This blog has reviewed six applications of stratification. The postings are: An Automatic Call Distribution System, Machine Shop Rejections, Molding Operation for a Wooden Entry Door, Resin Output Quantity Variation, Molding Operation Producing Plastic Switches, and Uneven Thickness of Steel Sheets. We have emphasized stratification. However, the Interrelationship Digraph Source posting lists the Seven Management and Planning Tools and the Seven Major SPC Tools (Magnificent Seven). The Memory Jogger Plus+ features the Seven Management and Planning Tools, and Douglas Montgomery proposed the Seven Major SPC Tools. Stratification is not listed in either of these tool lists. Have we over-emphasized stratification?

Stratification is the classification and analysis of data by a factor such as the day of the week, an operation, or a machine. The initial purpose is to divide a heterogeneous population in homogeneous subpopulations or strata. Then we can focus on the strata having the poorest quality.

We think that stratification is under-emphasized. Consider Ishikawa’s (1990) statements. On page 211, he states: “Neither improvement nor control is possible without stratification. I have emphasized over and over that stratification is needed for control, for detecting problems, and for studying improvement measures, …”. Shewhart (1931) on page 304 describes Criterion I for detecting lack of control. He recommends dividing a set of data into rational subgroups and testing whether the variation over the aggregated data set is greater than the variation within the subgroups. By variation, I mean the estimate of the standard deviation, i.e., σ. That is, is the estimate of the standard deviation for the aggregate data set significantly larger than the average of the standard deviations within each subgroup? If the test indicates that it is, then one has detected lack of control. Dividing the data into rational subgroups is stratification.

Clearly, the Statistical Thinking references by Britz, Emerling et al (2000) and Hoerl and Snee (2002) feature stratification. See the following postings: Analyze Common-Cause Variation, Analyze Common-Cause Variation Examples (Stratification), Analyze Common-Cause Variation (Disaggregation), and Analyze Common-Cause Variation A.

1. Ishikawa, Kaoru, Translated by John Loftus, (1990). Introduction to Quality Control , 3A Corporation, Tokyo, Japan.
2. Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product, D. Van Nostrand Company, New York.
3. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
4. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

The primary purpose Exploratory Data Analysis (EDA) is to identify the key variables that affect the quality measures. Two principles, mentioned by De Mast and Trip (2007), are helpful in identifying these variables. They are:

• Display the distribution of the data
• Display the distribution within individual stratum

Chang and Lu (1995) provide an example illustrating these principles. A steel sheet metal manufacturer had customers complaining about uneven thickness. The specification was 4.5 ± .5 mm. The production manager had data collected from 120 sheets giving the thickness measurements on the left, middle and right sides of the sheets. Employees selected five sheets at shift times of 0900, 1100, 1400 and 1700 over a period of five days. The histogram appearing below shows 13% of the sheet thickness measurements below the lower specification limit of 4.0 mm. Also, the mean is lower than 4.5 mm.

After discussions with shop-floor personnel, they stratified by position on the sheet and by time. Histograms for the two stratifications appear below. The stratification by position did not show distributions much different than the aggregate distribution. However, the stratification by time showed higher frequencies of thin measurements at 1100 and 1700. Twenty four of the 26 values in the histograms below 4 mm, 24 of them were at 1100 and 1700.

Discussions with shop-floor personnel identified mold wear out, build up of chips in a work holding device, and operator fatigue as possible causes. The corrective action was to take a 10 minute break at 1030 and 1630 each day and have maintenance performed during the breaks. The corrective action produced a substantial reduction in thin sheets.

References

1. Chang, P.-L. and K.-H. Lu (1995). "The Construction of the Stratification Procedure for Quality Improvement." Qualilty Engineering 8(2): 237-247.
   Chang, P.-L. and K.-H. Lu (1995). "The Construction of the Stratification Procedure for Quality Improvement." (2): 237-247.
2. De Mast, Jeroen and Albert Trip (2007). “Exploratory Data Analysis in Quality-Improvement Projects”, Journal of Quality Technology, 39(4): 301-311.

De Mast and Trip (2007) specify that the purpose of Exploratory Data Analysis (EDA) is to identify the dependent (Y) and independent (X) variables that may help understand or solve a quality problem. However, they point out that EDA can only identify variables that vary in the collected data set. If the EDA can not identify key variables affecting the system performance, available options include:

1. Collecting additional data and revising the variables recorded
2. Analyzing the available information, designing experiments, and conducting the experimental design

Option 1
The Pease Industries example illustrates the first option. A team wanted to reduce an 11% defect rate in glass inserts for a wooden entry door. They thought that humidity and temperature variations were the cause. They collected data and did a regression analysis where the dependent variable was the number of defects and the independent variables were temperature and humidity. They found no correlation. Then the team collected additional data, and they examined defect occurrence as related to part type, monthly occurrence and day of the week. They found that the defect rate varied with the day of the week. After investigating why the day of the week was important, they determined that dirty molds caused the elevated defect rate.

Option 2
The Cause and Effect Diagram posting describes a case study illustrating the second option above. A company was experiencing excessive variation in its grinding operation. A team conducted a brainstorming session to identify key factors causing the variation in the grinding operation. The brainstorming session produced a Cause & Effect diagram. The Grinding Process Example posting describes an experimental design conducted to determine which factors were most significant. The Part 4 posting describes the analysis of the experimental results. The company improved the grinding process performance index from .49 to 1.25.

References

1. De Mast, Jeroen and Albert Trip (2007). “Exploratory Data Analysis in Quality-Improvement Projects”, Journal of Quality Technology, 39(4): 301-311.

De Mast and Trip (2007) list the following three steps in performing Exploratory Data Analysis.

1. Display the Data
2. Identify salient features
3. Interpret salient features

Part B of the resin output variation example posting illustrates these steps. The Ricoh team constructed a histogram of the output quantity (Display the data), noticed the bimodal nature of the output quantity (Identify salient features), and this bimodal distribution suggested that the output distributions from lines A and B were different (Interpret salient features). Histograms of line A and line B output confirmed this conclusion. Another salient feature of the histograms was the excessive variation in output quantity. This feature motivated establishment of lower and upper limits and a target value.

References

1. De Mast, Jeroen and Albert Trip (2007). “Exploratory Data Analysis in Quality-Improvement Projects”, Journal of Quality Technology, 39(4): 301-311.

The Resin Output Variation Example postings (Part A, Part B, Part C and Part D) illustrate Statistical Thinking and the Hoerl-Snee Process Improvement Strategy. This example also makes extensive use of Exploratory Data Analysis.

Ricoh’s Numazu plant made raw material used as ingredients for copy machine toner. The company had a team which monitored the process in order to achieve continual quality improvement.

• The team examined a Resin Yield Run Chart and noted that the yield ratio sometimes exceeded 1.0 which was theoretically impossible. The run chart indicated the presence of an assignable cause which they eliminated by preventing a drop in air pressure. However, yields still exceeded 1.0. They decided that this result was due to variation and suspected that this variation would degrade finished product quality. The team started an effort to discover the source of variation and to make changes to eliminate it.
• The team collected output quantity data and constructed histograms. The Part B posting shows the histograms. The overall output quantity histogram clearly shows two peaks indicating a combination of two component distributions. After the second batch processing step, the process splits a batch into two parts which are processed on two separate lines, i.e., line A and line B. The Part B posting shows histograms for the output quantities of the two lines. Comparing the line A and line B histograms shows that the two lines have different distributions for their output quantities.
• Next the team constructed a Cause & Effect Diagram to show potential causes of the output quantity variation and the differences between the two lines. The Part B posting presents this Cause & Effect Diagram. The Part C posting describes the identification and elimination of two variation causes. They investigated the procedure for dividing the resin after the second processing step. They discovered that some material remained in the reaction tank after sending material to the two lines. This mean that line B had less input and thus less output than line A. They changed the dividing procedure and found no significant difference between the output quantities of the two lines.
• The second potential cause described in the Part C posting involved the solvent feed ratio. They constructed a scatter plot showing that increasing solvent feed ratio was correlated with increasing output. This correlation was inconsistent with the team’s understanding of the physical process. They found that the ratio measurement was affected by the time the solvent was in the tank. They changed the procedure to insure the solvent had stabilized prior to measurement. Examination of a control chart showed that the variation in output quantity was still excessive.
• The Part D posting describes the elimination of a third cause, and the posting shows a control chart. This control chart clearly shows a significant reduction in output variation.

The exploratory data analysis included examination of four different graphical displays. They are a run chart, histograms, a scatter plot, and several control charts. De Mast and Trip (2007) points out that Good (1983); Hoaglin, Mosteller et al (2000); and Bisgaard (1996) note that graphical presentations are preferred in Exploratory Data Analysis. They are more effective is showing an individual what he did not expect to see.

References

Bisgaard (2006) gives us an example where Exploratory Data Analysis leads us to narrow the scope of the quality improvement investigation. The example involves the production of small outboard motors by an assembly line. Monthly quality reports showed an unacceptable number of defective motors that caused costly rework. The Vice President of Manufacturing formed a team for the purpose of reducing the number of defects.

The first task performed by the team was to flowchart the assembly line. This is consistent with the first step in the Hoerl-Snee Process Improvement Strategy (See the Understand the Process posting). After the base motors were painted and dried, the motors traveled on a ten station line for the purpose of installing accessory components. These accessory components included the carburetor, brackets, the propeller, and electrical systems. Next, the team examined tables specifying defects and their type. The team found the tables difficult to analyze. To assist the analysis the team constructed Pareto charts specifying the defects by type of defect. For example, missing fasteners, loose fasteners, and missing operations. These Pareto Charts did not suggest principal causes. The team decided to categorize the defects by the station on the line where the defect originated. For example, a loose fastener on the carburetor, the defect originated at station 3. An incorrectly mounted spark plug wire would have occurred at station 9. The Pareto Chart categorizing defects by station appears below.

The team focused on station 9. The workers on the line revealed that design of the motors had changed leaving station 9 with more work than the other stations. The team redesigned the assembly line reducing the work load at station 9. A number of other changes were made such as improved lighting. The result was a dramatic reduction in the occurrence of defects.

De Mast and Trip (2007) claim that this example illustrates the use of exploratory data analysis change the focus of the problem from “too many defects” to “too many defects from station 9”. They state that the example illustrates the use of exploratory data analysis to identify a KPOV.

