Joint statistical design of double sampling and s charts

European Journal of Operational Research 168 (2006) 122–142

www.elsevier.com/locate/ejor

Stochastics and Statistics

Joint statistical design of double sampling X and s charts

David He *, Arsen Grigoryan

Department of Mechanical and Industrial Engineering, The University of Illinois at Chicago,

842 West Taylor Street, 3049 ERF, Chicago, IL 60607, USA

Received 12 August 2003; accepted 19 April 2004

Available online 1 July 2004

Abstract

In statistical quality control, usually the mean and variance of a manufacturing process are monitored jointly by two

statistical control charts, e.g., a X chart and a R chart. Because of the efficiency of double sampling (DS) X charts in

detecting shifts in process mean and DS s charts in process standard deviation it seems reasonable to investigate the

joint DS X and s charts for statistical quality control. In this paper, a joint DS X and s chart scheme is proposed.

The statistical design of the joint DS X and s charts is defined and formulated as an optimization problem and solved

using a genetic algorithm. The performance of the joint DS X and s charts is also investigated. The results of the inves-

tigation indicate that the joint DS X and s charts offer a better statistical efficiency in terms of average run length (ARL)

than combined EWMA and CUSUM schemes, omnibus EWMA scheme over certain shift ranges when all schemes are

optimized to detect certain shifts. In comparison with the joint standard, two-stage samplings and variable sampling

size X and R charts, the joint DS charts offer a better statistical efficiency for all ranges of the shifts.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Statistical quality control; Double sampling; Joint statistical design; Genetic algorithm

1. Introduction

In statistical quality control, usually the mean and variance of a manufacturing process are monitored

jointly. Over the past years, most of the research on joint monitoring of process mean and process variance

has focused on the joint X and R charts (see, for example, Jones and Case, 1981; Rahim, 1989; Saniga,

1991; Costa, 1993, 1998; Costa and Rahim, 2000).

0377-2217/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2004.04.033

* Corresponding author. Tel.: +1 312 996 3410; fax: +1 312 413 0447.

E-mail address: [email protected] (D. He).

mailto:[email protected]

D. He, A. Grigoryan / European Journal of Operational Research 168 (2006) 122–142 123

Particularly, Saniga (1991) presented a joint statistical design of X and R chart. A procedure coded in

FORTRAN enables users to determine the control limit parameters and sample size of a jointly designed

X and R chart for a specified statistical criterion. This criterion can be stated in terms of average run length

(ARL), Type I and Type II error probabilities, or average time to signal (ATS).

Gan (1995) developed control schemes for joint monitoring of process mean and variance by usingexponentially weighted moving average (EWMA) control charts. The first scheme, EEU was obtained by

running a two-sided EWMA mean chart and a high-side EWMA variance chart simultaneously. A two-

sided mean chart (Crowder, 1987a,b, 1989; Lucas and Saccuci, 1990) is obtained by plotting sample statistic

Qt ¼ ð1� kMÞQt�1 þ kMX t against sample number t for t=1, 2, . . . A high-side variance chart (Crowder and

Hamilton, 1992) is obtained by plotting sample statistic Qt ¼ maxf0; ð1� kV ÞQt�1 þ kV logðs2t Þg against

sample number t for t=1, 2, . . ., where s2t is the sample variance. This is a special case of the generalized

charting procedure introduced by Champ et al. (1991). The second scheme, EE consists of a two-sided

EWMA mean chart and a two-sided EWMA variance chart. A two-sided EWMA variance chart is ob-tained by plotting Q0 ¼ Eflogðs2t Þg ¼ �0:270 (when process variance r2 is equal to target process variance

r20) and Qt ¼ fð1� kV ÞQt�1 þ kV logðs2t Þg against sample number t for t=1, 2, . . ..In Gan (1995), the EEU and EE schemes were compared with a combined two-sided CUSUM (CC)

scheme and a omnibus EWMA chart scheme (Domangue and Patch, 1991). The results of the comparison

showed that the omnibus charts and EEU scheme are ineffective in detecting out-of-control situations char-

acterized by a shift in process mean with simultaneous decrease in variance. For adequate joint monitoring,

it was suggested that a combined scheme like EE and CC should be used. The schemes like EE and CC are

very similar in performance and they are often more sensitive than the omnibus charts in detecting otherout-of-control situations. The omnibus charts have difficulties in meaningful interpretations of the out-

of-control signals when an out-of-control signal is issued since an omnibus chart does not indicate whether

the signal is due to the mean shift or the variance shift.

The CC scheme is obtained by plotting sample statistics St ¼ maxf0; St�1 þ X t � kMg and T t ¼minf0; T t�1 þ X t þ kMg against sample number t for the mean chart and by plotting V t ¼ maxf0; V t�1 þlogðs2t Þ � kVUg and Kt ¼ minf0;Kt�1 þ logðs2t Þ � kVLg against sample number t for the variance chart.

The omnibus EWMA scheme proposed by Domangue and Patch (1991) is based on statistic

Ai ¼ kjZija þ ð1� kÞAi�1 for i=1, 2, . . ., where 0<k 6 1. The authors chose the EWMA of jZja for some abecause the proposed procedure was shown to be sensitive to changes in location when rP r 0 and to in-

creases in dispersion. The authors mentioned that the proposed chart would not be effective for detection

of a small change in mean when r<r0. For simplicity they considered only cases with a=0.5 and a=2. A

‘‘signal’’ to stop or inspect the process is given wheneverAiPE [Ai]+Lvar[Ai ]1/2. The quantity L is specified

by the user. For ifi1, E½Ai� ¼ ð2a

p Þ1=2C½ða þ 1Þ=2�, and var½Ai� ¼ 2ak

ð2�kÞp ½ffiffiffip

pC½a þ 0:5� � ðC½ða þ 1Þ=2�Þ2�. The

omnibus EWMA schemes with different design parameters were compared with other schemes, particularly

CUSUM of jZja proposed by Hawkins (1981) and Healy (1987), which is referred to as omnibus CUSUM

procedure. This procedure involves keeping a single sum given by Y i ¼ max½0; jZija � k þ Y i�1�. The processstops whenever control statistic Yi exceeds h, which is referred to as the decision value.

The comparison results in Domangue and Patch (1991) show that for all possible scenarios the omnibus

EWMA and CUSUM charts are better than the joint X and R chart, the individual moving range (IMR)

charts, the EWMA for mean, the X , and the X with warning limits.

Chen et al. (2001) proposed a new EWMA (MaxEWMA) chart which combines two sample statistics,

i.e., mean and variance into one. Let Ui be the ith standardized sample mean and

V i ¼ U�1fHððni�1Þs2ir2 ; ni � 1Þg, where U(z)¼P(Z 6 z) for Z N(0,1), the standard normal distribution,

U�1(�) is the inverse function of U(�), and H(w,m)¼P(W 6 wjv) for W v2m . Ui and Vi are independent

because X and si are independent. Since both Ui and Vi have the same distribution then a single chart could

be constructed to monitor both the process mean and the process variability. First, they defined two

EWMA charts Yi=(1� k)Yi�1+kUi and Zi=(1�k)Zi�1+kVi, i=1, 2, . . .. Then the two charts are combined

124 D. He, A. Grigoryan / European Journal of Operational Research 168 (2006) 122–142

into one sample statistic Mi given by maxfjY ij; jZijg. The statistic Mi will be large when the process mean

has drifted away from the target and/or when the process variability has increased or decreased. The com-

parison with standard X and R chart showed that MaxEWMA chart showed a better performance for small

shifts. The comparison with a combination of an EWMA mean chart and an EWMA variability chart

showed that MaxEWMA chart is better when the variability increases and there is shift in the process mean.The diagnostic abilities of the MaxEWMA chart are also better than the competitive charts.

