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Mathematical Model of Taguchi L18

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mathematics, taguchi designs and analysis
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  Post Number #1  
Old 7th August 2011, 06:01 AM
Dobby266

 
 
Total Posts: 3
Please Help! Mathematical Model of Taguchi L18

Hi peeps!

I am quite confused about Taguchi plans an their analysis. I hope that anyone of you can help me. I appreciate for any kind of help!

For a test plan I have an orthogonal array L18 with two noise factors in the outer array so that there are two response values for each run.
The analysis of this array with minitab is easy, but I have to know how the mathematical model is built.

As I know, minitab creates this model from the response values to make a prediction.
But how is the model made? Does minitab use a regression analysis or analysis of variance?

Is there anybody who can give me an answer or at least a link to literature?

Thanks a lot for your help!

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  Post Number #2  
Old 7th August 2011, 06:19 AM
Barbara B's Avatar
Barbara B

 
 
Total Posts: 390
Re: Mathematical model of Taguchi L18

There was a similar question answered in this thread a few days ago. Maybe these informations are sufficient to answer your question.

If not, it would be helpful to see your design (there are different types of design for L18) and the response(s) you want to evaluate. It would also be appreciated if you can provide the data.

Regards,

Barbara

(greetings to Kiel from Unna )
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  Post Number #3  
Old 7th August 2011, 08:11 AM
Dobby266

 
 
Total Posts: 3
Re: Mathematical model of Taguchi L18

Barbara, thanks for your quick answer!

I put my L18 in the attachment!
It consists of 8 control factors (1x2 levels and 7x3 levels) and two Noise levels in the outer array. These Noise levels N1 and N2 are compounds of factor settings for a good or a bad response. In this case they were determined in a screening experiment. Response is the time to start an engine. N1 causes a fast start an N2 a slow start.

In this taguchi design there are 3 level factors. Is the analysis the sme as you described in the welding problem?

Vielen Dank und schöne Grüße von der Küste!
Hannes
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  Post Number #4  
Old 7th August 2011, 09:48 AM
Barbara B's Avatar
Barbara B

 
 
Total Posts: 390
Re: Mathematical model of Taguchi L18

Quote:
In Reply to Parent Post by Dobby266 View Post

Is the analysis the sme as you described in the welding problem?
Thanks for the data. The analysis for a dynamic Taguchi design differs from the methods used in a static Taguchi design. Unfortunately the documentation of these methods is a little bit small within the Minitab help files.

The analysis for the slope (used for dynamic designs instead of the mean in static designs) depends on the settings in "Analyze Taguchi Design > Options". If you choose "Fit lines with no reference point" or "Fit each line through the average response at" (Signal reference value 1 or 2) there, the coefficients are the same like those in a general regression or general linear model with response variable N2-N1 (difference of results obtained with noise level 1 (N1) and noise level 2 (N2)). For the other option ("Fit all lines through a fixed reference point") I can't figured out what the respective regression model is.

Further informations on Taguchi methods are provided in the literature from the Minitab knowledgebase:
ID1246: References for Taguchi methods
or you can contact your local Minitab distributor (for Germany: Additive GmbH, Support: 06172-5905-20 / minitab@additive-net.de, give them my best ).

May I ask another question: Are you sure that you want to analyze your process using Taguchi methods? Just take a look into Kleppmann "Taschenbuch der Versuchsplanung", "9.4 Anmerkungen zu "Orthogonalen Feldern" u.ä." S. 170ff. (sorry, only available in German) for a detailed explanation what the pitfalls of the Taguchi method are and how this could be circumvented by the use of other doe methods.

Best regards,

Barbara
  Post Number #5  
Old 8th August 2011, 06:37 PM
Dobby266

 
 
Total Posts: 3
Re: Mathematical model of Taguchi L18

Hi Barbara,

I am not sure whether I understodd what you posted. What do you mean by static and dynamic Taguchi design? I think it is a static problem where S/N ratio is calculated by "Nominal the best" (in this case).

