A model for the statistical design of multivariate control charts with multiple control regions

Applied Mathematics and Computation 109 (2000) 73±91 www.elsevier.nl/locate/amc

A model for the statistical design of multivariate control charts with multiple control regions Joel K. Jolayemi Department of Mathematical Sciences, University of Zululand, Private Bag X1001, Kwadlangezwa, 3886, KwaZulu-Natal, South Africa

Abstract This paper develops a statistical model for the design of multivariate control charts with multiple control regions (MCCMCR). The model produces the sample size and values of the control limits needed for the operations of MCCMCR. It allows the consumers and the producers to specify desired values for the risks and power functions that have the greatest e€ects on the eciency of the chart. Numerical examples are given to illustrate the model and to study the properties of MCCMCR. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Multivariate control charts; Sample size; Control-limit constants; Assignable causes; Computational algorithm

1. Introduction When a process is a€ected by more than one assignable cause that lead to di€erent out-of-control states of the process and di€erent restoration procedures, neither the traditional X -chart nor the ordinary multivariate control chart (see Refs. [1±6]) is adequate. X -charts with multiple control regions ± or multiple-regions x-charts (see Refs. [7,9]) ± is more appropriate when product quality is measured on a single quality characteristic. It is much more appropriate to use a multivariate control chart with multiple control regions (MCCMCR) when product quality is measured on many quality characteristics (x1 ; x2 ; . . . ; xv ). A region of MCCMCR represents an area where a process shift caused by the occurrence of a particular assignable cause can be caught. That is, every 0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 0 2 4 - 7

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J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

region of an MCCMCR is associated with a single assignable cause and a required level of adjustment or restoration of the process. The ordinary MCC does not have this special feature and consequently, unlike MCCMCR, it is dicult to identify the particular assignable cause that causes a shift. Hence, unlike with MCCMCR, a timely application of an appropriate corrective action may not be possible with ordinary MCC. Despite the great need for the use of control charts with multiple control regions in industries, published research works in the area of its economic and statistical designs are scanty. We have only come across very few published models in this area and the few are for the design of multiple-regions x-charts (cf. Refs. [9,7]). We have not come across any model ± statistical or economic ± that is developed for the design of MCCMCR. In this research, we shall develop a statistical model for the design of MCCMCR. The basic works in the development of the model will ®rst be based on MCC with two control regions before it is extended to cover MCC with three control regions. Numerical examples will be given to illustrate the model and to study the properties of MCCMCR. 2. MCC with three control regions 2.1. The operations of the chart MCC with three control regions is for monitoring processes that are subject to the occurrence of two assignable causes, A1 and A2 , say (Fig. 1). The chart has one in-control region denoted region 0 and two out-of-control regions, denoted regions 1 and 2. Region 1 is expected to catch process shifts caused by the occurrence of A1 while region 2 is for catching those caused by the occurrence of A2 . Thus, the chart discriminates between two assignable causes. To operate the chart, samples of size n are taken at regular intervals of time  ÿ l † plotted. If the plotted point falls in  ÿ l †T Rÿ1 …X and the quantity n…X 0 0 region 0, the process is in control and production continues. If the plotted point falls in region 1, the process is said to have shifted from state 0 to state 1 due to the occurrence of assignable cause A1 (the ®rst assignable cause). If the point falls in region 2, then the process has shifted from state 0 or 1 to state 2 due to the occurrence of assignable cause A2 (second assignable cause). When a process shifts from a lower state to a higher state, it will never move back to the lower state unless a restoration procedure is undertaken. There are two levels of restoration procedures. The ®rst level restores the process from state 1 to state 0. The second level is a more costly restoration procedure that restores the process from state 2 to state 1. An error occurs in

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

75

Fig. 1. MCC with three control regions.

the operation of the chart when the jth level of restoration procedure is triggered when, in fact, the actual state of the process is i; i; j ˆ 0; 1; . . . ; s …s P 2†: A major advantage of the chart is that its operation does not involve any tedious search for an assignable cause since every assignable cause that occurs is isolated by the region of the chart associated with it. 2.2. Assumptions, de®nitions and model developments 2.2.1. Assumptions The development of the model is based on the following assumptions: (i) that the process is inspected on v quality characteristics X1 ; X2 ; . . . ; Xv , which are multivariate normal. (ii) that the process is characterized by an in-control state l0 ˆ …l01 ; l02 ; . . . ; l0V †; where l0 can be speci®ed by management to satisfy a particular objective or can be obtained from a large body of past data. (iii) that the process has two out-of-control states, designated states 1 and 2. (iv) that if the process is in state 2, the ®rst level of restoration procedure ± a minor corrective action ± will not be able to restore the process back to the in-control state, and the process will remain in state 2. (v) that the second level of corrective action ± a major and more costly restoration ± will always bring the process back to state 0, regardless of whether the process has been in state 1 or 2.

