Process Capability Indices under Non-Normality Conditions using Johnson Systems

Process capability indices (PCIs) quantify the ability of a process to produce on target and within specifications performances. Basic indices designed for normal processes gives flawed results for non-normal process. Numerous methods have been proposed for non-normal processes to estimate PCIs in which some of them are based on transformation methods. The Johnson system comprising three types that translate a continuous non-normal distribution to normal. The aim of this paper is to estimate four basic indices for non-normal process using Johnson system with single straightforward procedure. The efficacy of the proposed approach can be assessed for all three Johnson Curves   L U B S S S , , but result for SU is presented in this paper. PCIs for a data set are estimated and percentiles are obtained by our proposed exact method based on selected Johnson density function which was earlier based on approximate methods without any prior knowledge of density function of non-normal process. We compare our results with other existing methods to estimate PCIs for non-normal process. From statistical analysis we have noted that this modification improve process capability indices. Keywords—Johnson curve; percentiles; simulation; exact method


I. INTRODUCTION
Process capability analysis is conducted for a process which necessitate two conditions; first a controlled process and second the requirements pre-determined by product designer.Much of work has been performed for estimating PCIs for normal processes but these PCIs are inappropriate for non-normal because the estimators of PCIs for normal processes are not enough sufficient to characterize non-normal processes and deceive results.Several methods have been developed for PCIs of non-normal processes.Examples are Bates [1] discussed robustness of indices based on normal population, Pyzdek [2,3] showed that for non-normal processes standard process capability indices will give false process fallout rates and misled the results.Rivera [4] discussed C pk index estimation using transformation technique.Safdar and Ahmed [5,6] proposed procedures for estimating PCIs for non-normal processes and obtained capability estimates based on Weibull shape parameter.Piña-Monarrezet.al[7] extended their work for estimating indices for Weibull and Lognormal distributions.
More precisely two basic approaches are discussed so far in published literature to estimate PCIs for non-normal processes.First, translate non-normal process to normal using transformation and estimate PCIs from any existing method designed for normal processes.Few popular transformation methods are Burr percentile method [8], Johnson method [9] Box-Cox power transformation method [10] and Montgomery square root transformation method [11] .Second approach to estimate PCIs is based on percentile methods and most applied is Clements percentile method [12].His method is based on percentiles as process parameters to estimate C p and C pk indices for non-normal processes.Pearn and Kotz [13] extended his work for C pm and C pmk indices.Pearn and Chan [14] generalized a superstructure for all of these four indices.Zwick [15] Schneider and Pruett [16] and Chen [17][18][19][20] suggested various PCIs for non-normal processes.
In this present article we estimated PCIs for non-normal process under Johnson system of distribution.We presented one distribution S U for estimating PCIs for non-normal data set of 100 measurements for a hypothetical process [21].Our prime focus in this paper is to propose a uniform straightforward and easy approach to estimate PCIs for nonnormal processes.We estimated these PCIs from Pearn and Chan [14] superstructure using Johnson system and the percentiles (0.00135, 0.5 and 0.99865) of non-normal processes for the superstructure are obtained from best fitted Johnson density function whereas Pearn and Chan [14] estimated these percentiles as the percentage points of nonnormal processes.
We also emphasis on three assumptions of statistical process control (SPC) tools; First to manage the process in statistical control; second the parameters of the selected Johnson density function adequately fits the data and last the normality assumptions for the selected Johnson curve should not violated.
The paper structure is as follows: Section II presents existing PCIs for non-normal processes.Section III and IV briefly explains Johnson system of distribution and existing PCIs under Johnson distributions.In Section IV new capability computation procedures for non-normal distribution under Johnson system are proposed.Section VI comprehensively gives steps for obtaining modified PCIs using Johnson system with a brief flow chart.Section VII illustrates the proposed method explain with one example.
In this superstructure  and  are the process mean and process standard deviation respectively, T is target value, d= (USL-LSL)/2 is half length of specification interval and m= (USL+LSL)/2 is the midpoint between upper and lower specification limits.
. WhereF 0.135 , M and F 0.99865 are the 0.135 th , 0.5 th and 0.99865 th percentage points of non-normal processes under Pearsonian system respectively.Zwick [15] and Schneider et al. [16] considered two generalizations of C p and C pk but they used process mean  rather than process median M. Chang and Lu [23] applied Clements method to obtain percentiles for indices.Extending their methods Pearn and Chan [14] constructed another superstructure and obtained percentiles based on order statistics; In this setting, the notation [R] is defined as the greatest integer less than or equal to the number R and   i x is defined as i th order statistic.
Pearn and Kotz [13] and Pearn and Chen [24] applied Clements [12] method to estimate PCIs of non-normal processes.The PCIs in which those estimators correspond to can be expressed as and 1) to ( 4) are the existing superstructures to estimate PCIs for wide range of processes.Now we briefly describe the Johnson system of distributions and existing methods to estimate PCIs for non-normal processes in Section III.

III. JOHNSON SYSTEM OF DISTRIBUTION
The Johnson system of frequency curve was first developed by Johnson [9].Farnum [21] has given a detailed description on the use of Johnson Curves.Chou et al [25] and Polansky et al. [26] proposed Johnson system of distribution to transform non-normal data sets.For complete description of this system, see Bowman Shenton [27], Johnson, Kotz and Balakrishnan [28] ; Stuart and Ord [29] and Kendall and Stuart [30] Briefly, there are three distributions   Where    , ; x k i are chosen to cover a wide range of possible shapes and z is a standard normal variable.Johnson suggested these following functions for each distribution: The three well known methods of estimation the parameters of Johnson System are the moment matching method by Draper [31], the percentile matching method by Slifker and Shapiro [32] and the quantile estimation method by Wheeler [33].
for X (Johnson variate) and (Standard normal Johnson variates).In Section IV we summarize the earlier developed methods to estimate PCIs using Johnson system for non-normal distributions.

