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Statistical_Quality_Control

- 2. W. Edward Deming advocated the implementation of a statistical quality management approach. His philosophy behind this approach is ‘reduce variation’- fundamental to the principle of continuous improvement and the achievement of consistency, reliability, and uniformity. It helps in trustworthiness, competitive position, and success. Statistical Quality Control Statistics: data sufficient enough to obtain a reliable result. Quality: relative term and can be defined as totality of features and characteristics of a product or service that bear on its ability to satisfy stated or implied need (ISO). Control: The operational techniques and activities (a system for measuring and checking) used to fulfill the requirements for quality. It incorporates a feedback mechanism system to explore the causes of poor quality or unsatisfactory performance and takes corrective actions. also suggests when to inspect, how often to inspect, and how much to inspect. Basic Concept
- 3. Statistical Quality Control A quality control system using statistical techniques to control quality by performing inspection, testing and analysis to conclude whether the product is as stated or designed quality standard. Relying on the probability theory, SQC evaluates batch quality and controls the quality of processes or products It makes the inspection more reliable and less costly. The basis of the measurement is the performance indicator, either individual, group or departmental calculated over time (hourly, daily, or weekly). These performance measures are plotted on a chart. Pattern obtained from plotting these measures are basis of taking appropriate actions so that The process variation in minimized and Major problems are prevented in future. The timing and type of, and responsibility for, these actions depends on whether the causes of variation is controlled or uncontrolled Basic Concept ….Cont’d
- 4. Statistical Quality Control In repetitive manufacture of a product, even with refined machinery, skilled operator, and selected material, variations are inevitable in the quality of units produced due to interactions of various causes. Variation may be due to Common or random causes of variation (as a result of normal variation in material, method, and so on that causes natural variation in product or process quality) resulting in stable pattern of variation. Special causes (changes in men, machine, materials or tools, jigs and fixture and so on) resulting in a shift from the stable pattern of variation. SQC assists in timely identification and elimination of the problem with an object of reducing variations in process or product. The application of statistical method of collecting and analyzing inspection and other data for setting the economic standards of product quality and maintaining adherence to the standards so that variation in product quality can be controlled Basic Concept ….Cont’d
- 5. Statistical quality control (SQC) is the term used to describe the set of statistical tools used by quality professionals SQC encompasses three broad categories of; Descriptive statistics used to describe quality characteristics and relationships. the mean, standard deviation, and range. Acceptance sampling used to randomly inspect a batch of products to determine acceptance or rejection of entire lot based on the results. Does not help to identify and catch the in-process problems Statistical process control (SPC) Involves inspecting the output from a process Quality characteristics are measured and charted Helpful in identifying in-process variations Three SQC Categories
- 6. Variability: Sources of Variation Variation exists in all processes. Variation can be categorized as either; Common or Random causes of variation, or Random causes that cannot be identified Unavoidable: inherent in the process Normal variation in process variables such as material, environment, method and so on. Can be reduced almost to zero only through improvements in the process variables. Assignable causes of variation Causes can be identified and eliminated e.g. poor employee training, worn tool, machine needing repair Can be controlled by operator but it needs attention of management.
- 7. Traditional Statistical Tools Descriptive Statistics include Measure of accuracy (centering) Measure of central tendency indicating the central position of the series. A measure of the central value is necessary to estimate the accuracy or centering of a process. The Mean- simply the average of a set of data Sum of all the measurements/data divided by the number of observations. The Median- simply the value of middle item if the data are arranged in ascending or descending order. Applies directly if the number in the series is odd. It lies between two middle numbers if the number of the series is even. The Mode- value that repeat itself maximum number of times in the series. Shape of Distribution of Observed Data A measure of distribution of data Normal or bell shaped Skewed n x x n 1 i i 1 K j j X K
- 8. Distribution of Data Also a measure of quality characteristics. Symmetric distribution - same number of data are observed above and below the mean. This is what we see only when normal variation is present in the data Skewed distribution – a disproportionate number of data are observed either above or below the mean. Mean and median fall at different points in the distribution Centre of gravity is shifted to oneside or other.
- 9. Traditional Statistical Tools …cont’d Measure of Precision or Spread Reveals the extent to which numerical data tend to spread about the mean value. The Range- the simplest possible measure of dispersion. Difference between largest and smallest observations in a set of data. o Depends on sample size and it tends to increase as sample size increases. o Remains the same despite changes in values lying between two extreme values. Standard Deviation- a measure deviation of the values from the mean. Small values >> data are closely clustered around the mean Large values >> data are spread out around the mean. 1 n X x σ n 1 i 2 i
- 10. Statistical Process Control Process Control Refers to procedures or techniques adopted to evaluate, maintain and improve the quality standard in various stages of manufacture. A process is considered satisfactory as long as it produces items within designed specification. Process should be continuously monitored to ensure that the process behaves as it is expected. Salient features of process control Controling the process at the right level and variability. Detecting the deviation as quickly as possible so as to take immediate corrective actions. Ultimate aim is not only to detect trouble, but also to find out the cause. Developing an efficient information system in order to establish an efficient system of process control.
- 11. Statistical Process Control Statistical Process Control (SPC) Statistical evaluation of the output of a process during production. Goal is to make the process stable over time and then keep it stable unless the planned changes are made. Statistical description of stability requires that ‘pattern of variation’ remains stable over time, not that there be no variation in the variable measured. In statistical process control language: A process that is in control has only common or random cause variation - an inherent variability of the system. When the normal functioning of the prosess is disturbed by some unpredictable events, special cause variation is added to common cause variation. Applying SPC to service Nature of defect is different in services Service defect is a failure to meet customer requirements One way to deal with service quality is to devise quantifiable measurement of service elements Number of complaints received per month, Number of telephone rings before call is answered
- 12. Hospitals timeliness and quickness of care, staff responses to requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and checkouts Grocery Stores waiting time to check out, frequency of out-of-stock items, quality of food items, cleanliness, customer complaints, checkout register errors Airlines flight delays, lost luggage and luggage handling, waiting time at ticket counters and check-in, agent and flight attendant courtesy, accurate flight information, passenger cabin cleanliness and maintenance Fast-Food Restaurants waiting time for service, customer complaints, cleanliness, food quality, order accuracy, employee courtesy Catalogue-Order Companies order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time Insurance Companies billing accuracy, timeliness of claims processing, agent availability and response time Statistical Process Control
- 13. Statistical Process Control: Control Chart Control Chart A graphical display of data over time (data are displayed in time sequence in which they occurred/measured) used to differentiate common cause variation from special cause variation. Control charts combine numerical and graphical description of data with the use of sampling distribution normal distribution is basis for control chart. Goal of using this chart is to achieve and mainatain process stability A state in which a process has displayed a certain degree of consistency Consistency is characterized by a stream of data falling within the control limits. Basic Components of a Control Chart A control chart always has a central line usually mathematical average of all the samples plotted; upper control and lower control limits defining the constraints of common variations or range of acceptable variation; Performance data plotted over time. Lines are determined from historical data.