A fully adaptive X and R chart is presented in Costa (1998) and Costa (1999). When the joint X and R

charts with variable parameters (Vp) are used, the sample means are plotted on the X control chart with

warning limits l0 � wrX , and action limits l0 � krX , where 0<w<k and rX is the standard deviation of

the sample means. The ranges from samples of size ni, i=1,2, are plotted on the R control chart with warn-

ing and action limits given by wR(ni )r0 and kR(ni)r0, respectively, where wR(ni)<kR(ni). Here wR (ni)

and kR(ni) are numbers of standard deviations of relative range defined by Rr. The size of the sample, the

sampling interval, and the width of X and R chart action limits depend on the results of the preceding sam-ple statistic. If sample statistic of both charts is in the central region, then the size of the next sample de-

noted by n1 should be small and sampling interval large. If the sample statistic of either of the charts is

within warning action limits, then the next sample size, denoted by n2, should be large and sampling interval

small.

Costa and Rahim (2002) developed a joint X and R chart with two-stage sampling (TSS). As in the case

with Shewhart charts, samples of size n0 are taken from the process at regular time intervals. At the first

stage, one item of the sample is inspected. If its X value is close to the target then the sampling is inter-

rupted. Otherwise the sampling goes on to the second stage, and the remaining n0�1 items are inspectedand the X and R values are computed based on the whole sample size n0.

Other related joint charts include the X and variance charts by Rahim et al. (1988) and Lorenzen and

Vance (1986).

The work of Daudin (1992) and He and Grigoryan (2002, 2003) has provided an incentive for the devel-

opment of the joint DS X and s charts. In Daudin (1992), a DS X chart was developed to improve the

statistical efficiency (in terms of ARL) without increased sampling, or alternatively, to reduce the sampling

without reducing the statistical efficiency. Daudin compared the statistical efficiency of the DS X chart with

other procedures. The results of the comparison are summarized as follows. In comparison with the She-whart X chart, the DS X chart showed a large improvement in efficiency. The results of comparing with the

VSI chart showed that when the time required to collect and measure the samples is negligible, the DS

scheme is more efficient; otherwise the VSI chart is better. In comparison with the EWMA chart, the

DS chart has a greater efficiency than the EWMA chart with r=0.75. However, when r=0.5 or 0.25,

the EWMA chart is better for detecting small shifts, and the DS chart is better for detecting large shifts.

In comparison with the two-sided combined Shewhart CUSUM schemes, the DS chart provides a better

protection against large shifts while the Shewhart CUSUM schemes are better for detecting small

shifts. Daudin suggested that the DS X chart could be substituted for the Shewhart X chart wheneverthe improvement in efficiency outweighs the administration cost associated with the sampling. He and

Grigoryan (2002, 2003) extended the concept of the DS X chart scheme developed by Daudin (1992) to

monitoring of process variability and developed a DS s chart scheme. The efficiency of the DS s charts

was compared with that of the traditional s charts. The results of the comparison show that for the whole

range of the investigated shifts (0.6 6 k 6 5) in process standard deviation the DS s charts result in a sig-

nificant reduction in average sample size without decreasing the out-of-control ARL in comparison with

the traditional s charts.

Because of the efficiency of the DS X charts in detecting shifts in process mean and the DS s charts inprocess standard deviation it seems reasonable to investigate the joint DS X and s charts for simultaneously

monitoring of process mean and variance. Up to today, no research on the investigation of the joint DS Xand s charts has been reported.


In general, three design criteria are used for joint statistical design of X and R charts: statistical design,

economic design, and heuristic design (Saniga, 1991). As pointed out by Woodall (1986), the economic

design method is often not effective in producing charts that can quickly detect small shifts before substan-

tial losses can occur. Also discussed in Montgomery (1980), one of the problems associated with the eco-

nomic design is the difficulty in estimating the costs. For these reasons, the statistical design method ischosen in this paper for the joint design of the DS X and s charts. In joint statistical design of the DS charts,

we are interested in finding the control limits of the joint DS X chart and s charts as well as a joint sample

size such that if there is no shift in the process, the control charts are unlikely to signal an assignable cause,

while shifts of specified magnitude are likely to be detected.

2. The joint DS X and s charts

Throughout the paper, it is assumed that the joint DS X and s charts are used to monitor a process where

independent observations from the quality characteristic of interest, X, are normally distributed with a

mean l0 and a variance r20. Further, it is assumed throughout the paper that when the process is out of

control, it is due to a shift in mean from l0 to l1=l0±dr0, with d>0, and/or a shift in the process standard

deviation from r0 to r1=kr0. The proposed joint DS X and s chart scheme is presented in Fig. 1.

Before the joint chart procedure is presented, define:

c41 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2

n1 � 1

s !Cðn1=2Þ

C½ðn1 � 1Þ=2� ; coefficient for sample size of n1;

Fig. 1. Schematic representation of the joint DS X and DS s charts.


c4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

n1 þ n2 � 2

s !C½ðn1 þ n2 � 1Þ=2�C½ðn1 þ n2 � 2Þ=2� ; coefficient for sample size of n1 þ n2:

The joint DS X and s chart procedure:

(1) Take an initial sample of size n1 and calculate the sample mean X 1 and sample standard deviation s1.

(2) If X 1�l0r=ffiffiffin1

p lies in the range [�L1,L1] ands1�c41r

rffiffiffiffiffiffiffiffiffi1�c2

41

p lies in the range [�D1,D1], the joint scheme indicates that

the process is in control.


p lies in the range (�1,�L] and [L,+1) or s1�c41r


41

p lies in the range (�1,�D] and [D,+1), then

the joint scheme indicates that the process is out of control.


p lies in the intervals [�L,�L1] and [L1,L], and/or ifs1�c41r


41

p lies in the intervals [�D,�D1] and

[D1,D], take a second sample of size n2 and calculate the sample mean X 2 and the total sample mean

Y ¼ n1X 1þn2X 2

n1þn2, and/or the total sample standard deviation s12 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2

i¼1ðni�1Þs2iP2

i¼1ni�2

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn1�1Þs2

1þðn2�1Þs2

2

n1þn2�2

q.

(5) If Y�l0r=ffiffiffiffiffiffiffiffiffin1þn2

p lies in the interval [�L2,L2] and/ors12�c4r

rffiffiffiffiffiffiffi1�c2

4

p lies in the interval [�D2,D2], then the joint scheme

indicates that the process is in control; else the process is out of control.

The design of the joint DS X and s charts involves determining the values of the following parameters:

n1, n2 D, D1, D2, L, L1 and L2.

Note that the mean and the standard deviation of random variable s1 at the first stage are c41r and

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c241

p, the mean and the standard deviation of the random variable s12 at the second stage are c4r

and rffiffiffiffiffiffiffiffiffiffiffiffiffi1� c24

p, respectively.

