a review and interpretations of process capability indices

J.C. Baltzer AG, Science Publishers

A review and interpretations ofprocess capability indices

Kurt Palmera,H and Kwok-Leung TsuibaDepartment of Industrial and Systems Engineering, University of Southern California,

Los Angeles, CA 90089-0193, USA

E-mail: [email protected]

bDepartment of Industrial Engineering and Engineering Management,Hong Kong University of Science and Technology,

Clear Water Bay, Hong Kong

E-mail: [email protected]

Practitioners of industrial statistics are generally familiar with the common Cp and Cpkprocess capability indices. However, many additional indices have been proposed, andknowledge of these is less widespread. More importantly, information regarding the indicescomparative behavior is lacking. This paper compares the behavior of various indices undershifting process conditions. Both useful and misleading characteristics of the indices areidentified. We begin with a short history of process capability measures. Several processcapability indices are reviewed. Application areas for capability indices are also summarized.The indices are grouped according to the loss functions which are used in their interpretation.Characteristics of the various indices are discussed. Finally, recommendations are made forselection of indices at differing levels of process performance.

Keywords: capability index, loss function, process improvement

1. Introduction

While the concept of process capability was most likely developed by Americanstatisticians, use of process capability indices by the United States industrial commu-nity did not become popular until reports of Japanese manufacturing methods appearedin American trade and professional journals. One of the earliest domestically publisheddescriptions of capability indices was by Sullivan [20,21]. Kane [13] provided thefirst discussion of the indices sampling characteristics, and made some suggestions

H Corresponding author.

Annals of Operations Research 87(1999)3147 31

y A review and interpretations of process capability indices

for modifications to the indices. Kotz and Johnson [14] gave an exhaustive summaryof modified indices and their sampling characteristics. While use of the originalcapability indices has become widespread, many practitioners remain unfamiliar withthe detailed descriptions of the indices population characteristics. The motivationsfor and characteristics of the modified indices are even less well known.

The process capability indices which Sullivan reported represented an improve-ment over previous metrics that were used to describe process capability. Feigenbaum[4] and Juran [10] used 6s as a measure of process capability. They presented themeasure as a representation of the inherent variability of a process. This descriptiongave process capability an interpretation that was independent of customer specifica-tions. There was an implication of inevitability regarding process performance. Thenotion that the process doesnt know the specifications was emphasized. Whilethis is of course true, the interpretation seems to urge a passive approach to qualityimprovement.

Juran [11] created a stronger link between process variability and customer speci-fications by comparing 6s to the tolerance width as a method of determining the needfor process improvement activities. However, capability itself was still interpretedseparately from specifications. Finally, Juran and Gryna [12] proposed a capabilityratio, which provided the first metric that directly compared process variability tocustomer specifications:

Capability ratio=6s variation

tolerance width. (1)

As is the case with the capability ratio, all process capability indices explicitlylink process variability to customer specifications, and in so doing, they emphasizethe suppliers responsibility to satisfy those specifications. However, capability indicesalso have advantages over the capability ratio. Capability indices increase in value asthe process performance improves. This property may be of limited analytical value,but it does provide psychological value in that it reinforces the natural bigger isbetter predisposition. Furthermore, capability indices indicate the relative benefitsof improvements in both process location and variability.

This paper summarizes the univariate formulations of process capability indices,and reviews the basics regarding index interpretation and process improvement. Thepopulation characteristics for the indices are discussed in detail. Comparisons of theindices behavior under shifting process conditions are described. Finally, recommen-dations are given for the selection of indices to direct process improvement activitiesat various levels of process performance.

2. The original five capability indices

Sullivan [21] described five indices which he had observed in use at Japanesemanufacturing facilities. The indices were Cp, Cpk, k, Cpu, and Cpl.

32

2.1. Definitions of the original indices

While many indices now exist, all were developed from a common parent:

Cp =U S L- LSL

6s, (2)

where

USL is the upper specification limit,

LSL is the lower specification limit,

s is the process standard deviation.

The astute reader will notice that Cp is the reciprocal of Juran and Grynascapability ratio. It is not clear how Cp was developed. At the least, we believe that itis appropriate to credit Juran and Gryna with the development of the idea that led tothis family of metrics.

