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UCL UCL UCL LCL LCL Additional improvements LCL Process centered made to the process Process not centered and stable and not stable Quality Management: Total Quality Management and Statistical Process Control S. Cholette BUS786 1 Outline of Quality Module • Quality Management is very important for Operations Management. It is the entire focus of one DS class. • However, can only introduce many of the ideas and methods in our limited time. The emphasis in this class will be on Total Quality Management (TQM) and Statistical Process Control (SPC) • We skip around parts of 2 chapters of Stevenson: 1. Chapter 9: Management of Quality – Read all for background, but focus on Quality Tools – Chapter 10: Quality Control- read about Statistical Process Control – We will not study acceptance sampling or run tests • Focus is on identifying, creating and interpreting x, R, p and c charts 2 Introduction to Quality- Chapter 9 • What does the term quality mean? Quality is the ability of a product or service to consistently meet or exceed customer expectations 3 Dimensions of Quality 1. Performance - main characteristics of the product/service • Conformance - how well product/service conforms to specifications and to customer’s expectations 2. Aesthetics - appearance, feel, smell, taste 3. Special features - extra characteristics 4. Safety - Risk of injury 5. Reliability - consistency of performance 6. Durability - useful life of the product/service 7. Perceived Quality - indirect evaluation of quality (e.g. reputation) 8. Service after sale - handling of customer complaints or checking on customer satisfaction 4 Costs of Quality • Failure Costs - costs incurred by defective parts/products or faulty services – Internal Failure Costs • Costs incurred to fix problems that are detected before the product/service is delivered to the customer. – External Failure Costs • All costs incurred to fix problems that are detected after the product/service is delivered to the customer. • Appraisal Costs – All product and/or service inspection costs • Prevention Costs – All quality training, quality planning, customer assessment, process control, and quality improvement costs to prevent defects from occurring • In general, increasing spending (appropriately) on appraisal and prevention will lower failure costs 5 The Consequences of Poor Quality 1. Loss of business 2. Liability 3. Productivity – Rework consumes Cost of detection (dollars) capacity 4. Costs • Furthermore, the cost of poor quality increases with later detection Process Final testing Customer When defect is detected 6 Real Life Examples of Quality Failures: Thousands Given Wrong STD Results 10/30/03 Yahoo News • CRANBROOK, British Columbia - About 3,000 people got the wrong results when they were tested for gonorrhea and chlamydia over an 18-month period, health officials say. Because of a faulty diagnostic machine in this southeastern British Columbia town, test results for the two sexually transmitted diseases were reversed …. Officials were notified of the defective BD ProbeTec in July by the manufacturer, Becton, Dickenson and Co. of Paramus, N.J…. said. "The machine was flipping the tests results,… In other words, if you were a positive, you would have received a negative reading. If you were a negative, you would have received a positive reading." • About 3,000 people were tested between Nov. 1, 2000, and May 24, 2002, and about 83 were told they were clean when they actually had one of the diseases. Most of the 83 have been contacted but not all, Paine said. The rest of the 3,000 were told they were infected and were given treatment although they did not have the diseases, Paine said. 7 Ethics and Quality • Substandard work – Defective products – Substandard service – Poor designs – Shoddy workmanship – Substandard parts and materials Having knowledge of this and failing to correct and report it in a timely manner is unethical. 