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of Experiments (DOE) and the Taguchi Approach
learn about different topics in the technique by reading brief descriptions
in this page.
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If you are visiting this page for the first time, or you are just getting started to learn the subject, here are a few things for you to know.
Taguchi Approach - Do not be confused by the association of the name Taguchi with DOE. Dr. Genechi Taguchi has offered a standardized and a relatively simpler method of applying the DOE technique. This is why his name is associated with DOE. Bare in mind that for most simpler applications of the technique, you need not worry about the subtle differences. Just concentrate on learning on what the technique is all about and how to apply it in your projects, assuming that learning how to apply is your intention. For your purposes view them as the same.
It is a technique to lay
out experimental (investigation, studies, survey, tests, etc) plan in most
logical, economical, and statistical way. You can potentially benefit from it
when you want to determine the: most desirable design of
your product, best parameters combination for your process, most robust recipe
for your formulation, permanent solution for some of your production problems,
most critical validation/durability test condition, most effective survey/data
collection plan, etc.
What it does for you - The DOE is about option selection. It works the best when you already have a working design (product, process, system, plan, etc.) and you want wish to put the finishing touches. If you are after developing a new product or process, it is not the right time for DOE. You need to look for other means to determine the working parameters. It is only after you reached a workable condition that satisfy your objectives, and know your process/system well, you would benefit from DOE. You will apply DOE to determine the best among many good conditions. In other words, it is something that will help you hone-in the process to perfection or help you select something that will consistently produce what you want, all the time.
Why do DOE?
“I use DOE to help my clients optimize processes for
value-added products while minimizing production costs. In the manufacturing of
wood products like value-added oriented strand board panels and specialty
plywood panels, there are several parameters that affect the process. DOE is the
tool to deal with processes with so many variables.”
“Designed experiments can help untangle the nature of
complex and otherwise confusing relationships faster than many of the
alternatives. ‘Thinking DOE’ helps one think more systematically, regardless of
“Imagine the feeling of finding something you really
want when it is on sale at a deeply discounted price. A well thought-out
experiment allows you to find-out so much for relatively little time and effort;
you just can't beat it for economy, efficiency, and effectiveness. And it is so
beautiful, watching knowledge unfold like a flower.”
“When I need to adjust one thing to improve performance, or when the single source of problem is known, often I can arrive at the solution intuitively. But when I’m dealing with more than one factor, or looking for unknown sources of problem, DOE comes to help.”
“I use DOE to identify if the process parameters for enhancing the ceramic tensile strength. It saves me a lot of time by avoiding testing all the process probabilities. DOE/Taguchi method is an effective tool for me to study my process by experimental means.”
“Like all other quality tools, DOE is an important technique. But, the benefit is in the way one uses it. You have to learn how to apply first.”
“When comes to deciding what’s best for my product and process designs, opinion and judgments slow me down. When I make decisions based on DOE results, everyone agrees.”
“I consider DOE to be the tool to give finishing touch before settling on designs. I believe we gain a lot when we use DOE to fine tune product designs before release, and optimize processes before production begins.”
“I use DOE to solve production related problems when
basic disciplines (like 8D) do not offer the technical solution.”
“I use DOE in HAZOP (Hazard & Operability Studies)
and QRA (Quantitative Risk Analysis) of offshore structure/process platform, Oil
rigs and On-land oil installation like Group gathering stations etc. I am quite
well versed in Six Sigma techniques, and also of Dr Taguchi method of OA
(Orthogonal Arrays) as a tool in the analysis phase of Six Sigma as well as Dr.
Taguchi's concept of loss function for a robust design. “
“Simulation models of manufacturing systems involve many design
or operation parameters. The optimal settings for these must be determined by
running the model many times. DOE provides an efficient and effective way to
conduct experiments with the model of the system after the model has been
verified and validated. It allows the KPOV (key process output variables) to be
modeled in terms of the KPIV (key process output variables). In general, DOE
leads to a better understanding of the system and interactions among the design
variables or operational variables.”
“DOE helps to reduce product/process development time and hence
costs associated with product/process development process. It improves process
yield, reliability and process capability. It can be used to reduce product
performance sensitivity to various sources of noise (such as environmental
variations, manufacturing imperfections, product-to-product variations, machine
performance deterioration, etc.) “
|Your learning strategy - For
comprehensive knowledge of the technique, you would want to know about (1)
theory and math, (2) application methods, and (3) Philosophy and working
disciplines (planning). Do not spend too much in the theory and statistical
calculation. You need to focus on what they mean rather than how it is done. Try
to muster the application methods and standard experiment design techniques.
Understand the philosophy and follow the discipline well. This is what give s
you the most benefits. The theory and application methods are routine and same
for all projects, the experiment planning is what will be unique to your
project. Unfortunately, it is something you will not learn well by reading. To
know it well, learn from expert practitioners or learn as you go on applying.
