Higher National Certificate in Engineering

1. Higher National Certificatein Engineering LO 2.1.2 – Control Charts (Attribute)

2. Learning OutcomeLO2.1 • LO 2.1: select and group sample data based on variable inspection and attributableinspection and calculate appropriate control chart limits

3. Attribute Data • It is not always possible to measure variables. • Some defects such as scratches, tears, dents and holes are either there or they are not there i.e. the products either pass or they fail. • Assessing attributes is in general an easier task to perform than measuring variables and as a consequence it is easier to train someone to identify them. • Attributes however are not so sensitive a measure and thus detection of small changes becomes much less reliable

4. Attribute Data • When dealing with attribute data we need to understand more about the product (or service) and the ‘type’ of attribute we are wanting to ‘measure’. This leads us to use different types of control chart.

5. Types ofAttribute Data • Conforming or Non Conforming Units • Each of these can be wholly described as failing or notfailing, acceptable or not acceptable, present or not present. etc. • Conformities or non-conformities • Each of these may be used to describe a product or service such as the number of defects, errors, faults or positive values such as calls, truck deliveries, goals scored, etc.

6. Types ofAttribute Data • Thus a defective is an item or unit which contains one or more flaws, errors or defects. • A defect is an individual flaw or fault. • Consider 100-bearings. If two are defective, then 98 will be good. • Consider 1-windscreen. We can say that it has two bubbles in it but we cannot say how many bubbles it might have had!

7. Types ofAttribute Data • These two different types of data lead us to use two types of control chart… • Number of non-conforming units (or defectives) chart • Number of non-conformities (or defects) chart • These are each further split into two charts, one for the situation in which sample size(number of units) is constant and one of varying size

8. Types ofAttribute Data

9. Types ofAttribute Data • To ensure smooth control of a process using attribute data, it is often necessary to provide representative samples, photographs or other objectiveevidence to support the decision maker. • For instance a scratch on the paintwork on a table top may vary from a deep gouge to a slight mark hardly visible to the naked eye.

10. Types ofAttribute Data • By providing supporting evidence then attention and effort may be concentrated on improving the process rather than debating the issues surrounding the severity of non-conformities.

11. Attribute ProcessCapability • When a process has been shown to be in statistical control, the average level of events, errors, defects per unit or what ever, will represent the capability of the process when compared with the specification. • To improve process capability requires a systematic investigation of the whole process system – not just diagnostic examination of particular apparent causes of lack of control.

12. Attribute ProcessCapability • Such improvements will require management to improve contributing factors such as… • Operator performance, training and knowledge • Equipment performance, reliability and maintenance • Material suitability, conformity and grade • Methods, procedures and their consistent usage

13. np-Charts • Consider a process producing ball bearings, 10% of which are defective. Thus p, the proportion of which are defective is 0.1 • If we take a sample of 1-ball from the process, the chance (or probability) of finding a defective is 0.1 or p. • Similarly the probability of finding a non-defective is 0.9 or (1-p). [For convenience we represent this as q.] • Hence… p + q = 0.1 + 0.9 = 1.0 [1.0 represents certainty, either the ball bearing is defective or it is not]

14. np-Charts • If we increase the sample size to 2-ball bearings, the probability of finding two defectives becomes… p x p = 0.1 x 0.1 = 0.01 = p2 [This is one of the laws of probability – the multiplication law.] • When two or more events are required to follow consecutively, the probability of them all happening is the product of their individual probabilities. • Thus for A and B to happen, multiply the individual probabilities pA and pB.

15. np-Charts • For our sample size to 2-ball bearings, the probability of not finding two defectives becomes… q x q = 0.9 x 0.9 = 0.81 = q2 • Thus the probability of finding or not finding two defective units is… p2 + q2 = 0.01 + 0.81 = 0.82 • Since the probability of all events must be 1, it is therefore obvious that we have not considered all possibilities. There remains the possibility of picking one defective and one non-defective.

