Construction of double sampling s-control charts for agile manufacturing

QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL

Qual. Reliab. Engng. Int. 2002; 18: 343–355 (DOI: 10.1002/qre.466)

CONSTRUCTION OF DOUBLE SAMPLING s-CONTROL CHARTSFOR AGILE MANUFACTURING

DAVID HE∗ AND ARSEN GRIGORYAN

Intelligent Systems Modeling & Analysis Laboratory, Department of Mechanical and Industrial Engineering,The University of Illinois at Chicago, Chicago, IL 60607, USA

SUMMARYDouble sampling (DS)X-control charts are designed to allow quick detection of a small shift of process mean andprovides a quick response in an agile manufacturing environment. However, the DS X-control charts assume thatthe process standard deviation remains unchanged throughout the entire course of the statistical process control.Therefore, a complementary DS chart that can be used to monitor the process variation caused by changes inprocess standard deviation should be developed. In this paper, the development of the DS s-charts for quicklydetecting small shift in process standard deviation for agile manufacturing is presented. The construction of theDS s-charts is based on the same concepts in constructing the DS X-charts and is formulated as an optimizationproblem and solved with a genetic algorithm. The efficiency of the DS s-control chart is compared with that of thetraditional s-control chart. The results show that the DS s-control charts can be a more economically preferablealternative in detecting small shifts than traditional s-control charts. Copyright 2002 John Wiley & Sons, Ltd.

KEY WORDS: statistical quality control; double sampling s chart; agile manufacturing; genetic algorithm

1. INTRODUCTION

In any production process, regardless of how welldesigned or carefully maintained, a certain amountof inherent or natural variability or ‘backgroundnoises’ is always present. This variability in keyquality characteristics usually arises from threesources: improperly adjusted machines, operatorerrors and/or defective raw materials. Such variabilityusually represents an unacceptable level of processperformance. We refer to these sources of variabilitythat are not a part of the chance cause pattern as‘assignable causes’. Once a process is in a stateof statistical control, it keeps producing acceptableproducts for a relatively long period of time. However,occasionally assignable causes will occur, seeminglyat random, resulting in a ‘shift’ to an out-of-controlstate where a larger proportion of the process outputdoes not conform to the requirements. A majorobjective of statistical quality control is to quicklydetect the occurrence of assignable causes or processshifts so that investigation of the process andcorrective action may be undertaken before a largenumber of non-conforming units are manufactured.

∗Correspondence to: D. He, Intelligent Systems Modeling &Analysis Laboratory, Department of Mechanical and IndustrialEngineering, The University of Illinois at Chicago, Chicago,IL 60607, USA. Email: [email protected]

The control chart is an on-line process controltechnique widely used for this purpose.

Knowledge of the behavior of chance variations isthe foundation on which control chart analysis rests.If a group of data is studied and it is found thattheir variation conforms to a statistical pattern thatmight reasonably be produced by chance causes, thenit is assumed that no special assignable causes arepresent. The conditions which produced this variationare, accordingly, said to be under control. They areunder control in the sense that, if chance causes arealone at work, then the amount and character of thevariation may be predicted for large numbers and itis not possible to trace the variation of a specificinstance to a particular cause. On the other hand, ifthe variations in the data do not conform to a patternthat might reasonably be produced by chance causes,then it is concluded that one or more assignable causesare at a work. In this case the conditions producingthe variation are said to be out of control. In thecase of statistical process control using control charts,any group of data outside the control limits of thecontrol charts is taken as an out of control signal andactions need to be taken to discover and eliminate theassignable causes of the variation.

As today’s manufacturing firms are moving towardsagile manufacturing, quick and economic on-line

Received 27 June 2001Copyright 2002 John Wiley & Sons, Ltd. Revised 14 December 2001

344 D. HE AND A. GRIGORYAN

statistical process control solutions are in high de-mand. Double sampling (DS)X-control charts presenta good alternative for statistical process control inan agile manufacturing environment. The efficiencyof the DS X-control chart for detecting changes inprocess mean has been compared with other typesof control charts such as Shewhart, variable samplingintervals (VSIs), cumulative-sum (CUSUM) and ex-ponentially weighted moving averages (EWMAs) byDaudin [1]. When being compared to a fixed samplesize procedure, the DS procedure was chosen suchthat its average sample size when the process is incontrol is equal to the fixed sample size. The resultsof the comparison reported in [1] are summarized inthe following.

In comparing the DS control chart with the VSI, theaverage time to signal (ATS) was used. The ATS isdefined as the expected value of the time between thestart of the process and the time when the chart signals.The results of the comparison showed that when thetime required in collecting and measuring the sampleis negligible, the DS scheme is more efficient than theVSI one.

When being compared to a classic Shewhartchart, a DS chart showed better detection ofsmaller to moderate shifts. Detection of large shiftsis better accomplished with the Shewhart chart.Another advantage of using a DS chart over theShewhart chart is the dramatic decrease in averagesample size. When the process is in control, thisdecrease is nearly 50%. Based on the results ofthe comparison, Daudin [1] concluded that the DSX-chart might be substituted for the ShewhartX-chartwhenever the improvement in efficiency outweighs theadministration/trouble costs. In this paper, Daudin [1],defined the efficiency as the highest magnitude of theshift in process that the control chart scheme is capableof detecting while having a certain sample size.