8 Some “Quality” Humor • The emphasis on quality has changed over the past few decades, with Japan leading the West in terms of overall defect levels • Back in the 1970’s IBM Canada Ltd. of Markham, Ont., ordered some parts from a new supplier in Japan. The company noted in its order that acceptable quality allowed for 1.5 percent defects (a fairly high standard in North America at the time). The Japanese sent the order, with a few parts packaged separately in plastic. The accompanying letter said: "We don't know why you want 1.5 per cent defective parts, but for your convenience, we've packed them separately." 9 A Case for Quality • Many modern manufactured goods, especially computers and electronic goods, have hundreds or thousands of component parts • Assume that for the device to work: a) All of the components have to perform to specification, and b) All components have the same probability of failure, with the chance of failure for one component independent of all others • The table shows the effect of improving quality (lowering probability of defective parts) on overall product yield Probability a Chance of device Chance of part is Chance one # parts in with 100 parts # parts in device with 1000 defective part works a device working a device parts working Comments 1% defect rate generally acceptable 1.0000% 99.000% 100 36.60% 1000 0.004% in old-style manufacturing The minimum for a "capable" process- Cp = 1, where 99.74% of 0.2600% 99.740% 100 77.08% 1000 7.402% output in spec +/- 4-sigma, process is capable, 0.0032% 99.997% 100 99.68% 1000 96.881% but has not achieved 6-sigma Motorola's famous 6-sigma goal (Cp = 2), fewer than 3.4 defective parts 0.0003% 100.000% 100 99.97% 1000 99.661% per million 10 Total Quality Management TQM: A philosophy that involves everyone in an organization in a continual effort to improve quality and achieve customer satisfaction TQM Approach •Find out what the customer wants •Design a product or service that meets or exceeds customer wants •Design processes that facilitates doing the job right the first time •Keep track of results •Extend these concepts to suppliers 11 Continuous Improvement • Philosophy that seeks to make never-ending improvements to the process of converting inputs into outputs. • Commonly practiced in Japan before US – Kaizen: Japanese word for continuous improvement – However, the quality guru attributed with helping Japanese Manufacturers engage in these practices was American – And his name is… 12 Deming’s 14 Points 1. Create constancy of purpose 2. Adopt a new philosophy- we no longer have to accept defects as a way of life 3. Cease dependence on mass inspection 4. End the practice of buying on price alone (and buying from suppliers who can’t meet standards) 5. Use statistical methods to find problems and continually improve • SPC will be a tool in quest for continual improvement 6. Institute modern methods of on-the-job training 7. Improve supervision 8. Drive out fear 9. Break down departmental barriers 10. Eliminate numerical goals and slogans (at least arbitrary ones…) 11. Eliminate work standards that simply set quotas 12. Remove barriers between employees and their pride of workmanship 13. Institute a vigorous program of education and retraining 14. Structure management to continually support these points 13 Process Improvement and Tools • Process improvement - a systematic approach to improving a process – Process mapping – Analyze the process – Redesign the process • Tools – There are a number of tools that can be used for problem solving and process improvement – Tools aid in data collection and interpretation, and provide the basis for decision making 14 Basic Quality Tools • Flowcharts – useful for representing the process and identifying potential weak links • Check sheets – tallies by defect/time/attribute • Histograms • Pareto Charts – like histograms, except ordered by decreasing # of defects. Usually include a cumulative problem line. • Scatter diagrams – shows the correlation between two variables (remember Forecasting?) • Control charts (talk about later in Ch 10) • Cause-and-effect diagrams • Run charts- More general than Control Charts. – E.g. plot # of defects over time 15 Check Sheet Can make it multi-dimensionally by adding element of time or another variable Product Defects Defect type Tally Total A. Tears in fabric //// 4 B. Discolored fabric /// 3 C. Broken