My viewpoint is that this example illustrates the identification of a KPIV. That is, the assembly line station. Admittedly, the next phase of the improvement effort was clearly more focused on station 9. We must remember that quality improvement is often an iterative process. That is, successive Plan-Do-Check-Act (PDCA) cycles. Identifying a KPIV on a cycle may result in that KPIV being a KPOV on the next cycle.

1. Bisgaard, S. (1996). "Qualilty Quandaries: The Importance of Graphics in Problem Solving and Detective Work." Quality Engineering 9(1): 157-162.
2. De Mast, Jeroen and Albert Trip (2007). “Exploratory Data Analysis in Quality-Improvement Projects”, Journal of Quality Technology, 39(4): 301-311.

The purpose of Exploratory Data Analysis (EDA) is to generate hypotheses or clues that guide us in improving quality or process performance. Breyfogle (2003, pgs. 10-11) views Six Sigma as a murder mystery where we use a structured approach to uncover clues that lead us to improve process outputs. These clues are Key Process Input Variables (KPIVS) and process improvement strategies. As an example, he considers the process of traveling to work where the Key Process Output Variable (KPOV) is the arrival time. Examples of KPIVs are the setting of our alarm clock and our departure time. An alternative process improvement strategy might be a different travel route that is less subject to variation during congested time periods. Then, the route selected is another KPIV, and the travel time along that route is a function of both the route and departure time. Exploratory Data Analysis helps us identify these KPIVs.

De Mast and Trip (2007) state that the purpose of EDA from a quality improvement project viewpoint is to identify the dependent (Y) and independent (X) variables that may help understand or solve the quality problem. The dependent Y variables are KPOVs, and the independent X variables are KPIVs. Leitnaker (2000) gives an example of EDA to identify KPIVs. The example is a molding operation where:

• Yields are erratic
• Parts are produced that do not meet specifications
• Shipment schedules are not consistently met

A team studied a molding operation supplying plastic switches to industrial customers for use in assembled control pads. The operation has eight machines, each machine has two molds, and each mold has four cavities. To investigate the process capability, the team took a sample of size 5 from the output of one machine every 4 hours. The following control chart displays the results for a critical dimension.

The process is in control, and the range chart supported this conclusion. But the variation is large. Next the team investigated the effect of the cavities and molds on the measured dimension. To do this, they sampled one part from each of the four cavities of the two molds on one machine. Breaking down the data by cavity and mold is an example of stratification. Control charts for the individual cavities and molds showed that all cavities and molds appear to be in control. However, mold 2 cavities have larger averages than mold 1 cavities, and the averages for the cavities increases with cavity number. The following figure clearly shows this pattern.

The figure leads us to identify mold and cavities numbers as KPIVs. The exploratory data analysis produced a clue which generated a search for the reasons that molds and cavities produced different average dimensions. The team can proceed to reduce the variability in the measured dimension by reducing the differences in averages for the molds and cavities.

References

1. Breyfogle, F. W. (2003). Implementing Six Sigma. Hoboken, New Jersey, John Wiley & Sons, Inc.
2. De Mast, Jeroen and Albert Trip (2007). “Exploratory Data Analysis in Quality-Improvement Projects”, Journal of Quality Technology, 39(4): 301-311.
3. Leitnaker, M. G. (2000). Using the Power of Statistical Thinking, Special Publication of the ASQ Statistics Division, Summer 2000.

This posting describes the difference between Exploratory Data Analysis (EDA) and Confirmatory Data Analysis (CDA). Tukey (1977) distinguished between EDA and CDA. Confirmatory Data Analysis tests hypotheses and produces estimates with a specified precision. Regression analysis, Analysis of Variance, and Hypothesis Tests are examples of Confirmatory Data Analysis. Confirmatory Data Analysis requires hypotheses or assumptions to consider and evaluate.

Exploratory Data Analysis makes few assumptions, and its purpose is to suggest hypotheses and assumptions. Consider the OEM manufacturer described in the posting on 1/30/2008. The company was experiencing customer complaints. A team wanted to identify and remove causes of these complaints. They asked customers for usage data so the team could calculate defect rates. This started an Exploratory Data Analysis. The team plotted a control chart, and these charts identified a high defect rate in October, 1991. The investigation established that a supplier used the wrong raw material. Discussions with the supplier and team members motivated further analysis of raw material, and its composition. This decision to analyze raw material completed the Exploratory Data Analysis. The Exploratory Data Analysis used both data analysis and process knowledge possessed by team members. The supplier and company conducted a series of designed experiments which identified an improved raw material composition. Using this composition, the defect rate improved from .023% to .004%. The experimental design and its analysis was Confirmatory Data Analysis. Note that the experimental design required a hypothesis generated by the Exploratory Data Analysis.

Tukey states that EDA is detective work. He uses the criminal justice process as an analogue to illustrate the roles of EDA and CDA. A detective investigating a crime needs both tools and understanding. The detectives and other investigative units search for and produce evidence. The juries and judges evaluate the evidence’s strength. Exploratory Data Analysis uncovers statements or hypotheses for Confirmatory Data Analysis to consider. Experimental design and regression modeling are more effective if Exploratory Data Analysis uncovers precise statements or hypotheses. Admittedly, one can conduct experiments searching for hypotheses; however, our viewpoint is that preliminary Exploratory Data Analyses may reduce the costs of these experiments.

Exploratory and Confirmatory Data Analyses can be thought of as part of statistical thinking. De Mast and Trip (2007) present principles for more effective EDA in quality improvement projects. We will examine results from their paper in future postings. Their paper won the Nelson award for the paper having the greatest immediate impact for practitioners published during 2007 in the Journal of Quality Technology.

References

1. John W. Tukey (1977). Exploratory Data Analysis, Addison-Wesley Publishing Co.
2. de Mast, Jeroen and Albert Trip (2007). “Exploratory Data Analysis in Quality-Improvement Projects”, Journal of Quality Technology, 39(4): 301-311.

The following figures display graphically the relative significance of the six factors, i.e., A, B, C, D, AB and AC. The figures show the average response at the factor low (-1) and high (+1) values. Factors B and C are not nearly as significant as factors A and D since the average responses of B and C are nearly the same at their low and high values. That is, a change in the factor levels for factors B and C has little effect on the response. Also, the interaction factor AC is more significant than the interaction factor AB.

We can test the significance of the factors using an Analysis of Variance (ANOVA). Refer to Montgomery, Peck and Vining (2006). Let SS_T be the total sum of squares. That is:

where Y_i is the response on experiment i and ybar is the average response over the 8 experiments. That is, SS_T is the sum of the 8 squared deviations between the experiment responses and the average response. The value of ybar is 49.582, and the value of SS_T is 118.151. Then we partition SS_T into a sum of squares due to the estimated effects (SS_R) and a sum of squared deviations from the estimated effects (SS_RES). That is, SS_T = SS_R + SS_RES. The value of SS_R is the same as a sum of squares due to an estimated regression function when we have a two-level experiment. Consider the contribution of factor A to SS_R. The posting on 9/18/2008 gives the estimated effect of factor A to be -6.067. That is the difference between the average of the responses at the low values of factor A and the high values of factor A. Thus the estimated average response at the high values of factor A is ybar - 6.067/2 = 46.5485. Similarly, the estimated average response at the low values of factor A is ybar + 6.067/2 = 52.6155. The deviation between the mean response and the effect of A conditioned on whether A is high or low is 6.067/2. Since we have 8 experiments, the contribution of factor A to SS_R is 8*(6.067/2)² = 73.60788. For factor D and the interaction effect AC, the corresponding contributions to SS_R are 18.67308 and 11.38575. Thus, SS_R is 103.6667. The value of SS_RES is SS_T – SS_R = 14.48432. We can test whether these three factors are statistically significant using the F statistic. The F statistic assumes that the individual responses have a normal distribution. The F statistic is:

Higher values of the response S/N are desirable. Thus, the low value of factor A (feed rate of .0008 mm/Revolution) and the low value of factor D (wheel grade of A54) are preferred. Since the low value (-1) of the interaction effect AC is preferred, we select the high value of factor C which is a work speed of 360 RPM. For the insignificant factor, the team chose its low value ( a wheel speed of 2200 RPM).

The posting on 2/28/2008 reports that the preferred factor levels specified above improved the process performance index (P_pk) from .49 to 1.25. This is based on a sample of 40 parts. The posting on 5/1/2008 defines the process capability index C_pk. Process capability indices assume the process is stable. When we have insufficient evidence the process is stable, we call the capability index a performance index and use the same equation.

References

1. Montgomery, Douglas C., Elizabeth Peck, Geoffrey Vining (2006). Introduction to Linear Regression Analysis, John Wiley & Sons, p26.

This posting continues the grinding process case study (Gigo, 2008) that illustrates the use of design and analysis of experiments to reduce common-cause variation. We examine the properties of the experimental design reported by Gijo. The examination illustrates the potential for aliasing in an experimental design and shows how it can bias the results. The experimental design described by Gijo uses an orthogonal array which Taguchi recommended. We contrast the properties of that design with a standard fractional factorial.

The Design of Experiments: Grinding Process Example posting initiated the design of experiments portion of the case study. The primary purpose of the experimental design was to reduce the variation in the outer diameter produced by a grinding operation. That posting reports that the team was primarily interested in estimating the following effects:

A – Feed Rate
B – Wheel Speed
C – Work Speed
D – Wheel Grade
AB – Interaction between A and B
AC - Interaction between A and C

Gijo states that the experimental design was developed using an L₈ orthogonal array. He references Phadke (1989) for use of orthogonal arrays to construct designs. Taguchi made extensive use of orthogonal arrays in constructing robust designs. Hicks and Turner (1999, p381) give a table for using an L₈ orthogonal array to construct a design with the desired properties. That is, we do not want the A, B, C, D, AB, and AC effects aliased with each other. Two effects that have the same estimator are aliased. The previous posting on September 15 gives the design and estimates of the factor effects. Clearly the design meets the desired criterion since the factor effect estimates are all different.

However, consider the estimates of the of the BC, BD and CD interaction effects shown in the following table.

Note that the BC interaction effect is exactly equal to the negative of the D effect, the BD interaction effect is equal to the negative of the C effect and the CD interaction effect equals the negative of the B effect. That is true because the sequences of +1 and -1s in the BC, BD and CD columns are precisely the negatives of those in the D, C and B columns. With this design, the BC and D effects are aliased. That is, if the BC effect is not zero, then our estimate of the D effect is affected by the BC effect. Similarly, the BD effect estimate is aliased with the C effect, and the CD effect is aliased with the B effect. Then this design provides no information on whether the BC, BD and CD interaction effects are negligible. Also, this design can give a biased estimate of the D effect if the BC interaction defect is significant.