The comparison with the combined ShewhartCUSUM chart showed that the two-sided combinedShewhart CUSUM schemes are better for detectingsmall shifts and the DS chart is better for detectinglarge shifts. The efficiency of the DS chart is closeto that of the two-sided combined Shewhart CUSUMschemes with k = 1, but very different from thetwo-sided combined Shewhart CUSUM schemes withk = 0.25. Here, k is the reference value (or theallowance or the slack value) and is often chosen asabout halfway between the target value for the meanand the out-of-control value of the mean that we areinterested in quickly detecting [2]. Therefore, the DSchart may be appropriate when greater efficiency isrequired for small shifts than with the Shewhart chart

and protection against large shifts is also important.However, when large shifts are very unusual, theCUSUM or combined Shewhart CUSUM schemes arebest.

For comparison with the EWMA scheme, DS chartswere designed to match average run lengths (ARL)with corresponding EWMA charts when the processis in control. When λ = 0.75, the DS chart has agreater efficiency than the EWMA chart. However,when λ = 0.5 or 0.25, the EWMA chart is better fordetecting small shifts and the DS chart is better fordetecting large shifts. Here 0 � λ � 1 is a weightingfactor. As λ decreases, the weight on previous history(1 − λ) increases [3].

The DS X-control charts assume that the processstandard deviation remains unchanged throughout thewhole course of the process control. Therefore, the DSX-charts alone are not sufficient to monitor theprocess, as the process standard deviation could beshifted while the mean of the process stays the same.A change in process standard deviation with a fixedprocess mean could lead to a higher proportion ofnon-conforming parts from the process, although theprocess is under the control of the X-control chart.For quick detection of smaller shifts in the processstandard deviation, the double sampling s-controlchart, which was developed in this paper, could be agood complement to the DS X-control chart.

Nowadays there are a wide variety of chartsfor monitoring process variability. Eventually, thechoice will be in favor of more economical andstatistically sound at the same time. The methodproposed in this paper leads to a smaller averagesample size compared to traditional s-control charts,hence reducing the cost of the process monitoring.There are a few interesting papers published recentlyproposing new methods in economic design ofcontrol charts for variables. The work by Costa andRahim [4] of the economic design of X- and R-chartsfor simultaneously monitoring both the mean andvariance of a continuous production process is onesuch paper. The product variable quality characteristicis assumed to be normally distributed and theprocess is subject to two independent assignablecauses. One cause changes the process mean andthe other changes the process variance. It is alsoassumed that the occurrence times of the assignablecauses are described by a Weibull distribution withincreasing failure rate. They developed a cost modeland adopted a non-uniform sampling interval scheme.Then, by adding statistical constraints they extendedthe model to an economic–statistical design modelfor achieving desired levels of statistical performance

Copyright 2002 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2002; 18: 343–355

DOUBLE SAMPLING s-CONTROL CHARTS 345

while minimizing the expected cost. As expected,economic–statistical design is more costly thaneconomic design because of the added constraints.The study shows that the statistical performance of thecontrol charts with respect to the average time to signaldue to false alarm can be significantly improved by anadded cost.

The research work by Tolley and English [5]is a significant work in terms of studying theeconomic–statistical design (according to their def-inition, economic design of a constrained controlchart). They present a comparison between the costperformances of the exponentially weighted movingaverage (EWMA) and the combined EWMA and X-control chart schemes. In particular, they exploredthe impact of constraining the in-control average runlength on the optimal cost performances of bothschemes. This research suggests that the region ofgreatest chart discrimination occurs when the shiftis small, and the EWMA weighting parameter, λ, islarge. Examination of cost surfaces under in-controlaverage run length constraints revealed that for thecombined EWMA and X-charting schemes the costmodel is not a well-behaved function and presentsoptimization challenges. No significant differences inestimated total cost for the two charting schemes isrevealed though.

Because of the efficiency of DS X-charts indetecting a small shift in process mean, one couldnaturally extend the same principle of DSX-control todevelop a DS s-chart control. Up to now, no researchwork on the construction of DS s-chart has beenreported in the literature.

The remainder of the paper is organized asfollows. In Section 2, brief background informationon traditional s-control charts is provided. Section 3 isdevoted to the method of development for constructingthe DS s-control chart. In Section 4, the use ofa genetic algorithm for solving the DS s-chartconstruction problem is explained. In Section 5,the efficiency of the DS s-charts is compared withthat of the traditional s-charts and the results ofthe comparison are presented. Finally, Section 6concludes the paper.

2. BACKGROUND OF TRADITIONALs-CONTROL CHARTS

Let s be a sample standard deviation and σ be aprocess standard deviation. If samples are drawn froma population with a normal probability distribution,then the mean of the sample s is given as c4σ .It is also known that, as n increases, the distribution

of s becomes closer and closer to a symmetricaldistribution [3].

Theoretical knowledge of the distribution of s insamples from a normal universe is the basis for 3σlimits on the control charts for s. The sample standarddeviation s is not an unbiased estimator of σ . If theunderlying distribution is normal, then s actuallyestimates c4σ , where c4 is a constant that dependson the sample size n [2]. Furthermore, the standard

deviation of s is σ√

1 − c24. This information was used

to establish a traditional control chart on s. For thecase with known process standard deviation σ , sinceE(S) = c4σ , the centerline for the chart is c4σ .The control limits for an s-chart can be computed asfollows:

UCLS = c4σ + 3σ√

1 − c24

LCLS = c4σ − 3σ√

1 − c24

where:

c4 =(√

2

n− 1

)�(n/2)

�(n− 1/2)

and �(·) is a Gamma function.

3. DEVELOPMENT OF DS s-CONTROLCHARTS

Up to now, no effort on developing methods forconstructing DS s-charts has been reported in theliterature. In this section, the development of theDS s-chart is presented. Specifically, the followingprocedure is proposed for the chart construction.Two successive samples are taken without anyintervening time, which means that the samplesare coming from the same probability distribution.This could be achieved by collecting a master sampleof n1 + n2 units all at the same time. First, analyzethe first sample n1 and then decide whether to analyzethe remaining units. Figure 1 provides the graphicalrepresentation of the DS s-control chart. The controllimits of the chart are shown in the number of thestandard deviations of the estimated sample standarddeviation. All the steps involved in the DS s-controlchart could be summarized into a simple procedure.