How to acquire application skill
Design Of Experiments (DOE) is a powerful statistical technique introduced by R. A. Fisher in England in the 1920's to study the effect of multiple variables simultaneously. In his early applications, Fisher wanted to find out how much rain, water, fertilizer, sunshine, etc. are needed to produce the best crop. Since that time, much development of the technique has taken place in the academic environment, but did help generate many applications in the production floor.
As a researcher in Electronic Control Laboratory in Japan, Dr. Genechi Taguchi carried out significant research with DOE techniques in the late 1940's. He spent considerable effort to make this experimental technique more user-friendly (easy to apply) and applied it to improve the quality of manufactured products. Dr. Taguchi's standardized version of DOE, popularly known as the Taguchi method or Taguchi approach, was introduced in the USA in the early 1980's. Today it is one of the most effective quality building tools used by engineers in all types of manufacturing activities.
The DOE using Taguchi approach can economically satisfy the needs of problem solving and product/process design optimization projects. By learning and applying this technique, engineers, scientists, and researchers can significantly reduce the time required for experimental investigations. DOE can be highly effective when yow wish to:
- Optimize product and process designs, study the effects of multiple factors (i.e.- variables, parameters, ingredients, etc.) on the performance, and solve production problems by objectively laying out the investigative experiments. (Overall application goals)
- Study Influence of individual factors on the performance and determine which factor has more influence, which ones have less. You can also find out which factor should have tighter tolerance and which tolerance should be relaxed. The information from the experiment will tell you how to allocate quality assurance resources based on the objective data. It will indicate whether a supplier's part causes problems or not (ANOVA data), and how to combine different factors in their proper settings to get the best results (Specific Objectives).
Further, the experimental data will allow you
|Advantage of DOE Using Taguchi Approach -The
application of DOE requires careful planning, prudent layout of the experiment, and expert
analysis of results. Based on years of research and applications Dr. Genechi Taguchi has
standardized the methods for each of these DOE application steps. Thus, DOE using Taguchi
approach has become a much more attractive tool to practicing engineers and scientists.
Experiment planning and problem formulation - Experiment planning guidelines are consistent with modern work disciplines of working as teams. Consensus decisions about experimental objectives and factors make the projects more successful.
Experiment layout -High emphasis is put on cost and size of experiments... Size of the experiment for a given number of factors and levels is standardized... Approach and priority for column assignments are established... Clear guidelines are available to deal with factors and interactions (interaction tables)... Uncontrollable factors are formally treated to reduce variation... Discrete prescriptions for setting up test conditions under uncontrollable factors are described... Guidelines for carrying out the experiments and number of samples to be tested are defined
Data analysis -Steps for analysis are standardized (main effect, NOVA and Optimum)... Standard practice for determination of the optimum is recommended... Guidelines for test of significance and pooling are defined...
Interpretation of results - Clear guidelines about meaning of error term... Discrete indicator about confirmation of results (Confidence interval)... Ability to quantify improvements in terms of dollars (Loss function)
Overall advantage - DOE using Taguchi approach attempts to improve quality which is defined as the consistency of performance. Consistency is achieved when variation is reduced. This can be done by moving the mean performance to the target as well as by reducing variations around the target. The prime motivation behind the Taguchi experiment design technique is to achieve reduced variation (also known as ROBUST DESIGN). This technique, therefore, is focused to attain the desired quality objectives in all steps. The classical DOE does not specifically address quality .
"The primary problem addressed in classical statistical experiment design is to
model the response of a product or process as a function of many factors called model
factors. Factors, called nuisance factors, which are not included in the model, can also
influence the response... The primary problem addressed in Robust Design is how to reduce
the variance of a product's function in the customer's environment."
TAGUCHI METHOD REVIEW
The Taguchi method is used to improve the quality of products and processes. Improved quality results when a higher level of performance is consistently obtained. The highest possible performance is obtained by determining the optimum combination of design factors. The consistency of performance is obtained by making the product/process insensitive to the influence of the uncontrollable factor. In Taguchi's approach, optimum design is determined by using design of experiment principles, and consistency of performance is achieved by carrying out the trial conditions under the influence of the noise factors.
This is a necessary first step in any application. The session should include individuals with first hand knowledge of the project. All matters should be decided based on group consensus, (One person -- One vote).
- Determine what you are after and how to evaluate it. When there is more than one criterion of evaluation, decide how each criterion is to be weighted and combined for the overall evaluation.
- Identify all influencing factors and those to be included in the study.
- Determine the factor levels.
- Determine the noise factor and the condition of repetitions.
