Published on February 4, 2009
Chapter 10 Specification Limit Process Capability, and Gage Measurement
Specification limit • Specification limit : conformance boundary specified formal characteristic. • Type of Specification – Two-sided : upper and lower limits – One-sided : either an upper or lower limit
Statistical tolerance limits • Statistical tolerance limits: limits of the interval for which it can be stated with a given level of confidence that it contains at least a specified proportion of the population.
Setting Up Statistical Tolerance Limit • Use control chart to measure of process capability to established requirement • From observations of a given quality characteristics from one or more samples.
Statistical tolerance limits • When process is in control, data can be used to determine statistical tolerance limit. • Assume that a process involves a quality characteristic that follows a normal distribution with mean µ, and standard deviation, σ. The upper and lower natural tolerance limits or statistical tolerance limits of the process are UNTL = µ + 3σ LNTL = µ - 3σ
Statistical Tolerance Limits • Page 325 • Obtain a sample of n observation • Check for normality assumption • Calculate statistics parameters • Select the desired confidence level and population proportion to be cover by limits • Determine k2 • Compute limits
Types of specification conflict • Type I : distribution too wide for double specification • Type II : Distribution not centered correctly for single specification limit • Type III : Specification limits too wide for acceptable product
Type I : distribution too wide for double specification • Possible action – Changing the process – Changing the specification – Setting up an inspection/sorting operation to find, remove, or repair the units of product falling outside either specification limit – Adjusting centering of distribution to strike a balance among relative costs
Type II : Distribution not centered correctly for single specification limit • Possible action – Attempting to center the process at a value far enough from specification limit – Changing the specification limit – Setting up inspection/sorting operation to find and remove or repair the units of product that fall out side the specification limit
Type III : Specification limits too wide for acceptable product • Possible action – Conducting ore formal program of process experimentation to determine what range of value cause nonconformities – Tightening the specification limits – Establishing shop or manufacturing limits on a temporary basis until final values are determined
Process capability analysis • Process capability refers to the uniformity of the process. • Variability in the process is a measure of the uniformity of output. • Two types of variability: – Natural or inherent variability (instantaneous) – Variability over time • Process capability analysis is an engineering study to estimate process capability. • In a product characterization study, the distribution of the quality characteristic is estimated.
Major uses of data from a process capability analysis 1. Predicting how well the process will hold the tolerances. 2. Assisting product developers/designers in selecting or modifying a process. 3. Assisting in Establishing an interval between sampling for process monitoring. 4. Specifying performance requirements for new equipment. 5. Selecting between competing vendors. 6. Planning the sequence of production processes when there is an interactive effect of processes on tolerances 7. Reducing the variability in a manufacturing process.
Techniques used in process capability analysis 1. Histograms or probability plots 2. Control Charts 3. Designed Experiments
Process Capability Analysis Using a Histogram or a Probability Plot Using a Histogram • The histogram along with the sample mean and sample standard deviation provides information about process capability. The process capability can be estimated as x ± 3 s – – The shape of the histogram can be determined (such as if it follows a normal distribution) – Histograms provide immediate, visual impression of process performance.
Probability Plotting • Probability plotting is useful for – Determining the shape of the distribution – Determining the center of the distribution – Determining the spread of the distribution. σ= ˆ
Probability Plotting Cautions in the use of normal probability plots • If the data do not come from the assumed distribution, inferences about process capability drawn from the plot may be in error. • Probability plotting is not an objective procedure (two analysts may arrive at different conclusions).
Process Capability Ratios Use and Interpretation of Cp • Recall USL − LSL Cp = 6σ where LSL and USL are the lower and upper specification limits, respectively.
Use and Interpretation of Cp The estimate of Cp is given by ˆ = USL − LSL Cp 6σˆ Where the estimate σ can be calculated using the sample ˆ standard deviation, S, or R / d 2
Use and Interpretation of Cp One-Sided Specifications USL − µ C pu = 3σ µ − LSL C pl = 3σ These indices are used for upper specification and lower specification limits, respectively
Use and Interpretation of Cp Assumptions The quantities presented here (Cp, Cpu, Clu) have some very critical assumptions: 1. The quality characteristic has a normal distribution. 2. The process is in statistical control 3. In the case of two-sided specifications, the process mean is centered between the lower and upper specification limits. If any of these assumptions are violated, the resulting quantities may be in error.