3. Formulation of joint statistical design of the DS X and s charts

In this paper, the joint statistical design of the DS X and s charts is formulated as a design optimiza-

tion problem. Before the optimization model is presented, the following intervals and notations are

defined:

I11 ¼ l0 �L1rffiffiffiffiffin1

p ; l0 þL1rffiffiffiffiffin1

p�

;

I12 ¼ l0 �Lrffiffiffiffiffin1

p ; l0 �L1rffiffiffiffiffin1

p�

[ l0 þL1rffiffiffiffiffin1

p ; l0 þLrffiffiffiffiffin1

p�

;

I13 ¼ �1; l0 �Lrffiffiffiffiffin1

p

[ l0 þLrffiffiffiffiffin1

p ;þ1� �

;

I14 ¼ l0 �L2rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ n2

p ; l0 þL2rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ n2

p�

;

I15 ¼ �1; l0 �L2rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin þ n

p

[ l0 þL2rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin þ n

p ;þ1� �

;
1 2 1 2


I21 ¼ c41r � D1rffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c241

q; c41r þ D1r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c241

q� ;

I22 ¼ c41r � Drffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c241

q; c41r � D1r


q� [ c41r þ D1r


q; c41r þ Dr


q� ;

I23 ¼ �1; c41r � Drffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c241

q [ c41r þ Dr


q;1

� �;

I24 ¼ c4r � D2rffiffiffiffiffiffiffiffiffiffiffiffiffi1� c24

q; c4r þ D2r

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� c24

q� ;

I25 ¼ �1; c4r � D2rffiffiffiffiffiffiffiffiffiffiffiffiffi1� c24

q [ c4r þ D2r


q;þ1

� �:

aX =Probability of Type I error of X chart; aS=Probability of Type I error of s chart; bX =Probability of

Type II error of X chart; bS=Probability of Type II error of s chart.Mathematically, the optimization model can be written as follows:

Minn1;n2;L;L1;L2;D;D1;D2

ARL1 ð1Þ

Subject to : n1 þ n2 � fProbability of taking the second samplejl ¼ l0; r ¼ r0g6Nmax ð2Þ

Pr½DS X chart signalsjl ¼ l0; r ¼ r0�6 aX ð3ÞPr½DS s chart signalsjr ¼ r0�6 aS: ð4Þ

The objective function (1) is to minimize ARL1, i.e., the out-of-control average run length of the joint DS

X and s charts. Constraint (2) ensures that the average sample size of the joint DS X and s charts when the

process is in control is not greater than the prespecified value Nmax The average sample size is the sum of the

sample size taken at the first stage and the sample size taken at the second stage multiplied by the proba-

bility of taking the second sample. To conveniently compute the probability of taking the second sample thefollowing events are defined:

A ¼ ðX 1 2 I11Þ \ ðs1 2 I22Þ;B ¼ ðX 1 2 I12Þ \ ðs1 2 I22Þ;C ¼ ðX 1 2 I12Þ \ ðs1 2 I21Þ:

Then the probability of taking the second sample is equal to P ðA [ B [ CÞ, which could be expressed asfollow:

P ðX 1 2 I11ÞPðs1 2 I22Þ þ P ðX 1 2 I12ÞP ðs1 2 I22Þ þ P ðX 1 2 I12ÞP ðs1 2 I21Þ:
Constraints (3) and (4) ensure that the probability of making a false alarm is not greater than aX for the
DS X chart and aS for the DS s chart, respectively.

In addition to constraints (2)–(4), restrictions based on the feasibility of the solution are introduced on

L1 and D1 i.e., L1<L, D1<D. Integer constraints are imposed on n1 and n2.

Note that the overall Type I error probability of a joint chart is computed as: a ¼ aX þ aS � aXaS and

defined as the probability of concluding the process is out of control while there is no shift neither in process

mean nor in process standard deviation. Also note that the overall Type II error probability of a joint chart


is computed as b ¼ bXbS and defined as the probability of concluding the process is in control while there is

a shift in process mean and/or in process standard deviation.

Let us denote Pa1X and Pa2X as the probabilities that the DS X chart indicates that the process is in con-

trol at the first and second stages respectively, and Pa1s and Pa2s as the probabilities that the DS s chart

indicates that the process is in control at the first stage and second stages respectively. Note that sinceARL1 ¼ 1

1�bXbsthe optimization model (1)–(4) becomes:


1

1� Pa1X þ Pa2X jl ¼ l1; r ¼ r1

� Pa1s þ Pa2sjl ¼ l1; r ¼ r1ð Þ

ð5Þ

Subject to : n1 þ n2 � fPrðX 1 2 I11ÞPrðs1 2 I22Þ

þ PrðX 1 2 I12ÞPrðs1 2 I22Þ þ PrðX 1 2 I12ÞPrðs1 2 I21Þg6Nmax ð6Þ

1� Pa1X þ Pa2X

��l ¼ l0; r ¼ r0

� 6 aX ð7Þ

1� Pa1s þ Pa2sjl ¼ l0; r ¼ r0ð Þ6 aS ð8Þ

L1 < L;D1 < D: ð9Þ

Note, to solve efficiently model (5)–(9), lower and upper bound values can be imposed on L, L1, L2, D, D1

and D2. The lower and upper bound values could be set up as suggested by Daudin (1992) for practical

implementation of the DS X chart. Daudin (1992) recommended that L must be higher than the classicalvalues 3 or 3.09. A good choice is L=4 or 5. L1 must be lower than the classical value for warning limit,

which often is taken equal to 2. A good choice for L1 is between 1.3 and 1.8. The same bounds can also be

imposed on D, D1 and D2.

When there is a shift in process mean and process standard deviation, i.e., d>0 and k>1, Pa1X is com-

puted as:

Pa1X ¼ Pr½X 1 2 I11� ¼ U1

k

ffiffiffiffiffin1

pd þ L1ð Þ

� � U

1

k

ffiffiffiffiffin1

pd � L1ð Þ

� :

At the second stage, the probability that the DS X chart indicates that the process is in control, Pa2X is

equal to the probability that the total sample mean Y is in interval I14 and the mean of the first sample X 1 isin interval I12. Thus, when there is a shift in process mean and process standard deviation then Pa2X is com-

puted as (see the derivation of Pa2X in Appendix A):

Pa2X ¼Z

z2I�12

U cL2 þ rd

k

��

ffiffiffiffiffin1n2

rz

� � U c

�L2 þ rdk

��

ffiffiffiffiffin1n2

rz

� � �uðzÞdz;

where u(�)=standard normal density function.

I�12 ¼1

kð ffiffiffiffiffi

n1p

d � LÞ; 1kð ffiffiffiffiffi

n1p

d � L1Þ�

[ 1

kð ffiffiffiffiffi

n1p

d þ L1Þ;1

kð ffiffiffiffiffi

n1p

d þ LÞ�

:

The derivation of I�12 is shown in Appendix A.

Let PaX be the probability that the DS X chart indicates that the process is in control. Then it can be

computed as PaX ¼ Pa1X þ Pa2X . Consequently, the probability that the DS X chart indicates the process

is out of control is equal to: 1� PaX .

To conveniently compute probabilities Pa1s and Pa2s some transformations are needed. In order

to calculate the probabilities of the sample standard deviations falling into a specified interval, the


standard deviations of the first and second sample can be transformed into following Chi-square random

variables:

x1 ¼ðn1 � 1Þs21

r21

; with n1 � 1 degrees of freedom;

x2 ¼ðn2 � 1Þs22

r21

; with n2 � 1 degrees of freedom:

When there is a shift in process standard deviation, the probability that the DS s chart indicates that the

process is in control at the first stage is computed as:

Pa1s ¼Z

x12I�21

1

2n1�1

2

� C n1�1

2

� x n1�1

2

� �1

1 e�x1=2 dx1;

where

I�21 ¼ ðn1 � 1Þ r20

r21

c41 � D1


q �2

; ðn1 � 1Þ r20

r21

c41 þ D1


q �2" #

:

The derivation of I�21 is presented in Appendix B.

The probability that the DS s chart indicates that the process is in control at the second stage Pa2s is

equal to the probability that sample statistic s1 of the DS s chart at the first stage falls in interval I22and the combined sample statistic s12 at the second stage in interval I24.

Pa2s ¼Z

x12I�22

Zx22½A;B�

xn2�1

2

� �1

2 e�x2=2

2n2�1

2

� C n2�1

2

� dx224

35 x

n1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx1;
where
I�22 ¼ ðn1 � 1Þ r20

r21

c41 � Dffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c241

q �2

; ðn1 � 1Þ r20

r21

c41 � D1


q �2" #

[ ðn1 � 1Þ r20

r21

c41 þ D1


q �2

; ðn1 � 1Þ r20

r21

c41 þ Dffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c241

q �2" #

:

The derivation of Pa2s is presented in Appendix B. The derivation of I�22 is similar to that of I�21 shown in

Appendix B.

Then, the probability that the DS s control chart indicates that the process is in control can be computed

as Pas=Pa1s+Pa2s. Consequently, the probability that DS s control chart indicates that the process is out ofcontrol is equal to: 1�Pas.

Thus, model (5)–(9) can be written as:


1

1� bXbsð10Þ


where

bX ¼ U1

k

ffiffiffiffiffin1

pd þ L1ð Þ

� � U

1

k

ffiffiffiffiffin1

pd � L1ð Þ

�

þZ

z2I�12

U cL2 þ rd

k

��

ffiffiffiffiffin1n2

rz

� � U c

�L2 þ rdk

��

ffiffiffiffiffin1n2

rz

� � �uðzÞdz

bs ¼Z

x12I�21

xn1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx1 þZ

x12I�22

Zx22½A;B�

xn2�1

2

� �1

2 e�x2=2

2n2�1

2

� C n2�1

2

� dx224

35 x

n1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx1A ¼ r

1

k2c4 � D2


q �2

� x1

B ¼ r1

k2c4 þ D2


q �2

� x1

Subject to : n1 þ n2 � UðL1Þ � Uð�L1Þ½ � �Z

x12I�22

xn1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx10@

þf½Uð�L1Þ � Uð�LÞ� þ ½UðLÞ � UðL1Þ�g �Z

x12I�22

xn1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx1

þf½Uð�L1Þ � Uð�LÞ� þ ½UðLÞ � UðL1Þ�g �Z

x12I�21

xn1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx11A6Nmax ð11Þ

P r½False alarmjl ¼ l0; k ¼ 1�6 aX ; i:e:; 1� fU½L1� � U½�L1�g(

�Z

z2I�12

U cL2 �ffiffiffiffiffin1n2

rz

�� U �cL2 �

ffiffiffiffiffin1n2

rz

�� uðzÞdz

)6 aX ð12Þ

P r½False alarmjr ¼ r0�6 aS ; i:e:;

1�Z

x12I�21

xn1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx18<:

9=;

�Z

x12I�22

Zx22½A;B�

xn2�1

2

� �1

2 e�x2=2

2n2�1

2

� C n2�1

2

� dx224

35 x

n1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx1 6 aS ð13Þ

L1 < L;D1 < D: ð14Þ

4. Solving the optimization problem using genetic algorithm

As one can see that the joint DS X and s chart design optimization problem formulated by model (10)–

(14) is characterized by mixed continuous-discrete variables, and discontinuous and non-convex solution

space. Therefore, if standard nonlinear programming techniques are used for solving this type of optimi-

zation problem they will be inefficient and computationally expensive. Genetic algorithms (GAs) are well


suited for solving such problems, and in most cases, find a global optimum solution with a high probability

(Rao, 1996). Another motivation for using GAs to solve model (10)–(14) is that since the pioneer work by

Holland (1975), GAs have been developed into a general and robust method for solving all kinds of opti-

mization problems (e.g., Goldberg, 1989; Potgieter and Stander, 1998; Pham and Pham, 1999; Vinterbo and

Ohno-Machado, 2000) and computing software for applications of GAs have been commercially availablein the market. Thus solving model (10)–(14) using genetic algorithm provides a practical way for real-time

statistical process control implementation. The GAs have been used for the statistical design of double sam-

pling and triple sampling X control charts (He et al., 2002) and the DS s charts (He and Grigoryan, 2002,

2003), design and optimization of statistical quality control procedures (Hatjimihail, 1993), optimization of

univariate and multivariate exponentially weighted moving average control charts (Aparisi and Garcia-

Diaz, 2004).

The tool used for implementing the GAs in solving model (10)–(14) was a commercial software called

Evolver (Palisade, 1998). Evolver was used as an add-in program to the Microsoft Excel spreadsheet appli-cation. The optimization model (10)–(14) was set up in an Excel spreadsheet and solved by the genetic algo-

rithm in Evolver. The operation of the genetic algorithm involves following steps: (a) create a random

initial solution; (b) evaluate fitness, i.e., the objective function that minimizes the average run length;

(c) reproduction and mutation; (d) generate new solutions.

The quality of the solutions generated by the GAs depends on the setup of its parameters such as pop-

ulation size, crossover and mutation probability. During the implementation, the values of these parame-

ters were setup to obtain the best results.

Crossover probability determines how often crossover will be performed. If there is no crossover, an off-spring (new solution) will be an exact copy of the parents (old solutions). If there is a crossover, an offspring

(new solution) is made from parts of parents� chromosome. If crossover probability is 100%, then offspring

are made by crossover. Crossover is made up in hope that new chromosomes will have good parts of old

chromosomes and maybe the new chromosomes will be better. However it is good to leave some part of

population survive to next generation.

Mutation probability determines how often parts of the chromosomes will be mutated. If there is no

mutation, offspring will be taken after crossover (or copy) without any change. If mutation probability

is 100%, whole chromosome is changed. Mutation is made to prevent the search by genetic algorithm fall-ing into local extremes, but it should not occur very often, because then GA will in fact change to random

search.

In our computational experiment, the population size was set to 1,000. The crossover probability was

setup to 0.5 and mutation probability 0.06.

5. Performance of the joint DS X and s charts

5.1. Comparison with combined EWMA chart, combined CUSUM chart, and omnibus EWMA chart

In order to evaluate the performance of the joint DS X and s charts, we compared the joint DS charts

with the combined EWMA and CUSUM schemes presented in Gan (1995), and the omnibus EWMA

scheme proposed by Domangue and Patch (1991). Particularly, the comparison was conducted for sub-

group size of 5. The shifts in process mean in Gan (1995) were measured in number of standard devia-

tions of the subgroup mean distribution, i.e., D ¼ l1�l0r0

=ffiffiffin

por d ¼ D=

ffiffiffin

p, where r0 is the in-control

process standard deviation, and n is subgroup size, l1 and l0 are out and in-control process means,respectively. The parameters of the schemes used in the comparison are presented in Table 1. The

tabulated schemes selected for comparison have been optimized for detecting a shift with D=0.38 and

k=1.188, which corresponds to the actual shift in process mean with d=0.169 and process standard

Table 1

Parameters of the compared schemes

Scheme Scheme parameter

Joint DS-1 L1=1.72, L=3.40, L2=2.90, D1=1.603, D=3.833, D2=2.92, n1=3, n2=12

EE1 kM=0.134, hM=�0.345, HM=0.345, kV=0.106, hV=�0.867, HV=0.215

EE1U kM=0.134, hM=�0.345, HM=0.345, kV=0.043, HV=0.144

CC kM=0.224, hM=2.268, kVU=0.055, HV=4.006, kVL=0.666, hV=�5.054

O1 r=0.5, L=2.536

O2 r=0.025, L=1.587


deviation with k=1.188. The same optimal schemes of EE1, EE1U, the combined CUSUM scheme CC,

and the two omnibus EWMA schemes, O1 and O2 were selected for comparison with the DS schemes.

