OPER 3100 Chapter 13 Statistical Quality Control

Measuring Variation

Mean (x̄): The average value of a set of numbers x̄ = (Sum of observed values) / (Total Number of values) Standard Deviation(σ): A measure of the variation in a set of numbers σ = √(∑(xi - x̄)²) / n

Acceptance Sampling

Performed on goods that already exist to determine what percentage of the products conform to specifications Purpose is to... -Determine Quality Level -Ensure Qulaity is within predetermined level Executed through a sampling plan Resluts include accept, reject, or retest

Creating p-Charts

Calculate the sample proportions p for each sample ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓ Calculate the average of the sample proportions ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓ Calculate the standard deviation of the sample proportion ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓ Calculate the control limits ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓ Plot the individual sample proportions, the average of the proportions, and the control limits

Measuring Process Capability

Capable: the mean and standard deviation of the process are operating such that the upper and lower control limits are acceptable relative to the upper and lower specification limits Specification limits: range of variation that is considered acceptable by the designer or customer Process limits: range of variation that a process is able to maintain with a high degree of certainty

Process Control Procedures

Concerned with monitoring quality while the product or service is being produces -Statistical Process control: testing a sample of output to determine if the process is producing items within a preselected range -Attributes: quality characteristics that are classified as either conforming or not conforming -Variable: characteristics that are measured using an actual value

Acceptance Sampling Advantages

-Economy -Less handling damage -Fewer inspectors -Upgrading of the inspection job -Applicability to destructive testing -Entire lot rejection (motivation for improvement)

Acceptance Sampling Disadvantages

-Risks of accepting "bad" lots and rejecting "good" lots -Added planning and documentation -Sample provides less information than 100 percent inspection

Statistical Quality Control (SQC)

A number of different techniques designed evaluate quality from a conformance view. Involves sampling output from a process and using statistics to find when the process has changed in a nonrandom way.

X-Bar Charts - Using Sigma (σ)

If process standard deviation is σ or known -Upper control limits: UCL = x̄ + z (σx̄) LCL = x̄ - z (σx̄) where (σx̄) is the standard deviation of sample means: σx̄= σ/(√n) σ=Standard Deviation of the process distribution n= sample size z= number of standard deviations for a specific confidence level (typically, z=30 x̄= average of sample means

Process Capability Index (Cpk)

Ratio of the range of values produced divided by the range of values allowed Cpk= min[((x̄-LSL)/3σ),((x̄-LSL)/3σ)] standard = 1.33 good Cpk = 1 => 3 sigma quality => 99.74% is good -Shows how well the parts being produced fit into the range specified by the design specifications -Cpk larger than one indicates process is capable -When the two numbers are not close, indicates mean has shifted

Variable Measurement Process Control Charts: x̄ and R-Charts

Size of Sample: -Preferably to keep small (usually 4 or 5 units) Number of Samples: -Once chart set up, each sample compared to char -Use about 25 samples to set up chart Frequency of Sample: -Trade-off between cost of sampling and benefit of adjusting the system Control Limits -Generally use z=3

X-Bar Charts - Using Sigma (σ) STEPS

Step 1: Compute the mean of each sample Step 2: Find the average of sample means Step 3: Compute sample standard deviation Step 4: Find the control limits Step 5: Plot the X-bar control chart

Upper Specification

The maximum acceptable value for a characteristic

Lower Specification

The minimum acceptable value for a characteristic

Process Control with Attribute Measurements: Using p-Charts

Used when an item (or service) is either good or bad (a yes-no decision) 𝑝̅= (Total Number of defective units from all samples) / (Number of samples x Sample Size) Sp=√( (𝑝̅(1-𝑝̅)) / n ) UCL = 𝑝̅ + ZSp LCL = 𝑝̅ - ZSp or zero if equation evalutates to less than 0 Typically use z = 3

Process Control with Attribute Measurements: Using c-Charts

Used when an item(or service) may have multiple defects -Knotholes on lumber Underlying distribution is Poisson 𝑐̅= Average number of defects per unit Sc = √𝑐̅ UCL = 𝑐̅ + z√𝑐̅ LCL = 𝑐̅ - z√𝑐̅ or zero if equation evaluates to less than 0 Typically use Z=3

Assignable Variation

Variation is caused by factors that can be clearly identified and possible even managed

Common Variation

Variation that is inherent in the process itself --Also known as random variation

Understanding and Measuring Process Variation

When variation is reduced, quality is improved • It is impossible to have zero variation • Typical specification statement is 10.00 inches ± 0.02 inch • Upper specification: the maximum acceptable value for a characteristic • Lower specification: the minimum acceptable value for a characteristic

Control Charts for Variables

X-bar Charts (mean charts) -Used to monitor the central tendency of a process -TWO approaches to construct --1. Using the Standard Deviation of the process distribution σ --2. Using the sample range R R Charts (range charts) -Used to monitor the process dispersion

How to Construct x̄ and R-Charts

x̄-Chart control limits -UCL= x̄ + A₂R̄ -LCL= x̄ - A₂R̄ R-Chart control limits -UCL = D₄R̄ -LCL = D₃R̄ We are going to use these When the process standard deviation in unknown!

Acceptance Sampling - Designing a Sampling Plan

• Determine (1) how many units, n, to sample from a lot, and (2) the maximum number of defective items, c, that can be found in the sample before the lot is rejected • Acceptable quality level (AQL) --Maximum acceptable percentage of defectives defined by producer • Lot tolerance percent defective (LTPD) --Percentage of defectives that defines consumer's rejection point • α (producer's risk) -- The probability of rejecting a good lot • β (consumer's risk) -- The probability of accepting a bad lot