Qual. Reliab. Engng. Int. 2003; 19:101–110 (DOI: 10.1002/qre.513)
Research
Capability Measures for Processes
with Multiple Characteristics
K. S. Chen1, W. L. Pearn2,∗,†and P. C. Lin3
1Department of Industrial Engineering and Management, National Chin-Yi Institute of Technology, Taichung, Taiwan 2Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan
3Center of General Education, National Chin-Yi Institute of Technology, Taiwan
Process capability indices, such as Cp,Ca, andCpk, have been widely used in the manufacturing industry providing numerical measures on process precision, process accuracy, and process performance. Capability measures for processes with a single characteristic have been investigated extensively. However, capability measures for processes with multiple characteristics are comparatively neglected. In this paper, we consider a generalization of the yield indexSpk proposed by Boyles, for processes with multiple characteristics. We establish a relationship between the generalization and the process yield. We also develop a control chart based on the proposed general-ization, which displays all the characteristic measures in one single chart. Using the chart, the engineers can effectively monitor and control the performance of all process characteristics simultaneously. Copyright c 2003 John Wiley & Sons, Ltd. KEY WORDS: capability zone; MCPCA chart; multiple characteristics; process capability index; process
yield index
1.
INTRODUCTION
P
rocess capability indices, establishing the relationship between the actual process performance and the manufacturing specifications, have been the focus of recent research in quality assurance and capability analysis. Those capability indices quantifying process potential and process performance are essential to any successful quality improvement activities and quality program implementation. Some basic capability indices that have been widely used in the manufacturing industry include Cp, Ca, and Cpk, explicitly defined as follows1–3: Cp= USL− LSL 6σ Ca= 1 −|µ − m| d Cpk= min USL− µ 3σ , µ − LSL 3σ∗Correspondence to: W. L. Pearn, Department of Industrial Engineering and Management, National Chiao Tung University,
1001 TA Hsueh Road, Hsinchu, 30050 Taiwan.
†E-mail: [email protected]
where USL and LSL are the upper and the lower specification limits, respectively, µ is the process mean,
σ is the process standard deviation, m = (USL + LSL)/2 is the mid-point of the specification interval, and d = (USL − LSL)/2 is half the length of the specification interval.
The index Cp measures the overall process variation relative to the specification tolerance, therefore it only reflects process potential (or process precision). The index Ca measures the degree of process centering, which alerts the user if the process mean deviates from its target value. Therefore, the index Ca only reflects process accuracy. The index Cpktakes into account the magnitude of process variation as well as the degree of process centering, which measures process performance based on yield (proportion of conformities). For a normally distributed process with a fixed value of Cpk, the bounds on process yield, %Yield, are given by 2(3Cpk) − 1 ≤ %Yield < (3Cpk), where (·) is the cumulative distribution function of N(0, 1), the standard normal distribution. For example, if Cpk= 1.00, then it guarantees that the %Yield will be no less than 99.73%, or no greater than 2700 ppm (parts per million) of non-conformities. We note that the index Cpkonly provides an approximate rather than an exact measure of the process yield. To obtain an exact measure, Boyles4 considered a yield index, referred to as Spk, for normally distributed processes. The index Spkis defined as:
Spk= 1 3 −11 2 USL− µ σ +1 2 µ − LSL σ
where −1 is the inverse function of . For a process with Spk= c, we can obtain %Yield = 2(3c) − 1. Obviously, there is a one-to-one relationship between Spk and the process yield. Thus, the yield index Spk provides an exact measure of the process yield. For normally distributed processes, the number of non-conformities corresponding to a capable process with Spk= 1.00 is 2700 ppm, a satisfactory process with Spk= 1.33 is 63 ppm, an excellent process with Spk= 1.67 is 0.6 ppm, and a super process with Spk= 2.00 is 0.002 ppm.
2.
CAPABILITY MEASURE FOR MULTIPLE CHARACTERISTICS
Capability measures for processes with a single characteristic have been investigated extensively, see e.g. Kane1, Pearn et al.2,3, Boyles4, Chan et al.5, Choi and Owen6, Kotz and Johnson7, V¨annman8, Deleryd and
V¨annman9and Pearn and Lin10. However, capability measures for processes with multiple characteristics is comparatively neglected. For processes with multiple characteristics, Bothe11 considered a simple measure by taking the minimum measure of each single characteristic. For example, consider a ν-characteristic process with ν yield measures (percentage of conformities) P1, P2, . . . , and Pν. The overall process yield is measured as P = min{P1, P2, . . . , Pν}. We note that this approach does not reflect the real situation accurately. Suppose the process has five characteristics (ν = 5), with equal characteristic yield measures P1= P2= P3= P4= P5= 99.73%. Using the approach considered by Bothe11, the overall process yield is
calculated as P = min{P1, P2, P3, P4, P5} = 99.73% (or 2700 ppm of non-conformities). Assuming that the
five characteristics are mutually independent, then the actual overall process yield should be calculated as P = P1× P2× · · · × P5= 98.66% (or 134 273 ppm of non-conformities), which is significantly less than that
calculated by Bothe11.
