,Z-ls,. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ
(4.1.1)
(iv) Using (iii), the weighted probability of detecting a shift is given by
Ait
J
00
(u/2+2)ir(u/2+i) x2 X
0’2+i-1 dx. (4.1.2)
a.u
(V) Mt = ~~=lhjMj/CXj.
(vi) wt=Ef=1X
jwj/Zf= lx j’
(vii) Xt=~~=,Xj.
(~) it = ~~=lDix j/CX j’
Matching formulae (v) to (viii) are due to Duncan [3].
Substituting the matched parameter into (3.2.3), the matched singlecause
model becomes
X,M,B, + &AT + h,W , b + cn
ct =
l+ X,B,
+-
s ’
where
(4.1.3)
Bt = and
s x,s2
7t= - - -
2 12 l

7. Multivariate Control Charts 23
By putting zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
we have that
ct = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
X,M,+aT/s+h,W , b+cn zyxwvutsrqponmlkjihgfedcbaZY
1+x,
+-
s l
4.2. Solution Procedure and ComputationalAlgorithm
Since (4.1.5) reduces to a singkause model, the same
(4.1.4)
(4.1.5)
solution procedure
presented for a similar model in [7] will be applied to obtain the sample size
I&,the sampling interval s, and the upper control limit that make the total
operating cost a minimum. For ease of reference, we have included this
solution procedure and computational algorithm in Appendix A.
5. COMPUTATIONAL RESULTS, COMPARISONS,
AND CONCLUSIONS
Detailed discussions of the results of comparisons of the performances of
MCC and X-chart under a single-cause model have been given in [7] with
many examples. Our present model has been reduced to a singlecause model
through the matching procedure. We have limited ourselves to few examples
here, since the present matched singkause model is similar to the model
discussed in [7].
Following the procedures and computational algorithms presented in the
appendices, we have written a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI
FORTRAN program with IMSL’s MDCHI and
TABLE 2
THEINPUTP AMMETERS CORRESPONDING TO EACH OF
THE ASSIGNABLE CAUSES IN RUN 1
Al 0.4 0.5 .003 275 40 3.5
A, 2.5 2.0 .002 100 35 1.0
A, 1.5 1.5 .006 80 40 2.0
A, 3.0 3.0 a05 200 25 0.55
A, 1.8 1.8 .004 120 33 1.5
A, 0.55 0.55 .007 140 28 3.0

8. IBIG TO EACH OF
ASSIGl’UBLE CAUSES
i
8li
8
2j ‘j i zyxwvutsrqponmlkjihgfedcbaZYXWV
1 48 5
2 35 63 .2
3 27 2.8
4 20 1.75
le 5 show that in ge

9. TABLE 5
INPWAND zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
OPTIMUL PAR.AXE%RS FOR THE DESIGN OF A MULTIVARIATE CONTROL CHABT zyxwvutsrqponmlkjihgfedcbaZYXWV
un
Shifts
no. 6It t&t r A, Mt e Dt T Wt b c n* S* Xf.*,- a A” e* L”
1 1.502 1.502 0.5 .027 135.741 .05 2.01 .50 31.593 0.50 10 4 3.939 5.1 -0797 3209.57
0.5 .027 135.741 .05 2.0093 50 31.593 0.50 PO 2 2.6757 5.0656 .07935 2675.003
- 0.5 .027 :35.471 .05 2.0093 50 31.5926 0.50 10 1 1.79928 6.03612 .048989 2216.826
2 .85 1.5 0.0 .02 32.!% .8 8.824 35 60.786 1.5 12 3 10.6235 3.582 .16676 8.91’;s 9048 1334.841
3 .65 1.8 0.0 .0182 82.83 .l 1.54 35 84.67 0.8 5 3 10.6235 3.58 .16676 8.9175 9041 1334.858
TABLE 6
INPUT AND OPTIMAL “ARAMETERS FOR THE DESIGN OF AN &HART
1 : 92 .O27 135.741 .Q5 2.01 50 31.5926 .5 10 6 4.925 1.911 .056 3561.816
2a 0.85 .02 3x% .8 3.824 35 60.786999 .8 1.5 1 6.8413 0.856 .5572 .60372 1295.342
b 1.5 .OA SR.s4 .8 3.824 3,5 60.786 .8 1.5 5 14.136 1.592 .1113843 .961 1558.572
3 1.8 .OP82 s2.m .l 1.5478 35 84.67 .8 5 3 3.4305 1.952 .05336 .88213 1513.398