fiber board //// //// //// //// //// //// //// / 36 D. Ragged edges //// // 7 Total 50 16 Pareto Chart • 80% of the problems may be from 20% of the causes! • What percent of complaints are caused by the coffee machine under filling the cups? Coffee Vending Machine Complaints, Week1 140 Underfilled cup 50 Too Hot 36 120 Burnt Taste 21 100 Grainy Residue 15 80 Other 8 total 130 60 40 20 0 Underfilled cup Too Hot Burnt Taste Grainy Residue Other 17 Control Chart We will study these in greater detail in SPC – Chapter 10 1020 UCL 1010 1000 990 980 LCL 970 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 Tracking Improvements via Control Chart Week 1 Week 2 Week 7 UCL UCL UCL 1000ml 990ml LCL LCL Additional improvements LCL Process centered made to the process Process not centered and stable and not stable In a bottling plant that fills 1-liter bottles of soda, we can see that have successfully engaged in continuous improvement! 19 Tracking Improvements via Pareto Chart Coffee Vending Machine Complaints, Week1 Coffee Vending Machine Complaints, Week 2 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 Underfilled cup Too Hot Burnt Taste Grainy Residue Other Too Hot Burnt Taste Grainy cup overflows Other Residue • Improved complaint levels from 130/week to 80/week. • Predominant problem has changed • What should we focus on next? 20 Examples: Using TQM Tools • (# 2, Stevenson, p. 422) An air-conditioning repair shop is trying to determine the primary reasons for service calls in the previous week. Calls are recorded by whether they originate from residential or commercial customers and are categorized by problem type (Noisy, Warm, Odor, Failed) • Create a checklist by problem type and customer type, as well as a Pareto chart for each customer type. What problems should we look into? Call# problem r/c Call# problem r/c Call# problem r/c Call# problem r/c 301 f r 311 n r 321 f r 331 n r 302 o r 312 f c 322 o r 332 w r 303 n c 313 n r 323 f r 333 o r 304 n r 314 w c 324 n c 334 o c 305 w c 315 f c 325 f r 335 n r 306 n r 316 o c 326 o r 336 w r 307 f r 317 w c 327 w c 337 o c 308 n c 318 n r 328 o c 338 o r 309 w r 319 o c 329 o c 339 f r 310 n r 320 f r 330 n r 340 n r 341 o c 21 Examples: Using TQM Tools • Our checklist is organized by problem type and customer type. From here we can see that Residential customer complaints outnumber commercial complaints and that residential customers are more likely to complain about noise problems, where commercial customers are more likely to complain about odor. resident commerc r c total noisy n 10 3 13 failed f 7 2 9 odor o 5 7 12 warm w 3 4 7 total defect 25 16 41 22 Examples: Using TQM Tools • We create two pareto charts, one for each type of customer. The pareto charts show that these two different types of customers experience a different frequency of problems. (Note that the order of the attributes is different for each pareto chart) Complaints: Residential 30 25 Complaints: Commercial 20 20 # calls 15 15 # calls 10 10 5 5 0 0 o w n f n f o w type of complaint 23 type of complaint Chapter 10: Quality Control • As you might expect, this entails using statistical methods to assist with Quality Control • Two topics are traditionally covered 1. Inspection • Acceptance Sampling is in the supplement to Chapter 10 which we do not cover in this class 2. Statistical Process Control (SPC) 24 Inspection • How Much/How Often • Where/When • Centralized vs. On-site Inputs Transformation Outputs Acceptance Process Acceptance sampling control sampling 25 Inspection Costs Too little inspection and we incur costs from not catching defective items Too much inspection and incur large inspection costs - especially costly if need to use destructive sampling (fizz testing -beer cans) Cost Total Cost Cost of inspection Cost of passing defectives Optimal 26 Amount of Inspection Statistical Process Control • Definition: statistical evaluation of the output of a process during production – Take periodic samples and measure to a standard. If results are unacceptable, stop the process and take corrective action • The Control Process – Define – Measure – Compare to a standard – Evaluate – Take corrective action (if necessary) – Evaluate the corrective action (Did we fix it?) 27 Statistical Process Control • Variations and Control – Random variation: Natural variations in the output of process, created by countless minor factors and unable to be eliminated – Assignable variation: (Causal, non-random) A variation whose source can be identified and eliminated 28 Sampling Distribution • Because our sample has multiple observations, it will have a tighter distribution (less variation) than that of the process we are sampling from • Many factors determine how large the sample size should be, but for the purposes of this class, it will be provided • Also thanks to the Central Limit Theorem, the sampling distribution will be normal, even if the underlying process isn’t normally distributed Sampling distribution Process distribution 29 Mean Review: Normal Distribution Unlike in inventory management, we are now concerned about both being too high and too low. (Goldilocks Phenomenon.) • There is a .0228 probability a value falls 2 below the mean and a 1-.9772 = .0228 probability a value falls 2 above the mean. This results in a .0456 probability that a value falls outside the interval of [m-2,m+2] • If we expand the interval to [m-3,m+3], the probability of being outside drops to .0026 (an order of magnitude less probable!) Standard deviation -3 -2 +2 +3 Mean 95.44% 99.74% 30 Example: Using Normal Distributions for Quality Lard-Os potato chips guarantees that all snack-sized bags of chips are between 16 and 17 ounces. The machine that fills the bags has an output that can be approximated by a normal distribution, with a mean of 16.5 ounces and a standard deviation of .25 oz. What is the percentage of bags that violate Lard-O’s promise? • Given m16.5,.25 We are above/below specified weight if we go over/under it by .5 oz. • We can compute the Z-values, where z = (spec – m)/ (16-16.5)/.25 = -.5/.25 -> z = -2 (17-16.5)/.25 = .5/.25 -> z = 2 • Using a table lookup on p.888 for z = -2, yields .0228 = P(x<16oz) and a lookup for z = 2 gives .9772 Remember: (P(x>17) = 1 - P(x<17) = 1 - .9772 = .0228. Hint: If mean is centered between upper and lower bounds, you only need to determin z’s once and multiply by 2! • Therefore Lard=O’s violate their guarantee just under 5% of the time (2*.0228 = .0456) 31 Control Limits and Errors Type I error: Probability of searching for a cause when none exists UCL Process average LCL (a) Three-sigma limits 33 Control Limits and Errors Type I error: Probability of searching for a cause when none exists UCL Process average LCL (b) Two-sigma limits 34 Control Limits and Errors Type II error: Probability of concluding that nothing has changed when process has! UCL Shift in process average Process average LCL (a) Three-sigma limits 35 Control Limits and Errors Type II error: Probability of concluding that nothing has changed when process has! UCL Shift in process average Process average LCL (b) Two-sigma limits 36 Selecting Control Limits: Summary • Using wider control limits, a 3-sigma (3) instead of 2- sigma (2) range, will decrease chance of Type I errors. But it will increase chance of incurring Type II errors • In this class you will told which control limits to use. Typically 3-sigma limits will be used, but we may also look at 2-sigma 37 Control Chart Abnormal variation Out of due to assignable sources control UCL Mean Normal variation due to chance LCL Abnormal variation due to assignable sources 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Sample number 38 Control Charts for Variables • We use two different charts to monitor measurable variables • Examples: diameter of a tube, length of a piston rod, weight of a package, temperature of a cup of coffee • Ideally, we would like the process to be invariant. Deviations from specification in either direction are usually undesirable. (Too hot or too cold, Too small or too big, Too hard or too soft) 1. Mean control chart (X-bar, x ) measures the central tendency of a process 2. Range control chart (R) measures process dispersion • Later we will also discuss attribute charts (p-Charts and c- Charts) when we can’t measure • If any of the sample means fall outside the control limits, the process is considered Out of Control! 