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 allX fall within ± 2

Theoretical Basis of Control Charts Properties of normal distribution 99.7% of allX fall within ± 3

Skewness Lack of symmetry Pearson’s coefficient of skewness: Positive Skew > 0 Skewness = 0 Negative Skew < 0

Kurtosis Amount of peakedness or flatness Kurtosis = 0 Kurtosis < 0

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 LTL UTL 3 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 Mean: 1.505 Std. Dev. = 0.002 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 Process Specs 3 3 LTL UTL

Process Capability Tells how well parts produced fit into specs For our example: 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 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 Mix % Wrong wt. Wrong wt. Candy irregularity Make Candy Make Candy Make Candy Mix Package Put in big bags Make Candy Mix % Wrong wt. Wrong wt. Make Candy Make Candy Candy irregularity

Processes Involved Candy Manufacturing: Mixing: Individual packages: Are M&Ms uniform size & weight? Should be easier with plain than peanut Percentage of broken items (probably from printing) Mixing: Is proper color mix in each bag? Individual packages: 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 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

Peanut Candy Weights Avg. 2.18, stdv 0.242, c.v. = 0.111

Plain Candy Weights Avg 0.858, StDev 0.035, C.V. 0.0413

Peanut Color Mix website Brown 17.7% 20% Yellow 8.2% 20% Red 9.5% 20% Blue 15.4% 20% Orange 26.4% 10% Green 22.7% 10%

Plain Color Mix Class website Brown 12.1% 30% Yellow 14.7% 20% Red 11.4% 20% Blue 19.5% 10% Orange 21.2% 10% Green 21.2% 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 Graph of sample data plotted over time X UCL Process Average ± 3 LCL Time

Process Control Charts Graph of sample data plotted over time X UCL Assignable Cause Variation LCL Natural Variation Time

Definitions of Out of Control 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 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 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 # Defective Items in Sample i Sample i Size

p Chart Control Limits z = 2 for 95.5% limits; z = 3 for 99.7% limits # Defective Items in Sample i Sample i Size # Samples

p Chart Control Limits z = 2 for 95.5% limits; z = 3 for 99.7% limits # Defective Items in Sample i Sample i Size # Samples

p Chart Example You’re manager of a 500-room 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 No. No. Not Day Rooms Ready Proportion 1 200 16 16/200 = .080 2 200 7 .035 3 200 21 .105 4 200 17 .085 5 200 25 .125 6 200 19 .095 7 200 16 .080

p Chart Control Limits

p Chart Control Limits 16 + 7 +...+ 16

p Chart Solution 16 + 7 +...+ 16

p Chart Solution 16 + 7 +...+ 16

p Chart UCL LCL

R Chart Type of variables control chart Shows sample ranges over time Interval or ratio scaled numerical data Shows sample ranges over time 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 500-room 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 5.55 2 4.60 8.70 7.60 4.43 7.62 3 5.98 2.92 6.20 4.20 5.10 4 7.20 5.10 5.19 6.80 4.21 5 4.00 4.50 5.50 1.89 4.46 6 10.10 8.10 6.50 5.06 6.94 7 6.77 5.08 5.90 6.90 9.30

R &X Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55 5 Sample Mean =

R &X Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 Largest Smallest 7.30 - 3.45 Sample Range =

R &X Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.70 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20 5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70 2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

R Chart Control Limits From Exhibit 6.13 Sample Range at Time i # Samples

Control Chart Limits

 R Chart Control Limits R 3 . 85  4 . 27    4 . 22 R    3 . k  R i 3 . 85  4 . 27    4 . 22 R  i  1   3 . 894 k 7

 R Chart Solution R 3 . 85  4 . 27    4 . 22 R    3 . 894 k 7 1   3 . 894 k 7 UCL  D  R  (2.11) (3.894)  8 . 232 R 4 From 6.13 (n = 5) LCL  D  R  (0) (3.894)  R 3

R Chart Solution UCL

X Chart Control Limits Sample Mean at Time i Sample Range at Time i # Samples

X Chart Control Limits From Table 6-13

X Chart Control Limits From 6.13 Sample Mean at Time i Sample Range at Time i # Samples

Exhibit 6.13 Limits

R &X Chart Hotel Data Sample Day Delivery Time Mean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.70 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20 5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70 2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

X Chart Control Limits k  X i 5 . 32  6 . 59    6 . 79 X  i  1   5 . 813 k 7 k  R i 3 . 85  4 . 27    4 . 22 R  i  1   3 . 894 k 7

X Chart Control Limits k  X i 5 . 32  6 . 59    6 . 79 X  i  1   5 . 813 k 7 k  R i 3 . 85  4 . 27    4 . 22 From 6.13 (n = 5) R  i  1   3 . 894 k 7 UCL  X  A  R  5 . 813  . 58 * 3 . 894  8 . 060 X 2

X Chart Solution   X 5 . 32  6 . 59    6 . 79 X    5 . 813 k From 6.13 (n = 5)  R i 3 . 85  4 . 27    4 . 22 R  i  1   3 . 894 k 7 UCL  X  A  R  5 . 813  (0 . 58) (3.894) = 8.060 X 2 LCL  X  A  R  5 . 813  (0 . 58) (3.894) = 3.566 X 2

X Chart Solution* ` X, Minutes 8 UCL 6 4 2 LCL 1 2 3 4 5 6 7 Day

Thinking Challenge You’re manager of a 500-room 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