Approximate Tolerance Limits for Cp Capability Chart Based on Range - Kuo

8/13/2019 Approximate Tolerance Limits for Cp Capability Chart Based on Range - Kuo

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ISSN 0974-3235ProbStat Forum, Volume 03, October 2010, Pages 145157

Approximate tolerance limits forCp capability chart based on Range

Hsin-Lin KuoDepartment of Business Administration, Chung-Yu Institute of Technology

40, Yi-7 Road, Keelung, Taiwan 201, R.O.Ce-mail: [email protected]

Paper received on 30 January 2010; revised, 15 June 2010; accepted, 24 July 2010.

Abstract

Kotz and Johnson [10], Deleryd [1] indicated that there was a gap betweentheory and practice of process capability studies. Then how to reduce the gapand the variation between theory and practice of process capability studies has

been become a serious problem.Most of results obtained so far regarding the distributional properties of

estimated capability indices are based on the assumption of a simple sample ofobservations from the normally distributed process, which is in-control. To useestimators based on several small subsamples and then interpret the resultsas if they were based on a single sample may result in incorrect conclusions.For the sake of use past in-control data from subsamples to make decisionregarding process capability, the distribution of the estimated capability indexwith subgrouping should be considered.

And then, we know that all the information gathered from the productinspection process in the manufacturing industries can be used as the rationalsubgroup data in real. Therefore, we consider estimator that naturally occurwhen using an X-chart together with aR-chart in quality control. In addition,we make use of the Patnaiks [13] approximation to the central Chi-squareddistribution, to construct a procedure approximate tolerance limits for thesampling distribution of the Cpto assess the performance and enhance practicalvalue for the capability chart based on range.Keywords. Process Capability Chart; Approximate Tolerance Limits; Range;Relative Efficiency.

1 IntroductionProcess capability indices are widely used to measure whether the productquality meets to customers requirements. They can help companies to pro-mote marketing sales, retain customers and reduce the process variability. Thesetting and communication becomes much simpler and easier by using processcapability indices to express process capability between manufacturers and


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146 Hsin-Lin Kuo

customers. The use of these indices provides a unitless language for evaluat-ing not only the actual performance of production processes, but the potentialperformance as well. The indices are intended to provide a concise summary ofimportance that is readily usable. Engineers, manufacturers, and suppliers cancommunicate with this unitless language in an effective manner to maintain

high process capabilities and enable cost savings.Since capability analysis of a process and effectiveness of control charts are

directly related, therefore, Kuo, et al. [5, 6, 7]provided a confidence bound anda test of hypothesis for Cpm andCpp based on subsamples. Not only we makeuse of a better estimator of for moderate large sample size, which was thatintroduction by Kirmani, et al. [9] and Derman and Ross [2], respectively,but also consider with variable sample size, to construct a procedure lowerconfidence bounds in some detail in connection with minimum values for Cp.

Montgomery [12] and Juran and Gryna [8] pointed out that control charts,in addition to monitoring a process, provide estimates of the process param-

eters that are useful in capability studies. No matter how, not only we makeuse of a link between capability indices and tolerance limits, and we propose toutilize the information gathered by control charts to estimate tolerance limitsof a process on a continuous basis, and also construct an approximate tolerancecharts for Cp based on range. The procedures have been included.

Montgomery [12] pointed out that control chart is an on-line process con-trol technique widely used for this purpose. Control chart is an importantpart of the magnificent seven major tool of SPC (Statistical Process Control).Moreover, control charts are of great use in the analysis and control of man-ufacturing processes, so as to produce quality that is satisfactory adequate,dependable and economic. Nevertheless, the general characteristics of control

charts, and their usefulness, and emphasized two general purposes:(a) Analysis of past data for control. (b) Process control versus given

standards.

