A Review and Interpretations of Process Capability Indices

A nn a ls o f O p e ra ti on s Re se ar c h 87 (1 9 99 )3 1 –4 7

31

A review and interpretations of process capability indices Kurt Palmer a,  and Kwok-Leung Tsui b a

Department Depart ment of Indust I ndustrial rial and Systems S ystems Engineering Engin eering , Universit Uni versityy of South ern Califor Ca lifornia, nia, Los An geles, CA 90089-0 90 089-0193, 193, USA E-mail: [email protected] b

Department Depart ment of In dustrial dustri al Enginee En gineering ring a nd Engineeri Eng ineeri ng Managem Ma nagement, ent, Hong Ko ng University Uni versity of Science Sci ence and a nd Technolog Tec hnology, y, Clear Water Bay, Hong Kong E-mail: [email protected]

Practitioners of industrial statistics are generally familiar with the common C p and C pk process capability indices. However, many additional indices have been proposed, and knowledge of these is less widespread. More importantly, information regarding the indices’ comparative behavior is lacking. This paper compares the behavior of various indices under shifting process conditions. Both useful and misleading characteristics of the indices are identified. We begin with a short history of process capability measures. Several process capability 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 for selection of indices at differing levels of process performance. Keywords: Keywords : capability index, loss function, process improvement

1.

I n t r o d u c t i on

While the concept of process capability was most likely developed by American statisticians, use of process capability indices by the United States’ industrial community did not become popular until reports of Japanese manufacturing methods appeared in American trade and professional journals. One of the earliest domestically published descriptions of capability indices was by Sullivan [20,21]. Kane [13] provided the first discussion of the indices’ sampling characteristics, and made some suggestions 

Corresponding author.

© J.C. Baltzer AG, Science Publishers

32

K. Palmer, K.-L. Tsui  A review and interpretations of process capability indices

for modifications to the indices. Kotz and Johnson [14] gave an exhaustive summary of modified indices and their sampling characteristics. While use of the original capability indices has become widespread, many practitioners remain unfamiliar with the detailed descriptions of the indices’ population characteristics. The motivations for and characteristics of the modified indices are even less well known. The process capability indices which Sullivan reported represented an improvement over previous metrics that were used to describe process capability. Feigenbaum [4] and Juran [10] used 6 σ as a measure of process capability. They presented the measure as a representation of the inherent variability of a process. This description gave process capability an interpretation that was independent of customer specifications. There was an implication of inevitability regarding process performance. performance. The notion that “the process doesn’t know the specifications” was emphasized. While this is of course true, the interpretation seems to urge a passive approach to quality improvement. Juran [11] created a stronger link between process variability and customer specifications by comparing 6σ to the tolerance width as a method of determining the need for process improvement activities. However, capability itself was still interpreted separately from specifications. Finally, Juran and Gryna [12] proposed a capability ratio, which provided the first metric that directly compared process variability to customer specifications: 6σ variation Capability ratio = . (1) tolerance width As is the case with the capability ratio, all process capability indices explicitly link process variability to customer specifications, and in so doing, they emphasize the supplier’s responsibility to satisfy those specifications. However, capability indices also have advantages over the capability ratio. Capability indices increase in value as the process performance improves. This property may be of limited analytical value, but it does provide psychological value in that it reinforces the natural “bigger is better” predisposition. Furthermore, capability indices indicate the relative benefits of 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. The population characteristics for the indices are discussed in detail. Comparisons of the indices’ behavior under shifting process conditions are described. Finally, recommendations are given for the selection of indices to direct process improvement activities at various levels of process performance. 2.

Thee ori Th origi gina nall fiv fivee cap capab abil ilit ity y ind indic ices es

Sullivan [21] described five indices which he had observed in use at Japanese manufacturing facilities. The indices were C p , C pk , k , C pu , and C pl .


33

2.1. 2.1. Definiti Definitions ons of of the origin original al indices indices

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

=

U S L− L S L

6σ

,

(2)

where US L

is the upper specification limit,

LS L

is the lower specification limit,

σ

is the process standard deviation.