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J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

2.2.2. De®nition of symbols De®ne: aij

a1 a2 aij

di R l0 Xa21 ;v Xa22 ;v 2 Xv;g n

the probability that the jth level of restoration procedure is triggered when, in fact, the actual state of the process is i; i; j ˆ 0; 1; . . . ; s …s P 2†; the type 1 error or producer's risk associated with region 0, the type 1 error, or producer's risk associated with region 1, the shift in the process that triggers the ith level of corrective action with respect to the jth quality characteristic. Here i ˆ 1; 2 and j ˆ 1; 2; . . . ; V ; the vector of shifts that trigger the ith level of corrective action, dTi ˆ …di1 ; di2 ; . . . ; div †; i ˆ 1; 2; the variance±covariance matrix of all the quality characteristics when the process is operating in the in-control state, the vector of means of the process when it is operating in the in-control state lT0 ˆ …l01 ; l02 ; . . . ; l0V †; the upper limit for region 0 (also the lower limit for region 1), the upper limit for region 1 (also the lower limit for region 2), the non-central chi-square random variable with degree of freedom v and non-centrality parameter g, sample size.

2.3. The development of the model (Model 1) To develop the model, we ®rst develop expressions for the di€erent categories of risks and powers associated with the chart as follows: Producer's risks or type I errors. The ®rst among these risks is the probability that the ®rst level of corrective action is triggered by a sample observation when, in fact, the process is in-control. Using the symbol aij de®ned earlier, this error is given by a01 ˆ P …Xv2 > Xa21 ;v j l ˆ l0 † ÿ P …Xv2 > Xa22 ;v j l ˆ l0 †:

…1†

The second is the probability that the second level of corrective action is triggered by the sample observation when, in fact, the process is in-control and this is given by a02 ˆ P …Xv2 > Xa22 ;v j l ˆ l0 †:

…2†

The consumer's risks or type II errors. First, we have the probability that no corrective action is triggered when, in fact, the process is in state 1. This error is given by

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91 2 a10 ˆ P …Xv;g < Xa21 ;v j l ˆ l0 ‡ d1 †; 1

77

…3†

where g1 ˆ dT1 Rÿ1 d1 :

…4†

The second is the probability that no corrective action is triggered when, in fact, the process is out-of-control and is operating in state 2. The error is 2 < Xa21 ;v j l ˆ l0 ‡ d2 †; a20 ˆ P …Xv;g 2

…5†

where g2 ˆ ndT2 Rÿ1 d2 :

…6†

The other types of risks. The probability that the second level of restoration procedure is triggered when, in fact, the ®rst level should have been indicated. This probability is given by 2 > Xa22 ;v j l ˆ l0 ‡ d1 †: a12 ˆ P …Xv;g 1

…7†

The probability that the ®rst level of corrective action is triggered when, in fact, the second level should have been indicated. This is given by 2 > Xa21 ;v j l ˆ l0 ‡ d2 † a21 ˆ P …Xv;g 2 2 ÿ P …Xv;g > Xa22 ;v j l ˆ l0 ‡ d2 †: 2

…8†

Powers of the chart. The power of a chart is the probability that the chart will catch a shift in a process when it occurs. These are powers associated with the two regions (regions 1 and 2) that indicate out-of-control situation. These powers are as follows. The probability that the ®rst level of restoration procedure is rightly indicated when a shift to state 1 occurs ± the power of the chart with respect to region 1 ± is given by 2 > Xa21 ;v j l ˆ l0 ‡ d1 † a11 ˆ P …Xv;g 1 2 > Xa22 ;v j l ˆ l0 ‡ d1 †: ÿ P …Xv;g 1

…9†

The probability that the second level of restoration procedure is triggered when there is a shift in the process to state 2 ± the power of the chart with respect to region 2 ± is given by 2 > Xa22 ;v j l ˆ l0 ‡ d2 †: a22 ˆ P …Xv;g 2 2

…10†

Eqs. (1) and (2) follow the ordinary X -distribution while Eqs. (3), (5), (7)±(10) follow the non-central chi-square distribution. It should be noted that they have been written in probability notations to save space. For example, if the notation for the full non-central chi-square distribution is used in Eq. (3), a10 will be given by

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J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

a10 ˆ exp

1  ÿg X 1

2

jˆ0

g1J …v=2‡2† j!2 C…v=2 ‡ j†

Z

Xa2

1 ;v

ÿ1

X …v=2‡jÿ1† eÿX dx

…11†

X …v=2‡jÿ1† eÿX dx:

…12†

and a12 is given by a12 ˆ exp

1  ÿg X 1

2

jˆ0

g1J j!2…v=2‡2† C…v=2 ‡ j†

Z

1

Xa2

2 ;v

Normal approximation. Using Sankaran's normal approximation to the noncentral chi-square distribution (see Refs. [8,6]), Eqs. (3), (5), (7)±(10) can be written as Z u1 1 2 eÿz =2 dz; …13† a10 ˆ p 2p ÿ1 Z u2 1 2 a20 ˆ p eÿz =2 dz; …14† 2p ÿ1 Z 1 1 2 a12 ˆ p eÿz =2 dz; …15† 2p u3 Z 1 Z 1 1 2 2 a21 ˆ p eÿz =2 dz ÿ eÿz =2 dz; …16† 2p u2 u4 Z 1 Z 1 1 2 ÿz2 =2 e dz ÿ eÿz =2 dz; …17† a11 ˆ p 2p u1 u3 Z 1 1 2 a22 ˆ p eÿz =2 dz; …18† 2p u4 where u1 ˆ ‰…Z1 …v ‡ g1 †ÿ1 †h1 ÿ E…y1 †Š=

p V …y1 †;

