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Unit 1 (part 3): The Taguchi Method: Which Input Has the Greatest Effect? Concepts: The Taguchi method allows us to predict our ideal combination of independent variables (or input variables) to give the best result on a dependent variable. It can also be used to find which input variable has the greatest effect. Objectives: Students should explore the use of the Taguchi method to find which independent variable has the greatest effect on the dependent variable. Equipment/Materials: Computers with Internet access and Microsoft Excel Useful websites: Many Excel tutorials are online: for help with a specific Excel command or formula, it’s usually helpful to type the term into www.Google.com for quick, efficient tips. One good introductory Excel tutorial on Youtube can be seen at: http://www.youtube.com/watch?v=FNPNS6MT1Kk&feature=related (others are also available: go to www.youtube.com and search for “introduction to excel”) The Michigan Chemical Process Dynamics and Controls Open Text Book: http://controls.engin.umich.edu/wiki/index.php/Design_of_experiments_via _taguchi_methods:_orthogonal_arrays Wikipedia: Genichi Taguchi: http://en.wikipedia.org/wiki/Genichi_Taguchi A more complex site with many array choices: http://www2.research.att.com/~njas/oadir/index.html © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron Taguchi’s Robust Design Method (see “Extend”) http://www.mne.psu.edu/simpson/courses/ie466/ie466.robust.handout.pdf) Activity Sheets: Taguchi Exercise: Input with the Greatest Effect Material: Computers with Internet access Computers with Microsoft Excel Engage: Review the concepts of experiments, dependent and independent variables and defining the number of levels of each variable. Review the answers from the previous class to the following questions: 1. Why would it be important to identify the variable that causes the most change in the dependent variable we’re studying? 2. Can you see any information in the ice cream example from the last lesson that gives a hint about which variable causes the biggest change? If the students in class have little or no experience in using Microsoft Excel, you may want to review the video tutorials in “Useful websites” in class, and have the students work through the tutorials along with the video. Explore: 1. Review the ‘ice cream experiments’ from the teacher background with students in the classroom. Lead the students through the final values of signal-to-noise ratios and identification of the variable with the greatest effect. 2. During the ice cream example, emphasize that the Taguchi method was also useful to reduce the number of experiments we had to do. Had we done full-factorial experiments, this procedure would also work, but there would be many more calculations to perform. © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron 3. Have students work through the questions on the “Taguchi Exercise” worksheet. 4. Working in pairs, ask each student team discuss signal to noise ratio. For instance, what ‘noise’ could be present in a tank with algae? (Examples: a breeze could be blowing on a tank, contaminants could be introduced while testing, etc.). As the list grows, discuss which variables could be controlled and tested (i.e., airstone in the tank vs. no airstone in the tank) and those that cannot be controlled, and can only be considered noise (unexpected contaminants). Explain: 1. Discuss the results of the ice cream experiment. 