The astute reader will notice that C p is the reciprocal of Juran and Gryna’s capability ratio. It is not clear how C p was developed. At the least, we believe that it is appropriate to credit Juran and Gryna with the development of the idea that led to this family of metrics. Conceptually, C p compares the allowable process spread to the actual process spread. It can be thought of as indicating the potential of the process to produce conforming material. A common interpretation of C p assumes that the process output is normally distributed. This interpretation leads to the oft quoted proportions conforming that are given in table 1. Table 1 Proportion conforming after centering. Assumed distribution No rma l N on e

C p value

0. 0.50

0.75

1.00

1.25

86.639%

97.555%

99.730%

99.982%

> 55 .556%

> 80.24 7%

> 88.88 9%

> 92. 889%

A more general interpretation does not require normality. Chebyshev’s inequality can be used to determine a lower bound for the proportion conforming, regardless of the actual process distribution. These values are also shown in table 1. Since there is no guarantee that a process will be normally distributed, the Chebyshev values should be kept in mind as worst-case senarios. (The proportions conforming are given as lower bounds because Chebyshev’s inequality makes no assumptions regarding the shape of the distribution.) It is critical to note two points: (1)

potent ial proportion conforming, due to the The me metric C p can only indicate the potential centering assumption. The interpretations of proportion conforming given above represent maximum values provided that the process mean is located at the specification midpoint.


34

(2) (2)

A state of statistical statistical contro controll is required required of the process process in order order for for the index index value to have any long-term meaning. If the process is not stable, then any conclusions regarding the capability of the process will need to be modified as its performance varies.

To deal with violations of the centering assumption, the following pair of indices was developed: min(U S L− µ, µ − L S L) , (3) C pk = 3σ k =

where

µ−m (U S L− L S L)

2

(4)

,

µ

is the process mean,

m

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

The index k represents represents a measure of the distance that the process lies off-center and C pk demonstrates the reduction in process capability caused by the lack of centering. If the assumption LS L < µ < US L is made, then | k | ≤ 1 and there is a simple relationship among C p , C pk , and k . The relation is C pk = (1 – | k | ) C p . This relation leads to the concept of C p as an upper limit for C pk , and reinforces the description of C p as poten tial capability. The C the potential pk index will have a maximum value equal to the C p value when the process mean is at the specification midpoint. As the process mean shifts away from the specification midpoint, the value of C pk decreases linearly until it reaches a value of zero, when the process mean is equal to one of the specification limits. It should be noted that both Sullivan [21] and Kane [13] describe k as as an absolute value. We believe that it is useful for k to to retain its sign for applications such as the process improvement roadmap, which will be presented below. Sometimes, a customer will provide a one-sided specification. specification. The indices C pu and C pl were developed for these situations. For processes with upper specification limit only: C pu

=

U S L− µ

3σ

.

(5)

For processes with lower specification limit only: C pl

=

µ − L S L . 3σ

(6)

The index C pu compares the distance between the process mean and the upper specification limit to the upper half-width of the distribution. Similarly, C pl compares the distance between the process mean and the lower specification limit to the lower


35

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 C pu and C pl also provide insight into the formulation of C pk . Often, the relation C pk = min( C pu , C pl ) is used. Therefore, C pk is also a measure of the remaining size of the working margin, as a result of a shift in the process mean away from the center and towards one of the specification limits. Since the measure of the half-width, 3σ, is the same for the two indices, C pu and C pl , this definition of C pk implies an assumption that the process distribution is symmetric about its mean. 2.2. 2.2. Applica Application tionss of capabi capability lity indice indicess

Kane [13] described six application areas for capability indices, which will be paraphrased here: p erformance : A minimally desirable (1) A static goal for performance desirable index value is set set in order to avoid nonconforming output, to maintain customer acceptance, or to define contractual obligations. The stated objective is to achieve the minimal level. This application is typical of customer–supplier relationships. cont inuous improvement i mprovement : The index value is monitored (2) A measure of continuous monitored over time as an indication of relative improvement. The stated objective is to perpetually increase the index value. This technique applies to both individual processes and the distribution of index values for a collection of processes.