…19†

ÿ2

E…y1 † ˆ 1 ‡ h1 …h1 ÿ 1†…v ‡ 2g1 †…v ‡ g1 †

ÿ h1 …h1 ÿ 1†…1 ÿ 3h1 †…v ‡ 2g1 † V …y1 † ˆ

2h21 …v ‡ 2g1 † …v ‡ g1 †

2

" 1ÿ

2

…v ‡ g1 † 2

ÿ4

;

…1 ÿ h1 †…1 ÿ 3h1 †…v ‡ 2g1 † …v ‡ g1 †

2 ÿ2 h1 ˆ 1 ÿ …v ‡ g1 †…v ‡ 3g1 †…v ‡ 2g1 † ; 3

2

…20† # ;

…21† …22†

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

Z1 ˆ Xa21 ;v ; ÿ1 h

u2 ˆ ‰…Z1 …v ‡ g2 † † 2 ÿ E…y2 †Š=

79

p V …y2 †;

…23†

ÿ2

E…y2 † ˆ 1 ‡ h2 …h2 ÿ 1†…v ‡ 2g2 †…v ‡ g2 †

…v ‡ g2 † 2 " # 2h2 …v ‡ 2g2 † …1 ÿ h2 †…1 ÿ 3h2 †…v ‡ 2g2 † V …y2 † ˆ 2 1ÿ ; 2 …v ‡ g2 † …v ‡ g2 †2 ÿ h2 …h2 ÿ 1†…2 ÿ h2 †…1 ÿ 3h2 †…v ‡ 2g2 †2

2 ÿ2 h2 ˆ 1 ÿ …v ‡ g2 †…v ‡ 3g2 †…v ‡ 2g2 † ; 3 p ÿ1 h u3 ˆ ‰…Z2 …v ‡ g1 † † 1 ÿ E…y1 †Š= V …y1 †; p ÿ1 h u4 ˆ ‰…Z2 …v ‡ g2 † † 2 ÿ E…y2 †Š= V …y2 †

ÿ4

;

…24† …25† …26† …27† …28†

and Z2 ˆ Xa22 ;v : 2.4. Identi®cation of critical risks and power functions We have identi®ed a02 ; a10 and a22 as the most critical risks and power function for which suitable values should be speci®ed by a producer and, if need be, a consumer whenever an MCCMCR is to be designed. They are important for the following reasons: a02 represents the probability that the second level of corrective action is triggered by a false alarm. Since the second level involves a major and more costly intervention ± which is designed to restore the process from state 2 back to state 0 ± a producer will like to specify a suitable value for a02 to reduce the total cost of operating an MCCMCR. A consumer will not like to tolerate a high level of defects in the products purchased ± however minor the type of defect: He will therefore like a small value to be speci®ed for a10 . The larger the value of a22 the smaller the consumer's risks a20 and a21 . Therefore, specifying a value for a22 is equivalent to specifying values for a20 and a21 . A producer and a consumer would like to specify a suitable value for a22 . 2.5. The model Let the speci®ed value of a02 be a02 . On substituting this value for a02 in (2), we have

80

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91 2 a02 ˆ P …Xv2 > Xa;v j l ˆ l0 † ˆ a02 ; Z 1 1 X v=2ÿ1 eÿX dx ˆ a02 ; ) v=2 2 C…v=2† Xa2 ;v

…29† …30†

2

) a2 ˆ a02 :

…31†

Hence, Z2 ˆ Xa22 ;v ˆ Xa20 ;v 2

(Note that Xa20 is the calculated value of Z2 or Xa22 ;v ). 2 Substituting the speci®ed value b10 for a10 in Eq. (13), we have Z u1 1 2 p eÿz =2 dz ˆ b10 : 2p ÿ1 This gives u1 ˆ z0b ;

…32†

…33†

…34†

0

where b ˆ b10 and zb0 < 0: Substituting p22 for a22 in Eq. (18), we have Z 1 1 2 p eÿz =2 dz ˆ p22 : 2p u4 This gives u4 ˆ ÿz0p , or p ÿ1 h ‰…Z2 …v ‡ g2 † † 2 ÿ E…y2 †Š V …y2 † ˆ ÿzp0 ;

…35†

…36†

0

where p ˆ p22 . 2.6. Solution method To solve Eqs. (34) and (36) simultaneously for the values of the second control limit z1 ˆ Xa21 ;v and the sample size n, we ®rst solve Eq. (34) for z1 in terms of other input parameters. Substituting for u1 from Eq. (19) into Eq. (34), we have ‰…Z1 …v ‡ g1 †ÿ1 †h1 ÿ E…y1 †Š=V …y1 † ˆ Zb0 ; p ÿ1 h ) ‰…Z1 …v ‡ g1 † † 1 ÿ E…y1 †Š ˆ Zb0 V …y1 †; p ÿ1 h ) …Z1 …v ‡ g1 † † 1 ˆ E…y1 † ‡ Zb0 V …y1 †;  p ) Z h1 ˆ E…y1 † ‡ Zb0 V …y1 † …v ‡ g1 †h1 ;  p1=h1 …v ‡ g1 †: ) Z1 ˆ E…y1 † ‡ Zb0 V …y1 †

