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**Statistical Process Control**

6 SUPPLEMENT PowerPoint presentation to accompany Heizer and Render Operations Management, Eleventh Edition Principles of Operations Management, Ninth Edition PowerPoint slides by Jeff Heyl © 2014 Pearson Education, Inc.

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**Learning Objectives Apply quality management tools for problem solving**

Identify the importance of data in quality management 6S–2

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**Statistical Quality Control**

Introduction Statistical Quality Control Statistical Process Control (SPC) Acceptance Sampling (AS) Statistical process control is a statistical technique that is widely used to ensure that the process meets standards. Acceptance sampling is used to determine acceptance or rejection of material evaluated by a sample. 6S–3

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**Preparing the clay for throwing**

Introduction Pottery Making Process Pinching pots Preparing the clay for throwing Wedging Throwing Painting Firing 6S–4

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Introduction 6S–5

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**Statistical Process Control Chart (SPC)**

Variability is inherent in every process. Natural variation – can not be eliminated Assignable variation -- Deviation that can be traced to a specific reason: machine vibration, tool wear, new worker. Variation Natural Variation Assignable Variation 6S–6

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**Statistical Process Control Chart (SPC)**

The essence of SPC is the application of statistical techniques to prevent, detect, and eliminate defective products or services by identifying assignable variation. 6S–7

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**Statistical Process Control Chart (SPC)**

A control chart is a time-ordered plot obtained from an ongoing process Abnormal variation due to assignable sources Out of control UCL LCL Natural variation due to chance Mean Abnormal variation due to assignable sources 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Sample number 6S–8

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**Statistical Process Control Chart (SPC)**

Control Charts for Variable Data Control Charts for Attribute Data -charts (for controlling central tendency) R-charts (for controlling variation) p-charts (for controlling percent defective) c-charts (for controlling number of defects) Variable Data (continuous): quantifiable conditions along a scale, such as speed, length, density, etc. Attribute Data (discrete): qualitative characteristic or condition, such as pass/fail, good/bad, go/no go. 6S–9

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**Statistical Process Control Chart (SPC)**

Take random samples Calculate the upper control limit (UCL) and the lower control limit (LCL) Plot UCL, LCL and the measured values If all the measured values fall within the LCL and the UCL, then the process is assumed to be in control and no actions should be taken except continuing to monitor. If one or more data points fall outside the control limits, then the process is assumed to be out of control and corrective actions need to be taken.

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**x-Charts Upper control limit (UCL) = x + A2R**

Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor from Table S6.1(page 241) x = average of the sample means 6S–11

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**x-Charts Range=18-13=5 Range=17-14=3 R = (5+3)/2 = 4 Hour 1 Hour 2**

Box Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 5 17 6 16 7 15 8 17 9 16 Hour 2 Box Weight of Number Oat Flakes 1 14 2 16 3 15 4 14 5 17 6 15 7 15 8 14 9 17 Range=18-13=5 Range=17-14=3 R = (5+3)/2 = 4 6S–12

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**x-Charts Upper control limit (UCL) = x + A2R**

Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor from Table S6.1 (page241) x = average of the sample means 6S–13

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**x-Charts x = (16.11+15.22)/2 = 15.665 Hour 1 Hour 2**

Box Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 5 17 6 16 7 15 8 17 9 16 Hour 2 Box Weight of Number Oat Flakes 1 14 2 16 3 15 4 14 5 17 6 15 7 15 8 14 9 17 Average=(17+13+…+16)/9=16.11 Average=(14+16+…+17)/9=15.22 x = ( )/2 = 6S–14

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**x-Charts Upper control limit (UCL) = x + A2R**

Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor from Table S6.1 (page241) x = average of the sample means 6S–15

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**x-Charts Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3**

6S–16

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**x-Charts Upper control limit (UCL) = x + A2R**

Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor from Table S6.1 (page241) x = average of the sample means 6S–17

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x-Charts Example S6.1: Eight samples of seven tubes were taken at random intervals. Construct the x-chart with 3- control limit. Is the current process under statistical control? Why or why not? Should any actions be taken? Sample size = n = 7 A2 = ? 6S–18

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**x-Charts Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3**

6S–19

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x-Charts Example S6.1: Eight samples of seven tubes were taken at random intervals. Construct the x-chart with 3- control limit. Is the current process under statistical control? Why or why not? Should any actions be taken? A2 = 0.42 6S–20