Process Capability Ratio an Off-Center Process • Cp does not take into account where the process mean is located relative to the specifications. • A process capability ratio that does take into account centering is Cpk defined as Cpk = min(Cpu, Cpl)
Normality and the Process Capability Ratio • The normal distribution of the process output is an important assumption. • If the distribution is nonnormal, Luceno (1996) introduced the index, Cpc, defined as USL − LSL C pc = π EX−T 6 2
More About Process Centering • Cpk should not be used alone as an measure of process centering. Cpk depends inversely on σ and becomes • large as σ approaches zero. (That is, a large value of Cpk does not necessarily reveal anything about the location of the mean in the interval (LSL, USL)
More About Process Centering • An improved capability ratio to measure process centering is Cpm. USL − LSL C pm = 6τ where τ is the squre root of expected squared deviation from target: T =½(USL+LSL), [ ] τ = E (x − T ) = σ 2 + (µ − T) 2 2 2
More About Process Centering • Cpm can be rewritten another way: (same as 10.8) USL − LSL C pm = 6 σ 2 + (µ − T) 2 Cp = 1+ ξ2 where T−µ ξ= σ
More About Process Centering • A logical estimate of Cpm is: ˆ Cp ˆ C pm = 1+ V2 where T−x V= S
Confidence Intervals and Tests on Process Capability Ratios Cpk • Ĉpk is a point estimate for the true Cpk, and subject to variability. An approximate 100(1-α) percent confidence interval on Cpk is 1 1 1 1 ˆ 1 + Z pk 1 − Z α / 2 ≤ C pk ≤ C pk + + α/2 ˆ ˆ 9 n C pk 2( n − 1) 9 n C pk 2 ( n − 1)
Confidence Intervals and Tests on Process Capability Ratios Example n = 20, Ĉpk = 1.33. An approximate 95% confidence interval on Cpk is 1 1 1 1 1.33 1−1.96 + ≤ Cpk ≤1.33 1+1.96 + 9(20)1.33 2(19) 9(20)1.33 2(19) • The result is a very wide confidence interval ranging from below unity (bad) up to 1.67 (good). Very little has really been learned about actual process capability (small sample, n = 20.)
Confidence Intervals and Tests on Process Capability Ratios Cpc • Ĉpc is a point estimate for the true Cpc, and subject to variability. An approximate 100(1-α) percent confidence interval on Cpc is ˆ ˆ C pc C pc ≤ C pc ≤ sc sc 1+ t α 1− t α , n −1 c n , n −1 c n 2 2 where 1n c = ∑ xi − T n i =1
Process Capability Analysis Using a Control Chart • If a process exhibits statistical control, then the process capability analysis can be conducted. • A process can exhibit statistical control, but may not be capable. • PCRs can be calculated using the process mean and process standard deviation estimates.
Gage and Measurement System Capability Studies Control Charts and Tabular Methods • Two portions of total variability: – product variability which is that variability that is inherent to the product itself – gage variability or measurement variability which is the variability due to measurement error σ Total = σ 2 + σ gage 2 2 product
Control Charts and Tabular Methods X and R Charts • The variability seen on the X chart can be interpreted as that due to the ability of the gage to distinguish between units of the product • The variability seen on the R chart can be interpreted as the variability due to operator.
Control Charts and Tabular Methods Precision to Tolerance (P/T) Ratio • An estimate of the standard deviation for measurement error is R σgage = ˆ d2 • The P/T ratio is 6σgage ˆ P/T = USL− LSL
Control Charts and Tabular Methods • Total variability can be estimated using the sample variance. An estimate of product variability can be found using σTotal = σ2 + σgage 2 2 product S2 = σ2 ˆ product + σgage ˆ2 σ =S −σ 2 2 2 ˆ ˆ product gage
Control Charts and Tabular Methods Percentage of Product Characteristic Variability • A statistic for process variability that does not depend on the specifications limits is the percentage of product characteristic variability: σ gage ˆ × 100 σ product ˆ
Control Charts and Tabular Methods Gage R&R Studies • Gage repeatability and reproducibility (R&R) studies involve breaking the total gage variability into two portions: – repeatability which is the basic inherent precision of the gage – reproducibility is the variability due to different operators using the gage.
Control Charts and Tabular Methods Gage R&R Studies • Gage variability can be broken down as σ =σ =σ +σ 2 2 2 2 measuremen error t gage reproducibility repeatabil ity • More than one operator (or different conditions) would be needed to conduct the gage R&R study.
Control Charts and Tabular Methods Statistics for Gage R&R Studies (The Tabular Method) • Say there are p operators in the study • The standard deviation due to repeatability can be found as R σ repeatabil ity = ˆ d2 where R1 + R 2 + + Rp R= p and d2 is based on the # of observations per part per operator.
Control Charts and Tabular Methods Statistics for Gage R&R Studies (the Tabular Method) • The standard deviation for reproducibility is given as Rx σ reproducibility = ˆ d2 R x = x max − x min where x max = max(x1, x 2 ,…x p ) x min = min(x1, x 2 ,…x p ) d2 is based on the number of operators, p
Methods Based on Analysis of Variance • The analysis of variance can be extended to analyze the data from an experiment and to estimate the appropriate components of gage variability. • For illustration, assume there are a parts and b operators, each operator measures every part n times.
Methods Based on Analysis of Variance • The measurements, yijk, could be represented by the model i = 1,2,...a = µ + τ i + β j + (τβ) ij + ε ijk j = 1,2,..., b y ijk k = 1,2,..., n where i = part, j = operator, k = measurement.
Methods Based on Analysis of Variance • The variance of any observation can be given by V( y ijk ) = σ 2 + σ β + σ 2 + σ 2 2 τ τβ σ 2 , σ β , σ 2 , σ 2 are the variance components. 2 τ τβ
Methods Based on Analysis of Variance • Estimating the variance components can be accomplished using the following formulas σ 2 = MS E ˆ MS AB − MS E σ τβ = ˆ2 n MS B − MS AB σβ = 2 ˆ an MS A − MS AB στ = 2 ˆ bn
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