…37†

…38†

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

81

Next, we use the following computational procedure to obtain the values of other design parameters, risks and powers of the chart. Step 0: Obtain the values of Zb0 , Zp0 and Xa22 ;v ˆ …Z2 † from the tables of chisquare and normal distributions. Put n ˆ 1 and go to step 1. Step 1: Use this value of n to compute the values of g2 , h2 ; V …y2 † and E…y2 † from Eqs. (6), (26), (25) and (24), respectively. Substitute these values into Eq. (36) to obtain the value of u4 and go to step 2. Step 2: If u4 ˆ ÿZp0 ; put g2 ˆ g2 ; h2 ˆ h2 ; V …y2 † ˆ V …y2 †; E…y2 † ˆ E…y2 †; u4 ˆ  u4 ; n ˆ n and go to step 3. Otherwise put n ˆ n ‡ 1 and go to step 1. Step 3: Use the value of n to obtain the value of g1 ; h1 ; V …y1 † and E…y1 †, respectively, from Eqs. (4), (22), (21) and (20). Use these values together with the value of Zb0 to compute the value of Z1 in Eq. (38). Denote all the computed values by g1 ; h1 ; V …y1 †; E…y1 † and Z1 , respectively, and go to step 4. Step 4: Substitute g2 ; h2 ; E…y2 †; V …y2 † and Z1 into Eq. (23) to obtain the value of u2 ; and substitute g1 ; h1 ; E…y1 †; V …y1 † and Z2 into Eq. (27) to obtain the value of u3 . Denote u2 by u2 and u3 by u3 and go to step 5. Step 5: Noting that u01 ˆ Zb0 , use Eqs. (13)±(18) and the table of normal distributions to calculate the values of a10 ; a20 ; a12 ; a21 ; a11 and a22 , respectively. Note that the computed values of the second control limit Xa21 ;v , the third control limit Xa2 ;v and the sample size n are Z1 ; Z2 and n , respectively. 3. Extension to MCC with four control regions The MCC with four control regions discriminates among three assignable causes, say A1 ; A2 and A3 . The model developed in Section 2 will be extended to obtain a new model for its design. In doing this, the same de®nitions and procedures used in the section will be used to derive expressions for the value of the various risks and power functions associated with the operations of the chart (see Fig. 2). 3.1. Risks and power functions The expressions for the various risks and power functions are as follows: The producer's risks or type I errors. The producer's risks or type I errors are as follows: a01 ˆ P …Xv2 > Xa21 ;v j l ˆ l0 † ÿ P …Xv2 > Xa22 ;v j l ÿ l0 †;

…39†

a02 ˆ P …Xv2 > Xa22 ;v j l ˆ l0 † ÿ P …Xv2 > Xa23 ;v j l ÿ l0 †

…40†

a03 ˆ P …Xv2 > Xa23 ;v j l ˆ l0 †:

…41†

and

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J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

Fig. 2. MCC with four control regions.

The consumer's risks. The consumer's risks are a10 ˆ P …Xv2 ; g2 < Xa21 ;v j l ˆ l0 ‡ d1 †;

…42†

a20 ˆ P …Xv2 ; g2 < Xa21 ;v j l ˆ l0 ‡ d2 †

…43†

a30 ˆ P …Xv2 ; g3 < Xa21 ;v j l ˆ l0 ‡ d3 †:

…44†

and

The other risks. The other risks are a12 ˆ P …Xv2 ; g1 > Xa22 ;v j l ˆ l0 ‡ d1 † ÿ P …Xv2 ; g1 > Xa23 ;v j l ˆ l0 ‡ d1 †; a13 ˆ P …Xv2 ; g1 > Xa23 ;v j l ˆ l0 ‡ d1 †;

…45† …46†

a21 ˆ P …Xv2 ; g2 > Xa21 ;v j l ˆ l0 ‡ d2 † ÿ P …Xv2 ; g2 > Xa22 ;v j l ˆ l0 ‡ d2 †;

…47†

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

a23 ˆ P …Xv2 ; g2 > Xa23 ;v j l ˆ l0 ‡ d2 †;

83

…48†

a31 ˆ P …Xv2 ; g2 > Xa21 ;v j l ˆ l0 ‡ d3 † ÿ P …Xv2 ; g3 > Xa22 ;v j l ˆ l0 ‡ d3 †

…49†

and a32 ˆ P …Xv2 ; g3 > Xa22 ;v j l ˆ l0 ‡ d3 † ÿ P …Xv2 ; g3 > Xa23 ;v j l ˆ l0 ‡ d3 †:

…50†

The powers of the chart. The powers of the chart are a11 ˆ P …Xv2 ; g1 > Xa21 ;v j l ˆ l0 ‡ d1 † ÿ P …Xv2 ; g1 > Xa22 ;v j l ˆ l0 ‡ d1 †;