Conceptually, Cp compares the allowable process spread to the actual processspread. It can be thought of as indicating the potential of the process to produceconforming material. A common interpretation of Cp assumes that the process outputis normally distributed. This interpretation leads to the oft quoted proportions conform-ing that are given in table 1.

Table 1

Proportion conforming after centering.

Assumed Cp value

distribution 0.50 0.75 1.00 1.25

Normal 86.639% 97.555% 99.730% 99.982%

None > 55.556% > 80.247% > 88.889% > 92.889%

A more general interpretation does not require normality. Chebyshevs inequalitycan be used to determine a lower bound for the proportion conforming, regardless ofthe actual process distribution. These values are also shown in table 1. Since there isno guarantee that a process will be normally distributed, the Chebyshev values shouldbe kept in mind as worst-case senarios. (The proportions conforming are given aslower bounds because Chebyshevs inequality makes no assumptions regarding theshape of the distribution.)

It is critical to note two points:

(1) The metric Cp can only indicate the potential proportion conforming, due to thecentering assumption. The interpretations of proportion conforming given aboverepresent maximum values provided that the process mean is located at the speci-fication midpoint.

K. Palmer, K.-L. Tsui y A review and interpretations of process capability indices33

(2) A state of statistical control is required of the process in order for the indexvalue to have any long-term meaning. If the process is not stable, then any conclu-sions regarding the capability of the process will need to be modified as itsperformance varies.

To deal with violations of the centering assumption, the following pair of indiceswas developed:

Cpk =min(U S L- m , m - L S L)

3s,

k =m - m

(U S L- LSL) 2,

(3)

(4)

where

m is the process mean,

m is the specification midpoint, i.e. m = (USL + LSL) 2.

The index k represents a measure of the distance that the process lies off-centerand Cpk demonstrates the reduction in process capability caused by the lack of center-ing. If the assumption LSL < m < USL is made, then jkj 1 and there is a simplerelationship among Cp, Cpk, and k. The relation is Cpk = (1 jkj)Cp. This relation leadsto the concept of Cp as an upper limit for Cpk, and reinforces the description of Cp asthe potential capability. The Cpk index will have a maximum value equal to the Cpvalue when the process mean is at the specification midpoint. As the process meanshifts away from the specification midpoint, the value of Cpk decreases linearly untilit reaches a value of zero, when the process mean is equal to one of the specificationlimits.

It should be noted that both Sullivan [21] and Kane [13] describe k as an absolutevalue. We believe that it is useful for k to retain its sign for applications such as theprocess improvement roadmap, which will be presented below.

Sometimes, a customer will provide a one-sided specification. The indices Cpuand Cpl were developed for these situations.

For processes with upper specification limit only:

Cpu =U S L- m

3s. (5)

For processes with lower specification limit only:

Cpl =m - L S L

3s. (6)

The index Cpu compares the distance between the process mean and the upperspecification limit to the upper half-width of the distribution. Similarly, Cpl comparesthe distance between the process mean and the lower specification limit to the lower


half-width of the distribution. The indices show the relative size of the working margin,i.e. the closeness of the distribution to the specification limit.

The definitions of Cpu and Cpl also provide insight into the formulation of Cpk.Often, the relation Cpk = min(Cpu, Cpl) is used. Therefore, Cpk is also a measure of theremaining size of the working margin, as a result of a shift in the process mean awayfrom the center and towards one of the specification limits. Since the measure of thehalf-width, 3s , is the same for the two indices, Cpu and Cpl, this definition of Cpkimplies an assumption that the process distribution is symmetric about its mean.

2.2. Applications of capability indices

Kane [13] described six application areas for capability indices, which will beparaphrased here:

(1) A static goal for performance: A minimally desirable index value is set in orderto avoid nonconforming output, to maintain customer acceptance, or to definecontractual obligations. The stated objective is to achieve the minimal level. Thisapplication is typical of customersupplier relationships.

(2) A measure of continuous improvement: The index value is monitored over timeas an indication of relative improvement. The stated objective is to perpetuallyincrease the index value. This technique applies to both individual processes andthe distribution of index values for a collection of processes.

