Quality. Statistical Process Control: Control Charts Process Capability DEG/FHC 1

Transcription

1 Quality Statistical Process Control: Control Charts Process Capability DEG/FHC 1

2 SPC Traditional view: Statistical Process Control (SPC) is a statistical method of separating variation resulting from special causes from natural variation in order to eliminate the special causes, and to establish and maintain consistency in the process, enabling process improvement. A broader view is (Goetsch): SPC does not refer to a particular technique, algorithm or procedure SPC is an optimisation philosophy concerned with continuous process improvements, using a collection of (statistical) tools for: data and process analysis making inferences about process behaviour decision making SPC is a key component of Total Quality initiatives In either view, Variation is a key issue DEG/FHC 2

3 SPC Ultimately, SPC seeks to maximise profit by: improving product quality improving productivity streamlining process reducing waste reducing emissions improving customer service, etc. DEG/FHC 3

4 SPC-Tools Commonly used tools in SPC include: Flow charts Run charts Pareto charts and analysis Cause-and-effect diagrams Frequency histograms Scatter diagrams Control charts Process capability studies These tools are usually used to complement each other, rather than employed as stand-alone techniques. DEG/FHC 4

5 Statistical control Processes that are not in a state of statistical control: show excessive variations exhibit variations that change with time A process in a state of statistical control is said to be statistically stable. Control charts are used to detect whether a process is statistically stable or not. Control charts differentiates between variations: that are normally expected in the process due chance or common causes (many, small, unavoidable, chance causes jointly affecting the system) that change over time due to special causes (assignable cause, unexpected). DEG/FHC 5

6 Variation Variations due to common causes: have (usually) small effect on the process are inherent to the process because of: the nature of the system the way the system is managed the way the process is organized and operated can only be removed by making modifications to the process changing the process hence its removal are mainly the responsibility of higher management (e.g., equipment/technology, ) DEG/FHC 6

7 Variation Variations due to special causes are: localised in nature exceptions to the system considered as abnormalities often specific to a: certain operator certain machine certain batch of material, etc. Investigation and removal of variations due to special causes are key to process improvement Sometimes, the frontier between common and special causes may not be very clear due the complexity of the systems/process. DEG/FHC 7

8 1-Control Charts: how they work The principles behind the application of control charts are very simple and are based on the combined use of: Run Charts Hypothesis testing The procedure is: sample the process at regular intervals. plot the statistic (or some measure of performance), e.g. mean, range, number of defects, etc. check (graphically) if the process is under statistical control (if the process is not under statistical control, do something about it). DEG/FHC 8

9 Control Charts Control charts make assumptions about the plotted statistic, namely: it is independent, i.e., a value is not influenced by its past value and will not affect future values. it is normally distributed, i.e., the data has a normal probability density function. The assumptions of normality and independence enable predictions to be made about the data. DEG/FHC 9

10 DEG/FHC 10

11 Control charts provide a graphical means for testing hypotheses about the data being monitored. Consider the commonly used Shewhart Chart as an example.. DEG/FHC 11

12 Common (chance) and special (assignable) causes DEG/FHC 12

13 General model for a control chart General model for a control chart UCL = μ + kσ CL = μ LCL = μ kσ where μ is the mean of the variable, and σ is the standard deviation of the variable. UCL=upper control limit; LCL = lower control limit; CL = center line where k is the distance of the control limits from the center line, expressed in terms of standard deviation units. When k is set to 3, we speak of 3-sigma control charts. Historically, k = 3 has become a standard* in industry. *some industries use higher values; a trend is the 6-sigma approach. DEG/FHC 13

14 Types of control charts Different charts are used depending on the nature of the charted data. Commonly used charts are: for Variables The data is measured along a continuous scale such as length, time, weight, viscosity, breaking strength, temperature, etc. Provide more information than discrete data. for Attributes (countable) An attribute is an intrinsic property of a given item that either does or does not exist. Hence, the data assumes only two values: good-bad, pass-fail, etc, and is measured by counting. Being of qualitative nature, discrete data do not provide the degree to which a quality characteristic is nonconforming. DEG/FHC 14

15 I-Control Charts for Variables Of the several CC, we will address the Shewhart sample mean ( --Chart) Shewhart sample range (R-Chart) DEG/FHC 15

16 I-Control Charts for Variables X -Chart: deals with the average value in a process (monitors the central tendency) R-Chart: takes into count the range of the values (reflects process variability). Both are used together R-chart usually applies when the number of observations is less or equal to 10. For larger number observations the S-chart (using the standart deviation instead) may provide better information. DEG/FHC 16

17 Control charts for the mean and the range Step 1: Using a preselected sampling scheme and sample size (number of samples and number of instances per sample), record measurements of the selected quality characteristic on the appropriated forms. Step 2: For each sample calculate the sample mean and the range Step 3: Obtain the center line and trial control limits (and perhaps the warning limits) DEG/FHC 17