To overcome the problem, we propose the following overall capability index, referred to as SpkT :
SpkT =1 3 −1ν j =1 (2(3Spkj) − 1) + 1 2
where Spkj denotes the Spk value of the j th characteristic for j = 1, 2, . . . , ν, and ν is the number of characteristics. The new index, STpk, may be viewed as a generalization of the single characteristic yield index, Spk, considered by Boyles4.
Table I. Various SpkT values and the corresponding
process yield SpkT Process yield 1.00 0.997 300 204 1.24 0.999 800 777 1.33 0.999 933 927 1.50 0.999 993 205 1.67 0.999 999 456 2.00 0.999 999 998 Given SpkT = c, we have 1 3 −1ν j =1 (2(3Spkj) − 1) + 1 2 = c Hence, ν j =1 [2(3Spkj) − 1] = 2(3c) − 1
A one-to-one correspondence relationship between the index SpkT and the overall process yield P can be established as: P = ν j =1 Pj= ν j =1 [2(3Spkj) − 1] = 2(3SpkT ) − 1.
Hence, the new index STpkprovides an exact measure of the overall process yield. For example, if SpkT = 1.00, then the entire process yield would be exactly 99.73%. Table Idisplays various commonly used capability requirements and the corresponding overall process yield. For a process with ν characteristics, if the requirement for the overall process capability is SpkT ≥ c0, a sufficient condition (which is minimal) for the requirement to
each single characteristic can be obtained by the following. Let cbe the minimum Spkvalue required for each single characteristic, then
1 3 −1ν j =1 (2(3Spkj) − 1) + 1 2 ≥1 3 −1ν j =1 (2(3c) − 1) + 1 2 Hence, if 1 3 −1ν j =1 (2(3c) − 1) + 1 2 ≥ c0 i.e. c≥ 1 3 −1√ν2(3c0) − 1 + 1 2 then we have SpkT =1 3 −1ν j =1 (2(3Spkj) − 1) + 1 2 ≥ c0
Table II. Capability zones for multiple characteristics
Spkjfor single characteristic Characteristic
number ν Lower bound sL Upper bound sU
1 1.000 1.333 2 1.068 1.387 3 1.107 1.417 4 1.133 1.439 5 1.153 1.455 6 1.170 1.468 7 1.183 1.479 8 1.195 1.489 9 1.205 1.497 10 1.214 1.505 11 1.222 1.511 12 1.230 1.518 13 1.236 1.523 14 1.243 1.528 15 1.248 1.533
Thus, if the requirement for each single characteristic
Spkj≥ 1 3 −1 ν √ 2(3c0) − 1 + 1 2 , for all j = 1, 2, . . . , ν
is satisfied, then the overall capability requirement STpk ≥ c0 would be satisfied. For example, if c0 is set to be 1.00 with ν = 5, i.e. the overall process yield is set to be no less than 0.9973. The overall capability requirement SpkT ≥ 1.00 would be satisfied if each single characteristic yield is no less than (0.997 300 204)1/5= 0.999 459 50 (equivalent to 540 ppm of non-conformity items), and the capability for all the five characteristics is Spkj≥ 1 3 −1√52(3) − 1 + 1 2 = 1.153, for j = 1, 2, . . . , 5
If the requirement of the overall process capability is c1≤ SpkT ≤ c2 for a process with ν characteristics. The requirement would be satisfied, if the capability of j th characteristic satisfies sL≤ Spkj≤ sU for all
j = 1, 2, . . . , ν, where the lower bound sLand the upper bound sU on each Spkjcan be calculated, respectively, as sL= 1 3 −1√ν2(3c1) − 1 + 1 2 and sU= 1 3 −1√ν2(3c2) − 1 + 1 2
TableIIdisplays the lower bound sL and upper bound sU on Spkj if the requirement of the overall process capability is 1.000 ≤ SpkT ≤ 1.333 for ν = 1(1)15 characteristics. For example, suppose the requirement of the overall process capability is 1.000 ≤ SpkT ≤ 1.333 for process with five characteristics (ν = 5), we can obtain the lower bound sL= 1.153 and the upper bound sU= 1.455 on all the five Spkj values.