(3) A comm c ommon on proces pro cesss perf p erform ormanc ancee l anguag ang uagee: The index value is used to communicate process performance to those who may not have a detailed knowledge of the process. Since the value of any given index always carries the same interpretation of the relative relationship between process performance and customer specifications, specifications, the indices provide consistent frames of reference regardless of the specific process they describe. (4) A criterion crit erion for prioriti pri oritization zation : The index is used in process-to-process comparisons to determine either relative need for improvement or relative benefit for investment. roa dmarker er to t o direct di rect process proc ess improveme impro vement nt activ a ctiviti ities es: Pairwise comparisons (5) A roadmark of selected capability indices determine the relative benefit of adjusting process location versus reducing process variability.

(6) An indicat in dicator or of quality quali ty system sy stem defic d eficienci iencies es: Index values obtained during quality system audits are compared to process records to identify deficiencies in sampling, measurement, process control, etc. Shewhart [18], Juran [10] and Gryna [5] all discussed the use of process capability information to determine specification limits. In general, they recommended that the tolerance width should not be tighter than 6 σ. Feigenbaum [4] and Juran [10] referred to the use of process capability information to assign jobs to machines. Jobs requiring tight tolerance widths would be assigned to the most capable machine in

36


order to minimize in-process waste. While the applications given above are certainly certainly among the most common, we do not intend to suggest that this represents an exhaustive listing. An application of particular interest is the use of indices to direct process improvement activities. It is common for process capability indices to be used individually for most of the applications presented above. In order to direct process adjustments, however, capability indices must be considered collectively. The indices represent one such grouping. Collectively, these indices signal the need C p , C pk , and k represent for deliberate process location location adjustments and or process process variability variability reductions. reductions. Applying the steps sequentially, the roadmap operates as follows: Obtain simultan simultaneou eouss estima estimates tes of C p , C pk , and k . Step 1 . Obtain Step 2 . If C pk < C p , then evaluate k . (a) If k > > 0, then adjust the process locati on to decrease the process mean until C pk = C p . (b) If k < < 0, then adjust the process location to increase the process mean until C pk = C p . Step 3 . If C p < 1.0, then identify and remove sources of process variability.

The roadmap calls for adjustments to the process location prior to variability reduction because process mean adjustments are considered to be relatively simple to accomplish. As such, process mean adjustments, when necessary, can produce immediate improvements in process performance relative to the specifications. It is assumed that 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, are generally considered to be a more difficult task than that of adjusting the process location. Also, variability reductions may someti mes produce an unintendended shift in the process mean. So, it may be necessary to revisit step 2 following improvements made during step 3. 3.

Inte Interp rpre reta tati tion onss of cap capab abil ilit ity y indi indice cess

Over the years, two broad classes of indices have been developed. These categories are defined by the underlying loss function which is used to interpret the index. The categories are: stepwise loss function interpretation and quadratic loss function interpretation. Figure 1 displays the two types of loss functions. It will be shown that C p has a consistent meaning under either interpretation. The index C p is the only index which enjoys membership in both categories. 3.1. 3.1. Stepwis Stepwise e loss function function interpr interpretat etation ion

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


37

Figure 1. Loss functions used to interpret capability indices.

function is that it has a value of zero over the entire specification range, and has a value greater than zero outside the specification limits. The stepwise loss function is implied whenever process performance is interpreted in terms of yield or proportion conforming. All of the original indices belong to this category. The indices C p , C pk , C pu , and C pl each can be thought of as indicating the proportion of the process distribution which falls within the specification limits. The translation of the index value to a proportion conforming conforming is a simple matter in the cases of C pu and C pl . Once the type of process distribution has been assumed, the index value clearly communicates the performance of the process relative to the appropriate specification limit. However, the interpretation of proportion conforming is not quite so clear for C p and C pk . Consider the situation presented in figure 2. The figure shows two processes with the same C conforming. Obviously, the pk value that have differing proportions conforming. C pk value alone is not sufficient to determine the proportion conforming. The C pk value only communicates process performance relative to one of the two specification limits. Similarly, the C p value alone is not sufficient to communicate the current proportion conforming. While the C p value does communicate the potential of the process after centering, it is not appropriate to assume that the process is currently centered. In fact, it is necessary to know values for both C p and C pk in order to determine the current proportion conforming. Consider the process described by the second row of table 2. The C p value indicates the width of the specification range as a multiple of the process standard deviation, but the location is unknown. The C pk value provides the missing information. The steps below demonstrate the logic of the current proportion conforming calculation: C p = 0.75

⇒ US L – LS L = 4.5σ ,


38

Figure 2. Two processes with C pk = 0.50.