2. DESIGNING EXPERIMENTS
Using the factors and levels determined in the brainstorming session, the experiments now can be designed and the method carrying them out established. To design the experiment, implement the following:
- Select the appropriate orthogonal array.
- Assign factor and interaction to columns.
- Describe each trial condition.
- Decide order and repetitions of trial conditions.
3. RUNNING EXPERIMENT
Run experiments in random order when possible.
4. ANALYZING RESULTS
Before analysis, the raw experimental data might have to be combined into an overall evaluation criterion. This is particularly true when there are multiple criteria of evaluation.
Analysis is performed to determine the following:
- The optimum design.
- Influence of individual factors.
- Performance at the optimum condition & confidence interval (C. I.).
- Relative influence of individual factors. etc.
5. RUNNING CONFIRMATION EXPERIMENTS)
Running the experiments at the optimum condition is the necessary final step.
SUGGESTED TOPICS OF DISCUSSIONS: 1. OBJECTIVES OF STUDY AND EVALUATION CRITERIA - What are the criteria of evaluation? - How are each of these criteria measured? - How are these criteria combined into a single number? - What is the common characteristic of these criteria? - What is the relative influence these criteria exhibit? 2. FACTORS - What are the factors that influence the performance criteria? - Which factors are more important than others? 3. NOISE FACTORS - Which factors can't be controlled in real life? - Is the performance dependent on the application environment? 4. FACTOR LEVELS - What are the ranges of values the factors can assume within practical limits? - How many levels of each factor should be used for the study? - What is the tradeoff for a higher level? 5. INTERACTION BETWEEN FACTORS - Which factors are most likely to interact? - How many interactions can be studied? 6. SCOPES OF STUDIES - How many experiments can we run? - When do we need the results? - How much does each experiment cost? 7. ADDITIONAL ITEMS - What do we do with factors that are not included in the study? - In what order do we run these experiments? - Who will do these experiments? etc.
Quality Characteristic (QC) generally refers to the measured results of the experiment. The QC can be single criterion such as pressure, temperature, efficiency, hardness, surface finish, etc. or a combination of several criteria together into a single index. QC also refers to the nature of the performance objectives such as "bigger is better", "smaller is better" or "nominal is the best". In most industrial applications, QC consists of multiple criteria. For example, an experiment to study a casting process might involve evaluating cast specimens in terms of (a) hardness, (b) visual inspection of surface and (c) number of cavities. To analyze results, readings of evaluation under each of these three criteria for each test sample can be used to determine the optimum. The optimum conditions determined by using the results of each criterion may or may not yield the same factor combination for the optimum. Therefore, a weighted combination of the results under different criteria into a single quantity may be highly desirable. While combining the results of different criteria, they must first be normalized and then made to be of type 'smaller is better' or 'bigger is better'. When quality characteristic (QC) consists of, say, three criteria, an overall evaluation criteria (OEC) can be constructed as: OEC = (X1/X1ref.)W1 + (X2/X2ref.)W2 + (X3/X3ref.)W3 where X = evaluation under a criterion Xref = a reference (maximum) value of reading W = weighting factor of the criterion (in %) Use of OEC as the result of an experimental sample instead of several readings from all criteria, offers an objective method of determining the optimum condition based on overall performance objectives. When there are multiple criteria of evaluation, the experimenter can analyze the experiments based on readings under one category at a time as well as by using the OEC. If the individual outcomes differ from each other, the optimum obtained by using OEC as a result should be preferred.
FACTORS AND LEVELS
FACTORS ARE: - design parameters that influence the performance. - input that can be controlled. - included in the study for the purpose of determining their influence and control upon the most desirable performance. Example: In a cake baking process the factors are; Sugar, Flour, Butter, Egg, etc. LEVELS ARE: - Values that a factor assumes when used in the experiment Example: As in the above cake baking process the levels for sugar and flour could be: 2 pounds, 5 pounds, etc. (Continuous level) type 1, type 2, etc. (Discrete level) LIMITS: Number of factors: 2 -63, number of levels: 2, 3, and 4.
INTERACTION BETWEEN FACTORS
Two factors (A and B) are considered to have interaction between them when one has influence on the effect of the other factor respectively. Consider the factors "temperature" and "humidity" and their influence on comfort level. If the temperature is increased by, say 20 degrees F, the comfort level decreases by, say 30% when humidity is kept at 90%. On a different day, if the temperature is raised the same amount at a humidity level of 70%, the comfort level is reported to drop only by 10%. In this case, the factors "temperature" and "humidity" are interacting with each other. Interaction: - is an effect (output) and does not alter the trial condition. - can be determined even if no column is reserved for it. - can be fully analyzed by keeping appropriate columns empty. - affects the optimum condition and the expected result.