18 Step 2 Y- axis of a control chart Formulas For a Xbar-chart: Averages of the quality characteristic over a sample of n units X n i 1 n X i For a R-chart: Ranges of measures in a sample of n units R MAX ( X, i 1.. n) MIN( X, i 1.. n) i i DEG/FHC 18

19 Step 3 Instead of computing σ x from the raw data we can use the relation between the process standard deviation σ and the mean of the ranges. Multiplying factors (A 2, D 4, D 3, d 3,.., see table ) can be used to calculate the center line and the control limits. Central line Control limits Definition Formula Upper limit Lower limit Xbarchart R-chart Average on m samples of the quality characteristic averages over n units tested per sample Average on m samples of the ranges calculated for each sample of n units X m j 1 R m j 1 m m X R j j UCL X X A 2 UCL R D 4 R R LCL X X A 2 R LCL R D 3 R DEG/FHC 19

20 Factors for computing the Control limits, X-Charts, R-Charts (example of a partial table) n A2 D3 D4 d DEG/FHC 20

21 X and R charts COEFFICIENTS DEG/FHC 21

22 Run Rules Run Rules are rules that are used to indicate out-of-statistical control situations. Typical run rules for Shewhart X-charts with control and warning limits are: a point lying beyond the control limits. 2 consecutive (or 2 out 3 consecutive) points lying beyond the 2-sigma warning limits (0.025x0.025x100 = 0.06% chance of occurring). 4 out of 5 outside 1-sigma limit. 7 or more consecutive points lying on one side of the mean ( 0.57x100 = 0.8% chance of occurring and indicates a shift in the mean of the process). 5 or 6 consecutive points going in the same direction (indicates a trend). Other run rules can be formulated using similar principles. DEG/FHC 22

23 Examples of ooc: Points outside, cycles, trends, shifts (clockwise) DEG/FHC 23

24 II-Control Charts for Attributes Inlude: p-chart: a chart of the proportion defective in each sample set. np-chart: a chart of the number of defectives in each sample. A defect is a non-conformance to a specific standard. A defective part or product is one that exhibits one or more nonconformances (it has one or more defects). DEG/FHC 24

25 p-chart A process control chart that measures a proportion of defective or nonconforming items within a sample or population. An element or item under inspection may have one or more definable attributes (an attribute is an intrinsic property of a given item that either does or does not exist). If any one of the inspected attributes is nonconforming, the entire item is counted as nonconforming. The number of items in the sample that are determined to be nonconforming are summed and a proportion of the total is evaluated. DEG/FHC 25

26 P-Chart Although the distribution of sample information follows a binomial distribution, that distribution can be approximated by a normal distribution (the sample size is large, n >= 25) with a: mean of p standard deviation of p The 3s control limits are s p = [( p )(1 p)] / n UCL p = p + 3 s p LCL p = p - 3 s p DEG/FHC 26

27 p-chart Example Control Chart UCL (0.368) Proportion Observations (Sample Number) LCL (0.030) Proportion p-bar (0.199) UCL (0.368) LCL (0.030) DEG/FHC 27

28 p-chart (Example Krishnamoorthi) DEG/FHC 28

29 Collecting a dataset for a p-chart The data required for a p-chart should meet the following criteria: Subgroup Sample Size (n) 50 Sample size may be up to 100 or more, but between 50 and 100 is often adequate. Number of subgroups (number of samples taken) 25 When gathering data in the subgroup samples, it is preferable (but not mandatory) that the sample sizes be the same. If sample sizes are not the same, a different calculation will be required. DEG/FHC 29

30 p-chart with equal sample sizes With equal sample sizes, the first step requires calculating the mean subgroup proportion. This is accomplished by averaging all of the proportions calculated from each sample set: p k i 1 k P i Mean Subgroup Proportion (Equal Sample Sizes) where: P i = Sample proportion for subgroup i k = Number of samples of size n Business Statistics, 5th Edition, Groebner, et al DEG/FHC 30

31 Example dataset for a p-chart (Equal Sample Sizes; partial data:15 samples shown only) Sample Nonconforming Subgroup Sample Size Proportion The proportion of defective or nonconforming items in each sample is calculated by dividing the number defective by the sample size DEG/FHC 31

32 Example dataset for a p-chart (Equal Sample Sizes) For this example, there are 50 subgroups/samples (k) (only 15 shown on previous slides ) Applied Formula: Estimate of the sample error for subgroup proportions : where p = Mean subgroup proportion n = Common Sample Size DEG/FHC 32

33 Example dataset for a p-chart (Equal Sample Sizes) The standard error will be used to calculate the upper and lower control limits in the next step DEG/FHC 33

