International Journal of Physical Sciences Full Length Research Paper On enhanced control charting for process monitoring

The information on auxiliary characteristics helps significantly in increasing the efficiency of control charts for detecting shifts in process parameters. In this study we proposed Shewhart type control charts, namely the r A chart, the r D chart and the r K chart, which utilizes information on two auxiliary characteristics (X and Z) for improved monitoring of process location parameter with respect to a single quality characteristic of interest (Y). Assuming trivariate normality of (Y, X, Z), a general control chart structure is developed in the form of the three sigma and the probability limits. The performance of the proposed charts is compared with the usual Shewhart Y chart, the r M chart of Riaz (2008a), the control limits of Zhang (1984) and Wade and Woodall (1993). It has been observed that the proposed charts perform superior, in terms of discriminatory power, as compared to the above mentioned counterparts, depending upon the correlation structure among the auxiliary characteristics and the quality characteristic of interest. The said superiority zone of the correlation structures, favoring the proposed charts, needs to be identified very carefully to apply it in a given situation.


INTRODUCTION
A process is generally described by its characteristics and out of these some are of main concern and others are of a supplementary nature.The characteristics of main concern are termed as quality characteristics of interest while the other characteristics are termed as auxiliary characteristics.The auxiliary characteristics need to be identified very carefully along with the characteristics of interest.An auxiliary characteristic may be an early measurement in a process, crude but simple to obtain measurement, a property that would be monitored etc.The quality characteristic of interest may be any current variable of major interest which needs to be monitored for example, weight of a machine component, diameter of a shaft, spinning speed of wheel etc.
It is a common practice to take benefit out of the information available on auxiliary variable(s), along with the main study variable(s) of interest, in order to improve the efficiency in statistical terms.There is a variety of literature available in this regard for example, Kiregyera (1984), Mukerjee et al. (1987), Singh (2001), Singh et al. (2004), Kadilar and Cingi (2005), Joarder (2009), (2011) and Omar and Joarder (2011).This idea of using information on some additional characteristic(s) (for example, auxiliary characteristic(s)) which are associated with the main quality characteristic of interest has also been used in quality control literature.Particularly in control charting methodology, it has been used in the form of cause-selecting and regression-adjusted control charts (for example, Mandel (1969); Zhang (1984); Hawkins (1991Hawkins ( , 1993)); Wade and Woodall (1993); Shu et al. (2005) and auxiliary information based control charts *Corresponding author.E-mail: sabb025@aucklanduni.ac.nz, saddamabbasi@yahoo.com.
(for example, Riaz (2008a, b); Riaz and Does (2009)) for the sake of an improved process monitoring with respect to the quality characteristic of interest.Riaz (2008a) proposed a location control chart, namely the r M chart, in which he used the information of single auxiliary characteristic on regression pattern.Riaz (2008b) and Riaz and Does (2009) gave proposals for variability control charts based on a single auxiliary characteristic in which they used the auxiliary informationon regression and ratio patterns respectively.There may be many real situations when more than one auxiliary characteristics are available to be used e.g. to monitor the inner diameter of a shaft its outer diameter and weight may be the two possible auxiliary characteristics.For the sake of simplicity we assume that there are two auxiliary characteristics available along with the quality characteristics of interest.
The rest of this article is organized as follows: The design structure of the proposed charts, performance evaluation measures and a comparison of the proposed charts with the usual Shewhart Y chart and the r M chart of Riaz (2008a), the steps for using the proposed chart with help of simulated examples and finally, the paper ends with conclusions.

THE PROPOSED CHARTS
Suppose that the quality characteristic of interest is denoted by Y and the two auxiliary characteristics by X and Z .We write these three variables of this study in the form of a triplet as Y by exploiting the information on X and Z .We consider here the case when the auxiliary characteristics X and Z remain stable over time.Also we assume that the spread parameters of Y is in-control so the only parameter we are concerned with is the location parameter of Y , that is, y  .In a broader sense we can consider any process without cascade property (cf.Hawkins (1993)) where each variable may undergo a distributional change without affecting the other variables of the process.
To monitor the behaviour of

X and Z
, given as: The regression type estimator given in Kadilar and Cingi (2005), defined as: The ratio type estimator proposed by Abu-Dayyeh et al. (2003), defined as: The ratio type estimator proposed by Kadilar and Cingi (2003), defined as: where y , x and z are the sample means of Y , X and Z respectively.The quantities Y , X and Y , Z , respectively.We used the optimal choices of 1 2 1 ,,    and 2  as reported in Abu- Dayyehh et al. (2003) and Kadilar and Cingi (2005), respectively, that is: The population means of the two auxiliary characteristics that is,

Some distributional results
A general control chart structure that can be used with any of the three estimators will develop here.
First we define a pivotal quantity G , which is based on the sample statistic, say T, given as: Here T represents any of the sample statistic given in  2 to 4, respectively for the r A chart.The standard errors for the simulated results (reported in Tables 1 to 4) are less than 1%, which is acceptable in control chart studies (Schaffer and Kim, 2007).These simulation results are based on 1000 repetitions of 10,000 Monte Carlo simulations.Similar results can be easily obtained for r D and r K charts.

