Long wait times is an increasing problem in the United States, and visits to Hospital EDs has been increasing. From 1999 to 2009, it had increased by 32% to 136 million annual visits (Hing, Bhuiya 2012). That is, it increased at annual rate of 2.8%. For some hospitals, this increase has resulted in crowding and longer wait times to see a provider. Between 2003 and 2009, the mean wait time to be examined by a provider increased by 25% to 58.1 minutes. However, the distributions of wait times are highly skewed since more serious conditions are treated more quickly. The median wait time increased by 22% to 33 minutes.

The National Academy of Engineering and the Institute of Medicine prepared a report presenting the importance of systems engineering tools in improving health care processes (Reid, Compton, Grossman et all 2005). They emphasized the use of simulation. A discrete-event simulation of patient flow through an ED represents the ED as it evolves over time. The simulation’s state is stochastic since the processes such as patient arrival times, patient severity, and treatment times are stochastic and represented by random variables. Thus one must replicate the simulation model to estimate performance measures such as the average waiting time, the histogram of waiting times for a specified set of ED resources such as number of beds, doctor availability and nurse availability.

**References**

- Clark, Gordon (2016). “Statistics for Quality Improvement”
__ASQ Statistics Division Digest__,**35**(2): 22-26. - E. Hing, F. Bhuiya (2012). “Wait Time for Treatment in Hospital Emergency Departments: 2009”
__National Center for Health Statistics Data Brief__, No. 102, August 2012. - P. P. Reid, W. D. Compton, J. H. Grossman et al (2005).
__Building a Better Delivery System: A New Engineering/Health Care Partnership__, National Academies Press, Washington, DC.

Yeniay and Goktas (2002) used a real data set to predict the performance of the gross domestic product per capita (GDPPC) in Turkey. They compared the performance of PLS regression, ridge regression (RR), principal component regression (PCR) and ordinary least squares (OLS). The previous posting gives a brief description of these regression methods. The data consisted of 80 observations, and each observation represents one of the 80 provinces in Turkey. The data set included 26 predictor variables, and 22 of these variables are highly correlated. They estimated the predictive capability of the models using the leave-one-out approach to estimating the variability of predictions. PLS regression had the smallest prediction variability, but it was only slightly better than PCR. The statistical significance of the difference was not examined. However, the prediction variability of PLS regression and PCR was much smaller than the variability of RR and OLS.

Dumancas and Bello (2015) compared the performance of 12 predictive approaches using machine learning. The objective was to use lipid profile data to predict 5 year mortality after adjusting for confounding demographic variables. The approaches included PLS discriminant analysis, artificial neural network, ridge regression and logistics regression. PLS discriminant analysis (PLS-DA) is used when Y is a categorical variable like 1 when a person dies in 5 years. The dataset consisted of 726 individuals of which 121 died in the five year period. The total dataset was divided into a training set (483 individuals) and a test set (243 individuals). The results ranked PLS-DA first among the 12 approaches for predictive accuracy. However, the difference with PLS-DA was not statistically significant for artificial neural network, and logistics regression.

Dormann, Elith et al. (2013) evaluated methods for dealing with multicollinearity using simulation experiments. They created training and test data sets that had 1000 cases and 21 predictors. The condition number is a measure of the degree of collinearity. A condition number of 10 is approximately equivalent to |r| = .7. The condition number is the square root of the ratio between the largest and smallest eigenvalue of **X**. On page 30, the statement is made that several of the latent variable methods were only marginally better than Multiple Linear Regression (MLR) delaying the degeneration of model performance from a condition level of 10 to 30. MLR involves two or more explanatory variables when fitting a linear equation. PLS regression uses latent variables. However, the paper abstract states that latent variable methods did not outperform the MLR method. That conclusion did not apply when the condition number was less than 30.

Our conclusion is that evidence exists that PLS regression can outperform MLR in many situations with multicollinearity. However, severe multicollinearity can degrade PLS regression prediction performance.

