DOI 10.1007/s00170-004-2314-1 O R I G I N A L A R T I C L E
Kun-Lin Hsieh · Lee-Ing Tong
Incorporating process capability index and quality loss function into analyzing
the process capability for qualitative data
Received: 13 January 2004 / Accepted: 22 June 2004 / Published online: 13 April 2005 ©Springer-Verlag London Limited 2005
Abstract Process capability analysis (PCA) is frequently em-ployed to evaluate a product or a process if it can meet the cus-tomer’s requirement. In general, process capability analysis can be represented by using the process capability index (PCI). Until now, the PCI was frequently used for processes with quantitative characteristics. However, for process quality with the qualitative characteristic, the data’s type and single specification caused lim-itations of using the PCI. When the product can not meet the target, even if it lies in the specified range, it should lead to the corresponding quality loss. Taguchi developed a quadratic qual-ity loss function (QLF) to address such issues. In this study, we intend to construct a measurable index which incorporates the PCI philosophy and QLF concept to analyze the process ca-pability with the consideration of the qualitative response data. The manufacturers can not only employ the proposed index to self-assess the process capability, but they also can make com-parisons with the other competitors.
Keywords Process capability analysis (PCA)· Process capability indexes (PCIs)· Qualitative data · Quality loss function (QLF).
1 Introduction
Process capability analysis (PCA) [1, 4, 5, 11] is frequently em-ployed by the manufacturers to evaluate if the capability of pro-cess can meet the customer’s requirement. Propro-cess capability in-dexes (PCIs) [1, 4] are a quantitative measurement of the process K.-L. Hsieh (u)
Department of Information Management, National Taitung University,
684 Chunghua Rd., Sec. 1, Taitung, Taiwan, R.O.C. E-mail: klhsieh2644@mail200.com.tw
Tel.: +886-89-318855 ext. 2656 Fax: +886-89-321981 L.-I. Tong
Department of Industrial Engineering and Management, National Chaio Tung University,
HsinChu, Taiwan, R.O.C.
capability in most manufacturing industries. PCIs, such as Ca, Cpand Cpkare commonly used for most manufactures [3, 7–11], can frequently measure the process capability for the quantita-tive response. Herein, Caevaluates the related scale of the process mean with the tolerance specification (i.e. the difference between the upper tolerance limit and the lower tolerance limit). Cp eval-uates the related scale of the specification’s tolerance with pro-cess’s tolerance. While Cpksimultaneously evaluates the center-ing degree and the dispersion degree. These PCIs will make some adjustments if there are necessary particulars like the unilateral specification [10]. For the quantitative type, the theories on PCA and PCIs are well developed [1, 8, 11]. The qualitative data type may exist during the manufacturing environment, e.g. the inte-grated circuit (IC) manufacturing industry uses the defect count on a wafer to analyze their product’s yield and control their pro-cess, the process capability analysis for qualitative data will be an important issue to study. However, most studies only focus on the PCA application for the quantitative response data, and the qual-itative response data is seldom mentioned [6, 7]. Several difficul-ties can be mentioned as: (1) the target of the qualitative data may lead to unobvious centering evaluation, e.g. the target will be set as zero defect, (2) the limitation of the unilateral specification, espe-cially only the upper specification exist, e.g. the defect rate may be less than 1% and (3) the quantitative data utilizes the process mean (µ) and process deviation (σ) to compute the PCIs, however, the qualitative data can not directly utilize them to compute the PCIs. Under the global market environment, to realize the pro-cess capability comparison with other competitions can provide helpful information for enhancing organizational competence or making strategic decisions. Especially, the PCA for the different manufacturers will be a significant factor to seek for the col-laborators during the consideration of supply chain management (SCM). In this study, we intend to construct a process capability index, the PCI on qualitative response data, to evaluate the pro-cess capability for the qualitative response data. The logical idea is to combine the PCI philosophy and QLF concept. The rest of this study is organized as follows. Sect. 2 clearly demonstrates the construction procedure of the quantitative measurement we proposed. Sect. 3 will employ the numerical examples to
demon-strate the effectiveness of the proposed approach. Concluding remarks are finally made in Sect. 4.
