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
Theoretical Basis of Control Charts Properties of normal distribution 95.5% of allX fall within ± 2
Theoretical Basis of Control Charts Properties of normal distribution 99.7% of allX 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
Heteroskedasticity Sub-groups with different variances
Design Tolerances Design tolerance: Determined by users’ needs USL -- Upper Specification Limit LSL -- Lower Specification Limit Eg: specified size +/- 0.005 inches No connection between tolerance and completely unrelated to natural variation.
Process Capability LSL USL Capable LSL USL Not Capable LSL USL LSL USL
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 LSL = 1.5 – 0.01 = 1.49 USL = 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 (Cp) will tell the position of the control limits relative to the design specifications. Cp>= 1.0, process is capable Cp< 1.0, process is not capable
Process Capability, Cp Tells how well parts produced fit into specs 3 3 LSL USL
Process Capability Tells how well parts produced fit into specs For our example: Cp=0.02/0.012 = 1.667 1.667>1.0 Process not capable
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 See if process is “in control” Process should show random values No trends or unlikely patterns Visual representation much easier to interpret Tables of data – any patterns? Spot trends, unlikely patterns easily
NFL Control Chart?
Control Charts Values UCL avg LCL Sample Number
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
Control Charts Normal Too Low Too high 5 above, or below Run of 5 Extreme variability
Control Charts UCL 2σ 1σ avg 1σ 2σ LCL
Control Charts 2 out of 3 in the outer third
Out of Control Point? Is there an “assignable cause?” Or day-to-day variability? If not usual variability, GET IT OUT Remove data point from data set, and recalculate control limits If it is regular, day-to-day variability, LEAVE IT IN Include it when calculating control limits
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)
Normality
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
Mean and Range - 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 =
Hotel Data – Mean and Range 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 Sample Mean at Time i Sample Range at Time i # Samples
X Chart Control Limits A2 from Figure 13.10
Figure 13.10 Limits Sample Size (n) A2 D4 D5 2 1.88 3.27 3 1.02 2.57 4 3.27 3 1.02 2.57 4 0.73 2.28 5 0.58 2.11 6 0.48 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74
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
X Chart Solution* ` X, Minutes 8 UCL 6 4 2 LCL 1 2 3 4 5 6 7 Day
R Chart Control Limits Figure 13.10, p.402 Sample Range at Time i # Samples
Figure 13.10 Limits Sample Size (n) A2 D4 D5 2 1.88 3.27 3 1.02 2.57 4 3.27 3 1.02 2.57 4 0.73 2.28 5 0.58 2.11 6 0.48 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74
R Chart Control Limits
R Chart Solution UCL
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 z = 2 for 95.5% limits; z = 3 for 99.7% limits # Samples
p Chart Example You’re manager of a 1,700 room hotel. For 7 days, you collect data on the readiness of all of the rooms that someone checked out of. Is the process in control (use z = 3)? © 1995 Corel Corp.
p Chart Hotel Data # Rooms No. Not Proportion Day n Ready p 1 1,300 130 130/1,300 =.100 2 800 90 .113 3 400 21 .053 4 350 25 .071 5 300 18 .06 6 400 12 .03 7 600 30 .05
p Chart Control Limits
p Chart Solution
Hotel Room Readiness P-Bar