3.
S
pkMULTI-CHARACTERISTIC PROCESS CAPABILITY ANALYSIS CONTROL
CHART
Based on the yield index Spk, Boyles4developed a tool called the Spkcontour plot which is a contour plot of index Spkas a function of the process parameters (µ, σ ) for monitoring and controlling process performance.
Figure 1. The SpkMCPCA control chart with various departure control zones
In fact, the Spk contour plot is a useful tool for evaluating multiple processes, as we can obtain the process yield and the process departure ratio by checking the location of the index value falling on the contour plot. For multiple processes with the same specification limits, the contour plot not only shows the process capability for multiple processes simultaneously, but also provides a quick reference to the parameters that should be targeted for process improvement. Note, however, that the Spk contour plot is only applicable for multiple processes with the same specification limits on each single process, which may not be used on processes with multiple characteristics where the characteristic specifications are not the same.
To extend the applicability of the contour plot for processes with multiple characteristics, we apply the method developed by Deleryd and V¨annman9who introduced a process capability plot, called the (δ, γ )-plot, which is an adjustment of Boyles’ (µ, σ )-plot where δ = (µ − T )/d, γ = σ/d. We rewrite the definition of Spkas below, which can be expressed as a function of Cdr= (µ − T )/d and Cdp= σ/d. Note that Cdrmeasures the departure ratio, and Cdpmeasures the variation relative to the specification tolerance. If µ < T , we have Cdr< 0; if µ > T , we have Cdr> 0; and if µ = T , we have Cdr= 0 (the process is on target in this case). Obviously, if µ = LSL, then Cdr= −1; if µ = USL, then Cdr= 1. We will focus on the case where the specification interval is two-sided with the target value T at m, which is most common in practical situations. Henceforth we will assume that T = m. Spk= 1 3 −11 2 USL− µ σ +1 2 µ − LSL σ =1 3 −11 2 1− (µ − m)/d σ/d +1 2 1+ (µ − m)/d σ/d =1 3 −11 2 1− Cdr Cdp +1 2 1+ Cdr Cdp
Therefore, using Cdr as the x-axis and Cdp as the y-axis, we can plot the following point set forming the curve of Spk(bold curves in Figure1) on the (Cdr, Cdp) coordinates,
(Cdr, Cdp) 13−1 1 2 1− Cdr Cdp +1 2 1+ Cdr Cdp = Spk
Note that the process capability plot is invariable irrespective of the value of the specification limits. Processes with multiple characteristics having different characteristic specification limits can thus be plotted
Table III. Various control regions for process departure and improvement suggestions
Control region Cdrvalue Process improvement suggestion I1 ±0.25 Characteristic departure is tolerable
I2 ±0.50 Characteristic departure is abnormal,
need to investigate and improve I3 ±1.00 Characteristic departure is serious,
need to recheck overall process
Figure 2. The MCPCA chart with contours of Spk= 1.0, 1.33, 1.5, 1.67, 2.0 (from top to bottom)
simultaneously on a single chart. We shall call this control chart the multi-characteristic process capability analysis (MCPCA) chart. In Figure1, the top curve is the capability plot for processes with Spk= 1.00, and the bottom curve is the capability plot for processes with Spk= 1.33.
As we pointed out earlier, Spk is a yield-based index. However, according to today’s modern quality improvement theories, reduction of variation from the target is as important as increasing the process yield (meeting the specifications). Therefore, three pairs of control limits I1, I2, and I3 are drawn on the proposed
MCPCA chart (see Figure1) to monitor the variation from the target of each characteristic. Corresponding to the three pairs of control limits I1, I2, and I3, the values of Cdr= (µ − T )/d are −1.0, −0.5, −0.25, 0.25, 0.5, and 1.0 from left to right, respectively.
Under the six-sigma quality improvement program formulated by Motorola (see Noguera and Nielsen12) assuming d = 6σ , the three pairs of control limits I1, I2, and I3 correspond to|µ − T | = 1.5σ, 3σ and 6σ.
Research has shown that a typical process is likely to deviate from its natural centering condition by approximately 1.5σ at any given moment in time. Under the six-sigma quality improvement program, the process mean is allowed to shift as much as 1.5σ , i.e. all the pairs (Cdrj, Cdpj) of the j th characteristic should not locate outside of the pair of control limits I1. Note that six-sigma technically means having no more than
3.4 ppm of non-conformities by assuming that the specification limits are 6σ away from the target. Therefore, the three pairs of control limits I1, I2, and I3form various process accuracy (the degree of centering) control zones.
