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Process capability indices (PCIs), Cp, Ca, Cpk, Cpm, and Cpmk have been developed in certain manufacturing industry as capability measures based on various criteria, including process consistency, process departure from a target, process yield, and process loss. It is noted in certain recent quality assurance and capability analysis works that the three indices, Cpk, Cpm, and Cpmk provide the same lower bounds on the process yield. In this paper, we investigate the behavior of the actual process yield, in terms of the number of non-conformities (in ppm), for processes with fixed index values of Cpk=Cpm=Cpmk, possessing different degrees of process centering. We also extend Johnson's [1992. The relationship of CpmCpm to squared error loss. Journal of Quality Technology 24, 211–215] result formulating the relationship between the expected relative squared loss and PCIs. Also a comparison analysis among PCIs is carried out based on various criteria. The result illustrates some advantages of using the index Cpmk over the indices Cpk and Cpm in measuring process capability (yield and loss), since Cpmk always provides a better protection for the customers. Additionally, several extensions and applications to real world problem are also discussed. The paper contains some material presented in the Kotz and Johnson [2002. Process capability indices—a review, 1992–2000. Journal of Quality Technology 34(1), 1–19] survey but from a different perspective. It also discusses the more recent developments during the years 2002–2006.

Understanding the structure of a process and quantifying process performance no doubt are essential for successful quality improvement initiatives. Process capability analysis has become—in the course of some 20 years—an important and well-defined tool in applications of statistical process control (SPC) to a continuous improvement of quality and productivity. The relationship between the actual process performance and the specification limits (or tolerance) may be quantified using suitable process capability indices. Process capability indices (PCIs), in particular Cp, Ca, Cpk, Cpm and Cpmk, which provide numerical measures of whether or not a manufacturing process is capable to meet a predetermined level of production tolerance, have received substantial attention in research activities as well as an increased usage in process assessments and purchasing decisions during last two decades. By now (2006) there are several books (on different levels) cited in the references, which provide discussions of various PCIs. A number of authors have promoted the use of various process capability indices and examined (with a various degree of completeness) their properties. The first process capability index appearing in the engineering literature was presumably the simple “precision” index Cp (Juran, 1974; Sullivan, 1984 and Sullivan, 1985; Kane, 1986). This index considers the overall process variability relative to the manufacturing tolerance as a measure of process precision (or product consistency).1 Another index Ca, a function of the process mean and the specification limits, referred to as an “accuracy” index, is geared to measure the degree of process centering relative to the manufacturing tolerance (see, e.g., Pearn et al., 1998). This index is closely related to an earlier measure originally introduced in the Japanese literature (see Section 3). Formally: where μ is the process mean, σ is the process standard deviation, USL and LSL are the upper and the lower specification limits, d=(USL−LSL)/2 is the half specification width related to the manufacturing tolerance and m=(USL+LSL)/2 is the midpoint between the upper and lower specification limits. Due to its simplicity, Cp cannot provide an assessment of process centering (targeting). The index Cpk, on the other hand, takes both the magnitude of process variance and the process departure from the midpoint m into consideration. It may be written as Cpk=Cp×CaCpk=Cp×Ca a product of the two basic indices Cp and Ca. The standard definition is As alluded above the index Cpk was developed because Cp does not adequately deal with cases where process mean μ is not centered (the mean does not equal to the midpoint m). However, Cpk by itself still cannot provide an adequate measure of process centering. That is, a large value of Cpk does not provide information about the location of the mean in the tolerance interval USL−LSL. The Cp and Cpk indices are appropriate measures of progress for quality improvement situations when reduction of variability is the guiding factor and process yield is the primary measure of a success. However, they are not related to the cost of failing to meet customers’ requirement of the target. A well-known pioneer in the quality control, G. Taguchi, on the other hand, pays special attention on the loss in product's worth when one of product's characteristics deviates from the customers’ ideal value T. To take this factor into account, Hsiang and Taguchi (1985) introduced the index Cpm, which was also later