Montgomery (2005, p. 288) gives a standard one-half fraction of the 2⁴ factorial design. Call it the 2^4-1 design. This design uses 8 experiments and has four factors. The properties of this design are:
· Estimates of the main effects are not aliased with any two-factor interactions.
· Estimates of the main effects are aliased with three factor interactions.
· Every two factor interaction is aliased with another two factor interaction. That is AB=CD, AC=BD and BC=AD.

The 2^4-1 design might be superior to the one described by Gijo. Estimates of the A, B, C and D effects are not aliased with any two factor interaction. Also, estimates of the AB and AC effects are not aliased with a main effect.

The next posting will present results from the experimental design.

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.
2. Hicks, Charles R. and Kenneth V. Turner Jr. (1999). Fundamental Concepts in the Design of Experiments, Oxford University Press.
3. Montgomery, Douglas C. (2005). Design and Analysis of Experiments, 6^th Edition, John Wiley & Sons, Inc.
4. Phadke, Madhav S. (1989). Quality Engineering Using Robust Design, Prentice Hall.

The response variable was a measure of the variability of the outer diameter of the machined components. One could use the estimated variance, i.e. s², for each set of experimental conditions. That is, one would replicate the experiment for each set of experimental conditions and estimate s². Gijo chose to use -10*ln(s²). He lets the symbol S/N represent the -10*ln(s²). Could S/N mean that the response is a Taguchi signal-to-noise ratio? Montgomery (2005, p. 469) discourages the use of signal-to-noise ratios. He states that a more effective approach is to model the mean and variance separately. Hunter (1987) comes to the same conclusion. Gijo does not justify the use of S/N other than a reference to the 3^rd edition of Montgomery’s book.

A response variable that has a constant variance over the set of experimental conditions facilitates regression analyses of the results. Montgomery (2005, p. 83) recommends the use of the logarithmic transformation when the standard deviation of the response is proportional to its mean. Let’s proceed by assuming the team used S/N since they wanted to estimate the contribution of the selected factors to the variance of the outer diameter and the standard deviation was roughly proportional to the mean.

The following table gives the experimental design and the observed response for each experiment. The team replicated the experiment twice for each set of experimental conditions. From the two observed outer diameters, they calculated a variance estimate, i.e., s², and from that computed the response value S/N. The -1 and +1 symbols represent the lower and higher levels of the respective factors.

Montgomery (2005, p208) shows how to calculate the average factor effects using the -1 and +1 coding. For a single factor effect, we sum the products of the factor coding times the experiment response over all experiments. Then we divide the sum by the number of -1, +1 pairs. In this experiment, the number of pairs is 4. The last row in the above table shows the estimated factor effects. For an interaction effect, we multiply the experiment coding for each factor to get a coding for the interaction effect.

Notice that the estimated AB and AC interaction effects are larger than the single factor B and C effects.

The next posting will examine the properties of the experimental design.

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.
2. Hunter, J. S. (1987). "Signal-to-Noise Ratio Debated." Quality Progress 20(5): 7-9.
3. Montgomery, Douglas C. (2005). Design and Analysis of Experiments, 6^th Edition, John Wiley & Sons, Inc.

This posting describes a grinding process case study to illustrate the use of design and analysis of experiments to study cause and effect and reduce common-cause variation. We continue the Cause and Effect Diagram case study reported by Gijo (2005). That posting describes the construction of a cause-and-effect diagram by a team in an engineering organization identify potential causes of low grinding machine capability. The team selected four factors for further analysis based on designed experiments. These factors were Feed Rate, Wheel Speed, Work Speed, and Wheel Grade. The team chose to perform experiments using two levels for each factor. The following table shows the levels and factors selected for experimentation. The levels with an * were existing operating levels.

Experimental design terminology defines the effect of a factor as the change in the response produced by a change in the level of the factor. Assume that the response in this experiment is the variance of the outer diameter measurements. For example, if increasing the feed rate from .0008 to .0010 mm/revolution increases the variance of the outer diameter by .003 mm² then the feed-rate (factor A) effect is .003 mm². When the difference in response to a factor level change is not the same at all levels of another factor, an interaction effect exists between the factors. The factor A effect might be .003 mm² when the wheel speed is 2200 rpm and .005 mm² when the wheel speed (factor B) is 2400 rpm, then an interaction effect exists between factors A and B. The magnitude of the interaction effect is the average difference between the two A effects. Thus the AxB interaction effect is (.005-.003)/2 = .001 mm².

The team selected an experimental design the enables them to estimate the effects of the four factors in the above table. They also wanted to estimate two interaction effects: 1. (AxB) between Feed Rate and Wheel Speed (AxB) and 2. (AxC) between Feed Rate and Work Speed. The linear graph shown below depicts the effects the experimental design must be capable of estimating. That is, the A, B, C and D effects, the AxB and AxC interaction effects and the error variance.

The next posting will describe the experimental design.

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.

Davies (2007) describes a case study involving the treatment of minor injuries and medical problems in an emergency department in England. Receptionists route arriving patients with minor injuries or medical conditions are routed to the “Minors” department. The standard processing procedure has receptionists in the Minors department assign patients to a queue for triage nurses who assess the patient condition and needs. Then the triage nurse routes the patients to a doctor or nurse for treatment. The nurses are qualified to assess and treat minor injuries but not to handle minor medical conditions which are handled by doctors. These nurses are Emergency Nurse Practitioners (EPNs). Call this procedure “See” and “Treat”. The UK national health service recommended that emergency departments skip the triage nurse step. The health service recommended that receptionists route patients to a doctor or ENP for diagnosis and treatment. Call this procedure “See & Treat”. The intent was to reduce patient system time by eliminating a step and its associated queuing time. The following figure depicts the “See & Treat” patient flow.

Davies describes a simulation model for comparing the two procedures. This model represents the processing of individual patients, their waiting times, and individual task processing times. Inputs to the model would include distributions for task times, distributions for times between patient arrivals, and the numbers of doctors and EPNs. The following figure presents some of the simulation results. The new procedure “See & Treat” that eliminates the triage step gives the lowest system time.

References

1. Davies, R. (2007). "See and Treat" or "See" and "Treat" in an Emergency Department. 2007 Winter Simulation Conference. Washington, DC.

The Multi-Vari Chart graphically shows variation of a quality characteristic for multiple factors. The purpose of the chart is to permit identification of the factor or factors having the greatest effect on variability.

Recall the example in the previous posting taken from Breyfogle (2003, page 389). An injection molding process produced plastic cylindrical connectors. The example included data from a sample of two parts collected hourly from four mold cavities for three hours consisting of measurements at three locations on the parts. The three locations are bottom, middle, and top. We want to display the variability by location, cavity and part. The following figure shows averages over the three hours by location, cavity and part. The figure shows that cavities 2,3 and 4 had larger diameters at the ends (top and bottom) while cavity 1 had a taper. Thus, cavity and location have an interacting effect.

References

1. Breyfogle, F. W. (2003). Implementing Six Sigma. Hoboken, New Jersey, John Wiley & Sons, Inc.

To illustrate the Box Plot, we refer to an example given by Breyfogle (2003, page 389). An injection molding process produced plastic cylindrical connectors. Breyfogle presents data from a sample of two parts collected hourly from four mold cavities for three hours consisting of measurements at three locations on the parts. The Box Plot for the aggregated data appears below.

1. Breyfogle, F. W. (2003). Implementing Six Sigma. Hoboken, New Jersey, John Wiley & Sons, Inc.

GOAL/QPC, an educational consulting company, noticed a new book proposing seven new QC tools. This book (Mizuno, 1988) was eventually translated into English. GOAL/QPC created the Memory Jogger Plus+ (Brassard, 1989) featuring these new tools. They called these new tools the ‘Seven Management and Planning Tools’ to differentiate them from the ‘Seven Major SPC Tools’. The Seven Management and Planning Tools are:

1. Affinity diagram
2. Interrelationship digraph
3. Tree diagram
4. Prioritization matrices
5. Matrix diagram
6. Process decision program chart (PDPC)
7. Activity network diagram

Montgomery (2005, page 148) identifies ‘Seven Major SPC Tools’. He calls them the ‘Magnificent Seven’. They are:

1. Histogram (Resin Example and Address Special Causes and Evaluate Capability postings) or stem-and-leaf plot
2. Check sheet
3. Pareto chart (Pareto Chart and Analyze Common-Cause Variation postings)
4. Cause and effect diagram (Cause and Effect Diagram posting)
5. Defect concentration diagram
6. Scatter diagram (Resin Example Part C posting)
7. Control chart (Customer-Complaint-Process Example, Example Illustrating Four Variation Types, Distribution Center On-Time Delivery Example, and Resin Example Part D postings)

Earlier, Ishikawa (1985) identified ‘Seven Major TQM’ (Total Quality Management) tools. They are:

1. Histogram
2. Flowchart
3. Pareto chart
4. Cause and effect diagram
5. Run charts and graphs
6. Scatter diagram
7. X-bar and R control charts

One could say that Montgomery replaced the ‘flowchart’ and ‘run charts and graphs’ with the ‘check sheet’ and ‘defect concentration diagram’. Montgomery also generalized the X-bar and R control charts with all control charts.

References

1. Brassard, M. (1989). The Memory Jogger Plus+^â. Salem, NH, Goal/QPC.
2. Mizuno, S. (1988). Management for Quality Improvement: The Seven New QC Tools. Cambridge, Productivity Press.

This posting gives an example of an Interrelationship Digraph which is a tool for use in the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy. The quality issue is the potential causes or factors contributing to late deliveries. We take our example from Benbow and Kubiak (2005). The interrelationship digraph appears below.

1. ‘Poor scheduling practices’ (6 outgoing arrows),
   
2. ‘Late order from customer’ (5 outgoing arrows), and
   
3. ‘Equipment breakdown (3 outgoing arrows).

A concern with a large number of input arrows is affected by a large number of other concerns. Thus, it could be a source of a quality or performance metric. ‘Poor scheduling of the trucker’ has 4 input arrows. A measure of poor scheduling performance of the trucker could indicate the magnitude of system problems causing late delivery.

