Process capability for variables and its measurement 267 ᭹ Cpk = 1.67 Promising, non-conforming output will occur but there is a very good chance that it will be detected. ᭹ Cpk = 2 High level of confidence in the producer, provided that control charts are in regular use (Figure 10.2a). 10.4 The use of control chart and process capability data The Cpk values so far calculated have been based on estimates of ␴ from R, obtained over relatively short periods of data collection and should more properly be known as the Cpk (potential) . Knowledge of the Cpk (potential) is available only to those who have direct access to the process and can assess the short-term variations which are typically measured during process capability studies. An estimate of the standard deviation may be obtained from any set of data using a calculator. For example, a customer can measure the variation within a delivered batch of material, or between batches of material supplied over time, and use the data to calculate the corresponding standard deviation. This will provide some knowledge of the process from which the examined product was obtained. The customer may also estimate the process mean values and, coupled with the specification, calculate a Cpk using the usual formula. This practice is recommended, provided that the results are interpreted correctly. An example may help to illustrate the various types of Cpks which may be calculated. A pharmaceutical company carried out a process capability study on the weight of tablets produced and showed that the process was in statistical control with a process mean (X ) of 2504 mg and a mean range (R ) from samples of size n = 4 of 91 mg. The specification was USL = 2800 mg and LSL = 2200 mg. Hence, ␴ = R/d n = 91/2.059 = 44.2 mg and Cpk (potential) = (USL – X )/3␴ = 296/3 ϫ 44.2 = 2.23. The mean and range charts used to control the process on a particular day are shown in Figure 10.6. In a total of 23 samples, there were four warning signals and six action signals, from which it is clear that during this day the process was no longer in statistical control. The data from which this chart was plotted are given in Table 10.1. It is possible to use the tablet weights in Table 10.1 to compute the grand mean as 2 513 mg and the standard deviation as 68 mg. Then: Cpk = USL – X 3␴ = 2800 – 2513 3 ϫ 68 = 1.41. Figure 10.6 Mean and range control charts – tablet weights Process capability for variables and its measurement 269 The standard deviation calculated by this method reflects various components, including the common-cause variations, all the assignable causes apparent from the mean and range chart, and the limitations introduced by using a sample size of four. It clearly reflects more than the inherent random variations and so the Cpk resulting from its use is not the Cpk (potential) , but the Cpk (production) – a capability index of the day’s output and a useful way of monitoring, over a period, the actual performance of any process. The symbol Ppk is sometimes used to represent Cpk (production) which includes the common and special causes of variation and cannot be greater than the Cpk (potential) . If it appears to be greater, it can only be that the process has improved. A record of the Cpk (production) reveals how the production performance varies and takes account of both the process centring and the spread. The mean and range control charts could be used to classify the product and only products from ‘good’ periods could be despatched. If ‘bad’ product is defined as that produced in periods prior to an action signal as well as any periods prior to warning signals which were followed by action signals, from Table 10.1 Samples of tablet weights (n = 4) with means and ranges Sample number Weight in mg Mean Range 1 2501 2461 2512 2468 2485 51 2 2416 2602 2482 2526 2507 186 3 2487 2494 2428 2443 2463 66 4 2471 2462 2504 2499 2484 42 5 2510 2543 2464 2531 2512 79 6 2558 2412 2595 2482 2512 183 7 2518 2540 2555 2461 2519 94 8 2481 2540 2569 2571 2540 90 9 2504 2599 2634 2590 2582 130 10 2541 2463 2525 2559 2500 108 11 2556 2457 2554 2588 2539 131 12 2544 2598 2531 2586 2565 67 13 2591 2644 2666 2678 2645 87 14 2353 2373 2425 2410 2390 72 15 2460 2509 2433 2511 2478 78 16 2447 2490 2477 2498 2478 51 17 2523 2579 2488 2481 2518 98 18 2558 2472 2510 2540 2520 86 19 2579 2644 2394 2572 2547 250 20 2446 2438 2453 2475 2453 37 21 2402 2411 2470 2499 2446 97 22 2551 2454 2549 2584 2535 130 23 2590 2600 2574 2540 2576 60 270 Process capability for variables and its measurement the charts in Figure 10.6 this requires eliminating the product from the periods preceding samples 8, 9, 12, 13, 14, 19, 20, 21 and 23. Excluding from Table 10.1 the weights corresponding to those periods, 56 tablet weights remain from which may be calculated the process mean at 2503 mg and the standard deviation at 49.4 mg. Then: Cpk = (USL – X)/3␴ = (2800 – 2503)/(3 ϫ 49.4) = 2.0. This is the Cpk (delivery) . If this selected output from the process were despatched, the customer should find on sampling a similar process mean, standard