NOISE FACTORS AND OUTER ARRAYS
Noise factors are those factors: - that are not controllable. - whose influences are not known. - which are intentionally not controlled. To determine robust design, experiments are conducted under the influence of various noise factors. An "Outer Array" is used to reduce the number of noise conditions obtained by the combination of various noise factors. For example: Three 2-level noise factors can be combined using an L-4 into four noise conditions(4 repetitions). Likewise seven 2-level noise factors can be combined into eight conditions(8 repetitions) using an L8 as an outer array. When trial conditions are repeated without the formal "Outer Array" design, the noise conditions are considered random.
SCOPE AND SIZE OF EXPERIMENT
The scope of the study, i.e., cost and time availability, are factors that help determine the size of the experiment. The number of experiments that can be accomplished in a given period of time, and the associated costs are strictly dependent on the type of project under study. The total number of samples available divided by the number of repetitions yields the size of the array for design. The array size dictates the number of factors and their appropriate levels included in the study. Example: A number of factors are identified for an optimization study. - Time available is two weeks during which only 25 tests can be run. - Three repetitions for each trial condition is desired. - Array size 25/3 --> 8 L-8 array. - Seven from the identified 2-level factors can be studied.
ORDER OF RUNNING EXPERIMENTS
There are two common ways of running experiments. Suppose an experiment uses an L-8 array and each trial is repeated 3 times. How are the 3x8=24 experiments carried out? REPLICATION - The most desirable way is to run these 24 in random order. REPETITION - The most practical way may be to select the trial condition in random order then complete all repetitions in that trial. NOTE: In developing conclusions from the results of designed experiments and assigning statistical significance, it is assumed that the experiments were unbiased in any way, thus randomness is desired and should be maintained when possible. MINIMUM REQUIREMENT - A minimum of one experiment per trial condition is required. Avoid running an experiment in an upward or downward sequence of trial numbers.
REPETITIONS AND REPLICATIONS
REPETITION: Repeat a trial condition of the experiment with/without a noise factor (outer array). Example: L-8 inner array and L-4 outer array. 8x4 = 32 samples. Select a trial condition randomly and complete all 4 samples. Take the next trial at random and continue. REPLICATION: Conduct all the trials and repetitions in a completely randomized order. In the above example, select one sample at a time in random order from among the 24 (8x4). NOTE: Results from replication contain more information than those from repetition. Since replication requires resetting the the same trial condition, it captures variation in results due to experimental set up.
AVAILABLE ORTHOGONAL ARRAYS
The following Standard Orthogonal Arrays are commonly used to design experiments: 2-Level Arrays: L-4 L-8 L-12 L-16 L-32 L-64 3-Level Arrays: L-9 L-18 L-27 (L-18 has one 2-level column) 4-Level Arrays: L-16 & L-32 Modified
TRIANGULAR TABLE/LINEAR GRAPHS
TRIANGULAR TABLE OF INTERACTIONS (2-LEVEL COLUMNS) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (1) 3 2 5 4 7 6 9 8 11 10 13 12 15 14 (2) 1 6 7 4 5 10 11 8 9 14 15 12 13 (3) 7 6 5 4 11 10 9 8 15 14 13 12 (4) 1 2 3 12 13 14 15 8 9 10 11 (5) 3 2 13 12 15 14 9 8 11 10 (6) 1 14 15 12 13 10 11 8 9 (7) 15 14 13 12 11 10 9 8 (8) 1 2 3 4 5 6 7 (9) 3 2 5 4 7 6 (10) 1 6 7 4 5 (11) 7 6 5 4 (12) 1 2 3 (13) 3 2 (14) 1 (15) (Interaction tables for 3-level and 4-level factors are not shown here) LINEAR GRAPHS - Linear graphs are graphical representations of certain readings of the Triangular table for convenience of experiment designs. The graphs consist of combination of a line with circles/balls at the ends. The end points represent the columns where the interacting factors are assigned and the number associated with the line indicate the column number for the interaction. Example: For L-4 Orthogonal array, 1 x 2 => 3, which will be shown in graph form as 3 1 o-------------------------o 2 Complicated Linear Graphs for higher order arrays are not shown here.
UPGRADING A COLUMN
COLUMN MODIFICATIONS: PREPARING A 4-LEVEL COLUMN - Select 3 2-level columns that are naturally interacting. Pick two and discard the third. Use the two columns to generate a new column. Follow these rules to combine the new columns: Old Columns New Column 1 1 -------> 1 1 2 -------> 2 2 1 -------> 3 2 2 -------> 4 Example: Suppose factor A is a 4-level factor. Using columns 1 2 3 of an L-16, a new 4-level column can be prepared and factor A assigned. PREPARING AN 8-LEVEL COLUMN - An 8-level column can be prepared from three of the seven interacting columns of an L-16. (Use columns 1 2 & 4, discard 3 5 6 & 7.) Follow these rules: Old Columns New Column ____________________________ 1 1 1 -------> 1 1 1 2 -------> 2 1 2 1 -------> 3 1 2 2 -------> 4 2 1 1 -------> 5 2 1 2 -------> 6 2 2 1 -------> 7 2 2 2 -------> 8 Note: An eight level factor/column is not supported by QUALITEK-4 software. The above information is for user reference only.