References:

1. Benbow, D. W. and T. M. Kubiak (2005). The Certified Six Sigma Black Belt Handbook. Milwaukee, Wisconsin, ASQ Quality Press.

Form a team of knowledgeable individuals with respect to this quality issue. The team will select a number, e.g., from six to twelve, of the potential causes from the Cause & Effect diagram. Call these potential causes concerns. The process for generating the Interrelationship Digraph will construct causal relationships among the concerns. The word digraph is a combination of the two words diagram and graph. The resulting digraph reflects the collective judgment of the team.

Benbow and Kubiak (2005, page 40) specify a procedure for constructing the digraph. List the concerns on a sheet of easel paper or a whiteboard. Pick a pair of concerns. Ask the team to specify whether the first concern influences the second, the second concern influences the first, or whether there is no influential relationship between the concerns. If the team decides there is an influential relationship, draw an arrow from the most influential concern to the other concern. Does the first concern influence the second more than the second concern influences the first? If so, draw an arrow from the first concern to the second. Repeat this assessment for all possible pairs. A good way to proceed is to arrange the concerns in an approximate circular pattern. Start with the concern in the 12 o’clock position and call it the first concern. Compare it with the concern in the next clockwise position. Then, move clockwise and select another concern to compare with the first concern. Repeat this process until all possible combinations of concerns have been compared by the team.

The next posting will illustrate the construction of an interrelationship digraph.

• Experimental Design. A systematic planned variation of input factors for an actual process. The experimenter observes the effect of these variations on important quality characteristics. The Customer-Complaint-Process Example posting mentions the use of designed experiments by an OEM manufacturer to determine an improved raw material composition. The Cause and Effect Diagram posting discusses the effort by a company to reduce the rejection rate at one of its machine shops. Based on a Cause & Effect diagram, project members selected four factors for further analysis based on designed experiments. These factors were Feed Rate, Wheel Speed, Work Speed, and Wheel Grade. Analysis of the experimental results identified “optimum” levels for the four factors.
• Model Building. One could construct a model of a process that predicts quality performance based on input variables. The Service Time Flowchart posting describes the actions of a Midwest manufacturing firm to reduce time delays experienced by customers contacting their order processing center. They constructed a simulation model of the order-taking process. Using the simulation model they determined the staffing level of customer service representatives by the hour of a work day to meet time-delay objectives. Why don’t software companies use simulation models to specify technical support personnel requirements?

Subsequent postings will illustrate the use of experimental design and model building to Study Cause and Effect.

This posting discusses the seventh step, Study Cause and Effect, of the Hoerl-Snee Process Improvement Strategy. Refer to the figure in the Hoerl-Snee Process Improvement Strategy posting for an overview of the process. Use Britz et al (2000) and Hoerl and Snee (2002) as references.

The previous step analyzed common-cause variation to identify the source (s) of variation. If the previous step did not identify the source or if knowing the source does not reveal the root cause, we proceed to study cause and effect.

Some of the tools we might use in this step are:

• Scatter plot. A plot of a quality characteristic versus a potential explanatory variable. See the plot in Part C of the Resin Example posting showing the effect of solvent feed ratio on output weight.
• Cause & Effect Diagram. A diagram portraying the potential causes of an effect. See the diagram in the Cause and Effect Diagram posting showing the potential causes of rejections at the grinding operations. Frequently, the Cause & Effect diagram summarizes the results of a brainstorming session. However, some improvement efforts will use data to substantiate the cause and effect diagram.
• Box Plot. Box Plots depict the relationship between a discrete variable, such as location on a part, and the distribution of continuous variable, such as a dimension.
• Multi-Vari Charts. Multi-Vari charts display variations in categories that aid in identifying causes.
• Interrelationship Digraphs. Teams construct cause and effect relationships from a list of issues.

The next posting will summarize additional tools for this step. Subsequent postings will give examples of Box Plots, Multi-Vari Charts and Interrelationship Digraphs.

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

Histogram – Stratification. The Rosen Yield Example posting describes statistical thinking by a team at Ricoh’s Numazu plant. The plant makes raw material used as ingredients for copy machine toner. The team wanted to reduce variation in output quantity which indicated a lack of control of the underlying process. After removing a special cause, the team constructed a histogram of the output quantity. The histogram clearly displayed excessive variation and two peaks. The process flow chart showed a split after phase 2 into 2 separate lines, i.e., line A and line B. Separate histograms for the two lines showed the output from line B was consistently lower that line A. Constructing separate histograms for the two lines illustrates stratification by line. Next, the team conducted a brainstorming session to formulate their collective thinking about the causes of excessive variation and the differences between the two lines. They documented the results with a cause and effect diagram. The brainstorming session and the construction of a cause and effect diagram illustrate step 7, Study Cause & Effect.

Stratification requires identifying a stratification factor, such as time of the day, and the partitioning of this factor into logical categories. What tools may we use to aid in the selection of a stratification factor? The team in the example above noticed two peaks in a histogram. Breyfogle (2003) provides some guidance for this question.

1. On page 220, Breyfogle states that patterns on a control chart may suggest the need for stratification. A sequence of points with small up and down variation relative to the control limits may suggest that the sequence of points comes from a single strata. The opposite situation where a sequence of points that do not have values near the center line may indicate the combination of two strata.
   
2. On page 385, Breyfogle suggests dividing the data into categories based on posing basic questions such as who, what, when and where.

Disaggregation may be aided by constructing a process map such as the one used in the posting on 2/21/08. The process map (Breyfogle, 2003, p. 103) is a flowchart with key process input variables listed for each step in the process.

1. Breyfogle, F. W. (2003). Implementing Six Sigma. Hoboken, New Jersey, John Wiley & Sons, Inc.

This posting gives two additional examples illustrating the Analyze Common-Cause Variation step, step 6, in the Hoerl-Snee process improvement strategy. Refer to the posting describing this step. Both examples include disaggregation as a tool.

· Disaggregation – Stratification. The posting on Service Time Flowchart describes statistical thinking by a Midwest manufacturing firm to reduce waiting times by customers. The company’s goal was to have 95% of incoming customer calls answered by a customer service representative in less than 2 minutes. Based on a process flowchart, team collected service time data for each step in the process. That is disaggregation. The team also collected data for estimating the distribution of incoming calls by time of the day. That is stratification by the time of day. They used these data as inputs to a simulation of the call answering process. They used the simulation construct staffing levels by the hour of the day. The construction and use of the simulation illustrates step 7, Study Cause & Effect.
· Disaggregation – Regression Analysis. The posting on Flow Chart and Process Map describes statistical thinking by a manufacturer of automotive door frames. The purpose was to eliminate a problem meeting dimensional specifications of the finished product. Shop floor personnel thought that variations in the incoming raw material characteristics caused the problem meeting dimensional specifications. The team defined important quality characteristics for each step in the process. They included quality characteristics of the incoming material. The manufacturer collected data listing the important quality characteristics as well as the final part dimensions. A regression analysis showed no effect by the incoming material characteristics. Moreover, it identified several quality characteristics having a significant effect on finished product dimensions. The regression analysis also showed that the left and right door frames had significantly different variation for two quality characteristics. These results motivated corrective action and eliminated the need for rework. In this example, the team did not need to employ step 7, Study Cause & Effect.

This posting gives two examples illustrating the Analyze Common-Cause Variation step, step 6, in the Hoerl-Snee process improvement strategy. Refer to the previous posting for a description of this step.

· Stratification – Pareto Chart. The Pareto Chart posting describes statistical thinking by a company experiencing a high rejection rate in one of its machine shops. In order to determine the root cause of these rejections they stratified by classifying the rejections with respect to machine type causing the rejections. Then they created a Pareto Chart ranking the frequency of rejections by machine type. They found that 60% of the rejections were due to grinding problems. This finding did not give them the root cause of the rejections, but it allowed them to focus on grinding operations. Their next step was to construct a cause and effect diagram and then to design experiments to determine improved grinding procedures. This next step illustrates the implementation of the Study Cause & Effect step, step 7 in the Hoerl-Snee process improvement strategy.
· Regression Analysis – Stratification. The posting on the Application of Is-Is Not Analysis describes statistical thinking by Pease Industries to reduce the defect rate of decorative glass inserts for a wooden entry door. The prevailing opinion was that humidity and temperature variations in the mold department were the root cause. The team collected data and did a regression analysis using temperature and humidity as independent variables and the number of defects as the dependent variable. The result was no correlation between the independent variables and the number of defects. They collected more data and stratified the data by part type, month of occurrence and day of week. They were surprised by the result showing day of the week strongly affecting the defect rate. A Chi-Square test showed the day of the week was statistically significant. The next step was to construct a Cause-and-Effect diagram and do a Is-Is Not analysis. This step illustrates the Study Cause and Effect step, step 7.

In both of the above examples, the use of Cause-and-Effect diagrams, designed experiments and the Is-Is Not analysis required the previous results from the Analyze Cause and Effect steps. One needs to identify the effects prior to studying the effects.

This posting discusses the sixth step, Analyze Common Cause Variation, of the Hoerl-Snee Process Improvement Strategy. Refer to the figure in the Hoerl-Snee Process Improvement Strategy posting for an overview of the process. Use Britz et al (2000) and Hoerl and Snee (2002) as references.

Common-cause variation affects all of the data which distinguishes this step from the Address-Special-Causes step. The purpose of the Analyze-Common-Cause-Variation step is to identify sources of variation. Locating the sources of variation might also reveal its root cause without significant additional analysis. On other occasions, knowing a source of common-cause variation might require further analysis to determine its root cause. This additional analysis is performed in the next step, Study Cause and Effect.

Some of the tools we might use in this step are:

• Stratification. Define a stratification factor such as the day of the week or machine. Partition the factor into logical categories. Compare the data for each category to highlight differences.
• Disaggregation. Define quality measures for sub-processes or individual process steps. Study the variation in the individual sub-processes. How does it contribute to the overall process variation?
• Pareto Chart. Classify defects into categories. Highlight the categories having the most frequent occurrences.
• Histogram. Plot the distribution of quality measures. One or more peaks might indicate the presence of categories that could be examined by stratification.
• Regression Analysis. Existing opinion might suggest one or more input variables that influence the output quality measure. A regression analysis might verify this opinion or indicate that these variables have negligible effect.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

The following figures illustrate two problems with the C_pk index.
1. In the first figure, two processes with identical C_pk values (1.5) have significantly different means and standard deviations. Possibly changing the mean is easier to accomplish than the standard deviation.
2. In the second figure, two processes with identical C_pk values (1.0) have different distributions. One is normal and the other lognormal. For the normal distribution, the probability of being below the lower spec limit is .00135, and the probability of exceeding the upper spec limit has the same value. For the normal distribution, the total probability of not meeting the spec limit is .0027. For the lognormal distribution, the probability of the quality measure being below the lower spec limit is approximately zero, while the probability of being greater than the upper spec limit is .007915. For the lognormal, the probability of not meeting the spec limits is almost three times the corresponding value for the normal distribution.

