On Line Computation of Process Capability Indices

Statistical process control tools and techniques are widely used in assessing and monitoring manufacturing process. Effective implementation of statistical process control tools will improve the productivity and quality; reduce the wastage and improve the business together with various other benefits. Process capability indices are very well used to assess the performance of the production process. However the problem is to keep the produced items in the inventory until to take a decision on the acceptance of the lots based on the outcome of the process capability analysis. An attempt has been made in this paper to overcome such difficulties. As a result we have presented a simple method to assess the process capability indices on line itself, which helps the process as well as the quality engineers to take an instant decision on the manufacturing process whether the process has to be allowed further or to be modified. The method is explained with the help of illustrations.


Introduction
Th e p res ent s cen ario o f total qu ality managemen t demands the effective use of statistical tools for analysing quality problems and controlling, mon itoring, maintaining and improving the performance of manufacturing process. Many of the quality characteristics are measurable in nature and can be expressed in terms of numerical measurements and are to be monitored and controlled during the production. When dealing with measurable quality characteristics, it is usually necessary to monitor the behaviour of the mean value of the quality characteristics and its variability. To study the behaviour of the production process and to take a necessary action on the p rocess, there are several statistical too ls available wh ich include frequency distribution, histogram, Pareto chart, scatter d iagram, contro l charts, reg ression analysis, process capability analysis, design of experiments and Taguch i methods and are to be effectively used for further imp rovements on quality and productivity. Among the statistical too ls ment ioned above, the use o f p rocess capab ility analys is has been increasing ly recogn ised by industrial sectors, particularly the automobile manufacturing industries. The world g iant manufacturer of automob iles namely Ford, Chrysler, and General Motors have together developed a Quality System namely QS 9000 standards and advised their suppliers to fo llo w the QS 9000 -Quality Systems Requirements and to get certified for the same to continue as one of the potential suppliers to the above automobile manufacturing giants.
The process capability analysis as discussed by Montgomery [10] is a vital part of an overall quality improvement program. Several PCIs have been developed among which the basic indices are Cp, Cp k and Cp m (refer to Kane[8]; Chan et.al. [5]). A detailed review of the PCIs can be seen in Kotz and Johnson [9] and Yu m and Kim [13]. The statistical data collected for assessing the suitability of the manufacturing processes and or for computing the process capability indices can also be used to:  Predict how well the process will hold the tolerances  Assist process engineers in selecting a process  Assist product engineers in modify ing a product  Assist in establishing the frequency of sampling  Assess the capability of a new equip ment  Select co mpeting vendors  Reduce variat ions  Predict the outgoing quality level etc., However the problem encountered in the present day manufacturing systems are as follo ws:  The process capability ind ices are co mputed only after a considerable period.
 There is no method available to assess the capability of the process on line  It requires frequent interference of the process  It requires storing the produced items for a considerable period before taking the decision to accept or reject or rework based on the outcome of the process capability study.
 In case of allowing the process before getting the process capability indices, there may be a situation, one has to call back the produced items already passed on.
 It requires 100% inspection and sorting out of the items produced in non capable process The process capability indices Cp, Cpk and Cpm were investigated by many authors. For examp le, the 95% confidence limits for the capability indices Cp and Cpk were constructed by Chou et al. [6]. As their limits on Cpk can produce 97% or 98% lower confidence limits (instead of 95%) making them conservative, an approximation presented by Bissell [3] is reco mmended. Boyles [4] provided an approximate method for finding lo wer confidence limits for the index Cpm. A ll these lower confidence limits are calculated based on the assumption that the process is normally distributed. Franklin and Wasserman [7] constructed the confidence limits fo r some basic capability indices and examined their behaviours when the underlying process is either normal, skewed or heavy tailed. Balamurali and Kalyanasundaram [2] constructed bootstrap confidence limits for the capability indices Cp, Cpk and Cpm based on lognormal and chi-squared distributions. Balamu rali [1] considered the above capability indices under short run production processes and also constructed the bootstrap confidence limits.
An attempt is made in this paper to provide online computation of process capability indices which helps to assess the capability of the process on line itself. As a result a simp lified procedure is presented for co mputing the process capability indices fo r g iven samp le size, sample range and tolerance instantaneously without actually computing the standard deviations and without using the formulae for computing the process capability indices Consequently we have proposed a new manufacturing system called " Capability Based Manufacturing System", wh ich avoids unnecessary disturbance to the process; 100 % inspection; sorting and segregation; rework and scrap etc. The results are presented in the Table 1 to Table 10 and are given in the appendix. The Tables 1 to 9 can be used to determine the p C value for the given sample size, tolerance and the mean o f the sample ranges. On the other hand, it can also be used to determine the manufacturing range to ach ieve the required p C value. Table 10 can be used to determine the pk C value for the given p C and the delta (the proportion of shift in the mean to the targets value). The Table 10 can also be used to know the delta value be allowed to achieve the required p C and pk C values. For want of space we have not presented all the tables developed for various values of p C , Tolerance and Delta not included in the Table 1 to Tab le 10. If anyone interested to get other tables are advised to contact the author by email or post.