2. Review the solution of the Taguchi Exercise: Input with the Greatest Effect worksheet. Extend: The analysis done here is sometimes represented graphically, with a series of plots showing the values of SN for each factor. An example of this analysis can be seen in section 32.4.2 of Taguchi’s Robust Design Method (available at http://www.mne.psu.edu/simpson/courses/ie466/ie466.robust.handout.pdf) Evaluate: 1. Use student solutions to the Taguchi Experiment Exercise. © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron Teacher Notes: Design of experiments (DoE) or experimental design is very commonly used to figure out how a number of variables work together toward some result. We have seen that we can minimize the number of experiments we need to perform, and we can use the results to predict the best combination of variables. The data collected while using the Taguchi Method can be used to find the input variable in which a change results in the biggest change in the final, dependent variable. The basic process is: - - Collect data using the Taguchi method o You will need to collect 3 or more results from each experimental setup (in other words, run each experiment 3 or more times). Follow the instructions to set up an Excel spreadsheet. Types of experimentation: (Review) - Running the Experiments: First, determine how you will measure the dependent variable. In most cases, you want a maximum or minimum number: which experiment grew the most algae, which experiment resulted in the lowest cost. For our ice cream example, you may have to assign the deliciousness of the ice cream cone a scale of 1 – 100. Use the input variable values indicated in the Taguchi chart to run each experiment. For example, on the above ice cream experiment, you would eat: Vanilla soft-serve on a sugar cone, with dip, Chocolate, regular ice cream on a sugar cone with sprinkles, Vanilla, regular ice cream on a sugar cone (plain), etc. … and record the deliciousness of the ice cream cone in each of the 9 cases. To find the input variable with the greatest effect, we will need to do multiple tests of each case – 3 or more is recommended. Analyzing the results: Let’s assume we made 3 ice cream cones testing each of our test conditions, and collected the following data (and averaged all of the experiments for each setup): The average is shown. The Excel formula for calculating the average is: =average( ) © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron After typing =average( block the group of 3 numbers using the mouse ) (in our first row, select 88, 86 and 86). L9 (filled in, with random values in blank boxes) Each level is named 1, 2 or 3 (for the next table) Exp Cone 1 Sugar 1 Ice cream Topping Texture Vanilla 1 Dip 1 Soft 1 1 88 2 86 3 avg 86 86.67 2 Sugar 1 Chocolate 2 Sprinkles 2 Regular 2 78 77 75 76.67 3 Sugar 1 Vanilla 1 None 3 Regular 2 82 84 85 83.67 4 Regular 2 Vanilla 1 Sprinkles 2 Soft 1 77 81 76 78.00 5 Regular 2 Chocolate 2 None 3 Soft 1 57 54 52 54.33 6 Regular 2 Vanilla 1 Dip 1 Regular 2 80 80 81 80.33 7 Sugar 1 Vanilla 1 None 3 Regular 2 82 84 85 83.67 8 Regular 2 Chocolate 2 Dip 1 Soft 1 68 62 57 62.33 9 Regular 2 Vanilla 1 Sprinkles 2 Soft 1 77 81 76 78.00 Finding the signal to noise ratio (SN): A basic explanation of the signal-to-noise ratio is: a ratio of the change in output due to the changing variable vs. changes in things we cannot control. For example, if we recorded speed of a bike based on speed of the pedals, we expect that, the faster we pedal, the faster the bike travels. However, if we record this data in a very hilly area, the rider might coast, shift gears, or maybe a strong wind would blow and make him/her speed up or slow down. If we took a lot of data, we expect we could show: the faster you pedal, the faster your bike goes. But the data would not be perfect – there was some noise (hills, coasting, gears, wind) that had some effect. The signal (pedal speed) was affected by the noise (everything else). The signal-to-noise (SN) ratio is found by: Where: i tells us which row we are working with (which experiment). is the mean (or average) of the row (shown in the data table above) is the standard deviation of the row We can find each of these values using Excel. © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron Step 1: Our goal is to find a value for SN for each row of our table. We will need to calculate the average and the standard deviation before we can determine the SN. To find the average of a group of numbers (in our case, 3 numbers) use To find the standard