The two omnibus EWMA schemes were the ones with the highest and lowest smoothing parameter pre-

sented in Domangue and Patch (1991). Since a shift in process mean with D=0.38 for a subgroup size of

5 in Gan (1995) corresponds to the shift with d=0.169, the smallest positive d value used in the compar-

ison was set to 0.169. The rest of the shifts were defined on an equal incremental basis from d=0.169.

The ARL of the selected schemes was calculated using computer simulation, and 10,000 independent runs

were used for each ARL calculation. The model (10)–(14) was implemented in Excel spreadsheets andconnected with the genetic algorithm. The numerical integration was performed using Simpson�s rule

(Gerald and Wheatley, 1999). The obtained joint DS chart, denoted as Joint DS-1, was optimized for

detecting the shift with d=0.169 and k=1.188. The comparison results are presented in Table 2. All tab-

ulated data for joint DS X and s charts were confirmed by using Monte Carlo simulation with

MATLAB.

Note that all sample statistics have value of 0 at time 0.

The ARL values presented in the Table 2 were obtained for shifts in process mean with 0 6 d 6 1.0 and

shifts in process standard deviation with 1.0 6 k 6 2.0. From Table 2 we can observe that the proposedjoint DS scheme outperforms the rest of the schemes for shifts in process mean with d P 0.75 and shifts

in process standard deviation with k P 1.3.

Also from Table 2, we can see that except for the shift with d=0.5 and k=1.2, the combined EWMA and

CUSUM schemes outperform the DS scheme for shifts with d 6 0.5 and k 6 1.2.

However, as discussed in Yashchin (1987) and Lowry et al. (1992), the EWMA and CUSUM charts

could have a delays in reacting to shifts in a worst-case scenario in which the shift in process mean to

an opposite direction suddenly occurs. This delay is called the inertia problem. Thus, the advantage of

the combined EWMA and CUSUM schemes over the joint DS scheme in detecting relatively small shiftsholds only if the inertia problems can be ignored. For example, if a shift in process mean with d=0.169

occurs alone, then it takes on average 53.094 samples for an EE1U to detect the shift. Letting the practical

expediency of this performance aside, if a shift in the opposite direction would happened over this time

span, then the ARL would increase dramatically to detect the new shift in the process mean. Because

for the joint DS scheme samples are collected from the same population, then the DS scheme is free of

the inertia problems.

5.2. Comparison with joint STD, VSS, and TSS charts

The joint DS X and s charts were also compared to the standard joint X and R chart (STD), variable

sampling size (VSS), and two-stage sampling (TSS) schemes. The tabulated comparison data of the

STD, VSS, and TSS schemes were taken directly from Costa and Rahim (2002). The parameters of the

schemes in the comparison are presented in Table 3. The design parameters of all the schemes were chosen

Table 2

ARL values of the DS, EE1, EE1U, CC, O1, O2 schemes (n=5)

Scheme d k

1.00 1.10 1.20 1.3 1.40 1.50 2.00

Joint DS-1 0.00 250.0 69.70 25.54 12.03 6.89 4.57 1.79

EE1 250.0 67.69 25.64 14.80 10.44 8.12 4.30

EE1U 250.0 57.20 22.63 13.02 9.124 7.00 3.65

CC 250.0 71.44 26.65 14.92 10.34 8.02 4.14

O1 250.0 151.59 72.50 42.18 27.05 19.15 7.06

O2 250.0 81.70 42.30 28.37 21.08 16.93 9.07

Joint DS-1 0.169 131.32 50.18268 21.48 10.88 6.48 4.39 1.77

EE1 52.898 34.68 20.23 13.26 9.78 7.87 4.251

EE1U 53.094 31.80 18.14 11.94 8.62 6.82 3.62

CC 61.061 37.75 20.87 13.38 9.82 7.77 4.08

O1 205.55 94.68 52.31 32.14 22.75 16.72 6.77

O2 103.26 50.52 32.29 23.95 18.83 15.59 8.71

Joint DS-1 0.25 74.41 35.73 17.70 9.71 6.02 4.18 1.75

EE1 26.90 21.84 16.15 11.78 9.21 7.54 4.19

EE1U 26.84 20.84 14.72 10.61 8.19 6.61 3.58

CC 30.70 23.43 16.92 12.02 9.15 7.45 4.10

O1 112.77 61.73 36.88 25.70 18.59 14.27 6.35

O2 54.21 35.37 25.52 20.14 16.71 14.09 8.53

Joint DS-1 0.50 13.56 10.44 7.77 5.67 4.22 3.27 1.65

EE1 8.59 8.47 8.12 7.48 6.76 6.06 3.94

EE1U 8.63 8.40 7.81 7.05 6.29 5.49 3.42

CC 8.84 8.58 8.26 7.58 6.74 6.03 3.81

O1 19.26 15.38 12.80 10.80 9.21 8.15 5.02

O2 14.03 13.00 12.06 11.23 10.41 9.71 7.19

Joint DS-1 0.75 4.19 3.93 3.59 3.18 2.77 2.41 1.52

EE1 5.02 5.02 5.01 4.95 4.45 4.62 3.58

EE1U 4.99 4.99 4.98 4.84 4.64 4.35 3.18

CC 5.01 5.08 5.00 4.95 4.82 4.61 3.47

O1 6.19 5.87 5.56 5.25 5.04 4.76 3.80

O2 7.02 7.04 6.99 6.95 6.85 6.67 5.82

Joint DS-1 1.00 2.14 2.15 2.13 2.06 1.95 1.83 1.39

EE1 3.65 3.64 3.61 3.60 3.59 3.57 3.16

EE1U 3.59 3.60 3.57 3.60 3.56 3.49 2.88

CC 3.55 3.59 3.61 3.59 3.58 3.54 3.31

O1 3.22 3.24 3.21 3.18 3.16 3.12 2.88

O2 4.73 4.80 4.87 4.92 4.89 4.91 4.74


such that the in-control ARL was set equal to 433, and the average sample size while in control was set

equal to 5.

Two joint DS charts were chosen for the comparison. The first chart, denoted as Joint DS-2, was opti-

mized for detection of shifts with d=0.5 and k=1.1. The second chart, denoted as Joint DS-3, was opti-

mized for detection of shifts with d=0.75 and k=1.5. The results of the comparison are provided in

Table 4.

