Control Charts
OUT-OF-CONTROL SIGNAL
A difference of four or more standard deviations Between one sample and the next sample -Sample values (usually six) steadily increasing or decreasing. And using these signals is notwithout risk- Distribution of natural variation in samples, Type I error, Type II error -Not centered Process-Both processes have the same process capability ratio (Cp) but because one is centered and one is not, they will have different process capability indices (Cpk)
Mean Chart with Known Variation
But How Does Z set the risk? Upper and lower control limits set with z = 1.35 Producer's risk(combined area of the tails):1.0 - 0.8230 = 0.1770 = 17.7% This entire area within the limits is 0.4115 x 2 = 0.8230.82.3% of samples can be expected to fall within the limits when the process is in control Producer's risk(combined area of the tails):1.0 - 0.8230 = 0.1770 = 17.7%
operating characteristic curve
a graph of the likelihood of accepting a batch, given an increasing proportion of defects within that batch -acceptable quality level- proportion of defects a consumer considers acceptable -lot tolerance percent defective- maximum proportion of defects that a consumer can tolerate -average outgoing quality- estimate of the proportion of defects that pass an acceptance sampling plan
control chart
a tool for diagnosing the outbreak of assignable variation -provides a statistical tool when assignable causes are present -random variation (natural variation or common cause)- affects virtually all production processes variation. the expected amount of variation. not a cause for concern -assignable variation (special cause of variation)- from a specific cause, you can determine a defect. if you find the cause, you find the defect. some change in the process (generally). variations can be traced to a specific cause -why do we care? if your process only exhibits natural variation, it is said to be "in control". if your process exhibits assignable variation, it has gone "out of control". out of control -> stop the process, find the cause and fix it when assignable causes are present, eliminate the bad causes and incorporate the good causes. use a control chart to determine when assignable causes are present
mean chart
control chart used in monitoring the central tendency of some characteristic within a sample, also known as an x bar chart in reference to the plotting of averages -r-chart- control chart used in monitoring the range of some characteristic within a sample -p-chart- control chart used in monitoring the proportion of some characteristic within a sample -c-chart- control chart used in monitoring the count of some characteristic within a sample -acceptance sampling- estimating the quality of conformance of large batches through inspection of smaller samples -sampling plan- defined procedure for conducting acceptance sampling, including the criteria that determines rejection of the batch being samples
process capability
given we have a control chart can we know if we have any hope of staying between the lines? Cp(process capability ratio)= design specification width / natural width of the process or =design specification width/ 6*sigma design specification width= upper spec limit - lower spec limit manufacturer of light bulbs ex. USL=1200 hours, LSL=800 hours, avg life=900 hours, sigma= 48 hours Cp= (1200-800)/6*(48)= 1.39 but would that catch the problem if we were drifting toward one of the limits in particular? process ability index Cpk= min[(USL- mean)/3*sigma, (mean- LSL)/3*sigma] =min[(1200-900)/3(48), (900-800)/3(48)] =min[2.08, .69] -> Cpk= .69 (smaller of the 2) Cpk, Cp < 1 -> process is not capable Cpk, Cp= 1 -> just capable Cpk, Cp >1 -> capable
defects ex
if UCL= 29, LCL=11 for # of defects and data is out of control=5, then data continued below the LCL- can move limits. "good" because less defects- incorporate it bc you've improved the process
determine the difference between random and assignable variation
if data falls outside upper control limit and lower control limit- out of control -if values are between 3 st dev of mean- variation due to natural causes -if values are close to UCL or LCL- "close to control line" -if values go above and below the mean (up and down like parabolas)- "run" -if it goes way above and below mean like a parabola- "erratic behavior" what are the major limitations of this approach? (tactical quality control) -assumes cost of conformance is simple taguchi loss function- further away from target- higher loss. within 3 st dev of mean- good. outside 3 st dev- bad. field goal quality (as long as a value is between the "uprights" its good
Traditional Quality Control
-Focus on conformance -Belief in an Ideal Amount of Inspection -Reliance on Inspection. Vulnerability of Traditional QC: Taguchi Loss Function- Traditional QC assumes loss caused by lack of conformance is proportional to the distance from target. Traditional loss function above target specification- positive deviation from target, below target specification- negative deviation from target -Taguchi Loss Function suggests large penalties for all but the smallest of mistakes.
