11/23/2015ENGM 720: Statistical Process Control1 ENGM Lecture 08 P, NP, C, & U Control Charts

11/23/2015 ENGM 720: Statistical Process Control 2 Outline Assignment Discrete Distributions and Probability of Outcomes Examples of discrete distributions Hypothesis Testing to Control Charts P- & NP-Charts C- & U-Charts Summary of Control Chart Options Using the Control Chart Decision Chart

11/23/2015 ENGM 720: Statistical Process Control 3 Assignment: Reading: Chapter 6 Finish reading Chapter 7 Sections 7.1 and 7.2 through p.313 Sections 7.3 through p.325 Sections and 7.5 Assignments: Obtain the Control Chart Factors table from Materials Page Access Excel Template for X-bar, R, & S Control Charts: Download Assignment 5 for practice Use the data on the HW5 Excel sheet to do the charting, verify the control limits by hand calculations Access Excel Template for P, NP, C, & U Control Charts

11/23/2015 ENGM 720: Statistical Process Control 4 Process for Statistical Control Of Quality Removing special causes of variation Hypothesis Tests Ishikawa’s Tools Managing the process with control charts Process Improvement Process Stabilization Confidence in “When to Act” Reduce Variability Identify Special Causes - Good (Incorporate) Improving Process Capability and Performance Characterize Stable Process Capability Head Off Shifts in Location, Spread Identify Special Causes - Bad (Remove) Continually Improve the System Statistical Quality Control and Improvement Time Center the Process LSL  0 USL

11/23/2015 ENGM 720: Statistical Process Control 5 Review Shewhart Control charts Are like a sideways hypothesis test (2-sided!) from a Normal distribution UCL is like the right / upper critical region CL is like the central location LCL is like the left / lower critical region When working with continuous variables, we use two charts: X-bar for testing for change in location R or s-chart for testing for change in spread We check the charts using 4 Western Electric rules

11/23/2015 ENGM 720: Statistical Process Control 6 Continuous & Discrete Distributions Continuous Probability of a range of outcomes is area under PDF (integration) Discrete Probability of a range of outcomes is area under PDF (sum of discrete outcomes) 35.0  (  ) 41.4 (  +2  ) 32.6 (  -2  ) 43.6 (  +3  ) 30.4 (  -3  ) 39.2 (  +  ) 34.8 (  -  ) 35.0  (  )

11/23/2015 ENGM 720: Statistical Process Control 7 Discrete Distribution Example Sum of two six-sided dice: Outcomes range from 2 to 12. Count the possible ways to obtain each individual sum - forms a histogram What is the most frequently occurring sum that you could roll? Most likely outcome is a sum of 7 (there are 6 ways to obtain it) What is the probability of obtaining the most likely sum in a single roll of the dice? 6  36 =.167 What is the probability of obtaining a sum greater than 2 and less than 11? 32  36 =.889

11/23/2015 ENGM 720: Statistical Process Control 8 Continuous & Attribute Variables Continuous Variables: Take on a continuum of values. Ex.: length, diameter, thickness Modeled by the Normal Distribution Attribute Variables: Take on discrete values Ex.: present/absent, conforming/non-conforming Modeled by Binomial Distribution if classifying inspection units into defectives (defective inspection unit can have multiple defects) Modeled by Poisson Distribution if counting defects occurring within an inspection unit

11/23/2015 ENGM 720: Statistical Process Control 9 Discrete Variables Classes Defectives The presence of a non-conformity ruins the entire unit – the unit is defective Example – fuses with disconnects Defects The presence of one or more non- conformities may lower the value of the unit, but does NOT render the entire unit defective Example – paneling with scratches

11/23/2015 ENGM 720: Statistical Process Control 10 Binomial Distribution Sequence of n trials Outcome of each trial is “success” or “failure” Probability of success = p r.v. X - number of successes in n trials So: where Mean: Variance:

11/23/2015 ENGM 720: Statistical Process Control 11 Binomial Distribution Example A lot of size 30 contains three defective fuses. What is the probability that a sample of five fuses selected at random contains exactly one defective fuse? What is the probability that it contains one or more defectives?

