Outline 2.008 1. Optimization 2. Statistical Process Control 3. In-Process Control Process Control 2.008 - Spring 2004 1 2.008 - Spring 2004 2 Variation: Common and Special Causes What is quality? Pieces vary from each other: Size Size Size Size But they form a pattern that, if stable, is called a distribution: Size 2.008 - Spring 2004 3 Common and Special Causes (cont’d) Size 2.008 - Spring 2004 4 Common and Special Causes (cont’d) If only common causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable: Distributions can differ in… Location Size Shape Spread Prediction Size Size Size …or any combination of these Tim e Size 2.008 - Spring 2004 5 2.008 - Spring 2004 6 1 Common and Special Causes (cont’d) Variation: Run Chart Specified diameter = 0.490” +/-0.001” If special causes of variation are present, the process output is not stable over time and is not predictable: 0.4915 Diameter of part 0.4910 Prediction 0.4905 0.4900 0.4895 0.4890 0.4885 Tim 0.4880 e 0 10 20 Size Process change 2.008 - Spring 2004 7 30 40 50 Run number 2.008 - Spring 2004 Process Control 8 Statistical Process Control 1. Detect disturbances (special causes) In Control (Special causes eliminated) Tim e 2. Take corrective actions Out of Control (Special causes present) Size 2.008 - Spring 2004 9 2.008 - Spring 2004 Central Limit Theorem Shewhart Control Chart • A large number of independent events have a continuous probability density function that is normal in shape. Average (of 10 samples ) Diameter 1.010 • Averaging more samples increases the precision of the estimate of the average. Upper control limit 1.005 1.000 Lower control limit 0.995 0.990 0 2.008 - Spring 2004 10 11 2.008 - Spring 2004 10 20 30 40 50 Run number 60 70 80 90 100 12 2 Sampling and Histogram Creation Sampling and Histogram Creation (cont’d) 20 10 1 9 2 10 8 6 10 4 0 9 1 2 3 4 5 6 7 Outcome 8 9 8 Taking ∞ random samples, the resulting histogram would look like this 4 6 2 10 3 7 5 4 6 2 3 4 5 6 Outcome 7 8 9 13 2.008 - Spring 2004 Probability distribution 0.5 0 25 20 15 10 5 0 1 2 3 4 1 2 3 4 5 6 7 Outcome 8 9 10 5 Take 10 random samples, calculate their average, and repeat ∞ times, the resulting histogram would approach the continuous distribution shown 5 6 7 8 9 10 14 Normal Distribution Underlying distribution 1.5 1 30 2.008 - Spring 2004 Distribution resulting from averaging of 2 random values Underlying D istribution 35 Take 10 random samples, calculate their average, and repeat 100 times, the resulting histogram would resemble 10 Uniform Distributions 1.5 40 3 7 1 Probability distr ibution 1 2 Frequency 8 Frequency Taking 100 random samples, the resulting histogram would look like this Wheel of Fortune: Equal probability of outcome 1-10, P=0.1 14 12 Frequency 18 16 Wheel of Fortune: Equal probability of outcome 1-10, P=0.1 Distribution resulting from the average of 2 random values 1 0.5 0 0.5 0 1 Distribution resulting from the average of 3 random values 0 0.5 0 0 1 Distribution resulting from the average of 10 random values 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0 15 Precision Not precise 0 Distribution resulting from the average of 3 random values 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.008 - Spring 2004 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Distribution resulting from the average of 10 random values 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.008 - Spring 2004 16 Shewhart Control Chart Precise • Upper Control Limit (UCL), Lower Control Limit (LCL) Not accurate • Subgroup size (5 < n < 20) Accurate 2.008 - Spring 2004 17 2.008 - Spring 2004 18 3 Shewhart Control Chart (cont’d) Shewhart Control Chart (cont’d) Control chart based on 100 samples Control chart based on average of 10 samples, note the step change that occurs at run 51 1.010 Upper control limit Average (of 10 samples ) Diameter Average (of 10 samples ) Diameter 1.010 1.005 1.000 Lower control limit 0.995 1.005 Upper control limit 1.000 Lower control limit 0.995 Step disturbance Step disturbance 0.990 0.990 10 0 20 30 40 50 Run number 60 70 80 90 100 2.008 - Spring 2004 10 0 19 20 30 40 50 Run number 60 70 80 90 100 2.008 - Spring 2004 Control Charts 20 Setting the Limits Idea: Points outside the limits will signal that something is wrong- Upper control limit an assignable cause. We want limits set so that assignable causes Process average Lower control limit are highlighted, but few random causes are highlighted accidentally. 