Now that we know all existing process capability indices have some weak-nesses. Deleryd [1, Figure 1] indicated that Cp react to changes in processdispersion but not to change of process location. Cpk reacts to changes both inprocess dispersion and location. Cpm and Cpmk react more strongly to changesboth in dispersion and location than Cpk. And Cpmk is more sensitive thanCpm to deviations from the target value T. The index Cpp is useful to evalu-ate process capability for a single product in common situation, it cannot be

applied to evaluate the multi-process capability. The index Cpp is a simpletransformation from the index Cpm, and provided individual information con-cerning the process accuracy and process precision. Capability indices are keymeasures in the context of never-ending improvement in quality. The processmeasures are estimated based on a single random sample of observations fromthe normally distributed and in statistical control.

Extensive studies have been conducted to determine the effects of non-


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Approximate tolerance limits for 147

normality on the carious capability indices since Gunter [3] bemoaned the manydrawbacks ofCpk in particular. Several methods for handling non-normal datahave been suggested. Standard measurement-system analysis criteria assumethe gauge measures a single variable. However, in some kind of manufacturing,measurement systems take data for many quality characteristics, to support

using these data as a multivariate response. Majeske [11] develops multivariateextensions of gauge-approval criteria precision to tolerance ratio, percent R&R,and signal-to-noise ratio.

A successful implementation of process capability studies require that properresources are allocated, which is a managerial responsibility. The proper ed-ucation and training can be provided. So we can give proper education andtraining every co-worker knows the method. Then the conservative personalattitudes may be changed and the prerequisites for handling the practical prob-lems are much more promising.

To bridge some of the gap between theory and practice, loss function can

be used. In general, Spring, et al. [14] and Vannman [15] think that there is aneed for simple graphical tools to bridge some of the gap between practice andtheory in capability studies. And the visual impact of a plot is more effectivethan numbers, for example, estimates or confidence limits.

The capability of a process and effectiveness of control charts are directlyrelated. The gap between theoreticians and practitioners is, we believe andhope, mainly through software, but there still remain numerous instances oflack of understanding of the purpose and usage of Process capability indices(PCIs) and process performance.

In this paper, we use the Cp index, but also consider various sample size toconstruct the approximate tolerance limits for Cp, based on range. To assess

the capability of a process, it is proposed to consider estimated tolerance lim-its in capability analysis, along with control charts for monitoring the processmean and process standard deviation, respectively, and come from a normaldistribution and are independent. The capability chart used in conjunctionwith the traditional Shewhart variables charts will provide evidence of im-provement. It may also assist in ending the unfortunate practice of includingspecification limits on the Xchart andR-chart will incorporate the limits intothe calculation of process capability.

2 Approximate tolerance limits for

Cp based

on range

In any production or manufacturing process, regardless how well it was de-signed or carefully maintained, a certain amount of inherent or natural vari-ability will always exists. In the framework of statistical quality control, thisnatural variability is often called a stable system of chance causes. A process


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148 Hsin-Lin Kuo

Table 1: The R.E. of the range method to S2

n 2 3 4 5 6 7 10R.E. 1.000 0.992 0.975 0.955 0.930 0.910 0.850

that is operating with only chance causes of variation is said to be in statisti-cal control. Other kinds of variability (assignable causes) may occasionally bepresent in the output of a process is said to be out of control.

A quality characteristic that is measured on a numerical scale is called avariable. Usually it needs to be monitored both the mean value of the qualitycharacteristic and its variability. The most commonly used types of controlcharts for variable are Xcharts related to the process level, and the standarddeviationSchart and the rangeR chart related to the process variability.

LetXi1, Xi2, . . . , X in, i = 1, . . . ,m, be m preliminary independent randomsamples of sizenfrom normal distribution with meanand standard deviation

. The ith subsample mean is

Xi=(Xi1+ Xi2+ + Xin)

n (2.1)

In most cases, both and are unknown. Therefore, they need to be esti-mated from the preliminary sample or subsamples from process which is instatistical control. These estimates are usually based on at least 20 to 25 sub-samples. Suppose that m subsamples are available, each subsample containsn observations on the quality characteristic. Then, a reasonable estimator of, the process mean, is the grand average, say

X=(X1+ X2+ + Xm)m

(2.2)

To construct the confidence bounds for some process capability indices, weneed an estimate of the standard deviation . We may estimate either bythe sample standard deviation or the range ofm subsamples means. Mont-gomery [12] point out, the relative efficiency (R.E.) of the range method to thesample variance S2 for various sample size based on a single sample is shownin Table 1.