11. 27
u_ [z(o+A)-i]h-E(y)
__
l/"(Y)
and x is an zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
N(0, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
1) random variable. Subgtuting X, and At for X and A
respectively, we have that
-r2
Pr= medx.
I a-
*t ‘CT
t
W)
Other parameters in (A.2) and (A.3) are defined as‘ zyxwvutsrqponmlkjihgfedcbaZYXWVU
follows:
-h(h-1)(2-h)(b3h)(v+2h)
e(u+n)_’
2 , (A.6)
v(y)=
2h2(v +2A)
(v-t A)”
i-
(1- h)(l- 3h)(v +2A)
~(v+A)~
9 - (A 7)
.
/r = 1- ;( v + &‘I)(v +311)(0 +2A) -2, (A@
*I, = n6;9?6,, (A-9)
‘:‘= (S~~a~;S~~u~,*0~,Gi~ui,~~~,6,~u~)* (A.lO)
The remarkable accuracy of this approximation has been attested to by Alt
[ 11 and Johnson and Kotz [6].
APPENIDX B. MINIMIZATION OF C
We have
,C zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED
X,M,B, + &AT + X,w,
=
r
l+ X;B,
, uw
, .

12. 28
where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
JOEL K. JOLAYEMI AND JULIO N. BERREITONI
s AS2
7t
E--P zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK
2 12 l
On putting
and noting that
X,AT=X,
(B.l) becomes
X,M,+ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM
aT/S + AtWt zyxwvutsrqponmlkjihgfedcbaZYXWVUT
b + cn
ct=
1+x,
+-
s l
(B2)
.
@3)
.
(B-5)
Differentiating
(B.5) partiallywith respect to 2 gives
act Mt
az zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
-=
5(l+Xt)+;$(l+Xt)-(XtMt+$+At14$)~
az (I+ xt)”
.
After equating act /ik to zero and simplifying,we have that
aT ’ ax,
1
T aa
S+X,W,-Mt ~=O+xtJ
~~* @*7)

13. Multivariate Control Charts zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC
The partial derivatives of C with respect to s gives
29 zyxwvutsrq
b+cn
+-
s2 l
(B-8)
Equating to zero and simplifying gives
aT
Ad,----X,M,
S
$++x,)$+ (b+m)s!l+xg)2. (B-9)
Differentiating X, in (B.4) with respect to z g&es
ax, 1 W,
-= -pzaz x,s.
az ( 1
t
(B.10)
Using Leibniz’s method, the derivatives of P with respect to z [where P is
given by the approximation in (A.511is given by
a?t
z=
--(V+h~)-1h[z(v+AI)-1]h-1e-u2
_~l. (B 11)
=
ma-
.
From (3.2.1), zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
i?a 1
aZ.=- 2”‘2r( v/2)
z(o/2)-l~-z = - e2_
Substituting (B.ll) into (B.lO) gives
(B.12)
(B.13)
Equating the values of 8X,/k from (B.7) and (B.13) and substituting for

14. 30 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
JOEL K. JOLAYEMIAND JULIO N. BERRETTONI
ih,Gr from (B.12), we have
- P+ m-+2 wh
=- zyxwvutsrqponmlkjihgfedcbaZYXWV
S(aT/s+XtW t- M,) F: ’
(B.14)
Differentiating X, with respect to S in (B.4), equating the derivatives to
those obtained in (B.9), and simplifying, we have
x (l+X,)[aT+(l+X,)(b+cn)]
(M, - AW ,)S2 - aTS
. (B.15)
Rearranging (B.14), we have
1+x, zyxwvutsrqponmlkjihgfedcbaZYXWVU
( Mt - A;W ,)S2 - aTS l
(B.16)
Two quadratic equations can be obtained from (B.15) and (B.16). The first is
obtained by substituting the vaiue of (i + X,)/[(M, - Xiwt )S2 - CYTS]
from
(B.16) and of X, from (B.4) into (B.15) and simplifying:
qys’c( ;-;)(b+cn)At-yq
z)s
+(A,en)+A,D~+l(b+cn)-
(;-i)(z)
lYF:+ crT= 0. (B.17)
The second quadratic equation is obtained by substituting the value of X,
from (B.4) into (B.16) and simplifying:
h(Mt - woh
-
4T
G2
4 pt2
-X,T zyxwvutsrqponmlkjihgfedcbaZY
1
S
+ [ - T&en + AtO, + l)e2] = 0.
Eq-uations (B.17) and (B.18) can be written as
PltS2 + B2tS + P3t = 0
(B.18)
(B.19)