Various control regions (zones) for process departure and the improvement suggestions are summarized in Table III. The practitioners can judge the degree of centering of characteristic j by checking the location of the corresponding plotted point on the MCPCA chart. For example, the departure measure Cdr value is −0.5 for characteristic A in Figure 1. The departure ratio is considered to be significant which calls for an
immediate check to find abnormal changes in the parameter settings. Using the MCPCA chart, the practitioners can effectively control and monitor both the variation and the departure from its target for each single process characteristic.
In the MCPCA chart (see Figure 2), we use the standardized measures on process departure and process variation. Therefore, the MCPCA chart can be used for processes with multiple characteristics where the individual characteristic specifications may not be identical. The MCPCA chart displays all the characteristic index values on one chart, and indicates the characteristic yield based on the Spk contours. It can provide information instantly about the locations and spreads of all the studied characteristics by their (Cdr, Cdp)-coordinate values. The MCPCA chart also displays the relative magnitudes of process variation and the process departure in terms of the standardized measures (Cdr, Cdp) which can be used to control and monitor all characteristics simultaneously. Therefore, the MCPCA chart provides in-time information so that the practitioners can understand the quality level of the product easily. In addition, the MCPCA chart also provides a clear direction on which parameter needs to be targeted for quality improvement.
4.
AN APPLICATION
Development of the new index is novel and deals with a common and practical problem (yield measurement) occurring in the manufacturing industry. To the best of our knowledge, none of the existing methods published in the literature provides the same function as the new index SpkT . The new index is useful to the engineers/practitioners in measuring the yield, particularly for processes with multiple characteristics (a problem the industry must continue to face). We note that the exact sampling distribution of the estimated Spk, which is a special case of the proposed new index with ν = 1 (for single characteristic), has never been investigated. Statistical properties of the special case with ν = 1 are in fact mathematically intractable. Lee et al.13obtained an approximate distribution of the estimated Spkusing the Taylor expansion. Statistical properties of the new index, SpkT , are expected to be considerably more difficult to deal with than that of Spk(for a single characteristic).
Pearn and Kang14 considered the sampling distribution of the estimated Spk, and conducted a simulation to evaluate its accuracy. Their simulation results show that the estimated Spkoverestimates the true value of Spk. However, with a sample size n greater than 150, the relative bias is less than 0.01, and the sampling error becomes negligible. For practical purposes, we suggest that a minimal value of n = 150 must be used in the applications to maintain a good accuracy.
For each single characteristic the Cdrj and Cdpj values are calculated and the (Cdrj, Cdpj) pair is plotted on the chart for j = 1, 2, . . . , ν. In real applications Cdrjand Cdpj, representing the j th characteristic (with target value Tj), are replaced by their natural estimators:
ˆCdrj=( ¯Xj− Tj)
dj , ˆCdpj= sj dj
where dj is the half length of the j th characteristic’s specification interval, ¯Xj= n
i=1xij/n and sj2= n
i=1(xij− ¯Xj)2/(n − 1) are the sample mean and the sample variance of the j th characteristic, respectively. Hence, the natural estimators ˆSpkjof index Spkjcorresponding to the j th characteristic are
ˆSpkj = 1 3 −1 1 2 1− ˆCdrj ˆCdpj +1 2 1+ ˆCdrj ˆCdpj , for j = 1, 2, . . . , ν
To illustrate the application of the MCPCA chart, we consider a real example taken from an electronic thermos manufacturer, located in Taiwan, adapting the six-sigma quality improvement program. One special type of thermos investigated has five target-the-best quality characteristics with unequal manufacturing specifications.
The quality requirement for the final product is 1.00 ≤ SpkT ≤ 1.33, i.e. the requirement for the process yield is no less than 99.73%. For ν = 5, we can obtain the lower bound sL= 1.153 and the upper bound sU= 1.455 for
Table IV. Calculations for the electronic thermos process capability j LSL T USL ¯Xj sj ( ˆCdrj, ˆCdp) ˆSpkj 1 5.598 6.220 6.842 5.909 0.124 (−0.500, 0.199) 0.915 2 606.5 680.0 753.5 683.3 17.13 (−0.045, 0.233) 1.406 3 0.279 0.310 0.341 0.332 0.0076 (0.710, 0.245) 0.521 4 31.5 35.0 38.5 34.48 0.525 (−0.149, 0.150) 1.931 5 30 40 50 43.5 0.80 (0.35, 0.08) 2.737
Figure 3. An application of the SpkMCPCA chart
all five Spkj values from TableII. Hence, a process is capable if 1.153 ≤ Spkj≤ 1.455 and the pair (Cdrj, Cdpj) of the j th characteristic must locate between the pair control limits I1for all j = 1, 2, . . . , 5.