deviation, use =average( ) =stdev( ) The table below shows row 3 (i=3): the data is highlighted. The average of these 3 values is 83.67, and the standard deviation is 1.53. Exp 1 Cone Sugar 1 Ice cream Topping Vanilla 1 Dip 1 Texture Soft 1 1 88 2 86 3 avg 86 86.67 st dev 1.15 2 Sugar 1 Chocolate 2 Sprinkles 2 Regular 2 78 77 75 76.67 1.53 3 Sugar 1 Vanilla 1 None 3 Regular 2 82 84 85 83.67 1.53 4 Regular 2 Vanilla 1 Sprinkles 2 Soft 1 77 81 76 78.00 2.65 5 Regular 2 Chocolate 2 None 3 Soft 1 57 54 52 54.33 2.52 6 Regular 2 Vanilla 1 Dip 1 Regular 2 80 80 81 80.33 0.58 7 Sugar 1 Vanilla 1 None 3 Regular 2 82 84 85 83.67 1.53 8 Regular 2 Chocolate 2 Dip 1 Soft 1 68 62 57 62.33 5.51 9 Regular 2 Vanilla 1 Sprinkles 2 Soft 1 77 81 76 78.00 2.65 =average( ) =stdev() We now have all of the info we need to find the signal-to-noise ratio. The formula for the SN ratio is: =10*LOG((J18*J18)/(K18*K18)) … where J18 = the cell address of the average, and K18 is the cell address of the standard deviation. Be careful to type the parenthesis in correctly! For our example, this formula gives us: = 10 * log ((83.67*83.67) / (1.53*1.53)) = 34.76 (note: using Excel may give slightly different answers: for example, the result shown below is 34.771. Excel keeps track of the digits beyond those we see in the chart) © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron The table below shows each row with - average - standard deviation - Signal to Noise ratio Exp Cone Ice cream Topping Texture 1 2 3 avg st dev SN 1 2 3 4 5 6 7 8 9 1 1 1 2 2 2 1 2 2 1 2 1 1 2 1 1 2 1 1 2 3 2 3 1 3 1 2 1 2 2 1 1 2 2 1 1 88 78 82 77 57 80 82 68 77 86 77 84 81 54 80 84 62 81 86 75 85 76 52 81 85 57 76 86.67 76.67 83.67 78.00 54.33 80.33 83.67 62.33 78.00 1.15 1.53 1.53 2.65 2.52 0.58 1.53 5.51 2.65 37.508 34.012 34.771 29.391 26.685 42.869 34.771 21.075 29.391 Step 2: After calculating the SN ratio for each experiment, the average SN value is calculated for each factor and level. We will want to find: SN cone,1 SN cone,2 SN icecream,1 SN icecream,2 SN topping,1 SN topping,2 SN topping,3 SN texture,1 SN texture,2 Let’s look at SN values for texture: we’ll average each value for which texture was choice 1 (soft – shown in blue), then for each where it was 2 (regular – shown in gold): © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron SN texture,1 = (37.508 + 29.391 + 26.685 + 21.075 + 29.391) / 5 = 28.81 SN texture,2 = (34.012 + 34.771 + 42.869 + 34.771) / 4 = 36.61 Next, let’s find SN values for topping: SN topping,1 = (37.508 + 42.869 + 21.075) / 3 = 33.82 SN topping,2 = (34.012 + 29.391 + 29.391) / 3 = 30.931 SN topping,2 = (34.771 + 26.685 + 34.771) / 3 = 32.076 … the complete list is shown below: SN cone,1 35.266 SN cone,2 29.882 SN icecream,1 34.784 SN icecream,2 27.258 SN topping,1 33.817 SN topping,2 30.931 SN topping,3 32.076 SN texture,1 28.810 SN texture,2 36.606 © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron Step 3: Find the variable with the greatest change in signal-to-noise ratio We will find the difference between the highest and lowest SN for each variable. The most basic way is to fill in a table with the above values as shown: - - Cone Ice cream Topping Texture 1 2 3 35.226 29.882 34.784 27.258 33.817 30.931 32.076 28.81 36.606 Δ rank 5.344 3 7.526 2 2.886 4 7.796 1 Excel note: To automatically find Δ, you can use the following formula: =MAX(O18:O20)-MIN(O18:O20)… where O18:O20 would be replaced by the values in your Excel table. This automatically finds the maximum value in the range and subtracts the minimum. The values in the table are the SN values we’ve just calculated. The values for Δ (the Greek letter ‘delta’, which typically means ‘change in value’ in mathematics) are the maximum value – the minimum value of SN’s The rank is a ranking of largest to smallest value of delta. The ranking shows us that the texture of the ice cream has the greatest effect on the score for taste, while the topping has the smallest effect! © 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron