Statistical Process Control Operations Management Dr. Ron Lembke Designed Size 10 11 12 13 14 15 16 17 18 19 20 Natural Variation 14.5 14.6 14.7 14.8 14.9 15.0 15.1 15.2 15.3 15.4 15.5 Theoretical Basis of Control Charts Properties of normal distribution 95.5% of allX fall within ± 2 X Theoretical Basis of Control Charts Properties of normal distribution 99.7% of allX fall within ± 3 X 16 14 12 10 8 6 4 2 0 Skewness Lack of symmetry Pearson’s coefficient of skewness: 3( x Median ) s Positive Skew > 0 16 14 12 10 8 6 4 2 0 Skewness = 0 Negative Skew < 0 16 14 12 10 8 6 4 2 0 0.45 Kurtosis 0.4 0.35 0.3 0.25 Amount of peakedness or flatness 4 (x x) ns 4 Kurtosis < 0 0.2 0.15 0.1 0.05 0 -6 -4 -2 0 2 Kurtosis = 0 Kurtosis > 0 4 6 Design Tolerances Design tolerance: Determined by users’ needs UTL -- Upper Tolerance Limit LTL -- Lower Tolerance Limit Eg: specified size +/- 0.005 inches No connection between tolerance and completely unrelated to natural variation. Process Capability and 6 LTL UTL 3 LTL UTL 6 A “capable” process has UTL and LTL 3 or more standard deviations away from the mean, or 3σ. 99.7% (or more) of product is acceptable to customers Process Capability Capable Not Capable LTL UTL LTL UTL LTL UTL LTL UTL Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002 Are we in trouble? Process Capability Specs: 1.5 +/- 0.01 LTL = 1.5 – 0.01 = 1.49 UTL = 1.5 + 0.01 = 1.51 Mean: 1.505 Std. Dev. = 0.002 LCL = 1.505 - 3*0.002 = 1.499 UCL = 1.505 + 0.006 = 1.511 Process Specs 1.49 1.499 1.51 1.511 Capability Index Capability Index (Cpk) will tell the position of the control limits relative to the design specifications. Cpk>= 1.0, process is capable Cpk< 1.0, process is not capable Process Capability, Cpk Tells how well parts produced fit into specs C pk X LTL UTL X min or 3 3 Process Specs LTL 3 X 3 UTL Process Capability Tells how well parts produced fit into specs C pk X LTL UTL X min or 3 3 For our example: 1.505 1.49 1.51 1.505 C pk min or 0.006 0.006 Cpk= min[ 0.015/.006, 0.005/0.006] Cpk= min[2.5,0.833] = 0.833 < 1 Process not capable Process Capability: Re-centered If process were properly centered Specs: 1.5 +/- 0.01 LTL = 1.5 – 0.01 = 1.49 UTL = 1.5 + 0.01 = 1.51 Mean: 1.5 Std. Dev. = 0.002 LCL = 1.5 - 3*0.002 = 1.494 UCL = 1.5 + 0.006 = 1.506 Process Specs 1.49 1.494 1.506 1.51 If re-centered, it would be Capable C pk C pk 1.5 1.49 1.51 1.5 min , 0.006 0.006 0.01 0.01 min , 1.67 0.006 0.006 Process Specs 1.49 1.494 1.506 1.51 Packaged Goods What are the Tolerance Levels? What we have to do to measure capability? What are the sources of variability? Production Process Make Candy Make Candy Make Candy Mix Package Put in big bags Wrong wt. Wrong wt. Make Candy Mix % Make Candy Make Candy Candy irregularity Processes Involved Candy Manufacturing: Mixing: Is proper color mix in each bag? Individual packages: Are M&Ms uniform size & weight? Should be easier with plain than peanut Percentage of broken items (probably from printing) Are same # put in each package? Is same weight put in each package? Large bags: Are same number of packages put in each bag? Is same weight put in each bag? Weighing Package and all candies Before placing candy on scale, press “ON/TARE” button Wait for 0.00 to appear If it doesn’t say “g”, press Cal/Mode button a few times Write weight down on form Candy colors 1. 2. 3. 4. 5. 6. 