For the above reasons and others, Breyfogle (2003) recommends the use of estimated parts per million (ppm) beyond specification limits rather than process capability estimates.

Due to sampling variability, Hare (2007) recommends estimating process capability indices using at least 100 values.

Reference

1. Hare, Lynne B. (2007). “The Ubiquitous C_pk”, Quality Progress, pp. 72-73.
2. Breyfogle III, Forrest W. (2003). Implementing Six Sigma – Smarter Solutions^â Using Statistical Methods, John Wiley & Sons, Inc., pp 296-299.
   

This posting discusses the fourth and fifth steps in the Hoerl-Snee Process Improvement Strategy. Refer to the figure in the posting introducing the Hoerl-Snee Process Improvement Strategy for an overview of the process. Use Britz et al (2000) and Hoerl and Snee (2002) as references.

The approach for addressing special causes is different than the Process Improvement Strategy. Addressing special causes uses the Problem Solving Strategy which will be described in future postings.

The Evaluate Capability step compares process specifications (targets) and observed variation. The motivation is to determine whether the process can consistently meet established specifications and/or goals.

The histogram is an informative graphical method for assessing process capability. The posting for Part B of the Resin Example showed three histograms displaying resin output variation and two of them gave upper and lower limits for the output quantities. These histograms clearly showed excessive variation. That is, output quantities were frequently less than the lower limit and greater than the upper limit. One advantage of the histogram is that one does not have to assume a theoretical distribution to estimate the rate of non-conformances. Also, the histogram shape may suggest a theoretical distribution. For example a bell shaped histogram suggests a normal distribution. If the histogram displays unexpected patterns, it may suggest corrective action. For example, the resin output variation histogram showed two peaks suggesting difference between the two production lines. Also, a histogram that is shifted towards a specification limit (upper or lower) suggests that centering the process mean may reduce non-conformances.

Another popular measure of process capability is a process capability index such as C_p or C_pk. Let USL be the upper specification limit and LSL be the lower specification limit. Then C_p = (USL-LSL)/(6*sigma) where sigma is the process standard deviation. If the process quality characteristic has a normal distribution, then a C_p of 1.0 means that .27% of the items produced are non-conforming. For a C_p of 1.33 the non-conforming percentage is .00636. For one-sided specifications and calculation of C_pk, we define:

C_pu = (USL-mu)/(3*sigma) for the upper limit,

C_pl = (mu-LSL)/(3*sigma) for the lower limit,

C_pk = Min(C_pu, C_pl) where mu is the process mean.

If we think of three standard deviations as the process spread around its mean, then C_pk is the ratio between the allowable spread and the actual spread. For short term performance, a C_pk of 2.0 is the target standard for a Six Sigma project. In the past, C_pk of 1.33 had been required of suppliers in the automotive industry.

Important observations are:

• In order for C_p and C_pk to have any validity, the process must be stable.
• Both the Assess Stability and Evaluate Capability steps are important in estimating the amount of improvement needed for a project.
• Probability plots are another tool one can use in evaluating process capability.

The next posting will discuss problems in using process capability indices.

References
1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

This posting discusses the third step in the Hoerl-Snee Process Improvement Strategy. Refer to the figure in the posting introducing the strategy for an overview of the process. Use Britz et al (2000) and Hoerl and Snee (2002) as references.

After collecting data on key measures, the next step is to analyze process stability based on that data. First we define a stable process or one that is in-control. Shewhart (1931, p. 6) states: “..a phenomenon will be said to be controlled when, through the use of past experience, we can predict, at least within limits, how the phenomenon may be expected to vary in the future.” More recently, Montgomery (2005, p. 148) states: “In any production process, …., a certain amount of inherent or natural variability will always exist. …. this natural variability is often called a ‘a stable system of chance causes.’ A process that is operating with only chance causes of variation present is said to be in statistical control. … We refer to those sources of variability that are not part of the chance cause pattern as ‘assignable causes.’ A process that is operating in the presence of assignable causes is said to be out of control.” Montgomery references Shewhart for the terminology chance and assignable causes. He states that many now use the terminology common cause rather than chance cause and special cause rather than assignable cause.

An important characteristic of a stable or in-control process is that it is predictable. This comes from Shewhart’s definition. That is, one can predict future behavior from past behavior. Breyfogle (2003, p. 1109) and Wheeler (1993, p. 124 and 128) state that an in-control process is predicable whereas a process that is not in-control is unpredictable. This means that statistical methods such as t tests and ANOVA are inappropriate for unstable processes.

The definitions stated above immediately suggest methods for identifying whether a process is in-control. They include run charts and SPC control charts. A run chart is a time plot of quality characteristic and a control chart is a run chart with control limits. Using the points on these charts that signal lack of control, we can conduct investigations to determine what caused these points to be different. Two previous postings that do that are:

• Rosin yield run chart in the Hoerl-Snee Example. The team found that their cause was a drop in air pressure.
• Customer Complaint Process Example. The investigation identified the high defect rate in October, 91 was due to a supplier using the wrong material.

Two major reasons for assessing stability and removing assignable causes prior to addressing common-cause variation are:

• Detecting and removing assignable causes is easier than reducing the variation due to common causes. Remove the low hanging fruit first.
• The analysis approach to removing assignable causes is different than reducing common-cause variation. Common causes affect variation in all data points for a stable process. When searching for the root cause of variation in specific observations due to an assignable cause, one can focus on the differences between these observations and the others.

Consider the possibility of wasting effort when a process is in-control (stable) but some results do not meet targets. Managers could pressure staff to find the cause of specific results not meeting targets. That is, managers could direct staff to find causes for specific undesirable outcomes when the variation is present in all outcomes.

References
1. Breyfogle, Forrest W. (2003). Implementing Six Sigma: Smarter Solutions Using Statistical Methods, Second Edition, John Wiley & Sons, Inc.
2. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
3. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.
4. Montgomery, Douglas C. (2005). Introduction to Statistical Quality Control, John Wiley & Sons, Inc.
5. Shewhart, Walter A. (1931). Economic Control of Quality of Manufactured Product, D. Van Nostrand Company, Inc.
6. Wheeler, Donald J. (1993). Understanding Variation: The Key to Managing Chaos, SPC Press, Inc.

This posting discusses the second step in the Hoerl-Snee Process Improvement Strategy. Refer to the figure in the previous posting for an overview of the process. Use Britz et al (2000) and Hoerl and Snee (2002) as references.

After understanding and documenting the process, the next step is to collect data on key process and output measures. These key measures can include the overall process performance measure(s) and measures derived from the inputs and outputs of each process step.

For example, the Ricoh team in the Resin Output Variation example was concerned with the product yields being greater than theoretical expectations so they collected yield-ratio data. In the Pease Industry example the company team wanted to improve quality of their residential entry doors so they collected defect-rate data from their customers. In the automotive door frame example, the manufacturer wanted to improve the quality of critical dimensions on the welded door frame. They collected data from incoming material and after each processing step, i.e., roll mill, bender and saw. These data consisted mainly of dimensional measurements.

The process may be a sequence of steps required to perform a task with a cycle-time principal performance measure. For example, the process might be the activities required to fill a prescription in a hospital. For each order submitted, the data might include the submittal time, the arrival time at each processing step, the actual step processing time, the completion time for each processing step, and the drug prescribed. In addition, one would need the number of servers at each processing step.

Breyfogle (2003) on page 10 introduces several terms that are useful in identifying important process variables. A Key Process Output Variable (KPOV) in an important output for a process. Another name for this variable is a Critical to Quality Characteristic (CTQ). Key Process Input Variables (KPIVs) are process inputs that affect the KPOVs.

One might ask: how do we select the data to collect? We use a combination of the output of the previous step, Understand the Process, and existing process knowledge. Also, a previous iteration of the Process Improvement Strategy (see the posting on April 4) may have identified some KPIVs. The author has found in his consulting experience that manufacturers of process equipment may have important information regarding the sensitivity of their equipment to process variables. Also, do not forget the internet. A search may reveal research reports indicating the sensitivity of equipment to process variables.

References
1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.
3. Breyfogle, Forrest W. (2003). Implementing Six Sigma: Smarter Solutions Using Statistical Methods, Second Edition, John Wiley & Sons, Inc.

This posting discusses the first step in the Hoerl-Snee Process Improvement Strategy. Refer to the figure in the previous posting for an overview of the process. Use Britz et al (2000) and Hoerl and Snee (2002) as references.

The first step is to develop a common understanding of the process by recording and documenting it. In the Monthly-Billing-Cycle Time example key participants did not have the same understanding of the principal process steps. In the Ricoh Resin example, Hoerl-Snee Example, the team created a flowchart of the process which is the usual method for documenting the process. We start by documenting the process as it is currently performed. The flowchart serves as a reference and it facilitates communication. Sometimes the flowchart is called a process map.

The flowchart is particularly important when on can not visually observe the process flow. Many administrative or service processes have this property. The Service-Time example, Service Time Flowchart, may have had this property.

The flowchart may immediately suggest areas for improvement. Some steps in the flowchart may represent non-value added activity. For example, customers waiting in the Automatic Call Distribution (ACD) System Queue depicted in the service time example represent non-value added activity.

A flowchart or process map may have a number of variations. The Ricoh resin example flowchart is a high-level flowchart. Hoerl and Snee (2002) describe other more detailed flowcharts on pages 195-199. Our flowchart of the automotive door frame example, Flowchart and Process Map, displayed key quality characteristics for the process steps.

References
1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

This posting describes the Hoerl-Snee Process-Improvement Strategy. This strategy was originally described in Hoerl-Snee (1995), and it also appears in Britz et (2000) and Hoerl-Snee (2002). Prior to implementing the Process Improvement Strategy, one should define the scope and objectives for the improvement effort. The following figure displays a flowchart of the improvement strategy steps and lists some example tools to perform the corresponding steps.