proposed independently by Chan et al. (1988). The index is motivated by the idea of squared error loss and this loss-based process capability index Cpm, sometimes called the Taguchi index. The index is geared towards measuring the ability of a process to cluster around the target, and reflects the degrees of process targeting (centering). The index Cpm incorporates the variation of production items relative to the target value and the specification limits which are preset in a factory. The index Cpm is defined as where as above USL−LSL is the allowable tolerance range of the process, d=(USL−LSL)/2 is the half-interval length, and τ=[σ2+(μ-T)2]1/2τ=[σ2+(μ-T)2]1/2 is a measure of the average product deviation from the target value T. This index Cpm can also be expressed as a function of the two basic indices Cp and Ca, explicitly Cpm=Cp/{1+[3Cp(1-Ca)]2}1/2Cpm=Cp/{1+[3Cp(1-Ca)]2}1/2. The quantity τ2=E[(X-T)2]τ2=E[(X-T)2] combines two variation components: (i) variation relative to the process mean (σ2) and (ii) deviation of the process mean from the target ((μ-T)2(μ-T)2). Pearn et al. (1992) proposed the process capability index Cpmk, which combines the features of the three earlier indices Cp, Cpk and Cpm. The index Cpmk (motivated by the structure of Cpk(1)′) alerts the user whenever the process variance increases and/or the process mean deviates from its target value. The index Cpmk has been referred to as the third-generation capability index, and is defined as Comparing the pair of indices (Cpmk,Cpm)(Cpmk,Cpm), analogously to (Cpk,Cp)(Cpk,Cp), we obtain the relation Cpmk=Cpm×Ca=(Cpm×Cpk)/Cp. Consequently, Cpmk can be expressed as Cpmk=CpCa/{1+[3Cp(1-Ca)]2}1/2Cpmk=CpCa/{1+[3Cp(1-Ca)]2}1/2 in terms of the “elementary indices”. More recently, Vännman (1995) has proposed a superstructure Cp(u,v)=(d-u|μ-m|)/{3[σ2+v(μ-T)2]1/2}Cp(u,v)=(d-u|μ-m|)/{3[σ2+v(μ-T)2]1/2} of capability indices for processes based on normal distribution, which includes Cp, Cpk, Cpm and Cpmk as particular cases. By setting u, v=0 and 1, we obtain the four indices View the MathML sourceCp(0,0)=Cp, Cp(1,0)=CpkCp(1,0)=Cpk, View the MathML sourceCp(0,1)=Cpm, and View the MathML sourceCp(1,1)=Cpmk. These indices are effective tools for process capability analysis and quality assurance. Two basic process characteristics: the process location in relation to its target value, and the process spread (i.e. the overall process variation) are combined to determine formulas for these capability indices. The closer the process output is to the target value and the smaller is the process spread, the more capable the process is. The first feature (closeness to the target) is reflected in the denominator while the second one (the process spread) appears in the numerators of these four indices. In other words, the larger the value of a PCI, the more capable is the process. In this paper, all derivations are carried out assuming that the process is in a state of statistical control and the characteristics under investigation arise from a normal distribution. Moreover, the target value is taken to be the midpoint of the specification limits: T=m (which is common in practical situation) unless stated otherwise. During the last two decades many authors have promoted the use of various PCIs and examined them with a different degree of completeness. These contributions include (in the chronological order): Chan et al. (1988), Chou et al. (1990), Boyles (1991), Pearn et al. (1992), Kushler and Hurley (1992), Rodriguez (1992), Kotz and Johnson (1993), Vännman and Kotz (1995), Bothe (1997), Kotz and Lovelace (1998), Franklin (1999), Palmer and Tsui (1999), Wright (2000), Jessenberger and Weihs (2000), Pearn and Shu (2003), Vännman and Hubele (2003), Pearn and Wu (2005), Wu (2007) as well as references therein. Applications of these indices range over a great variety of situations and productions such as manufacturing of semiconductor products (Hoskins et al., 1988), head gimbals assembly for memory storage systems (Rado, 1989), jet-turbine engine components (Hubele et al., 1991), flip-chips and chip-on-board (Noguera and Nielsen, 1992), rubber edge (Pearn and Kotz, 1994), wood products (Lyth and Rabiej, 1995), aluminum electrolytic capacitors (Pearn and Chen, 1997a), audio-speaker drivers (Chen and Pearn, 1997), Pulux Surround (Pearn and Chang, 1998), liquid crystal display module (Chen and Pearn, 2002), and couplers and wavelength division multiplexers (Wu and Pearn, 2005a). Kotz and Johnson (2002) provided a compact survey (with interpretations and comments) of some 170 publications on PCIs, during 1992–2000. Spiring et al. (2003) consolidated the research findings of process capability analysis and provide a bibliography of papers for the period 1990–2002. We shall attempt to describe, in an organized manner the interconnection between the PCIs described above and (i) the process yield, in an organized manner, (ii) the process loss, (iii) the process departure from target and (iv) process variability. This may clarify the role of the index Cpmk which is still the least understood by practitioners.