- 14. Control Chart …Cont’d When to use a control chart? Controlling ongoing processes by finding and correcting problems as they occur. Predicting the expected range of outcomes from a process. Determining whether a process is stable (in statistical control). Analyzing patterns of process variation from special causes (non-routine events) or common causes (built into the process). Determining whether the quality improvement project should aim to prevent specific problems or to make fundamental changes to the process. Control Chart Basic Procedure Choose the appropriate control chart for the data. Determine the appropriate time period for collecting and plotting data. Collect data, construct the chart and analyze the data. Look for “out-of-control signals” on the control chart. When one is identified, mark it on the chart and investigate the cause. Document how you investigated, what you learned, the cause and how it was corrected. Continue to plot data as they are generated. As each new data point is plotted, check for new out-of-control signals
- 15. Control Chart …Cont’d Interpretation of control chart Points between control limits are due to random chance variation One or more data points above an UCL or below a LCL mark statistically significant changes in the process A process is in control if No sample points outside limits Most points near process average About equal number of points above and below centerline Points appear randomly distributed A process is assumed to be out of control if Rule 1: A single point plots outside the control limits; Rule 2: Two out of three consecutive points fall outside the two sigma warning limits on the same side of the center line; Rule 3: Four out of five consecutive points fall beyond the 1 sigma limit on the same side of the center line; Rule 4: Nine or more consecutive points fall to one side of the center line; Rule 5: There is a run of six or more consecutive points steadily increasing or decreasing Time period Measured characteristics
- 16. Control Chart …Cont’d Setting Control Limits Type I error Concluding a process is not in control when it actually is. Type II error Concluding a process is in control when it is not. In control Out of control In control No Error Type I error (producers risk) Out of control Type II Error (consumers risk) No error Mean LCL UCL /2 /2 Probability of Type I error Mean LCL UCL /2 /2 Probability of Type I error
- 17. General model for a control chart UCL = μ + kσ CL = μ LCL = μ – kσ where μ is the mean of the variable σ is the standard deviation of the variable UCL=upper control limit; LCL = lower control limit; CL = center line. k is the distance of the control limits from the center line, expressed in terms of standard deviation units. When k is set to 3, we speak of 3-sigma control charts. Historically, k = 3 has become an accepted standard in industry. Control Chart …Cont’d
- 18. Control Chart …Cont’d Suggested Number of Data Points More data points means more delay Fewer data points means less precision, wider limits A tradeoff needs to be made between more delay and less precision Generally 25 data points judged sufficient Use smaller time periods to have more data points Fewer cases may be used as approximation Sample Size Attribute charts require larger sample sizes 50 to 100 parts in a sample Variable charts require smaller sample sizes 2 to 10 parts in a sample
- 19. Control Chart …Cont’d Types of the control charts Variables control charts Variable data are measured on a continuous scale. For example: time, weight, distance or temperature can be measured in fractions or decimals. Applied to data with continuous distribution Attributes control charts Attribute data are counted and cannot have fractions or decimals. Attribute data arise when you are determining only the presence or absence of something: success or failure, accept or reject, correct or not correct. For example, a report can have four errors or five errors, but it cannot have four and a half errors. Applied to data following discrete distribution
- 20. Variable control charts X-bar (mean chart) R chart (range chart) S chart (sigma chart) Individual or run chart i-chart Moving range chart Median chart EWMA (exponentially weighted moving average chart) General formulae for a control chart UCL or UAL = μ + kσx k = 3 ; Accepted Standard UWL = μ + 2/3 kσx CL = μ LWL = μ – 2/3 kσx LCL or LAL = μ – kσx Control Chart …Cont’d m i i X X m X n
- 21. Control Chart …Cont’d Mean control charts Used to detect the variations in mean of a process. X-bar chart Range control charts Used to detect the changes in dispersion or variability of a process R chart System can show acceptable central tendencies but unacceptable variability or System can show acceptable variability but unacceptable central tendencies Use X-bar and R charts together Sample size : 2 ~ 10 Use X-bar and S charts together Sample size : > 10 Use i-chart and Moving range chart together Sample size : 1 or one-at-time data Interpret the R-chart first: If R-chart is in control -> interpret the X-bar chart -> (i) if in control: the process is in control; (ii) if out of control: the process average is out of control If R-chart is out of control: the process variation is out of control -> investigate the cause; no need to interpret the X-bar chart
- 22. Control Chart …Cont’d Constructing a X-bar Chart: A quality control inspector at the Cocoa Fizz soft drink company has taken three samples with four observations each of the volume of bottles filled. If the standard deviation of the bottling operation is 0.2 ounces, use the below data to develop control charts with limits of 3 standard deviations for the 16 oz. bottling operation. Centerline and 3-sigma control limit formulas Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 16.0 16.1 15.9 Observation 3 15.8 15.8 15.9 Observation 4 15.9 15.9 15.8 Sample means (X- bar) 15.875 15.975 15.9 Sample ranges (R) 0.2 0.3 0.2 3 X X UCL X 3 X X LCL X X CL X m i i X X m X n Where, m: # of sample mean n: # of observations in each sample
- 23. Control Chart …Cont’d Centerline (x-double bar): Control limits for±3σ limits: Control Chart Plot the sample mean in the sequence from which it was generated and interpret the pattern in the control chart. 15.875 15.975 15.9 x 15.92 3 x x x x .2 UCL x zσ 15.92 3 16.22 4 .2 LCL x zσ 15.92 3 15.62 4
- 24. Control Chart …Cont’d Second Method for X-bar Chart using Range and A2 factor Use this method when standard deviation for the process distribution is unknown. Control limits solution: Center line and 3-sigma control Fomulas: 1 2 k i i x n R R k R R or d d n ; ;& 2 2 2 2 3 3 x x x CL X R UCL X X A R d n R LCL X X A R d n
- 25. Control Chart …Cont’d OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 Calculate limits for X-bar chart using sample range
- 26. Control Chart …Cont’d R-Chart: Always look at the Range chart first. The control limits on the X-bar chart are derived from the average range, so if the Range chart is out of control, then the control limits on the X-bar chart are meaningless. Look for out of control signal. If there are any, then the special causes must be eliminated. There should be more than five distinct values plotted, and no one value should appear more than 25% of the time. If there are values repeated too often, then you have inadequate resolution of your measurements, which will adversely affect your control limit calculations. Once the effect of the out of control points from the Range chart is removed, look at the X-bar Chart. Standard Deviation of Range and Standard Deviation of the process is related as: Centerline and 3-sigma Control Limit Formulas: Where 3 3 2 R d d R d 3 3 4 2 2 3 3 3 2 2 3 1 3 3 1 3 R R R CL R d d UCL R R R D R d d d d LCL R R R D R d d ( ) ( ) d D d 3 4 2 1 3 max( , ) d D d 3 3 2 0 1 3
- 27. Control Chart …Cont’d OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 OBSERVATIONS(SLIP-RINGDIAMETER,CM) SAMPLEk 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 Calculate limits for R-chart
- 28. Control Chart …Cont’d S-Chart The sample standard deviations are plotted in order to control the process variability. For sample size (n>12), With larger samples, the resulting mean range does not give a good estimate of standard deviation the S-chart is more efficient than R-chart. For situations where sample size exceeds 12, the X-bar chart and the S-chart should be used to check the process stability. Centerline and 3-sigma Control Limit Formulas: Where s s s CL S c UCL S S B S c c LCL S S B S c 2 4 4 4 2 4 3 4 1 3 1 3 max( , ) c B c c B c 2 4 4 4 2 4 3 4 1 1 3 1 0 1 3 ( ) & k n j j i j i j S x x S S n k 2 1 1 1
- 29. Changing Sample Size on the X-bar and R Charts In some situations, it may be of interest to know the effect of changing the sample size on the X-bar and R charts. Needed information: = average range for the old sample size = average range for the new sample size nold = old sample size nnew = new sample size d2(old) = factor d2 for the old sample size d2(new) = factor d2 for the new sample size Centerline and 3-sigma Control Limit Formulas: old R new R Control Chart …Cont’d ( ) ( ) ( ) ( ) old old x chart d new UCL x A R d old d new LCL x A R d old 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) max , ( ) old new old old R chart d new UCL D R d old d new CL R R d old d new LCL D R d old 2 4 2 2 2 2 3 2 0
- 33. Control Chart: Interpreting the Patterns Patterns A nonrandom identifiable arrangement of plotted points on the chart. Provides sufficient reasons to look for special causes. Causes that affect the process intermittently and can be due to periodic and persistent disturbances Natural pattern No identifiable arrangement of the plotted points exists No point falls outside the control limit; Majority of the points are near the centerline; and Few points are close to the control limits These patterns are indicative of a process that is in control. One point outside the control limits Also known as freaks and are caused by external disturbance Not difficult to identify the special causes for freaks. However, make sure that no measurement or calculation error is associated with it, Sudden, very short lived power failure, Use of new tool for a brief test period or a broken tool, incomplete operation, failure of components
- 34. Interpreting the Patterns …cont’d Sudden shift in process mean A sudden change or jump in process mean or average service level. Afterward, the process becomes stable. This sudden change can occur due to changes- intentional or otherwise in Process settings e.g. temperature, pressure or depth of cut Number of tellers at the Bank, New operator, new equipment, new measurement instruments, new vendor or new method of processing. Gradual shift in the process mean Such shift occurs when the process parameters change gradually over a period of time. Afterward, the process stabilizes X-bar chart might exhibit such shift due to change in incoming quality of raw materials or components over time, maintenance program or style of supervision. R-chart might exhibit such shift due to a new operator, decrease in worker skill due to fatigue or monotoy, or improvement in incoming quality of raw materials.
