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BUSA 3110 Statistics for Business Statistical Process Control Kim I. Melton, Ph.D. Transition Population Process We have a fixed We have an on-going population (a group of process that produces interest) output We want to use We want to assess if information from a the observed output sample from the from the process can existing group to draw be considered as a a conclusion about the sample from a entire group. population and Goal We want to be able to Describe— “What” “infer” to future output from the process Goal Explain— “Why?” Population and Sample Population: All of the items/measurements of Inference is back interest to the population that produced the sample Sample: The items/measurements actually obtained Process and “Sample” Two issues arise: Changes can occur in an on- going process while you are collecting data—i.e., you don’t know if all of your data is coming from the same population Although describing past output may be useful, this is descriptive (history). You really want to be able to know what to expect in the future—i.e., you aren’t trying to make an inference about the process as it existed while you were collecting data. Classical Statistical Inference? Suppose you could obtain data on the process of delivering shipments to a customer (at much cost in terms of dollars and time). You would like to say something about the expected delivery time for the near future. How can you use the data? Looking at the SAME Data as Output from a Process… ? ? ? ? ? ? Process Based Studies We don’t know if there is a population. Conditions may not be the same for repeated observations from the process. The goal is to make decisions that will relate to the future output from the process; to do this we must understand the underlying cause system. If the process is stable, the history of the process (combined with subject matter knowledge) can be used to predict the future output. No numerical measure of confidence can be calculated! Aims: Monitoring and Improvement Monitoring Determine if conditions were similar enough over the entire period to make some kind of descriptive statement about output from the process Determine if the variation in the process comes from the design of the process or from a combination of the design of the process and some other sources of variation Improvement Determine what type of action is called for to influence future outcomes Improvement comes from Changes to the process (not just to the numbers) To change the average To reduce variation To make the process more robust to variation Innovation (change the problem!) SIPOC Model Suppliers provide Inputs to the Process which produces Output for Customers Voices of the Process Voice of the Customer Voice of the Product/Service Voice of the Process Voice of the System Example: Bottle Closures Voice of the Customer –What the customer wants (may be the end customer or the bottler) Fits e.g., The bottler may provide specs to the closure manufacturer: E dimension - target: 1.003; LSL = .995; USL 1.011 Clean Safe e.g., The FDA says that no more than 3 in 1000 TE bands can fail Reusable (can be taken off and put back on) Easy to open Doesn’t leak Stays on Bottle Closure Voice of the Product – Data from output from the process about the product that relates to what the customer wants: Fits actual results on various diameters and other dimensions Clean color and lack of debris Safe actual number of TE bands that fail Reusable (can be taken off and put back on) Threads Easy to open actual torque required to open Doesn’t leak data on liners or clearances Stays on results of heat tests Bottle Closure Voice of the Process The factors that can be controlled during the production process that might impact the product/service characteristic Fit/Reusable/Doesn’t leak/Stays on dimensions Machine settings: temperatures, pressure, time Resin: type, color Clean free from contamination Screening of resin prior to injection into the mold Safe TE Band Amount of resin forced into the mold (to control the size of the “bridges”) Doesn’t leak Liner Temperature of extruder head Voice of the System Research and Design (design of closure and molds) Hiring practices Maintenance practices Training Stability (aka: Statistical Control) Stability depends on the consistency of output (in terms of the range of variation seen); a stable process could consistently produce product that is too small, too large, or within the acceptable range. Variation is considered stable when there are only common causes of variation present; if there are both common and special causes of variation present the process is not considered stable (i.e., it is not in a state of statistical control) Stability, Acceptability, and Desirability Stability: What is consistently happening A process is called stable if the variation in the outputs is predictable (may be predictably small or predictably large—the issue is that the range can be predicted) Acceptability: What is considered “OK” from the process Output that meets the requirements set by the customer is considered “acceptable.” Desirability: The ideal output from the process A process that is producing acceptable output can still be improved by reducing variation around a target. Stability, Accuracy, Precision Stability, Accuracy, Acceptable vs. Precision