Control chart design
Based on the results given above, we are now in a position to define the design structure of a general control chart namely the r T Chart.From the expressions given in (4) and ( 5) above, we have the following: we can define the control limits of the r T charts, using the three sigma limit and the probability limit approaches, as: Three Sigma Limits , 3 and 3

Probability Limits
, and  6).Here  is a pre-specified false alarm rate which is equally divided on both the tails to define the probability limits.
In probability limits approach it is preferable to replace the CL by ML as we did in (8).The control limits given in ( 7) and ( 8) are the specified parameters versions of the three sigma and probability limits structures of the

PERFORMANCE EVALUATION AND COMPARISONS
To evaluate the performance of a control chart discriminatory power is a very popular measure.We evaluate here the performance of our proposed control charts using the same performance measure.Let the in control value of Now the ability of the three sigma and the probability limits based design structures of the proposed charts is given by the following power expressions:     In brief, the proposed charts has shown better performance for the case when the auxiliary variables ( X and Z ) are not much correlated with each other but strongly/moderately correlated with main variable ( Y ).To be specific, the correlation between X and Z (that is,  Zhang (1984); Wade andWoodall (1993) Zhang (1984) proposed an improvement over the separate use of the usual Shewhart's Y charts.Later Wade and Woodall (1993) gave an improvement over Zhang (1984) by proposing their prediction limits.Riaz (2008a) proved superiority of the r M chart over the control limits of Zhang (1984); Wade and Woodall (1993).We have shown from above that the charts proposed in this study perform better than the r M chart.Hence we can indirectly draw the conclusion that the proposed , rr ADand r K charts will perform better than the control limits of Zhang (1984); Wade and Woodall (1993).

ILLUSTRATIVE EXAMPLES
In this section we will illustrate the application procedure for one of the three proposed charts that is, the Example 2: First 20 samples are from 3 0.0 1.0 0.5 0.6 0.0 , 0.5 1.0 0.1 0.0 0.6 0.1 1.0 and the last 10 samples are from 3 1.0 1.0 0.5 0.6 0.0 , 0.5 1.0 0.1 0.0 0.6 0.1 1.0 . We have simulated the datasets for the above mentioned two mechanisms of examples 1 and 2 so that we can have datasets with the known characteristics.The resulting datasets are used to calculate the required quantities for the proposed chart.The sample statistic

Conclusions
In this study we proposed three Shewhart type control charts, namely the Y chart and the proposals of Zhang (1984); Wade and Woodall (1993); Riaz (2008a).The comparisons revealed that the r D chart is best among all the charts investigated in this study.All the new proposals outperformed the said existing counterparts in terms of discriminatory power.This superiority of the proposed chart demands the auxiliary characteristics to be highly/moderately correlated with the main quality characteristic of interest but not having high correlations with each other.The proposals of this article are of Shewhart type, focusing on an improved monitoring of the location parameter.However EWMA and CUSUM type structures can also be devised by using these location estimators,  for efficient detection of small deviations from the parameter value or alternatively some extra sensitizing rules can also be carefully planned to be used with its control structure.
r T chart.In case of unspecified parameters the estimated versions may used by replacing y  and y  by their estimators ˆy  and ˆy  , where ˆy  may be obtained from some spread control chart e.g. the R chart and ˆy  = r T based on an initial set of say m samples from the process in a stable situation.The choices of T as ,

y
be denoted by 0 y  and the shifted value by 1 y  .We define here 1 y  in terms of 0 y  and the distributional results given above.It is a common practice to evaluate such expressions by varying the values of  and we do the same here in this article.By varying the values of  from 0.0 to 3.0, we have evaluated the power expressions for different n by fixing the false rate at  level.It is to be mentioned that for power evaluation, the shifts are taken in y  in terms of y  units.The resulting powers/power curves are provided in the following sub-sections along with the comparison with other charts.We expect that the proposed control charts should detect any shift in the parameter of interest with a high probability, keeping the false alarm rate at fixed low level.We compare the yz

M
and Y charts.We have evaluated the power expression (11) of the proposed , powers are plotted against  and the resulting power curves are shown in the Figures 1 to 9 whereas Figure 10 presents power curves for a specific case when 25 n  , to investigate the effect on sample size on the performance of the proposed chart.We have varied the values of yx Riaz (2008a) and the usual Shewhart Y chart are also provided in Figures 1 to 10.The symbols used in Figures 1 to 10 are defined as: Y refers to the power curve of the usual Y chart;

Figure 1 .Figure 2 .Figure 3 .Figure 5 .
Figure 1.Power curves of , , , r r r Y M A D and r K charts for


keeps relaxing with an increase in the value of n .This can be examined by looking at Figures 5 and 10.One can easily see the gaps among the curves in Figures 10 are more than those of in Figures 5.It means that the conditions on the correlation structures xz  , yx  and yz  have been relaxed with the increase in sample size n .rA Chart vs.
time points and the sample on each time point comes from one of the following two mechanisms:Example 1: In all the 30 samples as a whole, 90% of the observations

Figure 6 .
Figure 6.Power curves of , , , r r r Y M A D and are calculated for both the examples and are provided in the following Table5.The control limits for the proposed r A chart are calculated using 0.01 for the two examples under discussion.The two sets of limits are given as: charts exploit the information of two auxiliary characteristics for the sake of an improved monitoring of the location parameter of the quality characteristic of interest.The design structure of the proposed charts has been developed in the form of the three sigma and the probability limits based on trivariate normality assumption.Comparison of the proposals has been made with the existing counterparts including the usual

Figure
Figure 11.r A chart for Example 1.

Table 1 .
Control chart coefficients 2 r T chart and the results for r T  s are given in (

Table 2 .
Quantile points of the distribution of G when

Table 3 .
Quantile points of the distribution of G when

Table 4 .
Quantile points of the distribution of G when yz 

Table 5
r A chart, we received out-of-control signals at time points

Table 5 .
Sample statistics r y R fro the simulated data sets.y  ). 