Before the DS s-control chart procedure ispresented, let us define the coefficient for sample sizeof n1:

c41 =(√

2

n1 − 1

)�(n1/2)

�[(n1 − 1)/2]Copyright 2002 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2002; 18: 343–355


D

D1 D2

-D1 -D2

-D

0 0

1st stage 2nd stage

In control In control

Take 2nd sample

Take 2nd sample

Out of control

Out of control

Out of control

Out of control

Figure 1. Graphic view of the DS s-control chart

and the coefficient for sample size of n1 + n2 (see thederivation of c4 in Appendix A):

c4 =(√

2

n1 + n2 − 2

)�[(n1 + n2 − 1)/2]�[(n1 + n2 − 2)/2]

Then the mean of the random variable s1 at thefirst stage is c41σ , the standard deviation of s1 is

σ

√1 − c2

41, the mean of the random variable s12 atthe second stage is c4σ , the standard deviation of s12

is σ√

1 − c24. It is also assumed that s1 and s12 are

normally distributed.

DS s-control chart procedure

(1) Take an initial sample of size n1. Calculate thestandard deviation of the sample, s1.

(2) If (s1 − c41σ)/(σ

√1 − c2

41

)lies in the range

[−D1,D1], the process is considered to be undercontrol.

(3) If (s1 − c41σ)/(σ

√1 − c2

41

)lies in the range

(−∞,−D] and [D,+∞) the process is considered tobe out of control.

(4) If (s1 − c41σ)/(σ

√1 − c2

41

)lies in the

intervals [−D,−D1] and [D1,D], take a secondsample of size n2 and calculate the total samplestandard deviation

s12 =√√√√∑2

i=1 (ni − 1)s2i∑2

i=1 ni − 2

=√(n1 − 1)s2

1 + (n2 − 1)s22

n1 + n2 − 2

(5) If (s12 − c4σ)/(σ

√1 − c2

4

)lies in the interval

[−D2,D2], then the process is in control, otherwisethe process is out of control.

The design of a DS s-chart involves determining thevalues of the following five parameters:

• n1, sample size of the first sample;• n2, sample size of the second sample;• D1 and −D1, the limits on the first sample within

which the process is said to be in control;• D and −D, the limits on the first sample beyond

which the process is said to be out of control;• D2 and −D2, the limits on the second sample

within which the process is said to be in control.

In this paper, the optimal design of a DS s-controlchart is formulated as an optimization problem.The objective function of the optimization problemis to minimize the average number of samples duringthe normal operation of the process. The optimizationproblem has two constraints. The first constraint isthat the probability of concluding that a normalprocess is out of control is less than a constant α,which is normally specified as the manufacturer’srisk. The second constraint is that the probability ofconcluding that an abnormal process is in controlis less than β, which is normally specified as theconsumer’s risk.

Let us define the intervals

I1 =[c41σ −D1σ

√1 − c2

41, c41σ +D1σ

√1 − c2

41

]I2 =

[c41σ −Dσ

√1 − c2

41, c41σ −D1σ

√1 − c2

41

]∪[c41σ +D1σ

√1 − c2

41, c41σ +Dσ√

1 − c241

]I3 =

(−∞, c41σ −Dσ

√1 − c2

41

]∪[c41σ +Dσ

√1 − c2

41,∞)

I4 =[c4σ −D2σ

√1 − c2

4, c4σ +D2σ

√1 − c2

4

]I5 =

(−∞, c4σ −D2σ

√1 − c2

4

]∪[c4σ +D2σ

√1 − c2

4,+∞)

Mathematically the optimization problem can bewritten as follows:

Minn1,n2,D,D1,D2

n1 + n2 Pr[s1 ∈ I2 | σ = σ0] (1)

Subject to:

Pr[Out of Control | σ = σ0] � α (2)

Pr[In Control | σ = σ1] � β (3)

The objective function (1) in the optimization modelis to minimize the average sample size subject to two



Formula 1

Pa1 = Pr[s1 ∈ I1]

=∫s1∈I1

exp

(−((s1 − c41σ0)

/(σ0

√1 − c2

41

))2 /2

)√

2π σ0

√1 − c2

41

ds1

Formula 2

Pa2 = Pr[s12 ∈ I4]= Pr

[c4σ −D2σ

√1 − c2

4 � s12 � c4σ +D2σ

√1 − c2

4

]

= Pr

c4σ −D2σ

√1 − c2

4 �

√(n1 − 1)s2

1 + (n2 − 1)s22

n1 + n2 − 2� c4σ +D2σ

√1 − c2

4

= Pr

[(c4σ −D2σ

√1 − c2

4

)2�(n1 − 1)s2

1 + (n2 − 1)s22

n1 + n2 − 2�(c4σ +D2σ

√1 − c2

4

)2]

= Pr

[σ 2

(c4 −D2

√1 − c2

4

)2�(n1 − 1)s2

1 + (n2 − 1)s22

n1 + n2 − 2� σ 2

(c4 +D2

√1 − c2

4

)2]