However, for the small sample size, n= 4, 5, or 6, the range chart methodis often employed and it is entirely satisfactory. But, for moderate value of

n, say, n 10, the range method loses its efficiency rapidly since it ignoresall the information in the sample between Xmax = max{X1, X2, . . . , X n} andXmin = min{X1, X2, . . . , X n}. In this case, theSchart is preferred to the Rchart.

As stated before, in this paper, a common practice in process control is toestimate the process capability indices by analyzing the past in control data.Suppose that m subsamples, each of size n, are available, and then we can


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estimate by X, given in (2.2). Since Xis equal to the average of all of themndata values, it is the natural estimator of.

Let X1, X2, . . . , X n be a random sample of size n drawn from a normalpopulation with mean and standard deviation . The range of this singlesample is defined by

R= XmaxXmin, (2.3)where Xmax= max{X1, X2, . . . , X n} and Xmin= min{X1,X2, . . . , X n}.

Suppose the total samples are grouped into m subsamples such that eachsubsample containsn observations. The mean of the m ranges will be denotedby Rm,n and the range of a single sample of size n is denoted by R1,n.

When E(R) =d2 and Var(R) =2d2

3, the mean and variance ofRm,n/

are given byE(Rm,n/) =E(R1,n/) =d2, (2.4)

and

Var (Rm,n/) = Var(R1,n/)/m= d

2

3/m, respectively. (2.5)Then Rm,n/d2 is an unbiased estimator of, where d2 and d3 are constants(see Hartley and Pearson [4]).

According to Patnaik [13], it has been shown that Rm,n/is approximately

distributed as c

2

. That is,

(Rm,n

)2 c22

, (2.6)

and (Rm,n

)2 c2 2, (2.7)

where 2denotes a chi-square distribution with degrees of freedom, and candare constants which are functions of the first two moments of the rangevariable, given by

= 1

(2 + 2

1 + 2(d3/d2)2/m), (2.8)

and c = d2 /2 (/2)/((+ 1)/2) d2(1 + 1/(4)). (2.9)

Using these relations, the values ofc and can be easily obtained for any nandm.

Assume that the process measurement follows N(, 2), the normal dis-tribution, the index and reasonable estimator ofCp are given as following,respectively,

Cp = (USL LSL)/(6), (2.10)and Cp = (USL LSL)/(6), (2.11)


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150 Hsin-Lin Kuo

where [LSL,USL] is the specification interval, is the process mean, isthe process standard deviation (overall process variability), under stationarycontrolled conditions.

From (2.10) and (2.11), we obtain

Cp

Cp

2=

2

, (2.12)

where = R/d2 is an unbiased estimator of, and R indicates either Rm,n

or R1,n, based on rang. Thus,

2R = C2

p/C2

p d22/c2 = (R/)2 /c2 2v. (2.13)

Apply a simple approximation procedure based on range we can obtain thetolerance limits of the estimator ofCp.

The 100(1)% approximate tolerance limits for Cp, together with

XRcharts

1 = P(21/2 2R 2/2)

=P(J1Cp Cp J2Cp),(2.14)

where

J1=d2c

2/2and J2=

d2c

21/2

, (2.15)

and 2/2() is the upper /2 quantile of the chi-squared distribution with

degrees of freedom. So, the 100(1 )% approximate tolerance limits for Cpbased on range, is given by

(J1Cp, J2Cp). (2.16)

When using XRcharts, the mean line, denoted Cp, is given by (2.17) form,as follows

Cp=USL LSL

6(Rd2

), (2.17)

resulting in upper and lower limits of the form U1= J2Cp and L1= J1Cp.

