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ENGM 620: Quality Management Session 8 – 23 October 2012 Control Charts, Part I –Variables

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Statistical Thinking All work occurs in a system of interconnected processes All process have variation Understanding variation and reducing variation are important keys to success

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Variability A certain amount of variability is inescapable Therefore, no two products are identical The larger the variability, the greater the probability that the customer will perceive its existence

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Sources of Variability Include: Differences in materials Differences in the performance and operation of the manufacturing equipment Differences in the way the operators perform their tasks

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Variability and Statistics Variability is difference from the target Characteristics of quality must be measurable Therefore, Variability is described in statistical terms We will use statistical methods in our quality improvement activities

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Recall: Types of Errors Type I error –Producers risk –Probability that a good product will be rejected Type II error –Consumers risk –Probability that a nonconforming product will be available for sale Type III error –Asking the wrong question

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Types of Errors H O H A HOHAHOHA Truth Accept No Error Type I Type II No Error

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A Parable Where should we put the additional armor?

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Data on Quality Characteristics Attribute data –Discrete –Often a count of some type Variable data –Continuous –Often a measurement, such as length, voltage, or viscosity

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Terms Specifications Target (or Nominal) Value Upper Specification Limit Lower Specification Limit Random Variation Non-random Variation Process stability

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Terms Nonconforming: failure to meet one or more of the specifications Nonconformity: a specific type of failure Defect: a nonconformity serious enough to significantly affect the safe or effective use of the produce or completion of the service

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Nonconforming vs. Defective A nonconforming product is not necessarily unfit for use A nonconforming product is considered defective if if it has one or more defects

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Classroom Exercise For a product or service in your job: –Name a quality characteristic –Give an example of a nonconformity that is not a defect –Give an example of a defect

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Types of Inspection Receiving In Process Final None One Hundred Percent Acceptance Sampling

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Quality Design & Process Variation 6080100120140 60140 60 Lower Spec Limit Upper Spec Limit Acceptance Sampling Statistical Process Control Experimental Design

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Variation and Control A process that is operating with only common causes of variation is said to be in statistical control. A process operating in the presence of special or assignable cause is said to be out of control.

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Finding Trends and Special Causes Inspection does not tell you about a problem until it becomes a problem We need a mechanism to help us spot special causes when they occur We need mechanism to help us determine when we have a trend in the data

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Statistical Process Control Originally developed by Walter Shewhart in 1924 at the Bell Telephone Laboratories Late 1920s, Harold Dodge and Harry Romig developed statistically based acceptance sampling Not recognized by industry until after World War II

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Definition Statistical Process Control (SPC): –“a methodology for monitoring a process to identify special causes of variation and signal the need to take corrective action when it is appropriate” (Evans and Lindsay)

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Statistical Process Control Tools The magnificent seven The tool most often associated with Statistical Process Control is Control Charts

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Common Causes Special Causes

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Histograms do not take into account changes over time. Control charts can tell us when a process changes

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Control Chart Applications Establish state of statistical control Monitor a process and signal when it goes out of control Determine process capability Note: Control charts will only detect the presence of assignable causes. Management, operator, and engineering action is necessary to eliminate the assignable cause.

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Capability Versus Control Control Capability Capable Not Capable In Control Out of Control IDEAL

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Commonly Used Control Charts Variables data –x-bar and R-charts –x-bar and s-charts –Charts for individuals (x-charts) Attribute data –For “defectives” (p-chart, np-chart) –For “defects” (c-chart, u-chart)

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Control Charts We assume that the underlying distribution is normal with some mean and some constant but unknown standard deviation . Let x x n i i n 1

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Distribution of x Recall that x is a function of random variables, so it also is a random variable with its own distribution. By the central limit theorem, we know that where, xN x (,) x n x

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Control Charts x x x x

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Control Charts x UCL LCL UCL & LCL Set at Problem: How do we estimate & ? 3 x

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Control Charts x x m m i i 1 )( 1 f m R R m i

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x x RALCL 2 x x RAUCL 2 RDLCL R 3 RDUCL R 4 x x m m i i 1 )( 1 f m R R m i

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Example Suppose specialized o-rings are to be manufactured at.5 inches. Too big and they won’t provide the necessary seal. Too little and they won’t fit on the shaft. Twenty samples of 2 rings each are taken. Results follow.

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X-Bar Control Charts X-bar charts can identify special causes of variation, but they are only useful if the process is stable (common cause variation).

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Control Limits for Range UCL = D 4 R = 3.268*.002 =.0065 LCL = D 3 R = 0

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37 Why Monitor Both Process Mean and Process Variability? X-barR R R Process Over TimeControl Charts

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38 Teminology Causes of Variation: –Assignable Causes Keep the process from operating predictably Things that we can do something about –Common / Chance Causes Random, inherent variation in the process Meaning of Control: –In Specification Meets customer constraints on product –In Statistical Control No Assignable Causes of variation present in the process

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Typical Out-of-Control Patterns Point outside control limits Hugging the center line Hugging the control limits Instability Sudden shift in process average Cycles Trends

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Shift in Process Average

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Identifying Potential Shifts

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Cycles

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Trend

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Western Electric Sensitizing Rules: One point plots outside the 3-sigma control limits Two of three consecutive points plot outside the 2-sigma warning limits Four of five consecutive points plot beyond the 1-sigma limits A run of eight consecutive points plot on one side of the center line

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Additional sensitizing rules: Six points in a row are steadily increasing or decreasing Fifteen points in a row with 1-sigma limits (both above and below the center line) Fourteen points in a row alternating up and down Eight points in a row in both sides of the center line with none within the 1-sigma limits An unusual or nonrandom pattern in the data One of more points near a warning or control limit

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Special Variables Control Charts x-bar and s charts x-chart for individuals

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X-bar and S charts Allows us to estimate the process standard deviation directly instead of indirectly through the use of the range R S chart limits: –UCL = B 6 σ = B 4 *S-bar –Center Line = c 4 σ = S-bar –LCL = B 5 σ = B 3 *S-bar X-bar chart limits –UCL = X-doublebar +A 3 S-bar –Center line = X-doublebar –LCL = X-doublebar -A 3 S-bar

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X-chart for individuals UCL = x-bar + 3*(MR-bar/d 2 ) Center line = x-bar LCL = x-bar - 3*(MR-bar/d 2 )

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Developing Control Charts 1.Prepare –Choose measurement –Determine how to collect data, sample size, and frequency of sampling –Set up an initial control chart 2.Collect Data –Record data –Calculate appropriate statistics –Plot statistics on chart

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Next Steps 3.Determine trial control limits –Center line (process average) –Compute UCL, LCL 4.Analyze and interpret results –Determine if in control –Eliminate out-of-control points –Re-compute control limits as necessary

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Final Steps 5.Use as a problem-solving tool –Continue to collect and plot data –Take corrective action when necessary 6.Compute process capability

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Final Steps 5.Use as a problem-solving tool –Continue to collect and plot data –Take corrective action when necessary 6.Compute process capability

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Next Class Homework –Ch. 11 Disc. Questions 5, 7 –Ch. 11 Problems 6, 11 Preparation –Chapter 11, Process Capability

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