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**x-Charts Control Chart for sample of 7 tubes 6.43 = UCL 6.36 = Mean**

6.29 = LCL 6.36 = Mean | | | | | | | | | | | | Sample number It is assumed that the central tendency of process is in control with 99.73% confidence. No actions need to be taken except to continuously monitor this process. 6S–21

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**Steps in Creating Charts**

Take samples from the population and compute the appropriate sample statistic Use the sample statistic to calculate control limits Plot control limits and measured values Determine the state of the process (in or out of control) Investigate possible assignable causes and take actions 6S–22

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**R-Charts Upper control limit (UCL) = D4R**

Lower control limit (LCL) = D3R where R = average range of the samples D3 and D4 = control chart factors from Table S6.1 (Page 241) 6S–23

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**R-Charts Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3**

6S–24

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**R-Charts Example S6.2 Average range R = 5.3 pounds Sample size n = 5**

From Table S6.1 D4 = ? D3 = ? 6S–25

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**R-Charts Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3**

6S–26

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**R-Charts Example S6.2 Average range R = 5.3 pounds Sample size n = 5**

From Table S6.1 D4 = 2.12, D3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0 UCLR = D4R = (2.12)(5.3) = 11.2 pounds LCLR = D3R = (0)(5.3) = 0 pounds 6S–27

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R-Charts Example S6.3: Refer to Example S6.1. Eight samples of seven tubes were taken at random intervals. Construct the R-chart with 3- control limits. Is the current process under statistical control? Why or why not? Should any actions be taken? n=7 D3 =? D4 = ? 6S–28

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**R-Charts Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3**

6S–29

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R-Charts Example S6.3: Refer to Example S6.1. Eight samples of seven tubes were taken at random intervals. Construct the R-chart with 3- control limits. Is the current process under statistical control? Why or why not? Should any actions be taken? D3 =0.08, D4 = 1.92 6S–30

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**R-Charts Control Chart for sample of 7 tubes 0.33 = UCL 0.17 = R**

0.01 = LCL 0.17 = R | | | | | | | | | | | | Sample number The variation of process is in control with 99.73% confidence. 6S–31

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Mean and Range Charts (a) The central tendency of process is in control, but its variation is not in control. R-chart (R-chart detects increase in dispersion) UCL LCL x-chart (x-chart does not detect dispersion) UCL LCL 6S–32

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Mean and Range Charts (b) The variation of process is in control, but its central tendency is not in control. UCL (R-chart does not detect changes in mean) R-chart LCL x-chart (x-chart detects shift in central tendency) UCL LCL 6S–33

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R-Chart and X-Chart Example S6.4: Seven random samples of four resistors each are taken to establish the quality standards. Develop the R-chart and the x-chart both with 3- control limits for the production process. Is the entire process under statistical control? Why or why not? n = 4 D3 = 0, and D4 = 2.28 R = ( … + 4)/7 = 3.0 6S–34

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**R-Chart and X-Chart Control Chart for sample of 4 resistors 6.84 = UCL**

0 = LCL | | | | | | | | | | | | Sample number The variation of process is in control with 99.73% confidence. 6S–35

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**R-Chart and X-Chart n = 4, A2 = 0.73 R = (3 + 2 + … + 4)/7 = 3.0**

6S–36

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**R-Chart and X-Charts Control Chart 101.98 = UCL 99.79 = Mean**

97.6 = LCL | | | | | | | | | | | | Sample number The central tendency of process is not in control with 99.73% confidence. In conclusion, with 99.7% confidence, the entire resistor production process is not in control since its central tendency is out of control although its variation is under control. 6S–37

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EX 1 in class A part that connects two levels should have a distance between the two holes of 4”. It has been determined that x-bar chart and R-chart should be set up to determine if the process is in statistical control. The following ten samples of size four were collected. Calculate the control limits, plot the control charts, and determine if the process is in control No. of Sample Mean Range 1 4.01 0.04 2 3.98 0.06 3 4.00 0.02 4 3.99 0.05 5 6 3.97 7 4.02 8 9 10 6S–38

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**R-Chart and X-Chart # of sample Readings of Resistance (ohms) 1 99 100**