Desirable Bottle Filling Process All three process could be stable Only the third process might be 1 considered acceptable Even the third process could be made more 2 desirable (by filling every bottle to the reported fill volume) 3 Zero Defects vs. Continual Improvement E Dimension on a closure Operational Definition identifies how “deep” in the closure to measure, how to orient the closure on the measurement machine, how to obtain the “diameter,” and where to start and stop measuring (the threads spiral down the closure) Acceptable (Specifications): Closure: LSL .995 USL 1.011 Bottle: LSL .972 USL .997 Desirable (Nominal): Closure: 1.003 Bottle: .984 If all product were made to nominal, clearance would be .019 If “meeting specs” is good enough Clearances could range from -.002 to .039 [Some closures won’t fit on the bottles!!!] ____ Limits Spec Limits (specifications) Measurements that are “acceptable” to the customer Control limits Statistically calculated limits used to assess stability and estimate the range of values for the characteristic plotted on a control chart Natural Process Limits Range of measurements for individual items expected to be seen from a process (depends on stability) Run Charts Measurements for a process characteristic are plotted in time order Patterns in the data indicate special causes of variability time ==> Trends Clusters A repeating pattern No variability in plotted points Run Charts If the process is stable all of the following will be true: • Most points will plot near some central value • Some variation will exist • Individual points will not be predictable, but the overall clustering of points will be predictable • No patterns will show up Control Charts A Run Chart with statistically calculated limits Limits are based on data collected from the process If points plot within the limits and show no patterns, the process is said to be "in control" or stable If the process is considered stable, limits represent the time ==> expected range of variation for the value plotted A process that is in control is predictable (it may or may not be producing desirable output) Setting Up Control Charts Step Determine question(s) to be answered 1: Step Design data collection plan and collect data 2: Step Plot run chart and look for obvious patterns 3: Step If no patterns, calculate control limits (using 4: formulas for the appropriate type of chart) Step 5: Conduct runs tests Step 6: Interpret the chart Step 7: Determine appropriate type of action and take steps to accomplish this Runs Tests-Melton A control chart fails to show stability if any of the following occur: • at least one point plots A outside the control limits B • two of three consecutive C points in the same A zone C • fifteen consecutive points B plot in the C zones A • more than seven consecutive points on the same side of the center line • seven or more consecutive increases (or decreases) Note: Other runs tests • fifteen consecutive points alternating up and down are available in other books. LCL = 6.387 CL = 15.467 UCL = 24.547 Width = 3.02666 Insurance Quotes An insurance company staffs quote lines so that independent agents can call in for quotes on insurance. The following data represent the time (in seconds) for one operator to respond to five consecutive calls from the same state. One subgroup (of five observations) is collected each day. The Data Day Obs. 1 Obs. 2 Obs. 3 Obs. 4 Obs. 5 1 197 190 162 159 194 2 200 192 177 227 180 3 186 178 209 197 190 4 206 168 209 208 182 5 182 175 158 207 226 6 195 179 216 213 193 7 197 195 213 198 217 8 208 248 193 158 177 9 184 166 224 186 180 10 203 185 212 214 161 11 189 183 207 176 207 12 223 175 196 213 200 13 200 168 193 233 164 14 186 161 179 155 203 15 199 218 211 217 230 X-bar and R charts X-Bar and R Charts Location and Spread Control Limits If the process appears to be stable, then: Using Excel to Calculate X-Bar and R Assume that the data for the first subgroup are in cells B2, C2, D2, E2, and F2 If you want to put X-Bar for this subgroup in cell I2, then click on cell I2 and type: =average(B2:F2) If you want to put R for this subgroup in cell J2, then click on cell J2 and type: =max(B2:F2)-min(B2:F2) Copy the formulas down the column to find the corresponding values for each subgroup X-Bars and Rs X-Bar R 180.4 38 195.2 50 192 31 194.6 41 189.6 68 199.2 37 204 22 196.8 90 188 58 195 53 192.4 31 201.4 48 191.6 69 176.8 48 215 31 Control Chart Constants n D3 D4 A2 d2 2 None 3.267 1.880 1.128 3 None 2.574 1.023 1.693 4 None 2.282 .729 2.059 5 None 2.114 .577 2.326 6 None 2.004 .483 2.534 7 .076 1.924 .419 2.704 R Chart Calculating Control Limits CL: 47.667 UCL: 2.114*47.667 = 100.767 LCL: none (NOTE: this is different from 0) Conducting Runs Tests Width of zones: (100.767 – 47.667)/3 = 17.7 Heights: 100.767 83.067 65.367 47.667 29.967 12.266 R Chart R Chart Runs: 100 80 60 R (n=5) 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Subgroup # R Chart (What does this tell us?) Remember that R looks at within subgroup variation. R does not address the magnitude of the observations—just the difference between the highest and lowest in a subgroup. For example, the following two subgroups would have the same range: 5, 10, 15, 20 105, 110, 115, 120 R addresses stability with respect to spread (or within subgroup variation) Lack of stability relates to