Statistical Process Control (SPC) Graduate School of Business University of Colorado Boulder, CO 80309-0419 Professor Stephen Lawrence

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 Cause & effect diagrams Scatterplots Histograms Control charts

Check Sheets 27 15 19 20 28 Check sheets explore what and where an event of interest is occurring. Attribute Check Sheet Order Types 7am-9am 9am-11am 11am-1pm 1pm-3pm 3pm-5-pm Emergency Nonemergency Rework Safety Stock Prototype Order Other 27 15 19 20 28

Run Charts measurement time

SCATTERPLOTS Variable A Variable B x x x x x x xx x x x x xx x x x

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

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

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

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

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)

Product Specification and Process Variation 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!

Process Capability LSL Spec USL 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 Capability 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

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

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 + 0.05 = 2.05 ounces LSL = 2 - 0.05 = 1.95 ounces Cp = (USL - LSL) / 6 = (2.05 - 1.95) / 6(0.02) = 0.1 / 0.12 = 0.85

Process Centering LSL Spec USL 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 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! _____ of production will be out of specification!

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

Process Capability Index -- 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 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 !

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!

Process Control Charts Upper Spec Limit Upper Control Limit 6 Target Spec 3 Lower Control Limit Lower Spec Limit

Process Control Charts UCL Target LCL Time

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

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

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

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

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)

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

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

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

Problems with Continuous Variables “Natural” Process Distribution Mean not Centered Increased Variability Target

Range (R) Chart Choose sample size n Determine average in-control sample ranges R-bar where R=max-min Construct R-chart with limits:

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:

x & R Chart Parameters

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

Control Chart Error Trade-offs Setting control limits too tight (e.g.,  ± 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.,  ± 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.