11/23/2015 ENGM 720: Statistical Process Control 12 Poisson Distribution Let X be the number of times that a certain event occurs per unit of length, area, volume, or time So: where x = 0, 1, 2, … Mean:Variance:

11/23/2015 ENGM 720: Statistical Process Control 13 Poisson Distribution Example A sheet of 4’x8’ paneling (= 4608 in 2 ) has 22 scratches. What is the expected number of scratches if checking only one square inch (randomly selected)? What is the probability of finding at least two scratches in 25 in 2 ?

11/23/2015 ENGM 720: Statistical Process Control 14 Moving from Hypothesis Testing to Control Charts Attribute control charts are also like a sideways hypothesis test Detects a shift in the process Heads-off costly errors by detecting trends – if constant control limits are used 00 22 22 00 22 22 2-Sided Hypothesis TestShewhart Control ChartSideways Hypothesis Test CLCL LCL UCL Sample Number

11/23/2015 ENGM 720: Statistical Process Control 15 P-Charts Sample Control Limits: Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard:  Tracks proportion defective in a sample of insp. units  Can have a constant number of inspection units in the sample

11/23/2015 ENGM 720: Statistical Process Control 16 P-Charts (continued) Mean Sample Size Limits: Approximate 3σ limits are found from sample mean: Variable Width Limits: Approximate 3σ limits vary with individual sample size:  More commonly has variable number of inspection units  Can’t use run rules with variable control limits

11/23/2015 ENGM 720: Statistical Process Control 17 NP-Charts Sample Control Limits: Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard:  Tracks number of defectives in a sample of insp. units  Must have a constant number of inspection units in each sample  Use of run rules is allowed if LCL > 0 - adds power !

11/23/2015 ENGM 720: Statistical Process Control 18 C-Charts Sample Control Limits: Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard:  Tracks number of defects in a logical inspection unit  Must have a constant size inspection unit containing the defects  Use of run rules is allowed if LCL > 0 - adds power !

11/23/2015 ENGM 720: Statistical Process Control 19 U-Charts Mean Sample Size Limits: Approximate 3σ limits are found from sample mean: Variable Width Limits: Approximate 3σ limits vary with individual sample size:  Number of defects occurring in variably sized inspection unit  (Ex. Solder defects per 100 joints joints in board = 3.5 insp. units)  Can’t use run rules with variable control limits, watch clustering!

11/23/2015 ENGM 720: Statistical Process Control 20 Steps for Trial Control Limits Start with 20 to 25 samples Use all data to calculate initial control limits Plot each sample in time-order on chart. Check for out of control sample points If one (or more) found, then: 1.Investigate the process; 2.Remove the special cause; and 3.Remove the special cause point and recalculate control limits. If can’t find special cause - drop point & recalculate anyway

11/23/2015 ENGM 720: Statistical Process Control 21 Summary of Control Charts Continuous Variable Charts Smaller changes detected faster Apply to attributes data as well (by CLT)* Require smaller sample sizes Attribute Charts Can cover several defects with one chart Less costly inspection  Use of the control chart decision table.

11/23/2015 ENGM 720: Statistical Process Control 22 Use p-Chart No, varies Yes, constant Use np-Chart Individual Defects Poisson Distribution Use c-Chart Use u-Chart No, varies Discrete Attribute What is the inspection basis? Is the size of the inspection unit fixed? Yes, constant Is the size of the inspection sample fixed? Continuous Variable Range Standard Deviation Which spread method preferred? Use X-bar and R-Chart Use X-bar and S-Chart Kind of inspection variable? Defective Units (possibly with multiple defects) Binomial Distribution Control Chart Decision Table

11/23/2015 ENGM 720: Statistical Process Control 23 Control Chart Sensitizing Rules Western Electric Rules: 1. One point plots outside the three-sigma limits; 2. Eight consecutive points plot on one side of the center line (run rule!); 3. Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line (zone rule!); 3. Four out of five consecutive points plot beyond one-sigma warning limits on the same side of the center line (zone rule!). If chart shows lack of control, investigate for special cause

11/23/2015 ENGM 720: Statistical Process Control 24 Attribute Chart Applications Attribute control charts apply to “service” applications, too. Number of incorrect invoices per customer Proportion of incorrect orders taken in a day Number of return service calls to resolve problem

11/23/2015 ENGM 720: Statistical Process Control 25 Questions & Issues