Prof. Indrajit Mukherjee, School of Management, IIT Bombay VISUALIZING VARIATION Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Capability Versus Control Statistical Control In Control Out of Control Capability IDEAL Capable Not Capable Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Process Capability Ratio – A Standard Measure of How Good a Process Is. A simple ratio: Specification Width Actual “Process Width” Generally, the higher the better. Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Process Capability Ratio Nominal value Process distribution Lower specification Upper specification 20 30 Minutes 25 Process is capable Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Process Capability Nominal value Process distribution Lower specification Upper specification 20 30 Minutes 25 Process is capable Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Mean Spec range=6σ USL LSL Spec Range 6σ Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Variability on Process Capability Effects of Reducing Variability on Process Capability Nominal value Six sigma Four sigma Two sigma Lower specification Upper specification Mean Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay LSL LCL µ UCL USL LCL µ UCL LSL USL 3σ 3σ 3σ 3σ LCL LSL µ USL UCL 3σ 3σ Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay PCR>1 Process Capability (a) µ LSL USL 3σ 3σ PCR=1 Nonconforming Units Nonconforming Units (b) µ 3σ 3σ Figure Process Fallout and the process capability ratio (PCR). USL LSL PCR<1 Nonconforming Units Nonconforming Units (c) LSL µ USL 3σ 3σ Prof. Indrajit Mukherjee, School of Management, IIT Bombay

LSL of Pressure Strength =200 psi Glass container strength data (psi) Sample Data x R 1 265 205 263 307 220 252 102 2 268 260 234 299 215 255.2 84 3 197 286 274 243 231 242.2 89 4 267 281 214 318 269 104 5 346 317 22 258 276 287.8 6 300 208 187 264 271 246 113 7 280 242 321 228 266.2 93 8 250 293 273.4 49 9 254 294 223 263.4 71 10 308 235 283 277 272.6 73 11 200 328 296 261 128 12 290 266.8 55 13 221 176 248 227.8 87 14 334 272 286.8 69 15 262 245 301 268.8 56 16 253 287 270.4 34 17 337 18 278 275 28 19 270 298 272.2 48 20 257 210 251 253.4 70 264.06 77.3 Process Capability Analysis using Control Charts LSL of Pressure Strength =200 psi Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Vane-opening Measurements sample number x1 x2 x3 x4 x5 x r s 1 33 29 31 32 31.6 4 1.67332 2 35 37 33.4 6 2.60768 3 34 36 1.58114 30 32.2 1.64317 5 33.8 0.83666 38 39 40 38.4 1.14018 7 1.51658 8 36.8 10 4.38178 9 28 43 15 5.43139 2.54951 11 29.8 1.78885 12 1.73205 13 27 3.80789 14 34.8 35.6 16 30.8 2.48998 17 2.0000 18 19 25 28.2 3.42053 20 2.38747 33.32 5.8 2.345 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Not capable Cp<1 T O W E R D S X C L N e Cp=1 Merely capable Acceptable Cp=1.33 4σ level Cp=1.66 Performing 6σ level Tolerance Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Process Fallout (In defective ppm) PCR One-Sided Specification Two-Sided Specification 0.25 226,628 453,255 0.5 66,807 133,614 0.6 35,931 71,861 0.7 17,865 35,729 0.8 8,198 16,395 0.9 3,467 6,934 1 1,350 2,700 1.1 484 967 1.2 159 318 1.3 48 96 1.4 14 27 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Recommended Minimum Values of the Process Capability ratio Two-sided Specification One-sided Existing processes 1.33 1.25 New processes 1.50 1.45 Safely, strength, or critical parameter, existing process Safely, strength, or critical parameter, new process 1.67 1.60 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Reasons for Poor Process Capability σ LSL µ USL (a) σ LSL µ USL (b) Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay LSL USL σ=2 • Cp does not take process centering into Account • It is a measure of potential capability, not actual capability (a) 38 44 50 56 62 σ=2 (b) 38 44 50 56 62 σ=2 (c) 38 44 50 56 62 σ=2 (d) 38 44 50 56 62 σ=2 (e) 38 44 50 56 62 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Process Capability What it is 1. CAPABILITY Short term in control Pooled std dev 2. PRFORMANCE Long term Not in control Total/overall std dev compares spec range (tolerance) to process width How much variability a how close process centre is to nearest spec limit How centered b or ENTITLEMENT is the theoretical best hat a process can achieve. Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Bursting Strength for 100 Glass Containers 265 197 346 280 200 221 261 278 205 286 317 242 254 235 176 262 248 250 263 274 260 281 246 271 307 258 321 294 328 245 270 220 231 276 228 223 296 301 337 298 268 267 300 334 257 208 299 308 264 210 234 187 269 253 214 283 272 287 215 318 293 277 290 275 251 Specification=300-220 Calculate Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Vane-opening Measurements sample number x1 x2 x3 x4 x5 x r s 1 33 29 31 32 31.6 4 1.67332 2 35 37 33.4 6 2.60768 3 34 36 1.58114 30 32.2 1.64317 5 33.8 0.83666 38 39 40 38.4 1.14018 7 1.51658 8 36.8 10 4.38178 9 28 43 15 5.43139 2.54951 11 29.8 1.78885 12 1.73205 13 27 3.80789 14 34.8 35.6 16 30.8 2.48998 17 2.0000 18 19 25 28.2 3.42053 20 2.38747 33.32 5.8 2.345 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Batch Viscosity Percent solids Lb.Gal 1 74 64.2 13.3 22 62.7 2 69 62.6 13.4 23 70 62.9 3 79 63.7 24 75 13.2 4 63.6 25 65 62.8 5 62 62.4 26 6 63.4 27 7 73 63.2 28 72 8 63 29 77 9 68 63.8 13.5 30 10 63.5 31 11 32 12 63.1 33 62.5 13 34 14 35 66 62.3 13.1 15 36 16 76 63.3 37 17 64.1 38 18 39 19 40 20 63.9 41 21 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Statistical Analysis - MSA Terminology – Measurement system: the collection of operations, procedures, gauges and other equipment, software and personnel used to assign a number to a characteristic being measured; the complete process used to obtain a measurement. Material Man Method MEASUREMENT