**References**

- Clark, Gordon (2016). “Quality Improvement Using Big-Data Analytics”
__ASQ Statistics Division Digest__,**35**(1): 25-29. - Dormann, C. F., J. Elith, et al. (2013). “Collinearity: A Review of Methods to Deal with It and a Simulation Study Evaluating Their Performance.”
__Ecography__**36**(1): 27-46. - Dumancas, G. G. and G. A. Bello (2015).
__Comparison of Machine-learning Techniques for Handling Multicollinearity in Big Data Analytics and High-performance Data Mining__. Supercomputing 2015: The International Conference for High Performance Computing, Networking, Storage and Analysis. - Yeniay, O. and A. Goktas (2002). “A Comparison of Partial Least Squares Regression with Other Prediction Methods.”
__Hacettpe Journal of Mathematics and Statistics__31: 99-111.

]]>

Montgomery, Peck and Vining 2006 outline several approaches for dealing with multicollinearity. One is to collect additional date to reduce the multicollinearity. This is not always possible. Another approach is to respecify the model. That could include defining a new set of predictors that are a function of the original predictors that reduces multicollinearity. Another option is to eliminate a predictor that doesn’t contribute much explanatory capability. Another approach is to develop predictions of the dependent variables that are slightly biased but have smaller variances. Three of these approaches are:

- Ridge Regression. The ridge estimator is a linear transformation of the least squares predictors with a biasing parameter.
- Principal Component (PC) Regression. Define a new set of orthogonal predictors called principal components that are linear functions of the original predictors. Then delete those that have the least effect on the overall variance of the predictors.
- Partial Least Squares (PLS) Regression (Adbi 2010). PC regression finds a set of orthogonal predictors that relate to the original predictors. PLS regression finds a set of orthogonal predictors, called latent vectors, that explain the covariance between the predictors and the dependent variables.

The next posting will review studies the compare the performance of the above three approaches to dealing with multicollinearity.

**References**

- Abdi, Herve. (2010). “Partial Least Squares Regression and Projection on Latent Structure Regression (PLS Regression)”
__Wiley Interdisciplinary Reviews: Computational Statistics__**2**(1): 97-106. - Clark, Gordon (2016). “Quality Improvement Using Big-Data Analytics”
__ASQ Statistics Division Digest__,**35**(1): 25-29. - Snee, R. D. (2015). “A Practical Approach to Data Mining: I Have All These Data; Now What Should I Do?”
__Quality Engineering__**27**(4): 477-487. - Montgomery, D. C., E. A. Peck and G. G. Vining (2006).
__Introduction to Linear Regression Analysis Fourth Edition__, John Wiley & Sons, New York.

Is use of Big-Data Analytics a significant factor in helping companies perform more effectively? The *MIT Sloan Management Review* and the IBM Institute for Business Value conducted a survey of more than 3000 business executives to help answer that question (LaValle, Lesser, et al. 2011). The respondents were located in 108 countries and in more than 30 industries. The results indicated a widespread belief that analytics offers value.

- Half of them stated that improvement of information and analytics was a top priority in their organizations.
- More than one if five stated that they were under intense or significant pressure to adopt advanced information and analytic approaches.
- Respondents specified whether their organizations substantially outperformed (top performers) or underperformed industry peers.
- Organizations that strongly agreed the use of business information and analytics differentiates them within their industry were twice as likely to be top performers.

Interest in big-data analytics is rapidly increasing (Davenport 2013). Data from Google Trends shows that searches using the term “analytics” 2012 is more than 20 times greater than it was in 2005. Starting in 2010, searches using the term “big data” increased at a higher rate than searches using “analytics”.

Netflix uses big-data analytics to recommend movies to customers using a system called Cinematch (Davenport and Harris 2007). Cinematch defines clusters of movies, connects customers to the clusters, and recommends movies the customers will like. It also recommends movies based on customer evaluations of movies they viewed. I personally use Netflix, and I value these recommendations since they significantly reduce the time and energy required to identify desirable movies. Netflix’s revenue increased from $5 million in 1999 to about $1 billion in 2006. A major reason for its success is that it uses analytics.

Brigham Hospital in the Boston area uses big-data analytics to reduce *Adverse Drug Events* (Davenport 2013). Brigham hospital established a Computerized Order Entry (CPOE) system for doctors to input online orders for drugs, tests and other treatments. This system could also check whether a particular order made sense for individual patients. For example, the CPOE could check whether:

- The prescribed drug is consistent with best-known medical practices.
- Did the patient have past adverse reactions to it?
- Had the same test been prescribed multiple times before with no apparent benefit?

