2.1 Process Capability Indices
There are many capability indices proposed to use for evaluating a supplier’s process capability. The first process capability index in the literature was Cp. It was introduced by Juran et al. (1974), but did not gain considerable acceptance until the early 1980s. It is defined as:
6
LSL Cp USL
,where USL is the upper specification limit, LSLis the lower specification limit, and is the process standard deviation. The index measures capability in terms of process variation only and did not take process location into consideration.
Pearn et al. (1998) introduced an accuracy index
C
a to measure the magnitude of process centering. It is defined as:| |
The index
C
a measures the centering tendency. User can get alerts from it if the process mean is deviate form the midpoint. Kane (1986) proposed the capability index Cpk, considered process location of mean and process variation. The index Cpk determines process ability of reproducing items within the specified manufacturing tolerance. It is defined as: | |
Based on the expression of process yield, Boyles (1994) considered the yield index Spk for normal process, as defined in the following:
1
11 1
where is the cumulative density function (c.d.f) of the standard normal distribution N
(0,1)
. Hsiang and Taguchi (1985) introduced the index Cpm , independently proposed by Chan et al. (1988). The index Cpm focuses on the product loss when one of its characteristics departs from the target value T. It is defined as:2.2 Process Yield Based on Spk
2.2.1 Process Yield
In the past, we have to count the number of nonconforming items from a sample to calculate the yield. However, the fraction of non-conformities now is less than 0.01%, and we usually use parts per million (ppm) to express.
Traditional methods for calculating the fraction nonconforming are no longer work since all reasonable sample sizes will probably have no defective items.
These methods are substituted for capability indices.
Process yield has long been a standard criterion used in the manufacturing industry as a common measure on process performance. Process yield is defined as the percentage of processed product passing inspection. That is, the product characteristic must fall within the manufacturing tolerance. It can be calculated as:
Yield
F USL( )
F LSL( )
,where USL and LSL are the upper and lower specification limits, and F x
( )
is the cumulative distribution function of the process characteristic. If the process characteristic is normal distributed, then the process yield can be expressed as:Yield (
USL ) (
LSL)
,where is the process mean, is the process standard deviation, and
( )
x is the cumulative distribution function of the standard normal distribution(0,1)
N .
2.2.2 Yield Assurance Based on Spk
For normal distributed process, the relationship between the process yield and the index Cpk is
Yield 2 (3
Cpk) 1
. Thus, the index Cpk provides us with an approximate, rather than exact, measure of the actual process yield.Based on the expression of process yield, Boyles (1994) considered the yield index Spk for normal process. This index Spk provides an exact measure on the process yield. If Spk
c , then the process yield can be expresses asYield 2 (3 ) 1
c
. There is a one-to-one correspondence between Spk and the process yield. Table 1 summarizes the process yield, nonconformity (in ppm) as a function of the index Spk=1.00, 1.33, 1.50, 1.67, and 2.00. For example, if a particular process the yield measure Spk=1.67, then the corresponding value of nonconformities is 0.544 ppm.Table 1. Some Spk values and the corresponding values of fraction yield and nonconformities (ppm).
Spk Yield Nonconformities
2.3 Supplier Selection Problems based on PCIs
Because of the process mean and the process variance
are not
2 known in real world. In order to calculate the estimator, however, data must be collected to calculate the index value, and a great degree of uncertainty may be introduced into capability assessments due to sampling errors.The most common methods to assess the process capability are to utilize the interval estimation and hypotheses testing. Consequently, these estimating methods must be performed by using their sampling distributions. Kotz and Johnson (2002) presented a thorough review for the PCI developments during the years 1992 to 2000. Spiring et al. (2003) consolidated the research findings of process capability analysis for the period 1990–2002. Lee et al. (2002) considered an asymptotic distribution for an estimate
S
pk of the process yield index Spk. A useful approximate distribution of
S
pk was furnished. Pearn and Chuang (2004) investigated the accuracy of the natural estimator of Spk computationally, using a simulation technique to find the relative bias and the relative mean square error for some commonly used quality requirements. Chen (2005) considered to use the bootstrap simulation technique to find four approximate lower confidence limits for index Spk. The simulation results show that the SB method significantly outperforms than other three methods. But, these studies considered the assessment of capability for a single process or supplier.
In a review of the problems for supplier selection based on PCIs, Tseng and Wu (1991) considered the problem for K available manufacturing processes based on the precision index Cp under a modified likelihood ratio (MLR) selection rule. Chou (1994) designed testing procedures for comparing two processes or suppliers in terms of Cp, Cpl, and Cpu when sample size are equal. Huang and Lee (1995) considered the supplier selection problem based on the index Cpm and developed a mathematically approximation method for selecting a subset containing the process associated with the smallest s2 (m T)2 from K given independent processes. Pearn et al. (2004) provided useful information regarding
the sample size required for a designated selection power. A two-phase selection procedure was developed to select a better supplier and to calculate the magnitude of the difference between two suppliers. Chen and Chen (2004) used four approximate confidence interval methods to present and compare for index Cpm. One based on the statistical theory given in Boyles (1991), and three based on the bootstrap (referred to as standard bootstrap, percentile bootstrap, and biased-corrected percentile bootstrap) for selecting a better supplier. However, the testing procedure for supplier selection based on Spk has not been done today. In this thesis, because the exact sampling distribution of Spk is analytical intractable, we will use bootstrap method to compare two processes based on
Spk.