c chart (attribute chart)
-used to monitor defects/unit -used when the # of occurrences can be counted, non-occurances cannot be counted what if there is something you want to track, but you cant sample for it exactly? what do you do if you are counting the number of times something happens? (poisson dist) ex. a random sample of 100 modern art dining room tables that came off the firms assembly line is examined. careful inspection reveals a total fo 200 blemishes. what are the 99.73% upper and lower control limits for the # of blemishes? if one table had 42 blemishes, should any special action be taken? 1. calculate the avg # of incidents (occurrences) c bar= 2000/100= 20 2. draw chart UCL= c bar + z*sqrt(c bar)= 20 + 3*sqrt(20) LCL= c bar - z*sqrt(c bar)= 20 - 3*sqrt(20) 42 blemishes- should special action be taken?- 42>33.42 so yes- assignable cause- investigate
r charts cont
4. draw your charts! now we should maintain two charts because it is a good idea to track sample ranges as well as sample means. UCL= x double bar + Ac*Rbar= 12.00375 + .21(.1)= 12.025 oz LCL= x double bar - Ac*Rbar= 12.00375 - .21(.1)= 11.983 oz range chart- new concern: we use the range to construct the x bar chart UCLr= D4*Rbar= 1.64(.1)= .164 oz LCLr= D3*Rbar= .36(.1)= .036 oz mean chart can show "in control but if the range is out of control, z bar chart is based on out of control data, so do range chart first -if range of sample fell out of range, the process is out of control (bad) is it bad to have small differences between the 2 most extreme # of a sample? -would need to investigate and see if there is a way to improve the process where we can drive the variability of our process down
Average outgoing quality (AOQ)
=Pac x p x ((N-n)/N) Pac= Probability of accepting a batch at level p curve) (obtain from the plan's OC) p=Actual proportion defective N = batch size n =sample size
To conduct acceptance sampling, you must have a sampling plan... this planspecifies
How large are the samples? (n) How large is the batch? (N) How many samples will be taken? What is the rule to accept or reject the batch? How many defects will cause rejection? (c) -The larger your sample size (n), the morelikely you will make the right call
type I error
in quality control, concluding the process is out of control when in fact it is not -producers risk- the likelihood of a type I error -type II error- in quality control, concluding the process is in control when in fact it is not -consumers risk- the likelihood of a type II error -process capability- the natural variation in an existing process, stated relative to the allowable variation specified in a products design -scientific management- a methodology stressing the use of data collection and analysis to redesign processes and improve efficiency -outcome bias- a tendency to assume a process is acceptable if its output is acceptable -functional organizational structure- an organization of specialists grouped not distinct depts
tactics
means to pursue strategic goals with available resources -total quality management- simultaneous and continuous pursuit of improvement in both the quality of design and conformance through the involvement of the entire org -DMIAC- define, measure, analyze, improve and control. emphasize the various phases of the six sigma methodology -ISO 9000- a certification of compliance with an internationally recognized set of quality management standards -tolerance- allowable variation from a standard -taguchi loss function- a proposed model of the cost of nonconformance that penalizes even small degrees of deviation from a larger specification
another example
suppose we want sample means to fall within the limits of 90.1% of the time, given the process is in control. where do we out the limits find z associated w .4505 z=1.65 UCL= 12.00375 + 1.65(.0125)= 12.02438 LCL= 12.00375 - 1.65(.0125)= 11.98313 another example suppose we will only stop the process if a sample mean falls outside the control limits. we are willing to accidentally stop (stop when nothing is wrong) 3% fo the time. what Z should we use for the control limits? .5 (because lower/upper half of curve is 50%)= .485 or 1-.03=.97/2= 4.85 (look up z) z=2.17 UCL= 12.00375 + 2.17(.0125)= 12.03088 LCL= 12.00375 - 2.17(.0125)= 12.03088
quality of conformance
the degree to which the output of an operation meets the producers expectations -defect- a single identifiable deviation from acceptable conformance -natural variation- the randomness inherent in a process, also known as random variation -assignable variation- deviations with a specific cause or source -statistical process control- the monitoring of overall conformance through the ongoing evaluation of samples -control chart- graph illustrating observed values in relationship to the allowable limits on those values -control limit- a control chart boundary, where values observed beyond this limit signal the process is not in control
choosing Z
there is always the possibility that a sample mean will fall outside the control limits, even thought he process is in control. what is that possibility- producers risk (type I error) 1. look up your Z on the Z table (pg 567) z=3- p= .4987 (area under normal curve) 2. double the area you get from the table. that is the probability that a sample mean will fall inside the control limits, given that the process is in control .4987 x 2=.9974 ex. z=3 99.74% change samples will fall within the limits when under control there is (1-.9974)% change of falling outside the control limits (type I error)
mean charts with unknown variation (r charts)
we are monitoring the central tendency of a process, but we dont know sigma, the st dev of random process variability. when you dont know the random variation, you must estimate it -variance, st dev, range, interquartile range 1. when gathering your samples, also record the range of each sample range= largest observation - smallest observation n=16 sample #1 range- .1, avg weight- 12.01, #2 r= .14, aw= 12.025, #3 r=.03 aw=11.995, #4 r= .1 aw= 11.985 2. calculate the average range r bar= .1+.15+.05+.1/4= .1 3. look up the factor A2 from table (pg 461) -w corresponding sample size=16 -never look up # of samples A2= .21, D1= .36, D4= 1.64 -have to have this table- z value I missing- table we use z=3
mean charts with known variation (x bar charts)
we are monitoring the central tendency of a process. (average). we assume we know sigma, the standard deviation of random process variability 1. gather several samples of size n. calculate the average of each sample ex. filling 12 oz can n=16, sigma= .05 sample #1- 12.01, #2- 12.025, #3- 11.995, #4- 11.985 2. calculate the mean of sample means x double bar= 12.01 + 12.025 + 11.995 + 11.985/4= 12.00375 3. calculate the standard deviation of the sample means not the same thing as st dev of the process itself. why?- the aves will not vary as much as individuals so we adjust it downward sigma xbar- st dev of sample means st dev of sample means= st dev/ sqrt(n)= .05/sqrt(16)= .0125 4. select "Z" and draw the chart UCL= x double bar + z*st dev of sample means= 12.00375+ 3(.0125)= 12.04125 LCL= x double bar- z* st dev of sample means= 12.00375 + 3(.0125)= 11.96625 -all sample data is between UCL and LCL so process is operating well
p charts
what if you dont measure each item in a sample, but just answer a "yes" or "no" question about it? this is an attribute. the "P" in p charts is short for proportion -observations fall into 2 categories (ex. good/bad) ex, the results of an inspection of DNA samples taken over the past 10 days are given in the table below. n=100 day 1- # of defectives= 7, proportion of defectives= 7/100= .7, day 2- # of defectives= 6, prop of defectives= 6/100= .6 1. calculate the avg proportion of defective samples p bar= sum of proportion of defectives/ # of samples 2. calculate the st dev in the proportion of defective samples =sqrt((p bar*(1-p bad))/n) 3. draw chart (z=3) UCL= p bar + Z*st dev of prop LCL= P bar - Z*st dev of prop
