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\title{Process Capability Analysis using Curve Fitting Methods}
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             \author[1]{Dr. John J.  Flaig}

             \affil[1]{  California Southern University}

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\date{\small \em Received: 15 December 2012 Accepted: 31 December 2012 Published: 15 January 2013}

\maketitle


\begin{abstract}
        


This paper offers a new approach to process capability measurement based on techniques for estimating the fraction nonconforming in the tails of the observed process distribution. The methodology proposed can provide significantly more accurate results and applicability to a much larger class of process distributions than the standard approaches of calculating capability indices based on making parametric assumptions or using standard curve fitting techniques.The tail probability estimation approach to capability analysis can significantly enhance the practitioners understanding of the processes they are attempting to develop and/or control because it provides a more robust and accurate measure of capability. This clearer insight into process performance becomes increasingly important as the allowable fraction nonconforming levels decline.

\end{abstract}


\keywords{: process capability, fraction nonconforming estimation}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
1. Empirical: Based on sampling the process to determine the number of conforming items divided by the total number of items sampled. This is the relative frequency approach to capability assessment and in the limit it would provide a true measure of capability assuming that the process is stable. Unfortunately, many real world processes are not stable. 2. Parametric: Based on the assumption that the observed values come from some theoretical distribution. This top-down approach is the classical method used by many practitioners to assess process capability \hyperref[b8]{[Somerville, 1997]}. The parametric assumption might be given credibility because the nature of the process may "a priori" give rise to the theoretical distribution or it might be supported by goodness-of-fit tests. This approach has two risks, the first is the assumption of stability and the second is the subjective nature of the assumed distribution. 3. Modeling: Based on curve-fitting techniques such as polynomial regression or Johnson curves. This is a bottom-up approach \hyperref[b7]{[Pyzdek, 1992}\hyperref[b2]{] [Farnum, 1996]}. This approach also assumes process stability. However, model selection is less subjective because it is based upon the limited set of choices typically offered by the computer program. The problem is that the limited set may not include a "good" fitting distribution. 
\section[{Year 2013}]{Year 2013}\par
Volume ( D D D D ) J t is critical in process development and ongoing monitoring to have an understanding of how capable the process is of meeting requirements. These requirements reflect internal and external demands that are expressed in terms of specifications. Historically, assessment of process capability using the indices such as Cp and Cpk became popular in the early 1980's. Engineers used these indices to determine if a process should be released to production (i.e., during qualification) and customers demanded that suppliers provide them as measures of their process performance. Clearly, important decisions were based on these indices. Then, around 1990, questions began to be raised about their validity in industrial applications where the assumptions underlying the calculation of the indices were often not met. Numerous papers have been published that discuss the shortcomings of capability indices \hyperref[b4]{[Gunter, 1989 and}\hyperref[b5]{1991]} \hyperref[b8]{[Somerville, 1997]}. However, we still find today that capability indices are the primary tool for assessing and communicating the process capability. 
\section[{I}]{I}\par
If process capability can be defined as the ability of a process to produce products or services that meet the specified requirements  {\ref [ASQC, 1983]} [ \hyperref[b1]{Duncan, 1986]}, the question then becomes; how can this ability be measured? A reasonable approach might be to try to estimate the probability that the product or service falls within the acceptance region defined by the specifications. There are three common methods for generating this estimate: descriptive statistics and specifications limits. Based on our experience using the fraction conforming as an estimator of process capability offers several advantages over the use of capability indices because it is more intuitive. That is, most people have an intuitive understanding of what percent nonconforming means (i.e., high yield implies the process is capable and low yield implies that it is incapable) whereas there is no such intuitive understanding for an abstract numeric capability index. In addition, there are so many capability indices (in excess of one hundred) that it is difficult to recall the merits of each. 
\section[{III.}]{III.} 
\section[{Alternate Methods}]{Alternate Methods}\par
There are a number of different approaches to estimating the fraction nonconforming. For example, Pyzdek and Farnum discussed using Johnson curves to estimate the fraction nonconforming \hyperref[b7]{[Pyzdek, 1992}\hyperref[b2]{] [Farnum, 1996]}. Other researchers have expressed concerns with the curve fitting approach because of accuracy issues \hyperref[b10]{[Wheeler, 1995]}. The author's agree with Wheeler that curve fitting methods will typically not be able to resolve nonconformance rates down to low levels unless there is a relatively large amount of data available.\par
If the process distribution and specification limits are reasonably well structured (i.e., the process distribution is mostly within the specification limits), then the problem of determining process capability becomes one of estimating tail probabilities. A major criticism of the curve fitting approach is that a single function is probably not sufficient to fit the observed data in the tails and in the middle of the distribution simultaneously. This follows from the observation that least squares regression analysis will tend to fit the bulk of the data (i.e., the central mass) and miss-fit the limited amount of data in the tails. Because of this, attempts to fit parametric distributions such as Normal, Johnson, or Weibull to mound shaped empirical data sets will give rise to tail fit errors. This problem is further complicated because real world processes are generally dynamicmeaning that the data may not be coming from a single or static distribution generator.\par