34 Example dataset for a p-chart (Equal Sample Sizes) With the Mean Subgroup Proportion, standard error, and upper / lower control limits determined, fill out the table with the calculated data: Partial table: Sample Nonconforming Sample Size Proportion UCL (0.359) p-bar (0.192) LCL (0.025) DEG/FHC 34

35 Example dataset for a p-chart (cont.) DEG/FHC 35

36 Evaluating the p-chart Four conditions or trends which warrant immediate attention: Five sample means in a row above or below the target or reference line. Six sample means in a row that are steadily increasing or decreasing (trending in one direction). Fourteen sample means in a row alternating above and below the target or reference line. Fifteen sample means in a row within 1 standard error of the target or reference line. DEG/FHC 36

37 Summary on control charts (in Krishnamoorthi) DEG/FHC 37

38 2-Process Capability Process capability represents the performance of a process in a state of statistical control. It is determined by the total variability that exists because of al common causes present in the system. In some sense, is a measure of the uniformity of a quality characteristic of interest. It is an index of the uniformity of the output. A common measure is given by 6σ, also called the process spread. If a normal distribution for the output quality characteristic can be assumed, 99,74% of the distribution will be 3σ to either side of the mean. Process capability can also be viewed as the variation in the product quality characteristic that remains after all special causes have been removed: the product s performance is predictable. This allows determining the ability of the product to meet customer s expectattions. DEG/FHC 38

39 Capability vs Control In Control Control Out of Control Capability Capable IDEAL Not Capable DEG/FHC 39

40 Process Capability Analysis (PCA) Process capability analysis estimates process capability. It involves estimating the process mean and standard deviation of the quality characteristic. Also, it allows estimating the form of the relative frequency distribution of the characteristic of interest. If the specification limits are known a PCA will also estimate the proportion of of nonconforming product. Natural tolerance limits (or process capability limits) are influenced or established by the process itself. represent the inherent variation in the quality characteristic of the individual items produced by a process in control. Are estimated based on the population values or, more commonly from large representative samples. DEG/FHC 40

41 PCA Assuming a normal distribution of the quality characteristic, the upper and lower natural limits are, respectively: UNTL= µ + 3σ, LNTL= µ - 3σ, where µ is the process mean, σ the process standard deviation. Technically, there might not any relationship between the process capability limits and the specification limits (USL and LSL). The former is inherent to the process; the latter is influenced or determined by the customer. DEG/FHC 41

42 PCA Three cases may occur: Process spread < specification spread, the proceses is capable. Process spread = specification spread, we have an acceptable or adequate situation (in which there is no room for error!). If the process goes out of control (change of mean, and/or of the standard deviation), then a proportion of the product will be nonconforming. Process spread > specification spread, even though the process is under control, this is an undesirable situation. The process is not capable. DEG/FHC 42

43 Process capability index Cp The process capability index is an easily measure of the goodeness of the process performance. The following indices are nondimensional (do not depend on a specific process). The Cp index: Cp = (USL-LSL) / 6σ when σ is unknown it is replaced by its estimate. The sample standard deviation s is one estimate of σ (standart deviation of the process). Cp > 1, the process is quite capable. Cp = 1, the process is adequate Cp < 1, the process is not capable Sigma is a capability estimate typically used with attribute data (i.e., with defect rates). Motorola s number of sigmas (6σ) DEG/FHC 43

44 Process capability index Cpk However, the Cp index does not tackles the lack of centering. It only measures the process variability in comparison to the specification spread. It represents the process potential. The Cpk index takes into account the lack of centering in the process. It represents the actual capability of the process with existing parameters values; it measures the process performance. Cpk= Min {(USL- µ ) / 3σ, (µ - LSL) / 3σ} Cpk > 1.33 (quite capable) Cpk = 1.00 to 1.33 (capable) Cpk < 1.00 (not capable) Desirable values are Cpk > 1 (usually, customers require 1.33 or above to cover possible drift in the process center) DEG/FHC 44

45 Example Cp,Cpk (Krishnamoorthi) DEG/FHC 45

46 Notes on Accuracy and Precision (Measurement Systems Analysis) The measurement system analysis (MSA) is done to make sure adequate measuring instruments are available for measurements to be made. And to make certain that the measuring instruments will measure the product characteristics and process parameters truthfully. Definitions: Bias: Difference between the average of readings and the true value of the measurement. Small bias, good Accuracy Precision: Measured by the std. dev. of the readings, which measures the lack of Precision Accuracy: exatidão, grau de proximidade c/ o valor real Precision: grau de variação DEG/FHC 46

47 Concepts of accuracy and precision (a note) DEG/FHC 47

48 Main references Read: First Course in Quality Engineering-Krishnamoorthi (Chaps 4,5) Quality Mgt-Goetsch/Davies (Chap 18) The Certified Manager of Quality Handbook-Westcott DEG/FHC 48