2 Construction of PCI for the qualitative response
data
2.1 Construction concept of PCI
The product can not meet the target, even if it lies in the speci-fied range, and it should lead to quality loss. Taguchi developed a quadratic quality loss function to address such a case [2]. The quadratic quality loss function is defined as follows:
L(y) = k(y − T)2 (1)
and the expected quality loss can be described as:
Q L= E[L(y)] = E[k (y − T)2]
= kE[(y − µ + µ − T)2] = k((µ − T)2+ σ2) (2)
where y denotes the response data, T denotes the target value or the nominal value, k denotes the constant of the quality loss when the process is within the allowable tolerance,µ denotes the process mean andσ denotes the process deviation.
For the qualitative response data set, the target situation should be zero defect (T= 0) or not non-conforming. Hence, ap-plying it into the quality loss function, Eq. 2 can be modified as:
Q L= k(µ2+ σ2) (3)
Then, the quality loss function can be represented as Q L=
Q L(θ) if the process parameter θ is involved. For the
qualita-tive data, according to the concept of the quality loss function, the quantitative measurement of the process capability can be constructed. That is, we can take the ratio of the customer’s allowable quality loss and the actual quality loss. Hence, the gen-eralized PCI of the attribute data can be defined as:
PCI= Q L(θc)
Q L(θ) (4)
Fig. 1. The diagram for Ber-noulli distribution with dif-ferent pc, p and PCI
whereθ denotes the process parameter of the actual process and θcdenotes the process parameter of the customer’s expectation. In fact, the qualitative data can be described well by several distributions like the binomial and Poisson distributions. In this section, we will clearly describe the PCI value of both distri-butions, and the features hidden in the proposed PCI also are explained in the next section.
2.2 Binomial distribution
First, let y∼ Ber(1, p), Ber(1, p) denotes the Bernoulli’s trial, θc= pc, θ = p, µ = p, σ2= p(1 − p), hence, PCI= Q L(θc) Q L(θ) = K[p2c+ (pc− p2c)] k[p2+ (p − p2)] = pc p (5)
where p denotes the non-conforming rate (the parameter of the binomial distribution), pcdenotes the acceptable quality level of the customer for the non-conforming rate and n denotes inspec-tion count.
The features include:
1. When pc< p, it means the capability of the process can not meet the customer’s requirement; that is, it is a “bad” pro-cess, PCI< 1.
2. When pc= p, it means that the capability of the process ex-actly meets the customer’s requirement, PCI= 1.
3. When pc> p, it means that the capability of the process ab-solutely satisfies the customer’s requirement, PCI> 1. 4. Figure 1 graphically depicts the relationship between process
parameter p and the PCI value with different pc.
Then, let y∼ Ber(1, p), and if the lot count is n, the related parameters can be denoted as:
D= n i=1 yi ∼ i.i.d.B(n, p), θc= pc, θ = p, µ = np, σ2= np(1 − p)
where B(n, p) will denote the binominal distribution. Hence, the
PCI formula for binominal distribution can be represented as
follows: PCI= Q L(θc) Q L(θ) = K[(n pc)2+ n(pc− p2c)] k[(np)2+ n(p − p2)] = [(n − 1)p2 c+ pc] [(n − 1)p2+ p] (6) Features:
1. When pc< p, it means the capability of the process can not meet the customer’s requirement; the PCI will decrease with respect to the inspection count n.
2. When pc= p, it means that the capability of the process ex-actly meets the customer’s requirement.
3. When pc> p, it means that the capability of the process ab-solutely satisfies the customer’s requirement; the PCI will increase with respect to the lot count n.
4. From Fig. 2a, when the process capability can meet the cus-tomer’s requirement, the PCI will increase with respect to the lot count n; while the process capability can not meet the cus-tomer’s requirement, the PCI will decrease with respect to
Fig. 2. a The PCI curve dia-gram for different inspection count when pc= 0.01 b The PCI curve diagram for different inspection count when pc= 0.1
the lot count n; the pc= p will be the saddle point in such cases. Furthermore, from Fig. 2a and b, we can clearly realize the change of the PCI curve when the customer’s allowable parameter changes.