16. np-Charts • For our sample size to 2-ball bearings, the probability of finding one defective and one non defective becomes… p x q = 0.1 x 0.9 = 0.09 = pq • For our sample size to 2-ball bearings, the probability of finding one non-defective and one defective becomes… q x p = 0.9 x 0.1 = 0.09 = qp • Thus… pq + qp = 0.09 + 0.09 = 0.18 = 2pq

17. np-Charts • Thus the probability of a chance of an event (i.e. selecting 2-ball bearing from a 10 and ascertaining whether you’ve selected a defective unit) happening, is the sum of all the possible chances… p2 + 2pq + q2 = 0.01 + 0.18 + 0.81 = 1.0 • Thus… (p + q)2 = 1 • This is a binomial expression and it can be written in a general form… (p + q)n = 1

18. np-Charts (p + q)n = 1 • n = sample size (number of units) • p = proportion of defects or ‘non-conforming units in the population from which the sample is drawn • q = proportion of non-defectives or ‘conforming units’ in the sample population = (1 – p)

19. np-Charts • Eg: (p + q)4= 1 p4 + 4p3q + 6p2q2 + 4pq3 + q4 p4 = prob. of 4-defectives in the sample 4p3q = prob. of 3-defectives 6p2q2 = prob. of 2-defectives 4pq3 = prob. of 1-defectives q4= prob. of zero-defectives

20. np-Charts • This general equation may be written in the form… P(x) = n px(1-p)n-x x where n = n! x (n – x)!x! Where n! = 1 x 2 x 3 x … x n And x! = 1 x 2 x 3 x … x x

21. np-Charts • Example: Find the probability of having two defect bearings in a sample size of 5 taken from a process that produces 10% defectives? p = 0.1 q = 0.9 n = 5 x = 2 P(x) = npx(1-p)n-x = 5! 0.12x0.93 = 0.0729 x (5-2)!2!

22. np-Charts • Thus on average about 7 out of every 100 samples of 5-bearings taken from a process will have two defectives in them. • The average number of defectives present in a sample of 5 will be 0.5 • Thus if we calculate the probability of exceeding a certain number of defectives in a sample, we can draw action lines and warning lines on charts to enable us to control a process

23. np-Charts • In order to do this, the pre-requisite is that we must know the proportion of defective units being produced by the process. • This may be discovered by taking a reasonable number of samples – say 50 – over a typical period and recording the number of defectives (non conforming units) in each. • Exercise 1

24. np-Charts • Note: σ = √n.p(1 - p) UCLp = n.p + 3√n.p(1 - p) _ _ _ _ _

25. np-Charts np chart showing the effect of a moving average

26. np-Charts It should be noted that for each percentage defect, the run length, that is the number of samples which needed to be taken before the action line is crossed following the increase in process defective is… Thus clearly this chart is not as sensitive as an x-bar and R chart for detecting changes in process defective.

27. np-Charts For this reason, the UCL and UWL on attribute charts are modified (lowered) – i.e. they provide higher probabilities of detection: approximately 1 in 200 (UCL) and 1 in 20 (UWL) There is a danger in doing this however in as much as there will also be an increase in incorrect action signals. As inspection for attributes say by using a go/no-go gauge is usually less costly than the measurement of variables an increase in the amlount of re-sampling may be tolerated.

28. np-Charts If the probability of an event is say 0.25, then on average it will occur every fourth time as the average run length (ARL) is simply the reciprocal of the probability. Hence in our pen cartridge case, if the probability defective is 3% (p=.03) and the action line is set between 6 and 7, the probability of finding 7 or more defectives may be calculated or derived from the binominal expansion as 0.0312 (n=100). Thus ARL(3%) = 1/(P(>7) = 1/0.0312 = 32

29. np-Charts For a process producing 5% defectives, ARL(5%) = 1/(P(>5) = 1/0.0234 = 4 The conclusion from the run length value is that given time the np-chart will detect a change in the proportion of defectives being produced. If the change is an increase of approximately 50%, the np-chart will be very slow to detect it on average. If the changes is a decrease of 50% the chart will not detect it because in the case of a process with 2% defective there is no lower limit.

30. p-Charts • p-charts are used for proportion defective or non-conforming units • In cases where it is not possible to maintain a constant sample size for attribute control, the p-chart (Proportion defective or non-conforming units chart) may be used. • It is possible and quite acceptable to use the p-chart instead of the np-chart even when the sample size is constant - however plotting directly the number of defectives in each sample onto an np-chart is simple and usually more convenient than having to calculate the proportion defective.

31. p-Charts • The data required for the design of a p-chart are identical to those of an np-chart, both the sample size and the number of defectives need to be observed. • Exercise 2 – Firstly determine the proportion defective p = Σ x / Σ n Where Σ x is the total number of defective items and Σ n is the total number of items inspected. -

32. p-Charts • If a constant sample size is being inspected, the p-chart limits would remain the same for each sample. • When p-charts are being used with samples of varying sizes, the standard deviation and control limits change with n (sample size) and unique limits should be calculated for each sample size. • However for practical purpose an average sample size n-bar may be used to calculate action and warning lines.