…51†

a22 ˆ P …Xv2 ; g2 > Xa22 ;v j l ˆ l0 ‡ d2 † ÿ P …Xv2 ; g2 > Xa23 ;v j l ˆ l0 ‡ d2 †

…52†

and a33 ˆ P …Xv2 ; g3 > Xa23 ;v j l ˆ l0 ‡ d3 †:

…53†

Normal approximation. Using the normal approximation, as done in Section 2, we have Z u5 1 2=2 eÿz dz; …54† a30 ˆ p 2p ÿ1 Z 1 Z 1 1 1 2=2 ÿz2=2 a12 ˆ p e dz ÿ p ez dz; …55† 2p u3 2p u6 Z 1 1 2=2 a13 ˆ p eÿz dz; …56† 2p u6 Z 1 1 2=2 a23 ˆ p eÿz dz; …57† 2p u7 Z 1 Z 1 1 1 2=2 2=2 eÿz dz ÿ p ez dz; …58† a31 ˆ p 2p u8 2p u9 Z 1 Z 1 1 1 2=2 ÿz2=2 a32 ˆ p e dz ÿ p ez dz dz; …59† 2p u9 2p u10 Z 1 Z 1 1 1 2=2 2=2 a22 ˆ p eÿz dz ÿ p ez dz …60† 2p u4 2p u7

84

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

and 1 a33 ˆ p 2p where

Z

1

u10

2=2

eÿz dz;

ÿ1 h

u5 ˆ ‰…Z1 …v ‡ g3 † † 3 ÿ E…y3 †Š=

…61† p V …y3 †;

…62†

E…y3 † ˆ 1 ‡ h3 …h3 ÿ 1†…v ‡ 2g3 †…v ‡ g3 †ÿ3 ÿ4 2 …v ‡ g3 † ; ÿ h3 …h3 ÿ 1†…2 ÿ h3 †…1 ÿ 3h3 †…v ‡ g3 † 2 " # 2h3 …v ‡ 2g3 † …1 ÿ h3 †…1 ÿ 3h3 †…v ‡ 2g3 † 1ÿ v…y3 † ˆ ; 2 …v ‡ g3 † …v ‡ g3 †

2 ÿ2 h3 ˆ 1 ÿ …v ‡ g3 †…v ‡ 3g3 †…v ‡ 2g3 † ; 3 g3 ˆ nd3 Rÿ1 d3 ; dT3 ˆ …d31 ; d32 ; . . . ; d3v †;

…63† …64† …65† …66† …67†

p u6 ˆ ‰…Z3 …v ‡ g1 †ÿ1 †h1 ÿ E…y1 †Š= V …y1 †;

…68†

Z3 ˆ Xa23 ;v ;

…69†

p ÿ1 h u7 ˆ ‰…Z3 …v ‡ g2 † † 1 ÿ E…y2 †Š= V …y2 †; p ÿ1 h u8 ˆ ‰…Z1 …v ‡ g3 † † 3 ÿ E…y3 †Š= V …y3 †; p ÿ1 h u9 ˆ ‰…Z2 …v ‡ g3 † † 3 ÿ E…y3 †Š= V …y3 †; p ÿ1 h u10 ˆ ‰…Z3 …v ‡ g3 † † 3 ÿ E…y3 †Š= V …y3 †;

…70† …71† …72† …73†

g1 ; g2 ; h1 ; h2 ; E…y1 †; E…y2 †; V …y1 †; V …y2 †; Z1 and Z2 are as de®ned in Section 2. The normal approximation for the values of a10 ; a20 ; a21 and a11 are also as given in Section 2. 3.2. The model (Model 2) Since we have the values of four unknown parameters Z1 ; Z2 ; Z3 and n to determine, we need to develop a model consisting of four di€erent equations that can be solved simultaneously to obtain these values. Based on the same arguments given earlier in Section 2, we specify values a03 ; b10 ; p22 and p33 for a03 ; a10 ; a22 and a33 , respectively, in Eqs. (41), (42), (59) and (60) to obtain P …XV2 > Xa23 ;V j l ˆ l0 † ˆ a03 ;

…74†

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

1 p 2p

Z

u1

ÿ1

2=2

eÿz dz ˆ b10

85

…75†

and this gives u1 ˆ Zb0 or ÿ1 h

‰…Z1 …v ‡ g1 † † 1 ÿ E…y1 †Š=

p V …y1 † ˆ Zb0

(where b0 ˆ b10 and Zb0 < 0†: Z 1 Z 1 1 1 2=2 2=2 p eÿz dz ÿ p eÿz dz ˆ p22 ; 2p u4 2p u7 Z 1 1 2=2 p eÿz dz ˆ p33 2p u10 and this gives u10 ˆ Zp00 or ÿ1 h

‰…Z3 …v ‡ g3 † † 3 ÿ E…y3 †Š=

p V …y3 † ˆ Zp00 ;