= Pr

[(c4 −D2

√1 − c2

4

)2�(n1 − 1)s2

1/σ2 + (n2 − 1)s2

2/σ2

n1 + n2 − 2�(c4 +D2

√1 − c2

4

)2]

constraints. Constraint (2) ensures that the probabilityof making a false alarm is not greater than α (Type Ierror) and constraint (3) ensures that the probabilityof not detecting a shift in process standard deviationis not greater than β (Type II error). In additionto constraints (2) and (3), lower and upper boundsare imposed on D, D1 and D2. The values of thelower and upper bounds are set up as suggested byDaudin [1] for practical implementation of DS charts.Daudin [1] recommended that D must be higher thanthe classical values 3 or 3.09. A good choice is D =4 or 5. D1 must be lower than the classical valuefor the warning limit, which often is taken to beequal to 2. A good choice for D1 is between 1.3and 1.8. Also, integer constraints are imposed on n1and n2. In order to solve the optimization problem(1)–(3), the probabilities in constraints (2) and (3)should be defined in terms of the decision variablesn1, n2, D, D1 and D2. Therefore, the derivationof probabilities Pr[Out of Control | σ = σ0] andPr[In Control | σ = σ1] is provided next.

Because the control chart involves two stages, thedecision whether the process is in control or notshould be based on analyzing both stages. Let Pa1

and Pa2 be the probabilities that the process is incontrol at the first and second stage, respectively.Then, the probability that the process is in control canbe computed as Pa = Pa1 + Pa2. Consequently, theprobability that the process is out of control is equal to1 − Pa .

Let us first expand constraint (2). It could berepresented as

1 − (Pa1 + Pa2) � α

When there is no shift in process standard deviation,the probability that the process is in control at the firststage is computed as shown in Formula 1.

The probability that the process is in control atthe second stage when there is no shift in theprocess standard deviation can be derived as shown inFormula 2.

Let

r = n1 + n2 − 2

x = (n1 − 1)s21

σ 2, y = (n2 − 1)s2

2

σ 2

Random variables xand y follow chi-square distribu-tions with n1 − 1 and n2 − 1 degrees of freedom



Formula 3

1 −

∫s1∈I1

exp

(−((s1 − c41σ0)

/(σ0

√1 − c2

41

))2/2

)√

2π σ0

√1 − c2

41

ds1

−∫x∈I ∗

2

[∫y∈[A,B]

y((n2−1)/2)−1 e−y/2

2(n2−1)/2�((n2 − 1)/2)dy

]x((n1−1)/2)−1 e−x/2

2(n1−1)/2�((n1 − 1)/2)dx � α

respectively, i.e.

x ∼ χ2n1−1, y ∼ χ2

n2−1

Therefore, the conditional probability is computed asfollows:

Pr[s12 ∈ I4 | s1 = x]= Pr

[r(c4 −D2

√1 − c2

4

)2 − x

� y � r(c4 +D2

√1 − c2

4

)2 − x]

Let

A = r(c4 +D2

√1 − c2

4

)2 − x

B = r(c4 −D2

√1 − c2

4

)2 − xThe chi-square probability density function is

f (x) = 1

2(n1−1)/2�((n1 − 1)/2)x((n1−1)/2)−1 e−x/2,

x > 0

f (y) = 1

2(n2−1)/2�((n2 − 1)/2)y((n2−1)/2)−1 e−y/2,

y > 0

Pa2 =∫x∈I ∗

2

{Pr[s12 ∈ I4 | s1 = x]}f (x) dx

=∫x∈I ∗

2

[ ∫y∈[A,B]

y((n2−1)/2)−1 e−y/2

2(n2−1)/2�((n2 − 1)/2)dy

]

× x((n1−1)/2)−1 e−x/2

2(n1−1)/2�((n1 − 1)/2)dx

where I∗2 can be written as follows (see the derivation

of I∗2 in Appendix B):

I∗2 =

[(n1 − 1)

(c41 −D

√1 − c2

41

)2,

(n1 − 1)(c41 −D1

√1 − c2

41

)2]

∪[(n1 − 1)

(c41 +D1

√1 − c2

41

)2,

(n1 − 1)(c41 +D

√1 − c2

41

)2]

Thus constraint (2) can be written as:Pr[Out of Control | σ = σ0] � α, i.e. as shownin Formula 3.

Constraint (3) ensures that the probability ofdeciding that there is no shift in process, when thereis actual shift in process standard deviation less thanor equal to a predefined acceptable probability:

Pa1 + Pa2 � β

When there is a shift in the process standarddeviation from σ to σ1, then the probability that theprocess is in control at the first stage is equal to

Pa1 = Pr[s1 ∈ I1]

=∫s1∈I1

exp(−((s1−c41σ1)

/(σ1

√1−c2

41

))2/2)

√2πσ1

√1−c2

41

ds1

When there is a shift in the process standard deviationfrom σ to σ1, then the probability that the process isin control at the second stage is computed as shown inFormula 4.

Define

A1 = rσ 2

σ 21

(c4 +D2

√1 − c2

4

)2 − x

B1 = rσ 2

σ 21

(c4 −D2

√1 − c2

4

)2 − x

Thus, the conditional probability that random variables12 will fall into interval I4 given that random variables1 belongs to interval I2 is computed as

Pa2 = Pr[s12 ∈ I4 | s1 ∈ I2]=∫x∈I ∗∗

2

{Pr[s12 ∈ I4 | s1 = x]}f (x) dx



Formula 4

Pr

[σ 2

σ 21

(c4 −D2

√1 − c2

4

)2�(n1 − 1)s2

1/σ21 + (n2 − 1)s2

2/σ21

n1 + n2 − 2� σ 2

σ 21

(c4 +D2

√1 − c2

4

)2]

= Pr

[rσ 2

σ 21

(c4 −D2

√1 − c2

4

)2 − x � y � r σ2

σ 21

(c4 +D2

√1 − c2

4

)2 − x]

Formula 5

∫s1∈I1

exp

(−((s1 − c41σ1)