Our proposed approach to fitting the process distribution differs from the classical curve fitting methodology in two ways: 1. The distribution is divided into three parts (left, middle, and right) and the tails are fit separately. 2. A well-known and very flexible modeling approach is used to fit the tails of the distribution so that the left and right tails are approximated by unique functions.\par
The first point focuses attention and statistical techniques where they should be --on the tail probabilities and not the bulk of the distribution. Johnson, Kotz and Pearn proposed a somewhat similar analysis approach  {\ref [Johnson, 2006]}. However, they divided the process distribution in half, which is an improvement but still has the central mass fitting issue.\par
The second point allows the practitioner to fit the observed data in a realistic way. For example, there are distributions where the observed data is increasing and then decreasing in the tail, so the fitting function should have this property. The classical approach of assuming a Normal distribution (which goes to zero in the tail) is clearly unrealistic. Bounded or truncated distributions offer another example, where the standard approaches do not work very well. For example, fitting a Johnson curve to a bounded distribution gives rise to a function (SB type) that goes to zero in the tail whereas the bounded function may have no tail area (e.g., if the LSL is less than the lower bound).\par
IV. 
\section[{Analytic Methodology}]{Analytic Methodology}\par
Techniques from reliability analysis will be used to fit various functions to the tail distributions of data drawn at random from known distributions \hyperref[b9]{[Tobias, 1995]}. The fitted curve results will be compared with the true results from the actual distribution and contrasted with the results of using the classical assumption of normality.\par
The first class of distributions to be considered are the bounded type (i.e., the domain (t) of the function is bounded on one or both sides and the range (y) does not go to zero on at least one side). A triangular distribution defined by, y = -2t+2 on the interval [0, 1] will be used in this example. This function was selected because it offers a challenging test of the classical normality assumption and its ability to yield a realistic assessment of process capability.\par
The analysis is carried out as follows and displayed in Table  {\ref 1} and Figure \hyperref[fig_0]{1:} 1. One thousand data points are generated at random from the triangular (Tri) distribution 2. The data is sorted smallest to largest 3. The first one hundred (left tail) and last one hundred values (right tail) are selected 4. Normal (Nor), Johnson (Jon), and Weibull (Wei) distributions are fitted to the tail values 5. The PDF and CDF functions for each distribution are generated and graphed 6. Several estimates of forecast accuracy are generated so that the results can be compared.  
\section[{Global Journal of Researches in}]{Global Journal of Researches in} 
\section[{Analysis}]{Analysis}\par
It can be seen from Table \hyperref[tab_0]{2} that the CDF errors for Weibull and Normal were about equal in the right tail and both were significantly better than Johnson. For the left tail the Weibull error is significantly less than Normal or Johnson. Thus, the Weibull estimates more accurately reflect the true tail probabilities than does the Normal or Johnson curve for this triangular distribution. Left Tail (0, .1) Right Tail (.9, 1) The second type of distribution to be considered is the unbounded type. The data for this distribution arose as part of a real world study at a semiconductor equipment manufacturer. Five hundred and sixty nine readings were taken and the distribution formed by this data is displayed in  The most disconcerting part of this study is the realization that many practitioners are currently basing their process capability analysis and conclusions on the assumption of normality, which can be seen, in this example, to yield very unrealistic results. 
\section[{VI.}]{VI.} 
\section[{Summary}]{Summary}\par
The Weibull tail fitting approach to capability analysis has been shown to offer good accuracy in estimating the fraction nonconforming when compared with two other common fitting distributions in the examples tested. The use of capability indices for measuring process capability seem weak because they offer limited intuitive communication ability and they do not map one-to-one into an accurate estimate of the fraction nonconforming which is what management is interested in knowing. The standard curve fitting approach is handicapped by the attempt to force a single function to fit the entire distribution (which may be a mixture of several distributions) when only the tails are generally of interest in capability analysis.\par
The merits of this new approach are: 1. It attempts to estimate an intuitively reasonable measure of process capability (i.e., the fraction conforming which is one minus the fraction nonconforming). 2. It separates the data distribution into three parts (left tail, middle, and right tail) so that the analysis can be focused where it should be --on the tails. This approach results in significantly increased accuracy in estimating the tail probabilities. \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Figure 2 :*}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Figure 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-5.png}
\caption{\label{fig_3}3 .}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{figure6}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-7.png}
\caption{\label{figure7}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-8.png}
\caption{\label{figure8}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_0}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.26741573033707866\textwidth}P{0.3151685393258427\textwidth}P{0.26741573033707866\textwidth}}
\tabcellsep Left Tail\tabcellsep Right Tail\\
Observed\tabcellsep 14,060\tabcellsep 12,302\\
Weibull\tabcellsep 14,261\tabcellsep 11,367\\
Johnson\tabcellsep 23,053\tabcellsep 385\\
Normal\tabcellsep 39,754\tabcellsep 237\end{longtable} \par
 
\caption{\label{tab_1}Table 3 :}\end{figure}
 			\footnote{© 2013 Global Journals Inc. (US) Process Capability Analysis using curve Fitting Methods} 			\footnote{© 2013 Global Journals Inc. (US)} 			\footnote{Process Capability Analysis using curve Fitting Methods} 		 		\backmatter  			 
\subsection[{Global Journals Inc. (US) Guidelines Handbook 2013}]{Global Journals Inc. (US) Guidelines Handbook 2013}\par
www.GlobalJournals.org			 			  				\begin{bibitemlist}{1}
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\end{document}