Table 2 Examples of proportion conforming for normally distributed processes.

P ( x li lie s bet wee n µ and far spec)

Current P ( x li lies be tw tw ee een LS L and US L )

Poten tial P ( x lies between LS L and US L )

C pk

C p

P ( x li l ie s bet wee n µ and near spec)

0.50

0.50

43.319%

43.319%

86.639%

86.639%

0.50 0.50 0.75

0.75 1.00 1.00

43.319% 43.319% 48.778%

49.865% 49.999% 49.991%

93.184% 93.318% 98.769%

97.555% 99.730% 99.730%

1.00

1.00

49.865%

49.865%

99.730%

99.730%

C pk = 0.50

⇒

min(U S L− µ,

µ − L S L) = 1.5σ

⇒ max(U S L− µ, µ − L S L) = 3.0σ ⇒ P( L S L< x < U S L) = P(0 < z < 1.5) + P(0 < z < 3.0). The above variable x represents the process output and z has a standard normal distribution. The potential proportion conforming is given directly by C p (see table 1). Table 2 also demonstrates how the stepwise loss function interpretation has led to the use of capability indices to direct process improvement activities. For the process represented by the second row of the table, the current proportion conforming is 93.184%. The comparison C pk < C p indicates that the process mean is off-center and


39

conformance may be improved by adjusting the mean. When the process is centered, C pk = C p = 0.75, and the proportion conforming becomes 97.555%. Since C p < 1.0, reductions in process variation will be required to attain a proportion conforming of 99.730. Since the assumptions surrounding the original capability indices have been challenged over the years, several modifications have been proposed in order to perpetuate the stepwise loss function interpretation. For example, the above discussion relies heavily upon the assumption that the process distribution is normal. The interpretation of the original indices, in terms of proportion conforming, differs from that described above when the process distribution is not normal. Clements [3] presented generalized versions of C p , C pk , C pu , and C pl which are applicable to non-normal ′ , C pu′ , and C pl′ : distributions. His formulations are denoted here as C p′ , C pk C p′

=

U S L− L S L P0.99865

C pk ′ = min C pu ′

=

C pl ′

=

 

− P0.00135

U S L− P0.5 P0.99865

U S L− P0.5 P0.99865

− P0.5

− L S L P0.5 − P0.00135 P0.5

,

− P0.5

(7)

,

− L S L  , − P0.00135 

P0.5 P0.5

(8)

,

(9)

,

(10)

where Pα is the 100 α 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 µ ± 3σ in i n the normal distribution. As a result, the formulations reduce to the original indices ′ , C pu′ , when the process distribution is normal. The use of percentiles provides C p′, C pk and C pl′ applicability to any distribution for which the percentiles can be estimated, including both symmetric and asymmetric non-normal distributions. Clements’ method for estimating the percentiles involved calculation of the skewness and kurtosis from stable process data, which could then be compared to tabulated values for the Pearson family of distributions. These indices can directly replace the original indices in most applications. However, the proportions conforming only match those of the normal distribution at index values of 1.0. Another modified index which follows the same line of thought is that of Johnson et al. [9]. They proposed the index C p (θ), remarking that θ should be chosen so as to represent a similar proportion conforming over many distributions. To accomplish this end, they recommended θ = 5.15, which achieves a stable proportion conforming of 0.9900 over several chi-squared distributions:

40


C p (θ )

=

U S L− L S L

θσ

.