/(σ1

√1 − c2

41

))2 /2

)√

2π σ1

√1 − c2

41

ds1

+∫x∈I ∗∗

2

[∫y∈[A1,B1]

y((n2−1)/2)−1 e−y/2

2(n2−1)/2�((n2 − 1)/2)dy

]x((n1−1)/2)−1 e−x/2

2(n1−1)/2�((n1 − 1)/2)dx � β

where I∗∗2 can be written as follows (see the derivation of I∗∗

2 in Appendix B):

I∗∗2 =

[(n1 − 1)

σ 2

σ 21

(c41 −D

√1 − c2

41

)2, (n1 − 1)

σ 2

σ 21

(c41 −D1

√1 − c2

41

)2]

∪[(n1 − 1)

σ 2

σ 21

(c41 +D1

√1 − c2

41

)2, (n1 − 1)

σ 2

σ 21

(c41 +D

√1 − c2

41

)2]

Formula 6

Minn1,n2,D,D1,D2

n1 + n2

∫s1∈I2

exp

(−((s1 − c41σ0)

/(σ0

√1 − c2

41

))2 /2

)√

2π σ0

√1 − c2

41

ds1 (4)

Subject to

1 −

∫s1∈I1

exp

(−((s1 − c41σ0)

/(σ0

√1 − c2

41

))2 /2

)√

2π σ0

√1 − c2

41

ds1

−∫x∈I ∗

2

[∫y∈[A,B]

y((n2−1)/2)−1 e−y/2

2(n2−1)/2�((n2 − 1)/2)dy

]x((n1−1)/2)−1 e−x/2

2(n1−1)/2�((n1 − 1)/2)dx � α (5)

∫s1∈I1

exp

(−((s1 − c41σ1)

/(σ1

√1 − c2

41

))2 /2

)√

2π σ1

√1 − c2

41

ds1

+∫x∈I ∗∗

2

[∫y∈[A1,B1]

y((n2−1)/2)−1 e−y/2

2(n2−1)/2�((n2 − 1)/2)dy

]x((n1−1)/2)−1 e−x/2

2(n1−1)/2�((n1 − 1)/2)dx � β (6)



=∫x∈I ∗∗

2

[∫y∈[A1,B1]

y((n2−1)/2)−1 e−y/2

2(n2−1)/2�((n2 − 1)/2)dy

]

× x((n1−1)/2)−1 e−x/2

2(n1−1)/2�((n1 − 1)/2)dx

Constraint (3) can be written as (for a givenintended change in process standard deviation)

Pr[In Control | σ = σ1] � β

i.e. as shown in Formula 5.Then, the optimization problem (1)–(3) can be

written as shown in Formula 6.

4. SOLVING THE OPTIMIZATION PROBLEMUSING A GENETIC ALGORITHM (GA)

As one can see that the minimization problem for-mulated by model (4)–(6)) is characterized by mixedcontinuous-discrete variables and discontinuous andnon-convex solution space. Therefore, if standard non-linear programming techniques are used for solvingthis type of optimization problem, they will be inef-ficient and computationally expensive. Genetic algo-rithms (GAs) are well suited for solving such problemsand, in most cases, find a global optimum solution witha high probability [6]. Another motivation in usingGAs to solve model (4)–(6) is that since the pioneeringwork by Holland [7], GAs have been developed intoa general and robust method for solving all kindsof optimization problems (e.g. Goldberg [8], Porti-eter and Stander [9], Pham and Pham [10], Vinterboand Ohno-Machado [11]) and computing software forapplications of GAs have been made commerciallyavailable in the market. Thus solving model (4)–(6)using GAs provides a practical way for real-timestatistical process control implementation. The GAshave been used for the statistical design of DS andtriple sampling X-control charts [12].

The tool used for implementing the GAs insolving model (4)–(6) is a commercially available GAsoftware called Evolver [13]. Evolver is used as anadd-in program to the Microsoft Excel spreadsheetapplication. The optimization model (4)–(6) is setup in an Excel spreadsheet and solved by the GAin Evolver. The operation of the genetic algorithminvolves the following steps:

(a) create a random initial solution;(b) evaluate fitness, i.e. the objective function that

minimizes the average sample size when theprocess is in control;

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

The quality of the solutions generated by theGAs depends on the setup of its parameters such aspopulation size, crossover and mutation probability.During the implementation, the values of theseparameters are setup to obtain the best results.

Crossover probability determines how oftencrossover will be performed. If there is no crossover,an offspring (new solution) will be an exact copy ofthe parents (old solutions). If there is a crossover, anoffspring (new solution) is made from parts of parents’chromosome. If crossover probability is 100%, thenoffspring are made by crossover. Crossover is madein the hope that new chromosomes will have the goodparts of the old chromosomes and maybe will bebetter. However, it is good to leave some part of thepopulation to survive to the next generation.

Mutation probability determines how often parts ofthe chromosomes will be mutated. If there is no muta-tion, offspring will be taken after crossover (or copy)without any change. If mutation probability is 100%,the whole chromosome is changed. Mutation is madeto prevent the search by the GA falling into localextremes, but it should not occur very often, becausethen the GA will in fact change to random search.

In our computational experiment, the populationsize is set to 1000. The crossover probability is setupto 0.5 and the mutation probability is 0.06.

The integrations in (5) and (6) are computed withnumerical integration using Simpson’s rule [14].

5. PERFORMANCE EVALUATION OF THE DSs-CONTROL CHART

In order to evaluate the performance of the developedDS s-chart, its efficiency (measured by the averagenumber of samples inspected to detect a shift of acertain magnitude) was compared with that of thetraditional one. For any control chart for variables, theaverage number of samples before a shift of a certainmagnitude is detected should be as small as possible.The average number of samples before the false alarmshould be as large as possible. The ability of thes-charts to detect a shift in process quality is describedby their operating-characteristic (OC) curves.