SYSTEM Machine (Time) Environment Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Sources of Variation Product Variability (Actual variability) Measurement Variability Total Variability (Observed variability) Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Possible Source of Variation Observed Process Variation Actual Process Variation Measurement Variation Long-term Variation Short-term Variation Variation within a Sample Variation due to Gage Variation due to Operators Accuracy/Bias Repeatability Linearity Stability Reproducibility Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Types of Measurement Error that Affect Location Measurement System Bias Accuracy Linearity Stability Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay What is GR&R? Measurement Systems Analysis How good is our measurement system? Total Variance Measurement System Variability — Determined through “R&R Study” Process Variance Measurement Variance Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Gauge R&R Allows Control of the Measurement System Components of Measurement System Variation Variation due to gage Variation due to operators Operator Operator by part Reproducibility Repeatability Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Types of Measurement Error that Affect Spread Precision/Repeatability Reproducibility Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay BIAS Is Definition BIAS— the difference between the observed average of the measurement and the reference value. The reference-value is the serves agreed-upon Reference value that as an Value reference. The reference value can be determined by averaging several measurements with a higher level (e.g., metrology lab) of measuring equipment. ACCURACY IS THE SAME AS BIAS Reference value Observed Average Value Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Stability (drift) Definition Stability — Is the total variation in the average value measurement obtained with a measurement system (test / gage ) on the same master parts when measuring a single characteristic over an extended period. Time-2 time Time-1 Magnitude time Points to the frequency of mean center Calibration Stability Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Repeatability Definition Repeatability — The variation in measurements obtained with one measurement instrument when used several times by one appraiser while measuring the identical characteristic on same part. REPEATABILITY Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Part number Measurements 1 2 X R 21 20 20.5 24 23 23.5 3 4 27 5 19 18 18.5 6 22 7 21.5 8 17 9 10 11 25 12 13 14 15 29 30 29.5 16 26 25.5 22.3 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Part Measurement Data (Single Operator or p=1) Measurement Part number 1 2 X-bar R 21 20 20.5 24 23 23.5 3 4 27 5 19 18 18.5 P=1 For n=2 is 1.128, USL=60; LSL=5 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Reproducibility Definition Reproducibility — Is the variation in the average of the Operator measurements made by different appraisers using the same measuring instrument when measuring the identical characteristic on the same part. Operator-B Operator-C Operator-A Reproducibility Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Repeatability-Equipment Variation (EV) Trials 2 4.56 3 8.05 = x 4.56 = n=number of parts R=number of trials Reproducibility-Appraiser Variation (AV) Appraisers 2 3.65 3 2.70 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Repeatability &Reproducibility (R&R) Parts 2 3.65 3 2.7 4 2.3 5 2.08 6 1.93 7 1.82 8 1.74 9 1.67 10 1.62 Part variation (PV) Total variation(TV) Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Source of Variation % Contribution Total Gage Repeatability & Reproducibility Repeatability Reproducibility Part-to-Part Total Variation Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Gage R&R, Distinct Categories Number of Distinct Categories = Eight • This is the number of distinct categories this measurement system can distinguish. • The number of groups within your process data that your measurement system can discern. Source of Variation Varcomp Total Gage R&R 1.90E-04 Repeatability 8.73E-05 Reproducibility 1.02E-04 Name Part-to-Part 6.45E-03 Total Variation 6.64E-03 Distinct categories Good Gage >10 Marginal Gage 4 to 9 Reject Gage <4 Dist Categories = Round down Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Assignment (Data are from Vardeman & Jobe (1999)) A study was conducted to investigate the precision of measuring the heights of 10 steel punches (in 10-3 inches) using a certain micrometer caliper. Three operators were randomly selected for the the study. Ten parts with steel punches are randomly selected. Each operator measured each punch three time. Here are the data. Row Punch Oper A Oper B Oper C 1 496 497 97 2 499 498 3 4 5 6 500 7 8 9 10 11 12 13 14 501 15 Row Punch Oper A Oper B Oper C 16 6 499 500 498 17 18 497 19 7 503 20 21 502 22 8 501 23 24 25 9 26 27 28 10 496 29 494 30 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Check This in Minitab Thermal Impedance Data (c/w*100)for the Gauge R&R Experiment Part Number Inspector 1 Inspector 2 Inspector 3 Test 1 Test 2 Test 3 1 37 38 41 40 42 2 43 3 30 31 29 28 4 5 6 45 44 46 7 25 26 27 8 9 10 35 34 Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Prof. Indrajit Mukherjee, School of Management, IIT Bombay Attribute Gage R&R Effectiveness SCORING REPORT Attribute Legend 1.Pass 2.Fail DATE: 3/10/1996 NAME: Allied Employee PRODUCT: 3313 Spark Plug SBU: TEST CONDITIONS: F&SP Known Population Operator #1 Operator #2 Operator #3 Y/N Sample # Attribute Try #1 Try #2 Agree 1 Pass Fail N 2 3 4 Y 5 6 7 8 9 10 11 12 13 14 15 _ 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Prof. Indrajit Mukherjee, School of Management, IIT Bombay