(11)

A modification which appears to have a motivation similar to that underlying C p′ and C p ( θ) was proposed by Chan et al. [1]. They suggested the use of tolerance intervals, rather than percentiles or multiples of σ, as an alternative representation representation of the process variability. In theory, this method can be applied to any process distribution because no assumptions are made regarding the shape of the distribution. However, in practice, the sample size must be very large in order to estimate the interval with sufficient certainty. A sample size of 1000 or more measurements is required to estimate an interval covering 99.73% of the distribution (Kotz and Johnson [14]). Each of the modifications to this point mai ntains the property that an increasing index value indicates an increasing proportion conforming. The specific proportions for 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 develop modifications of the original indices in order to extend the stepwise loss function interpretation. The stepwise loss function shown in figure 1 not only assumes that all process output falling within the specification limits is equally good, it also assumes that all results falling outside the specification limits are equally bad (a symmetric loss function). What would happen if, for some reason, it was preferable for nonconforming process output to either fall below the lower specification limit or above the upper specification limit? Kane [13] observed that Springer [19] had shown the specification midpoint would not be the most economical point to position the process mean if an asymmetric loss function existed. Hunter and Kartha [8] and Nelson [15] had also developed target values for the one-sided specification situation. Kane [13] proposed modifications to * * * * * C p , C pk , k , C pu , and C pl for these cases, denoted here as C p , C pk , k , C pu , and C pl :

 

* = min  C p * C pk *

* ), = min(C pl* , C pu

k

=

* C pl

=

*

T − L S L U SL − T  ,  , 3σ 3σ

C pu =

(13)

µ − T

,

(14)

µ|  , T − L S L

(15)

| T − µ |  , U S L− T 

(16)

min(T − L S L, U S L− T ) T − L S L

3σ

1− 

U S L− T 

3σ

where T is the process target value.

1 − 

(12)

| T −


41

While these indices may share some conceptual features with the original indices, they do not share all of the original indices’ characteristics. The index C p* represents the relative size of the smaller semi-tolerance, rather than the relative size of the entire specification range per C p . (The semi-tolerances are the distances between the target and the specification limits.) The potential of the process is now interpreted in terms of 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 µ = T . (The indices C pl and C pu do not have * * upper limits.) The upper limit for C pl i s ( T – – LS L ) 3σ . The upper limit for C pu i s * (US L – T ) 3σ . Note that these values are the arguments of the min function in C p . As

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

the mean shifts away from the target, the index value decreases linearly. When the mean 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 the * smaller value throughout the specification range. Therefore, C pk has a maximum value * * equal to C p , which is attained when µ = T . Also, C pk will have a value of zero whenever the mean is shifted away from the target by a distance equal to or exceeding the smaller semi-tolerance. The index C pk only has a value of zero when the mean is at or outside the specification limits. The index k * indicates the direction that µ lies away from the target and the magnitude of the required adjustment, relative to the size of the smaller semi-tolerance.

42


As was the case for k , we believe that it is valuable for k * to retain its sign, rather than being defined as an absolute value according to Kane. The result of these modifications it that the centering assumption has been * * replaced 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 the direction of improved conformance, will produce a decrease in the index value. The * potential of the process, represented by C p , is assumed to be realized when µ = T . This assumption, however, begins to violate the spirit of the stepwise loss function interpretation. * * * The indices C p , C pk , and k may be used as direct replacements for C p , C pk , and k in in the roadmap of section 2.2. However, if the process standard deviation is large in comparison to the specification width, then an appreciably greater proportion conforming will be realized by adjusting the process mean to the specification midpoint, rather than the target (assuming a normal process distribution). The effect of adjusting the * * process to make C pk = C p will be to trade off a fraction of the maximum possible proportion conforming against the cost of producing noncomforming output in the less desirable region. 3.2. 3.2. Quadra Quadratic tic loss function function interpr interpretat etation ion

The quadratic loss function interpretation holds that there exists an ideal target value for each process and any deviation from the target value is detrimental, even within the specification limits. Large deviations from the target are considered to be worse than small deviations. The defining characteristic of the penalty function is that it only takes on a value of zero when process output is at the target; otherwise, the penalty is proportional to the square of the deviation from the target. All indices that follow the quadratic loss function interpretation include terms which are related, more or less strongly, to the expectation of the squared deviations from the process target value: E X [( X − T ) 2 ]

= σ 2 + (µ − T )2 .