The figure does not show the entire process improvement flow. Eliminating special causes involves the Problem Solving Strategy. Future postings will describe this strategy.

Note that the resin output variation case study clearly illustrated the above features of the process improvement strategy. Improvement occurred in sequential cycles involving planning, implementing and collecting data. Also, the first improvement action by the resin team was to determine whether special causes were present and then to correct them. After that they moved on to reduce the variation contributed by common causes.

The overall process had two weighing processes. The first was an in-process manual method, and the second method was a final, automatic scale. The manual method had individuals reading a line on a scale. They observed that individuals of different heights read the line from different viewpoints. Thus, they produced different readings. The team changed the presentation of the line so people of different heights had the same view point. This change reduced in-process measurement variation.

Next the team investigated the automatic scale and found significant measurement errors. They reduced these errors by:

1. Redesigning the scales protective cover.

2. Establishing procedures for checking the alignment on a periodic basis.

The following figure presents a control chart showing the results for this project. The difference between the final upper and lower control limits was less than the team objective of ± 5 kg. However, the resulting average was 4292 kg which is less than the original target of 4300 kg. Given the reduction in output variability, management regarded the results as more than adequate. The improvement also resulted in reduction in the variation of resin viscosity. This verified the team’s motivation to reduce variation of finished product quality by reducing the output quantity variation. To maintain the results, the team created procedure manuals and established a schedule adjusting the automatic weighing process.

The overall improvement process consisted of four Plan-Do-Check-Act (PDCA) cycles. This posting describes the last one, the previous post describes two of them and the first Hoerl-Snee example posting describes the first one. That one focused on finding and correcting special causes. This process is different than that suggested by a serial DMAIC process. Our next posting will present the Hoerl-Snee process improvement strategy which has an overall PDCA approach.

References
1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

1. The procedure for dividing the resin after phase 2 (potentially the cause of differences between line A and B).
2. The solvent feed ratio.
3. The weighing process, i.e., final (automatic) and in-process (manual).

After attacking the first potential cause, the team found that some resin remained in the reaction tank after sending the materials to the two lines. That meant that line B had less input and therefore less output. After changing the dividing procedure, the team found no significant difference between the outputs of the two lines.

The output quantities still had too much variation. The team turned to the second potential cause, i.e., the solvent feed ratio. The following figure shows a scatter plot indicating that increasing solvent feed ratio is correlated with increasing output. In calculating the regression line the team regarded the high output occurring at a feed ratio slightly less than 1, as an outlier. This correlation violated the team’s knowledge of the underlying process. They investigated the measurement of the feed ratio, and they found that the ratio measurement was affected by the length of time the solvent was in the tank. They changed the procedure to insure that the solvent had stabilized prior to measurement. They collected more data to measure the impact of this change and found less variation in the measured feed ratio and no correlation between the measured feed ratio and the output quantity.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

Having removed the special cause, the Ricoh team focused on output quantity variability. A histogram displays this variability, and the following figure shows recent output data. This histogram displays an unexpected pattern indicating a combination of two underlying distributions for the output quantity. Notice the peaks at 4284 and 4308 kg.

The process flowchart appearing in the previous posting suggested that these two component distributions were due to the split after phase 2 into two separate lines, i.e., lines A and B. The following histograms shown below confirmed this difference. The output from line B was consistently lower than line A. Based on the needs of their customers, the team established the limits shown in the histograms, i.e., 4300 kg ± 5 kg.

Clearly, the variation in output quantity is excessive. Next the team conducted a brainstorming session to document their collective thinking on potential causes of excessive variation and differences between the two lines. The following cause and effect diagram shows the result of this session.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.

Resin Output Variation Example (Part A)

Ricoh’s Numazu plant made raw material used as ingredients for copy machine toner. The tolerances for the raw material ingredients were measured in ten-thousands of a gram. They had a team which continually monitored the process to achieve continuous improvement. The team noticed a problem with actual output results compared to theoretical output quantities. The yield ratio frequently exceeded 1.0. That is, the actual output quantity for a batch divided by the theoretical output quantity sometimes exceeded 1.0. The following figure shows a run chart displaying these results. These values were technically impossible, so the team attributed these results to undesirable variation somewhere in the process. Their experience indicated that this variation would degrade finished product quality. They wanted to know the source of the variation and how to eliminate it.

Notice that their first steps were to document (understand) the problem and look for special causes.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Imai, M. (1986). Kaizen, The Key to Japan’s Competitive Success. New York: Random House Business Edition.

A set of fundamental principles define Statistical Thinking, and these principles appear above in the introductory statements to this blog. A number of different approaches exist for applying Statistical Thinking to improve quality. Call these approaches Process Improvement Strategies. The Statistics Division has promoted one originally defined by Hoerl and Snee (1995). This is the same process improvement strategy described in detail by Britz, Emerling, et al (2000) and Hoerl and Snee (2002). Call this process improvement strategy the Hoerl-Snee strategy.

Six Sigma has another process improvement strategy. Six Sigma uses the DMAIC steps which are Define, Measure, Analyze, Improve and Control. The DMAIC steps differ from the Hoerl-Snee process improvement strategy. Our blog posting on Background and Motivation points out that Statistical Thinking is a crucial concept in Six Sigma. Clearly Six Sigma regards work as a system of interconnected processes, looks for variation in all processes, and regards understanding and reducing variation as keys to success.

Each element in the Hoerl-Snee strategy maps to an element in the DMAIC strategy. However, the author thinks that the Hoerl-Snee strategy is more explicit and easier to understand.

The Shainin System^TM or Statistical Engineering has another approach to quality improvement. See Shainin (1995) for an overview or Steiner and MacKay (2005) for improvements to Statistical Engineering. Statistical Engineering does use Statistical Thinking. Its process improvement strategy places more emphasis on finding and eliminating a dominant cause (The Red X) than the Hoerl-Snee and Six Sigma strategies. Statistical Engineering does not differentiate between special and common causes. Also, it places less emphasis on advance planning prior to data gathering. In addition, Statistical Engineering does not explicitly separate special causes from common causes so that it more effectively identifies the causes and eliminates them.

Approach in Subsequent Postings

First, we will specify the Hoerl-Snee strategy. This strategy will be illustrated by example applications which we will present next. After that we will discuss the differences between the three strategies mentioned above. Case studies will illustrate the differences.

References

1. Hoerl, R. W. and R. D. Snee (1995). Redesigning the Introductory Statistics Course. Madison, Wisconsin, University of Wisconsin, Center for Quality and Productivity Improvement.
2. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
3. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxbury.
4. Shainin, R. D. (1995). A Common Sense Approach to Quality Management. 49th Annual Quality Congress Proceedings.
5. Steiner, S. H. and R. J. MacKay (2005). Statistical Engineering: An Algorithm for Reducing Variation in Manufacturing Processes. Milwaukee, Wisconsin, ASQ Quality Press.

We illustrate that display of the analysis results using knowledge gained by the author’s consulting experience.

Peach Pit Example

Packers located in the vicinity of peach farms purchase raw peaches and process them for use as ingredients in food products. Their processing includes removal of the skin and pits. They use high-speed machines to remove the pits. We will call these machines pitters. However, the best pitters leave some pits and pit fragments. Some packers use inspectors downstream of the pitters to remove pits and pit fragments left by the pitters. The picture on the left shows inspectors removing pits and pit fragments.

The following table shows the results of the Is-Is not analysis.

1. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxburry

Britz, Emerling, et al (2000, p 118) and Hoerl and Snee (2002, p 203) describe the Is–Is Not Analysis which helps narrow the search for a root cause. It does that by documenting the problem. The analysis documents where, what, when and who are associated with the problem. In addition, the analysis documents where, what, when and who are not associated with the problem symptoms. This documentation suggests further action to discover the root cause.

Pease Industries Example

Smith and Adams (2001) give an example of Is-Is Not analysis being important in identifying a root cause when other approaches failed. Pease Industries is a large Midwestern manufacturer of residential entry doors. This was done as part of a Lean Six Sigma project. A line that was placing rosin and glass inserts for more expensive residential entry doors had a 16% defect rate. They formed a team consisting of operators, managers, and consultants. The work flow used a batch processing system. The team reduced non value-added activity, eliminated batch processing, and re-assigned operators who were no longer needed. The line now had one-piece flow. The defect rate decreased to 11% from 16%. In addition, line productivity increased by 62%.

This improvement still left an 11% defect rate in the decorative glass inserts for a wooden entry door. The defect was a consistent hairline imperfection where liquid resin should have met the edge. They called this defect a “shrink line.” Engineers and managers felt that humidity and temperature variations in the mold department were the root cause. The team collected data and did a regression analysis. The dependent variable was the number of defects and the independent variables were temperature, humidity, and an interaction term involving both temperature and humidity. (I hope they used Poisson regression rather than ordinary least squares.) The result was no correlation between the independent variables and the number of defects.

Team members including engineers, quality managers, an operator, and a consultant went to the shop floor to personally collect data. Using recorded data, they examined defect occurrence by the following factors: part type, monthly occurrence, and day of the week. Stratification is the analysis of data by these factors. To their surprise the following figure shows that defect occurrence was highest on Monday and declined through the week. A Chi-Square test showed the day of week was statistically significant.

Next, the team constructed a Cause-and-Effect diagram giving all possible causes of the defect. Then the team performed an Is-Is Not analysis. The used the data they collected to do the analysis. Their statistical analysis of the data supported not only what circumstances were associated with the defect but the circumstances that were not associated with the defect. For example, resin, swirls, bad resin mixes and laminations were not problems. They examined what the problem was versus what it was not, when it happened and when it did not, and where it happened versus where it did not. They went back to the Cause-and-Effect diagram and eliminated possible causes.

Their conclusion was that they had dirty molds. Molds were cleaned on certain days. After implementing controls, they estimated the annual savings to be $1,050,000.

The example illustrates the potential benefits of using an Is-Is Not analysis. However, a strict application of Statistical Thinking would have employed the Is-Is Not analysis prior to doing regression analyses.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxburry.
3. Smith, B. and E. Adams (2001). LeanSigma: Advanced Quality. Annual Quality Congress.

The Cause and Effect Diagram graphically portrays the potential causes of an effect. The causes are grouped into categories. Common categories are manpower (personnel), materials, methods and machines. When the diagram uses these specific categories we might call the diagram a 4M diagram. Depending on the effect, the diagram might display other categories. The diagram is also known as an Ishikawa diagram since Dr. Ishikawa devised its first use of the diagram. Another name for the diagram is a Fishbone diagram because of its appearance. Recording the results of a brainstorming session is a typical use for the diagram. A project might use a brainstorming session to generate a list of potential causes of an effect or a quality problem.

