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Statistical Process Control Used to determine whether the output of a process conforms to product or service specifications. We use control charts to detect production of defective products or services or to indicate that the production or service process has changed. We can measure items such as: Increases in defective production Decrease in service complaints Consistently low measure of units Decline in re-work or scrap

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Common Causes of Variation Purely random, unidentifiable sources of variation that are unavoidable with the current process. For example, weigh 100 bags of M&M’s. The results on a scatter gram tend to have the shape of a bell curve. Symmetric: same number of points above and below the mean. Skewed: preponderance of observations either above or below the mean.

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Common Causes Characteristics of distributions Mean—the average observation Spread—the dispersion of observations around the mean Shape—whether the observations are symmetrical or skewed Common cause variation is normally distributed (symmetrical) and stable (the mean and spread do not change over time).

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Assignable Causes of Variation Any cause of variation that can be identified and eliminated. Such as an employee that needs training or a machine that needs repair. Change in the mean, spread, or shape of a process distribution is a symptom that an assignable cause of variation has developed. After a process is in statistical control, SPC is used to detect significant change, indicating the need for corrective action.

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The Inspection Process Use of inspection to simply remove defectives is improper. It does nothing to prevent defects. We cannot “inspect” quality into a part. Quality measurements Variables—a characteristic measured on a continuous scale: such as weight, length, volume, or time. Advantage: if defective, we know by how much—the direction and magnitude of corrections are indicated. Disadvantage: precise measurements are required.

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The Inspection Process Quality measurements Attributes—a characteristic that can be counted (yes-no, integer number). Used to determine conformance to complex specifications, or when measuring variables is too costly Advantages - Quickly reveals when quality has changed, provides an integer number of how many are defective. Requires less effort and fewer resources than measuring variables. Disadvantages: Doesn’t show by how much they were defective, the direction and magnitude of corrections are not indicated, Requires more observations, as each observation provides little information

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The Inspection Process Sampling Complete inspection Used when costs of failure are high relative to costs of inspection Inspection is automated Some defects are not detected because of Inspector fatigue or Imperfect testing methods

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The Inspection Process Sampling plans Used when Inspection costs are high or Inspection destroys the product Some defectives lots may be purchased and some good lots may be rejected when The sample does not perfectly represent the population Testing methods are imperfect Sampling plans include Sample size, n random observations Time between successive samples Decision rules that determine when action should be taken

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The Inspection Process Sampling distributions Sample means are usually dispersed about the population mean according to the normal probability distribution (reference the central limit theorem described in statistics texts). Control charts Used to judge whether action is required A sample characteristic measured above the upper control limit (UCL) or below the lower control limit (LCL) indicates that an assignable cause probably exists.

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Inspection station location Purchased input materials Could use acceptance sampling Work in process Not after every process Before it is covered up Before costly, irreversible, or bottleneck operations so that resources are used efficiently Final product or service Before stocking or shipping to the customer Customers often play a major role in final inspection of services

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Normal Distribution –3 –2 –1 +1 +2 +3 Mean 68.26% 95.44% 99.74% = Standard deviation

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Control Charts UCL Nominal LCL Assignable causes likely – may also indicate improvement 1 2 3 Samples Common Causes

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Using Control Charts for Process Improvement Measure the process When changes are indicated, find the assignable cause Eliminate problems, incorporate improvements Repeat the cycle

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Indicators of out of control conditions A trend in the observations (the process is drifting) A sudden or step change in the observations A run of five or more observations on the same side of the mean (If we flip a coin and get “heads” five times in a row, we become suspicious of the coin or of the coin flipping process.) Several observations near the control limits (Normally only 1 in 20 observations are more than 2 standard deviations from the mean.)

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Control Chart Examples Nominal UCL LCL Sample number Variations In Control

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Control Chart Examples Nominal UCL LCL Sample number Variations Run – generally 5 or more, take action, even if not outside control limits

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Control Chart Examples Nominal UCL LCL Sample number Variations Sudden change – last 4 are unusual

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Control Chart Examples Nominal UCL LCL Sample number Variations Upward run

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Control Chart Examples Nominal UCL LCL Sample number Variations Out of control

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Control Limits and Errors LCL Process average UCL Three-sigma limits – the cost of searching for assignable cause is large relative to the cost of not detecting a shift in the process average. Type I error: Probability of searching for a cause when none exists Type II error: Probability of concluding that nothing has changed Shift in process average

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Control Limits and Errors Type I error: Probability of searching for a cause when none exists Two-sigma limits – the cost of not detecting a shift in the process exceeds the cost of searching for assignable causes. With smaller control limits, a shift in process average will be detected sooner. Type II error: Probability of concluding that nothing has changed Shift in process average UCL LCL Process average

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Sample Number1234 10.50140.50220.50090.5027 20.50210.50410.50240.5020 30.50180.50260.50350.5023 40.50080.50340.50240.5015 50.50410.50560.50340.5047 Special Metal Screw Control Charts for Variables

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Sample Number1234R x 10.50140.50220.50090.50270.00180.5018 20.50210.50410.50240.5020 30.50180.50260.50350.5023 40.50080.50340.50240.5015 50.50410.50560.50340.5039 0.5027 – 0.5009=0.0018 (0.5014 + 0.5022 + 0.5009 + 0.5027)/4=0.5018 0.5009 + 0.5027)/4=0.5018 Special Metal Screw _ Control Charts for Variables