(17)

This is the quadratic loss function which was popularized by Taguchi. Because each of the indices in this category is scaled by the quadratic loss function, rather than simply the variance, none of them have a clear interpretation in terms of the proportion conforming, with the exception of C p . As an alternative to the notion of proportion conforming, the quadratic loss is interpreted as a measure of economic penalty. The ideal process (all output lying exactly on target) is assumed to be the most economically advantageous. When process output varies from the target, an economic loss is assumed to exist. The loss can be viewed either in terms of extraneous manufacturing cost or in terms of extraneous utilization cost for the product. From this point of view, the quadratic loss is the predominant quality measure of the process. Process improvement becomes a continuous effort to reduce loss, rather than an effort to achieve 100% conformance to


43

specification. The use of process capability indices, which appeal to the specificati on limits, emphasizes standardization of the loss for process to process comparisons. The first index to be developed that explicitly used a loss function formulation was C pm . It was proposed independently by Hsiang and Taguchi [7] and by Chan et al. [2]: U S L− L S L . (18) C pm = 6 σ 2 + ( µ − T )2 This index attains its maximum value when µ = T , and will decrease in value symmetrically, in a bell-shaped pattern, as the process mean shifts away from the target value. See figure 4. If the process mean is at the target value, then C pm = C p . So, C p is the upper limit for C pm; and C p retains its meaning as the process potential under the quadratic loss function interpretation.

Figure 4. Comparisons of C pk (solid), C pm (dotted), and C pmk (dashed).

44


Unlike C pk , the value of C pm does not go to zero at the specification limits. The value of the index C pm is independent of the closeness of µ to the specification limits. Only the distance between µ and the target is considered. The entire curve for C pm shifts 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, as long as σ and the specification width remain fixed, the shape of the C pm curve will remain fixed too. The fact that C pm indicates the reduction in process capability due to shifts in the process mean away from the target suggests that the pair of indices C p and C pm could be used to direct process improvement improvement activities in a manner similar to the roadmap of section 2.2. The revised roadmap would be as follows: Obtain simultan simultaneou eouss estima estimates tes of C p and C Step 1 . Obtain pm . Step 2 . If C p < 1.0, then identify and remove sources of process variability. Step 3 . If C pm < C p , then evaluate

µ.

(a) If µ > T , then adjust the process location to decrease the process mean until C pm = C p . (b) If µ < T , then adjust the process location to increase the process mean until C pm = C p . This version of the roadmap calls for variability reduction prior to adjustments in process location. The order of the activities has been reversed because the overriding concern, under the quadratic loss function interpretation, is variability reduction. It is assumed that process location adjustments are a relatively simple matter. Since process mean shifts may occur during variability reduction efforts, the process mean is only adjusted after C p achieves its desired value. Furthermore, immediate adjustments to the process location will not necessarily result in improved performance relative to the specifications. specifications. If C p < 1.0 and the target is not the specification midpoint, then adjusting the process location to the target may result 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 LS L < µ < US L is true, variability reduction will increase the proportion conforming. Of course, it is usually the case that the target value for a two-sided specification is the specification midpoint. So, the important point to focus upon is that the quadratic loss function interpretation considers the ideal process to be one which is invariably on-target. The primary goal is to reduce variability. This is the motivation for developing a loss function which penalizes off-target process ouput that falls within the specification limits, and the predominant reason for the change in the roadmap. An index which attempts to harmonize the characteristics of C pm and C pk is C pmk , which was proposed by Pearn et al. [16]:


C pmk =

min( µ 3

− L S L, U S L− µ ) . σ 2 + (µ − T )2

45

(19)

