#### Transcript Statistical Process Control

Operations Management Supplement 6 – Statistical Process Control PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e © 2006 Prentice Hall, Inc. Hall, Inc. © 2006 Prentice S6 – 1 Outline Statistical Process Control (SPC) Control Charts for Variables Setting Mean Chart Limits (x-Charts) Setting Range Chart Limits (R-Charts) Process Capability Process Capability Ratio (Cp) Process Capability Index (Cpk ) © 2006 Prentice Hall, Inc. S6 – 2 Statistical Process Control (SPC) Variability is inherent in every process Natural or common causes Special or assignable causes SPC charts provide statistical signals when assignable causes are present SPC approach supports the detection and elimination of assignable causes of variation © 2006 Prentice Hall, Inc. S6 – 3 Inspection Inspection is the activity that is done to ensure that an operation is producing the results expected (post production activity). Where to inspect At point of product design At point of product production (source) At point of product assembly At point of product dispatch to customer At point of product reception by customer © 2006 Prentice Hall, Inc. S6 – 4 What to Inspect Variables of an entity Degree of deviation from a target (continuum scale) Lifespan of device Reliability or accuracy of device Attributes of an entity Classifies attributes into discrete classes such as good versus bad, or pass or fail Maximum weight of bag at airport Minimum height for exit seat © 2006 Prentice Hall, Inc. S6 – 5 Data Used for Quality Judgments Variables Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables, e.g. weight, length, duration, etc. © 2006 Prentice Hall, Inc. Attributes Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables S6 – 6 How to Inspect: Methods Visual inspection Manual inspection (weigh, count ) Mechanical inspection (machine-based) Testing of device © 2006 Prentice Hall, Inc. S6 – 7 Disadvantages of Inspection Most inspections are not done at the source Errors are discovered after its too late No link between error and the cause of errors Errors are often too costly to correct As products grow in number more staff are needed for inspection As products grow in number, more time is needed for inspection Most inspections involve the inspection of good parts as well as bad ones CHALLENGE: How could one achieve high product quality without having to inspect all goods produced? © 2006 Prentice Hall, Inc. S6 – 8 Statistical Process Control A statistics-based approach for monitoring and inspecting results of a process, through the gathering, structuring, and analyses of product variables/attributes, as well as the taking of corrective action at the source, during the production process. © 2006 Prentice Hall, Inc. S6 – 9 Class Example What can we learn from the results of the bodyguards? © 2006 Prentice Hall, Inc. S6 – 10 Common Cause Variations Also called natural causes Affects virtually all production processes Generally this requires some change at the systemic level of an organization Managerial action is often necessary The objective is to discover avoidable common causes present in processes Eliminate (when possible) the root causes of the common variations, e.g. different arrival times of suppliers, different arrival times of customers, weight of products poured in box by machine © 2006 Prentice Hall, Inc. S6 – 11 Assignable Variations Also called special causes of variation Generally this is caused by some change in the local activity or process Variations that can be traced to a specific reason at a localized activity The objective is to discover special causes that are present Eliminate the root causes of the special variations, e.g. machine wear, material quality, fatigued workers, misadjusted equipment Incorporate the good process control © 2006 Prentice Hall, Inc. S6 – 12 SPC and Statistical Samples To conduct inspection using the SPC approach, one has to compute averages for several small samples instead of using data from individual items: Steps Figure S6.1 © 2006 Prentice Hall, Inc. Frequency (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Each of these represents one sample of five boxes of cereal # # # # # # # # # # # # # # # # # # # # # # # # # # Weight S6 – 13 SPC Sampling To measure the process, we take samples of same size at different times. We plot the mean of each sample for each point in time Figure S6.1 © 2006 Prentice Hall, Inc. Frequency (b) After enough samples are taken from a stable process, they form a pattern called a distribution The solid line represents the distribution Weight S6 – 14 Attributes of Distributions (c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Central tendency Variation Shape Frequency Figure S6.1 © 2006 Prentice Hall, Inc. Weight Weight Weight S6 – 15 (d) If only common causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable Frequency Identifying Presence of Common Sources of Variation Prediction Weight Figure S6.1 © 2006 Prentice Hall, Inc. S6 – 16 Identifying Presence of Special Sources of Variation Frequency (e) If assignable causes are present, the process output is not stable over time and is not predicable ? ?? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? Prediction Weight Figure S6.1 © 2006 Prentice Hall, Inc. S6 – 17 Central Limit Theorem Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve 1. The mean of the sampling distribution (x) will be the same as the population mean m 2. The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n © 2006 Prentice Hall, Inc. x=m sx = s n S6 – 18 Interpreting SPC Charts Frequency Lower Control Limit (a) In statistical control and capable of producing within control limits Upper Control Limit (b) In statistical control but not capable of producing within control limits Size (weight, length, speed, etc.) © 2006 Prentice Hall, Inc. (c) Out of statistical control and incapable of producing within limits Figure S6.2 S6 – 19 Population and Sampling Distributions Three population distributions Distribution of sample means Mean of sample means = x Beta Standard deviation of s the sample = sx = n means Normal Uniform | | | | -3sx -2sx -1sx x | | +1sx +2sx +3sx 95.45% fall within ± 2sx 99.73% of all x fall within ± 3sx © 2006 Prentice Hall, Inc. | Figure S6.3 S6 – 20 Sampling Distribution Sampling distribution of means Process distribution of means x=m (mean) © 2006 Prentice Hall, Inc. Figure S6.4 S6 – 21 Steps In Creating Control Charts 1. Take representative sample from output of a process over a long period of time, e.g. 10 units every hour for 24 hours. 2. Compute means and ranges for the variables and calculate the control limits 3. Draw control limits on the control chart 4. Plot a chart for the means and another for the mean of ranges on the control chart 5. Determine state of process (in or out of control) 6. Investigate possible reasons for out of control events and take corrective action 7. Continue sampling of process output and reset the control limits when necessary © 2006 Prentice Hall, Inc. S6 – 22 In-Class Exercise : Control Charts 6/15 6/16 8AM 10AM 12AM 2PM 8AM 5 6 8 6.5 6.5 5.5 8 6.5 6 7 6.5 6.5 7.5 6.5 7 8 6.5 6.5 6 6 8 7.5 6.5 7 6.5 Calculate X bar and R’s for new data Calculate X double bar and R bar figures for new data Draw X bar chart Calculate LCL and UCL for X bar chart Draw lines for LCL and UCL and for X double bar in chart © 2006 Prentice Hall, Inc. S6 – 23 Control Charts for Variables For variables that have continuous dimensions Weight, speed, length, strength, etc. x-charts are to control the central tendency of the process R-charts are to control the dispersion of the process These two charts must be used together © 2006 Prentice Hall, Inc. S6 – 24 Setting Chart Limits For x-Charts when we know s Upper control limit (UCL) = x + zsx Lower control limit (LCL) = x - zsx where © 2006 Prentice Hall, Inc. x = mean of the sample means or a target value set for the process z = number of normal standard deviations sx = standard deviation of the sample means = s/ n s = population standard deviation n = sample size S6 – 25 Setting Control Limits Hour 1 Sample Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 n=9 5 17 6 16 7 15 8 17 9 16 Mean 16.1 s= 1 © 2006 Prentice Hall, Inc. Hour 1 2 3 4 5 6 Mean 16.1 16.8 15.5 16.5 16.5 16.4 Hour 7 8 9 10 11 12 Mean 15.2 16.4 16.3 14.8 14.2 17.3 For 99.73% control limits, z = 3 UCLx = x + zsx = 16 + 3(1/3) = 17 ozs LCLx = x - zsx = 16 - 3(1/3) = 15 ozs S6 – 26 Setting Control Limits Control Chart for sample of 9 boxes Variation due to assignable causes Out of control 17 = UCL Variation due to natural causes 16 = Mean 15 = LCL | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 Sample number © 2006 Prentice Hall, Inc. Out of control Variation due to assignable causes S6 – 27 Setting Chart Limits For x-Charts when we don’t know s Upper control limit (UCL) = x + A2R Lower control limit (LCL) = x - A2R where © 2006 Prentice Hall, Inc. R = average range of the samples A2 = control chart factor found in Table S6.1 x = mean of the sample means S6 – 28 Control Chart Factors Sample Size n Mean Factor A2 Upper Range D4 Lower Range D3 2 3 4 5 6 7 8 9 10 12 1.880 1.023 .729 .577 .483 .419 .373 .337 .308 .266 3.268 2.574 2.282 2.115 2.004 1.924 1.864 1.816 1.777 1.716 0 0 0 0 0 0.076 0.136 0.184 0.223 0.284 Table S6.1 © 2006 Prentice Hall, Inc. S6 – 29 Setting Control Limits Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 © 2006 Prentice Hall, Inc. S6 – 30 Setting Control Limits Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCLx © 2006 Prentice