We will continue the case study reported by Gijo (2005) to illustrate the Cause and Effect diagram. The previous post presented a Pareto chart for a machine shop showing that the grinding operations generated most of the rejections experienced by the shop. They estimated grinding machine capability based on a sample of 40 parts. The estimated P_pk for this sample was .49. This result verified the lack of grinding machine capability.

The posting on evaluating capabiliity defines the process capability indices C_p and C_pk. Process capability indices assume the process is stable. When we have lack evidence that the process is stable, we call the capability index a performance index and use the same equation. The index P_pk is a process performance index.

Selected individuals participated in a brainstorming session to generate a set of potential causes of grinding machine rejections. The following figure shows the resulting causes.

After further study, project members selected four factors for further analysis based on designed experiments. These factors were Feed Rate, Wheel Speed, Work Speed, and Wheel Grade. Analysis of the experimental results identified “optimum” levels for the four factors. The estimated P_pk at the optimum factor levels was 1.25 based on a sample of 40 parts. This showed significant improvement.

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.

We use Pareto Charts to rank problems or causes with respect to their frequency of occurrence. The charts highlight those causes which result in the most quality problems.

Pareto charts get their name from Vilfredo Pareto (1848 – 1923) who was an economist. He analyzed and studied the unequal distribution of wealth. Dr. Juran in the 1940s stated a principle of the “vital few” and the “trivial many” (see Juran and Godfrey (1999)). That is, in many situations a few problem categories (about 20%) will produce the most problems (about 80%). Juran called this principle “Pareto’s principle of unequal distribution.”

We illustrate the application of Pareto Charts using a case study taken from Gijo (2005). A company was experiencing a high rejection rate in one of its machining shops. They did not know the root causes of these rejections nor how to reduce their occurrence. They started by examining existing records, and they classified the defects by the individual operations causing the defect. The analysis of data by this classification is called stratification. Using the results, they constructed a Pareto chart. The following figure presents the chart.

The chart shows that 60% of the rejections were due to grinding problems. Based on the Pareto Chart they started a study improve grinding operations. This study resulted in designed experiments to determine improved grinding operating procedures. The resulting analyses lead to operating procedures that significantly reducing rejections and rework due to grinding operations.

References

1. Juran, J. M. and A. B. Godfrey (1999). Juran’s Quality Handbook, 5^th Edition, New York, McGraw-Hill.
2. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.

This post illustrates the Statistical Thinking tool, the flowchart or process map, using an example taken from the author’s consulting experience. A flowchart of a process is sometimes referred to as a process map. A manufacturer produced automotive door frames, depicted in the following figure. The door frame consists of four parts which were joined by a welding operation. The shape and finished product dimensions were important quality characteristics of the finished product. However, they had a problem meeting dimensional specifications on the assembled final product. As a result they did considerable rework to insure satisfactory quality for the finished product.

The manufacturer formed a team to recommend corrective action to reduce rework costs and the time to meet shipment schedules. Shop floor personnel thought that variations in incoming raw material caused the quality problems. An analysis showed that the header was the primary quality problem.

The following figure gives the flowchart or process map for producing a header. The roll mill takes sheet metal, cuts the input material to the proper length, forms the two parts for a header, and spot welds them together. The bender bends the header to the proper shape punches two holes which will be used to position the part in subsequent operations. The saw forms the proper angles at the two ends of the header. The data on the flow chart below each operation specify important quality characteristics. The symbols h₁, h₂, g, D1, D2, D3 and SC 4 through SC20 specify dimensions.

The manufacturer collected data for the team for relating the quality characteristics on the flowchart to finished part dimensions. Collecting and analyzing data for individual steps in the flowchart is an example of disaggregation. A regression analysis resulted in the following conclusions:

1. Variation in material characteristics has little effect on quality characteristics.
2. D1, D2 and D3 have considerable variation and affect finished product quality
3. The left and right headers have significantly different variation for D2 and D3.

The above conclusions motivated corrective action, and the manufacturer eliminated the need for rework. This example reinforces the conclusion that data-driven decision making gives Statistical Thinking a significant advantage over expert opinion.

This post starts a series of posts to present the use of Statistical Thinking Tools in applying Statistical Thinking. The Statistical Thinking Tool illustrated by this example is a flowchart. We can have flowcharts for processes having service time objectives as well as processes processes producing a physical product. Jeffries and Sells (2004) present this example and describe the use of “statistical tools” to meet company service time objectives. We regard their use of statistical tools as an application of Statistical Thinking.

A Midwest manufacturing firm processes orders for its 6 manufacturing plants and 12 warehouses. Originally, each plant and warehouse had its own order processing service staffed by a total of 36 customer service representatives. To improve customer service and reduce costs, the company president directed a team to develop a centralized customer service center located at corporate headquarters. The president made this decision after the team surveyed customers and found that they were adamant that they did not want to wait for a customer service representative to answer a phone call and they were not very interested in personalized service provided by a plant or warehouse representative.

The team established a goal where 95% of incoming calls would wait less than 2 minutes for a customer service representative. The team acquired an Automatic Call Distribution (ACD) system to route customer calls to customer service representatives. The call center would operate from 7:00 am to 7:30 pm Central Time. The following figure gives a flowchart specifying the process of answering incoming customer calls.

The team collected data giving the distributions of incoming calls by time of day and the service times of the customer service representatives to answer the calls. Recording and analyzing data for individual steps in the process flow chart is an example of disaggregation. Classifying and analyzing data by a factor such as time of the day is an example of stratification.

The customer service center staffing levels by hour of the day is a crucial system design parameter. Wait times will be long without adequate staff. On two occasions in the past two months, I have had to wait more than an hour for technical service support personnel to answer my calls. I know that this happens because the companies involved have allocated inadequate staffing to handle the incoming calls.

The team developed staffing levels throughout the day using a simulation of the process represented by the figure above. Constructing a simulation requires a flowchart. Refer to Jeffries and Sells (2004) for additional details.

The next post will illustrate the use of a flowchart for a process producing a physical product.

References

1. Jeffries, R. D. and P. R. Sells (2004). Managing Customer Service Using Statistical Tools: A Case Study. Annual Quality Congress Proceedings.
   

The previous post described Part A of the Distribution Center On-Time Delivery Example. It illustrated Off-Target and Common-Cause Variation. This part of the example illustrates Special-Cause and Structural variation. The control chart in Figure 2 of Part A shows a Lower Control Limit (LCL) of about 88% on-time deliveries. This means that Common-Cause variation would rarely result in a weekly on-time delivery percentage lower than 88%.

Shawn formed a team to study the process and improve it. While monitoring the process the weekly on-time delivery percentage fell to 73%. Something had happened to increase the variation. Figure 3 shows this drop in on-time delivery performance. The team reviewed the distribution center activities and found that a division supplying the distribution center announced a price increase. Customers responded by submitting additional orders to avoid the price increase. This one-time increase in volume caused the distribution center to fall behind in filling orders because it was unprepared to handle the additional work. Also, this one-time increase in volume is an example of Special-Cause variation.

Figure 3

The team responded by developing an improvement policy. They determined that one could predict the weekly work load given the state of orders on Wednesday. The improvement policy provided for overtime hours based on the orders received and remaining work on Wednesday. Figure 4 shows the result. The average on-time percentage rose to 98.5%. The new LCL became 97%. However, subsequent results showed two weeks, Special-Cause weeks, where the volume was so heavy even the improvement policy could not handle the heavy demand. These were end-of-quarter weeks. Figure 4 shows them as weeks 39 and 52.

Figure 4

These end-of-quarter weeks shown in Figure 4 are examples of Structural variation. Structural variation is a blend of common and predictable special causes. Structural variation is due to causes that operate as an inherent part of the system as common causes do. However, on a control chart, they appear to be due to special causes. But their occurrence is predictable.

The four types of variation defined by Britz, Emerling et al (2000, p. 34) are:

· Off-Target variation occurs when the process average is not equal to its target value.

· Common-Cause variation is the variation exhibited by the process while operating in its best manner.

· Special-Cause variation results from the intervention of causes that are unplanned and undesirable.

· Structural variation is variation inherent in the system but appears to be due to special causes on a control chart. However, the causes of Structural variation are predictable.

References
1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.

The Customer-Complaint-Process Example illustrated two types of variation, i.e., special and common cause. The example, taken from Britz, Emerling, et (2000, p. 29), in this post illustrates four types of variation, i.e.,

· Off-target

· Common Cause

· Special Cause

Off-target variation occurs when the process average does not meet the organization’s desired target. Structural variation occurs when causes occur in a predictable manner. For example, the waiting line for a table at a restaurant might be longer on Saturday evenings than on other days.

Distribution Center On-Time Delivery Example

Shawn was perplexed when she examined Figure 1 showing a plot of weekly on-time deliver percentages at her distribution center. The corporation’s goal was to deliver 97.5% of orders each week in a timely manner. During the past quarter, the center had only met that goal twice. In addition, a review to the center’s activities during the two satisfactory weeks did not reveal any unusual behavior.

The overall average of weekly on-time delivery percentages was 94% which was significantly below the corporate goal of 97.5%. The average of weekly on-time percentages must be greater than 97.5% in order for the center to consistently meet its goal of 97.5%. If the average of all weekly on-time delivery percentages exactly equaled 97.5% then about half of the weeks would have on-time delivery percentages less than the goal of 97.5%. Assume that a target of 99% on-time deliveries would permit the center to consistently meet the goal of 97.5% for each week. This gap between the target (99%) and the weekly averages of 94% is Off-target variation.

Figure 1

Figure 2 suggests that the variation in on-time delivery percentages is due to common-cause variation. One reason is that all of the plotted points are less than the Upper Control Limit (UCL) and greater than the Lower Control Limit (LCL). Factors contributing to Common-Cause Variation are:

· Number and complexity of orders in each week

· Truck schedules

· Personnel availability

The conclusion is that an analysis of the actions during the two weeks where the center met the goal of 97.5% would be an inefficient approach to improving the system. Analyzing all of the weeks where the same common-causes are active would be more effective in identifying process improvements.

The next post will illustrate special-cause and structural variation.

Figure 2

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.