39 X-Bar / Mean Charts • We have two different choices for computing control limits 1. If process std. deviation, is known (or can be estimated from historical data) UCL x + z x LCL x - z x Where: x /√n (standard deviation of distribution of sample means) n = sample size z = standard normal deviate x = mean of sample means or target process value -sometimes called x-bar-bar. - if we know the true process mean we would use that instead 40 Example: X-bar Charts When is Known • Stevenson, #12: A hotel is investigating the speed of the check- in process. Traditionally, the process standard deviation () has been.146 minutes. Below are 39 samples (each sample has 14 observations) Is the process in control (using 3-sigma control limits)? sample avg time sample avg time sample avg time 1 3.86 14 3.81 27 3.81 2 3.9 15 3.83 28 3.86 3 3.83 16 3.86 29 3.98 4 3.81 17 3.82 30 3.96 5 3.84 18 3.86 31 3.88 6 3.83 19 3.84 32 3.76 7 3.87 20 3.87 33 3.83 8 3.88 21 3.84 34 3.77 9 3.84 22 3.82 35 3.86 10 3.8 23 3.89 36 3.8 11 3.88 24 3.86 37 3.84 12 3.86 25 3.88 38 3.79 13 3.88 26 3.9 39 3.85 41 Example: X-bar Charts When is Known • Given we know , we can use method 1 • We have been given the control limits of 3-sigma, so z = +/- 3 • Determine the mean of means (x-bar-bar) = 3.85 • Now since we know x-bar-bar, , and N, we use the formulae: UCL x + z x = 3.85 + 3*.146/(sqrt(14)) = 3.97 LCL x - z x = 3.85 - 3*.146/(sqrt(14)) = 3.73 • Remember to factor in that sqrt(N), for N = sample size. – We are actually taking into account 14*39 = 546 data points! 42 Example: X-bar Charts When is Known • Plotting the observations against the control limits shows that sample 29 is above the UCL. Therefore we would conclude the process is out of control. 3- Sigma Control Chart: Check-in Timing 4.00 3.95 3.90 3.85 LCL 3.80 data 3.75 UCL 3.70 3.65 3.60 1 4 7 10 13 16 19 22 25 28 31 34 37 43 Example extension: 2 vs. 3 Control Limits • 1 observation out of 39 might not suggest process was greatly out of control, but if we had used stricter limits (2- sigma) we would be even more alert. for 2 sigma, UCL = 3.85 + 2*.146/(sqrt(14)) = 3.93 LCL = 3.85 - 2*.146/(sqrt(14)) = 3.77 2-sigma Control Limits for Check-in Process 4.000 3.950 3.900 lcl 3.850 data 3.800 ucl 3.750 3.700 3.650 44 1 4 7 10 13 16 19 22 25 28 31 34 37 X-Bar / Mean Charts, Revisited 2. If process standard deviation is unknown, (perhaps the process is new or recently revised) or if we are satisfied with using the Average Range (works fine for sample sizes up to 10), we can use these, instead: UCL x + A2 R LCL x - A2 R Where x average of sample means or target value R average of sample Ranges (max - min) A2 control factor (see chart, p. 429) Note: this approach assumes that the process dispersion is in control • if not, our control limits would be too loose 45 • to make sure, we would graph the forthcoming R-chart first Mean and Range Charts (process mean is shifting upward) Sampling Distribution UCL x-Chart Detects shift LCL UCL Does not R-chart detect shift LCL 46 Mean and Range Charts Sampling Distribution (process variability is increasing) UCL x-Chart Does not reveal increase LCL UCL R-chart Reveals increase LCL 47 Range Chart • Range control limits are more straightforward to compute: UCL D4 R LCL D3 R Where R average of sample Ranges (max - min) D3 , D4 control factors (see chart) • For small sample sizes, (n < 7), note that LCL will be 0 • Generally we are more concerned about increased dispersion. (Exceeding UCL) – However if process variability decreases below LCL, it is worth investigating – perhaps to use as a process improvement 48 Bookmark: Factors for 3-Sigma Control Limits Also found in Table 10-3 (I will provide a copy for the final) Note that the values result in tighter CLs as sample size increases n A2 D3 D4 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 Etc…. 