The C pmk index does have a value of zero when µ is at the specification limits, like C pk ; so, it will indicate closeness of µ to the specification limits. This is a desirable characteristic that is not shared by C pm . But as figure 4 demonstrates, demonstrates, when the target target value is not the specification midpoint, the maximum value for C pmk is not the same as the maximum for C pk and C pm , namely, the C p value. In addition, the process location where the maximum is attained is not readily determined. These facts suggest that the ideal process characteristics underlying C pmk cannot be easily stated. Therefore, it is not practical to use this index for directing process improvement activities. The undesirable characteristics presented by C pmk are caused by the decision to modify the numerator of C pm in order to force the index to a value of zero when µ is at a specification limit. It is possible to nearly achieve this desirable result by modifying the denominator of C pm . The rate at which the value of C pm decreases, as µ moves away from the target, depends upon the relative size of the process variance in comparison to the square of the process bias ( µ – T )2. The rate can be controlled by adjusting the weight of the bias term in the index. This is the concept employed in the formulation of the index C pm (a), an adapted version of the index proposed by Kotz and Johnson [14]: C pm (a)

=

U S L − L S L

6

σ 2 + a(µ − T )2

.

(20)

The C pm (a) index will always have C p as an upper limit, and will always attain its maximum value when µ = T . In addition, its formulation allows the investigator unlimited flexibility in defining the relative contribution of the process variance and process bias components of the loss function. The value of a may be selected in accordance with the location of the target, so that the index value approaches zero when µ is near the specification limits. As a result, for the purpose of directing process * improvement activities, C pm ( a) presents an attractive alternative to both C pk and C pmk . 4.

R e c o m m e n d a t i on s

The selection of a practical set of capablility indices, for use in directing process improvement activities, should be based upon the stated requirements of the customer and the current ability of the process to satisfy those requirements. Below, we give some guidelines for selection. (1)

If the the customer customer has has given given a one-si one-sided ded specif specificat ication, ion, use use C pl or C pu , as appropriate.

(2) (2)

If the custome customerr has given given a two-side two-sided d specificati specification on and the the current current proporti proportion on conforming is less than 95%, use C p , C pk and k .

(3) (3)

If the custome customerr has given given a two-side two-sided d specificati specification on and the the current current proporti proportion on conforming is between 95% and 99%, use C pm (a ) and C p .

46

(4) (4)


If the custome customerr has given given a two-side two-sided d specificati specification on and the the current current proporti proportion on conforming is greater than 99%, use C pm and C p .

Among the indices formulated for one-sided specification situations, C pl and C pu are the easiest to implement and interpret. For a process with a stable σ , they will provide a measure of the remaining working margin that will change linearly as the process mean shifts. As the mean is adjusted in the correct direction, C pl and C pu will * * increase monotonically (unlike C pl and C pu , which decrease in value after the mean ′ is not justified passes the target). Finally, the effort required to generate C pl′ and C pu by the advantage of all process distributions indicating the same proportion conforming when the index value is 1.0. In the case of two-sided specifications, the selection of indices is dictated by the nature of the current process improvement challenge. When the current proportion conforming is less than 95%, the overriding concern should be to increase the proportion conforming. Among the indices in the stepwise loss function grouping, the original indices are the easiest to use. Even though application of the original indices is straightforward, the introduction of new operating procedures always creates some level of confusion in the workplace. The possible benefits of using more complex formulations (such as C p′ , C pk ′ , or C p (θ)) are outweighed by the additional confusion that they may create, even for non-normal process distributions. When the current proportion conforming is between 95% and 99%, process performance is in a region where the level of conformance is becoming a given, on the percentage scale. It is becoming more important to concentrate on a targeting approach. However, it is also necessary to maintain some awareness of working margins. The index C pm ( a) can be set up, by appropriate selection of a, to indicate the size of the working margin, similar to C pk , while beginning to introduce the characteristics of the * quadratic loss function. The index C pk is a less desirable substitute for C pm (a) because * * * C pk requires C p to indicate appropriate adjustment of the mean, and C p is a less readily interpreted indicator of potential process performance relative to the specification limits than C p is. When the current proportion conforming is greater than 99%, conformance on the percentage scale is virtually guaranteed, if the process remains on target. Targeting should therefore become the overriding philosophy. The index C pm is the simplest of the quadratic loss function based indices available. The index C pm (a) is a potential replacement if the weighting of bias versus variance becomes an important consideration. Use of C pmk as a replacement for C pm introduces unneccessary complexity in both application and interpretation. References [1]

L.K. Chan, S.W. Cheng and F.A. Spiring, The robustness of process capability index C p to departures from normality, in: Statistical Theory and Data Analysis, II , ed. K. Matusita, North-Holland, Amsterdam, 1988, pp. 223–229.