This method allows a 3-level column to be made into a 2 -level column or a 4-level column into a 3-level column (e.g. levels 1 2 3 to 1 2 1'). The notation 1' is used to keep track of the changed status only. For level assignment 1'=1. The selection of the level to be treated is arbitrary. Example: Three 3-level factors and one 2-level factor. - Use an L-9. Dummy treat any column and assign the 2-level factor.
RESULTS OF MULTIPLE CRITERIA
Frequently, your experiment may involve evaluating results in terms of more than one criteria of evaluations. For example, in a cake baking experiment, the cakes baked under different recipes (trial conditions) may be evaluated by taste, looks and moistness. These criteria may be subjective and objective in nature. The best recipes can be determined by analyzing results of each criterion separately. The recipes for optimum conditions determined this way may or may not be the same. Thus, it may be desirable to combine the evaluations under different criteria into one single overall criteria and use them for analysis. To combine readings under different evaluation criteria, they must first be normalized (unitless), then combined with proper weighting. Furthermore, all evaluations must be of the same quality characteristic, i.e., either bigger or smaller is a better type. When an evaluation is of the opposite it can be subtracted from a larger constant to conform to the desired characteristic [(X2ref. - X2) instead of X2]. For the purpose of combining all evaluations into a single criterion, Assume: X1 = Numeric evaluation under criterion 1 X1ref = Highest numerical value X1 can assume Wt1 = Relative weighting of criterion 1 Then an Overall Evaluation Criterion (OEC) can be defined as: OEC = (X1/X1ref)xWt1 + (X2/X2ref)xWt2 + .......
SIGNAL TO NOISE RATIOS (S/N) FOR STATIC AND DYNAMIC SYSTEMS
MSD AND S/N RATIOS NOTES AND RECOMMENDATION ON USE OF S/N RATIOS (Static condition) Recommendation: If you are not looking for a specific objective, then SELECT S/N ratio based on Mean Squared Deviation (MSD). MSD expression combines variation around the given target and is consistent with Taguchi's quality objective. S/N based on variance is independent of target value and points to variation around the target. S/N based on variance and mean combines the two effects with target at 0. The purpose is to increase this ratio ((Vm-Ve)/(nxVe)) and thus a + sign is used in front of Log() for S/N. Also, since for an arbitrary target value, (Vm-Ve) may be negative, target=0 is used for calculation of Vm. Expressions for all types of S/N ratios are shown on the next screen. RELATIONSHIPS AMONG OBSERVED RESULTS, MSD AND S/N RATIOS (Static condition) MSD = ( (Y1-Y0)^2 + (Y2-Y0)^2 + .... (Yn-Y0)^2 )/n for NOMINAL IS BEST Variance: Ve = (SSt - SSm)/(n-1) ................. for NOMINAL IS BEST Variance and Mean = (SSm - Ve)/(n*Ve) (with TARGET=0) where SSt = Y1^2 + Y2^2 and SSm = (Y1 + Y2 +..)^2/n MSD = ( Y1^2 + Y2^2 + ................... Yn^2 )/n for SMALLER IS BETTER MSD = ( 1/Y1^2 + 1/Y2^2 + ............. 1/Yn^2 )/n for BIGGER IS BETTER S/N = - 10 x Log(MSD)................. for all characteristics S/N = + 10 x Log(Ve or Ve and Mean) .. for NOMINAL IS BEST only. Note: Symbol (^2) indicates the value is SQUARED. DYNAMIC CHARACTERISTIC (Conduct of experiments and analysis of results) Reference text: Taguchi Methods by Glen S. Peace, Addison Wesley Publishing Company, Inc. NY, 1992 (Pages 338-363) WHAT IS DYNAMIC CHARACTERISTIC? A system is considered to exhibit dynamic characteristics when the strength of a particular factor has a direct effect on the response. Such a factor with a direct influence on the result is called a SIGNAL factor. SIGNAL FACTOR- is an input to the system. Its value/level may change. CONTROL FACTOR - is also an input to the system. Values/level is fixed at the optimum level for the best performance. NOISE FACTOR - is an uncontrollable factor. Its level is random during actual performance. STATIC SYSTEM GOAL - is to determine combination of control factor levels which produces the best performance when exposed to the