changes within a subgroup. X-Bar Chart Calculating Control Limits CL: 194.133 LCL: 194.133 - .577(47.667) = 166.63 UCL: 194.133 + .577(47.667) = 221.637 Conducting Runs Tests Width of zones: (221.637 – 194.133)/3 = 9.168 Heights: 221.637 212.469 203.301 194.133 184.965 175.798 166.63 X-Bar Chart X-Bar Chart Runs: 230 220 210 200 X-Bar (n=5) 190 180 170 160 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Subgroup # X-Bar Chart (What does this tell us?) Remember that the X-Bar Chart plots subgroup averages. For the limits on the X-Bar Chart to have meaning, you need stability with respect to spread—the average R needs to be meaningful. The X-Bar chart assesses if there are additional sources of variation between subgroups (above and beyond the variation captured within the subgroups). Interpreting Results When the Process is Judged to be Stable Recall that control limits deal with the expected range of variation for the characteristic plotted on the control chart. Therefore, the limits on the X-bar chart deal with the expected range of variation for subgroup averages. Natural process limits, expected ranges for the individual values, can be calculated as: *based on Empirical Rule (and assumes mound shaped distribution for individual measurements) Can we use the Empirical Rule? Stability allows Calculating Natural us to consider Process Limits appears the observations reasonable. as coming from a single distribution. Histogram of Individual Obs. 16 14 12 LNPL=194.133-3(20.493) 10 = 132.654 Frequency 8 6 4 UNPL=194.133+3(20.493) 2 0 = 255.612 9 9 9 9 9 9 9 9 9 9 15 16 17 18 19 20 21 22 23 24 Time in Seconds (max in bin given) Limits Control Limits (LCL and UCL) calculated from data collected from the process used to assess stability of the process related to the characteristic plotted (e.g., X-Bar or R) Natural Process Limits (LNPL and UNPL) only calculated if the process appears to be stable related to measurements for individual items provides information about range of measurements for individual items that can be expected Based on the Empirical Rule (so assumes mound shaped distribution of individual measurements Specification Limits - “Specs” (LSL and USL) determined by the user based on desired or needed measurements for an item related to desired (acceptable) range of measurements for individual items Call Center Goals Suppose that management of the insurance company wants quotes times to be between two minutes and four minutes. (They believe any times below two minutes will be rushed and unfriendly, and times above four minutes would discourage future business.) With the current process, what proportion of the calls would you expect to take more than four minutes? How are we doing? The manager’s expectation that calls will be between 2 and 4 minutes provides “specs” that we will write in terms of seconds: LSL = 120; USL = 240 We just found Natural Process Limits of: LNPL = 132.654; UNPL = 255.612 120 240 132.654 255.612 PROBLEM: The process is expected to produce some calls that are too long. How many calls will be too long? Based on a comparison of our Natural Process Limits and the Specs, we would expect to find a few calls that are too long. Calculating, we see P(X > 240) = =1-NORMDIST(240,194.133,20.493,1) = .0126 About one and a quarter percent (1.26%) of the calls will be too long. How many calls will be too short? We can compare our Natural Process Limits and the specs to estimate that very few calls will be too short. To find the proportion of calls that would be expected to be too short, we find: P(X < 120) = =NORMDIST(120,194.133,20.493,1) = .00015 Less than .02% would be expected to be too short. Call Center Ads Suppose the manager wants to create an ad telling people how quickly they can provide a quote. The manager wants to include a statement along the lines of: “If you call us, our agent will provide you with a quote in less than ________ minutes.” What number should be placed in the blank? Graphically… Logically, the advertised length will be high enough that “most” of the calls will be handled in the advertised time. The shaded in area represents the calls that are completed within the advertised time. The advertised time will depend on the proportion of calls that Where z is the value of z that we want to complete has an area to the left and is within the claim found with number of seconds. =normsinv() Process Capability To talk about capability of a process, we must have stability Capability refers to the “voice of the process” The capability of the process tells the range of values that can be expected for the measurements of some process characteristic Specifications (specs) provide a “voice of the customer.” Capability indexes are a fairly common way of communicating the relationship between specifications and process performance. Capability indexes attempt to compare the “voice of the process” with the “voice of the customer.” Two Approaches to Talking about Capability Approach 1: Comparison of Engineering Tolerances to Natural Tolerances Engineering Tolerances refer to the specifications for the characteristic ET = USL - LSL Natural Tolerances refer to the natural process limits for the characteristic NT = UNPL - LNPL (where natural process limits are calculated as m ± 3s and s is estimated by Rbar/d2 from a stable process) If NT < ET, we say the process