…76†

…77† …78†

…79†

where p00 ˆ p33 and u10 < 0: 3.3. Solution method The solution method involves the derivation of mathematical expressions for the values of Z1 and Z2 before developing a computational algorithm for obtaining the values of other design parameters, risks and power functions. The derivation of the values of the second and third control limits Z1 and Z2 (or Xa21 ;v and Xa22 ;v ). Using the results in Section 2, Eq. (75) is solved for Z1 to obtain  p1=h1 : …80† Z1 ˆ …v ‡ g1 † E…y1 † ÿ Zb0 …V …y1 † Now, let Z 1 1 2=2 p eÿz dz ˆ w: 2p u7 Then, from Eq. (75), Z

1

u4

2=2

eÿz dz ˆ w ‡ p22 ˆ w0 ;

…81†

86

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

) u4 ˆ Zw0 ; ) ‰…Z2 …v ‡ g2 †ÿ1 †h2 ÿ E…y2 †Š=

…82† p V …y2 † ˆ Zw0 :

…83†

Using a similar procedure used for deriving the expression for Z1 in Section 2, we solve Eq. (82) for Z2 to obtain  p1=h2 : Z2 ˆ …v ‡ g2 † E…y2 † ‡ Zw0 V …y2 †

…84†

With the formulas for the values of Z1 and Z2 as given in Eqs. (79) and (83) and with the value of Z3 being Xa200 ;v (from Eq. (74)) where a00 ˆ a03 is known. The computational procedure developed in Section 2.6 have been modi®ed to develop a procedure for determining the values of the second, third, and fourth control limits Xa21 ;v ˆ Z1 ; Xa22 ;v ˆ Z2 ; Xa23 ;v ˆ Z3 , and the sample size n that simultaneously satisfy Eqs. (75)±(77). The procedure is as follows. Step 0: Obtain the values of Zb0 and Xa23 ;v ˆ Z3 from the tables of the chisquare and normal distributions. Put n ˆ 1 and go to step 1. Step 1: Use the speci®ed value of n to compute the value of g1 from Eq. (4). Using g1 , obtain the values of h1 ; V …y1 † and E…y1 † in Eqs. (22), (21) and (20), respectively. Use these values together with the values of v and Zb0 to obtain the value of Z1 in Eq. (79) and go to step 2. Step 2: Using the value of n compute the value of g2 from Eq. (6) and substitute the value of g2 and v into Eqs. (24) and (25) to obtain the values of E…y2 † and V …y2 †, respectively. Use these values together with the known values of h1 ; Z3 and v to calculate the value of u7 in Eq. (69). Using the table of the normal distribution, obtain the area under the normal curve that corresponds to the value of the standard normal deviate u7 . This area is the value of w in Eq. (80). Go to step 3. Step 3: With the known value of w from step 2, the value w0 ˆ w ‡ p22 in Eq. (82) is now known. Use the normal distribution table to obtain the value of the standard normal deviate that corresponds to w0 . Denote this value by Zw0 and go to step 4. Step 4: With the values of z0w ; v; h2 ; g2 ; E…y2 † and v…y2 † as determined in the previous steps, compute the value of z2 from Eq. (83) and go to step 5. (Note that the current value of n satis®es Eqs. (74)±(76).) Step 5: Use the value of n to compute the value of g3 from Eq. (65), and the values of g3 and the known values of g3 and v to compute the values of h3 ; V …y3 † and E…y3 † from Eqs. (64), (63) and (62), respectively. Use these computed values to obtain the value of u10 : If u10 ˆ ÿZP00 ; put Z1 ˆ Z1 ; Z2 ˆ Z2 ; Z3 ˆ Z3 and n ˆ n : Thus, the optimal values Z1 ; Z2 ; Z3 and n of the second, third and fourth control limits and the sample size have been determined for the chart. Otherwise, put n ˆ n ‡ 1 and go to step 1.

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

87

4. Numerical examples 4.1. Numerical examples on Model 1 Di€erent values of the input parameters (see Table 1) have been used to compute various values of the design parameters for MCC with three control regions. Corresponding to each set of data in Table 1, values of design parameters are computed with respect to correlation coecients of 0.5, 0.0, and ÿ0.5, respectively (see Table 2). The table (Table 2) shows the following results: (i) In each of the seven examples, the sample size decreases as the value of the correlation coecient decrease negatively from 0.5 to ÿ0.5. This result is similar to those obtained by Jolayemi [3±6] for the ordinary MCC. (ii) The second control limit Z1 …ˆ Xa21 ;v † decreases as the correlation coecient decreases negatively from 0.5 to ÿ0.5, while a01 ; a11 and a21 all increase and a12 decreases. The reason for increases in the values of a02 is that as Z1 decreases, region 1 of the chart becomes wider and region 0 gets narrower. These situations increase the chance of a plotted point falling in region 1 while decreasing the chance of its falling in either regions 0 or 2. (iii) Contrary to the expectation, the values of a02 ; a10 and a22 in the example are not equal to their speci®ed values a02 ; b10 and P22 , respectively. The reason for this is that since n is discrete, it is impossible to obtain values of n that will produce values of these risks that are exactly equal to their speci®ed values. (iv) When the speci®ed value of a22 (p22 to be exact) is increased from 98 to 99 in Example 2 and all other input parameters remain as they are in Example 1, the sample sites and values of the second control limit Z1 are larger than their corresponding values in Example 1 while the values of a11 are smaller. An increase in the power p22 of the chart with respect to region 2 increases the sample size and Z1 . The increase in the value of Z1 decreases the size of region 1 and, consequently, a11 decreases. (v) As can be expected, the risk a20 (the probability that the second level of restoration procedure is not triggered when there is a shift in the process to Table 1 Input data for the numerical examples on the design of MCC with three control limits Data no.