Another way to evaluate the decisions regardingsample size and sampling frequency is throughthe average run length (ARL) of the control chart.Essentially, ARL is the average number of points thatmust be plotted before a point indicates an out-of-control condition [2]. For any Shewhart control chart,ARL can be expressed as ARL = 1/P (one point plotsout of control) or ARL0 = 1/α for the in-control ARLand ARL1 = 1/(1 − β) for the out-of-control ARL.



In order to compare the DS s-control chart withthe traditional one a procedure similar to that usedby Daudin [1] for comparing DS X-control carts withShewhart charts is used. In Daudin’s procedure, theefficiency of a DS X-chart is compared with that ofa corresponding Shewhart X-chart that is matchedwith the same ARL0 and ARL1 as the DS X-chart.The efficiency is measured by the number of samplescollected on average to detect the shift of a givenmagnitude.

In our procedure, for a given pair of ARL1 pointsand a given shift in process standard deviation, aDS s-chart and traditional s-chart were constructed.Then, the efficiency of the two charts was compared.Here a shift in process standard deviation is expressedas a ratio of the shifted process standard deviation σ1to the normal process standard deviation σ0:

λ = σ1

σ0

In our computational experiment, the DS s-chartsand traditional s-charts were constructed over a rangeof λ values from 1.2 to 6.0. The two ARL points usedin the comparison were: ARL0 = 370.4 (α = 0.0027)and ARL1 = 1.222 (β = 0.1817).

Since for a traditional s-chart with 3σ control limits,ARL0 = 370.4, hence we try to determine its samplesize n such that ARL1 � 1.222, i.e.

β = Pr{LCL � s � UCL | σ = σ1} � 0.1817

For constructing a corresponding DS s-controlchart, the two ARL points (α and β values) and thegiven σ1 value (obtained from given λ and known σ0)were used to solve the optimization model (4)–(6) witha GA.

It should be noticed that after the determinationof two matched charts, for shifts less than λ the DSchart has a lower ARL and for shifts larger than λthe DS has a higher ARL. For example, a plot of theARL of a traditional s-chart with a sample size of18 and the corresponding ARL of a DS s-chart withan average sample size E(N) = 13.68 is shown inFigure 2.

From Figure 2 we see that when λ � 1.8 the DS ismore efficient and when λ � 1.8 the traditional chartis more efficient.

Having constructed the charts, the efficiency of thecharts is compared in terms of the number of samplesrequired to detect a shift of a certain magnitude.The results are provided in Table 1.

From Table 1, one can observe that the results couldbe grouped into two major patterns. In comparisonwith the traditional s-control chart, the computational

results show that for a relatively small shift in processstandard deviation (1.2 � λ � 2.1) there isa significant reduction in the average sample sizewhen the DS s-control charts are used. For thosewho are interested in the detection of shifts lessthan the corresponding λ, when λ changes from 1.2to 2.0, the DS s-control chart could be used tosignificantly reduce the cost associated with collectingand inspecting samples.

When the shift ratio λ is greater than 2.1, thecomputational results show that the DS s-controlchart is required to have the same sample size ofn1 = 6 and n2 = 16 to detect those shifts, while fortraditional s-control charts the sample size required todetect certain shift is decreasing as shift magnitudeis increasing. The reason for such a behavior is thatwhen the sample size is relatively small (n1 � 5),the s-control chart becomes one-sided and the normalapproximation is not valid anymore. Indeed, whenthe chart is one-sided the area under the normaldistribution curve becomes less than 1. While samplesize decreases the area under the normal distributioncurve decreases accordingly. With probability lessthan unity, the normal approximation cannot providesatisfaction of the constraints. In particular, forconstraint (2), there is no such a combination ofdesign variables which could ensure the probabilityof a false signal less than 0.0027, because thedifference between unity and the combined probabilityof mistakenly deciding that process is out of controlfor both stages (which is less than 1 by morethan 0.0027) is always going to be more than0.0027.

For the case where λ < 1 the model behaves asexpected, i.e. a larger average sample size is neededto detect a smaller shifts and smaller average samplesize is needed to detect larger shifts. The results alsoshow that for the same shift, the DS s-charts require asmaller average sample size.

From the statistical theory, when the samples comefrom a normal population we know the expected meanand standard deviation of s in samples, but not thetype of distribution. What kind of distribution does shave for small sample sizes? For example, for samplesof size five or less while setting up the limits fortraditional s-chart, the lower control limit is set up to

zero, because c4σ −3σ√

1 − c24 is becoming negative.

The reason is that the real distribution differs fromthe normal distribution significantly, that is why theLCL implied to be 3σ away from the mean becomesa negative number (the s distribution does not extendfor 3σ from the left side of the mean, it starts less



Figure 2. The ARL of a traditional s-control chart with a sample size of 18 and the corresponding ARL of a DS s-control chart with an averagesample size E(N) = 13.68

Table 1. The results of comparison between the DS s-control charts and the Shewhart type s-control charts

DS chartn n1 n2 λ = σ1/σ0 ARL0 ARL1 D1 D D2 E(N)