Main Results
The following are the various computational fo rmulae involved in constructing the control charts and for the computation of process capability indices, wh ich will be used to derive the simplified procedure for the computation of online process capability indices: : Mean of i-th sample Range of i-th samp le : , which can be written as Similarly, one may obtain the control limits for the R-Charts.
Similarly, two Capability Indices p C and pk C are used for measuring the Process Capability, where p C will take only the variability into consideration and will not consider the centrality of the process for assessing the process capability. Whereas the capability index pk C will take into account both, the variation and centrality of the process. For a detailed discussion on the definition and the derivation of various formu lae involved in control charts and process capability analysis one may refer to Montgomery [10], Subraman i [11,12] and the references cited there in.

Computati on of Average Range, Maximum and Mi ni mum Range Values Vs Process Capability Index C p Values
Consider the process capability index where T is the Tolerance.
Further assume that L R and U R are the maximu m and the min imu m range values in which range the producer can operate the manufacturing process so as to meet the customer requirements including the process capability indices pk p C and C values. That is, the maximu m allo wable range value should be in between L R and U R . The limit of these range values can be obtained from the R values obtained in the above equation as follows: Fro m the above equations one can easily determine the required average range value and the maximu m and minimu m values to be operated so as to meet the customer requirements. That is, for any given value of p C , samp le size n and the manufacturing tolerance T, as per the requirements of the customer the producer can determine the range values in which he has to operate the manufacturing process. For want of space and for the sake of convenience we have tabulated respectively the sample size, tolerance and process capability index values in Tab les 1 to 9. These tables can also be used to determine the value of process capability index p C for the range values, average range in which the process is operating and the sample size by looking at the appropriate co lu mns corresponding to the tolerance and range values by not actually computing the process capability index. This is the advantageous of this study. The above can be explained with the help of the following illustrations.
Suppose that the customer wants to get products produced in a manufacturing process with tolerance 20 and process capability index p C with 1.7. Let us assume that the producer measures the process capability with a sample size of 5 then we use the Table 4 to determine the manufacturing range values. If we look at the Table 4 for n =5, it gives that for the given sample size 5, the tolerance 20 and p C value 1.7 the average range value is 4.56 and the upper value of the range is 9.65. Hence to achieve the customer needs, the producer has to operate the manufacturing process with manufacturing range of 0 to 9.65 units.
On the other hand the Tables from 1 to 9 can also be used to determine the process capability index p C without actually computing the σ value and p C value by using the formulae as detailed below: Let the given specificat ion be n=5, T=50 and the operating Range of R value is 20. By looking at the Tab le 4 with sample size n=5, in the row of T=50 and inside the Row the value close to 20 we obtain the p C value must be in between 1 and 1.1.

Computati on of Process Capability Index C pk for the shift in the Process Average Vs Process Capability Inde x C p Values
Consider the capability index pk C is defined as pk C =Minimu m of { pu C , pl C }, Fro m the above expressions one can obtain , the proportion of shift in the process average to the total tolerance. If we mult iply the ∂ by 100 one may get the percentage of the shift in the process average to the total tolerance. For want of space and for the sake of convenience of the readers, we have tabulated pk C values for the given values of ∂ (in percentage to the total tolerance) in the range 100 0 ≤ ∂ ≤ with increments of 2 and the process capability index values in the range with the increments of 0.1 respectively in Table 10. The Tab le 10 can be used to determine the value of p rocess capability index pk C for the delta values (in percentage to the tolerance) and the p rocess capability index p C by looking at the appropriate colu mns corresponding to the delta values and the p C values by not actually computing the process capability index pk C . The above can be exp lained with the help of the following illustrations. Table 10 can be used to determine the value of pk C for the given value of p C and the process shift fro m the target value. For examp le with the process capability p C in the range of 1.7 with the delta of 30 the process capability index pk C value will be 1.36 only. Similarly the Table 10 can also be used to decide the amount of shift in process average from the target value to maintain the required process capability indices p C and pk C for the given tolerance.

Conclusions
The present scenario of total quality management requires effective use of statistical tools and techniques for a continual improvement on the production processes. Particularly, the auto mobile sectors require the use of process capability indices to assess the suitability of the manufacturing process as well as to assess the products quality. After the init ial approval of the manufacturing process, the future performance will be assessed in terms of the process capability indices, for which the data are collected for a longer period. In case, the computed process capability indices have not satisfied the required conditions, the manufactured products will be screened and called back for some times. In such situations the decision will affect the smooth flow of the production line and also increase the cost of production due to withdrawal of some products; screening of the production lot; rework, rejection and inventory of some produced items. To avoid all these problems, we have suggested a method to assess the process capability indices while the production process is on. As a result we reduce the inventory, recall, rework and reject ion of produced items. All these features have been explained with the help of numerical examp les. Table 1. RL, RBAR and RU values for the given p C value and the Sample Size n=2  Table 2. RL, RBAR and RU values for the given p C value and the Sample Size n=3        Table 5. RL, RBAR and RU values for the given p C value and the Sample Size n=6      Table 7. RL, RBAR and RU values for the given p C value and the Sample Size n=8

Tolerance
Process Capability Indexp C values