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Sample Number1234R x 10.50140.50220.50090.50270.00180.5018 20.50210.50410.50240.50200.00210.5027 30.50180.50260.50350.50230.00170.5026 40.50080.50340.50240.50150.00260.5020 50.50410.50560.50340.50470.00220.5045 Special Metal Screw _

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Sample Number1234R x 10.50140.50220.50090.50270.00180.5018 20.50210.50410.50240.50200.00210.5027 30.50180.50260.50350.50230.00170.5026 40.50080.50340.50240.50150.00260.5020 50.50410.50560.50340.50470.00220.5045 R =0.0021 x =0.5027 Special Metal Screw = _ Control Charts for Variables

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Control Charts – Special Metal Screw R -Charts R = 0.0021 UCL R = D 4 R LCL R = D 3 R Control Charts for Variables

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Control Chart Factors Control Chart Factors Factor for UCLFactor forFactor Size ofand LCL forLCL forUCL for Samplex-ChartsR-ChartsR-Charts (n)(A 2 )(D 3 )(D 4 ) 21.880 03.267 31.023 02.575 40.729 02.282 50.577 02.115 60.483 02.004 70.419 0.0761.924 Control Charts for Variables

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Control Charts—Special Metal Screw R -Charts R = 0.0021 D 4 = 2.282 D 3 = 0 UCL R = 2.282 (0.0021) = 0.00479 in. LCL R = 0 (0.0021) = 0 in. UCL R = D 4 R LCL R = D 3 R Control Charts for Variables

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Control Charts—Special Metal Screw X -Charts UCL x = x + A 2 R LCL x = x - A 2 R = = R = 0.0021 x = 0.5027 = Control Charts for Variables

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Control Charts—Special Metal Screw x -Charts UCL x = 0.5027 + 0.729 (0.0021) = 0.5042 in. LCL x = 0.5027 – 0.729 (0.0021) = 0.5012 in. UCL x = x + A 2 R LCL x = x - A 2 R = = R = 0.0021 A 2 = 0.729 x = 0.5027 = Control Charts for Variables

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Measure the process Find the assignable cause Eliminate the problem Repeat the cycle Control Charts for Variables

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Control Charts for Variables Using UCL x = 5.0 + 1.96(1.5)/ 6 = 6.20 min UCL x = 5.0 – 1.96(1.5)/ 6 = 3.80 min UCL x = x + z x LCL x = x – z x x = / n== Sunny Dale Bank x =5.0 minutes =1.5 minutes n =6 customers z =1.96 =

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Control Charts for Attributes Hometown Bank UCL p = p + z p LCL p = p – z p p = p (1 – p )/ n

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Hometown Bank SampleWrong NumberAccount Number 115 212 319 4 2 519 6 4 724 8 7 910 1017 1115 12 3 Total 147 Total defectives Total observations p = n = 2500 147 12(2500) p = p = 0.0049

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Hometown Bank UCL p = p + z p LCL p = p – z p p = 0.0049(1 – 0.0049)/2500 n = 2500 p = 0.0049 Control Charts for Attributes

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Hometown Bank p = 0.0014 n = 2500 p = 0.0049 UCL p = 0.0049 + 3(0.0014) LCL p = 0.0049 – 3(0.0014) Control Charts for Attributes

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p -Chart Wrong Account Numbers Measure the process Find the assignable cause Eliminate the problem Repeat the cycle

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p-Chart comments Two things to note: The lower control limit cannot be negative. When the number of defects is less than the LCL, then the system is out of control in a good way. We want to find the assignable cause. Find what was unique about this event that caused things to work out so well.

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c = 20 z = 2 UCL c = c + z c LCL c = c – z c Control Charts for Attributes

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c = 20 z = 2 UCL c = 20 + 2 20 LCL c = 20 – 2 20 Control Charts for Attributes UCL c = 28.94 LCL c = 11.06

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Control Charts for Attributes

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Process Capability Nominal value 80010001200 Hours Upper specification Lower specification Process distribution (a) Process is capable

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Process Capability Nominal value Hours Upper specification Lower specification Process distribution (b) Process is not capable 80010001200

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Process Capability Lower specification Mean Upper specification Six sigma Four sigma Two sigma Nominal value

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Upper specification = 1200 hours Lower specification = 800 hours Average life = 900 hours = 48 hours Process Capability Ratio Cp = Upper specification - Lower specification 6 Cp = 1200 – 800 6(48) Cp = 1.39

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Process Capability Lightbulb Production Upper specification = 1200 hours Lower specification = 800 hours Average life = 900 hours = 48 hours Cp = 1.39 Cpk = Minimum of ProcessCapabilityIndex Upper specification – x 3 x – Lower specification 3 , = =

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Process Capability Lightbulb Production Upper specification = 1200 hours Lower specification = 800 hours Average life = 900 hours = 48 hours 1200 – 900 3(48) 900 – 800 3(48) ProcessCapabilityIndex, Cpk = Minimum of Cp = 1.39

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Process Capability Lightbulb Production Upper specification = 1200 hours Lower specification = 800 hours Average life = 900 hours = 48 hours Cp = 1.39 Cpk = 0.69 ProcessCapabilityIndexProcessCapabilityRatio

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Taguchi's Quality Loss Function Loss (dollars) LowerNominalUpper specificationvaluespecification

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