IV. EXISTING PCIS FOR NON-NORMAL DISTRIBUTION
USING JOHNSON SYSTEM For normal processes the indices C p and C pk are defined as; Pyzdek [2] worked on measurements of a hypothetical process to illustrate the use of Johnson transformation and fitted S B curve.He used  (the scale parameter of S B curve) as process spread  6 with the reason that difference between lower  and upper bound  Farnum [21] obtained C p and C pk using Johnson system and fitted S U curve.He replaced process spread  6 by p p L U  for non-normal process and found p L and p U by putting z = -3 and z = 3 in equation of x for S U curve (as displayed in Table I).These two indices are defined as: Extending Equation ( 8) the two other PCIs for non-normal processes under Johnson distribution are Where M is the average of (L p ,U p ) and T is target value specified by product designer.In Section V standardized method to estimate   as process parameters for nonnormal processes with new capability computation procedure is presented.

NEW CAPABILITY CALCULATION FOR NON-NORMAL DISTRIBUTION UNDER JOHNSON SYSTEM
A uniform SPC based procedure is presented to estimate PCIs for non-normal processes under Johnson system.We used superstructure form of Pearn and Chan [14] to estimate PCIs based on Johnson distributions.
For our new capability calculations we estimate these percentiles using density function  

VI. STEPS FOR OBTAINING MODIFIED PCIS USING JOHNSON SYSTEMS
The following steps are made for the proposed method.
Step 1: For the given data set obtain proportion nonconforming NC, process yield% and Vannman PCIs from Equation (1).
Step 2: Check whether process parameters   Step 7: Estimate indices using Johnson system named as JPCI based on our computation procedure using superstructure from Equation (3) and percentiles for superstructure from Equation (10), John (Z) by Pyzdek and Farnum method from Equations ( 8) and (9).We also estimates PCIs by existing methods under Pearson distribution to check the efficacy of our method; Clements from Equation ( 2), Pearn and Chan from Equation (3), Pearn-Kotz-Chan from Equation (4).

VII. ILLUSTRATION OF PROPOSED METHOD
For illustration of proposed method the data consist of 100 measurements from a process earlier presented by Farnum [21] is taken.See Table II.
First we estimate the PCIs assuming that observations come from normal distribution for the given hypothetical process.
From Table III, we observe that PCIs assuming normal population are not satisfactory and misleading.We must use PCIs designed for non-normal processes.The best fitted equation as Farnum [21] found for data set is S U  (11) The S U distribution is an unbounded distribution so both transformed limits are possible for each sample.Now we constructed histogram for the data set and superimpose the fitted Johnson curve.Fig. 1 shows that fitted Johnson curve adequately fits the original data.
For each simulated sample histograms and normal probability plots NPP are drawn to graphical examine the adequacy of model and normality assumption, see Fig. 2. For statistical test of normality Shapiro-Wilk normality test is performed and results are comprised in Table IV.
For original data and each sample transformed limits for S U , proportion of non-conforming (NC) and 0.135 th , 0.5 th and 0.99865 th percentiles (for Farnum method and our proposed method) are obtained.The results are given in Table V.
From Table VI, it is observed that capability calculations proposed under Johnson system for S u distribution improve indices with those obtained by Pyzdek and Farnum theories using Johnson system and other existing methods under Pearsonion system.The aim of this Paper is to focus attention on PCIs and their estimators and to emphasize their original basic purpose that of controlling the expected proportion outside specification limits based on engineering consideration.Earlier the percentiles of PCIs which are designed-modified for non-normal processes using Johnson system are estimated by approximate methods which do not require the knowledge of the density function of the data.We simulate the samples based on Johnson density function from the Johnson parameters estimated for the original non-normal process and then obtain the capability calculation for the indices.The percentiles of the PCIs are estimated by our modified method because the density function of the simulated samples is known.
In this paper we not only compare the results of PCIs based on Johnson systems with other existing methods for non-normal populations but list a program that can comprises the complete analysis step-by-step for choosing any of the Johnson curve.This program initially make the process in statistical control with the given specification limits, plot the density curve on original data to check the adequacy, and estimate PCIs based on new probability calculation as we execute the script for any non-normal or even normal data sets.All the computations are performed in R-console.
shape (  and ), one location    and one scale    real parameters.S B cover bounded distributions as gamma, beta and other distributions.It is bounded on lower end by  , upper end by    or both.S U are unbounded and cover t and normal distributions.S L covers log-normal family and bounded only lower side by  .These three distributions are generated by transformations of the form www.ijacsa.thesai.org distribution which is found to be the best fit for the given nonnormal data set.on our new capability calculations we named our capability indices as JPCI in computation tables.Now we illustrate our procedure in steps to estimate PCIs easily.

Fig. 1 .
Fig. 1.Original and Best Fitted Johnson Curve with Probability Histogram.

2 Z
www.ijacsa.thesai.orgVIII.DISCUSSION AND CONCLUSION After illustration of the proposed procedure a conclusion is made.www.ijacsa.thesai.org

TABLE IV .
SHAPIRO-WILK NORMALITY TEST