influence of the varying levels of noise factors. DYNAMIC SYSTEM GOAL - is to find the combination of control factor levels which produces different levels of performances in direct proportion to the signal factor, but produces minimum variation due to the noise factors at each level of the signal. Example: Fabric dyeing process Control factor: Types of dyes, Temperature, PH number, etc. Signal factor: Quantity of dye Noise factor: Amount of starch CONDUCTING EXPERIMENTS WITH DYNAMIC CHARACTERISTICS When carrying out experiments, proper order and sequence of samples tested under each trial condition must be maintained. The number of samples required for each trial condition, will depend on the number of levels of signal factor, noise conditions and repetitions for each cell (a fixed condition of noise and signal factor). Step 1. Design experiment with control factors by selecting your design type (manual or automatic design) from the main screen menu. Step 2. Print description of trial conditions by selecting the PRINT option. Step 3. Enter in your descriptions and experiment notes on the DYNAMIC CHARACTERISTICS screen. * You will need to describe signal and noise factors and their levels. You will also have to decide on the number of levels of signal and noise factors. BUT MOST IMPORTANTLY, you will have to choose the nature of the ideal function (Straight line representing the behavior Response vs. Signal) applicable to your system. Step 4. Strictly follow the prescribed test conditions. Step 5. Enter results in the order and locations (run#) prescribed using the RESULTS option from the main menu. SIGNAL-TO-NOISE RATIO EQUATIONS (alternate dynamic characteristic equations) Signal factor may not always be clearly defined or known. For common industrial experiments, one or more attributes may be applicable: * TRUE VALUE KNOWN * INTERVAL BETWEEN FACTOR LEVELS KNOWN * FACTOR LEVEL RATIOS KNOWN * FACTOR LEVEL VALUES VAGUE Depending on the circumstances of the input signal values and the resulting response data, different signal-to-noise (S/N) ratio equations are available. ZERO POINT PROPORTIONAL - Select this response type of equation when response line passes through the origin. The signal may be known, unknown or vague. REFERENCE POINT PROPORTIONAL - This response type of equation should be the choice when the response line does not go through the origin but through a known value of the signal or when signal values are wide apart or far away from origin. When the signal values are known, zero point or reference point proportional should be considered first. If neither is appropriate, the linear equation should be used. LINEAR EQUATION - is based on the least I response squares fit equation and should be used where neither zero and reference point proportional equation are appropriate. Use it when signal values are close together and response does not pass through the origin. WHEN IN DOUBT plot the response as a function of the signal factor values on a linear graph and examine the y-intercept. If it passes through origin, use ZERO POINT. If the intercept is not through origin but the line passes through a fixed point, use REF. POINT. In all other situation use LINEAR EQUATION. S/N Ratio Equation and Calculation Steps y = m + Beta (M - Mavg) + e Linear Eqn. (L) y = Beta M Zero Point (Z) y = Beta (M - Mstd.) + ystd Ref. Point (R) Where y = system response (QC), M = Signal factor Beta = slope of the ideal Eqn. Mavg = Average of signals ystd. = avg. response under reference/standard signal Mstd = reference/standard value of the signal strength Notations * = multiplication, ^ = raised to the power / = division by Response Components for Each Trial Condition (Layout shown only for trial#1 below) S I G N A L F A C T O R Signal lev 1 Signal lev 2 Signal lev 3 _________N1_______N2__________N1______N2____________N1_______N2______ Trl#1| y11, y12, y13, y14.. y21, y22, y23, y24.. y31, y32, y33, y34. Step 1: Determine r (Start with trial# 1) ro = Number of samples tested under each SIGNAL LEVEL (Number of NOISE LEVELSxSAMPLES per CELL) M1, M2, M3,.. Mk. Signal