To prevent dangerous errors, Brigham set up its CPOE system in 1989. They also setup an outpatient electronic medical record system (EMR) at Brigham in 1989. The EMR contributed outpatient data to the CPOE. The combination of the EMR and CPOE was very effective in helping to prevent medical errors called *Adverse Drug Events*. In the U.S. 14 out of 1,000 inpatients experience Adverse Drug Events. Before the EMR and CPOE, at Brigham patients experienced about 11 events per 1,000 inpatients. After implementation of the EMR and CPOE, patients at Brigham experienced about 5 Adverse Drug Events per 1,000 inpatients. That is a 55% reduction.

However, Snee (2015) points out potential problems with big data. The data could be observational data not collected from a statistically designed experiment or survey. The next posting will address this problem and how to reduce its effects.

This posting has been adapted from “Quality Improvement Using Big-Data Analytics“ in the Statistics for Quality Improvement Column appearing in the ASQ Statistics Division Statistics Digest, October 2015.

**References**

- Davenport, T. (2013).
*Enterprise Analytics: Optimize Performance, Process and Decisions Through Big Data*. FT Press, Upper Saddle River, NJ. - Davenport, Thomas H., Harris, Jeanne, G. (2007).
*Competing on Analytics: The New Science of Winning*. Harvard Business School Publishing Corporation, Boston, Massachusetts. - LaValle, S., E. Lesser, et al. (2011). “Big Data, Analytics and the Path from Insights to Value.”
__MIT Sloan Management Review__**52**(2): 21-31. - Snee, R. D. (2015). “A Practical Approach to Data Mining: I Have All These Data; Now What Should I Do?”
__Quality Engineering__**27**(4): 477-487.

A |
B |
C |
D |
E |
S |

415 | 25 | 5 | 40 | 1.28 | 29.2 |

550 | 25 | 5 | 40 | 4.05 | 23.0 |

415 | 95 | 5 | 40 | 4.05 | 37.0 |

550 | 95 | 5 | 40 | 1.28 | 139.7 |

415 | 25 | 15 | 40 | 4.05 | 23.3 |

550 | 25 | 15 | 40 | 1.28 | 38.3 |

415 | 95 | 15 | 40 | 1.28 | 42.6 |

550 | 95 | 15 | 40 | 4.05 | 141.36 |

415 | 25 | 5 | 60 | 4.05 | 22.4 |

550 | 25 | 5 | 60 | 1.28 | 37.2 |

415 | 95 | 5 | 60 | 1.28 | 31.3 |

550 | 95 | 5 | 60 | 4.05 | 48.6 |

415 | 25 | 15 | 60 | 1.28 | 22.9 |

550 | 25 | 15 | 60 | 4.05 | 36.2 |

415 | 95 | 15 | 60 | 4.05 | 33.6 |

550 | 95 | 15 | 60 | 1.28 | 172.6 |

Minitab estimated the coefficients for a regression model estimating S from a linear sum of terms including a constant, A, B, C, D, E, and all of the two factor interactions. Minitab used a stepwise procedure where it starts with no significant terms and it adds a term at each step. A term is introduced based on the p-value of the estimated term coefficient. Assuming that the term true value of the coefficient is zero, a p-value is the probability of obtaining an estimate as large or larger than the one observed. In this estimate, the p-value had to be less than or equal to .015 for the term to be added.

The only statistically significant terms were those for the constant, A (pressure), B (temperature), and AB (pressure*temperature). The estimated regression model is:

Solubility = 80.0 – 0.144 Pressure – 3.36 Temperature + 0.00849 Pressure*Temperature

A surface plot of the regression model appears below, and it shows the effect of the two-factor interaction term.

The p-values for the model coefficients are listed in the following table.

Model Coefficient |
p-value |

Constant | 0.000 |

Pressure | 0.003 |

Temperature | 0.002 |

Pressure*Temperature | 0.012 |

This example clearly shows the advantage of a combined array design over the Taguchi crossed array design when two-factor interactions are present among the controllable variables. As mentioned in the previous posting, the standard Taguchi crossed array design would have the same number of experiments but the estimates of two-factor interactions would be aliased. If the experimenters were aware of the possibility of significant two factor interactions, they would have to have a larger experiment in order to use a Taguchi crossed array design.