Example S6.5: Resistors for electronic circuits are manufactured at Omega Corporation in Denton, TX. The head of the firm’s Continuous Improvement Division is concerned about the product quality and sets up production line checks. She takes seven random samples of four resistors each to establish the quality standards. Develop the R-chart and the chart both with 3- control limits for the production process. Is the entire process under statistical control? Why or why not? # of sample Readings of Resistance (ohms) 1 99 100 102 101 2 103 3 98 4 5 6 95 97 96 7 6S–39

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**R-Chart and X-Chart D3 =0 D4 = 2.28 n=4 # of Sample 1 2 3 4 5 6 7**

Sample range Sample mean 100.5 101.5 100.0 99.5 99.0 97.0 101.0 D3 =0 D4 = 2.28 n=4 6.84 = UCL variation of process is in control with 99.73% confidence. 3.0 = R 0 = LCL | | | | | | | | | | | | Sample number 6S–40

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**R-Chart and X-Chart A2 =0.73 n=4 X= (100.5 + … + 101.0)/7 99.8**

# of Sample 1 2 3 4 5 6 7 Sample range Sample mean 100.5 101.5 100.0 99.5 99.0 97.0 101.0 X= ( … )/7 99.8 A2 =0.73 n=4 102.0 = UCL central tendency of process is not in control with 99.73% confidence. 99.8 = X 97.6 = LCL Thus, entire process is not in control. | | | | | | | | | | | | Sample number 6S–41

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EX 2 in class A quality analyst wants to construct a sample mean chart for controlling a packaging process. Each day last week, he randomly selected four packages and weighed each. The data from that activity appears below. Set up control charts to determine if the process is in statistical control Day Package 1 Package 2 Package 3 Package 4 Monday 23 22 24 Tuesday 21 19 Wednesday 20 Thursday 18 Friday 6S–42

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**Statistical Process Control Chart (SPC)**

Control Charts for Variable Data Control Charts for Attribute Data -charts (for controlling central tendency) R-charts (for controlling variation) p-charts (for controlling percent defective) c-charts (for controlling number of defects) Variable Data (continuous): quantifiable conditions along a scale, such as speed, length, density, etc. Attribute Data (discrete): qualitative characteristic or condition, such as pass/fail, good/bad, go/no go. 6S–43

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**Control Charts for Attribute Data**

Categorical variables Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure Percentage of defects (p-chart) Number of defects (c-chart) 6S–44

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**P-Charts where p = mean percent defective overall the samples**

z = number of standard deviations = 3 n = sample size 6S–45

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P-Charts Example S6.6: Data-entry clerks at ARCO key in thousands of insurance records each day. One hundred records entered by each clerk were carefully examined and the number of errors counted. Develop a p-chart with 3- control limits and determine if the process is in control. Sample Number Percent Sample Number Percent Number of Errors Defective Number of Errors Defective Total = 80 6S–46

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**P-Charts n = 100 Because we cannot have a negative percent defective**

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**P-Charts Possible assignable causes present**

.11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – Sample number Percent defective UCL= 0.10 LCL= 0.00 p = 0.04 | | | | | | | | | | Possible good assignable causes present The process is not in control with 99.73% confidence. 6S–48

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C-Charts A c-chart is used when the quality cannot be measured as a percentage. Number of car accidents per month at a particular intersection Number of complaints the service center of a hotel receives per week Number of scratches on a nameplate Number of dimples found on a metal sheet 6S–49

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**C-Charts UCL = c + 3 c LCL = c - 3 c**

where c = mean number defective overall the samples 6S–50

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**C-Charts UCL = c + 3 c = 6 + 3 6 = 13.35 LCL = c - 3 c = 6 - 3 6**

Example S6.7: Over 9 weeks, Red Top Cab company received the following numbers of calls from irate passengers: 3, 0, 8, 9, 6, 7, 4, 9, 8, for a total of 54 complaints. Determine the 3- control limits of a c-Chart. | 1 2 3 4 5 6 7 8 9 Week Number Number of defect 14 – 12 – 10 – 8 – 6 – 4 – 2 – 0 – c = 54 / 9 = 6 complaints /week UCL = 13.35 LCL = 0 c = 6 UCL = c + 3 c = = 13.35 LCL = c - 3 c = = => 0 The process is in control with 99.73% confidence. Because we cannot have the negative number of defective records 6S–51

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**Managing Quality Summary**

Effective quality management is data driven There are multiple tools to identify and prioritize process problems There are multiple tools to identify the relationships between variables 6S–52

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