deviation and Cpk (delivery) and should be reasonably content. It is not surprising that the Cpk should be increased by the elimination of the product known to have been produced during ‘out-of-control’ periods. The term Csk (supplied) is sometimes used to represent the Cpk (delivery) . Only the producer can know the Cpk (potential) and the method of product classification used. Not only the product, but the justification of its classification should be available to the customer. One way in which the latter may be achieved is by letting the customer have copies of the control charts and the justification of the Cpk (potential) . Both of these requirements are becoming standard in those industries which understand and have assimilated the concepts of process capability and the use of control charts for variables. There are two important points which should be emphasized: ᭹ the use of control charts not only allows the process to be controlled, it also provides all the information required to complete product classification; ᭹ the producer, through the data coming from the process capability study and the control charts, can judge the performance of a process – the process performance cannot be judged equally well from the product alone. If a customer knows that a supplier has a Cpk (potential) value of at least 2 and that the supplier uses control charts for both control and classification, then the customer can have confidence in the supplier’s process and method of product classification. 10.5 A service industry example – process capability analysis in a bank A project team in a small bank was studying the productivity of the cashier operations. Work during the implementation of SPC had identified variation in transaction (deposit/withdrawal) times as a potential area for improvement. Process capability for variables and its measurement 271 The cashiers agreed to collect data on transaction times in order to study the process. Once an hour, each cashier recorded in time the seconds required to complete the next seven transactions. After three days, the operators developed control charts for this data. All the cashiers calculated control limits for their own data. The totals of the Xs and Rs for 24 subgroups (three days times eight hours per day) for one cashier were: ⌺ X= 5640 seconds, ⌺ R = 1 900 seconds. Control limits for this cashier’s X and R chart were calculated and the process was shown to be stable. An ‘efficiency standard’ had been laid down that transactions should average three minutes (180 seconds), with a maximum of five minutes (300 seconds) for any one transaction. The process capability was calculated as follows: X = ⌺X k = 5640 24 = 235 seconds R = ⌺R k = 1900 24 = 79.2 seconds ␴ = R/d n , for n = 7, ␴ = 79.2/2.704 = 29.3 seconds Cpk = USL – X 3␴ = 300 – 235 3 ϫ 29.3 = 0.74. i.e. not capable, and not centred on the target of 180 seconds. As the process was not capable of meeting the requirements, management led an effort to improve transaction efficiency. This began with a flowcharting of the process (see Chapter 2). In addition, a brainstorming session involving the cashiers was used to generate the cause and effect diagram (see Chapter 11). A quality improvement team was formed, further data collected, and the ‘vital’ areas of incompletely understood procedures and cashier training were tackled. This resulted over a period of six months, in a reduction in average transaction time to 190 seconds, with standard deviation of 15 seconds (Cpk = 2.44). (See also Chapter 11, Worked example 2.) Chapter highlights ᭹ Process capability is assessed by comparing the width of the specification tolerance band with the overall spread of the process. Processes may be classified as low, medium or high relative precision. ᭹ Capability can be assessed by a comparison of the standard deviation (␴) and the width of the tolerance band. This gives a process capability index. 272 Process capability for variables and its measurement ᭹ The RPI is the relative precision index, the ratio of the tolerance band (2T) to the mean sample range (R ). ᭹ The Cp index is the ratio of the tolerance band to six standard deviations (6␴). The Cpk index is the ratio of the band between the process mean and the closest tolerance limit, to three standard deviations (3␴). ᭹ Cp measures the potential capability of the process, if centred; Cpk measures the capability of the process, including its centring. The Cpk index can be used for one-sided specifications. ᭹ Values of the standard deviation, and hence the Cp and Cpk, depend on the origin of the data used, as well as the method of calculation. Unless the origin of the data and method is known the interpretation of the indices will be confused. ᭹ If the data used is from a process which is in statistical control, the Cpk calculation from R is the Cpk (potential) of the process. ᭹ The Cpk (potential) measures the confidence one may have in the control of the process, and classification of the output, so that the presence of non- conforming output is at an acceptable level. ᭹ For all sample sizes