]]>The purpose of the study implementing the experiment was provide relationships among factors affecting the recovery of peanut oil from peanut particles. These relationships would guide the design of oil extraction equipment (Goodrum and Kilgo, 1987). Carbon dioxide (CO_{2}) at very high pressures was to be used to extract the oil from peanuts. The oil would dissolve in the CO_{2}. Previous research indicated five factors that would affect the amount of oil that would dissolve in the CO_{2}. These factors are A = CO_{2} pressure, B = CO_{2} temperature, C = peanut moisture, D = CO_{2} flow rate, and E = peanut particle size. The experimental design as described by Kilgo (1988) did not consider that a factor might be a noise factor. So the analysis did not include the Robust Parameter Design viewpoint. We will assume that factor E, peanut particle size, is a noise or uncontrollable factor in the actual operation of the machine.

The 2^{5-1} fractional factorial design can be a resolution V design where every main effect is aliased with a four-factor interaction and every two-factor interaction is aliased with a three-factor interaction. Hicks and Turner (1999) in Example 13.2 analyzed the resolution of this design by Kilgo (1988) and came to the same conclusion. Thus, assuming the three and four factor interactions are negligible, we can estimate main effects and two-factor interactions among the controllable and noise factors using data produced by this design.

Contract this capability with the Taguchi crossed-array approach. Assume that both the crossed-array and combined-array designs have the same number of runs, i.e., 16. For four controllable factors, the inner array of the four controllable factors could be a 2^{4-1} fractional factorial design with 8 runs. Montgomery (2005) in Example 8-1 shows that the highest possible resolution for a 2^{4-1} design is IV. That means, no main effect is aliased with another main effect or with any two-factor interactions; however, the two-factor interactions are aliased with each other. Thus, the crossed-array approach can’t estimate two-factor interactions among controllable factors.

The next posting will present the results for a Robust Parameter Analysis of the data produced by the Kilgo fractional factorial experiment.

**References**

- Goodrum, J. W. and M. B. Kilgo (1987). “Peanut Oil Extraction with SC-CO
_{2}: Solubility and Kinetic Functions.”__Transactions of the ASAE__**30**(6): 1865-1868. - Charles R. Hicks and Kenneth V. Turner, Jr. (1999).
__Fundamental Concepts in the Design of Experiments__, Fifth Edition, Oxford University Press, New York. - Kilgo, M. (1988). “An Application of Fractional Factorial Experimental Designs.”
__Quality Engineering__**1**(1): 45-54. - Montgomery, D. (2005). Design and Analysis of Experiments, 6
^{th}Edition, John Wiley & Sons, Inc., New York.

An important objective for the combined array design is to be able to estimate factor main effects and two factor interactions. A resolution V design is one where no factor main effects or two-factor interactions are aliased with other main effects or two-factor interactions (Montgomery, 2012). However, two factor interactions can be aliased with three factor interactions.

The posting on December 8 describes a Robust Parameter design developed by a LSS team. The team identified six factors where four of them were controllable. If the team had done a combined-array design, they could have use a one-half fraction of a six-factor factorial design (a 2^{6-1} design). That would give them a resolution VI design. One can use software such as Minitab to generate the design and determine its resolution level. So the 2^{6-1} design exceeds the resolution V criterion. The 2^{6-1} design has 32 runs. The LSS team actually used a Taguchi crossed-array design having 36 runs. The inner array of the crossed array design was an orthogonal L_{9} for the four controllable factors. It is a resolution III design so it could not estimate two-factor interaction effects among the controllable factors. But the Taguchi design has three levels for the controllable factors so it could estimate some quadratic effects.

The next posting presents an example of a combined-array design.

**References**

- Douglas Montgomery (2012).
__Design and Analysis of Experiments, Eighth Edition__, John Wiley & Sons, New York, Chapter 12.

**Robust Parameter Designs**

When implementing Robust Parameter Designs, we have two types of factors, i.e., controllable and noise factors. Controllable factors are those whose values can be set when doing the experiments as well as in the field when operating the system. The system can be a process or a product. For the Taguchi example, the controllable factor were Tip Size, Feed Rate, Voltage, and Amperage in a process. The noise factors can have their values set during the experiments, but not during actual operation of the systems in the field. The noise factors in the Taguchi example were Air Pressure and Pierce Time.