What makes me confused is that I don't know which mathematical model sits behind this plan. The L18 consists of one Factor with 2 levels and 7 factors with 3 levels. The 3 levels provide the use of a quadratic model (y=b0+b1x1....) but what does this polynom look like in general? And what to do with the 2 level factor in this model?

I have the two books, Kleppmann and Klein "Versuchsplanung- DoE" right under my nose, but I can't find the answer. If it is too difficult to explain in english (or too difficult to understand for me ;o) ) can you answer in german?
Thanks a lot for your help! I really despair of this analysis.
  Post Number #6  
Old 9th August 2011, 02:05 PM
Barbara B's Avatar
Barbara B

 
 
Total Posts: 390
Re: Mathematical model of Taguchi L18

Quote:
In Reply to Parent Post by Dobby266 View Post

What do you mean by static and dynamic Taguchi design?
Those definitions are used within Minitab:
Static design = Taguchi design without signal (or noise) factor
Dynamic design = Taguchi design with signal (or noise) factor

In your design the signal factor "Noise" provides the settings for the combination of two factors (oil and fuel temperature) which are noise factors (hardly controllable).

Statisch und dynamisch sind Minitab Definitionen:
statisches Design = Taguchi Design ohne Signal- (bzw. Rausch-)Faktor
dynamisches Design = Taguchi Design mit Signal- (bzw. Rausch-)Faktor

Bei Dir steckt der Rausch-Faktor in "Noise" als Kombination zweier Faktoren (Öl- und Benzin-Temperatur), die schwer kontrollierbar sind.


Quote:
In Reply to Parent Post by Dobby266 View Post

I think it is a static problem where S/N ratio is calculated by "Nominal the best" (in this case).
There are three possible response variables which should be evaluated (following Taguchi's approach):
  1. signal-to-noise ratio
  2. mean (static design) / slope (dynamic design)
  3. standard variation
For signal-to-noise ratio, slope and standard deviation the basic design has to be replicated at least 2 times. The aim is to find factor settings for the controllable factors which provide minimal variation AND optimal outcome.

Mit der Methode von Taguchi werden 3 verschiedene Antwortvariablen untersucht:
  1. Signal-Rausch-Verhältnis
  2. Mittelwert (statisches Design) / Steigung (dynamisches Design)
  3. Standardabweichung
Um das Signal-Rausch-Verhältnis, die Steigung und die Standardabweichung berechnen zu können, muss das Basis-Design mindestens 2 Mal durchgeführt werden. Das Ziel bei der Auswertung ist es diejenigen Faktor-Einstellungen der Steuer-Faktoren zu finden, die eine möglichst kleine Streuung UND ein optimales Ergebnis liefern.


Quote:
In Reply to Parent Post by Dobby266 View Post

What makes me confused is that I don't know which mathematical model sits behind this plan. The L18 consists of one Factor with 2 levels and 7 factors with 3 levels. The 3 levels provide the use of a quadratic model (y=b0+b1x1....) but what does this polynom look like in general? And what to do with the 2 level factor in this model?
It's possible to fit quadratic terms for those factors with at least 3 levels (B, C, D, E, F, G, H), but the regression model becomes instable for the L18-design (see Wikipedia reference-linkmulticollinearity). Minitab provides only models with linear terms within the Taguchi menu, but you can fit a model with quadratic terms using other menus (see attachement for details). I won't recommend that for your plan

Es ist möglich, quadratische Effekte für die Faktoren anzupassen, die mindestens 3 Level haben (B, C, D, E, F, G, H), aber das Regressions-Modell wird für das L18-Design instabil (vgl. Wikipedia (de) Multikollinearität). Minitab arbeitet in dem Taguchi-Menü ausschließlich mit linearen Termen, aber Du kannst mit anderen Menüs ein Modell mit quadratischen Termen anpassen (s. Anhang). Ich würde das allerdings nicht empfehlen