a Cpk (potential) of 1 or less is unacceptable, since the generation of non-conforming output is inevitable. ᭹ If the Cpk (potential) is between 1 and 2, the control of the process and the elimination of non-conforming output will be uncertain. ᭹ A Cpk value of 2 gives high confidence in the producer, provided that control charts are in regular use. ᭹ If the standard deviation is estimated from all the data collected during normal running of the process, it will give rise to a Cpk (production) , which will be less than the Cpk (potential) . The Cpk (production) is a useful index of the process performance during normal production. ᭹ If the standard deviation is based on data taken from selected deliveries of an output it will result in a Cpk (delivery) which will also be less than the Cpk (potential) , but may be greater than the Cpk (production) , as the result of output selection. This can be a useful index of the delivery performance. ᭹ A customer should seek from suppliers information concerning the potential of their processes, the methods of control and the methods of product classification used. ᭹ The concept of process capability may be used in service environments and capability indices calculated. References Grant, E.L. and Leavenworth, R.S. (1996) Statistical Quality Control, 7th Edn, McGraw-Hill, New York, USA. Owen, M. (1993) SPC and Business Improvement, IFS Publications, Bedford, UK. Process capability for variables and its measurement 273 Porter, L.J. and Oakland, J.S. (1991) ‘Process Capability Indices – An Overview of Theory and Practice’, Quality and Reliability Engineering International, Vol. 7, pp. 437–449. Pyzdek, T. (1990) Pyzdek’s Guide to SPC, Vol. One – Fundamentals, ASQC Quality Press, Milwaukee WI, USA. Wheeler, D.J. and Chambers, D.S. (1992) Understanding Statistical Process Control, 2nd Edn, SPC Press, Knoxville TN, USA. Discussion questions 1 (a) Using process capability studies, processes may be classified as being in statistical control and capable. Explain the basis and meaning of this classification. (b) Define the process capability indices Cp and Cpk and describe how they may be used to monitor the capability of a process, its actual performance and its performance as perceived by a customer. 2 Using the data given in Discussion question No. 5 in Chapter 6, calculate the appropriate process capability indices and comment on the results. 3 From the results of your analysis of the data in Discussion question No. 6, Chapter 6, show quantitatively whether the process is capable of meeting the specification given. 4 Calculate Cp and Cpk process capability indices for the data given in Discussion question No. 8 in Chapter 6 and write a report to the Development Chemist. 5 Show the difference, if any, between Machine I and Machine II in Discussion question No. 9 in Chapter 6, by the calculation of appropriate process capability indices. 6 In Discussion question No. 10 in Chapter 6, the specification was given as 540 mm ± 5 mm, comment further on the capability of the panel making process using process capability indices to support your arguments. Worked examples 1 Lathe operation Using the data given in Worked example No. 1 (Lathe operation) in Chapter 6, answer question 1(b) with the aid of process capability indices. 274 Process capability for variables and its measurement Solution ␴ = R/d n = 0.0007/2.326 = 0.0003 cm Cp = Cpk = (USL – X ) 3␴ = (X – LSL) 3␴ = 0.002 0.0009 = 2.22. 2 Control of dissolved iron in a dyestuff Using the data given in Worked example No. 2 (Control of dissolved iron in a dyestuff) in Chapter 6, answer question 1(b) by calculating the Cpk value. Solution Cpk = USL – X ␴ = 18.0 – 15.6 3 ϫ 1.445 = 0.55. With such a low Cpk value, the process is not capable of achieving the required specification of 18 ppm. The Cp index is not appropriate here as there is a one-sided specification limit. 3 Pin manufacture Using the data given in Worked example No. 3 (Pin manufacture) in Chapter 6, calculate Cp and Cpk values for the specification limits 0.820 cm and 0.840 cm, when the process is running with a mean of 0.834 cm. Solution Cp = USL – LSL 6␴ = 0.84 – 0.82 6 ϫ 0.003 = 1.11. The process is potentially capable of just meeting the specification. Clearly the lower value of Cpk will be: Cpk = USL – X 3␴ = 0.84 – 0.834 3 ϫ 0.003 = 0.67. The process is not centred and not capable of meeting the requirements. Part 5 Process Improvement . 254 0 255 5 2461 251 9 94 8 2481 254 0 256 9 257 1 254 0 90 9 250 4 2 59 9 2634 2 59 0 258 2 130 10 254 1 2463 252 5 255 9 250 0 108 11 255 6 2 457 255 4 258 8 253 9 131 12 254 4 2 59 8 253 1 258 6 256 5 67 13 2 59 1 2644. 26 45 87 14 2 353 2373 24 25 2410 2 390 72 15 2460 250 9 2433 251 1 2478 78 16 2447 2 490 2477 2 498 2478 51 17 252 3 257 9 2488 2481 251 8 98 18 255 8 2472 251 0 254 0 252 0 86 19 257 9 2644 2 394 257 2 254 7 250 20. application 10 000 3 79 95 0 95 . 0 Phenol content >1% 9 600 3 89 55 0 97 .4 High iron content 5 000 3 95 550 98 . 65 Unacceptable chromatogram 4 50 0 399 050 99 . 75 Unacceptable abs. spectrum 95 0 400 000 100.0