a02

b10

p22

Zb

Zp0

d11

d21

d12

d22

v

1 2 3 4 5 6 7

0.005 0.005 0.05 0.005 0.05 0.005 0.05

0.08 0.08 0.08 0.15 0.15 0.08 0.08

0.98 0.999 0.999 0.999 0.999 0.98 0.98

ÿ1.41 ÿ1.41 ÿ1.41 ÿ1.04 ÿ1.04 ÿ1.41 ÿ1.41

2.06 3.10 3.10 3.10 3.10 2.06 2.06

1.0 1.0 1.0 1.0 1.0 0.5 0.5

1.0 1.0 1.0 1.0 1.0 0.5 0.5

2.0 2.0 2.0 2.0 2.0 1.0 1.0

2.0 2.0 2.0 2.0 2.0 1.0 1.0

2 2 2 2 2 2 2

Correlation

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

Examples

1

2

3

4

5

6

7

5 2 1

7 4 2

5 3 1

7 4 2

5 3 1

14 9 4

20 13 6

n

1.244 1.179 0.992

2.1215 2.04 1.811

3.221 2.823 1.722

4.925 4.1 4.1

2.1215 1.8111 0.9922

3.4991 2.786 2.786

2.12 1.811 0.9922

Z1

5.991 5.991 5.991

10.597 10.592 10.592

5.991 5.991 5.991

10.597 10.597 10.597

5.991 5.991 5.991

10.597 10.597 10.597

10.597 10.597 10.597

Z2

0.543 0.562 0.613

0.357 0.372 0.413

0.20 0.249 0.435

0.089 0.137 0.137

0.357 0.413 0.613

0.165 0.254 0.254

0.3569 0.402 0.613

a01

0.05 0.05 0.05

0.005 0.005 0.005

0.05 0.05 0.05

0.005 0.005 0.005

0.05 0.05 0.05

0.005 0.005 0.005

0.005 0.005 0.005

a02

Table 2 The design parameters corresponding to each set of input data in Table 1

0.0793 0.0793 0.0793

0.0793 0.0793 0.0793

0.1492 0.1492 0.1492

0.1492 0.1492 0.1492

0.0793 0.0793 0.0793

0.0793 0.0793 0.0793

0.0793 0.0793 0.0793

a10

0.4446 0.510 0.5573

0.4207 0.6227 0.6596

0.2177 0.2676 0.4379

0.4367 0.4534 0.4534

0.2876 0.3195 0.5078

0.4367 0.5233 0.5233

0.6192 0.6559 0.7761

a11

0.4761 0.4602 0.4129

0.50 0.298 0.2611

0.6331 0.5832 0.4129

0.484 0.3974 0.3974

0.6331 0.5832 0.4129

0.484 0.3974 0.3974

0.3015 0.2648 0.1446

a12

0.0002 0.002 0.002

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

a20

0.0291 0.1585 0.1585

0.0228 0.1587 0.1587

0.0228 0.0228 0.1587

0.228 0.0228 0.0228

0.0228 0.1587 0.1587

0.0228 0.0228 0.0228

0.0228 0.1587 0.5

a21

0.9772 0.8413 0.8413

0.9772 0.8413 0.8413

0.9772 0.9772 0.3413

0.9772 0.9772 0.9772

0.9772 0.8413 0.8413

0.9772 0.9772 0.9772

0.9772 0.8413 0.5

a22

88 J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

89

Table 3 Input data for the design of MCC with four control regions Data no.