- 35. Interpreting the Patterns …cont’d Trending pattern Trend represents changes that steadily increases or decreases. Trends do not stabilize or settle down X-bar chart may exhibit a trend because of tool wear, dirt or chip buildup, aging of equipment. R-chart may exhibit trend because of gradual improvement of skill resulting from on-the-job-training or a decrease in operator skill due to fatigue. Cyclic pattern A repetitive periodic behavior in the system. A high and low points will appear on the control chart X-bar chart may exhibit a cyclic behavior because of a rotation of operator, periodic changes in temperature and humidity, seasonal variation of incoming components, periodicity in mechanical or chemical properties of the material R-chart might exhibit cyclic pattern because of operator fatigue and subsequent energization following breaks, a difference between shifts, or periodic maintenance of equipment. Graph will not show cyclic pattern, if the samples are taken too infrequently
- 36. Interpreting the Patterns …cont’d Zones for Pattern Test UCL LCL Zone A Zone B Zone C Zone C Zone B Zone A Process average 3 sigma = x + A2R = 3 sigma = x - A2R = 2 sigma = x + (A2R) = 2 3 2 sigma = x - (A2R) = 2 3 1 sigma = x + (A2R) = 1 3 1 sigma = x - (A2R) = 1 3 x = Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 UCL LCL Zone A Zone B Zone C Zone C Zone B Zone A Zone A Zone B Zone C Zone C Zone B Zone A Process average 3 sigma = x + A2R = 3 sigma = x - A2R = 2 sigma = x + (A2R) = 2 3 2 sigma = x - (A2R) = 2 3 1 sigma = x + (A2R) = 1 3 1 sigma = x - (A2R) = 1 3 x = Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
- 37. Interpreting the Patterns …cont’d Performing a Pattern Test OBSERVATIONS (SLIP- RING DIAMETER, CM) SAMPLE k 1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15
- 38. Interpreting the Patterns …cont’d Performing a Pattern Test x x- - bar bar Chart Chart Example Example (cont.) (cont.) UCL = 5.08 LCL = 4.94 Mean Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – x = 5.01 = UCL = 5.08 LCL = 4.94 Mean Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – x = 5.01 = x = 5.01 =
- 39. Interpreting the Patterns …cont’d Performing a Pattern Test 1 1 4.98 4.98 B B — — B B 2 2 5.00 5.00 B B U U C C 3 3 4.95 4.95 B B D D A A 4 4 4.96 4.96 B B D D A A 5 5 4.99 4.99 B B U U C C 6 6 5.01 5.01 — — U U C C 7 7 5.02 5.02 A A U U C C 8 8 5.05 5.05 A A U U B B 9 9 5.08 5.08 A A U U A A 10 10 5.03 5.03 A A D D B B SAMPLE SAMPLE x x ABOVE/BELOW ABOVE/BELOW UP/DOWN UP/DOWN ZONE ZONE 1 1 4.98 4.98 B B — — B B 2 2 5.00 5.00 B B U U C C 3 3 4.95 4.95 B B D D A A 4 4 4.96 4.96 B B D D A A 5 5 4.99 4.99 B B U U C C 6 6 5.01 5.01 — — U U C C 7 7 5.02 5.02 A A U U C C 8 8 5.05 5.05 A A U U B B 9 9 5.08 5.08 A A U U A A 10 10 5.03 5.03 A A D D B B SAMPLE SAMPLE x x ABOVE/BELOW ABOVE/BELOW UP/DOWN UP/DOWN ZONE ZONE 1 1 4.98 4.98 B B — — B B 2 2 5.00 5.00 B B U U C C 3 3 4.95 4.95 B B D D A A 4 4 4.96 4.96 B B D D A A 5 5 4.99 4.99 B B U U C C 6 6 5.01 5.01 — — U U C C 7 7 5.02 5.02 A A U U C C 8 8 5.05 5.05 A A U U B B 9 9 5.08 5.08 A A U U A A 10 10 5.03 5.03 A A D D B B SAMPLE SAMPLE x x ABOVE/BELOW ABOVE/BELOW UP/DOWN UP/DOWN ZONE ZONE SAMPLE SAMPLE x x ABOVE/BELOW ABOVE/BELOW UP/DOWN UP/DOWN ZONE ZONE
- 40. Control Chart …Cont’d A process is assumed to be out of control if Rule 1: A single point plots outside the control limits; Rule 2: Two out of three consecutive points fall outside the two sigma warning limits on the same side of the center line; Rule 3: Four out of five consecutive points fall beyond the 1 sigma limit on the same side of the center line; Rule 4: Nine or more consecutive points fall to one side of the center line; Rule 5: There is a run of six or more consecutive points steadily increasing or decreasing
- 41. Control Chart for Attributes Attributes are discrete events: yes/no or pass/fail Construction and interpretation are same as that of variable control charts. Attributes control charts p chart Uses proportion nonconforming (defective) items in a sample Based on a binomial distribution Can be used for varying sample size. np chart Uses number of nonconforming items in a sample Should not be used when sample size varies c chart Uses total number of nonconformities or defects in samples of constant size. Occurence of nonconformities follows poisson distribution. u chart when the sample size varies, the number of nonconformities per unit is used as a basis for this control chart.