0 10 20 30 40 50 60 70 80 90 100 Run number 1. 2. 3. Convention for Control Charts: Collection: Gather data and plot on chart. • Control: Calculate control limits from process data, using simple formulae. • Identify special causes of variation; take local actions to correct. • Capability: • Quantify common cause variation; take action on the system. • Upper control limit (UCL) = x + 3σsg • Lower control limit (LCL) = x - 3σsg (Where σsg represents the standard deviation of a subgroup of samples) These three phases are repeated for continuing process improvement. 2.008 - Spring 2004 21 Setting the Limits (cont’d) 2.008 - Spring 2004 22 Benefits of Control Charts Properly used, control charts can: Convention for Control Charts (cont’d): • Be used by operators for ongoing control of a process • Help the process perform consistently, predictably, for quality and cost σsub group = σsg ≠ σprocess σsubgroup = σprocess/√n • Allow the process to achieve: – Higher quality UCL = x + 3σprocess/√n – Lower unit cost – Higher effective capacity LCL = x - 3σprocess/√n • Provide a common language for discussing process performance As n increases, the UCL and LCL move closer to the center line, • Distinguish special from common causes of variation; as a guide to local or management action making the control chart more sensitive to shifts in the mean. 2.008 - Spring 2004 23 2.008 - Spring 2004 24 4 Process Capability Process Capability Index In Control and Capable (Variation from common cause reduced) 1. Cp = Range/6σ Lower specification limit Upper specification limit Tim 2. Cpk e In Control but not Capable (Variation from common causes excessive) Size 2.008 - Spring 2004 25 Process Capability • Take an example with: Mean = .738 Standard deviation, σ = .0725 UCL = 0.900 LCL = 0.500 • Normalizing the specifications: UCL − x σ 26 Process Capability (cont’d) LCL 0.500 UCL 0.900 x 0.738 • Using the tables of areas under the Normal Curve, the proportions out of specification would be: PUCL= 0.0129 σ ZLCL ZUCL -3.28 2.23 PLCL = 0.0005 Ptotal = 0.0134 0.900 − 0.738 = 2.23 0.0725 LCL − x 0.500 − 0.738 Z LCL = = = −3.28 0.0725 σ ZMIN = 2.23 (on an absolute basis) Z UCL = 2.008 - Spring 2004 = • The Capability Index would be: CPK = Zmin/3 = 2.23 / 3 = 0.74 2.008 - Spring 2004 27 2.008 - Spring 2004 28 Process Capability (cont’d) Process Capability (cont’d) • If this process could be adjusted toward the center of the specification, the proportion of parts falling beyond either or both specification limits might be reduced, even with no change in Standard deviation. • For example, if we confirmed with control charts a new mean = 0.700, then: • The proportions out of specification would be: Z UCL = Z LCL = UCL − xnew σ LCL − xnew σ Zmin = 2.76 2.008 - Spring 2004 0.900 − 0.700 = 2.76 0.0725 0.500 − 0.700 = = −2.76 0.0725 = LCL 0.500 PUCL= 0.0029 PLCL = 0.0029 Ptotal = 0.0058 UCL 0.900 x 0.700 • The Capability Index would be: CPK = Zmin/3 = 2.76 / 3 = 0.92 σ ZLCL ZUCL -2.76 2.76 29 2.008 - Spring 2004 30 5 Improving Process Capability • To improve the chronic performance of the process, concentrate on the common causes that affect all periods. These will usually require management action on the system to correct. • Chart and analyze the revised process: – Confirm the effectiveness of the system by continued monitoring of the Control Chart 2.008 - Spring 2004 31 6