(3) A common process performance language: The index value is used to com-municate process performance to those who may not have a detailed knowledgeof the process. Since the value of any given index always carries the same inter-pretation of the relative relationship between process performance and customerspecifications, the indices provide consistent frames of reference regardless ofthe specific process they describe.

(4) A criterion for prioritization: The index is used in process-to-process compari-sons to determine either relative need for improvement or relative benefit forinvestment.

(5) A roadmarker to direct process improvement activities: Pairwise comparisonsof selected capability indices determine the relative benefit of adjusting processlocation versus reducing process variability.

(6) An indicator of quality system deficiencies: Index values obtained during qual-ity system audits are compared to process records to identify deficiencies insampling, measurement, process control, etc.

Shewhart [18], Juran [10] and Gryna [5] all discussed the use of process capabil-ity information to determine specification limits. In general, they recommended thatthe tolerance width should not be tighter than 6s . Feigenbaum [4] and Juran [10]referred to the use of process capability information to assign jobs to machines. Jobsrequiring tight tolerance widths would be assigned to the most capable machine in


order to minimize in-process waste. While the applications given above are certainlyamong the most common, we do not intend to suggest that this represents an exhaustivelisting.

An application of particular interest is the use of indices to direct processimprovement activities. It is common for process capability indices to be used indi-vidually for most of the applications presented above. In order to direct processadjustments, however, capability indices must be considered collectively. The indicesCp, Cpk, and k represent one such grouping. Collectively, these indices signal the needfor deliberate process location adjustments andor process variability reductions.Applying the steps sequentially, the roadmap operates as follows:

Step 1. Obtain simultaneous estimates of Cp, Cpk, and k.

Step 2. If Cpk < Cp, then evaluate k.(a) If k > 0, then adjust the process location to decrease the process mean

until Cpk = Cp.(b) If k < 0, then adjust the process location to increase the process mean

until Cpk = Cp.

Step 3. If Cp < 1.0, then identify and remove sources of process variability.

The roadmap calls for adjustments to the process location prior to variabilityreduction because process mean adjustments are considered to be relatively simple toaccomplish. As such, process mean adjustments, when necessary, can produce imme-diate improvements in process performance relative to the specifications. It is assumedthat adjustments in the process mean will have no effect upon process variability.

On the other hand, reductions in process variability, as required in step 3, aregenerally considered to be a more difficult task than that of adjusting the processlocation. Also, variability reductions may sometimes produce an unintendended shiftin the process mean. So, it may be necessary to revisit step 2 following improvementsmade during step 3.

3. Interpretations of capability indices

Over the years, two broad classes of indices have been developed. These cate-gories are defined by the underlying loss function which is used to interpret the index.The categories are: stepwise loss function interpretation and quadratic loss functioninterpretation. Figure 1 displays the two types of loss functions. It will be shown thatCp has a consistent meaning under either interpretation. The index Cp is the only indexwhich enjoys membership in both categories.

3.1. Stepwise loss function interpretation

The stepwise loss function interpretation holds that all process output which fallswithin the specification limits is equally good. The defining characteristic of the loss


function is that it has a value of zero over the entire specification range, and has avalue greater than zero outside the specification limits. The stepwise loss function isimplied whenever process performance is interpreted in terms of yield or proportionconforming.

All of the original indices belong to this category. The indices Cp, Cpk, Cpu, andCpl each can be thought of as indicating the proportion of the process distributionwhich falls within the specification limits. The translation of the index value to aproportion conforming is a simple matter in the cases of Cpu and Cpl. Once the type ofprocess distribution has been assumed, the index value clearly communicates theperformance of the process relative to the appropriate specification limit. However,the interpretation of proportion conforming is not quite so clear for Cp and Cpk.

Consider the situation presented in figure 2. The figure shows two processeswith the same Cpk value that have differing proportions conforming. Obviously, theCpk value alone is not sufficient to determine the proportion conforming. The Cpk valueonly communicates process performance relative to one of the two specification limits.

Similarly, the Cp value alone is not sufficient to communicate the current propor-tion conforming. While the Cp value does communicate the potential of the processafter centering, it is not appropriate to assume that the process is currently centered.In fact, it is necessary to know values for both Cp and Cpk in order to determine thecurrent proportion conforming.