Hall, Inc. = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces From Table S6.1 S6 – 31 Setting Control Limits Process average x = 16.01 ounces Average range R = .25 Sample size n = 5 UCLx LCLx © 2006 Prentice Hall, Inc. = x + A2R = 16.01 + (.577)(.25) = 16.01 + .144 = 16.154 ounces UCL = 16.154 = x - A2R = 16.01 - .144 = 15.866 ounces LCL = 15.866 Mean = 16.01 S6 – 32 R – Chart Type of variables control chart Shows sample ranges over time Difference between smallest and largest values in sample Monitors process variability Independent from process mean © 2006 Prentice Hall, Inc. S6 – 33 Setting Chart Limits For R-Charts Upper control limit (UCLR) = D4R Lower control limit (LCLR) = D3R where R = average range of the samples D3 and D4 = control chart factors from Table S6.1 © 2006 Prentice Hall, Inc. S6 – 34 Setting Control Limits Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D4 = 2.115, D3 = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds UCL = 11.2 LCLR LCL = 0 © 2006 Prentice Hall, Inc. = D3R = (0)(5.3) = 0 pounds Mean = 5.3 S6 – 35 Mean and Range Charts (a) (Sampling mean is shifting upward but range is consistent) These sampling distributions result in the charts below UCL (x-chart detects shift in central tendency) x-chart LCL UCL (R-chart does not detect change in spread) R-chart LCL Figure S6.5 © 2006 Prentice Hall, Inc. S6 – 36 Mean and Range Charts (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) UCL (x-chart does not detect the increase in mean) x-chart LCL UCL (R-chart detects increase in dispersion) R-chart LCL Figure S6.5 © 2006 Prentice Hall, Inc. S6 – 37 Automated Control Charts © 2006 Prentice Hall, Inc. S6 – 38 Control Charts for Attributes For variables that are categorical Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure Percent defective (p-chart) Number of defects (c-chart) © 2006 Prentice Hall, Inc. S6 – 39 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2006 Prentice Hall, Inc. Normal behavior. Process is “in control.” S6 – 40 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2006 Prentice Hall, Inc. One plot out above (or below). Investigate for cause. Process is “out of control.” S6 – 41 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2006 Prentice Hall, Inc. Trends in either direction, 5 plots. Investigate for cause of progressive change. S6 – 42 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2006 Prentice Hall, Inc. Run of 5 above (or below) central line. Investigate for cause. S6 – 43 Process Capability The natural variation of a process should be small enough to produce products that meet the standards required A process in statistical control does not necessarily meet the design specifications Process capability is a measure of the relationship between the natural variation of the process and the design specifications © 2006 Prentice Hall, Inc. S6 – 44 Process Capability Ratio Upper Specification - Lower Specification Cp = 6s A capable process must have a Cp of at least 1.0 Does not look at how well the process is centered in the specification range Often a target value of Cp = 1.33 is used to allow for off-center processes Six Sigma quality requires a Cp = 2.0 © 2006 Prentice Hall, Inc. S6 – 45 Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6s © 2006 Prentice Hall, Inc. S6 – 46 Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6s 213 - 207 = = 1.938 6(.516) © 2006 Prentice Hall, Inc. S6 – 47 Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6s 213 - 207 = = 1.938 6(.516) © 2006 Prentice Hall, Inc. Process is capable S6 – 48 Process Capability Index Upper Lower Cpk = minimum of Specification - x , x - Specification Limit Limit 3s 3s A capable process must have a Cpk of at least 1.0 A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes © 2006 Prentice Hall, Inc. S6 – 49 Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches © 2006 Prentice Hall, Inc. S6 – 50 Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches (.251) - .250 Cpk = minimum of , (3).0005 © 2006 Prentice Hall, Inc. S6 – 51 Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches (.251) - .250 .250 - (.249) Cpk = minimum of , (3).0005 (3).0005 Both calculations result in .001 Cpk = = 0.67 .0015 © 2006 Prentice Hall, Inc. New machine is NOT capable S6 – 52 Process Capability Comparison New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Cp = Upper Specification - Lower Specification Cp = .251 - .249 0.66 .0030 © 2006 Prentice Hall, Inc. 6s = New machine is NOT capable S6 – 53 Interpreting Cpk Cpk = negative number Cpk = zero Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Figure S6.8 © 2006 Prentice Hall, Inc. S6 – 54 Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Only screens lots; does not drive quality improvement efforts © 2006 Prentice Hall, Inc. S6 – 55