33. p-Charts • These have been found to be acceptable when individual sample or lot sizes vary from n-bar by no more than 25% each way. • For sample sizes outside this range, separate control limits must be calculated • Thus the next step for exercise 2 is to determine the average sample size and 25% either side. • For samples sizes within this range, the control chart lines may be calculated using a value for std dev. given by…

34. p-Charts - - - σ = [√ p (1 – p)] / √ n • Calculate the std. dev. (σ) for exercise 2. • Calculate the UCL, LCL, AWL and LWL for the exercise • Identify the sample sizes that fall outside the 25% limit. Calculate the control limits for each of these samples. • Plot the resulting p-chart

35. p-Charts • It is evident that the design, calculation, plotting and interpretation of a p-chart is more complex than that of an np-chart • It can be seen in exercise 2 that the process is out of control. • Note one of the out-of control conditions shows a point below the bottom control limit. There would be a need to investigate this as this is out-of-control to the good as it could lead to an on-going improvement in the process.

36. c-Charts • The control charts for attributes considered so far have applied to cases in which a random sample of definite size is selected and examined in some way. • The the process control of attributes there are situations where the number of events, defects, errors or non-conformities can be counted, but there is no information about the number of events, defects or errors which are not present. • Hence there is an important distinction between defectives and defects.

37. c-Charts • So far we have considered defectives where each item is classed as conforming or non-conforming (a defective). This gives rise to a binomial distribution. • In the case of scratches on a car door panel, or holes in a piece of fabric we know the number of defects but we don’t know the number of non-defects present. • In this case the binomial distribution cannot apply

38. c-Charts • This type of problem is described by the Poisson distribution. • As there is no fixed sample size when counting the number of events, defects, etc, theoretically the number could tail off to infinity. • Any distribution which does this must include something of the exponential distribution and the constant e (where e = 2.71828)

39. c-Charts Thus… e = 1/0! + 1/1! + ½! + 1/3! + ….+ 1/∞! The equation for Poisson’s Distribution is given by… P(x) = e-c(cx/x!) Where x is the number of defects observed and c is the average number of defects per unit being produced by the process - -

40. c-Charts • For example the probability of finding three bubbles in a windscreen from a process which is producing them which has an average of one bubble present is given by… P(3) = e-1 x 13/(3 x 2 x 1) = 0.0613 • As with the np-chart it is not necessary to calculate the probabilities in this way to determine control limits for the c-chart. • One again the UCL is set at 3 std. dev. Above the average number of events, defects, errors, etc.

41. c-Charts • Exercise 3 • Note in this case the length of the sample is constant • The std. dev. of a Poisson distribution is the square root of the process average… σ = √ c • The control limits for a c chart are at… UCL = c + 3√ c LCL = c - 3√ c And the warning limits are… UWL = c + 2√ c LWL = c - 2√ c _ _ _ _ _ _ _ _ _ _

42. U-charts • Where it is not always possible to maintain a constant number of units we replace ‘c-chart’ with the ‘u-chart’. • The length of a piece of material (steel strip), volume of substance or length of time may vary Sometimes it may be desirable to continue examination until a defect is found and then note the sample size.

43. U-charts • For example the average value of c in the polythene film process had fallen to 0.5, the values plotted on the chart would mostly be 0 and 1, with the occasional 2. • Control of a process by a whole number c-chart would be rather nebulous. • The u-chart is more suitable for controlling this type of process as it measures the number of events, defects or non-conformities per unit or time period and the sample size can be allowed to vary.

44. U-charts • The inspection of paint defects on a strip of steel for instance, the area examined may be allowed to vary and the u-chart will show the number of defects per unit area (e.g. per square metre). • The design of a u-chart is similar to the p-chart for proportion defective – the control lines will vary for each sample size, but for practical purposes may be kept constant if sample sizes remain with 25% either side of the average sample size n-bar.

45. U-charts • As with the p-chart, it is necessary to calculate the process average average defect rate, u-bar. u = number of defects total sample inspected = Σ x / Σ n • The defects found per unit (u) will follow a Poisson distribution the std. dev. (σ) being the square root of the process average u +/- 3√u/n _

46. U-charts _ _ _ _ _ _ UCL = u - 3√ u /√ n LCL = u - 3√ u /√ n UWL == u - 2√ u /√ n LCL = u - 2√ u /√ n _ _ _ _ _ _