2.2.1 Poisson distribution
Let y∼ P(λ), herein, P denotes the Poisson distribution, and the related distribution’s parameters can be represented as:θc=
λc, θ = λ, µ = λ, σ2= λ, PCI= Q L(θc) Q L(θ) = K[λ2c+ λc] k[λ2+ λ] = [λ2 c+ λc] [λ2+ λ] (7)
whereλ denotes the defect rate of the Poisson distribution for the actual process andλcdenotes the acceptable quality level of the customer for the defect rate of the Poisson distribution.
The features include:
1. Whenλc< λ, it means the process capability can not meet the customer’s requirement; that is, it is a “bad” process, that is, PCI< 1.
2. When λc= λ, it means that the process capability exactly meets the customer’s requirement; that is, PCI = 1. 3. Whenλc> λ, it means that the process capability absolutely
satisfies the customer’s requirement; that is, PCI> 1. 4. Figure 3 depicts graphically the relationship between the
pro-cess parameterλand PCI with different λc.
If the inspection count be m and it is denoted as the case of the average defect rate, y∼ P(λ), P denotes the Poisson distri-bution, and the related parameter can be denoted as:θc= λc, θ =
λ, µ =mλ, σ2=mλ2, and the PCI formula is: PCI= Q L(θc) Q L(θ) = K[λ2c m+λmc] k[λm2+mλ] = [λ2 c+ λc] [λ2+ λ] (8)
whereµ denotes the average defect rate and µcdenotes the ac-ceptable quality level of the customer for the average defect rate. We can find out that it has the same structure as the PCI of the Poisson distribution. Hence, the features are also the same as Poisson’s.
2.3 Construction of PCI for comparison with different competitors
We will make some integration according to the qualitative PCI we previously proposed for the case of comparison between different competitors. The concept is to substitute the manufac-turer’s quality loss by the competitors’ quality loss. Hence, the constructed comparison PCI is given as follows:
PCIcompetitor: manufacturer=
Q L(θcompetitor)
Q L(θmanufacturer)
(9) Features:
1. PCIcompetitor: manufacturer< 1, it means the competitor’s
pro-cess capability is better than the manufacturer’s propro-cess ca-pability.
2. PCIcompetitor: manufacturer= 1, it means the competitor’s
pro-cess capability is equal to the manufacturer’s propro-cess capabil-ity.
Fig. 3. The PCI curve dia-gram for the process parame-terλc, λ and PCI
3. PCIcompetitor: manufacturer> 1, it means the manufacturer’s
process capability is better than the competitor’s process ca-pability.
3 Numerical analysis and the conclusions
3.1 Illustrative example 1
A lead frame manufacturer in Taiwan expects to realize if their process capability can meet the customer’s requirement (the packaging fabrication). Several hundreds of lead frame types are produced in lead frame manufacturing. The packaging fabrica-tions expect the defect count of the lead frame in their in-line quality control (IQC) must be less than ten strips per 500 in-spection strips, and then the yield of the packaging product can be enhanced. Hence, the lead frame manufacturer plans to study their process capability and make a suitable compromise with the customers. The following data listed in Table 1 are collected for
Table 1. The non-conforming count
Day Non-conforming Day Non-conforming
count per 500 strips count per 500 strips
1 7 16 10 2 5 17 7 3 13 18 9 4 11 19 14 5 12 20 12 6 9 21 11 7 10 22 8 8 14 23 9 9 10 24 12 10 6 25 8 11 13 26 10 12 9 27 9 13 12 28 7 14 8 29 8 15 12 30 10
30 lots as the same QFP lead frame, and the constant k of the manufacturer is 180 (that is the necessary cost for rework). The quality (result) of the product can be divided into two categories: conforming and non-conforming. Each inspection is independent of the others. About 30 records were collected in Table 1. Obvi-ously, the data obey the binomial distribution.
First, we must make sure that the process is in-control. The data type is owing to the qualitative type. Hence, the np-chart is applied to process control (see Fig. 4). The total non-conforming count can be computed as 295. Hence, we can use the following formula to estimate the non-conforming rate p.