For a product, the noise factors could be environmental factors such as temperature or humidity that affect performance. Other components of the system using the product might have variable attributes that affect the product performance. For example, the electrical power source to the product might have variable properties. Consider a manufacturing process. The noise variables might be raw material properties.

Robust Parameter Design is used when systems have noise variables. The purpose of Robust Parameter design is to choose levels of controllable factors so that the mean and variance of the output response meets system objectives (Montgomery, 2012). However, Montgomery (2012) and other references present approaches to Robust Parameter Design that are often more effective than Taguchi designs. That is, in many cases, other Robust Parameter Designs could require fewer experiments and reveal more effective system solutions.

**Crossed Array Designs**

The Taguchi designs use crossed arrays between the controllable and uncontrollable factors. That is, the designs consists of an inner array containing the controllable factors and outer array containing the uncontrollable factors (Montgomery 2012 and Chen, Li and Cox 2009). The arrays are crossed because every treatment combination for the controllable factors are run for every treatment combination for the noise factors.

In the Taguchi designs, interactions among controllable factors may give misleading results. Hicks and Turner (1999) on page 398 point out that the inner arrays used by Taguchi are all resolution III designs when used with the maximum allowable factors. That is, no main effect are aliased with other main effects; however, main effects are aliased with two-factor interactions. Consider the Taguchi Design posting where the controllable factors include tip size and voltage. Assume that tip size and voltage interact where large values of tip size combined with large values of voltage give an additional bevel magnitude not predictable based on considering tip size and voltage alone. Then the estimate of the tip size and voltage main effects include this interaction effect in a resolution III design.

The crossed array designs could give large experiments. What if we wanted to expand the inner array in the Taguchi Design posting to estimate two-factor interaction effects. To do that we need a resolution V design (Montgomery, 2012). Then no main effects or two factor interactions are aliased with other main effects or two factor interactions. We could do that by using a complete factorial design. With four factors and three levels for each factor that would require a 3^{4} or 81 runs. For the complete crossed array, we would have 324 runs. If we reduced the inner array to only include two levels for each factor, the inner array would have 2^{4} or 16 runs. That would give 64 runs for the complete crossed array design.

The next posting will describe combined array designs which will usually achieve the desired resolution levels with fewer experimental runs.

**References**

- Chen, J.C., Y. Li, and R. A. Cox (2009). “Taguchi-based Six Sigma Approach to Optimize Plasma Cutting Process: An Industrial Case Study
__”. International Journal of Advanced Manufacturing Technology__**41**: 760-769. - Charles R. Hicks and Kenneth V. Turner, Jr. (1999).
__Fundamental Concepts in the Design of Experiments__, Fifth Edition, Oxford University Press, New York. - Douglas Montgomery (2012).
__Design and Analysis of Experiments, Eighth Edition__, John Wiley & Sons, New York, Chapter 12.

**Performance Measures**

The Taguchi design consists of an inner array specifying controllable factor levels and an outer array specifying uncontrollable factor levels. In this case, the inner array has 9 rows and is an orthogonal L_{9} array. The outer array has 4 rows and is an orthogonal L_{4} array. Let y_{ij} be an experimental result where i specifies the inner array row, and j specifies the outer array row. For each inner array row, an average value of y_{ij} is calculated for the 4 values of j. Taguchi recommends analyzing variation using a **signal-to-noise ratio (S/N)**. When small values of the performance measure are best, the S/N is:

where j = 4 in this case.

For both Bevel Magnitude and Circularity Measures the target value T is zero and smaller values of y_{ij} are preferred. A value of S/N is computed for each inner array row. Large values of the signal-to-noise ratio (S/N) are preferred.

Taguchi’s Parameter Design philosophy is to emphasize the reduction of variation rather than determining whether the experimental result is within specification limits. This is similar to the Six Sigma philosophy.

A Taguchi design produces two response variables for a performance measure. That is, the average value and the signal-to-noise ratio (S/N). Each response variable is calculated for the nine values in the inner L_{9} orthogonal array. That is, the four values of the outer L_{4} array of uncontrollable factor level experiments are used to calculate an average Bevel Magnitude and an average Circularity Measure.

