Créer une présentation
Télécharger la présentation

Télécharger la présentation
## ENGM 720 - Lecture 08

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**ENGM 720 - Lecture 08**P, NP, C, & U Control Charts ENGM 720: Statistical Process Control**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 ENGM 720: Statistical Process Control**Assignment:**• Reading: • Chapter 6 • Finish reading • Chapter 7 • Sections 7.1 and 7.2 throughp.313 • Sections 7.3 through p.325 • Sections 7.3.2 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 ENGM 720: Statistical Process Control**Statistical Quality Control and Improvement**Improving Process Capability and Performance Continually Improve the System Characterize Stable Process Capability Head Off Shifts in Location, Spread Time Identify Special Causes - Bad (Remove) Identify Special Causes - Good (Incorporate) Reduce Variability Center the Process LSL 0 USL 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” ENGM 720: Statistical Process Control**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 ENGM 720: Statistical Process Control**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 2.5 35.0 2.5 30.4 (-3) 34.8 (-) 39.2 (+) 43.6 (+3) 30 34 38 42 32.6 (-2) 37 () 41.4 (+2) 32 36 () 40 Continuous & Discrete Distributions ENGM 720: Statistical Process Control**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 ENGM 720: Statistical Process Control**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 ENGM 720: Statistical Process Control**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 ENGM 720: Statistical Process Control**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: ENGM 720: Statistical Process Control**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? ENGM 720: Statistical Process Control**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: ENGM 720: Statistical Process Control**Poisson Distribution Example**• A sheet of 4’x8’ paneling (= 4608 in2) 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 in2? ENGM 720: Statistical Process Control**UCL**0 CL LCL 0 Sample Number 2-Sided Hypothesis Test Sideways Hypothesis Test Shewhart Control Chart 2 2 2 2 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 ENGM 720: Statistical Process Control**Sample Control Limits:**Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard: P-Charts • Tracks proportion defective in a sample of insp. units • Can have a constant number of inspection units in the sample ENGM 720: Statistical Process Control**Mean Sample Size Limits:**Approximate 3σ limits are found from sample mean: Variable Width Limits: Approximate 3σ limits vary with individual sample size: P-Charts (continued) • More commonly has variable number of inspection units • Can’t use run rules with variable control limits ENGM 720: Statistical Process Control**Sample Control Limits:**Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard: NP-Charts • 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 ! ENGM 720: Statistical Process Control**Sample Control Limits:**Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard: C-Charts • 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 ! ENGM 720: Statistical Process Control**Mean Sample Size Limits:**Approximate 3σ limits are found from sample mean: Variable Width Limits: Approximate 3σ limits vary with individual sample size: U-Charts • Number of defects occurring in variably sized inspection unit • (Ex. Solder defects per 100 joints - 350 joints in board = 3.5 insp. units) • Can’t use run rules with variable control limits, watch clustering! ENGM 720: Statistical Process Control**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: • Investigate the process; • Remove the special cause; and • Remove the special cause point and recalculate control limits. • If can’t find special cause - drop point & recalculate anyway ENGM 720: Statistical Process Control**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 Summary of Control Charts • Use of the control chart decision table. ENGM 720: Statistical Process Control**Control Chart Decision Table**Is the size of the inspection sample fixed? Defective Units (possibly with multiple defects) Binomial Distribution Use p-Chart No, varies Use np-Chart Yes, constant Is the size of the inspection unit fixed? What is the inspection basis? Individual Defects Poisson Distribution Use c-Chart Discrete Attribute Yes, constant Kind of inspection variable? Use u-Chart No, varies Which spread method preferred? Range Use X-bar and R-Chart Continuous Variable Standard Deviation Use X-bar and S-Chart ENGM 720: Statistical Process Control**Control Chart Sensitizing Rules**• Western Electric Rules: • One point plots outside the three-sigma limits; • Eight consecutive points plot on one side of the center line (run rule!); • Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line(zone rule!); • 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 ENGM 720: Statistical Process Control**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 ENGM 720: Statistical Process Control**Questions & Issues**ENGM 720: Statistical Process Control