15. Multivariate Control Charts 31 zyxwvutsrq
and
where
(B.21)
TV2 $2
(b+cn)X,---
6 QrY
(B.22)
/?3t=(Xtm+X,Dt+l)(b+cn)-
+ aTp (B*23)
X,(M, - &wt)+, x2,T+2, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
Y1t =
P? -- 12 ’
wo
&W,
Y2t =
P2
(B.25) zyxwvutsrq
Y3t = - T(Xp + X,D+l)+2. (~.26)
The coefficients pt and yt are functions of n and the z’s only. We can
multiply Equation (B.19) by ylt and (B.20) by fi3t, solve simultaneously for S,
and simplify to obtain
s B3tYlt
- PltY3t
= &tYzt - P2th m
(B.27)
By multiplying (B.19) by y3t and (B.20) by /?3t, solving simultaneously for S,
and simpfying, we have
s fl2tY3t - b3tY2t
= PztY1t
- PltY3tl
(B.28)

16. 32 JOEL K. JOMYEMI AND JULIO N. BERRE’ITONI
Equating zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
(B.27) and (B.28) and simplifying, we have
APPENDIX C. COMPUTATIONAL ALGOIWHM
This algorithm determines IL, z, and s that satisfy equation (B.29). The
following are the steps:
step
0:
step1:
step 2:
step 3:
step 4:
steg 5:
St&p 6: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
De fine a suitable interval for fl. That is, let n take integer values
from 1 to Q where Q has been suitably chosen. Also, define a
suitable range for 2, and partition the range of 2 into iV points
(possibly N >,600) whose values are multiples of 0.01. Put i = 1 and
go to step 1.
For n = ni, substitute values of z obtained in step 0 into Equation
(3.2.1) to obtain the corresponding values of a. Substitute the
values of a and z into Equation (B.29), and obtain values of 2 that
satisfy f(n, z) = 0 as closely as desired.
Calculate S for each of the Zvalues, using Equation (B.27).
Using only the real positive Svalues from step 2 (say sl, s2,. . . , Si)
and the corresponding Zvalues, (say zr, z2,. . . , Zi), obtain the local
minimum lo!%
cost Cm:= minC( n,, Sj, Zi):=l:
Check the status of C< as a minimum by comparing values of C,.,:
at (n,, zi* + i) for some i a multiple of 0.001, where Zi* is the Zj
associated with CR*.
If ni = Q, go to step 6. Otherwise put i = i + 1 and go to step 1.
Determine the overall minimum loss cost C*, where C* is given by
REFERENCES
1 F. B. Alt, Multivariate economic control charts for the man, presented at Seventh
Annual Meeting, North East Regional Conference, American Inst. for Decision
Sciences, Washington, l-2 June 1978.
2 W K. Chiu and A. B. Wetherill, A simplified scheme for the economic design of
&harts, I. @u&y Technd. 6(2):63-69 (1974).

17. Multivariate Control Charts 33
3
4
5
6
7
8
9
10
11
A. J. Duncan, The economic design of kzharts when there is a multiplicity of
assignable causes, I. Amer. Statist. Assoc. 51(33): 107-121 (1971).
A. L. Goel and S. M. Klu, An algorithm for the determination of the economic
design of %hart based on Duncan’s model, 1. Amer. Statist. Assoc.
63(321):304-320 (1968).
R. G. Heikes, G. C. Montgomery, and J. Y. Yeung, Alternative process models in
the economic design of T’$quared control charts, AIIE Tram. 655-61 (1974).
Johnson and Katz, Continuous Univariate Distribution-& Houghton Miffin,
Boston, 1970.
J. K. Jolayemi and J. N. Berrettoni, Multivariate control charts: An optimization
approach to effective use and measure of performance, Appl. Math. Cornput.
D. C. Montgomery and P. J. Klatt, Economic design of T-square control charts to
main control of a process, Management Sci. 19:76-89 (1972).
D. C. Montgomery and P. J. Klatt, Minimum cost multivariate quality control
tests, AIIE Trans. 4(2):103- 110 (1972).
M. Sankaran, Approximation to the non-central chi-square distribution,
Biometiku 50:119-209 (1963).
J. Y. H. Yeung, ,Akemative Process Models for the Economic Design of T2
Control Charts, M.S. Thesis, Georgia Inst. of Technology, Atlanta, 1972.