Statistical Thinking gives a framework for learning and action to improve performance. We initiate the application of Statistical Thinking by identifying, documenting and defining the business process. The Monthly-Billing-Process Example began by flowcharting and defining the billing process. The team in the Customer-Complaint Process recognized that the process included raw material suppliers, the OEM manufacturer, and their customers. Statistical Thinking recognizes that reducing variation is the key to success. Often reducing variation involves recognizing the different types of variation. The team in the Customer-Complaint-Process Example recognized the difference between special-cause and common-cause variation.

Usually Statistical Thinking requires the collection and analysis of data to estimate and reduce variation. Statistical Thinking is data-driven decision making. However, we need to define the overall process including its customer before collecting and analyzing data. Also, the process definition includes available subject matter knowledge. In the Monthly-Billing-Process Example, the process definition created knowledge concerning the process that did not exist without the flowcharts. In the Customer-Complaint-Process example, the team recognized that it had to collect usage rates in order to estimate variation and identify special-cause outcome. This process definition allows us to collect the appropriate data and focus our analysis.

Britz, Emerling, et al (2000, p26) point out two key advantages of Statistical Thinking and data-driven decision making.

1. Managers react to the last outcome. If it is satisfactory, everyone is pleased and satisfied that the system is performing well. If it is unsatisfactory, the implication is that something needs correction. Results from common-causes are treated as resulting from special causes. The analysis to reduce common-cause variation is much more effective if results from multiple outcomes are used. Trying to find a special cause when one does not exist leads to frustration.
2. The lack of data makes everyone an expert. Individual opinions vary and conflict with each other.

The figure depicts the relationship among Statistical Thinking, data and statistical methods. Effective application of statistical methods occurs after performing Statistical Thinking. In the Customer Complaint Process Example, a control chart and designed experiments occur after Statistical Thinking. Lynne Hare points out in the reference by Britz, Emerling, et al (2000, p27) that he was successful in getting increased use of statistical tools only after explaining Statistical Thinking to managers. They would not permit employees to use tools when they did not understand their purpose.

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
   

The previous blog post describes an application of Statistical Thinking to increase customer satisfaction by examining the processes starting with the raw material supplier and ending with the customer use of the product. This example illustrates important features of Statistical Thinking described by Britz, Emerling et al (2000). These features include:

1. Reducing variation improves performance. The team had training in Statistical Thinking so they understood the various types of variation. One type is common-cause variation which is always present. Another type is special-cause variation which is due to isolated events. One way to reduce overall variation is to identify outcomes that have a significant special-cause variation component. For those outcomes, we can analyze the circumstances causing the increased variation. To reduce common-cause variation, we need to analyze all of the outcomes not affected by special causes. The team knew this so they collected data to help them identify outcomes due to special causes. A control chart which is a statistical method is a common approach for identifying outcomes affected by special causes.
2. The team had management support. An understanding of Statistical Thinking facilitates management support. Management had to support the tactic of going to the customer and convincing them to collect usage data so the team could estimate defect rates. The team explained the concepts of Statistical Thinking to the customer to gain their support. Note that management and the customer did not have to have a detailed understanding of the statistical methods. When the team discovered that the importance of raw material, they had to convince management to support designed experiments to reduce common-cause variation. My experience is that management can be very reluctant to approve designed experiments unless they appreciate the principles of Statistical Thinking.
3. Reducing variation is often a sequential process. The team went after special-cause variation and then discovered a potential contributor to common-cause variation.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
   

Britz, Emerling et al (2000, p52) describe an application of Statistical Thinking that illustrates the following: the first principle, “All work consists of interconnected processes”, two types of variation, and shows the application of statistical methods to improve quality. An OEM manufacturer responded to customer complaints by regarding them as isolated events. Their corrective action did little to improve quality for future products. They received training in Statistical Thinking and formed a team to improve the complaint handling process. The team wanted to analyze each complaint to determine if it was the result of an isolated event (a special cause) or if it resulted from a process that needed improvement (a common cause). Shewhart (1931) developed these terms which are basic to Statistical Quality Control. Common-cause variation is the natural variation of a process when it is operating in a stable manner, and special-cause variation is due to an unpredicable special event. Examples of special causes in manufacturing are improperly maintained machines, operator errors or defective raw material.

In order to categorize the causes, the company asked the customer for usage data so the team could calculate defect rates. The company explained Statistical Thinking concepts to their customers to convince them to supply usage data. The team plotted using the control chart shown in the following figure. The high defect rate in October 91 indicated a special cause. An investigation led to raw material. The raw material supplier used the wrong material. However, discussions with the supplier and within the team motivated further analysis of the raw material. The supplier and the company conducted a series of designed experiments which identified an improved raw material composition. They changed their standard operating procedure to use this new raw material specification. The control chart shows a defect rate improvement from .023% to .004%.

The significant reduction in the complaint rate required recognition of a process involving raw material suppliers, the OEM manufacturer, and their customers. The team also used two statistical methods: Statistical Process Control (SPC) and Designed Experiments. The team used SPC to identify the special cause, and they used Designed Experiments to reduce the common-cause variation.

References

1. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
2. Shewhart, W. A. (1931), Economic Control of Quality of Manufactured Product, Milwaukee, WI, American Society for Quality.
   

The principal performance measure in the Monthly Billing Time Example was the total cycle time to prepare customer bills at the end of a month. How important is variation in systems that have a flow-time or cycle-time performance measure? Assume we have a process that consists of a number of stations that must be performed in series. That is, a work item must be processed by station 1 and then after completing station 1, it must be processed by station 2, and so on. Also, assume the each station has a single server which can only process a single work item at a time. For example, in the billing time example, a work item might be a single task the corporation performed for a customer in the previous month. A station might determine the hours expended on that task and calculate its cost. Assume a clerk determines the hours and calculates its cost.

Reducing the mean time for the clerk to determine the hours expended and calculate a cost will reduce the mean flow time. What if we reduce the variation in the time a clerk takes to calculate a cost for a work item? For illustrative purposes, assume the mean time expended by the clerk is 9 minutes to calculate the cost of a work item. Also, assume the work item arrivals have a mean inter-arrival time of 10 minutes.

The first figure depicts the case where inter-arrival times and service times are constant, i.e., there is no variation. The number in the system is the number of work items being served and waiting for service. In this case no work items have to wait. Therefore, their time at the clerk’s station is a constant 9 minutes.

The second figure illustrates the case where inter-arrival times and service times have variation. For the five arrivals, their inter-arrival times are: 8, 7, 10, 12 and 13 minutes, for an average of 10 minutes. Their service times are: 12, 11, 9, 7 and 6 minutes, for an average of 9 minutes. Now the times at the work station are 12, 16, 15, 10 and 6 minutes for an average of 11.8 minutes. The variation in inter-arrival and service times increased the time at the work station by 31%. Thus, variation can have a significant effect on system performance when the performance measure is a flow time or cycle time.

The two figures illustrate a manual simulation for calculating the system time increase due to variation. When analyzing an actual system, one can predict the system time using a discrete-event simulation. Inputs to the simulation would include a flow diagram, service time distributions and system inter-arrival time distributions. Arrival times at the individual work stations would be calculated by the simulation.

The previous blog post illustrates several key features of Statistical Thinking. One of these features is that Statistical Thinking is a philosophy of learning and action. That is, learning how to best obtain information which forms the basis of effective action. In the example, an important first step was to create a systems map and flow chart. Next the team collected cycle-time observations. Statistical Thinking evaluates a process by collecting data in addition to past experiences and perceptions. These data may be numerical (cycle-time measurements) or simply process documentation. The systems map and the flowcharts are process documentation. Once we have this documentation, we can ask why we operate in that manner and how we can improve the system. These data allow us to advance beyond personal opinions expressed by individuals. Recall that the departments involved thought that the other departments caused the lengthy billing time. Snee (1986) points out that “Good Decisions are based on facts, not opinions and emotions. … without data everyone is an expert.”

One guideline for effective application of Statistical Thinking is to always flowchart the process. The flowchart shows the relationships among different people and functions. Examining the flowchart suggests opportunities for improvement and areas for further examination and data collection.

The acronym SIPOC depicts our systems view of a process. The following figure depicts the SIPOC components which are Suppliers, Inputs, Process, Outputs and Customers. One motivation in the Monthly Billing-Time Example was to satisfy customer interests.

References

1. Snee, R. D. (1986). "In Pursuit of Total Quality." Quality Progress 20(8): 25-31.

Britz et al (1996, p4), Britz et al (2000, p7), and Hoerl and Snee (2002, p3) all describe the same example of an actual application of Statistical Thinking at a large corporation. The corporation wanted to decrease the average monthly billing cycle time of about 17 days to the corporation target of 10 days. A shorter time would improve the corporation’s cash flow and satisfy customer needs to more rapidly close monthly books. Customers had complained that other competing corporations were not as tardy in submitting bills.

The initial review of the process revealed that three separate departments constituted the billing process. Each department worked independently to perform their billing functions. No one understood the entire billing process flow. The corporation did not have a standard bill process operating procedure. Members of three departments claimed that the billing time delays were due to the other departments.

The initial step to improve the billing system was to develop both a systems map and a flow chart. The systems map identified responsible groups and the information flow among the groups. The flow allowed the construction of a production schedule for the monthly billing cycle. The schedule:

· listed the specific activities that had to be performed

· identified the responsible group for each activity

· specified due dates for each activity

The next step involved creating cross-functional teams to improve the performance of individual activities. They recorded cycle times, and these cycle times highlighted problem areas. They developed solutions to minimize variations in the problem areas. They documented the entire process and the procedures. The documented process procedures helped reduce variation in the problem areas. The documentation also helped in training new employees.

The corporation assigned a process owner to insure that they continued to realize the performance improvements.

The result was a reduction in billing cycle time in a 5 month period from 17 day to 9-10 days. That was almost a 50% reduction. This result satisfied customers and generated annual savings of $2.5 million.

This example illustrates key features of Statistical Thinking.

· Regarding the system as a process

· Reducing variation

· Using data to determine improvements for the system

References

1. Britz, G., D. Emerling, et al. (1996). Statistical Thinking, ASQ Statistics Division Special Publication
2. Britz, G. C., D. W. Emerling, et al. (2000). Improving Performance Through Statistical Thinking. Milwaukee, WI, ASQ Quality Press.
3. Hoerl, R. and R. D. Snee (2002). Statistical Thinking - Improving Business Performance. Pacific Grove, CA, Duxburry.