49 Example: Mean and Range Charts • We wish to determine if screw production is in statistical control. We have no prior information, other than the 5 samples (where each sample has 4 observations) 1 2 3 4 s1 0.5014 0.5022 0.5009 0.5027 s2 0.5015 0.5031 0.5029 0.5012 s3 0.5018 0.5026 0.5035 0.5023 s4 0.5010 0.5034 0.5011 0.5012 s5 0.5041 0.5056 0.5034 0.5035 50 Example: Mean and Range Charts 1. First compute the sample range and mean for each of the 5 samples -e.g. s1 varies between .5009 to .5027, so the range for s1 = .0018 and the average of the 4 observations in S1 is .5018, so that is the sample mean 2. Next we can compute x or x-bar-bar, the average of sample means and R or R-bar, the average of sample ranges 1 2 3 4 Range Mean s1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018 s2 0.5015 0.5031 0.5029 0.5012 0.0019 0.5022 s3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026 s4 0.501 0.5034 0.5011 0.5012 0.0024 0.5017 s5 0.5041 0.5056 0.5034 0.5035 0.0022 0.5042 R-bar= 0.002 X-bar= 0.5025 51 Example: Mean and Range Charts • Since we have no prior information on process metrics, we cannot use the first method to establish control limits for the mean. • Before we can use the second method to set control limits for the mean, we first need to determine if process variation is in control with the R – Chart - Factor table lookup shows that sample size 4, D3 = 0, D4 = 2.28 R = 0.0020 UCLR = D4R = 2.282 (0.0020) = 0.00456 LCLR = D3R = 0 (0.0020) = 0 52 Example: Range Chart 0.005 UCLR = 0.00456 0.004 Range (in.) 0.003 0.002 R = 0.0020 0.001 0 LCLR = 0 1 2 3 4 5 6 Sample number 53 Example: Control Limits for X-Chart Given the Range is in control, we can use the second approach to find the control limits for the mean chart Control limits for x - Chart R = 0.0020 A2 = 0.729 x = 0.5025 UCLx = x + A2R LCLx = x - A2R UCLx = 0.5025 + 0.729 (0.0020) = 0.5040 LCLx = 0.5025 - 0.729 (0.0020) = 0.5010 54 Example: Process Out of Control (with Respect to Mean) 0.5050 0.5040 UCLx = 0.5040 Average (in.) 0.5030 x = 0.5025 0.5020 0.5010 LCLx = 0.5010 1 2 3 4 5 6 Sample number 55 Control Charts for Attributes • For some processes, we may need to count, rather than measure the samples. – Use the term Attributes instead of Variables 1. p-Chart - Control chart used to monitor the proportion of defectives in a process • Requires that all observations can be placed in 1 of 2 categories: i.e. fresh –vs.- spoiled 2. c-Chart - Control chart used to monitor the number of defects per unit • Use only when number of occurrences per unit of measure can be counted; non-occurrences cannot be counted. i.e. Number of accidents at an intersection in 1 month. 56 Use p-Charts When: 1. When observations can be placed into one of two categories – Good or bad – Pass or fail – Operate or don’t operate 2. When the data consists of multiple samples of several observations each • Most common example: Number of defects/errors in a fixed sample size. – 8 out of 200 apples rotten-> 4% of sample is bad 57 Control Limits for the p-Chart • Error rates technically should be modeled with Binomial distribution, but instead are approximated by Normal • p-bar = average fraction defective in observations = defects observed / total observations – if we know the true population tendency, p, we can use it in place of p-bar – We estimate the standard deviation of the sample, p UCLp = p + zp LCLp = p - zp p = p(1 - p)/n 58 Example: p-Chart • Modified from Stevenson, #5: In checking several samples (where a sample is 200 statements) of credit card statements for errors, an auditor found the following. (Statements were either classified as erroneous or correct): Sample 1 2 3 4 # erroneous statements | 4 2 5 9 • What is the fraction defective in each sample? 4/200 = .02 2/200 = .01 5/200 = .025 9/200 = .045 • If the true fraction is unknown, what is your estimate for it? – Mean of sample means = (.02+ .01 + .025 + .045)/4 = .025 • What is estimate of mean and standard deviation of sampling distribution of fractions defective for samples of this size? m= .025 and p = sqrt[(1-p)(p)/N] = sqrt(.975)(.025)/200) = .011 59 Example: p-Chart continued • What are the 2-sigma control limits Re-compute UCL/LCL for new m,p UCL = m+ z*p = .025 + 2*.011= .047 LCL = m- z*p = .025 - 2*.011 = .003 • Is the process in control given the 2-sigma control limits? Yes, because all 4 sample means fall within the interval of [.003, .047] • Now assume that the long-term fraction