Therefore, the 100(1 )% approximate upper and lower limits for Cp inconjunction with X

Rcharts based on Range are of the form.

Approximate upper tolerance limits: U1= J2Cp

center line: Cp

Approximate lower tolerance limits:

L1= J1Cp. (2.18)


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Table 2: The step of the approximate tolerance limits for Cp capability ChartStep Cp1. Determine the value ofith subsample mean Xi and range Ri.

2. a. Compute the grand average, X.

b. Calculate the mean of the m ranges, R.3. a. Compute = Rm,n/d2.

b. Calculate the value .

c. Calculate a series of the estimates Cp.4. a. Compute center line Cp.

b. Calculate the approximate upper tolerance limits U1.c. Calculate the approximate lower tolerance limits L1.

3 The procedure

As stated before, to check it the process meets the capability requirement, wefirst determine the ith subsample mean is Xi and the ith subsample range isRi. Second, we calculate the grand average, say

X, and the mean of the mranges, say R. Third, calculate an unbiased estimator of is = R/d2and achi-square distribution withdegrees of freedom, and a series of the estimatesofCp, Cp. Finally, we compute center line Cp, and the approximate upper andlower tolerance limits U1 andL1, respectively.

Otherwise, we do not have sufficient information to conclude that the pro-cess meets the present capability requirement. In sum, we summarize thesesteps shown in Table 2.

4 Numerical example

We use the data given in Table 3 to demonstrate this procedure. This exampleis about a manufacturing process with m = 20 subsamples, each subsampleconsists ofn = 5 samples, have been taken from the process when the processwas in control. A total of 100 observations were collected and are displayedin Table 3. The upper and lower specification limits are USL= 1.2 and LSL= 0.8, respectively.

Since each subgroup in the process provides a measure of location, Xi, and

a measure of variability either Ri, an estimator ofCp can be determined foreach subsample using either equation (2.11). The result is a series of estimatesfor Cp over the life the process.

Control charts can indicate whether or not statistical control is being main-tained and provide us with other signals from the data. If the process is in astate of statistical control, then the value ofCp at the subgroup level providesinformation regarding process capability. If the process is out-of-control, then


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Figure 1: X-Rcharts and a runs chart ofCp.


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154 Hsin-Lin Kuo

Table 4: Values of the vary from subgroup to subgroup for the Cpm/1 2 3 4 5 6 7 8 9 10

0.5743 0.7384 0.7753 0.7753 0.9122 0.7384 0.8615 0.7384 0.7384 0.775311 12 13 14 15 16 17 18 19 20

0.8161 0.8615 0.8161 0.7384 0.8615 0.8615 0.9122 0.7384 1.1076 0.8161

show in Table 1.To analysis these fluctuations, limits and a means line analogous to the

Shewhart limits and center line are required. Similar to Shewhart controlcharts, the upper and lower limits for Cp will represent the interval expectedto contain 99.73% of the estimates if the process has not been altered.

From the process data, we obtain that sample mean x = 1.1206, R =0.1950, A2 = 0.577, d2 = 2.326, d3 = 0.8641, D3 = 0, D4 = 2.114, and = 72.7080. The result is a series of estimates ofCp over the life of the

process, shows in Table 4.The upper and lower limits for Cp in the conjunction with X and R

charts are of the form U1 and L1, respectively, and center line as follows isApproximate upper tolerance limits: U1= J2Cp = 1.05205,

center line: Cp= 0.79521,

Approximate lower tolerance limits: L1= J1Cp= 0.63465.The limits and mean line have been showed in Figure 2.

Apparently, the process capability estimates vary from subgroup to sub-group. With the exception of subgroup 19 the fluctuations in Cp appear to

be due to random causes. In period 19 the process capability appears to haveincreased significantly and warrants investigation. Practioners would likely at-tempt to determine what caused the capability to rise significantly and recreatethat situation in the future. If the estimated process capability had droppedbelow L1 this would signals a charge in the process (for example, subgroup1), and the process capability was not at the level required by the customer,changes in the process would be required.