**Experimental Results**

Figure 1 Bevel Magnitude Results

Figure 1 gives the Bevel Magnitude results which were created based on data provided by Chen, Li and Cox (2009).. Small values of the average Bevel Magnitude are preferred. Thus, the small Tip Size, a Feed Rate of 83 inches per minute, a Voltage of 105 volts and an Amperage of 63 amps gave the best results based on average Bevel Magnitude. Large value of the S/N ratio are preferred. Thus, the small Tip size, the Feed rate of 93 inches per minute, the Voltage of 100 volts, and the Amperage of 53 amps gave the best results based on S/N ratio. For Bevel Magnitude, the only factor that had the best results for both the average Bevel Magnitude and the S/N ratio was the Tip Size.

Figure 2 gives the Circularity results also based on data in Chen, Li and Cox (2009). For Circularity, the small Tip Size, 93 inches per minute Feed Rate, 100 volts Voltage, and 63 amps Amperage gave the best results. Based S/N ratio, the same factor levels are best.

**Figure 2 Circularity Results**

To resolve this conflict among the results the LSS team decided to select the factor levels that had the highest number occurrences. The following table summarizes the results.

Selection Criteria |
Tip Size |
Feed Rate |
Voltage |
Amperage |

Average Bevel | Small | 83 in/min | 105 | 63 |

Bevel S/N Ratio | Small | 93 in/min | 100 | 53 |

Average Diameter Deviation | Small | 93 in/min | 100 | 63 |

Diameter S/N Ration | Small | 93 in/min | 100 | 63 |

Preferred Setting | Small | 93 in/min | 100 | 63 |

**Project Effectiveness**

The effectiveness of the Preferred Setting for the four factors was verified by 30 work pieces where each one of them met the quality requirements for subsequent assemblies (Chen, Li, and Shady (2010). The cycle time of the plasma cutter was reduced from 47 minutes to 30 minutes since the time spent of inspection and rework was reduced. That reduced the fabrication operation cycle time to 106.5 minutes so it was no longer the bottleneck operation. In addition, the output quality of the product was improved.

**Improved Approach and Controversy**

This case study clearly shows that Robust Parameter Design can improve system efficiency and effectiveness. It can improve the mean output and reduce variability. Montgomery (2012) notes that Taguchi Parameter Design was used in the 1980s by large corporations such as AT&T Bell Labs, Ford Motor Company and Xerox. However, later studies showed that the experimental procedures and data analysis methods advocated by Taguchi could be significantly improved. The next posting will describe problems in using crossed arrays in Robust Parameter Designs.

**References**

- Chen, J. C., Y. Li, and B. D. Shady (2010). “From Value Stream Mapping Toward a Lean/Sigma Continuous Improvement Process: An Industrial Case Study.” International Journal of Production Research
**48**(4): 1069-1086. - Chen, J.C., Y. Li, and R. A. Cox (2009). “Taguchi-based Six Sigma Approach to Optimize Plasma Cutting Process: An Industrial Case Study”. International Journal of Advanced Manufacturing Technology
**41**: 760-769. - Douglas Montgomery (1997). Design and Analysis of Experiments, Fourth Edition, John Wiley & Sons, New York, 622-641.
- Douglas Montgomery (2012). Design and Analysis of Experiments, Eighth Edition, John Wiley & Sons, New York, Chapter 12.

]]>

**Design Objectives**

The LSS team started by using the “5 Whys” method to identify the root cause of the long plasma cutter cycle time. The team noted that the plasma cutter was creating defects that needed to be reworked, and that inspection time was added to correct the defects. The root cause of the defects was that the plasma cutter was not operating at optimal parameter settings. These desired parameter setting were not known, and the team decided to conduct a DOE (Design of Experiment) study to identify effective parameter settings. The experimental design and results are presented by Chen, Li and Shady (2010), but Chen, Li and Cox (2009) give a more in depth presentation of the design and the analysis of results.