41 20 30 0.6 370.4 1.222 1.69 4.10 2.59 22.71175 82 174 0.8 370.4 1.222 1.58 4.15 2.62 101.73211 133 200 1.2 370.4 1.222 1.79 3.60 2.95 146.9299 65 80 1.3 370.4 1.222 1.79 3.30 2.99 70.5559 37 71 1.4 370.4 1.222 1.71 3.04 2.53 42.2540 25 54 1.5 370.4 1.222 1.75 3.03 2.64 29.1730 17 50 1.6 370.4 1.222 1.66 3.01 2.73 21.7622 15 26 1.7 370.4 1.222 1.80 3.01 2.65 16.8118 11 29 1.8 370.4 1.222 1.67 3.00 2.63 13.6815 10 20 1.9 370.4 1.222 1.79 3.03 2.61 11.4013 8 23 2.0 370.4 1.222 1.76 3.02 2.58 9.7512 8 12 2.1 370.4 1.222 1.79 3.02 2.66 8.8410 6 16 2.2 370.4 1.222 1.49 3.04 2.68 8.119 6 16 2.3 370.4 1.222 1.77 3.02 2.52 6.899 6 16 2.4 370.4 1.222 1.77 3.02 2.52 6.898 6 16 2.5 370.4 1.222 1.77 3.02 2.52 6.895 6 16 3.0 370.4 1.222 1.77 3.02 2.52 6.895 6 16 4.0 370.4 1.222 1.77 3.02 2.52 6.894 6 16 4.5 370.4 1.222 1.77 3.02 2.52 6.894 6 16 5.0 370.4 1.222 1.77 3.02 2.52 6.89



than 3σ away from the mean). It means that thereal distribution is asymmetric, with a shorter tailfrom the left of the mean and a longer tail from theright.

The assumption that s has a normal distributionis becoming unconvincing as the sample sizedecreases, because the real distribution becomes moreasymmetrical. On the other hand, we can assume that shas a normal distribution for moderate or large samplesizes because it is known that the real distributionbecomes symmetrical. A good indicator of whetherwe can assume the normality of the distribution ofs could be the LCL of the s-control chart. If it ispositive, it means that the left tail extends for at least3σ from the mean. If LCL is negative, it means thatthe s distribution is highly asymmetric and the smallerthe sample size the more asymmetric the distribution.For samples of size five or less, it is not reasonable toassume that s has a normal distribution, because themean is more shifted towards the left side. It meansthat the computational data for the shifts of more than2.1 are not valid and cannot be used for a comparisonof the two charts.

We can conclude that the assumption of thenormality of s is valid only for samples of size sixand more and we consider the samples of size five andless as having another distribution (non-symmetrical);we investigate this distribution more closely in otherresearch. Also, it could be a good topic of investigationto find out whether the loosing constraint (3) couldgive an economical advantage due to inspectingsmaller sample sizes or not.

6. CONCLUSIONS

DS X-control charts are designed to allow the quickdetection of a small shift of process mean andprovides a quick response in an agile manufacturingenvironment. However, DS X-control charts assumethat the process standard deviation remains unchangedthroughout the entire course of the statistical processcontrol. Therefore, a complementary DS chart thatcan be used to monitor the process variationcaused by changes in process standard deviationshould be developed. In this paper, the developmentDS s-chart for quickly detecting small shifts inprocess standard deviation for agile manufacturing ispresented. The construction of the DS s-charts is basedon the same concepts in constructing DS X-charts andis formulated as an optimization problem and solvedwith a GA. The efficiency of DS s-control charts iscompared with that of the traditional s-control chartsand the results obtained are presented. The results

of the comparison show that the DS s-control chartscan be a more economically preferable alternativein detecting small shifts than traditional s-controlcharts.

The work presented in this paper was an attemptto develop a quick and efficient control chart for thedetection of a small shift in the process variance.The results of the comparison show that the averagesample size required for detection of a small shiftis much less than for DS s-control charts. However,the assumption that s has normal distribution isbecoming invalid when sample size is less than five.That is why in this paper we did not consider thecomparison of the two charts for smaller samplesizes. A future research work could include developingDS s-charts without the assumption of a normaldistribution.

It is known that many practitioners still preferR-control charts to s or to s2, but for moderate valuesof sample size n, say n � 10, a R-chart rapidlyloses its efficiency, as it ignores all the informationin the sample between maximum and minimumvalues [2]. The results of the comparison between theDS s-control charts and the traditional s-control chartshave shown that the DS s-charts can be very efficientfor a sample size greater than ten. So our convictionis that the DS s-control charts could be used when thesample size is moderate or large.

ACKNOWLEDGEMENT

The authors of the paper would like to thankthe anonymous reviewers for suggestions on thederivation of c4 and the revision of the paper.

APPENDIX A. DERIVATION OF c4

Implicitly it is being assumed that the qualitymeasurement follows a normal distribution with amean µ and a standard deviation σ . As randomvariables

x = (n1 − 1)s21

σ 2and y = (n2 − 1)s2

2

σ 2

follow chi-square distributions with degrees offreedom n1−1 and n2−1, respectively, the sum of twoindependent chi-squares is a chi-square distributionwith degrees of freedom

(n1 − 1)+ (n2 − 1) = n1 + n2 − 2



It now follows that

s12 =√(n1 − 1)s2

1 + (n2 − 1)s22

n1 + n2 − 2

=√√√√ σ 2

n1 + n2 − 2

[(n1 − 1)s2

1

σ 2 + (n2 − 1)s22

σ 2

]

=√

σ 2

n1 + n2 − 2Y

= σ√n1 + n2 − 2

Y 1/2

where Y has a chi-square distribution with n1 +n2 −2degrees of freedom. Thus,

E(s) = µs

= σ√n1 + n2 − 2

E[Y 1/2]