Consider the process described by the second row of table 2. The Cp value indi-cates the width of the specification range as a multiple of the process standarddeviation, but the location is unknown. The Cpk value provides the missing infor-mation. The steps below demonstrate the logic of the current proportion conformingcalculation:

Figure 1. Loss functions used to interpret capability indices.

Cp = 0.75 USL LSL = 4.5s ,


The above variable x represents the process output and z has a standard normaldistribution. The potential proportion conforming is given directly by Cp (see table 1).

Table 2 also demonstrates how the stepwise loss function interpretation has ledto the use of capability indices to direct process improvement activities. For the processrepresented by the second row of the table, the current proportion conforming is93.184%. The comparison Cpk < Cp indicates that the process mean is off-center and

Table 2

Examples of proportion conforming for normally distributed processes.

Current PotentialP (x lies betweenP (x lies betweenP (x lies betweenP (x lies between

Cpk Cp m and near spec) m and far spec) LSL and USL) LSL and USL)

0.50 0.50 43.319% 43.319% 86.639% 86.639%0.50 0.75 43.319% 49.865% 93.184% 97.555%0.50 1.00 43.319% 49.999% 93.318% 99.730%0.75 1.00 48.778% 49.991% 98.769% 99.730%1.00 1.00 49.865% 49.865% 99.730% 99.730%

Cpk = 0.50 min(U S L- m , m - L S L) = 1.5s

max(U S L- m , m - LSL) = 3.0s

P(L S L< x < U S L) = P(0 < z < 1.5)+ P(0 < z < 3.0).

Figure 2. Two processes with Cpk = 0.50.


conformance may be improved by adjusting the mean. When the process is centered,Cpk = Cp = 0.75, and the proportion conforming becomes 97.555%. Since Cp < 1.0,reductions in process variation will be required to attain a proportion conforming of99.730.

Since the assumptions surrounding the original capability indices have beenchallenged over the years, several modifications have been proposed in order to per-petuate the stepwise loss function interpretation. For example, the above discussionrelies heavily upon the assumption that the process distribution is normal. The inter-pretation of the original indices, in terms of proportion conforming, differs from thatdescribed above when the process distribution is not normal. Clements [3] presentedgeneralized versions of Cp, Cpk, Cpu, and Cpl which are applicable to non-normaldistributions. His formulations are denoted here as C p, C pk, C pu, and C pl:

C p =U S L- L S L

P0.99865- P0.00135,

C pk = minU S L- P0.5

P0.99865- P0.5,

P0.5 - L S LP0.5 - P0.00135

,

C pu =U S L- P0.5

P0.99865- P0.5,

C pl =P0.5 - L S L

P0.5 - P0.00135,

(7)

(8)

(9)

(10)

where Pa is the 100a percentile.These indices are based upon the concept that a C p value of 1.0 should continue

to represent a proportion conforming of 0.9973, the proportion enclosed by m 3s inthe normal distribution. As a result, the formulations reduce to the original indiceswhen the process distribution is normal. The use of percentiles provides C p, C pk, C pu,and C pl applicability to any distribution for which the percentiles can be estimated,including both symmetric and asymmetric non-normal distributions. Clements methodfor estimating the percentiles involved calculation of the skewness and kurtosis fromstable process data, which could then be compared to tabulated values for the Pearsonfamily of distributions. These indices can directly replace the original indices in mostapplications. However, the proportions conforming only match those of the normaldistribution at index values of 1.0.

Another modified index which follows the same line of thought is that of Johnsonet al. [9]. They proposed the index Cp(q ), remarking that q should be chosen so as torepresent a similar proportion conforming over many distributions. To accomplishthis end, they recommended q = 5.15, which achieves a stable proportion conformingof 0.9900 over several chi-squared distributions:


A modification which appears to have a motivation similar to that underlying C pand Cp(q ) was proposed by Chan et al. [1]. They suggested the use of tolerance inter-vals, rather than percentiles or multiples of s , as an alternative representation of theprocess variability. In theory, this method can be applied to any process distributionbecause no assumptions are made regarding the shape of the distribution. However, inpractice, the sample size must be very large in order to estimate the interval withsufficient certainty. A sample size of 1000 or more measurements is required to esti-mate an interval covering 99.73% of the distribution (Kotz and Johnson [14]).

