Review of the Development in Process Capability Analysis

International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2471 ISSN 2229-5518

Vidhika Tiwari1, N.K.Singh2

Abstract— This review paper is devoted to the study for the analysis of the characteristics of the product may be interrelated. It is process capability of the manufacturing processes. The process capability indices necessary to develop the process capability measure for the Cp; C pk C pm , C pmk , C py and C pc are presented, related to process parameters and the practical applications of the conventional as well as some new indices in the quality characteristics related to the above mentioned manufacturing industries are provided. distributions and sampling distribution of their estimate. Under non normal distribution, some properties of the PCIs Keywords— Quality control, Process capability index, Non-normal distribution, and their estimators differ from those of normal distribution. Gamma and Weibull Distribution,Industrial application . To utilize the PCIs more reasonably and accurately in ——————————  —————————— assessing the lifetime performance of components, this study is conducted.

1. INTRODUCTION 2. PROCESS CAPABILITY INDICES FOR QUALITY Process capability compares the process output with the CHARACTERISTICS FOLLOWING VARIOUS customer’s specification. Two parts of process capability are: i) DISTRIBUTIONS Measure the variability of the output of a process, and ii) Compare that variability with requirement specification or 2.1. PCI For Normal Distribution product tolerance. Process capability analysis is the evaluation of a production Process capability analysis is based on some fundamental process to determine whether or not the inherent variability of assumptions that is, the process is stable and that the studied its output falls within the acceptable range and process characteristic is normally distributed. There are several capability index or process capability ratio is a statistical statistics that can be used to measure the capability of a measure of process capability. The concept of process process. According to the philosophy of the quality control capability holds meaning for processes that are in a state approach, process capability indices of any process can be of statistical control. Process capability indices measure how divided into capability indices of the first and second much "natural variation" a IJSERprocess experiences relative to its generation. The design of the first generation capability specification limits and allows different processes to be indexes (Cp, Cpk) is based on classical philosophy of the compared with respect to how well an organization controls statistical process control. According to that philosophy all them. measurement results within required tolerance interval are A process capability index uses both the process intended to be good. Measurements outside tolerance interval variability and the process specifications to determine whether are considered to be bad. Under these assumptions the two the process is capable. The process capability index (PCI) is a most widely used indices in industry are Cp and Cpk, where value which reflects real-time quality status. The PCI is Cp was presented by Juran (1974) and Cpk by Kane (1986). considered as one of the quality measurements tool. In practice, process capability indices (PCIs) are used as a means of measuring process potential and performance. Moreover, USL− LSL USL−−µµ LSL Cp = Cpk = min , σ σσ most PCIs have been developed or investigated under the 6 33 assumption that components have a lifetime with a normal distribution. In many processes, the quality characteristics may follow the non-normal distribution e.g. Weibull, Exponential, and Geometric distribution etc. In some cases, the

———————————————— • Mrs.Vidhika tiwari1 is currently pursuing PhD degree program in ME & MME Departmen in ISM Dhanbad, India,[email protected]. • Dr. N.K.Singh2 Associate professor in ME &MME Department in ISM Dhanbad.India,[email protected]. IJSER © 2013 http://www.ijser.org

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−−ξξ Ue−−11 Le = Cpk Min(,)σσ− ee33−−11

And estimated proportion of non-conforming items as φ{(logLU− ξσ ) /ˆˆ } +− 1 φ (log − ξσ ) / Where ξ = Scale parameter, σˆ = Estimated Standard deviation.

A superstructure or family of capability indices, containing Cp, Cpk, Cpm and Cpmk involving two parameters, is introduced by Vannman (1993) as:

du−−µ m Cp(,) u v = 3{σµ22+−vT ( ) } Fig.1 Normal distribution Where [LSL, USL] is the specification interval, μ is the process mean and σ is the process standard deviation of the in-control UL− UL+ process. Henceforth, we call the process mean μ and the Where d = and m = 2 2 process standard deviation σ for the process parameters. The capability index Cp relates the distance between the By letting u = 0 or 1 and v = 0 or 1 in the above equation, we specifications limits to the range over which the process is can obtain four basic indices i.e. Cp(0,0) = Cp, Cp(1,0) = Cpk, actually varying. Cp(0,1) = Cpm and Cp (1.1) = Cpmk. Chan and Mak (1993) propose a Process Capability indicator as: 2.2. PCI For Non Normal Distribution