Quote:
In Reply to Parent Post by Dobby266 View Post

I have the two books, Kleppmann and Klein "Versuchsplanung- DoE" right under my nose, but I can't find the answer.
There are several pitfalls and issues within the Taguchi DoE:
  • For all Taguchi designs:
    • Most Taguchi designs can not evaluate interactions (resolution III designs) if used in a standard manner. But interactions are very likely in the real world and often have a higher impact than main effects or even quadratic effects.
    • A random run order is not recommended. But if the run order isn't randomized (at least for the settings of the controllable factors), you can't distinguish between time-related effects (e.g. heating of the system with an effect on the response or attrition) and effects caused by the change of factor settings.
    • Taguchi designs aren't orthogonal which can lead to multicollinearity (and therefore instable models), especially for factors with more than 2 levels and quadratic terms.
  • for designs with more than 2 levels (for at least 1 factor)
    • Taguchi designs provide the evaluation of quadratic effects. But even for those designs Taguchi assessed 2-way interactions to be negligible.
    • Mixed level designs (like the L18) have a complex aliasing structure (confounding of main effects and interactions) and could therefore be misleading (see Montgomery "Design and Analysis of Experiments", 9.4 "Factorials with Mixed Levels", 6th or 7th edition, ISBN 978-0-47148-735-7 or 978-0-470-39882-1, and Anderson & Whitcomb "DOE simplified", 7 "General Factorial Designs", 2nd edition, ISBN 978-1-56327-344-5 for details).

    But on the other hand Taguchi's methods aren't a complete waste of time: They are better than every unstructured data gathering or the OFAT-principle (in most cases), so it is better to conduct a Taguchi DoE than no DoE at all.

    Back to your data:

    IMHO the results do not provide any reliable information about the process at all: No factor have a significant impact on the outcome, so every interpretation is like fishing in murky waters.

    Depending on the effort you can add to this study it could be necessary to go back to square one and select a design which has a more stable structure and provides estimation of the terms you're interested in. But this could means that a whole new doe has to be done.

Es gibt einige Stolpersteine und Schwierigkeiten bei der Taguchi Versuchsplanung:
  • Für alle Taguchi Designs:
    • Bei der Mehrheit der Taguchi Designs können Wechselwirkungen nicht untersucht werden (Auflösung III), wenn die Pläne in den Standardeinstellungen eingesetzt werden. Interaktionen (Wechselwirkungen) kommen allerdings sehr häufig im wahren Leben vor und haben oft einen stärkeren Effekt auf das Ergebnis als Haupt-Effekte und zum Teil auch als quadratische Effekte.
    • Die zufällige Reihenfolge der Versuche wird nicht empfohlen. Werden die Versuche aber nicht randomisiert (in einer zufälligen Reihenfolge durchgeführt), kann nicht zwischen zeit-abhängigen Veränderungen (z. B. durch Erwärmung des Systems oder Verschleiß) und Effekten durch Veränderungen der Faktoreinstellungen unterschieden werden.
    • Taguchi Versuchspläne sind nicht orthogonal. Dies kann zu Multikollinearität (und damit instabilen Modellen) führen, vor allem bei Faktoren mit mehr als 2 Leveln und quadratischen Termen.
  • Für Designs mit mehr als 2 Leveln (für mindestens 1 Faktor)
    • Mit Taguchi Designs können quadratische Terme modelliert werden. Aber auch für diese Designs hat Taguchi Wechselwirkungen als vernachlässigbar eingestuft.
    • Versuchspläne mit verschiedenen Stufen-Anzahlen (wie beim L18) haben eine komplexe Alias-Struktur (Vermischung von Haupteffekten und Interaktionien) und können daher falsche Schlussfolgerungen begünstigen (vgl. Montgomery "Design and Analysis of Experiments", 9.4 "Factorials with Mixed Levels", 6th or 7th edition, ISBN 978-0-47148-735-7 or 978-0-470-39882-1, and Anderson & Whitcomb "DOE simplified", 7 "General Factorial Designs", 2nd edition, ISBN 978-1-56327-344-5 und Kleppmann "Taschenbuch der Versuchsplanung", 8.2.4 "Was bedeutet Vermengung?" und 13.3 "Einsatzempfehlungen" (für mehrstufige Pläne), 6. Auflage, ISBN 978-3446-42033-5).

