# Statistical Process Control Operations Management Dr. Ron Lembke.

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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 95.5% of all  X fall within ± 2  Properties of normal distribution

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: Skewness = 0 Negative Skew < 0 Positive Skew > 0

Kurtosis  Amount of peakedness or flatness Kurtosis < 0 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 and 6   A “capable” process has USL and LSL 3 or more standard deviations away from the mean, or 3σ.  99.7% (or more) of product is acceptable to customers LSLUSL 33 66 LSLUSL

Process Capability LSLUSL LSL USL CapableNot Capable LSLUSL LSLUSL

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 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 1.499 1.511.491.511 Process Specs

Capability Index  Capability Index (C pk ) will tell the position of the control limits relative to the design specifications.  C pk >= 1.33, process is capable  C pk < 1.33, process is not capable

Process Capability, C pk  Tells how well parts produced fit into specs Process Specs 33 33 LSLUSL

Process Capability  Tells how well parts produced fit into specs  For our example:  C pk = min[ 0.015/.006, 0.005/0.006]  C pk = min[2.5,0.833] = 0.833 < 1.33 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 1.4941.511.491.506 Process Specs

If re-centered, it would be Capable 1.4941.511.491.506 Process Specs

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 PackagePut in big bags Make Candy Mix Mix % Candy irregularity Wrong wt.

Processes Involved  Candy Manufacturing: 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 1. Write Name on form 2. Write weight on form 3. Write Package # on form 4. Count # of each color and write on form 5. Count total # of candies and write on form 6. (Advanced only): Eat candies 7. 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%  Blue15.4%20%  Orange26.4%10%  Green22.7%10%

Classwebsite  Brown12.1%30%  Yellow14.7%20%  Red11.4%20%  Blue19.5%10%  Orange21.2%10%  Green21.2%10% Plain Color Mix

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 UCL LCL avg Values Sample Number

Definitions of Out of Control 1. No points outside control limits 2. Same number above & below center line 3. Points seem to fall randomly above and below center line 4. Most are near the center line, only a few are close to control limits 1. 8 Consecutive pts on one side of centerline 2. 2 of 3 points in outer third 3. 4 of 5 in outer two-thirds region

Control Charts NormalToo LowToo high 5 above, or belowRun of 5 Extreme variability

Control Charts UCL LCL avg 1σ1σ 2σ2σ 2σ2σ 1σ1σ

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

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)

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)

p Chart Control Limits # Defective Items in Sample i Sample i Size

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

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 # RoomsNo. NotProportion DaynReady p 11,300130130/1,300 =.100 2800 90.113 340021.053 435025.071 530018.06 640012.03 760030.05

p Chart Control Limits

p Chart Solution

R Chart  Type of variables control chart 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

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 Example

Hotel Data DayDelivery Time 17.304.206.103.455.55 24.608.707.604.437.62 35.982.926.204.205.10 47.205.105.196.804.21 54.004.505.501.894.46 610.108.106.505.066.94 76.775.085.906.909.30

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55 5 Sample Mean =

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 7.30 - 3.45Sample Range = LargestSmallest

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 24.608.707.604.437.626.594.27 35.982.926.204.205.104.883.28 47.205.105.196.804.215.702.99 54.004.505.501.894.464.073.61 610.108.106.505.066.947.345.04 76.775.085.906.909.306.794.22

R Chart Control Limits Sample Range at Time i # Samples Table 10.3, p.433

Control Chart Limits

R Chart Control Limits

R Chart Solution UCL

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

 X Chart Control Limits A 2 from Table 10-3

Table 10.3 Limits

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 24.608.707.604.437.626.594.27 35.982.926.204.205.104.883.28 47.205.105.196.804.215.702.99 54.004.505.501.894.464.073.61 610.108.106.505.066.947.345.04 76.775.085.906.909.306.794.22

 X Chart Control Limits

 X Chart Solution* 0 2 4 6 8 1234567  X, Minutes Day UCL LCL