levels (strengths) N1 & N2 are two levels of the noise factor k = number of signal levels Mavg = (M1 + M2 + .... Mk)/k r = ro [ (M1-Mavg)^2 + (M2-Mavg)^2 +... + (Mk-Mavg)^2] ... (L) r = ro [ (M1-Mstd)^2 + (M2-Mstd)^2 +... + (Mk-Mstd)^2] ... (R) r = ro ( M1^2 + M2^2 + M3^2 ... + Mk^2) ............. (Z) Step 2: Calculate of Slope Beta Beta = (1/r) [y1*(M1-Mavg) +y2*(M2-Mavg) +... + yk*(Mk-Mavg)] .. (L) Beta = (1/r) [y1*(M1-Mstd) +y2*(M2-Mstd) +... + yk*(Mk-Mstd)] .. (R) Beta = (1/r) ( y1*M1 + y2*M2 + ... + yk*Mk) ......... (Z) Step 3: Determine Total Sum of Squares St = Sum [Sum (yij - yavg)] i= 1,2 .. k. j=1,2,.. ro .. (L) yavg = ystd for (R), yavg = 0 for (Z) Step 4: Calculate Variation Caused by the Linear Effect Sbeta = r Beta^2 .... for all equations Sbeta = (1/r) [y1*(M1-Mavg) +y2*(M2-Mavg) +.. + yk*(Mk-Mavg)]^2 .. (L) Sbeta = (1/r) [y1*(M1-Mstd) +y2*(M2-Mstd) +.. + yk*(Mk-Mstd)]^2 .. (R) Sbeta = (1/r) ( y1*M1 + y2*M2 + ... + yk*Mk)^2 ....... (Z) Step 5: Calculate the Variations Associated with Error and Non-linearity Se = St - Sbeta ... for all equations Step 5: Calculate Error Variance Ve = Se / [ k*ro - 2 ] ...... (L) Ve = Se / [ k*ro - 1 ] ...... (R and Z) Step 6: Calculate S/N Ratio Eta = 10 Log (Sbeta - Ve) / (r*Ve) ... for all Eqns. Step 7: Repeat calculations for all other trials in the same manner. Example calculations: Case of LINEAR EQUATION (Expt. file: DC-AS400.QT4) The results of samples tested for trial#1 of an experiment with dynamic characteristic. There are three signal levels, two noise levels, and two repetitions per cell. M1 M2 M3 Noise 1 Noise 2 Noise 1 Noise 2 Noise 1 Noise 2 |_______________________|______________________|______________________ Trl#1| 5.2 5.6 5.9 5.8 | 12.3 12.1 12.4 12.5| 22.4 22.6 22.5 22.2 Signal strengths: M1 = 1/3, M2 = 1, M3 = 3 CALCULATIONS FOR S/N: Mavg = (1/3 + 1 + 3 ) / 3 = 1.444 ro = 4 (2 simple/cell * 2 noise levels) r = 4[(1/3 - 1.444)^2 + (1 - 1.444)^2 + (3 - 1.444)^2] ... (L) = 4( 1.2343 + 0.1971 + 2.421 ) = 15.41 y1 = 5.2 + 5.6 + 5.9 + 5.8 = 22.5 y2 = 12.3 + 12.1 + 12.4 + 12.5 = 49.3 y3 = 22.4 + 22.6 + 22.5 + 22.2 = 89.7 Beta = (1/r)[22.5*(1/3-1.444) + 49.3*(1-1.444) + 89.7*(3-1.444)] = (1/15.41) [ -24.9975 - 21.692 + 139.5732 ] = 92.8842/15.4101 = 6.01 Sbeta = r*Beta^2 = 15.4101 * 6.0274^2 = 556.82 yavg = [5.2 + 5.6 + ...... + 22.2]/12 = 161.5/12 = 13.46 St = (5.2 - yavg)^2 + (5.6 - yavg)^2 + .....+ (22.2 - yavg)^2 = 68.23 + 61.78 + 57.15 + 58.67 + 1.346 + 1.85 + 1.123 + .921 + 79.92 + 83.54 + 81.72 + 76.387 = 572.65 Se = St - Sb = 572.65 - 556.82 = 15.83 Ve = Se / ( 12 - 2 ) = 15.83 /10 = 1.583 Eta = 10 Log (Sbeta - Ve) / (r*Ve) ... for all Eqns. = 10 Log [(556.82 - 1.583)/(15.41*1.583)] = 10 Log(22.76) = 13.572 (S/N for the trial# 1 results ) Similarly, S/N ratios for all other trial conditions are calculated and analysis performed using NOMINAL IS THE BEST quality characteristic as normally done for the static systems.
Attractiveness of the Taguchi Approach
YOU NEED TO:
TAGUCHI VS. CLASSICAL DESIGN OF EXPERIMENTS (DOE)
CLASSICAL DOE Objective: Gather scientific knowledge about factor effects and their interactions. Weak main effects for random error.
LOSS FUNCTION The Loss Function offers a way to quantify the improvement from the optimum design determined from an experimental design study. Definitions: L = K (Y - Yo)^2 .... for a single sample. L = K (MSD) ........ for the whole population. where L = Loss in dollar. K = Proportionality constant. Yo = Target value of the quality characteristic. Y = Measured value of the quality characteristic. THE COST SAVINGS WHEN THE MEAN VALUE IS HELD AT A TARGET VALUE CAN BE CALCULATED WHEN THE FOLLOWING INFORMATION IS AVAILABLE : - TARGET VALUE OF QUALITY CHARACTERISTIC. - TOLERANCE OF QUALITY CHARACTERISTIC. - COST OF REJECTION AT PRODUCTION (PER UNIT). - UNITS OF PRODUCTION PER MONTH (TOTAL). - S/N RATIO OF THE OLD DESIGN. - S/N RATIO OF THE IMPROVED DESIGN. : Since the S/N ratio is a direct product of ANOVA, it is conveniently used for calculation of loss. However, the loss function requires MSD and must be calculated from the S/N ratio.