erroneous of the process is known to be 2%. What are the values of the mean and standard deviation of the sampling distribution? We now use p = .02 instead of p = .025 to calculate mand p m.02and p = sqrt(p(1-p)/N) = sqrt(.02*.98/200) = .0099 • What would the 2-sigma control limits now be, given the above info? Re-compute UCL/LCL for new m,p UCL = m+ z*p = .02 + 2*.0099 = .0398 LCL = m- z*p = .02 - 2*.0099 = .0002 • Is process in control given the new 2-sigma control limits? No. Last sample mean, (#4 at .045) is above UCL 60 Use of c-Charts • Use only when the number of occurrences per unit of measure can be counted • Non-occurrences cannot be counted – Scratches, chips, dents, or errors per item • Example: You lend you car out to different people every week. At the end of 5 weeks, the number of new dings per week was 3,4,2,9,2. Was the person in week 4 “out of control” or is this the sort of variation you might expect? (try for both 2 and 3 sigma CLs) – Cracks or faults per unit of distance – Breaks or Tears per unit of area – Bacteria or pollutants per unit of volume – Calls, complaints, accidents, failures per unit of time – Number of typos on a page of text 61 c-Chart • In theory, should use a Poisson distribution, but, in practice, this can be approximated by a normal distribution • UCL c + z c LCL c - z c Where c process mean number of defects (use sample mean c if c unknown) 62 Example: c-Chart • A history professor is grading a midterm essay question. To take her mind off the tedium, she’s decided to count the number of egregious grammar mistakes. (Note that as this is an essay, rather than a multiple choice test, she can’t count proportion right/wrong and then use a p-Chart) • Is the process in control, given 3-sigma control limits? test #errors test #errors 1 2 8 0 2 3 9 2 3 1 10 1 4 0 11 3 5 1 12 1 6 3 13 2 7 2 14 0 63 Example: c-Chart • Since we need to use a c-chart, and we have not been given any historical information for the process mean, c, we need to the sample mean instead. c-bar = average of 14 observations = 1.5 • Now compute UCL and LCL UCL = c-bar + z*sqrt(c-bar ) = 1.5 + 3*sqrt(1.5) = 5.17 LCL = c-bar - z*sqrt(c-bar ) = 1.5 - 3*sqrt(1.5) = -2.17 -> 0 Remember to truncate negative LCL’s to 0 • Looking back at all 14 observations, none exceed the 5.17 UCL, so process is in control. 64 More Practice with Control Charts • ( Part of a question from the 412 Midterm from another professor’s class) A clothing manufacturer is monitoring sock output. Although many things can go wrong in the process of making socks, they are only recording socks as either defective or non-defective. A sample is composed of 100 randomly selected socks. Here are the results: sample 1 2 3 4 5 6 #defective 4 7 2 3 2 6 65 More Practice with Control Charts • What is the sample size? How many samples are we taking? • 100 socks per sample, 6 samples in total • What type of control chart would you use and why? • p-Chart, (Are counting attributes and can sort each observation into 1 of 2 categories) • Determine the fraction defective for each sample • S1- .04 S2- .07 S3- .02 S4- .03 S5-.02 S6- .06 (= #defects / #observations!) • If the true fraction defective for the process is unknown, what is your estimate for it? • .04 (mean of means- either average the 6 numbers above, or tally total_defects/600) • If the true fraction defective is known to be 3%, what is the mean and standard deviation of the sampling distribution? • Mean = .03. Standard Dev. = sqrt(p(1-p)/N) = sqrt(.03*.97/100) = .017 • What would the three-sigma control limits for this process be? = m+/- 3* = .03 +/- 3*(.017) = 0 to .081 ( truncate -.02 to 0 for LCL) • Is this process under control, given the three-sigma control limits? • Yes (all of the “fraction defective” for each of the 6 samples is less than .081) 66 Examples: HW Problems • Modified from Ch10, #2: An automatic filling machine is used to fill 1-liter bottles of cola. The machine’s output is known to be approximately normal a process standard deviation of = 10ml. The output is monitored using means of samples of 25 observations. • Determine the upper and lower control limits that will include about 95.5% of the