Table 3

Parameters of the compared schemes

Scheme Scheme parameter

Joint DS-2 L1=1.552; L=3.922, L2=2.922; D1=1.613, D=4.396, D2=3.016; n1=2, n2=16

Joint DS-3 L1=1.664; L=3.907, L2=3.015; D1=1.659, D=4.171, D2=3.065; n1=3, n2=12

STD N=5; k=3.250; kR(n)=5.432

TSS1 n0=15; k0=3.100; kR(n0)=5.786; w0=1.068

TSS2 n0=20; k0=3.100; kR(n0)=5.829; w0=1.252

VSS1 n1=2; n2=15; k=3.250; kR(n1)=4.594; kR(n2)=6.188

VSS2 n1=2; n2=20; k=3.250; kR(n1)=4.594; kR(n2)=6.366


From Table 4 we can observe that the proposed joint DS X and s charts result in a better statistical per-

formance than the rest of the charts for almost all the shifts, except for the shifts with d=0 and k=1.1 and

with d=0 and k=1.2. For this particular shift the DS X and s performance is slightly worse than that of

TSS2.

The TSS chart procedure requires that at the first stage only a single unit is taken, and whether the sec-

ond sample is calculated depends on the results of the first sample. Theoretically, it is possible to follow the

same procedure also for the DS X and s charts, but in this case the flexibility in the construction of the joint

DS X and s charts will be lost. The joint DS X and s charts have more flexibility in determining the sizes ofsampling units at the first and second stages, which leads to more design space and better solutions, while

the TSS procedure implies that only a single sampling unit statistic is calculated at first stage. The STD

chart is known to be insensitive to the small shifts, because the standard X and R charts are not sensitive

to small disturbances in the process. The performance of the STD chart for detecting small shifts, as

expected, is the worst among the compared charts. The joint DS X and s charts also outperform VSS chart

mostly because of the following reason: whenever the sample statistic of the VSS chart falls within the

warning limits at the following sampling point the sampling units increase. In the joint DS charts procedure

the second sample is calculated at the same sampling point, thus the shifts can be detected with a smallerARL.

5.3. The distribution of the Type I error probability

The results of the comparison between the joint DS charts and other competing schemes presented in

Tables 2 and 4 were obtained when the Type I error probability was distributed evenly between aX and

aS. However, since the allocation of the Type I error probability can be chosen as desired then the question

regarding how the ARL performance of joint DS charts are impacted by different distributions of the TypeI error probability becomes interesting.

Over the past years, the question regarding the Type I error probability distribution has received little

attention in the development of the schemes for joint monitoring process mean and standard deviation.

Most researches on statistical design of joint charts assume equal distribution of Type I error probability

between aX and as. For example, in Gan (1995), the Type I error probability is distributed evenly between

the mean and the variance charts. In Costa and Rahim (2002), the frequency of false alarms of both charts

is also matched and no further discussion regarding the distribution of the Type I error is provided. Costa

(1998) presented a variable parameter (Vp) joint X and R chart. He recommended that when detection ofsmall shifts in process mean is important use Vp chart with false alarm for R chart equal to practically zero.

No further discussion about Type I error probability distribution is provided. For some statistically

designed joint charts such as omnibus EWMA and CUSUM charts presented in Domangue and Patch

(1991), as they cannot differentiate whether an out-of-control signal is due to a shift in process mean or

Table 4

ARL values for the joint DS, STD, TSS and VSS charts (n=5)

Scheme d k

1.00 1.10 1.20 1.3 1.40 1.50 2.00

Joint DS-2 0.00 433.0 112.50 37.99 16.83 9.30 6.05 2.33

Joint DS-3 433.0 106.59 35.31 15.43 8.38 5.34 1.91

STDa 433.0 133.5 53.64 26.24 14.89 9.46 2.64

TSS1a 433.0 111.9 39.37 17.72 9.65 6.09 2.06

TSS2a 433.0 103.7 35.19 15.67 8.58 5.50 2.08

VSS1a 433.0 138.7 51.62 22.14 11.18 6.67 2.24

VSS2a 433.0 140.6 51.95 21.67 10.66 6.29 2.26

Joint DS-2 0.50 13.49 10.85 8.45 6.38 4.85 3.82 2.01

Joint DS-3 15.10 11.77 8.87 6.48 2.12 3.64 1.73

STDa 56.59 32.65 20.21 13.23 9.12 6.62 2.41

TSS1a 17.25 13.10 9.81 7.24 5.40 4.14 1.93

TSS2a 14.60 11.39 8.70 6.52 4.93 3.85 1.98

VSS1a 21.41 14.87 10.54 7.58 5.60 4.31 2.11

VSS2a 16.14 11.95 8.92 6.69 5.11 4.04 2.15

Joint DS-2 0.75 4.25 4.03 3.74 3.37 2.98 2.63 1.76

Joint DS-3 4.28 4.06 3.76 3.37 2.94 2.56 1.57

STDa 16.94 12.43 9.44 7.33 5.80 4.67 2.17

TSS1a 4.85 4.52 4.16 3.74 3.31 2.92 1.81

TSS2a 4.46 4.17 3.86 3.51 3.15 2.81 1.87

VSS1a 5.02 4.64 4.24 3.83 3.43 3.06 1.98

VSS2a 4.55 4.22 3.90 3.58 3.26 2.96 2.03

Joint DS-2 1.00 2.34 2.33 2.29 2.20 2.09 1.97 1.55

Joint DS-3 2.10 2.13 2.13 2.08 1.99 1.88 1.42

STDa 6.39 5.49 4.79 4.20 3.69 3.25 1.92

TSS1a 2.44 2.43 2.40 2.35 2.26 2.16 1.68

TSS2a 2.57 2.51 2.45 2.38 2.29 2.19 1.77

VSS1a 2.79 2.73 2.66 2.58 2.48 2.36 1.85

VSS2a 2.95 2.83 2.72 2.61 2.50 2.39 1.91

Joint DS-2 1.25 1.71 1.73 1.73 1.71 1.67 1.62 1.41

Joint DS-3 1.46 1.50 1.53 1.54 1.53 1.50 1.30

STDa 3.07 2.92 2.78 2.63 2.48 2.33 1.69

TSS1a 1.76 1.78 1.80 1.80 1.78 1.76 1.57

TSS2a 1.99 1.97 1.95 1.93 1.90 1.87 1.67

VSS1a 2.17 2.15 2.12 2.08 2.04 2.00 1.73

VSS2a 2.38 2.32 2.26 2.20 2.14 2.08 1.80

Joint DS-2 1.50 1.39 1.42 1.45 1.45 1.44 1.42 1.31

Joint DS-3 1.21 1.24 1.27 1.29 1.30 1.29 1.21

STDa 1.84 1.85 1.85 1.84 1.81 1.77 1.50

TSS1a 1.49 1.52 1.53 1.55 1.55 1.55 1.48

TSS2a 1.67 1.68 1.69 1.69 1.69 1.68 1.58

VSS1a 1.89 1.87 1.86 1.83 1.81 1.79 1.63

VSS2a 2.05 2.02 1.99 1.96 1.92 1.88 1.70

a Data from Costa and Rahim (2002).



a shift in standard deviation, the distribution of the Type I error probability seems relatively unimportant.

For economic design of the joint charts, the issue of Type I error probability distribution seems more com-

plicated than statistical design. Since the main objective of economic design is to minimize the cost, thus the

obtained overall Type I error probability is a function of the cost and can be of any magnitude. The dis-

tribution of the Type I error probability between the charts depends also on cost considerations. In a paperon economic–statistical design of the joint X and R charts by Saniga (1989), two examples are discussed. In

the first example, a shift in process standard deviation is not expected. In the second example, a shift is in

process standard deviation is expected and a power constraint is imposed on the joint chart to detect the

shift. In the first case the optimal solution (i.e. minimal cost) occurs when the Type I error probability

of the R chart is equal approximately to 0. In the second case the optimal solution occurs when the Type

I error probability is distributed approximately equally. No further generalization about the optimal distri-

bution of the Type I error probability is provided however in the paper.