- 42. Control Chart: p chart Proportion nonconforming or defectives for each sample are plotted on the p-chart The chart is examined to determine whether the process is in control. Means to calculate center line and control limits No standard or target value of proportion nonconforming is specified It must be estimated from sample infromation and For each sample, proportion of nonconforming items are determined as The average of these individual sample proportion of nonconforming items is used as the center line (CLp): As true value of p is not known, p-bar is used as an estimate x p n m m i i i p p x CL p m nm ( ) ( ) p p p p UCL p n p p LCL p n 1 3 1 3
- 43. 20 samples of 100 pairs of jeans 20 samples of 100 pairs of jeans NUMBER OF NUMBER OF PROPORTION PROPORTION SAMPLE SAMPLE DEFECTIVES DEFECTIVES DEFECTIVE DEFECTIVE 1 1 6 6 .06 .06 2 2 0 0 .00 .00 3 3 4 4 .04 .04 : : : : : : : : : : : : 20 20 18 18 .18 .18 200 200 20 samples of 100 pairs of jeans 20 samples of 100 pairs of jeans NUMBER OF NUMBER OF PROPORTION PROPORTION SAMPLE SAMPLE DEFECTIVES DEFECTIVES DEFECTIVE DEFECTIVE 1 1 6 6 .06 .06 2 2 0 0 .00 .00 3 3 4 4 .04 .04 : : : : : : : : : : : : 20 20 18 18 .18 .18 200 200 UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 = 200 / 20(100) = 0.10 total defectives total sample observations p = UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 LCL = 0.010 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 = 200 / 20(100) = 0.10 total defectives total sample observations p = = 200 / 20(100) = 0.10 total defectives total sample observations p = = 200 / 20(100) = 0.10 total defectives total sample observations p = If the target or standard value is specified Center line is selected as that target value i.e. CLp= po where, po represent a standard value Control limits are also based on the target velue. If the lower control limit for p is turned out to be negative, LCL is simply counted as zero. Lowest possible value for proportion of nonconformng item is zero Control Chart: p chart …Cont’d
- 44. Control Chart: p chart …Cont’d Variable sample size Changes in sample size casues the control limits to change, although the center line remained fixed. Control limits can be constructed: For individual samples If no standard value is given and sample mean proportion nonconforming is p-bar, control limit for sample i with size ni are Using average sample size Where ( ) ( ) i i p p UCL p n p p LCL p n 1 3 1 3 ( ) ( ) p p UCL p n p p LCL p n 1 3 1 3 m i i n n m 1
- 45. Control Chart: c chart No standard given Average number of nonconformities per sample unit is found from the sample observation and is denoted by c-bar. The center line and control limits are: If lower control limit is found to be less than zero, it is converted to zero. Standard given if the specified target for the number of nonconformities per sample unit be co.. The center line and control limits are then calculated from: c c c CL c UCL c c LCL c c 3 3 c o c o o o o o CL c UCL c c LCL c c 3 3
- 46. Control Chart: c chart …Cont’d Number of defects in 15 sample rooms Number of defects in 15 sample rooms 1 12 1 12 2 8 2 8 3 16 3 16 : : : : : : : : 15 15 15 15 190 190 SAMPLE SAMPLE c c = = 12.67 = = 12.67 190 190 15 15 UCL UCL = = c c + + z z c c = 12.67 + 3 12.67 = 12.67 + 3 12.67 = 23.35 = 23.35 LCL LCL = = c c + + z z c c = 12.67 = 12.67 - - 3 12.67 3 12.67 = 1.99 = 1.99 NUMBER OF DEFECTS Number of defects in 15 sample rooms Number of defects in 15 sample rooms 1 12 1 12 2 8 2 8 3 16 3 16 : : : : : : : : 15 15 15 15 190 190 SAMPLE SAMPLE c c = = 12.67 = = 12.67 190 190 15 15 c c = = 12.67 = = 12.67 190 190 15 15 c c = = 12.67 = = 12.67 190 190 15 15 190 190 15 15 UCL UCL = = c c + + z z c c = 12.67 + 3 12.67 = 12.67 + 3 12.67 = 23.35 = 23.35 UCL UCL = = c c + + z z c c = 12.67 + 3 12.67 = 12.67 + 3 12.67 = 23.35 = 23.35 LCL LCL = = c c + + z z c c = 12.67 = 12.67 - - 3 12.67 3 12.67 = 1.99 = 1.99 NUMBER OF DEFECTS 3 3 6 6 9 9 12 12 15 15 18 18 21 21 24 24 Number of defects Number of defects Sample number Sample number 2 2 4 4 6 6 8 8 10 10 12 12 14 14 16 16 UCL = 23.35 LCL = 1.99 c = 12.67 3 3 6 6 9 9 12 12 15 15 18 18 21 21 24 24 Number of defects Number of defects Sample number Sample number 2 2 4 4 6 6 8 8 10 10 12 12 14 14 16 16 UCL = 23.35 LCL = 1.99 c = 12.67 3 3 6 6 9 9 12 12 15 15 18 18 21 21 24 24 Number of defects Number of defects Sample number Sample number 2 2 4 4 6 6 8 8 10 10 12 12 14 14 16 16 2 2 4 4 6 6 8 8 10 10 12 12 14 14 16 16 UCL = 23.35 LCL = 1.99 c = 12.67
- 47. Process Capability Analysis Process Variability Natural variation in the process. Tolerance or specification Range of acceptable values established design engineers or product design specialist. For the product to be considered acceptable, its quality characteristics must fall within this preset range. Process Capability Process variability relative to specification. Relationship between the process variability and the tolerance can be formulized by the consideration of standard deviation, σ of the process. In order to manufacture product within the specification, the distance between the upper specification limit (USL) and the lower specification limit (LSL) i.e. (USL-LSL) or 2T must be equal to or greater than width of the process variability defined by the control limits, i.e. 6σ Lower Specification Upper Specification 6σ
- 48. Process Capability Analysis …Cont’d Relationship between the process width and specification The relationship between (USL - LSL) and 6σ results in three levels of situations: Process variation is large relative to the specifications. A large percentage of the product will fall outside the specification. Process is not capable of meeting specifications all the time. The process cannot be considered capable regardless of the process centering. Process variation must be reduced drastically (a) Natural variation exceeds design specifications Design Specifications Process
- 49. Process Capability Analysis …Cont’d Process variability closely matches the predefined specification. This is the absolute minimum requirements for the process to be capable of producing the acceptable product. Almost all (99.74%) the output falls within the preset specifaction range. Process is capable of meeting specifications most of the time. The process must remain well centered for the process capability to be maintained at a tolerable level. Variation must be reduced and this will reduce the defectives per million, cost of quality and increase profitability. (b) Design specifications and natural variation are same Design Specifications Process
- 50. Process Capability Analysis …Cont’d Process variation is small relative to the specifications. The process mean can shift about without causing the process to degrade its capability Process is capable of meeting specifications all the time. This will reduce the defects per million (DPM), reduce the cost of quality (COQ), and hence increase profitability. Simply setting up control charts to monitor whether a process is in control does not guarantee process capability. (c) Design specifications greater than natural variation Design Specifications Process
- 51. Process Capability Analysis …Cont’d Process Capability Index A measure that relates the actual performance of a process to its specified performance. Calculated when a process is under statistical control (i.e. Only the random or common causes of variation are present). Process capability can be quantified by the calculation of several indices: Relative Precision Index (RPI) A quick and simple measure of the process potential. Process is considered to be centered over the specification range This index only deals with relative spread or variation. RPI is based on the ratio of mean sample range with the tolerance band or width. i.e. To avoid production of defective product, specification width must be greater than the process variation. Value of 6/dn is the minimum RPI T RPI R 2 n n T R T d T d R 2 6 6 2 2 6