GENERAL NOTES AND COMMENTS: HELPFUL TIPS ON APPLICATIONS
COMBINATION DESIGN This is a method to fit two "2-level" factors in a "3-level" column. Suppose you have factors A and B at two levels and factors C, D, and E at three levels. An L-9 has four "3-level" columns. Factors C, D & E can occupy three columns leaving one column for A and B. A and B form A1B1 A2B1 A1B2 & A2B2. Select any three of these four and assign them to the three levels of the respective columns reserved for A and B. DESIGNS TO INCLUDE NOISE FACTORS (OUTER ARRAY) This version (version 4.7) of the program simultaneously handles inner and outer arrays. The noise conditions for repetitions can be studied by describing the outer array following completion of experiment design (inner array). Whether an outer array is present or not, up to 35 repetitions of results (columns) can be entered and an analysis performed using this software.
Comments from Expert Users(Why or why not use the Taguchi Approach)
"To me, Taguchi is attractive because of two reasons:
The down side is that at some point of time, you should be bold enough to make the giant leap (or at least what seems like a giant leap) to implement the findings."
- Sogal, E-mail: firstname.lastname@example.org
"Having done some extensive research in the area of Experimental Design, there are no hard and fast rules for the choice of experimental design for a particular problem. It is not a good practice to stick to one approach for solving all process optimization problems using Taguchi methods of Experimental Design. However Taguchi approach is the best approach for those organizations who are new to experimental design area due to its statistical or mathematical simplicity (degree of statistics involved). It provides a systematic approach to experimentation so that you can study a large number of variables in a minimum number of experimental trials. This will have a knock-on effect on experimental budget and resources. Another reason why Taguchi approach is better over Classical approach is the concept of achieving robustness in the functional performance by inducing the presence of noise factors during the experiment. It is a good starting point towards continuous improvement of process/product performance. However it is simply not the best optimization technique available today. Taguchi would not be able to provide us the true optimal value of a factor setting. It merely tells us which is the best level for a factor setting from the levels chosen for experimentation. In my view, the choice of experimental design is based on :
1. the degree of optimization required for the response or quality characteristic
Hope this helps. You may add my contact name and e-mail address for further discussion.".
- Dr. Jiju Antony, International Manufacturing Center, University of Warwick
"The classical DOE is more concerned with statistics and model creating. Engineering solution is rather on behind. For this reasons, C-DOE is not generally accepted in industrial environment. In other words, engineers consider C-DOE as a too difficult tool for practice.
Taguchi DOE does not require deep and rigorous scientific and statistical background (knowledge), instead engineering solution is preferred. So this approach is more understandable for practical engineers. Method is relatively easy to implement and understand. Method gives good results in practice."
- Dr. Pavel Blecharz
"Response Surface Methods (RSM) and other approaches are quite suitable for eg. research studies where often the influence of the various factors to be investigated are not well known. Here often quite a number of experimental trials need to be done as one ventures into somewhat uncharted territory. On the other hand in practical engineering problems the problem under investigation often relates to "fine tuning" of a process where the people involved have a reasonable "feel" for the process. The Taguchi approach is quite suitable for this purpose. Often researchers make use of Taguchi Methods for screening a large number of factors to narrow it down for more intensive study by RSM.
Taguchi Methods are relatively easy to grasp by co-workers on the shop floor as compared to the more statistically intense alternatives and aids their buy-in."
- Dr. Wim Richter, South Africa
When approaching a comparison of two viable alternatives, you should always "appear" to take the high road while serving your own interests. Expound on areas where "both" methods are viable and comparable, but then identify areas where the "preferred" element is clearly MORE advantageous to the user, creating the both very good, but one obviously better illusion.
Use two different "obviously better" scenarios:
"Both are viable in this arena, but the specific advantages in this area are.....". By playing the odds, one half of the prospects will fall within the common field of use (the advantages there must be very specific - showing your expertise), but the other half falling outside of the portrayed common field of use, making the "assumptive" decision obvious to the reader/receiver when presented in this manner.
- Cliff Veach, "As a consultant and trainer in the areas of Statistics and Statistical Process Control I am confronted, on a regular basis, with this question of whether to suggest a Classical Design of Experiments or to use the Taguchi methods . The question becomes quit easy to answer. If the customer has minimal knowledge of their process with a large number of factors to investigate or has more than two levels of each factor to examine the answer is Taguchi.
There are more reasons but I'll keep this reply short." -Karyn Heydt