Auf der anderen Seite sind Taguchi Methoden nicht unbedingt immer völlige Zeitverschwendung: Sie sind besser als unstrukturierte Datensammlungen und ein-Faktor-zur-Zeit-Methoden (meistens jedenfalls), deshalb ist es besser Taguchi Versuchsplanung zu machen als gar keine Versuchsplanung.


Zurück zu Deinen Daten:

IMHO gibt es überhaupt keine verwertbaren Ergebnisse aus den Versuchen: Kein Faktor hat einen signifikanten Effekt auf das Ergebnis, womit jede Art der Interpretation dem Stochern im Nebel gleichkommt.

Je nachdem wie viel Aufwand Du noch in diese Untersuchung stecken kannst könnte es deshalb notwendig sein nochmal neu anzufangen und einen Versuchsplan auszuwählen, der eine stabilere Struktur hat und Dir ermöglicht die Euch interessierenden Terme zu bestimmen. Aber das würde bedeuten, dass Du eine komplett neue Versuchsplanung machen musst.


Hope this helps (nevertheless)

Barbara

Last edited by Barbara B; 10th August 2011 at 10:13 AM. Reason: attachment removed (corrected version see below)
  Post Number #7  
Old 10th August 2011, 06:12 AM
Ingo1966's Avatar
Ingo1966

 
 
Total Posts: 41
Re: Mathematical Model of Taguchi L18

Hallo Hannes

Attached you will find your L18 analysed with Minitab.
Because your Response should be minimized (time to start the Engine) the S/N Ratio Formula is used for "Smaller the Better" (keep in mind that this is related to your Response the S/N Ratio goal is maximize)

The Static Parameter Design has an inner array with Control Factors and an outer array for Noise Factors. (acc. Mr. Taguchi's Definition)
The goal is 1. to minimize Variation due to Noise Factors and 2. improve the response (maximize, nominal, smaller depends on what you want --> different S/N Formulars)

The Dynamic Parameter Design includes additional to the outer array of Noise Factors also a Signal (z.B. unterschiedliche Betriebszustände).
The calculation Formulars for S/N is different and additional there will be a slope Calculation to improve the response over the whole Signal Settings.

But let's come back to your L18 --> First it makes no really sense to calculate a Standard Deviation (Needed for S/N Ratio) out of 2 values.
You need minimum 3 repeated measurements for each Noise Level Setting.

But Minitab is just doing the Math Job without checking the sense.
Ok acc. what Minitab gives as Coefficiants B1 is only significant (p-value small) to use it to Maximize the S/N Ratio and if you check the Graphs you are in the lucky position that also the Mean can be reduced with B1.

But every Prediction of a DoE or PD has to be confirmed.
I hope the Factor B on Level 1 can really help to reduce Variation due to noise and reduce the Response.

An alternativ Approach would be the Inner outer Array PD.
The Interaction between Control Factors and Noise Factors can be investigated and used to minimize the Effect to the Response due to the Noise.
In this PD you do not need Repeated Measurements per Test Run. (I would not run any DoE without repeated Measurments but in some cases it is a Question of Money or Time)

Attached you will find a 2 Level L16 with 7 Control Factors and One Noise Factor.
(Response --> random number so just for Info)

Parameter Design ist sehr schön von Wilhelm Kleppmann erklärt.
In meiner Schulung zeige ich immer gerne das Beispiel der Ziegel Herstellung.

Schöne Grüsse aus Köln Ingo
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File Type: zip L18_INGO_SMALLER THE BETTER.zip (63.9 KB, 28 views)
File Type: zip Alternativ PD Approach.zip (203.7 KB, 36 views)

Last edited by Ingo1966; 10th August 2011 at 06:58 AM.
Thanks to Ingo1966 for your informative Post and/or Attachment!
  Post Number #8  
Old 10th August 2011, 11:09 AM
Barbara B's Avatar
Barbara B

 
 
Total Posts: 390
Re: Mathematical Model of Taguchi L18

Ingo,

thanks for your valuable input

Quote:
In Reply to Parent Post by Ingo1966 View Post

Attached you will find your L18 analysed with Minitab.
Because your Response should be minimized (time to start the Engine) the S/N Ratio Formula is used for "Smaller the Better" (keep in mind that this is related to your Response the S/N Ratio goal is maximize)