- 52. Process Capability Analysis …Cont’d Cp Index A measure of process potential to meet the specification limit. It is valuable index in measuring the process capability. It is computed as a ratio of specification width to the width of process variability when σ is unknown, it is obtained from its estimates Three possible ranges for Cp Cp = 1, process variability just meets specifications and hence, process is minimally capable. 0.26% of the product will not be acceptable. Cp < 1, process variability is outside the specification range and hence, process is not capable of producing within specifications Cp > 1, process variability is tighter than the specification range and process is capable of meeting the specification all the time. A way to reduce the generation of defective product is to increase the process capability. p USL LSL C 6 s R and c d 4 2
- 53. Process Capability Analysis …Cont’d Computing the Cp Value at Cocoa Fizz: three bottling machines are being evaluated for possible use at the Fizz plant. The machines must be capable of meeting the design specification of 15.8-16.2 oz. with at least a process capability index of 1.0 (Cp≥1) The table below shows the information gathered from production runs on each machine. Are they all acceptable? Solution: Machine A Machine B Cp = Machine C Cp = Machine σ USL-LSL 6σ A .05 .4 .3 B .1 .4 .6 C .2 .4 1.2 USL LSL .4 C = 1.33 6σ 6(.05) p
- 54. Process Capability Analysis …Cont’d Cp Index It has one shortcoming: Cp index assumes that the process is centered on the specification range. Process may be off-centered and because of this, a proportion of products will fall outside the specification range. Using only the Cp measure would lead to an incorrect conclusion
- 55. Process Capability Analysis …Cont’d CpK Index Another measure of process capability This measure accounts for the location of the process mean and is used when the process mean is not at the target value, which is assumed to be the halfway between the specification limits. The process capability of each half of the normal distribution is computed and minimum of the two is used. i.e. where µ = the mean of the process σ = the standard deviation of the process min( , ) pk USL LSL C 3 3
- 56. Process Capability Analysis …Cont’d Net weight specification = 9.0 oz 0.5 oz Process mean = 8.80 oz Process standard deviation = 0.12 oz Cp = = = 1.39 upper specification limit - lower specification limit 6 9.5 - 8.5 6(0.12) Net weight specification = 9.0 oz 0.5 oz Process mean = 8.80 oz Process standard deviation = 0.12 oz Cp = = = 1.39 upper specification limit - lower specification limit 6 9.5 - 8.5 6(0.12) Cp = = = 1.39 upper specification limit - lower specification limit 6 upper specification limit - lower specification limit 6 9.5 - 8.5 6(0.12) 9.5 - 8.5 6(0.12) Net weight specification = 9.0 oz 0.5 oz Process mean = 8.80 oz Process standard deviation = 0.12 oz Cpk = minimum = minimum , = 0.83 x - lower specification limit 3 = upper specification limit - x 3 = , 8.80 - 8.50 3(0.12) 9.50 - 8.80 3(0.12) Cpk = minimum = minimum , = 0.83 x - lower specification limit 3 = upper specification limit - x 3 = , x - lower specification limit 3 = upper specification limit - x 3 = , 8.80 - 8.50 3(0.12) 9.50 - 8.80 3(0.12) 8.80 - 8.50 3(0.12) 8.80 - 8.50 3(0.12) 9.50 - 8.80 3(0.12) 9.50 - 8.80 3(0.12) The Cp value leads us to conclude that the process is capable. Whereas, the Cpk value is less than 1, revealing that process is not capable. Reasons for the difference in measures is that process is not centered on specification range.
- 57. Process capability ratio, Cp = specification width process width Upper specification – lower specification 6 Cp = pk L USL- C =min or 3 3 LS If the process is centered use Cp If the process is not centered use Cpk Process Capability Analysis …Cont’d
- 58. Process Capability Analysis …Cont’d Conclusion: The process is centered (Cp = Cpk), and of low capability since the indices are only just greater than 1. Conclusion: Cp at 1.89 Indicates a potential for higher capability than the result (i). The low Cpk shows that this potential is not being realized as the process is not centered.
- 59. Acceptance Sampling Acceptance sampling is concerned with inspection and decision making regarding the entire lot/production/shipment. A form of inspection used to accept or reject entire lot based on the sample information. Not consistent with TQM or Zero Defects philosophy. producer and customer agree on the number of acceptable defects a means of identifying not preventing poor quality Typical Application of Acceptance Sampling A company receives a shipment of product from a vendor. This product is a raw material used in the company’s manufacturing process. A sample is taken from the shipment, and quality characteristic of the products in the sample is inspected. Based on the information in this sample, a decision is made regarding disposition of the shipment. Usually this decision is either to accept or to reject the entire shipment. Accepted lots are used into production, and Rejected lots are returned to vendor.
- 60. The decision to accept or reject the shipment is based on the following set standards: Lot size = N Sample size = n Acceptance number = c Defective items = d If d <= c, accept lot If d > c, reject lot Acceptance Sampling …Cont’d
- 61. Acceptance Sampling …Cont’d Three Important Aspects of Acceptance Sampling Purpose is to sentence the lot (accept or reject) rather than to estimate the lot quality. Acceptance sampling plan does not provide any direct form of quality control. It simply makes the decision whether to reject or accept the lot. Process controls are used to control and systematically improve quality Most effective use of acceptance sampling is Not to “inspect quality into the product,” but rather as audit tool to ensure that ‘output of process conforms to requirements’. Three Approaches to Lot Sentencing Accept with no inspection. 100% inspection – inspect every item in the lot, remove all defectives Defectives – returned to vendor, reworked, replaced or discarded Acceptance sampling Sample is taken from lot, a quality characteristic is inspected; then on the basis of information in sample, a decision is made regarding lot disposition.
- 62. Acceptance Sampling …Cont’d When is Acceptance Sampling Useful? Testing is destructive and time consuming. 100% inspection is not technologically feasible. 100% inspection error rate results in higher percentage of defectives being passed than is inherent to product. Cost of 100% inspection extremely high. Vender has excellent quality history so reduction from 100% is desired but not high enough to eliminate inspection altogether. Potential for serious product liability risks; program for continuously monitoring product required. When can Acceptance Sampling be Used? At any point in production The output of one stage is the input of the next At the input stage Prevents goods that don’t meet standards from entering into the process This saves rework time and money At the output stage Can reduce the risk of bad quality being passed on from the process to a consumer This can prevent the loss of prestige, customers, and money
- 63. Acceptance Sampling …Cont’d Advantages of Acceptance Sampling It is usually less expensive because there is less inspection. There is less handling of the product, hence reduced damage. It is applicable to destructive testing. Fewer personnel are involved in inspection activities. It often greatly reduces the amount of inspection error. The rejection of entire lots are opposed to the sample return of defectives often provides a stronger motivation to the vendor for quality improvements. Disadvantages of Acceptance Sampling Always a risk of accepting “bad” lots and rejecting “good” lots Producer’s Risk: chance of rejecting a “good” lot – Consumer’s Risk: chance of accepting a “bad” lot – Less information is generated about the product or the process that manufactured the product. Requires added planning and documentation of the acceptance sampling procedure – 100% inspection does not.