Each of the modifications to this point maintains the property that an increasingindex value indicates an increasing proportion conforming. The specific proportionsfor differing distributions, however, can only be made to match at a single index value,as was discussed in the case of C p.

Asymmetric loss functions have also offered researchers a motivation to developmodifications of the original indices in order to extend the stepwise loss functioninterpretation. The stepwise loss function shown in figure 1 not only assumes that allprocess output falling within the specification limits is equally good, it also assumesthat all results falling outside the specification limits are equally bad (a symmetricloss function). What would happen if, for some reason, it was preferable for non-conforming process output to either fall below the lower specification limit or abovethe upper specification limit?

Kane [13] observed that Springer [19] had shown the specification midpointwould not be the most economical point to position the process mean if an asymmetricloss function existed. Hunter and Kartha [8] and Nelson [15] had also developed targetvalues for the one-sided specification situation. Kane [13] proposed modifications toCp, Cpk, k, Cpu, and Cpl for these cases, denoted here as Cp

*, C*pk, k*, C*pu, and C

*pl:

Cp(q ) =U S L- LSLq s

. (11)

Cp* = min

T - L S L3s

,U SL - T

3s

,

Cpk* = min(Cpl

* ,Cpu* ),

k* =m - T

min(T - LSL, U S L- T),

Cpl* =

T - L S L3s

1 -jT - m jT - L S L

,

Cpu* =

U S L- T3s

1 -jT - m jU S L- T

,

where T is the process target value.

(12)

(13)

(14)

(15)

(16)


While these indices may share some conceptual features with the original indices,they do not share all of the original indices characteristics. The index Cp

* representsthe relative size of the smaller semi-tolerance, rather than the relative size of the entirespecification range per Cp. (The semi-tolerances are the distances between the targetand the specification limits.) The potential of the process is now interpreted in termsof the ability of the distribution half-width to fit within the smaller semi-tolerance.

The behavior of the indices C*pl and C*pu is demonstrated in figure 3. These indices

have maxima which are realized when m = T. (The indices Cpl and Cpu do not haveupper limits.) The upper limit for C*pl is (T LSL) 3s . The upper limit for C

*pu is

(USL T) 3s . Note that these values are the arguments of the min function in C*p. As

the mean shifts away from the target, the index value decreases linearly. When themean has shifted by a distance equal to or exceeding the respective semi-tolerance,the index value becomes zero.

The index C*pk is defined as the minimum of C*pl and C

*pu. Figure 3 demonstrates

that the index related to the smaller semi-tolerance, C*pu in this case, produces thesmaller value throughout the specification range. Therefore, C*pk has a maximum valueequal to Cp

*, which is attained when m = T. Also, C*pk will have a value of zero when-ever the mean is shifted away from the target by a distance equal to or exceeding thesmaller semi-tolerance. The index Cpk only has a value of zero when the mean is at oroutside the specification limits.

The index k* indicates the direction that m lies away from the target and themagnitude of the required adjustment, relative to the size of the smaller semi-tolerance.

Figure 3. Comparison of C*pl (solid) and C*pu (dashed).


As was the case for k, we believe that it is valuable for k* to retain its sign, rather thanbeing defined as an absolute value according to Kane.

The result of these modifications it that the centering assumption has beenreplaced by a targeting assumption. The indices C*pl, C

*pu, and C

*pk indicate relative

distance from the target. A shift in the mean away from the target, even if it is in thedirection of improved conformance, will produce a decrease in the index value. Thepotential of the process, represented by Cp

*, is assumed to be realized when m = T.This assumption, however, begins to violate the spirit of the stepwise loss functioninterpretation.

The indices Cp*, C*pk, and k

* may be used as direct replacements for Cp, Cpk, andk in the roadmap of section 2.2. However, if the process standard deviation is large incomparison to the specification width, then an appreciably greater proportion conform-ing will be realized by adjusting the process mean to the specification midpoint, ratherthan the target (assuming a normal process distribution). The effect of adjusting theprocess to make C*pk = Cp

* will be to trade off a fraction of the maximum possibleproportion conforming against the cost of producing noncomforming output in theless desirable region.