It is believed in this paper that the Type I error probability distribution in statistical design of a jointchart is an important issue. Investigation on optimal distribution of the Type I error probability deserves

further research investigation and is beyond the scope of this paper. However, in this paper we try to pro-

vide some insights on how the distribution of the Type I error probability between aX and aS affects the

ARL performance of the joint DS charts under certain scenarios.

In our investigation, the three joint DS charts presented in Tables 2 and 4 were studied. The first chart,

Joint DS-1, was optimized for detection of shifts with d=0.169 and k=1.188. The overall Type I error prob-

ability of this chart is 0.004 which corresponds to an in-control ARL of 250. The second chart, Joint DS-2,

was optimized for detection of shifts with d=0.5 and k=1.1. The third chart, Joint DS-3, was optimized fordetection of shifts of with d=0.75 and k=1.5. The overall Type I error probability for the second chart and

the third chart is 0.00230947 which corresponds to an in-control ARL of 433. By changing the distribution

of the Type I error probability between aX and aS, ARL values of each joint DS chart for detecting three

different shifts were computed. Among the three shifts in the investigation, one represents the shift each

chart was optimized to detect, for example, the shift with d=0.169 and k=1.188 for Joint DS-1 chart.

The other two shifts represent the two extreme cases, i.e., shift alone in process mean (d>0, k=1) and

shift alone in process standard deviation (d=0, k>1). The results of the investigation are presented in

Tables 5–7.Note that presented in Tables 5–7 also include the ARL values of each individual DS X chart and DS s

chart optimized for respective mean shifts and standard deviation shifts.

From Tables 5–7, we can see that the ARL value of the DS X chart decreases as aX increases. Similarly,

the ARL value of the DS s chart decreases as aS increases. These results are well expected as increasing

Table 5

The ARL performance of Joint DS-1 chart optimized for detection of shifts with d=0.169 and k=1.188 for different aX and aS values

Type I error probability ARL

a aX aS DS X(d=0.169)

DS s

(k=1.188)

Joint DS-1

(d=0.169, k=1.888)

Joint DS-1

(d=0, k=1.888)

Joint DS-1

(d=0.169, k=1)

0.004 0.000509 0.0035 140.16 34.22 27.66 30.96 182.30

0.001 0.003 91.41 37.61 26.86 31.10 161.74

0.001509 0.0025 68.17 38.58 24.87 29.68 142.10

0.002 0.002 58.70 38.76 23.58 28.41 131.32

0.0025 0.001509 51.88 43.69 23.97 29.36 125.15

0.003 0.001 47.13 54.44 25.51 31.70 125.81

0.0035 0.000509 41.36 87.65 28.32 37.89 113.37

Table 6



a aX aS DS X(d=0.5)

DS s

(k=1.1)

Joint DS-2

(d=0.5, k=1.1)

Joint DS-2

(d=0, k=1.1)

Joint DS-2

(d=0.5, k=1)

0.00230947 0.00031 0.002 16.12 118.48 14.30 106.53 19.25

0.00081 0.0015 13.40 153.15 12.40 115.38 15.86

0.0011554 0.0011554 11.53 168.51 10.85 112.51 13.49

0.0015 0.00081 11.10 187.57 10.53 108.78 13.15

0.002 0.00031 10.96 368.29 10.67 126.24 13.13

Table 7



a aX aS DS X(d=0.75)

DS s

(k=1.5)

Joint DS-3

(d=0.75, k=1.5)

Joint DS-3

(d=0, k=1.5)

Joint DS -3

(d=0.75, k=1)

0.00230947 0.00031 0.002 4.51983 5.47648 2.72968 5.18060 5.67095

0.00081 0.0015 4.02355 6.02457 2.67902 5.42634 4.82093

0.0011554 0.0011554 3.68554 6.139137 2.56395 5.34547 4.28464

0.0015 0.00081 3.47379 6.53993 2.52041 5.46896 3.99137

0.002 0.00031 3.34564 7.27650 2.53005 5.70698 3.88237


Type I error probability generally means tightening the control limits and hence increasing the chance of

giving out-of-control signals. This type of behavior observed on an individual DS chart (X chart or s

chart), i.e., the ARL value monotonously decreases as the Type I error probability (aX for X chart or

aS for s chart) increases, can also be observed on the joint DS chart when only mean shifts occur(d>0, k=1). In this case, the ARL value of the joint DS chart decreases monotonously as aX increases

regardless any changes in aS. This behavior observed on the joint DS chart can be explained by the fact

that the DS s chart is insensitive to mean shifts and only the DS X chart contributes to the detection power

of the joint chart when the process shift is caused by mean alone. However, the monotonous change of the

ARL value cannot be observed for the joint chart when the process shift is caused by both mean and

standard deviation (d>0, k>1) or by standard deviation alone (d=0, k>1). In the case where the process

shift is caused by both mean and standard deviation (d>0, k>1), the ARL value of the joint DS chart

decreases then increases as aX increases or aS decreases. Obviously, there must exist an optimal distributionof the Type I error probability between aX and aS that gives the minimum ARL value. In the case where

the process shift is caused by the standard deviation alone (d=0, k>1), the changes in ARL value of the

joint DS chart as aX increases or aS decreases seem more difficult to predict. This type of behavior can be

explained by the fact that a DS X chart can be also sensitive to a shift in process standard deviation and it

contributes more or less to the detection power of the joint DS chart when shifts in process standard devi-

ation occur alone.

In general, we can conclude from the results in Tables 5–7 that the most unfavorable scenario for a

joint DS X and s chart can happen when the Type I error probability is distributed to have a greaterprotection against shifts in process standard deviation (aS>aX ) and only shifts in process mean occur

(d>0, k=1).


6. Conclusions

In this paper, a joint DS X and s chart scheme is proposed and its joint statistical design presented.

The design of the joint DS X and s charts is formulated as an optimization problem which minimizes

the out-of-control ARL given the Type I error probability constraint and average sample size constraintwhen the process is in control.

The statistical performance of the proposed joint DS X and s chart scheme measured by the out-of-con-

trol ARL was compared with that of the combined EWMA and CUSUM schemes, the omnibus EWMA

scheme, and the joint STD, TSS and VSS X and R charts. The results of the comparison with the combined

EWMA and CUSUM schemes, the omnibus EWMA scheme show that the proposed joint DS X and s

chart scheme outperforms these schemes for shifts in process mean with d P 0.75 and shifts in process

standard deviation with k P 1.3. Except for shifts with d=0.5 and k=1.2, the combined EWMA and

CUSUM schemes outperform the DS scheme for smaller shifts in process mean with d 6 0.5 and shiftsin process standard deviation with k 6 1.2. However it is argued that the advantage of the combined

EWMA and CUSUM schemes over the joint DS scheme for detecting relatively small shifts holds only

if the inertia problems can be ignored. In comparison with the joint STD, TSS and VSS X and R charts,

the results show the proposed joint DS X and s chart scheme outperforms these schemes for all shifts in

process mean with 0<d 6 1.0 and shifts in process standard deviation with 1.0<k 6 2.0.