T − µ 6σ Second generation capability index (Cpm) is rising from new Cl = +w ,0 ≤≤w 1 UL− ()UL− approach to the quality improvement (Taguchi approach). It is () 2 not enough to know that measurements are so called good = DI + w (VI) (being within the tolerance interval) but important is T-m knowledge on how good they are. Such index enables to Where DI= Deviation Indicator = U-L determine whether the values of the searched quality index () IJSER 2 approach to the tolerance limits even when all measurement 61σ results fit within the tolerance. Since the indices Cp and C pk do VI = Variation Indicator = = ()U− L Cp not take into account of the differences between the processes mean and its target value, Chan (1988) and Pearn (1992) Most capability indexes assume that the quality characteristic is normally distributed. Clements (1989), Kotz and Johnson considered this difference to develop indices Cpm and Cpmk as follows: (1993), and Sommerville and Montgomery (1996) among others, comment in detail on the distortion in information

 provided by these indices when process distribution moves USL− LSL Cpm = 2 away from the assumed normal. Ahmad and saleh (1999) 2 6 σµ+−()T presented a generalization of Clements’ method with

asymmetric tolerances. These quantile-based indices, Cp and Cpk for non normal data are defined as follow: USL −−µµLSL Cpm = min 22, , 22 − 33σµ+−()TT σµ +−() USL LSL USL−− X50 X 50 LSL  Cp = Cpk = min, X99.865−− XX 50 50 X 0.135 XX99.865− 0.135

Where, T represents the target value of quality characteristics.

The process parameters μ and σ2 are estimated from the Wang (1996), Zimmer (1998), Kan and Yasici (2006) and sample mean variance for X and S2, when μ and σ2 are Mousa & Jaheen (2002) have presented a comprehensive unknown. Rodriguez (1992) proposed a process capability review of Burr distribution and its application to many non- index (Cpk) for Lognormal distribution as normal situations.Ahmed and Rahbar (2001) developed an

International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2473 ISSN 2229-5518 asymptotic theory for a capability index for arbitrary industry. Pyzdek (1992) mentioned that the distributions of populations. Nahar and Zimmer (2001) and Chang(2002) certain chemical processes, such as zinc plating in a hot-dip considered the process capability indices for skewed galvanizing process, are very often skewed. Choi et al. (1996) populations. Abbasi (2006) used simulated annealing presented an example of a skewed distribution in the ‘‘active algorithm to estimate three parameters of Weibull distribution. area’’ shaping stage of the wafer’s production processes. Process capability analysis when observations are Gamma distribution (skewed) covers a wide class of non- autocorrelated is investigated by Noorossana (2002) using normal applications, including the manufacturing of time series modeling and regression analysis. Luceno (1996) semiconductor products, head/gimbals assembly for memory suggested a process capability index which seems to be storage systems, jet-turbine engine components, flip-chips and insensitive to departure from the assumption of normal chip-on-board, audio-speaker drivers, wood products, and variability. A detailed discussion and references on dealing many others. with non-normality of the process distribution are given in The control charts are commonly used in many industries for Kotz and Johnson (2002). providing early warning for the shift in the process mean. For example, the cumulative sum chart is known to be effective on 2.3. PCI For Discrete Distribution detecting sustained shifts in the process mean (see e.g. Lucas and Crosier, 2000; Luceno and Puig-Pey, 2002; Lucas, 1976). If

The most widely used such indices are Cp,Cpk,Cpm, and Cpmk the control chart detects a process mean shift, then the process or their generalizations for non-normal processes, suggested is not under control. However, for momentary process mean by Clements (1989), Pearn and Kotz (1994), and Pearn and shifts, it may be beyond the control chart detection power. Chen (1995). Often, however, one is faced with processes Consequently, the undetected shifts may result in described by a characteristic whose values are discrete. overestimating process capability. If the process mean shifts Therefore, in such cases none of these indices can be used. are not detected, then unadjusted Cpk would overestimate the Perakis and Xekalaki (2004) proposed a new index, which can actual process yield. Bothe (2002) provided a statistical reason be used regardless of whether the examined process is discrete for considering such a shift in the process mean for normal or continuous. This index is defined as processes. However, if the capability indices are based on the assumption of a normal distribution of data but are used to 1− p0 Cpc = 1− p deal with non-normal observations, the values of the capability indices may, in the majority of situations, Where p and p0 denote the proportion of conformance (yield) and the minimum allowable proportion of conformance of the misrepresent actual product quality Ya-Chen Hsu, W.L. Pearn, examined process, respectively. As is well known, the term Pei-Ching Wu (2007) examines Bothe’s approach and finds proportion of conformanceIJSER refers to the probability of that the detection power of the control chart is less than 0.5 producing within the so-called specification area, i.e. the when data comes from gamma distribution. This shows that interval determined by L and U. If the tolerances are Bothe’s adjustments are inadequate when we have gamma unilateral, then the value of p is given by P(X > L), if only L processes. Then, the adjustments under various sample sizes has been set, and by P(X < U), if only U has been assigned. (n) and gamma parameters (N) with a fixed detection power of 0.5 are calculated. Finally, the process capability formula is adjusted to accommodate the undetected shifts. 2.4. PCI for Gamma And Weibull Distribution