7. Write Name on form Write weight on form Write Package # on form Count # of each color and write on form Count total # of candies and write on form (Advanced only): Eat candies Turn in forms and complete wrappers The effects of rounding 25.00 0.80 24.00 g - rounded oz - rounded 23.00 0.7 Ounces 0.70 Rounded Weight - Ounces Rounded Weight - grams 22.00 21 grams 21.00 20 grams 20.00 0.6 Ounces 0.60 19 grams 19.00 18 grams 18.00 17.00 0.50 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 Original Weight in grams 19.5 20.0 20.5 21.0 21.5 22.0 22.5 Peanut Candy Weights Avg. 2.18, stdv 0.242, c.v. = 0.111 Peanut Individuals 9 8 7 6 Count 5 4 3 2 1 0 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 Mass (g) 2.4 2.5 2.6 2.7 2.8 2.9 3 Plain Candy Weights Avg 0.858, StDev 0.035, C.V. 0.0413 Individual Plain Candies 16 14 12 10 8 6 4 2 0 0. 78 0. 79 0. 8 0. 81 0. 82 0. 83 0. 84 0. 85 0. 86 0. 87 0. 88 0. 89 0. 9 0. 91 0. 92 0. 93 0. 94 0. 95 0. 96 0. 97 Count Mass (g) Peanut Color Mix Brown 17.7% Yellow 8.2% Red 9.5% Blue 15.4% Orange 26.4% Green 22.7% website 20% 20% 20% 20% 10% 10% Plain Color Mix Brown Yellow Red Blue Orange Green Class 12.1% 14.7% 11.4% 19.5% 21.2% 21.2% website 30% 20% 20% 10% 10% 10% So who cares? Dept. of Commerce National Institutes of Standards & Technology NIST Handbook 133 Fair Packaging and Labeling Act Acceptable? Package Weight “Not Labeled for Individual Retail Sale” If individual is 18g MAV is 10% = 1.8g Nothing can be below 18g – 1.8g = 16.2g Goal of Control Charts collect and present data visually allow us to see when trend appears see when “out of control” point occurs Process Control Charts X Graph of sample data plotted over time 60 50 40 30 20 10 0 UCL LCL 1 2 3 4 5 6 7 8 9 10 11 12 Time Process Average ± 3 Process Control Charts Graph of sample data plotted over time X 60 Assignable Cause Variation Natural Variation 50 40 30 20 10 0 UCL LCL 1 2 3 4 5 6 7 8 9 10 11 12 Time Definitions of Out of Control 1. 2. 3. 4. No points outside control limits Same number above & below center line Points seem to fall randomly above and below center line Most are near the center line, only a few are close to control limits 1. 2. 3. 8 Consecutive pts on one side of centerline 2 of 3 points in outer third 4 of 5 in outer two-thirds region Attributes vs. Variables Attributes: Good / bad, works / doesn’t count % bad (P chart) count # defects / item (C chart) Variables: measure length, weight, temperature (x-bar chart) measure variability in length (R chart) Attribute Control Charts Tell us whether points in tolerance or not p chart: percentage with given characteristic (usually whether defective or not) np chart: number of units with characteristic c chart: count # of occurrences in a fixed area of opportunity (defects per car) u chart: # of events in a changeable area of opportunity (sq. yards of paper drawn from a machine) p Chart Control Limits UCLp p z p 1 p n k p X i i1 k n i1 i # Defective Items in Sample i Sample i Size p Chart Control Limits p 1 p UCLp p z n k k n # Samples n i1 k i p X i i1 k n i1 i z = 2 for 95.5% limits; z = 3 for 99.7% limits # Defective Items in Sample i Sample i Size p Chart Control Limits p 1 p UCLp p z n p 1 p LCLp p z n k k n # Samples n i1 k i p X i i1 k n i1 i z = 2 for 95.5% limits; z = 3 for 99.7% limits # Defective Items in Sample i Sample i Size p Chart Example You’re manager of a 500room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control (use z = 3)? © 1995 Corel Corp. p Chart Hotel Data Day 1 2 3 4 5 6 7 No. Rooms 200 200 200 200 200 200 200 No. Not Ready Proportion 16 16/200 = .080 7 .035 21 .105 17 .085 25 .125 19 .095 16 .080 p Chart Control Limits k n n i1 k i 1400 200 7 p Chart Control Limits k k n n i1 k i 1400 200 7 p X i i1 k n i1 16 + 7 +...+ 16 i 121 0.0864 1400 p Chart Solution k k n n i1 k i 1400 200 7 p X 16 + 7 +...