In this paper, we have improved SKC method by adjusting the functions of skewness and kurtosis in SKC control charts. Our MSKC method is also without assumptions on the form of the process distribution. When the skewness is significantly nonzero and the kurtosis is significantly nonnegative, the performance of the MSKC method is significantly better than those of SKC, SC, WV, SN, WSD and Shewhart methods.
If the process distribution and the parameters are unknown, it is necessary to deter-mine whether the process distribution is skewed or leptokurtic. Some tests about skewness and kurtosis can be found in Castagliola (1999) and Tadikamalla and Popescu (2007). The KC control chart is suggested if the skewness is not significantly nonzero and the kurtosis is significantly positive. Otherwise, the MSKC control chart will be recommended.
Finally, we note that the MSKC method can also be used to construct control charts for any Type I risk α. For example, if α = 0.005, then the simulation results show that δ1 and δ2 in (1.1) and (1.2) are closed to 0.1 and 20.8, respectively. Hence the corresponding MSKC control charts can be constructed as in Section 1.2.
2 Process Capability Measures with Multiple Char-acteristics
2.1 Introduction
Process capability indices display the relationship between performance of the actual process and manufacturing specifications, and are used for the rating of the process. Some process capability indices have been widely used in the manufacturing industry. Capability measures for processes with a single characteristic have been investigated extensively, see e.g. Kane (1986), Pearn et al (1992, 1998), Boyles(1994), Chan et al. (1998), Choi and Owen (1990), Kotz and Johnson (1993), V¨annman (1995), Deleryd and V¨annman (1999) and Pearn and Lin (2000). Let USL, LSL, µ and σ, respectively, be the upper specification limit, lower specification limit, mean and standard deviation of the process.
Also let T =(USL+LSL)/2 and d=(USL-LSL)/2 be the target value and half length of the specification interval. The index
Cp = USL − LSL 6σ
measures the spread of the specifications relative to the six-sigma spread in the process.
The index Cp reflects potential capability in the process. The index Ca= 1 − |µ − T |
d
measures the degree of the process centering, and it only reflects process accuracy. The index
Cpk = min
½USL − µ
3σ ,µ − LSL 3σ
¾
take into account the process variation and the degree of process centering. Therefore the index Cpk reflect more actual process capability than the indices Cp and Ca.
For a normally distributed process with a fixed value of Cpk, we have Φ (3C¯ pk) ≤ P ≤ 2 ¯Φ (3Cpk)
where P is the non-conforming rate of the process, Φ is the cumulative distribution function of a standard normal random variable and ¯Φ = 1 − Φ. It is noted that the index Cpk only provides an approximate measure rather than an exact measure for non-conforming rate of the process. Boyles (1994) proposed the index Spk to reflect an exact non-conforming rate of the process. The index Spk is defined as
Spk = 1
where Φ−1is the inverse function of the cumulative distribution function Φ. For a normally distributed process with a fixed value of Spk, we know that the non-conforming rate is P = 2 ¯Φ(3Spk). It is noted that there is a one-to-one relationship between the index Spk and the non-conforming rate P . For a normally distributed process, the index Spk can reflect non-conforming rate, but it is irrelevant to target value. The index
Cpm = USL − LSL 6
q
σ2+ (µ − T )2
can reflect how the process center deviates from the target value. Also for a normally distributed process with a fixed value of Cpm, as mentioned in Chen et al. (2006), it has that P ≤ 2 ¯Φ(3Cpm) when Cpm is sufficiently large. Thus it also only provides an approximate measure for non-conforming rate of process.
Bothe (1992), Chen et al. (2003) and Chen et al. (2006) extended the capability measures for the processes with a single characteristic to the processes with multiple characteristics. More precisely, assume that the process has v characteristics. Let Pj be the conforming rate of the jth characteristic, j = 1, 2, . . . , v, and P be the overall non-conforming rate of the process. Bothe (1992) gave a simple process capability measure.
It defined the measure 1 − P = min{1 − P1, 1 − P2, . . . , 1 − Pv} as a capability measure related to the overall conforming rate of process. But this measure does not reflect the real situation accurately. Considering that the characteristics of the process are mutually independent, Chen et al. (2003) proposed the following overall capability index, referred to as SpkI :
where Spkj denotes the Spk value of the jth characteristic for j = 1, 2, . . . , v, that is,
where USLj, LSLj, µj and σj, respectively, are the upper specification limit, lower speci-fication limit, mean and standard deviation of the jth characteristic, j = 1, 2, . . . , v. It can be seen that
Hence there is a one-to-one correspondence relationship between the index SpkI and the overall non-conforming rate P .
On the other hand, assume that the independence of v characteristics is unknown, Chen et al. (2006) gave an extension of capability index Cpm. They defined the index CpmT as the overall non-conforming rate P and the process capability index Cpmis P ≤ 2 ¯Φ(3CpmT ).
Following the line of Chen et al. (2006), an extension of capability index Spkcan be defined as
Again the relation between the overall non-conforming rate P and the process capability index SpkD is P = 2 ¯Φ(3SpkD).
Now considering some generalizations of the above process capability index SpkD, let a manufacturing process have v subprocesses, and for 1 ≤ i ≤ v, the ith subprocess has pi
characteristics. Assume that the independence of subprocesses and that of characteristics are unknown, Wang et al. (2009) proposed an overall process capability index to reflect the non-conforming rate of a process with v = 5 and pi = p = 3, ∀i = 1, 2, 3. In fact, the process capability index proposed by Wang et al. (2009) is equivalent to that of Chen et al. (2006) with pv = 15 characteristics.
In this chapter, the extended process capability indices for two different situations will be proposed and investigated. First in Section 2.2, considering that the all subprocesses of the process are mutually independent and the independence of the characteristics in each subprocess is unknown, an extended process capability index will be discussed. Next if the all characteristics of each subprocess are mutually independent, and the independence of subprocesses is unknown, the corresponding results will be given in Section 2.3. Finally in Section 2.4, some discussions are provided.