is capable If NT < ET and the natural process limits are within the specification limits, we say the process is capable and meeting spec. Capable but Not Meeting Specs??? When capability is described in terms of the amount of variation (without looking at location), a process with very little variation could consistently produce unacceptable product. Example: Nails are sold by weight, but builders need to know how many nails are contained in boxes of a given weight. Suppose a builder specifies that each box of nails should contain 990 to 1010 nails (i.e., 1000 ± 10). The producer has reduced the variation in the weight of nails to the point where there is only a difference of 1 to 4 nails from one box to another—but, boxes actually contain 983 to 987 nails. Capability (cont.) Approach 2: Capability Indexes Cp tells us if the natural variation is smaller than the allowed variation. Cp does not look at process location; therefore it is possible to have a ‘good’ Cp and be making large amounts of unacceptable product. Cpk tells us if the natural variation is ‘small enough’ and ‘far enough’ from the specifications for most product to meet specs for the characteristic. Cpk cannot be larger than Cp. Capability Formulas USL - LSL ET Cp = = 6s NT Cpk = min { USL - X 3s , X - LSL 3s } If the process is centered between the specs: Cpk = Selecting the Appropriate Type of Control Chart Since control limits are calculated from data collected from the process, we need to know which formulas to use! Control Charts–Variables Data X-bar and R charts (used together) X-bar (the average of n observations) attempts to assess location R (the range of n observations) attempts to assess spread X and Moving Range charts (used together) When there is no logical grouping, individual values are plotted on the X chart A moving range is used to assess spread Control Charts–Attributes Data p or np charts "n" items are studied each item is classified in one of two categories we are counting the number in one of the categories a p chart plots the proportion in one category an np chart plots the number in one category c or u charts an inspection unit (IU) is defined the number of occurrences are counted and plotted Determine characteristic to study Counting or counting Measuring measuring Attributes Data Variables Data Classifying How many no into two items per yes categories? subgroup? one 2 to 8 Constant Constant subgroup area of size? opportunity? Consider Consider X/mR X/R yes no yes no Consider Consider Consider Consider np or p p c u Examples Potholes per mile Time of delivery of the meal cart at lunch at a nursing home Complaints per day (lunch meal) at a nursing home 1:1s in a mental health facility Confiscated items per day at Atlanta Hartsfield Jackson Airport Late arriving flights per day by Delta at ATL Weight of contents of cans of tomato soup Food Service A nursing home serves meals to 100 patients each day. Meals are prepared in a central kitchen facility, served onto trays, placed in specially heat carts, and delivered to the staff on the units for distribution to patients. Lunch is scheduled to be served at noon. In order to start to address complaints about cold food being delivered to patients, some data was collected. The Data Day of Number of Time of Cart Day of Number of Time of Cart Week Complaints Delivery Week Complaints Delivery Mon. 39 11:30 Tues. 30 11:58 Tues. 31 11:35 Wed. 38 11:50 Wed. 30 12:00 Thurs. 24 12:20 Thurs. 27 12:15 Fri. 25 12:03 Fri. 38 11:45 Sat. 28 12:04 Sat. 33 11:32 Sun. 39 12:56 Sun. 35 12:00 Mon. 31 11:35 Mon. 26 12:15 Total number of complaints: 474 Complaints Data (actually data on the # of “complainers”) Day of Number of Day of Number of Week Complaints Week Complaints Mon. 39 Tues. 30 Tues. 31 Wed. 38 Wed. 30 Thurs. 24 Thurs. 27 Fri. 25 Fri. 38 Sat. 28 Sat. 33 Sun. 39 Sun. 35 Mon. 31 Mon. 26 Total number of complaints: 474 Formulas For this example: Formulas: Complaints 15 20 25 30 35 40 45 50 M on . Tu e s. W ed . Th ur s. Fr i . S at . Su n . M on . Tu e s. Day of Week W ed . # of Complaints Th ur s. Fr i . S at . S un . M on . np Chart for # of Complaints Using Data to Generate Theories Revised Complaint Data Suppose the data had been: Day # Complaints# Meals Served pi 1 39 95 .411 2 31 102 .304 3 30 87 .345 … … … … 15 31 110 .282 Totals 474 1500 p chart for Complaints Point 1: Formulas Point 2: p Chart for Complaints 1:1's in a Mental Hospital When patients 'act-out' in ways that can be hazardous to themselves or to others, they may be placed on one-to-one (1:1). When a patient is on 1:1 a staff member must remain within arm's distance of the patient at all times. This requires one staff member to have no other duties other than the supervision of this one patient. Obviously, this is very expensive. In an attempt to improve service and reduce costs, the number of 1:1's is being studied. To start, data from the past quarter were studied. The Data (Day Shift) Week 1:1’s Week 1:1’s 1 23 10 18 2 18 11 19 3 27 12 17 4 35 13 16 5 10 14 14 6 19 15 21 7 24 16 11 8 20 17 14 9 21 18 15 Total number of 1:1's: 342 Formulas for c Chart For this example: Formulas: c Chart for 1:1s 1:1s for Day Shift 40 35 30 25 1:1's 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Week Improvement? 1:1s for Day Shift 40 35 30 25 1:1's 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Week