[2] [2] [3] [4] [5] [6] [6] [7]

[8] [9] [9]

[10] [10] [11] [11] [12] [12] [13] [13] [14] [14] [15] [15] [16] [17] [18] [18] [19] [20] [21] [21]

47

L.K. Chan, Chan, S.W. S.W. Cheng Cheng and F.A. F.A. Spiring, Spiring, A new new measure measure of process process capabi capability lity:: C pm , Journal of Modeling and Simulation 11(1988)1–6. J.A. Clements Clements,, Process capabilit capability y calculations calculations for non-norma non-normall distributio distributions, ns, Quality Quality Progress Progress 22(9) (1989)95–100. A.V. A.V. Feig Feigen enba baum um,, Quality Control, McGraw-Hill, New York, 1951, pp. 383–389. F.M. Gryna, Quality Control Handbook , ed. J.M. Juran, McGraw-Hill, New York, 1988, pp. 16.14– 16.35, 4th ed. J.T. J.T. Herman, Herman, Capabil Capability ity index index – enough enough for proces processs industr industries? ies?,, ASQC Annu al Qu ality alit y Con gress Transactions , Toronto, 1989, pp. 670–675. T.C. Hsiang Hsiang and G. Taguchi, Taguchi, A tutorial tutorial on on quality control and and assurance assurance – the Taguchi Taguchi methods, methods, unpublished presentation given at the Annual Me etings of the Americ an Stati stical Assoc iation , Las Vegas, NV, 1985. W.G. Hunter Hunter and C.P. C.P. Kartha, Kartha, Determining Determining the most most profitable profitable target target value for a production production process, process, Journal of Quality Technology 9(1977)176–181. N.L. Johnso Johnson, n, S. Kotz and and W.L. Pearn, Pearn, Flexible Flexible proces processs capability capability indices indices,, Insti tute of S tatis tics Mimeo Series , No. 2072, Department of Statistics, University of North Carolina, Chapel Hill, NC, 1992. J.M. J.M. Jura Juran, n, Quality Control Handbook , McGraw-Hill, New York, 1951, pp. 253–279 and 404–409. J.M. J.M. Jur Juran, an, Quality Control Handbook , 2nd ed., McGraw-Hill, New York, 1962, pp. 11-14–11-27, J.M. Juran and F.M. F.M. Gryna, Gryna, Quality Planning and Analysis, 2nd ed., McGraw-Hill, New York, 1980, pp. 283–295. V.E. Kane, Process Process capability indices, indices, Journal of Quality Quality Technology 18(1986)41– 18(1986)41–52. 52. S. Kotz Kotz and N.J. N.J. Johns Johnson, on, Process Capability Indices , Chapman and Hall, London, 1993, pp. 121–127. L.S. Nelson, Best Best target value for a production production process, Journal Journal of Quality Technology Technology 10(1978) 88–89. W.L. Pearn, S. Kotz and N.L. Johnson, Johnson, Distributional and inferential properties of process control indices, Journal of Quality Technology 24(1992)216–231. R.N. Rodriguez, Rodriguez, Recent developments developments in process process capability capability analysis, analysis, Journal of Quality Quality Technology 24(1992)176–187. W.A. W.A. Shewhar Shewhart, t, Econo mic Contr ol o f Qu ality of M anuf actu red P rodu ct s, s , D. Van Nostrand, New York, 1931, pp. 249–272. C.H. Springer, Springer, A method for determining determining the most economical economical position position of a process mean, mean, Industrial Industrial Quality Control 8(1951)36–39. L.P. Sullivan, Sullivan, Reducing variability: variability: A new approach to quality, quality, Quality Progress Progress 17(7)(1984)15–21. 17(7)(1984)15–21. L.P. Sullivan, Sullivan, Letters, Letters, Quality Quality Progress Progress 18(4)(1985)7–8. 18(4)(1985)7–8.

A Review and Interpretations of Process Capability Indices

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