MONITORING PROCESS PERFORMANCE USING SELF-STARTING CUMULATIVE SUM CONTROL CHART

Statistical process control (SPC) charts are important tools for detecting process shifts. The control chart is an important statistical technique that is used to monitor the quality of a process. Shewhart control charts help to detect larger shifts in the process parameters, but Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) charts are expected for smaller and moderate changes. The CUSUM control chart is normally used in industry for the result of small and moderate shifts in process spot and disparity. It can be shown that if there are sharp, irregular changes to a process, these types of charts are highly effective. On the other hand, if one involved in a small, persistent shift in a process, other types of control charts may be chosen, for instance the CUSUM control chart, originally developed by Page (1954). In this article, we used CUSUM control chart for monitoring the moisture level of the paper sheet.


INTRODUCTION
A most important purpose for a product or a process control is to constantly look up its quality. This aim, in statistical terms, may be expressed as variability reduction. SPC is a notorious collection of methods aiming at this purpose and the control charts are considered as the main tools to detect shifts in a process. The most accepted control charts are the Shewhart charts, CUSUM charts and the EWMA charts. Shewhart type charts are used to detect large shifts in a process whereas CUSUM and EWMA charts are known to be fast in detecting small to moderate shifts. CUSUM control chart is a time-weighted control chart that displays the CUSUM of the variation of each sample value from the target value. A minor drifting in the process mean will direct to steadily rising or declining cumulative deviation value, Owing to the factor that it is cumulative. It was developed by E. S. Page of the University of Cambridge.
CUSUM control chart consider being more efficient in detecting small shift in the mean of a process. CUSUM control chart shows a better result than Shewhart control chart when it is required to detect smaller shift. Also, CUSUM control chart is comparatively slow to respond to large shift and firm to detect and analyse special trend patterns.

REVIEW OF LITERATURE
CUSUM control chart conceived by Page, E.S., 1954 and which have been developed by many authors; in particular, Ewan(1963), Page(1961), Gan (1991), Lucas(1976), Hawkins(1981Hawkins( , 1993a) and Woodall and Adams (1993).They have been proposed as a substitute to Shewhart control charts. Since they detect small shifts in the process level more swiftly, they provide an early signal of process change and they are more meaningful graphically as they point out areas needing attention.
The self-starting control chart has many other advantages that get better the engineer's ability to control a process. One such benefit is the ability to chart in real time and essentially with the first units of production. By using a self-starting chart the engineer can still determine if a shift has occurred from the situation obtained at process start-up without knowing the exact parameter values. The chart uses the past observations to calculate approximately the in-control process parameters. The most fundamental self-starting methodology that we will focus on is the self-starting Q chart (Quesenberry, 1991). Q charts are Shewhart-type self-starting chart techniques. Hawkins (1987) at first proposed a self-starting CUSUM scheme that utilizes two pairs of CUSUMs one for monitoring the location of the process and the other for monitoring the spread. Quesenberry (1991) proposes the self-starting Q chart for both the mean and variance which applies transformations to the quality statistic so it can be plotted on standard normal control charts. Thus so far we have seen the self-starting methodology in the univariate case.
However, if we want to chart two related quality statistics such as inner and outer diameters of a type of parts from the process start-up, multivariate self-starting quality control charts are the

METHODOLOGY
When the process remains in control with mean at , the cumulative sum is a random walk with mean zero. When the mean shifts upward with a value 0 such that  > 0 then an upward or positive drift will be developed in the cumulative sum. When the mean shifts downward with a value 0 such that  < 0 then a downward or negative drift will be developed in the CUSUM. There are two basic ways to present CUSUM control chart, which are tabular or algorithmic CUSUM and v-mask.

Tabular or Algorithmic CUSUM for Monitoring the Mean of the Process:
The tabular CUSUM limits are defined as C i where − and + are called one sided lower and one sided upper CUSUM respectively; K is the reference value and 0 is the targeted mean. If either − or + value exceeds the decision interval H, which usually defined as H=5  then the process is considered to be out of control.

First Initial Response (Fir):
The FIR feature gives a simple procedure for detecting an out-of-control situation at start-up more quickly. However, if the process is initially in control state, the FIR feature has little effect whereas if it is in an out-of-control state, a signal is given much more quickly. The FIR essentially just sets the starting value equal to non zero value typically H/2.

Standardized CUSUM:
A number of CUSUM prefers using standardized CUSUM control chart, which is defined as then the standardized CUSUM is calculated using Then becomes iid standard normal. Start with 0 The Self Starting CUSUM: The self starting CUSUM for the mean of a normally distributed random variables can be applied immediately without any need for a phase I sample to estimate the process parameters, the mean µ and the variance . Let ̅ The average of the first n observations and let w n = ∑ ( be the sum of squared deviations from the average of those observations. Formulas to update these quantities after each new observation x ̅ n = x ̅ n−1 +

RESULTS AND DISCUSSIONS
In this section, the moisture level in the paper sheet material is monitored using Self-Started CUSUM control chart. For monitoring the process, 30 samples were collected from the on-going process. The calculations of Shewhart control chart is given in the Table 1. From the

Cusum Chart
Upper Cusum C+ C-Lower Cusum  The calculation of the Self -Starting CUSUM control chart is given in the Table 3.

CONCLUSION
The CUSUM control chart is very authoritative chart for identifying changes in the process average. This paper discussed the problem based on monitoring the moisture level in the paper sheet material using the traditional and CUSUM control chart. From the standing theory, it is clear that CUSUM charts are valuable for monitoring the process performance, as they tend to detect process shifts on the 21 th observation itself, where the traditional charts showed no change. This will help the decision maker to proactively respond to the output and correct the problems in time. CUSUM chart is particularly adapted to detect the processes characterized by small shifts, whereas traditional chart is effective in detecting variation of the process response.
Also Shewhart chart is better in detecting an immediate abrupt (transient) change, the CUSUM chart is more effective in detecting more sustained changes. Hence it is clear that the CUSUM chart is far superior to the traditional control chart in detecting the small (1 σ) shift. The Self Starting -CUSUM chart is very easy to use and it quickly detects both small and large shifts in the process mean and /or standard deviation. This Self Starting-control chart has then additional advantage over other charts because it swiftly detects decreases in the process SD as well. It also shows graphically the parameters and the sample that triggers an alarm at the earliest, thus indicating that the process is out of control.