The acceptance sampling plans had been many researched. But as today’s modern quality improvement philosophy, suppliers require their products to be of high quality with a very low fraction of defectives. Acceptance sampling plans of traditional methods for calculating the fraction nonconforming no longer work.
An alternative method of measuring the fraction of defectives is to use process capability indices. Therefore, in this section, we will review these papers about acceptance sampling plans for index Cpk, Cpm and Cpmk.
2.1. Acceptance Sampling Plans Based on Cpk
Process capability analysis has become an important and integrated part in the applications of statistical process control for continuous improvement in productivity and quality assurance. Process capability indices (PCIs), establishing the relationship between the actual process performance and the manufacturing specifications, have been the focus of recent research in quality assurance and statistical literature. The most commonly used indices include Cp, Cpk, Cpm and Cpmk. The Cp and Cpk have been proposed for a long time and widely discussed in the paper of Kane (1986). The indices Cpm and Cpmk were originally developed by Chan et al. (1988) and Pearn et al. (1992), respectively.
Based on analyzing the PCIs, a production department can trace and improve a poor process so that the quality level can be enhanced and the requirements of the customers can be satisfied. Index Cp has been defined as Cp =(USL LSL / 6− ) σ, where USL and LSL are the upper and lower specification limit, respectively,
σ
is the process standard deviation. In process capability analysis, Cpk is the most popular index. It has been defined as:min ,
where USL is the upper specification limit, LSL is the lower specification limit, μ is the process mean, and
σ
is the process standard deviation.The Cpk index is an appropriate measure of progress for quality improvement paradigms in which reduction of variability is the guiding principle and process yield is the primary measure of success. Pearn and Wu (2007) provided an acceptance sampling plan for Cpk index as a quality benchmark for product acceptance. Since the quality characteristic is variable, the lower specification limit and the upper specification limit can be used to define the acceptable values of this parameter. It is easy to design a sampling plan with a specified OC curve. Let (AQL,1− ) and (LTPD,
α
β ) be the two points on the OC curve of interest.For processes with target value set to the mid-point of the specification limits (i.e. T =M ), the index may be rewritten as Cpk =( /d
σ ξ
− ) 3 , where( M)
ξ
=μ
−σ
, T is the target value, d =(USL−LSL) / 2 is the half length of the specification interval, m=(USL+LSL) / 2 is the midpoint of the specificationlimits. It’s noted, the sampling distribution of Cˆpk =(d− X M− )/3S is expressed in terms of mixture of the chi-square and the normal distributions.
Given Cpk =C, b d= /
σ
can be expressed as b=3C+ξ
. Thus, the probability of accepting the product can be expressed asπ
where is the cumulative distribution function of the chi-square distribution with degree of freedom n – 1,
( ) G ⋅
1
χn− , and φ( )⋅ is the probability density function of the standard normal distribution N (0, 1).
Therefore, the required inspection sample size n and critical acceptance value for the sampling plan are the solutions to the following two nonlinear simultaneous equations: 0 represent the capability requirements corresponding to the AQL and the LTPD based on
AQL LTPD
C >C CAQL CLTPD
Cpk index, respectively.
For practical application purposes, we calculate and tabulate the critical acceptance values ( ) and required sample sizes ( ) for the sampling plans, with commonly used 0
c n
, , CAQL and CLTPD
α β . The results obtained are useful to the practitioners in making reliable decisions.
2.2. Acceptance Sampling Plans Based on Cpm
Pearn and Wu (2006) developed the acceptance sampling plan for Cpm index. Hsiang and Taguchi (1985) introduced the index Cpm, which was also proposed independently by Chan et al. (1988). The index is related to the idea of squared error loss loss X( ) (= X T− )2. This loss based process capability index Cpm has also been called the Taguchi capability index. The index emphasizes on measuring the ability of the process to cluster around the target, which reflects the degrees of process targeting (centering). The index Cpm incorporates with the product variation with respect to the target value and the manufacturing specifications preset in the factory. The index Cpm is defined as
2 2
target value.
According to today’s modern quality improvement philosophy, customers do notice unit-to-unit differences in these characteristics, especially if the variance is large and /or the mean is offset from the target. With the increasing importance of clustering around the target rather than conforming to specification limits, the understanding of loss functions is the guiding principle to assess the process capability. Therefore, for this reason the Cpm index can be used as a quality benchmark for acceptance of a production lot.
The probability of accepting the lot can be expressed as:
π
probability density function of the standard normal distribution N (0,1). Therefore, the required inspection sample size and critical acceptance value of n c0 ˆC pm
for the sampling plans can be obtained by solving the following two nonlinear simultaneous equations:
and represent the capability requirements corresponding to the AQL and the LTPD based on
2 1/2
1 3 AQL(1 )
b = C +
ξ
b2 =3CLTPD(1+ξ2 1/2) CAQL >CLTPD CAQLCLTPD
Cpm index, respectively.
Pearn and Wu (2006) also tabulated the required sample size and the critical acceptance value for various
n
c0
α
-risks, β -risks, and the fraction of defectives of process that correspond to acceptable quality levels. Practitioners can determine the number of required inspection units and the critical acceptance value, and make reliable decisions.2.3. Acceptance Sampling Plans Based on Cpmk
The index Cpmk is constructed by combining the yield-based index Cpk and the loss-based index Cpm, taking into account the process yield (meeting the manufacturing specifications) as well as the process loss (variation from the target). So, the Cpmk index is defined as
where USL is the upper specification limit, LSL is the lower specification limit, μ is the process mean,
σ
is the process standard deviation, and T is the target value.When the process mean μ departs from the target value T, the reduced value of Cpmk is more significant than those of C Cp, , pk and Cpm. Hence, the index Cpmk responds to the departure of the process mean μ from the target value T faster than the other three basic indices C Cp, , pk and Cpm, while it remains sensitive to the changes of process variation (see Pearn and Kotz, 1994-1995). Thus, the index Cpmk indeed provides more quality assurance with respective to process yield and process loss to the customers than the two indices Cpk and Cpm.
According to today’s modern quality improvement theory, reduction of the process loss is as important as the process yield, Cpmk can be used as a quality benchmark for acceptance of a product lot. Therefore, Wu and Pearn (2008) provided the acceptance sampling plan for Cpmk index.
The probability of accepting the lot can be expressed as:
π
probability density function of the standard normal distribution N (0,1). Therefore, the required inspection sample size and critical acceptance value n c0 of ˆCpmk
for the sampling plans can be obtained by solving the following two nonlinear simultaneous equations: Here and represent the capability requirements corresponding to the AQL and the LTPD based on
AQL LTPD
C >C CAQL CLTPD
Cpmk index, respectively.
Wu and Pearn (2008) developed a method of acceptance sampling plan for obtaining the required sample size for inspection and the corresponding critical acceptance values based on the exact sampling distribution, which provides the desired levels of protection for both producers and consumers.