a03

b10

p2

p3

Zb

Zp00

d11

d21

d12

d22

d13

d23

V

1 2 3 4 5

0.0 0.0 0.005 0.005 0.0

0.05 0.05 0.05 0.1 0.1

0.2 0.55 0.4 0.4 0.4

1.0 0.999 0.999 0.98 0.999

ÿ1.64 ÿ1.64 ÿ1.64 ÿ1.28 ÿ1.28

5.0 3.10 3.10 2.06 3.10

0.8 0.8 0.8 0.8 0.8

0.8 0.8 0.8 0.8 0.8

1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0

2.0 2.0 2.0 2.0 2.0

2.0 2.0 2.0 2.0 2.0

2 2 2 2 2

state 2) has the least value among all other risks in the examples. This is followed by the risk a02 . (vi) Examples 6 and 7 show that small shifts require large sample sizes. 4.2. Numerical examples on Model 2 To study the properties of MCC with four control regions, we have used various values of input parameters (Table 3) to compute di€erent values of design parameters, risks and power functions ± just as in the case of Model 1. The results are shown in Table 4. A study of the table (Table 4) shows the following: (i) Many of the results obtained are identical to those in Table 2. For example, the sample size n and the second control limit Z1 decreases as the correlation coecient decreases negatively from 0.5 to ÿ0.5 in all the ®ve examples in the table. However, the second control limit Z2 …ˆ Xa22 ;v † shows some increases, though the increases are not so regular. (ii) The risks a02 ; a12 ; a13 and a23 decrease as the correlation coecient decreases negatively from 0.5 to ÿ0.5. But, unlike in the previous results in Table 2, a11 increases with decreases in the correlation coecient. The reason for this is easy to see. Z1 decreases and Z2 increases (though irregularly) as the correlation coecient decreases negatively. This makes the size of region 1 to get larger and, consequently, the probability that a shift in the process will be caught in the region becomes bigger. 5. Conclusion The model which is developed in this paper is an advancement on earlier models developed for the statistical design of the ordinary MCC (see Refs. [1,6]). It produces charts (MCCMCRs) with very low values of control-chart errors. The fact that the charts' operations do not involve any major search for assignable cause makes them (the charts) more cost-e€ective than the ordinary MCC. The charts will ®nd useful applications in both low and high technology environments.

14 9 4

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

0.5 0.0 ÿ0.5

1

2

3

4

5

8 5 2

5 3 1

7 4 2

8 5 2

n

Examples Correlation

2.56 2.34 1.71

1.33 1.16 0.698

1.29 0.99 0.99

1.63 1.46 0.99

4.1 3.85 3.19

Z1

8.11 8.14 8.17

3.81 5.04 4.8

4.31 5.51 5.51

3.69 5.81 6.44

12.6 10.1 10.96

Z2

15.0 15.0 15.3

10.6 10.6 10.6

10.6 10.6 10.6

15.0 15.0 15.0

15.0 15.0 15.0

Z3

a02

a11

a12

a13

a20

a21

a23

0.118 0.067 0.067

0.1587 0.0401 0.256 0.3085 0.1357 0.0436 0.292 0.2648 0.0749 0.0548 0.4095 0.1357

0.2611 0.0129 0.1031 0.484 0.209 0.0158 0.1868 0.3974 0.209 0.0158 0.1868 0.3974

0.1357 0.0122 0.1186 0.3192 0.117 0.0129 0.1531 0.284 0.0749 0.0158 0.2501 0.184

0.2621 0.0189 0.4199 0.3466 0.1335 0.0287 0.252 0.3192 0.495 0.0187 0.4636 0.3194 0.117 0.0301 0.2856 0.2843 0.6188 0.0184 0.5331 0.2922 0.0747 0.0359 0.380 0.1841

0.2934 0.448 0.416 0.348 0.6704 0.155

0.221 0.468 0.3943 0.347 0.3943 0.347

0.1662 0.1062 0.708 0.055 0.176 0.663 0.042 0.474 0.401

0.361 0.158 0.4836 0.079 0.6126 0.088

0.408 0.546 0.546

0.287 0.431 0.576

0.1345 0.0025 0.4261 0.1342 0.3897 0.0066 0.7668 0.7157 0.1506 0.007 0.3057 0.2546 0.3897 0.0069 0.7695 0.6897 0.199 0.0049 0.4221 0.2298 0.2981 0.0062 0.7223 0.5987

a01

Table 4 The design parameters corresponding to each set of input data in Table 3

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

a30

0.0 0.0 0.0

0.0 0.0 0.0

a32

0.0035 0.0194 0.0 0.0228 0.0 0.1587

0.0 0.228 0.0 0.1587 0.0036 0.4964

0.0045 0.0183 0.0045 0.0183 0.0 0.0228

0.0 0.0 0.0

0.0 0.0 0.0

a31

90 J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

J.K. Jolayemi / Appl. Math. Comput. 109 (2000) 73±91

91

References [1] F.B. Alt, Multivariate control charts for the mean, in: Proceedings of the Seventh Annual North-East Regional Conference of the American Institute for Decision Sciences, Washington, DC, 1978, pp. 109±112. [2] H. Hotelling, Techniques of Statistical Analysis, Eisenhart, in: Hastay, Wallis (Eds.), McGrawHill, New York, 1947, pp. 111±184. [3] J.K. Jolayemi, J.N. Berrettoni, Multivariate control charts: an optimization approach to e€ective use and measurement of performance, Applied Mathematics and Computation 32 (1) (1989) 1±16. [4] J.K. Jolayemi, J.N. Berrettoni, An optimal design of multivariate control charts in the presence of multiple assignable causes, Applied Mathematics and Computation 32 (1) (1989) 17±34. [5] J.K. Jolayemi, An optimal design of multivariate control charts with producer's risk speci®cation, Journal of the Nigerian Institute of Industrial Engineers 2 (1) (1992) 9±24. [6] J.K. Jolayemi, A power function model for determining sample sizes for the operations of multivariate control charts, Computational Statistics and Data Analysis 20 (1995) 633±646. [7] J.K. Jolayemi, Design of X -charts with multiple control regions when some risk-values are speci®ed, Walking paper, Department of Statistics, University of Ibadan, 1994. [8] W. Sankaran, Approximation to the non-central chi-square distribution, Biometrika 46 (1959) 235±237. [9] T. George, H.L. Lee, Economic design of control charts with di€erent control limits for di€erent assignable causes, Management Science 34 (11) (1988) 1347±1366.