A sample of 150 or more items is taken from the factory to inspect. The specification, target value, sample mean, sample standard deviation, estimated Cdrj, Cdpj, and Spkj are summarized in TableIV. First, we plot two contours corresponding to Spk= 1.153 and Spk= 1.455 on the MCPCA chart, shown as two bold curves in Figure3. At the same time, five pairs of the estimated Cdrjand Cdpjvalues are plotted on the chart. This chart clearly shows the status of each in-process characteristic. Since all of these in-process characteristics are plotted in one chart, it is easy to determine their relative status. From the MCPCA chart, we can quickly conclude that the process is incapable because there are two plotted points out of the contour of Spk= sL= 1.153. We note that the estimated SpkT = 0.5135. Based on the analysis of this chart, we can make the following conclusions and recommendations.
1. The plotted points corresponding to characteristics 1 and 3 are out of the contour of Spk= 1.153 which show both ˆSpk1 and ˆSpk3 are less than the lower bound value sL= 1.153 (the lower bound of quality requirement for all five Spkj). Hence, the process is considered to be ‘incapable’ for characteristics 1
and 3. In fact, ˆSpk1= 0.915 and ˆSpk3= 0.521. Since both ˆSpk1and ˆSpk3are significantly less than 1.00 (the lower bound of quality requirement SpkT for the final product), then both characteristics 1 and 3 are candidates for high-priority quality improvement efforts. Furthermore, the plotted points 1 and 3 are out of the control limits I1. Hence, both the variability and the deviation from target must be reduced to improve the process quality for characteristics 1 and 3.
2. The plotted point corresponding to characteristic 2 is between two contours corresponding to Spk= 1.153 and Spk= 1.455. At the same time, the plotted point 2 is located between the pair of the control limits I1. This shows that the process is satisfactory for characteristic 2. Note that the corresponding ˆSpk2= 1.406.
3. The plotted points corresponding to characteristics 4 and 5 are inside the contour of Spk= 1.455, which shows both ˆSpk4and ˆSpk5are greater than the upper bound value sU= 1.455 (the upper bound of quality requirement for all five Spkj). In fact, ˆSpk4= 1.931 and ˆSpk5= 2.737. Note that the plotted point 4 is located between the pair of the control limits I1, but the plotted point 5 is out of the control limits I1. A reduced sampling plan for characteristic 4 could be considered since the process is super. Under the six-sigma program, the quality improvement effort for characteristic 5 could be focused on the reduction of the process deviation from its target.
5.
CONCLUSIONS
Process capability indices have been widely used in the manufacturing industry, providing numerical measures on process precision, process accuracy, and process performance. Capability measures for processes with a single characteristic has been investigated extensively. However, capability measures for processes with multiple characteristics are comparatively neglected. In this paper, we proposed a generalized capability measure, called SpkT , based on the yield index Spk proposed by Boyles4, for processes with multiple characteristics. We established a relationship between the new measure and the process yield. We also developed a control chart MCPCA based on the proposed new measure, which displays all the characteristic measures in one single chart. Using the control chart MCPCA, the engineers can effectively monitor and control the performance of all process characteristics simultaneously.
Acknowledgements
The authors would like to thank the Editor, Professor Douglas Montgomery, for his helpful suggestions. The authors would also like to thank the anonymous referees for their careful reading of the paper and for helpful comments which improved the paper. This paper was supported in part by the National Science Council, Taiwan, under the contract NSC 90-2218-E-167-002.
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Authors’ biographies
K. S. Chen is a Professor in the Department of Industrial Engineering and Management, National Chin-Yi
Institute of Technology, Taichung, Taiwan. He received his MS degree in Statistics from Cheng Kung University, and his PhD degree in Quality Management from the National Chiao Tung University, Taiwan.
W. L. Pearn is a Professor of Operations Research and Quality Management at the Department of Industrial
Engineering and Management, National Chiao Tung University, Taiwan. He received his MS degree in Statistics and his PhD degree in Operations Research from the University of Maryland, College Park, MD, U.S.A. He worked for AT&T Bell Laboratories at Switch Network Control and Process Quality Centers before he joined the National Chiao Tung University.
P. C. Lin received his MS degree in Statistics from the National Chung Hsing University, and his PhD degree in
Quality Management from the National Chiao Tung University, Taiwan. Currently, he is an Associate Professor at the General Eduction Center, National Chin-Yi Institute of Technology, Taichung, Taiwan.