- 64. Acceptance Sampling …Cont’d Risks of Acceptance Sampling Producer’s Risk(α) Refers to the probability of rejecting a good lot Also refered to as Type I error. Acceptable Quality Level (AQL) The numerical definition of a ‘good lot’. ANSI/ASQC describes AQL as: The percentage level of defective or nonconforming items at which the customer is willing to accept a lot as good. Consumer’s Risk (β) Refers to the probability of accepting a bad lot Also refered to as Type II error Lot Tolerance Percentage Defective (LTPD) Numerical definition of a ‘bad lot’ ANSI/ASQC describes LTPD as: Upper limit on the percentage defectives that a customer is willing to accept
- 65. Acceptance Sampling …Cont’d Lot Formation Lots should be homogeneous Units in a lot should be produced by the same: machines, operators, from common raw materials, approximately same time If lots are not homogeneous – acceptance-sampling scheme may not function effectively and make it difficult to eliminate the source of defective products. Larger lots are preferred to smaller ones Lots should conform to the materials-handling systems in both the vendor and consumer facilities Lots should be packaged to minimize shipping risks and make selection of sample units easy
- 66. Acceptance Sampling …Cont’d Mr. Smith owns and operates a manufacturing plant. He receives a shipment of 1,000 sheets of glass. Of the shipment, Mr. Smith chooses to sample 50 sheets. If more than 2 are defective, he is sending back the entire shipment to the supplier. Mr. Smith observes 5 defective sheets of glass. What should Mr. Smith do in reference to the number of defective items observed???? Moonlight Jeans Moonlight Jeans store receives a shipment of 300 pairs of jeans from its warehouse. It is common practice for the store to sample 5% of the total received. The acceptance number under any and all circumstances for Moonlight Jeans is 10. Of the 15 pairs of jeans observed, 2 were defective. What conclusion should the store manager come to based on this information? The store manager has just found out that the clerk who inspected the samples made a huge mistake… The actual number of defective pairs of jeans sampled was 12. What type of risk is involved with the error made at Moonlight Jeans? How might this error affect the store and their customers?
- 67. Acceptance Sampling …Cont’d Random Sampling Important Units selected for inspection from lot must be chosen at random Should be representative of all units in a lot. Watch for Salting Vendor may put “good” units on top layer of lot knowing a lax inspector might only sample from the top layer. Suggested Technique Assign a number to each unit, or use location of unit in lot. Generate / pick a random number for each unit / location in lot. Sort on the random number – reordering the lot / location pairs. Select first (or last) n items to make sample.
- 68. Acceptance Sampling …Cont’d Acceptance Sampling Plan Specifies the lot size, sample size, number of samples and acceptance/rejection criteria for lot sentencing. Unless it is mentioned following convention is practiced Sampling is performed without replacement Sampling is a simple random sample Each item in the lot has equal probability of being in the sample. Designing Sampling Plan for Attributes Simpliest sample plan Based on binomial distribution (if sample is less than 20 units otherwise use poisson’s distribution) Requires large sample size Sampling plan involves Single sampling plan Double sampling plan, Multiple sampling plan
- 69. Acceptance Sampling …Cont’d Single Sampling Plan Quality characteristic is an attribute, i.e., conforming or nonconforming Define: N: lot size n: sample size, and c: Acceptance number Procedure: Take a sample of size n and inspect each of the items drawn If d ≤ c, accept lot; else reject, d: number of defective items in sample
- 70. Double Sampling Plan Define: n1: sample size of the first sample c1: acceptance number for the first sample n2: sample size of the second sample c2: acceptance number for the second sample Procedure: Take an initial sample of size n1 If number of defective items, d1 ≤ c1, accept the lot If number of defective items, d1 > c2, reject the lot If c1 < d1 ≤ c2, take second sample of size, n2 If the combined number of defective items (d1+d2) ≤ c2, accept the lot; otherwise reject the lot Acceptance Sampling …Cont’d
- 71. Characterizing attribute sampling plan Typically four graphs are used to characterize a sampling plan. Operating Characteristic (OC) curve The probability of acceptance for a given quality level. Average Sample Number (ASN) curve The expected number of items we will sample (most applicable to double, multiple, and sequential samples) Average Outgoing Quality (AOQ) curve The expected fraction nonconforming after rectifying inspection for a given quality level. Average Total Inspected (ATI) curve The expected number of units inspected after rectifying inspection for a given quality level. Acceptance Sampling …Cont’d
- 72. Operating Characteristics Curve OC curves are graphs which show the probability of accepting a lot for various proportions of defective items in the lot. X-axis shows percentage of items that are defective in a lot- “lot quality”. Y-axis shows the probability or chance of accepting a lot. As proportion of defective items increases, the chance of accepting lot decreases. Example: 90% chance of accepting a lot with 5% defectives; 10% chance of accepting a lot with 24% defectives Acceptance Sampling …Cont’d
- 73. Acceptance Sampling …Cont’d OC curve is typically used to represent the four parameters (Producers Risk, Consumers Risk, AQL and LTPD) of the sampling plan. AQL is the small % of defects that consumers are willing to accept; order of 1-2% LTPD is the upper limit on the percentage of defective items that consumers are willing to tolerate. Consumer’s Risk (α) is the chance of accepting a lot that contains a greater number of defects than the LTPD limit; Type II error Producer’s risk (β) is the chance a lot containing an acceptable quality level will be rejected; Type I error Probability Probability of of Acceptance Acceptance Percent Percent defective defective | | | | | | | | | 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 100 100 – 95 95 – 75 75 – 50 50 – 25 25 – 10 10 – 0 0 – Probability Probability of of Acceptance Acceptance Percent Percent defective defective | | | | | | | | | 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 100 100 – 95 95 – 75 75 – 50 50 – 25 25 – 10 10 – 0 0 – =0.05 =0.05 producer producer’’sriskforAQL sriskforAQL =0.05 =0.05 producer producer’’sriskforAQL sriskforAQL =0.10 =0.10 Consumer Consumer’’s s riskforLTPD riskforLTPD =0.10 =0.10 Consumer Consumer’’s s riskforLTPD riskforLTPD LTPD LTPD AQL AQL LTPD LTPD AQL AQL Badlots Badlots Indifference Indifference zone zone Good Good lots lots Badlots Badlots Indifference Indifference zone zone Good Good lots lots Discriminates between ‘good lot’ and ‘bad lot’
- 74. How to Compute the OC Curve Probabilities For a specified single sampling plan, the OC curve probability can be computed using a binomial or Poisson approximation. Assume that the lot size N is large (infinite) d - # defectives ~ Binomial(p,n) where p - fraction defective items in lot n - sample size Probability of acceptance: Acceptance Sampling …Cont’d n! P(d DEFECTIVES ) = ( ) d!(n-d)! ( ) P(d DEFECTIVES ) , , , ! d n d d np p p np e d d 1 0 1 2 ! ( ) ( ) !( )! c d n d a d n P P d c p p d n d 0 1 Binomial Approx. Poisson Approx.
- 75. Example: Acceptance Probability Suppose p =0.02, n = 60, and c =3. Acceptance Sampling …Cont’d . . . . ( | . ) . ! ( | . ) . ! ( | . ) . ! ( | . ) . ! ( | . ) . . . . . e prob x p e prob x p e prob x p e prob x p prob x p 0 1 2 1 1 2 2 1 2 3 1 2 1 2 0 0 02 0 3012 0 1 2 1 0 02 0 3614 1 1 2 2 0 02 0 2169 2 1 2 3 0 02 0 3012 3 3 0 02 0 3012 0 3614 0 2169 0 0867 0 9662 = probability of accepting the lot.