+ 16 i i1 k n 121 0.0864 1400 i i1 p z p 1 p 0.0864 1 0.0864 0.0864 3 n 200 p Chart Solution k k n n i1 k i 1400 200 7 p X 16 + 7 +...+ 16 i i1 k n 121 0.0864 1400 i i1 p z p 1 p 0.0864 1 0.0864 0.0864 3 n 200 0.0864 3* 0.01984 0.0864 0.01984 0.1460, and 0.0268 p Chart 0.15 P UCL 0.10 0.05 LCL 0.00 1 2 3 4 Day 5 6 7 R Chart Type of variables control chart Shows sample ranges over time Interval or ratio scaled numerical data Difference between smallest & largest values in inspection sample Monitors variability in process Example: Weigh samples of coffee & compute ranges of samples; Plot Hotel Example You’re manager of a 500room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control? Hotel Data Day Delivery Time 1 7.30 4.20 6.10 3.45 2 4.60 8.70 7.60 4.43 3 5.98 2.92 6.20 4.20 4 7.20 5.10 5.19 6.80 5 4.00 4.50 5.50 1.89 6 10.10 8.10 6.50 5.06 7 6.77 5.08 5.90 6.90 5.55 7.62 5.10 4.21 4.46 6.94 9.30 R &X Chart Hotel Data Day Delivery Time 1 7.30 4.20 6.10 3.45 5.55 Sample Mean Range 5.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55 Sample Mean = 5 R &X Chart Hotel Data Day Delivery Time 1 7.30 4.20 6.10 3.45 5.55 Largest Smallest Sample Mean Range 5.32 3.85 Sample Range = 7.30 - 3.45 R &X Chart Hotel Data Day 1 7.30 2 4.60 3 5.98 4 7.20 5 4.00 6 10.10 7 6.77 Delivery Time 4.20 6.10 3.45 8.70 7.60 4.43 2.92 6.20 4.20 5.10 5.19 6.80 4.50 5.50 1.89 8.10 6.50 5.06 5.08 5.90 6.90 5.55 7.62 5.10 4.21 4.46 6.94 9.30 Sample Mean Range 5.32 3.85 6.59 4.27 4.88 3.28 5.70 2.99 4.07 3.61 7.34 5.04 6.79 4.22 R Chart Control Limits UCLR D4 R From Exhibit 6.13 LCLR D3 R k R Ri i 1 k Sample Range at Time i # Samples Control Chart Limits n A2 D3 D4 2 1.88 0 3.278 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 R Chart Control Limits k R Ri i 1 k 3.85 4.27 4.22 3.894 7 R Chart Solution k R Ri i 1 k 3.85 4.27 4.22 3.894 7 UCLR D4 R (2.11) (3.894) 8.232 LCLR D3 R (0)(3.894) 0 From 6.13 (n = 5) R Chart Solution R, Minutes 8 6 4 2 0 1 2 UCL 3 4 Day 5 6 7 X Chart Control Limits UCLX X A2 R Sample Mean at Time i k X X i 1 k k i R R i 1 k i Sample Range at Time i # Samples X Chart Control Limits From Table 6-13 UCLX X A2 R LCLX X A2 R k X Xi i 1 k k R Ri i 1 k X Chart Control Limits From 6.13 UCLX X A2 R Sample Mean at Time i LCLX X A2 R k X Xi i 1 k k R Ri i 1 k Sample Range at Time i # Samples Exhibit 6.13 Limits n A2 D3 D4 2 1.88 0 3.278 3 1.02 0 2.57 4 0.73 0 2.28 5 0.58 0 2.11 6 0.48 0 2.00 7 0.42 0.08 1.92 R &X Chart Hotel Data Day 1 7.30 2 4.60 3 5.98 4 7.20 5 4.00 6 10.10 7 6.77 Delivery Time 4.20 6.10 3.45 8.70 7.60 4.43 2.92 6.20 4.20 5.10 5.19 6.80 4.50 5.50 1.89 8.10 6.50 5.06 5.08 5.90 6.90 5.55 7.62 5.10 4.21 4.46 6.94 9.30 Sample Mean Range 5.32 3.85 6.59 4.27 4.88 3.28 5.70 2.99 4.07 3.61 7.34 5.04 6.79 4.22 X Chart Control Limits k X Xi i 1 k 5.32 6.59 6.79 5.813 7 k R Ri i 1 k 3.85 4.27 4.22 3.894 7 X Chart Control Limits k X Xi i 1 k 5.32 6.59 6.79 5.813 7 k R Ri i 1 k From 6.13 3.85 4.27 4.22 3.894 (n = 5) 7 UCLX X A2 R 5.813 0.58 * 3.894 8.060 X Chart Solution k X Xi i 1 k k R Ri i 1 k 5.32 6.59 6.79 5.813 7 From 6.13 (n = 5) 3.85 4.27 4.22 3.894 7 UCLX X A2 R 5.813 (0 .58) (3.894) = 8.060 LCLX X A2 R 5.813 (0 .58) (3.894) = 3.566 X Chart Solution* X, Minutes 8 6 4 2 0 1 2 3 UCL LCL 4 Day 5 6 7 Thinking Challenge You’re manager of a 500room hotel. The hotel owner tells you that it takes too long to deliver luggage to the room (even if the process may be in control). What do you do? N © 1995 Corel Corp. Solution Redesign the luggage delivery process Use TQM tools Cause & effect diagrams Process flow charts Pareto charts Method People Too Long Material Equipment