Owing, in the never-ending improvement system, the process capabilityshould be under constant influence to increase. The capability chart used inconjunction with the traditional Shewhart variables charts will provide evi-dence of improvement.

5 Conclusion

Control chart is an important part of the magnificent seven major tool of SPC.Thereby, control charts are of great use in the analysis and control of man-ufacturing processes, so as to produce quality that is satisfactory adequate,


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Figure 2: X-R charts and Cp capability chart with upper and lower limits,and center line.


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156 Hsin-Lin Kuo

dependable and economic. And, the effect of any changes to the process willalso show up on the chart, thereby providing feedbacks to the practioner re-garding the effect changes to the process have on process capability.

The proposed chart is easily appended to X-Rcharts and makes easy judg-ments regarding the ability of a process to meet requirements, also providing

evidence of process performance. The capability chart represents a modifica-tion of the control charts that combines customer requirements and processperformance in a certain degree reflecting the needs of the customer as well asproviding information to the manager of the process.

Acknowledgements

The author is grateful to the editor and two anonymous referees for theirhelpful comments and suggestions which have improved the contents and styleof the article.

References

[1] Deleryd, M. (1998), On the gap between theory and practice of processcapability studies. International Journal of Quality and Reliability Man-agement. 15(2), pp. 178191.

[2] Derman, C., and Ross, S. M. (1995), An improved estimator ofin qualitycontrol. Probability in the Engineering and Informational Sciences, 9(3),411416.

[3] Gunter, B. H. (1989 a,b,c,d), The use and abuse of Cpk, Parts 14.QualityProgress. 22, (1) pp. 7273; (3) pp. 108109; (5) pp. 7980; (7) pp. 8687.

[4] Hartley and Pearson. (1951). Moment Constants for the Distribution ofRange in Normal Samples. Biometrika, 38(3/4), pp. 463464.

[5] Hsin-Lin Kuo.(2005). Dec. Practical Implementation of the Cpp Indexwhen using Subsamples. Management Sciences Research, Special Issue,pp.6575.

[6] Hsin-Lin Kuo, Chin-Chuan Wu, Chao-Hshien Wu, 2006. ApproximateConfidence Bounds for the Process Capability Index Cpm Based on the

Range. International Journal of Information and Management Sciences,17 (2), pp.5770.

[7] Hsin-Lin Kuo, Chin-Chuan Wu, Shu-Ching Hsu.(2008). ApproximateConfidence Bounds for the Process Capability Index Cpm Based on sub-samples.International Journal of Information and Management Sciences,19(4), pp. 601620.


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[8] Juran, J. M. and Gryan, F. M.(1980), Quality Planning and Analysis,McGraw-Hill, New York.

[9] Kirmani S. N. U. A., Kocherlakota K. and Kocherlakota S.(1991), Esti-mation of and the process capability index based on subsamples, Com-

munication in Statistic-Theory Method, 20(1), pp. 275291.

[10] Kotz, S. & Johnson, N. L. (2002), Process Capability Indices-A review,19922000 (with discussion), Journal of Quality Technology, Vol. 34, No.1, pp.253.

[11] Majeske, K. D. (2008), Approval Criteria for Multivariate MeasurementSystems, Journal of Quality Technology, 40(2), pp. 140153.

[12] Montgomery, D. C. (2001),Introduction to Statistical Quality Control(4thed.), New York.

[13] Patnaik, P. B.(1950), The use of mean range as an estimator of variancein statistical tests, Biometrika, Vol. 37, pp.7892.

[14] Spiring, F., Cheng, S., Yeung, A. and Leung, B. (2002), Discussion,Jour-nal of Quality Technology, 34(1), pp. 2327.

[15] Vannman, K. (2002), Discussion, Journal of Quality Technology, 34(1),pp. 4042.

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Approximate Tolerance Limits for Cp Capability Chart Based on Range - Kuo

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