¯p =total number of non-conforming unitstotal number inspected = 295
30× 500 = 0.0197 Then, the np-chart is constructed as follows.
UCL= n ¯p + 3n¯p(1 − ¯p) = 19.17 CL= n ¯p = 9.85
LCL= n ¯p − 3n¯p(1 − ¯p) = 0.53.
It is in-control for screening out the np-chart. Then, we will employ the proposed PCI formula to study the process capabil-ity of the lead frame manufacturer. As the customer’s require-ment, the parameter θc(pc) is 0.02, and the manufacture’s es-timated parameter θ (p) is 0.0197. When the inspection count is 500, the average count of the non-conforming unit is 9.85 (0.0197 × 500). Then, the proposed PCI value can be computed as follows: PCI= Q L(θc) Q L(θ) = K[(n pc)2+ n(pc− p2c)] k[(np)2+ n(p − p2)] =[(n − 1)p2c+ pc] [(n − 1)p2+ p] = 499× 0.022+ 0.02 499× 0.01972+ 0.0197 = 1.0393. We can find out that the PCI value exceeds 1, and it means the current process capability can meet the customer’s require-ment. However, the ratio is not significantly larger than 1, so the lead frame manufacturer still need to pay more attention to their process.
Fig. 4. The constructed NP-chart
3.2 Numerical example 2: Comparison with different competitors
For the same numerical example, besides, the manufacturer also collected the related competitor’s information. They also ex-pect to realize the difference between their process capability and competitors’ process capability. The collected information is given as follows.
For competitor A: the parameterθa(the non-conforming rate
pa) is 0.015, the inspection count is 600 strips and the constant k is 200.
For competitor B: the parameterθb(the non-conforming rate
pb) is 0.025, the inspection count is 400 strips and the constant k is 250.
For manufacturer: the parameterθb(the non-conforming rate
pb) is 0.0197, the inspection count is 500 strips and the constant
k is 180.
Compare with competitor A:
PCIA:M= Q LQ L(θ(θA) M) = kA[(nApA)2+ nA(pA− p2A)] kM[(nMpM)2+ nM(pM− p2M)] = 200[(0.015 × 600)2+ 600 × (0.015 − 0.0152)] 180[(0.0197 × 500)2+ 500 × (0.0197 − 0.01972)] = 17973 19202.22 = 0.936 < 1
Conclusion: PCIA:M< 1, it means the competitor’s process capability is better than the manufacturers’ process capability. That is, competitor A’s quality loss is less than the manufactur-er’s quality loss.
Compare with competitor B:
PCIB:M= Q LQ L(θ(θB) M) = kB[(nBpB)2+ nB(pB− p2B)] kM[(nMpM)2+ nM(pM− p2M)] = 250[(0.025 × 400)2+ 400 × (0.025 − 0.0252)] 18[(0.0197 × 500)2+ 500 × (0.0197 − 0.01972)] = 27437.5 19202.22 = 1.429 > 1
Conclusion: PCIB:M> 1, it means the competitor’s process
capability is worse than the manufacturer’s process capability. That is, competitor B’s quality loss is larger than the manufactur-ers’ quality loss.
According to the comparison, we can make the conclusion: “Competitor B’s process capability is worse than the manufac-turer, while competitor A’s process capability is significantly bet-ter than the manufacturer. Furthermore, the sequence of process capability from the best to the worst is Competitor A→ Man-ufacturer→ Competitor B”. Restated, the manufacturer should
work hard to enhance their process capability by performing the necessary quality improvement.
4 Concluding and remarks
In this study, we construct a quantitative measurement PCI for the qualitative response. The quantitative measurement is based on the Taguchi’s quality loss function philosophy and PCI con-cept. It is a ratio deriving from the customer’s quality loss with respect to the actual process’s quality loss. By employing the proposed PCI, the manufacturers can employ it to assess if the process capability can meet the customer’s requirement. Be-sides, the constructed PCI can also be employed to make the comparison between the manufacturer and the competitors. The
PCI formulas for different quality data obeying the binomial
distribution or Poisson distribution are proposed in this study. The other advantage is that the practitioners do not need com-plicated computation to obtain the attribute PCIs by using the proposed PCIs.
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