sample means when the process is in control. (z =2) • Given these sample means: 1005, 1001, 998, 1002, 995, 999, is the process in control? Why/why not? 67 More HW Problems • Ch10, #3, p. 462: Checkout time at a supermarket is monitored using a mean and a range chart. (i.e. we don’t know the standard deviation of the process) Six samples of n=20 observations have been obtained and the sample means and ranges computed (so we don’t have calculate these ourselves, like we will in #4!) sample mean range 1 3.06 .42 2 3.15 .50 3 3.13 .41 4 3.11 .46 5 3.06 .46 6 3.09 .45 • Using the factors in Table 10-2, determine upper and lower limits for mean and range charts • Is the process in control? 68 Practice Old Exam Problems • 1# Quiz : A program at SF State is concerned about their graduation rates. Every year they enroll 400 junior-level students, and they’ve been tracking the percent who manage to finish the program in a timely manner. Below are the data for the past 8 years. Express all calculations to 3 decimal places: ie. 12.3% or 0.123. – Calculate the proportion of the class that graduates ontime each year – What kind of control chart should be used in this situation? Justify Why. – Compute the 3 sigma control limits – Are we in control, give the control limits? Why/Why Not? year 2000 2001 2002 2003 2004 2005 2006 2007 ontime grads 230 215 250 225 245 220 235 205 69 More Test Problem Practice For her quality control project, a student considers the height of the women players in the basketball teams in her school’s league. Each of the 6 teams has exactly 8 players. She’s recorded both the average height and the range of heights in inches- and she has no other information available. – What type chart should she use to see if the differences in height ranges between these teams is within what would be expected with random variation? Compute the control limits. – What type chart should she use to see if the differences in average heights between these teams is within what would be expected with random variation? Compute the control limits. – Are there any school teams that appear to be outliers? Range Team Mean (inches) (inches) school1 70 15 school2 76 10 school3 69 13 school4 75 14 school5 71 11 70 school6 77 15 Yet More Old Test Questions • #3: Odwalla’s OJ is packaged in 250 ml bottles and has a process standard deviation of 10 ml. In monitoring the fill process, 6 samples (of 25 bottles each) were collected and averaged. Sample Sample mean • What is the total number of bottles s1 249 sampled? s2 252 • What is the standard deviation of the s3 253 distribution of sample means s4 248 • Compute the 3-sigma control limits. Is the process in control, given these control s5 245 limits? Why/Why not? s6 253 • Compute the 2-sigma control limits. Is the process in control, given these control limits? Why/Why not? 71 Last of the Test Problems • Exam question: An avant-garde clothing manufacturer runs a series of high profile, risqué ads on a billboard on Hwy101 and regularly collects protest calls from people who are offended by them. They have no idea how many people in total see the ad, but they have been collecting statistics on the number of phone calls from irate viewers. type Description #complaints R Offensive racially/ethnically 10 M Demeaning to men 4 W Demeaning to women 14 I Ad is Incomprehensible 6 O other 2 • Depict this data with a pareto chart. Also depict the cumulative complaint line. • What percent of the total complaints can be attributed to the most prevalent complaint? 72 HW Problem Continued • The ad agency also tracks the complaints by week received. week #complaints 1 4 2 5 3 4 4 11 5 3 6 9 • What type of control chart would we use to monitor this process? (a) X-bar chart (b) R-chart (c) p-chart (d) c-chart • And why? • What are the 3-sigma control limits for this process? Assume the historical complaint rate is unknown. • Is the process mean in control, according to the control limits? Y/N Why/Why not? • BONUS Assume now that the historical complaint rate has been 4 calls a week. What would the 3-sigma control limits for this process be now? Is the process in control, according to the 73 control limits? Y/N (Circle one)