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W5INSE6220 - 1 3 Process capability analysis(cont INSE 6220...

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1 INSE 6220 -- Week 5 Advanced Statistical Approaches to Quality Process capability More on Hypothesis Testing More on Statistical Inference More on Control Charts: X-bar, R, and S control charts Dr. A. Ben Hamza Concordia University 0 5 10 15 20 25 30 35 40 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 UCL LCL CL S Chart Sample Number Standard Deviation 2 Process capability analysis 1. Compute the mean of sample means ( X ). 2. Compute the mean of sample ranges ( R ). 3. Estimate the population standard deviation ( σ x ): σ x = R / d 2 4. Estimate the natural tolerance of the process: Natural tolerance = 6 σ x 5. Determine the specification limits: USL = Upper specification limit LSL = Lower specification limit 3 Process capability analysis (cont.) 6. Compute capability indices: Process capability potential C p = (USL – LSL) / 6 σ x Upper capability index C pU = (USL – X ) / 3 σ x Lower capability index C pL = ( X – LSL) / 3 σ x Process capability index C pk = min (C pU , C pL ) 4 Control Charts Suppose we have a general statistic W We plot W over time We specify control limits of the form A control chart based on a number of standard deviations of the statistic from the mean of the statistic is called a Shewart Control Chart Some commonly used W’s X bar: Average R: Range s: Standard deviation We can also specify control charts using probability limits W W W W W k LCL CL k UCL Mean of W Std. Dev. of W
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5 X-bar and R Charts R A x LCL x line Central R A x UCL Chart x 2 2 : 6 ~ 4 25 ~ 20 ... 2 1 n m m x x x x m 25 ~ 20 ... 2 1 min max m m R R R R x x R m R D LCL R line Central R D UCL Chart R 3 4 : A 2 , D 3 , D 4 =? To find the control limits, need to estimate the variance, or standard deviation Estimates process mean, µ 6 Control Charts for X-bar and s s s s s s k LCL CL k UCL If is a random sample from a population, then n X X X ,..., , 2 1 ) , ( 2 N ) ( but ) ( 2 2 s E s E 7 8
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9 Example 10 11 12
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13 S B ULC S CL S B LCL 4 3 ; c S ˆ 4 S A x ULC x CL S A x LCL 3 3 6 4 5 B ULC c CL B LCL R D ULC R CL R D LCL 4 3 R A x ULC x CL R A x LCL 2 2 2 2 1 D ULC d CL D LCL A ULC CL A LCL 2 d R ˆ X ˆ X ˆ known X bar chart R chart S chart Process Parameters X bar & R chart X bar & S chart Summary of Control Charts 14 Example: S charts with MATLAB This example plots an S chart of measurements on newly machined parts, taken at one hour intervals for 36 hours. Each row of the runout matrix contains the measurements for 4 parts chosen at random. The values indicate, in thousandths of an inch, the amount the part radius differs from the target radius.
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