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**Statistical Process Control (SPC)**

Stephen R. Lawrence Assoc. Prof. of Operations Mgmt Leeds.colorado.edu/faculty/lawrence

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Process Control Tools

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Process Control Tools Process tools assess conditions in existing processes to detect problems that require intervention in order to regain lost control. Check sheets Pareto analysis Scatter Plots Histograms Run Charts Control charts Cause & effect diagrams

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**Check Sheets 27 15 19 20 28 Check sheets explore what and where**

an event of interest is occurring. Attribute Check Sheet Order Types am-9am 9am-11am 11am-1pm pm-3pm 3pm-5-pm Emergency Nonemergency Rework Safety Stock Prototype Order Other

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Run Charts measurement time Look for patterns and trends…

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**SCATTERPLOTS Variable A Variable B**

x x x x x x xx x x x x xx x x x x x xx x xx x x x x xx x x x x x xx x x x xx xxx x x x x xx xx x x x x xx x x x xxx xx x x x x xxx x x xx x x xx xx x x x x x xx x x xxx xx xx xxx x x xx xxx x x x x x x xx x x x x x x xx x x xx x x xx x x x Variable A Larger values of variable A appear to be associated with larger values of variable B. Variable B

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**Frequency of Occurrences**

HISTOGRAMS A statistical tool used to show the extent and type of variance within the system. Frequency of Occurrences Outcome

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**A B C D E F G PARETO ANALYSIS H Percentage of Occurrences**

A method for identifying and separating the vital few from the trivial many. A B Percentage of Occurrences C D E F G H I J Factor

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**CAUSE & EFFECT DIAGRAMS**

Classification Error Inspection Pins not Assigned Defective Pins Received Damaged in storage CPU Chip BAD CPU Maintenance Equipment Condition Employees Procedures and Methods Training Speed

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**Example: Rogue River Adventures**

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Process Variation

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**Deming’s Theory of Variance**

Variation causes many problems for most processes Causes of variation are either “common” or “special” Variation can be either “controlled” or “uncontrolled” Management is responsible for most variation Categories of Variation Management Employee Controlled Variation Uncontrolled Variation Common Cause Special Cause

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**Causes of Variation What prevents perfection? Process variation...**

Natural Causes Assignable Causes Inherent to process Random Cannot be controlled Cannot be prevented Examples weather accuracy of measurements capability of machine Exogenous to process Not random Controllable Preventable Examples tool wear “Monday” effect poor maintenance Common causes: * Inherit to the process * Random * Not controllable by operators * Management is responsible (e.g., a filling cereal machine may be replaced by a better one) Assignable causes: * Not part of the process * Not random * Operators have control * Management is responsible for training operators. (e.g., a machine is not properly set)

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**Specification vs. Variation**

Product specification desired range of product attribute part of product design length, weight, thickness, color, ... nominal specification upper and lower specification limits Process variability inherent variation in processes limits what can actually be achieved defines and limits process capability Process may not be capable of meeting specification!

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Process Capability

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**Process Capability Capable process (Very) capable process**

Process variation LSL Spec USL Capable process (Very) capable process Out of control - The process is out of control because the distribution is not centered around the target value. The assignable cause may be a wrong setting for the particularned. Not capable - The process is in control but is not capable. This is determined by the large probability of producing parts that do not meet the engineering index (called Cp) is less than 1. Cp has a value of one when the range of the distribution (measure by standard deviations from the mean) equals the range of the tolerances. Capable process have a Cp value of at least 1.3. Capable - The process is now capable because the entire distribution falls within the tolerance limits. Improvements occur when the range of the distribution is reduce, increasing the probability for producing on target. Capability - It is the ability to meet customer’s specifications. In control - A process that does not show signs of assignable causes of variation. Process not capable

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**Process Capability 6 99.7% 3 LSL Spec USL**

Measure of capability of process to meet (fall within) specification limits Take “width” of process variation as 6 If 6 < (USL - LSL), then at least 99.7% of output of process will fall within specification limits LSL Spec USL 6 99.7% 3

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**Variation -- RazorBlade**

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**Process Capability Ratio**

Define Process Capability Ratio Cp as If Cp > 1.0, process is... capable If Cp < 1.0, process is... not capable

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**Process Capability -- Example**

A manufacturer of granola bars has a weight specification 2 ounces plus or minus 0.05 ounces. If the standard deviation of the bar-making machine is 0.02 ounces, is the process capable? USL = = 2.05 ounces LSL = = 1.95 ounces Cp = (USL - LSL) / 6 = ( ) / 6(0.02) = / 0.12 = Therefore, the process is not capable!

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**Process Centering Capable and centered Capable, but not centered**

LSL Spec USL Capable and centered Capable, but not centered Out of control - The process is out of control because the distribution is not centered around the target value. The assignable cause may be a wrong setting for the particularned. Not capable - The process is in control but is not capable. This is determined by the large probability of producing parts that do not meet the engineering index (called Cp) is less than 1. Cp has a value of one when the range of the distribution (measure by standard deviations from the mean) equals the range of the tolerances. Capable process have a Cp value of at least 1.3. Capable - The process is now capable because the entire distribution falls within the tolerance limits. Improvements occur when the range of the distribution is reduce, increasing the probability for producing on target. Capability - It is the ability to meet customer’s specifications. In control - A process that does not show signs of assignable causes of variation. Not capable, and not centered

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**Process Centering -- Example**

For the granola bar manufacturer, if the process is incorrectly centered at 2.05 instead of 2.00 ounces, what fraction of bars will be out of specification? 2.0 LSL=1.95 USL=2.05 Out of spec! 50% of production will be out of specification!