3.2. Quadratic loss function interpretation

The quadratic loss function interpretation holds that there exists an ideal targetvalue for each process and any deviation from the target value is detrimental, evenwithin the specification limits. Large deviations from the target are considered to beworse than small deviations. The defining characteristic of the penalty function is thatit only takes on a value of zero when process output is at the target; otherwise, thepenalty is proportional to the square of the deviation from the target.

All indices that follow the quadratic loss function interpretation include termswhich are related, more or less strongly, to the expectation of the squared deviationsfrom the process target value:

EX[(X - T)2] = s 2 + (m - T)2. (17)

This is the quadratic loss function which was popularized by Taguchi. Becauseeach of the indices in this category is scaled by the quadratic loss function, rather thansimply the variance, none of them have a clear interpretation in terms of the proportionconforming, with the exception of Cp.

As an alternative to the notion of proportion conforming, the quadratic loss isinterpreted as a measure of economic penalty. The ideal process (all output lyingexactly on target) is assumed to be the most economically advantageous. When processoutput varies from the target, an economic loss is assumed to exist. The loss can beviewed either in terms of extraneous manufacturing cost or in terms of extraneousutilization cost for the product. From this point of view, the quadratic loss is thepredominant quality measure of the process. Process improvement becomes a con-tinuous effort to reduce loss, rather than an effort to achieve 100% conformance to


specification. The use of process capability indices, which appeal to the specificationlimits, emphasizes standardization of the loss for process to process comparisons.

The first index to be developed that explicitly used a loss function formulationwas Cpm. It was proposed independently by Hsiang and Taguchi [7] and by Chanet al. [2]:

Cpm =U S L- L S L

6 s 2 + (m - T )2. (18)

Figure 4. Comparisons of Cpk (solid), Cpm (dotted), and Cpmk (dashed).

This index attains its maximum value when m = T, and will decrease in valuesymmetrically, in a bell-shaped pattern, as the process mean shifts away from thetarget value. See figure 4. If the process mean is at the target value, then Cpm = Cp. So,Cp is the upper limit for Cpm; and Cp retains its meaning as the process potential underthe quadratic loss function interpretation.


Unlike Cpk, the value of Cpm does not go to zero at the specification limits. Thevalue of the index Cpm is independent of the closeness of m to the specification limits.Only the distance between m and the target is considered. The entire curve for Cpmshifts with the target value, regardless of the actual locations of the specification limits.Recall that the target value is not necessarily the specification midpoint value. So, aslong as s and the specification width remain fixed, the shape of the Cpm curve willremain fixed too.

The fact that Cpm indicates the reduction in process capability due to shifts in theprocess mean away from the target suggests that the pair of indices Cp and Cpm couldbe used to direct process improvement activities in a manner similar to the roadmapof section 2.2. The revised roadmap would be as follows:

Step 1. Obtain simultaneous estimates of Cp and Cpm.

Step 2. If Cp < 1.0, then identify and remove sources of process variability.

Step 3. If Cpm < Cp, then evaluate m .(a) If m > T, then adjust the process location to decrease the process mean

until Cpm = Cp.(b) If m < T, then adjust the process location to increase the process mean

until Cpm = Cp.

This version of the roadmap calls for variability reduction prior to adjustmentsin process location. The order of the activities has been reversed because the overridingconcern, under the quadratic loss function interpretation, is variability reduction. It isassumed that process location adjustments are a relatively simple matter. Since processmean shifts may occur during variability reduction efforts, the process mean is onlyadjusted after Cp achieves its desired value.

Furthermore, immediate adjustments to the process location will not necessarilyresult in improved performance relative to the specifications. If Cp < 1.0 and the targetis not the specification midpoint, then adjusting the process location to the target mayresult in a reduced proportion conforming (assuming a normal process distribution).The immediate benefits of adjusting the process location, referred to in section 2.2,only occur when the target is the specification midpoint. However, as long as LSL

a review and interpretations of process capability indices

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