USL−− F0.5 AS 50σσ F 0.5 − AS 50 − LSL Since Motorola, Inc. introduced its Six Sigma quality initiative dynamic Cpk = min , { −−} in the 1980s; quality practitioners have questioned why the F0.99865 F 0.5 FF 0.5 0.00135 followers of this initiative have added a 1.5σ shift to the As a result, our adjustments provide significantly more process mean when estimating process capability. The accurate calculations of the capability of gamma processes. Ya- advocates of Six Sigma have claimed that such an adjustment Chen Hsu, W. L. Pearn and Chun-Seng Lu (2011) calculate the is necessary, but they have offered only personal experiences mean shift adjustments under various sample sizes n and and empirical studies as justification for this claim (see Bender, Weibull parameter, with the power fixed to 0.5. Then, we 1975;Evans, 1975; Gilson, 1951). By examining the sensitivity implement the adjustments to accurately estimate capability of control charts to detect changes of various magnitudes, index Cpk for Weibull processes with mean shift consideration. Bothe (2002) provided a statistically based reason for this With detection power of the Erto’s-Weibull control chart fixed claim. In his study, Bothe assumed that the process data is to 0.5, using the adjusted process capability formula, the approximately normally distributed. However, non-normal engineers could determine the actual process capability more processes occur frequently, in particular, in the semiconductor accurately.

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there were 10% of processes with capabilities between 2.5. PCI FOR POISSON DISCRETE DISTRIBUTION 1 and 1.33in a month and some of these improved to between 1.33 to 1.67 the next month, improvement Maiti(2011) have proposed a generalized process capability has occurred. This distributional shift can easily be index that is the ratio of proportion of specification monitored. conformance to proportion of desired conformance. • Cpy = p/p 0' Specifying performance requirement for new where p is the process yield that is F(U) –F(L),F(t) =P(X < t ) is equipment (Purchase Quality Control). the cumulative distribution function of X, and p0 is the • Selecting between competing vendors (Vendor desirable yield that is; Sourcing). p0 = F (UDL) – F (LDL). Bayesian estimation of the index has been considered under squared error loss function. Normal, • Planning the sequence of production processes when exponential nonnormal and Poisson discrete processes have there is an interactive effect of processes on tolerances been taken into account. (Manufacturing Planning / Production Planning)

2.6. PCI for Pearsonian Distribution • Reducing the variability in a manufacturing

process (Process Control) – For each characteristic, it When the process has a distribution of Pearsonian type, W. L. is meaningful to compare Cp and Cpk. If Cpk is too Pearn and K. S. Chen proposed an estimator for non-normal low, then Cp must be examined to determine whether Pearsonian populations (1995), CNp(u, v) by using Clements' the variability is unacceptably high. If Cp is close to method (Clements, 1989)as follows: Cpk, then process location is not a problem. The d−− uMˆ m CˆNp ( uv , ) = indices Cpu, Cpl and k provide an assessment of how 2 ULpp− 2 3+−vM ()ˆ T close the process mean is from the target mean. 6

Where d = (USL±LSL)/2, where USL and LSL are respectively In the electronics or microelectronics manufacturing the upper and lower specification limits, m = (USL + LSL)/2, industry Process capability measures convey critical the specification center, and T is the target value and M is the information regarding percentage of conforming items, median. meeting product design speciation limits, which is a basic Pearn and Kotz applied Clements' method to obtain first criterion used for judging whether the products are approximation estimators of PCIs for non-normal populations reliable from manufacturing perspective. A high value of to the two more advanced PCIs, Cpm and Cpmk developed by IJSERPCI implies a high quality of the product. The most Chan and Pearn. important application of Process Capability Analysis is in measurement Phase, and in semiconductor industry for 3. INDUSTRIAL APPLICATION OF PCI quality control. Other use of Process Capability Indices in Some of the application areas are described below:- the supplier certification process; in optimization of multi • Predicting how well the process will hold tolerances response problems for batch manufacturing processes and (Manufacturing) - For various types of machine and quality computation model of complex assembling process qualification trials, it is sometimes reasonable process using multivariate process capability index. to establish a benchmark capability. A typical bench mark is c pk = 1.33 which will make non conforming 4. FUTURE SCOPE OF THE WORK AND units unlikely in many situation (Hoffer 1985). CONCLUSION