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**Process Capability Index Cpk**

Std dev Mean m If Cpk > 1.0, process is... Centered & capable If Cpk < 1.0, process is... Not centered &/or not capable

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Cpk Example 1 A manufacturer of granola bars has a weight specification 2 ounces plus or minus 0.05 ounces. If the standard deviation of the bar-making machine is s = 0.02 ounces and the process mean is m = 2.01, what is the process capability index? USL = 2.05 oz LSL = 1.95 ounces Cpk = min[(m -LSL) / 3 , (USL- m) / 3 ] = min[(2.01–1.95) / 0.06 , (2.05 – 2.01) / 0.06 ] = min[1.0 , 0.67 ] = 0.67 Therefore, the process is not capable and/or not centered !

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Cpk Example 2 Venture Electronics manufactures a line of MP3 audio players. One of the components manufactured by Venture and used in its players has a nominal output voltage of 8.0 volts. Specifications allow for a variation of plus or minus 0.6 volts. An analysis of current production shows that mean output voltage for the component is volts with a standard deviation of volts. Is the process "capable: of producing components that meet specification? What fraction of components will fall outside of specification? What can management do to improve this fraction?

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**Process Control Charts**

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**Process Control Charts**

Statistical technique for tracking a process and determining if it is going “out to control” Establish capability of process under normal conditions Use normal process as benchmark to statistically identify abnormal process behavior Correct process when signs of abnormal performance first begin to appear Control the process rather than inspect the product!

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**Process Control Charts**

Upper Spec Limit Upper Control Limit 6 Target Spec 3 Lower Control Limit Lower Spec Limit

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**Process Control Charts**

Look for special cause ! In control Out of control ! Back in control! UCL Target LCL Time Samples Natural variation

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When to Take Action A single point goes beyond control limits (above or below) Two consecutive points are near the same limit (above or below) A run of 5 points above or below the process mean Five or more points trending toward either limit A sharp change in level Other erratic behavior

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**Samples vs. Population Sample Distribution Population Distribution**

Mean

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**Types of Control Charts**

Attribute control charts Monitors frequency (proportion) of defectives p - charts Defects control charts Monitors number (count) of defects per unit c – charts Variable control charts Monitors continuous variables x-bar and R charts

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**1. Attribute Control Charts**

p - charts Estimate and control the frequency of defects in a population Examples Invoices with error s (accounting) Incorrect account numbers (banking) Mal-shaped pretzels (food processing) Defective components (electronics) Any product with “good/not good” distinctions

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Using p-charts Find long-run proportion defective (p-bar) when the process is in control. Select a standard sample size n Determine control limits

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p-chart Example Chic Clothing is an upscale mail order clothing company selling merchandise to successful business women. The company sends out thousands of orders five days a week. In order to monitor the accuracy of its order fulfillment process, 200 orders are carefully checked every day for errors. Initial data were collected for 24 days when the order fulfillment process was thought to be "in control." The average percent defective was found to be 5.94%.

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**2. Defect Control Charts c-charts**

Estimate & control the number of defects per unit Examples Defects per square yard of fabric Crimes in a neighborhood Potholes per mile of road Bad bytes per packet Most often used with continuous process (vs. batch)

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Using c-charts Find long-run proportion defective (c-bar) when the process is in control. Determine control limits

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2. c-chart Example Dave's is a restaurant chain that employs independent evaluators to visit its restaurants as secret shoppers to the asses the quality of service. The company evaluates restaurants in two categories, food quality, and service (promptness, order accuracy, courtesy, friendliness, etc.) The evaluator considers not only his/her order experiences, but also evaluations throughout the restaurant. Initial surveys find that the total number of service defects per survey is 7.3 when a restaurant is operating normally.

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**3. Control Charts for Variables**

x-bar and R charts Monitor the condition or state of continuously variable processes Use to control continuous variables Length, weight, hardness, acidity, electrical resistance Examples Weight of a box of corn flakes (food processing) Departmental budget variances (accounting Length of wait for service (retailing) Thickness of paper leaving a paper-making machine

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**x-bar and R charts Two things can go wrong Two control solutions**

process mean goes out of control process variability goes out of control Two control solutions X-bar charts for mean R charts for variability

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**Problems with Continuous Variables**

“Natural” Process Distribution Mean not Centered Increased Variability Target

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**Range (R) Chart Choose sample size n**

Determine average in-control sample ranges R-bar where R=max-min Construct R-chart with limits:

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**Mean (x-bar) Chart Choose sample size n (same as for R-charts)**

Determine average of in-control sample means (x-double-bar) x-bar = sample mean k = number of observations of n samples Construct x-bar-chart with limits:

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x & R Chart Parameters

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**R and x-bar Chart Example**

Resistors for electronic circuits are being manufactured on a high-speed automated machine. The machine is set up to produce resistors of 1,000 ohms each. Fifteen samples of 4 resistors each were taken over a period of time when the machine was operating normally. The average range of the samples was found to be R-bar=21.7 and the average mean of the samples was x-double-bar=999.1.

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When to Take Action A single point goes beyond control limits (above or below) Two consecutive points are near the same limit (above or below) A run of 5 points above or below the process mean Five or more points trending toward either limit A sharp change in level Other statistically erratic behavior

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**Control Chart Error Trade-offs**

Setting control limits too tight (e.g., m ± 2) means that normal variation will often be mistaken as an out-of-control condition (Type I error). Setting control limits too loose (e.g., m ± 4) means that an out-of-control condition will be mistaken as normal variation (Type II error). Using control limits works well to balance Type I and Type II errors in many circumstances. 3s is not sacred -- use judgement.

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Video: SPC at Frito Lay

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**Statistical Process Control**

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