Statistical Process Control: Mastering Control Charts for Quality

Statistical Process Control SPC monitors a production process to detect and prevent poor quality. Does the process vary from what it is supposed to be doing? Key Tool is a Control Chart: • Periodically take a sample from a process • Calculate a statistic of interest from the sample • Plot the statistic on a control chart • Determine if the process is in control • Prevent quality problems Process Control Chart Signs of a process in control: a.No sample points outside limits b.Most points near process average c. About equal number of points above and below center line d.Points appear randomly distributed (no discernible pattern) Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 1 1 2 3 4 5 6 7 8 9 10 Sample number Process average Violations: b. c. d.

If the process is in control, the sampling distribution of the plotted statistic should be ~Normal (and the Empirical Rule applies) This class will always set the control limits 3 σ from the process average Types of Variation Observed Assess the chart to determine if the variation present is inherent to the process or if it needs to be removed Common Causes of Variation  ever-present factors that contribute to small, random shifts in output  difficult to track to a source Special Causes of Variation  identifiable factors that induce variation beyond the inherent variation in the system  can usually be tracked to a source Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 2  =0 1  2  3  -1  -2  -3 

Initial Construction of a Control Chart The control chart built for a process should be a benchmark for when the process is “in-control” Steps: • Decide what to measure or count Attribute Data product characteristic evaluated with a discrete choice Variable Data product characteristic that can be measured on a continuous scale • Collect the sample data • Calculate and plot the control limits & center line • Plot the samples on the control chart • Evaluate: if special-cause variation is present, discard the data and remove special-cause variation before collecting a new set of data Control Charts for Attribute Data p - charts percentage of defects found in a sample Example Sample 1: test 100 light bulbs and find that 4 don’t work c - charts count the number of defects found in an item Example Sample 1: inspect a roll of carpet and find 6 blemishes Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 3 0 . 0 8 0 . 0 7 0 . 0 6 0 . 0 5 0 . 0 4 0 . 0 3 0 . 0 2 0 . 0 1 1 2 3 4 5 Sample 8 7 6 5 4 3 2 1 1 2 3 4 5

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p - Chart Example p - Chart Formulas Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 5 total # defects total # observations p = 0.00 Sample 0 2 4 6 8 10 12 14 16 18 20 0.10 0.20 p p(1 – p) = p + Z n = p - Z p(1 – p) n n p(1 – p) Each day a sample of 100 towels is randomly selected. The number of towels found defective in the sample is recorded. Sample # # Defects 1 3 2 2 3 4 4 8 5 6 6 4 7 12 8 10 9 8 10 10 Sample # # Defects 11 13 12 11 13 14 14 11 15 9 16 16 17 12 18 15 19 18 20 18 Total = 204 UCL P LCL P where: Z = # standard deviations = standard deviation of the proportion sampling distribution = 0.1020 – 3 √ 0.1020 ( 1 − 0.1020 ) 100 = 0.01121

c - Chart Example c - Chart Formulas Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 6 0 2 4 6 8 10 12 14 0 Sample 8 16 4 12 c = c + Z c = c - Z c = 7.8 – 3 7.8 = – 0.5785 Each day a large roll of carpet is randomly selected. The number of defects found in the roll is recorded. Sample # # Defects Sample # # Defects 1 7 9 8 2 3 10 10 3 11 11 7 4 9 12 5 5 5 13 9 6 8 14 12 7 4 15 10 8 9 Total = 117 UCL C LCL C

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Control Charts for Variable Data x – Chart Uses central tendency of a sample R – Chart Uses amount of dispersion in a sample These two charts are typically used together because both the process average and process variability must be in control for the process to be in control. Washer Example R - Chart Formulas UCL R = D 4 R LCL R = D 3 R = 0 (0.062) = 0 Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 7 k R R   Observations (Washer Diameter, cm) Sample k 1 2 3 4 5 x R 1 2.95 2.95 2.91 2.94 2.92 2.934 0.04 2 2.95 2.96 2.98 2.91 2.92 2.944 0.07 3 2.94 2.94 2.90 2.89 2.94 2.922 0.05 4 2.96 2.89 2.95 2.93 2.88 2.922 0.08 5 2.91 2.89 2.96 2.97 2.95 2.936 0.08 6 2.92 2.98 2.98 2.92 2.92 2.944 0.06 7 2.97 2.95 3.00 2.92 2.94 2.956 0.08 8 2.99 3.00 2.94 2.94 2.99 2.972 0.06 9 3.02 3.00 2.98 2.99 3.01 3.000 0.04 10 2.95 2.93 2.99 2.98 2.94 2.958 0.06 Total 29.488 0.62