The plasma cutting machine produces holes on work pieces for installing hardware. Holes having excessive beveled edges and poor circularity cannot be used. A beveled edge is one where the hole is not perpendicular to the face of the switchboard. Figure 1 shows the Bevel Magnitude which is the bevel performance measure. Figure 2 shows the Smallest Diameter Deviation, |Dnormal – Dsmallest|, which is the circularity performance measure. Figures 1 and 2 are similar to Figures 5 and 6 in Chen, Li and Cox (2009).

**Factors**

The LSS team identified four controllable factors, i.e., Tip Size, Feed Rate, Voltage, and Amperage), and two uncontrollable noise factors, i.e., Air Pressure and Pierce Time. The manufacturer is unable to control the uncontrollable factors. They decided to run the experiments with three levels for the controllable factors and two levels for the uncontrollable (noise) factors. The following tables shows values appearing in Figure 8 of Chen, Li and Cox (2009).

Controllable Factors |
Level 1 |
Level 2 |
Level 3 |

Tip Size | Small | Medium | Large |

Feed Rate (inches per minute) | 83 | 93 | 103 |

Voltage (volts) | 100 | 105 | 110 |

Amperage (amps) | 43 | 53 | 63 |

Uncontrollable Factors |
Level 1 |
Level 2 |

Air Pressure (lbs/in^{2}) |
45 | 60 |

Pierce Time (seconds) | 0.70 | 1.40 |

**Taguchi Parameter Design**

The LSS team chose a Taguchi Parameters Design because they claimed it allowed for a reduction in the time and money to conduct the experiment. That is, the Taguchi design would require fewer experiments than a factorial design.

Taguchi Parameter Design is an example of a Robust Parameter Design, Montgomery (2012). Taguchi developed his experimental designs to:

- Develop products that are robust to external variability sources.
- Minimize variation about target values rather than conform to specifications limits.

A key objective for Taguchi Parameter Design is to reduce variability. Taguchi uses a loss function of the form L(y) = k(y – T)^{2 }, where T is the target value and y is the observed outcome. The loss function is used in identifying the preferred factor levels. Note the similarity between this loss function and the Six Sigma objective measure.

Montgomery (1997) describes the Taguchi Parameters Design, and gives an example similar to the one used by the LSS team with four controllable factors having three levels and three uncontrollable factors having two levels. The complete design consists of two arrays: an inner array containing the controllable factors and outer array containing the uncontrollable factors. The inner array is a L_{9} orthogonal array, and the outer array only needs to be an L_{4} orthogonal array since the LSS team only had two uncontrollable factors. The numbers 1, 2 and 3 in the following tables denote the factor levels. See Table 14-17 in Montgomery (1997) for a similar experimental design.

**L _{9} Orthogonal Array for Controllable Factors**

Run |
Tip Size |
Feed Rate |
Voltage |
Amperage |

1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 2 |

3 | 1 | 3 | 3 | 3 |

4 | 2 | 1 | 2 | 3 |

5 | 2 | 2 | 3 | 1 |

6 | 2 | 3 | 1 | 2 |

7 | 3 | 1 | 3 | 2 |

8 | 3 | 2 | 1 | 3 |

9 | 3 | 3 | 2 | 1 |

**L _{4} Orthogonal Array for Uncontrollable Factors**

Run |
Air Pressure |
Pierce Time |

1 | 1 | 1 |

2 | 1 | 2 |

3 | 2 | 1 |

4 | 2 | 2 |

Each of the 9 runs in the inner array was tested across the 4 runs in the outer array by the LSS team. That gave a total sample size of 36 runs. This type of design is called a **crossed array design, **Montgomery (2012).

The next posting will present the experimental results.

**References**

- Chen, J. C., Y. Li, and B. D. Shady (2010). “From Value Stream Mapping Toward a Lean/Sigma Continuous Improvement Process: An Industrial Case Study.” International Journal of Production Research
**48**(4): 1069-1086. - Chen, J.C., Y. Li, and R. A. Cox (2009). “Taguchi-based Six Sigma Approach to Optimize Plasma Cutting Process: An Industrial Case Study”. International Journal of Advanced Manufacturing Technology
**41**: 760-769. - Douglas Montgomery (1997). Design and Analysis of Experiments, Fourth Edition, John Wiley & Sons, New York, 622-641.
- Douglas Montgomery (2012). Design and Analysis of Experiments, Eighth Edition, John Wiley & Sons, New York, Chapter 12.