= σ√n1 + n2 − 2

×∫ ∞

0y1/2 y((n1+n2−1)/2)−1 e−y/2

�((n1 + n2 − 1)/2)2(n1+n2−1)/2dy

= σ�((n1 + n2 − 1)/2)2(n1+n2−1)/2

√n1 + n2 − 2�((n1 + n2 − 2)/2)2(n1+n2−1)/2

×∫ ∞

0y1/2 y((n1+n2−1)/2)−1 e−y/2

�((n1 + n2 − 1)/2)2(n1+n2−1)/2dy

=(√

2

n1 + n2 − 2

)�[(n1 + n2 − 1)/2]�[(n1 + n2 − 2)/2]σ

= c4σ

APPENDIX B. DERIVATION OF INTERVALS OFI∗

2 AND I∗∗2

Since s1 is in interval I2 and

I2 =[c41σ −Dσ

√1 − c2

41, c41σ −D1σ

√1 − c2

41

]∪[c41σ +D1σ

√1 − c2

41, c41σ +Dσ√

1 − c241

]then

c41σ −Dσ√

1 − c241 � s1 � c41σ −D1σ

√1 − c2

41

and

c41σ +D1σ

√1 − c2

41 � s1 � c41σ +Dσ√

1 − c241

For the case when there is no shift in the processvariability, i.e. λ = 1, by multiplying both sides of

the inequality signs by (n1 − 1)/σ 2, we obtain

(n1 − 1)(c41σ −Dσ

√1 − c2

41

)2

σ 2

�s2

1 (n1 − 1)

σ 2

�(n1 − 1)

(c41σ −D1σ

√1 − c2

41

)2

σ 2

and

(n1 − 1)(c41σ +D1σ

√1 − c2

41

)2

σ 2

�s2

1 (n1 − 1)

σ 2

�(n1 − 1)

(c41σ +Dσ

√1 − c2

41

)2

σ 2

Since x is defined as x = (n1 − 1)s21/σ

2 then

(n1 − 1)(c41 −D

√1 − c2

41

)2

� x � (n1 − 1)(c41 −D1

√1 − c2

41

)2

and

(n1 − 1)(c41 +D1

√1 − c2

41

)2

� x � (n1 − 1)(c41 +D

√1 − c2

41

)2

Thus, interval I∗2 can be written as

I∗2 =

[(n1 − 1)

(c41 −D

√1 − c2

41

)2,

(n1 − 1)(c41 −D1

√1 − c2

41

)2]

∪[(n1 − 1)

(c41 +D1

√1 − c2

41

)2,

(n1 − 1)(c41 +D

√1 − c2

41

)2]

When there is a shift in process standard deviationfrom σ to σ1, then the interval I∗∗

2 within which therandom variable x should vary, could be obtained asfollows. We have

c41σ −Dσ√

1 − c241 � s1 � c41σ −D1σ

√1 − c2

41

and

c41σ +D1σ

√1 − c2

41 � s1 � c41σ +Dσ√

1 − c241



By multiplying by both sides of the inequality signsby (n1 − 1)/σ 2

1 , we obtain

(n1 − 1)(c41σ −Dσ

√1 − c2

41

)2

σ 21

�s2

1 (n1 − 1)

σ 21

�(n1 − 1)

(c41σ −D1σ

√1 − c2

41

)2

σ 21

and

(n1 − 1)(c41σ +D1σ

√1 − c2

41

)2

σ 21

�s2

1 (n1 − 1)

σ 21

�(n1 − 1)

(c41σ +Dσ

√1 − c2

41

)2

σ 21

Since x is defined as x = (n1 − 1)s21/σ

21 then

(n1 − 1)σ 2

σ 21

(c41 −D

√1 − c2

41

)2

� x � (n1 − 1)σ 2

σ 21

(c41 −D1

√1 − c2

41

)2

and

(n1 − 1)σ 2

σ 21

(c41 +D1

√1 − c2

41

)2

� x � (n1 − 1)σ 2

σ 21

(c41 +D

√1 − c2

41

)2

Therefore, I∗∗2 can be written as

I∗∗2 =

[(n1 − 1)

σ 2

σ 21

(c41 −D

√1 − c2

41

)2,

(n1 − 1)σ 2

σ 21

(c41 −D1

√1 − c2

41

)2]

∪[(n1 − 1)

σ 2

σ 21

(c41 +D1

√1 − c2

41

)2,

(n1 − 1)σ 2

σ 21

(c41 +D

√1 − c2

41

)2]

REFERENCES

1. Daudin JJ. Double sampling X-bar charts. Journal of QualityTechnology 1992; 24(2):78–87.

2. Montgomery DC. Introduction to Statistical Quality Control.John Wiley and Sons: New York, NY, 2001.

3. Grant EL, Leavenworth RS. Statistical Quality Control.McGraw-Hill: New York, NY, 1996.

4. Costa AFB, Rahim MA. Economic design of X-bar and Rcharts under Weibull shock models. Quality and ReliabilityEngineering International 2000; 16(2):143–156.

5. Tolley GO, English JR. Economic design of constrainedEWMA and combined EWMA-X bar control schemes. IIETransactions 2001; 33(6):429–436.

6. Rao SS. Engineering Optimization: Theory and Practice. JohnWiley and Sons: New York, NY, 1996.

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Authors’ biographies:

David He, PhD (Industrial Engineering), MBA and BS(Metallurgical Engineering), is an Associate Professor inthe Department of Mechanical and Industrial Engineeringat The University of Illinois at Chicago. His main areasof interest are in agile manufacturing, quality control,reliability and safety analysis of manufacturing systems andmanufacturing scheduling.

Arsen Grigoryan, MS (Industrial Engineering), BS (CivilEngineering), is a PhD student at the Department ofMechanical and Industrial Engineering at The University ofIllinois at Chicago. His main area of interest is in statisticalquality control.


Construction of double sampling s-control charts for agile manufacturing

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