3σ Chart Factors x – Chart Formulas ¯ ¯ x = ∑ ¯ x k UCL X = x + A 2 R LCL X = x – A 2 R = 2.9488 – 0.58(0.062) = 2.9128 Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 9 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 1 2 3 4 5 6 7 8 9 10 Sample = = 0 Sample 0.0 3 0.0 6 0.0 9 0.1 20 0.1 5 1 2 3 4 5 6 7 8 9 10 R x-chart R-chart n A 2 D 3 D 4 2 1.88 0 3.27 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 0.31 1.69 x

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Control Chart Zones x – Chart Washer Example p - Chart Zone Example Every day, 150 units are randomly selected from the production line and determined defective or not defective. The following table shows the results from 10 samples. Sample # Defects Sample # Defects 1 14 6 16 2 17 7 13 3 12 8 14 4 15 9 11 5 14 10 15 What are the boundary values between zones A and B for this p-chart? ¯ p ± z √ ¯ p ( 1 − ¯ p ) n Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 10 2.9848 2.9488 2.9368 2.9248 2.9128 UCL LCL Zone A Zone B Zone C Zone C Zone B Zone A A B C C B A x + A 2 R = x + A 2 R = x + A 2 R = 2.9488 + 2/3 (0.58) (0.062) = 2.9728 2.9488 – 1/3 (0.58) (0.062) = 2.9488 – 2/3 (0.58) (0.062) = 2.9248 2.9488 + 1/3 (0.58) (0.062) = B/C Boundary: A/B Boundary: Control Limits: Total Defects = 141

Observations in Zones Examples Given: Collect 10 samples/day, 7 days/week (70 samples/week) If the process is in control, how often do you expect a data point in zone A? What is the probability of 15 points in a row in zone C? Event A: point 1 in zone C Event B: point 2 in zone C What is the probability of 5 points in a row in either zone A or B? Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 12 A B C C B A P(A  B) = P(A)P(B)

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Process Capability The natural variation of a process relative to the variation allowed by the design specifications. Bearing Example Bearings with Design Specification for the diameter = 1.250” ± 0.005” Process monitored with an x chart: x = 1.250” UCL= 1.256” LCL = 1.244” The Quality of Conformance can be measured by the # of standard deviations between the process average and the design specification Traditionally, a process was thought to have an acceptable level of quality if the there were at least 3σ between the xprocess average and the design specifications Since the Quality Revolution in the United States, many companies have been striving for higher levels of Quality. Such as 6σ Quality: design specifications are 6 σ from the process average Where do the control limits need to fall to have 6σ quality? Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 13 1 . 2 5 0 1 . 2 4 5 1 . 2 5 5 D es ig n S p ec s 1 . 2 5 0 1 . 2 4 5 1 . 2 5 5 D e s i g n S p e c s 1 . 2 5 0 1 . 2 4 5 1 . 2 5 5 D e s i g n S p e c s

Motorola definition: permits a process average drift of 1.5 standard deviations from the nominal value Process Capability Measures Process Capability Ratio: Potato Chip Bag Example Munchies snack food company packages potato chips in bags. The net weight of the chips in each bag is designed to be 9.0 oz with a tolerance of ± 0.5 oz. The packaging process results in bags with an average net weight ( x ) of 8.80 oz and a standard deviation of 0.12 oz. Process Capability Index: Copyright 2023 LL Clark Referenced Material : Operations & Supply Chain Management, 10 th Ed. by Russell & Taylor p. 14 upper specification limit - lower specification limit 6 C p = 9 . 0 8 . 5 5 D e s i g n S p e c s C pk = minimum 9.5 – 8.8 3 (0.12) UCL= x + 3 = LCL= x – 3 = = = = C pk = minimum upper specification limit - x 3  x - lower specification limit 3  , ,

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Q2-Statistical Process Control

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