Properties of the PCI index, suggested by various researchers • Assisting product designers / developers in selecting are examined under different distributional assumptions. The or modifying a process (Product and Process design). obtained results offer a useful approach for measuring process

capabilities on the basis of quantitative aspects since none of • Assisting in establishing an interval between the most broadly used capability indices can be used in sampling for process control (Quality Control connection with discrete type of data despite the fact that they Planning) – Within a department or plant it is often are quite frequently encountered in process control. The study useful to monitor continuous improvement, which of the properties of this index on the basis of such data under can be accomplished by observing the changing different distributional assumptions, for either continuous or distribution of process capabilities. For example, if

International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2475 ISSN 2229-5518 discrete processes that usually arise in applications, would be [14] W.L. Pearn, G.H. Lin On the reliability of the estimated an interesting issue for further research. incapability index Qual. Reliab. Engng. Internat., 17 (2001), pp. 279–290. Often the quality of a process is determined by several [15] W.L. Pearn, S. Kotz, N.L. Johnson Distributional and inferential correlated univariate variables. Various different multivariate properties of process capability indices J. Qual. Technol., 24 process capability indices (MPCI) have been developed for (1992), pp. 216–231. such a situation, but confidence intervals or tests have been [16] W.L. Pearn, G.H. Lin, K.S. Chen Distribution and inferential derived for only a handful of these. Our objective is to develop properties of the process accuracy and process precision indices new PCI and a decision procedure, based on a case study from Comm. Statist. Theory Methods, 27 (1998), pp. 985–1000. [17] R.N. Rodriguez Recent developments in process capability the industry, to be used to decide whether a process can be analysis J. Qual. Technol., 24 (1992), pp. 176–187. capable or not at a stated significance level. [18] S.E. Sommerville, D.C. Montgomery Process capability indices and non-normal distributions Qual. Engng., 9 (1996), pp. 305–316. [19] Bothe DR (2002). Statistical reason for the 1.5s shift. Qual. ACKNOWLEDGEMENT Eng.,14(3): 479-487. [20] Chan LK, Cheng SW and Spiring FA (1988). A new measure of We remain ever grateful to Professor (Retd.) S.P.Mukherjee process capability. J. Qual. Tech., 20(3): 162-175. former Centenary Professor of Statistics Calcutta University, [21] Chen KS. Pearn WL. An application of non-normal process Kolkata for their encouragement, guidance and suggestions capability indices. Qual. Reliab. Eng. Int., (1997) 13: 355-360. which made the submission of this paper possible in the [22] Choi KC, Nam KH, Park DH. Estimation of capability index based present form. on bootstrap method. Microelectron Reliab, (1996) 36(9): 141-153. [23] Erto P, Pallotta G. A New Control Chart for Weibull Technological Processes. Qual. Tech. Quant. Manag., (2007) 4(4): 553-567. [24] Hsu YC, Pearn WL, Pei CW. Capability adjustment for Gamma REFERENCES processes with mean shift consideration in implementing six [1] Bender A. Statistical tolerancing as it relates to quality control and sigma program. Eur. J. Oper. Res., (2008) 191(2): 517-529. the designer. Automotive Division Newsletter of ASQC(1975) . [25] Lu XM, Peng NF. The approximation distribution for the sum of [2] Ekvall, D.N. Juran, M.J., Manufacturing planning. In: Juran, J.M. independent and identical Weibull distributions. National Chiao (Ed.), Quality Control Handbook, third ed. McGraw-Hill, New Tung University, Taiwan(2003). York 1974. [26] Nichols MD, Padgett WD .A bootstrap control chart for Weibull [3] V.E. Kane Process capability indices J. Qual. Technol., 18 (1986), percentiles. Qual. Reliab. Eng. Int., (2006). 22: 141-151. pp. 41–52. [27] Shore H. A new approach to analyzing non-normal quality data [4] Y.M. Chou, D.B. Owen On the distributions of the estimated with application to process capability analysis. Int. J. Prod. Res., process capability indices Comm. Statist. Theory Methods, 18 (1998) 36(7):1917-1933 (1989), pp. 4549–4560. [5] L.K. Chan, Z. Xiong, D. Zhang On the asymptotic distribution of some process capability IJSERindices Comm. Statist. Theory Methods, . 19 (1990), pp. 11–18. [6] S.E. Ahmed, A.K.M.E. Saleh Improved nonparametric estimation of location vector in a multivariate regression model Nonparametric Statist., 11 (1999), pp. 51–78.

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