The study of the ARL performance of the DS joint chart with unevenly distributed Type I probability

shows that the worse case scenario of the joint DS chart scheme might occur when aS is greater than aX and

the process shift is caused by mean alone (d>0, k=1).

Appendix A

A.1. Derivation of Pa2X

Let r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ n2

p, Z1 ¼

ffiffiffiffiffin1

p ðX 1 � lÞ=r1, and Z2 ¼ffiffiffiffiffin2

p ðX 2 � lÞ=r1, c ¼ffiffiffiffiffiffiffiffiffin1þn2

n2

q.

The computation of the probability that Y is in interval I5 can be derived as follows:

Pr½Y 2 I5� ¼ Pr�L2r0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ n2

p 6n1X 1 þ n2X 2

n1 þ n2� l0 6

L2r0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 þ n2

p�

¼ Pr�L2r0

r6

n1ðX 1 � l1Þðn1 þ n2Þ

þ n2ðX 2 � l1Þðn1 þ n2Þ

� ðmu0 � l1Þ6L2r0

rr1

�

¼ Pr�L2r0

rr0r1

6n1ðX 1 � l1Þðn1 þ n2Þr0r1

þ n2ðX 2 � l1Þðn1 þ n2Þr0r1

� ðl0 � l1Þr0r1

6L2r0

rr0r1

�

¼ Pr�L2

rr1r0

% & 6

ffiffiffiffiffin1

pZ1

ðn1þn2Þr0r0

% &þ ffiffiffiffiffin2

pZ2

ðn1þn2Þr0r0

% &� dr1r0

% & 6L2

r1r0

% &24

35:

Since k ¼ r1=r0, then

Pr�L2

rk6

ffiffiffiffiffin1

pZ1

ðn1 þ n2Þþ

ffiffiffiffiffin2

pZ2

ðn1 þ n2Þ� d

k6

L2

rk

� ;

Pr�L2 þ rd

k6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

n1 þ n2

ffiffiffiffiffin1

pffiffiffiffiffin2

p Z1 þ Z2

�s6

L2 þ rdk

" #;


Pr c�L2 þ rd

k

�6

ffiffiffiffiffin1

pffiffiffiffiffin2

p Z1 þ Z2 6 cL2 þ rd

k

�� :

Thus, for the conditional probability, one obtains

Pr½Y 2 I4jX 1 ¼ x� ¼ Pr½Y 2 I4jZ1 ¼ z� ¼ Pr c�L2 þ rd

k

��

ffiffiffiffiffin1n2

rz6 Z2 6 c

L2 þ rdk

��

ffiffiffiffiffiffiffin1n2

zr�

:Z � � � �
Thus Pa2X ¼
z2I�12

U c L2þrdk

� �

ffiffiffin1n2

qz � U c �L2þrd

k

� �

ffiffiffin1n2

qz uðzÞdz.

A.2. Derivation of I�12ffiffiffiffiffip
z ¼ n1ðX 1 � l1Þ=r1;
X 1 ¼zr1ffiffiffiffiffin1

p þ l1:

From the control chart we have

l0 þL1r0ffiffiffiffiffi

n1p 6

zr1ffiffiffiffiffin1

p þ l1 6 l0 þLr0ffiffiffiffiffin1

p ;

ffiffiffiffiffin1

p ðl0 � l1Þ þ L1r0ffiffiffiffiffin1

p 6zr1ffiffiffiffiffin1

p 6

ffiffiffiffiffin1

p ðl0 � l1Þ þ Lr0ffiffiffiffiffin1

p ;

ffiffiffiffiffin1

p ðl0 � l1Þ þ L1r0 6 zr1 6ffiffiffiffiffin1

p ðmu0 � l1Þ þ Lr0;ffiffiffiffiffin1

p ðl0 � l1Þr1

þ L1r0

r1

6 z6ffiffiffiffiffin1

p ðl0 � l1Þr1

þ Lr0

r1

;

ffiffiffiffiffin1

pd

kþ L1

k6 z6

ffiffiffiffiffin1

pd

kþ L

k;

1

k

ffiffiffiffiffin1

pd þ L1ð Þ6 z6

1

k

ffiffiffiffiffin1

pd þ Lð Þ:

Similarly the other part of the interval I�12 is obtained as:

1

k

ffiffiffiffiffin1

pd � Lð Þ6 z6

1

k

ffiffiffiffiffin1

pd � L1ð Þ:

Thus

I�12 ¼1

kð ffiffiffiffiffi

n1p

d � LÞ; 1kð ffiffiffiffiffi

n1p

d � L1Þ�

[ 1

k

ffiffiffiffiffin1

pd þ L1ð Þ; 1

k

ffiffiffiffiffin1

pd þ Lð Þ

� :

Appendix B

B.1. Derivation of I�21

Since s1 is in interval I21 and

I21 ¼ c41r0 � D1r0


q; c41r0 þ D1r0


q� :


Then we have:

c41r0 � D1r0


qs1c41r0 þ D1r0


q:

By multiplying (n1�1) to and dividing by r21 both sides of the inequality signs, we obtain

ðn1 � 1Þðc41r0 � D1r0


pÞ2

r21

6s21ðn1 � 1Þ

r21

6ðn1 � 1Þðc41r0 þ D1r0


pÞ2

r21

:

Since x1 is defined as ðn1 � 1Þs21=r21 then

ðn1 � 1Þ r20

r21

c41 � D1


q �2

6 x1 6 ðn1 � 1Þ r20

r21

c41 þ D1


q �2

:

Therefore, I�21 can be written as:

I�21 ¼ ðn1 � 1Þ r20

r21

c41 � D1


q �2

; ðn1 � 1Þ r20

r21

c41 þ D1


q �2" #

:

For the case when there is no shift in process standard deviation, i.e., k=1, I�21 reduces to

¼ ½ðn1 � 1Þðc41 � D1


pÞ2; ðn1 � 1Þðc41 þ D1


pÞ2�:

B.2. Derivation of Pa2s

When there is a shift in process standard deviation from r0 to r1, then the probability that the combined

sample statistic s12 falls within the control limits is computed as:

Prr20

r21

c4 � D2


q �2

6

ðn1 � 1Þ s21

r21

þ ðn2 � 1Þ s22

r21

n1 þ n2 � 26

r20

r21

c4 þ D2


q �2

24

35

¼ Pr rr20

r21

c4 � D2


q �2

� x1 6 x2 6 rr20

r21

c4 þ D2


q �2

� x1

" #;

where r=n1+n2�2

Define:

A ¼ rr20

r21

c4 � D2


q �2

� x1;

B ¼ rr20

r21

c4 þ D2


q �2

� x1:

When there is no shift in the process standard deviation A and B reduce to:

A ¼ r c4 � D2


q �2

� x1;

B ¼ r c4 þ D2


q �2

� x1:

The probability that the process is in control at the second stage Pa2s is equal to the probability that sam-

ple statistic s1 of the DS s chart at the first stage is in interval I22 and the combined sample statistic s12 at the

second stage in interval I24.


Pa2s ¼Z

x12I�22

fPr½s12 2 I4jx1�gf ðx1Þdx1 ¼Z

x12I�22

Zx22½A;B�

xn2�1

2

� �1

2 e�x2=2

2n2�1

2

� C n2�1

2

� dx224

35 x

n1�1

2

� �1

1 e�x1=2

2n1�1

2

� C n1�1

2

� dx1:

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