- 76. How to Construct OC Curve OC curve is plot of Pa vs p Pa = P (Accepting Lot | true proportion of defective is p) p = lot fraction defective Suppose, n = 89 and c = 2. For each value of p, compute the probability of acceptance using a Binomial or Poisson approximation as given in the table: Acceptance Sampling …Cont’d p = fraction defective in lot Pa = P [Accepting Lot] 0.005 0.9897 0.010 0.9397 0.015 0.8502 0.020 0.7366 0.025 0.6153 0.030 0.4985 0.035 0.3936 Probability of Acceptance, Pa 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 Lot fraction defective, p Pa n=89 c=2
- 77. Acceptance Sampling …Cont’d Ideal OC Curve This characterizes the ideal sampling plan that discriminates perfectly between good and bad shipments If the supplier’s process average nonconforming is below the AQL, the consumer will accept all the shipped lots. If the supplier’s process average nonconforming is above the AQL, the consumer will reject all the shipped lots. Both α and β are zero it is obtainable by 100% inspection IF inspection are error free. ideal OC curve is unobtainable in practice
- 78. Acceptance Sampling …Cont’d Effect of n on OC Curve Precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases Increasing n (with c proportional) approaches the ideal OC curve. Operating Characteristic Curve 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% Percent nonconforming, p Probability of acceptance, Pa n=200, c=4 n=100, c=2 n= 50, c=1
- 79. Acceptance Sampling …Cont’d Effect of c on OC Curve Changing acceptance number, c, does not dramatically change slope of OC curve. Increasing c (with n constant) approaches the ideal OC curve. Operating Characteristic Curve 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% Percent nonconforming, p Probability of acceptance, Pa n=100, c=2 n=100, c=1 n=100, c=0
- 80. Acceptance Sampling …Cont’d Balancing the producer and consumer risks The true value of p is unknown Even if p known, sampling involves randomness, and we can still reject a lot even if p < c/n accept a lot even if p > c/n The OC curve gives an indication of the values of α and β. Producer’s risk- α Producer wants as many lots accepted by consumer as possible so Producer “makes sure” the process produces a level of fraction defective equal to or less than: p1 = AQL = Acceptable Quality Level α is the probability that a good lot will be rejected by the consumer even though the lot really has a fraction defective ≤ p1 Lot is rejected given that process has an acceptable quality level P Lot is rejected P p AQL
- 81. Acceptance Sampling …Cont’d Consumer’s risk- β Consumer wants to make sure that no bad lots are accepted Consumer says, “I will not accept a lot if percent defective is greater than or equal to p2” p2 = LTPD = Lot Tolerance Percent Defective β is the probability a bad lot is accepted by the consumer when the lot really has a fraction defective ≥ p2 We receive a shipment of 3000 items, AQL = 0.02, LTPD = 0.06, n = 60, c = 3. α = probability of rejecting a batch with defective rate p=0.02. = probability that four or more defective item = prob (x ≥4 | p = 0.02) = 1- prob (x ≤3 | p = 0.02) = 1-0.9662 = 0.0338 β = probability of acceptance a batch with p=LTPD=0.06 = prob (x ≤ 3 | p=0.06) = 0.5153 Lot accepted given that lot has unacceptable quality level P Lot accepted P p LTPD
- 82. Acceptance Sampling …Cont’d Designing a Sampling Plan with a Specified OC Curve Use a chart called a Binomial Nomograph to design a sampling plan Specify: p1 = AQL (Acceptable Quality Level) p2 = LTPD (Lot Tolerance Percent Defective) 1 – α = P [Lot is accepted | p = AQL] β = P [Lot is accepted | p = LTPD] Draw two lines on Nomograph Line 1 connects p1 = AQL to (1- α) Line 2 connects p2 = LTPD to β Pick n and c from the intersection of the lines Example: Suppose p1 = 0.01, α = 0.05, p2 = 0.06, and β = 0.10. Find the acceptance sampling plan.
- 84. Acceptance Sampling …Cont’d Designing a Sampling Plan using Tables For a given producer’s and consumer’s risks, various tables have been developed for constructing single and double sampling plans. Three alternatives for specifying sampling plans Producer’s Risk and AQL specified Consumer’s Risk and LTPD specified All four parameters specified Excerpt From a Sampling Plan Table with Producers Risk = 0.05 and Consumers Risk = 0.10 C LTPD/AQL n(AQL) n(LTPD) 0 44.89 0.052 2.334 1 10.946 .355 3.886 2 6.509 .818 5.324 3 4.89 1.366 6.68 4 4.057 1.97 7.992 5 3.549 2.613 9.274 6 3.206 3.286 10.535 7 2.957 3.981 11.772 8 2.768 4.695 12.996 9 2.618 5.426 14.205
- 85. Acceptance Sampling …Cont’d Producer’s Risk and AQL specified Consumer’s Risk and LTPD specified Choose the acceptance number and divide the appropriate column by the associated parameter to get the sample size. Ex I: Given a producer’s risk of 0.05 and an AQL of 0.015 determine a sampling plan. c = 1; n(AQL) = 0.355; n= 0.355/0.015 ≈ 24 c = 4; n(AQL) = 1.97; n = 1.97/0.015 ≈ 131 Ex II: Given a consumer’s risk of 0.10 and an LTPD of 0.08 determine a sampling plan. c = 0; n(LTPD) = 2.334 n = 2.334/0.08 ≈ 29 c = 5; n(LTPD) = 9.274 n = 9.274/0.08 ≈ 116
- 86. Acceptance Sampling …Cont’d All four parameters specified We must first find out a value close to the ratio LTPD/AQL in the table. Then find out the values of n and c that corresponds to that specified ratio. Example: Given producers risk of 0.05, consumers risk of 0.10, LTPD of 4.5%, and AQL of 1%, determine a sampling plan. Since the ratio of LTPD/AQL = 4.5/1 = 4.5 is in between c= 3 and c = 4; Using the n(AQL) column the sample sizes suggested are 137 and 197 respectively. Note: using n(AQL) column will ensure a producers risk of 0.05. using the n(LTPD) column will ensure a consumers risk of 0.10 For double sampling plan, use Grubbs’Tables (Table 10-6 and Table 10-7: Amitava Mitra, Fundamentals of Quality Control and Improvement, 2nd ed., Pages: 445-446)
- 87. Acceptance Sampling …Cont’d Rectifying Sampling All known defective units replaced with good ones, that is, If lot is rejected, replace all bad units in lot If lot is accepted, just replace the bad units in sample Such sampling program is known as rectifying inspection program Since such inspection activity affects the final quality of the outgoing product, two questions come to mind : How many items are inspected on average after rectifying inspection? What is the average outgoing quality after rectifying inspection?
- 88. Acceptance Sampling …Cont’d Average Outgoing Quality: AOQ Quality that results from application of rectifying inspection. The long-run ratio of expected number of defectives to expected number of items successfully passing through the inspection plan Average value obtained over long sequence of lots from process with fraction defective p: N - Lot size, n = # units in sample that, after inspection, contains no defectives. If the lot is accepted: number of defective units in the lot a P p N n AOQ N # units fraction remaining defective in lot p N n
- 89. Acceptance Sampling …Cont’d Average Outgoing Quality: AOQ Expected number of defective units: Average fraction defective or Average Outgoing Quality, AOQ: The maximum value or the worst possible value of AOQ over all possible values of p is called Average Outgoing Quality Limit (AOQL). If this is too high, then the sampling plan should be revised. Lot # defective Prob accepted units in lot a P p N n a P p N n AOQ N Average Outgoing Quality Curve 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% Percent nonconforming, p Average fraction nonconforming, outgoing lots The AOQ curve for N = 150, n = 20, c = 2 AOQL
- 90. Acceptance Sampling …Cont’d Average Total Inspected: ATI Total number of inspection per lot required for sampling plan. If the lot is fully conforming, p=0.0 (Pa=1.0), then no lot will be rejected and hence, inspect only the sample. If the lot is totally nonconforming, p=1.0 (Pa=0.0), then do 100% inspection on the lot. If the lot quality is 0 < p < 1, the average total number of inspection per lot: p = fraction defectives; Pa = Probability of accepting a lot. 1 a ATI n P N n Average Total Inspection Curve 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% Percent nonconforming, p Average total inspection (ATI) The ATI curve for N = 150, n = 20, c = 2
- 91. Design of Experiment Why Experiment? To increase knowledge about a process What good is increased knowledge? Increase the output of the process Reduce variability Produce a robust product that is not influenced by environmental factors Why Experimental Design? To obtain valid conclusions from an experiment To yield the maximum amount of useful information with the least amount of experimentation (experiments cost money)
- 92. Design of Experiment ….Cont’d Terminology Used Design: Specification of the number of experiments, the factor level and the number of replications for each experiment. Factorial design: An experimental design where all combinations of factor levels under consideration are tested at each replicate. Response Variable: Outcome of an experiment, which you are to measure during experiment e.g., throughput, response time Factors: Variables that have effects on the response variable Also called predictor variables or predictors Primary Factors: The factors whose effects need to be quantified. Secondary Factors: Factors whose impact need not be quantified e.g., the workloads and user educational level. Levels: Specific value of the factor Also called treatment Replication: Repetition of all or some experiments. Experimental Unit: Any entity that is used for experiments Interaction: Two factors interact if one shows dependence upon another.
- 93. Design of Experiment ….Cont’d Full Factorial Design Consists of all possible combinations of the factor levels. Design Matrix is the complete specification of the experimental test runs. Treatment Combination is a specific combination of the factor levels. Speed Octane Tire Pressure 55 85 30 60 85 30 55 90 30 60 90 30 55 85 35 60 85 35 55 90 35 60 90 35 Design Matrix A treatment combination Speed Octane Tire Pressure 55 85 30 60 85 30 55 90 30 60 90 30 55 85 35 60 85 35 55 90 35 60 90 35 Design Matrix A treatment combination
- 94. Design of Experiment ….Cont’d What makes up an experiment? Response Variable(s) Factors Randomization Repetition and Replication Response Variable The variable that is measured and the object of the characterization or optimization (the Y). Defining the response variable can be difficult. Often selected due to ease of measurement. Factors A variable which is controlled or varied in a systematic way during the experiment (the X) Tested at 2 or more levels to observe its effect on the response variable(s) (Ys)
- 95. Design of Experiment ….Cont’d Randomization Randomization can be done in several ways : Run the treatment combinations in random order Assign experimental units to treatment combinations randomly An experimental unit is the entity to which a specific treatment combination is applied Advantage of randomization is to “average out” the effects of extraneous factors (called noise) that may be present but are not controlled or measured during the experiment Repetition and Replication Repetition : Running several samples during one experimental setup (short- term variability) Replication : Repeating the entire experiment (long-term variability) Repetition and Replication provide an estimate of the experimental error, which is used to determine if observed differences are statistically significant.
- 96. Design of Experiment ….Cont’d Steps in Design of Experiment Five key steps in designing an experiment include: 1. Identify factors of interest and a response variable. 2. Determine appropriate levels for each explanatory variable. 3. Determine a design structure/matrix. 4. Randomize the order in which each set of conditions is run and collect the data. 5. Organize the results in order to draw appropriate conclusions. How to organize and draw conclusions for a specific type of design structure, the full factorial design.
- 97. Design of Experiment ….Cont’d Steps in Design of Experiment Problem Statement: A soft drink bottler is interested in obtaining more uniform heights in the bottles produced by his manufacturing process. The filling machine theoretically fills each bottle to the correct target height, but in practice, there is variation around this target, and the bottler would like to understand better the sources of this variability and eventually reduce it. The engineer can control three input variables during the filling process (each at two levels): Factor A: Carbonation with Levels: 10% and 12% Factor B: Operating Pressure in the filler with Levels: 25 and 30 psi Factor C: Line Speed with Levels: 200 and 250 bottles produced per minute (bpm) Unit: Each bottle Response Variable: Deviation from the target fill height The steps to design this experiment include: 1. Identify factors of interest and a response variable. 2. Determine appropriate levels for each explanatory variable.
- 98. Design of Experiment ….Cont’d 3. Determine a design structure Design structures can be very complicated. One of the most basic structures is called the full factorial design. This design tests every combination of factor levels an equal amount of times. To ensure each factor combination exactly once: 1st Column - alternate every (20) row 2nd Column - alternate every 2 (21) rows 3rd Column - alternate every 4 (=22) rows This is called a 23 full factorial design (i.e. 3 factors at 2 levels each will need 8 runs). Each row in this table gives a specific test run. For example, the first row represents a specific test in which the manufacturing process runs with A set at 10% carbonation, B set at 25 psi, and line speed, C, is set at 200 bpm.
- 99. Design of Experiment ….Cont’d 4. Randomize the order in which each set of test conditions is run and collect the data. In this example the tests are run in the following order: 7, 4, 1, 6, 8, 5, 2, 3. Data are the results obtained from each of these test conditions. 5. Organize the results in order to draw appropriate conclusions. Results are the observed deviation from the target fill height for each set of these test conditions. To start organizing the results, computing all the averages at low and high levels. To determine the effect of change in factor A on the results, calculate the average values of the test results for A- and A+.
- 100. Design of Experiment ….Cont’d Main Effects: A Main Effect is the difference between the factor average and the grand mean. From the table, Overall average of the test results (grand mean) is 2. Mean values for A- (factor A run at low level) and A+ (factor A run at a high level) are -1 and 5 respectively Likewise, calculate mean values of the test results for factors B and C Effect sizes determine which factors have the most significant impact on the results. ANOVA determine the significance of each factor based on these effect calculations.
- 101. Design of Experiment ….Cont’d Main Effects Plots: Effects plots are a quick and efficient way to visualize effect size. The grand mean is plotted as a horizontal line. The average result is represented by dots for each factor level. The Y axis is always the same for each factor in Main Effects Plots. Factors with steeper slopes have larger effects and thus larger impacts on the results. This graph shows that A+ has a higher mean fill height than A- B+ and C+ also have higher means than B- and C- respectively, and The effect size of factor A, Carbonation, is larger than the other factor effects.
- 102. Design of Experiment ….Cont’d Interaction Effects: It is often critical to identify how multiple factors interact in affecting the results. An interaction occurs when one factor affects the results differently depending on a second factor. To find the AB interaction effect, first calculate the average result for each of the four level combinations of A and B: Calculate the average when factors A and B are both at the low level (-4 + -1) / 2 = -2.5 Calculate the mean when factors A and B are both at the high level (5 + 11) / 2 = 8 Also calculate the average result for each of the levels of AC and BC.
- 103. Design of Experiment ….Cont’d Interaction Effect plots: Interaction plots are used to determine the effect size of interactions. For example, the AB plot below shows that the effect of B is larger when A is 12%. AB plot also shows that when the data is restricted to A+, the B effect is more steeper than when we restrict our data to A-. The bottom right plot shows the interaction (or 2-way effects) of all three factors: When the lines are parallel, no interaction effects between the factors. The more different the slopes, the more influence the interaction effect has on the results. This graph shows that the AB interaction effect is the largest.