The Static Parameter Design has an inner array with Control Factors and an outer array for Noise Factors. (acc. Mr. Taguchi's Definition)
The goal is 1. to minimize Variation due to Noise Factors and 2. improve the response (maximize, nominal, smaller depends on what you want --> different S/N Formulars)

The Dynamic Parameter Design includes additional to the outer array of Noise Factors also a Signal (z.B. unterschiedliche Betriebszustände).
The calculation Formulars for S/N is different and additional there will be a slope Calculation to improve the response over the whole Signal Settings.
Good point I've corrected the attachment.

Quote:
In Reply to Parent Post by Ingo1966 View Post

But let's come back to your L18 --> First it makes no really sense to calculate a Standard Deviation (Needed for S/N Ratio) out of 2 values.
You need minimum 3 repeated measurements for each Noise Level Setting.
Why 3 and not 4, 5, ...? The standard deviation can be calculated if 2 or more values are present, but the answer to the question "how many replications are necessary" depends on other knowledge as well (e.g. measurement system error, effect size, resolution of the measurement).

In the L18-design are three runs (run 2, 3 and 5) where exactly the same outcome is given for N1 and N2. For these runs the standard deviation is 0.

To evaluate variation a common method is to use the natural logarithm of the standard variation (ln(s)) as it has a nearly linear relationship to the factors. This method is provided as an option for Taguchi analyses in Minitab (see Analyze Taguchi Design > Options) and is used within the "Analyze Variability" menu for factorial designs. But ln(0) is not defined, because the natural logarithm can only be calculated for values >0. So there will be a lack of information from those runs with equal results (independent of the number of repeated measurements) in the analysis of variation.

Quote:
In Reply to Parent Post by Ingo1966 View Post

But Minitab is just doing the Math Job without checking the sense.
Ok acc. what Minitab gives as Coefficiants B1 is only significant (p-value small) to use it to Maximize the S/N Ratio and if you check the Graphs you are in the lucky position that also the Mean can be reduced with B1.

But every Prediction of a DoE or PD has to be confirmed.
I hope the Factor B on Level 1 can really help to reduce Variation due to noise and reduce the Response.
I won't be too optimistic that an optimization depending on B=1 will provide a better result. If all non-significant terms in the model are removed (A, C, D, E, F, G, H) and only B remains (Analyze Taguchi Design > Terms > Selected Terms: B:B), the coefficient of determination is R²=31% (for SN-ratio) and 37% (for Mean). That's really sparse, especially for a doe.

Quote:
In Reply to Parent Post by Ingo1966 View Post

An alternativ Approach would be the Inner outer Array PD.
The Interaction between Control Factors and Noise Factors can be investigated and used to minimize the Effect to the Response due to the Noise.
In this PD you do not need Repeated Measurements per Test Run. (I would not run any DoE without repeated Measurments but in some cases it is a Question of Money or Time)
Imho this is a better approach, but to evaluate the mean and the variation as response variables (and optimize the process to achieve a specific target for the mean and a minimal variation) the design has to be done at least 2 times (or more, depending on the certainty of measurement system, etc.)

If you're "only" interested in the vital factors which have an impact on the outcome (not the variation), it would be sufficient to do only 1 replication. As a rule of thumb in a screening experiment only 20% of the factors do have a real (significant) impact on the response (see Anderson & Whitcomb), so imho the number of replications can be set to 1 here (depending on the certainty of measurement system, etc.)

Quote:
In Reply to Parent Post by Ingo1966 View Post

Attached you will find a 2 Level L16 with 7 Control Factors and One Noise Factor.
(Response --> random number so just for Info)
Shouldn't that be a design with 8 control factors (A-H) and 1 noise factor (Noise) to be comparable to the original factor structure? Imho a fractional factorial design with 9 factors (2**(9-4), resolution IV) and 32 runs would be sufficient. And I would recommend to add some centerpoints for a curvature test (see attached xls-file with 4 centerpoints).

To separate time-dependent effects and factor-setting effects, the order should be randomised as much as possible (see first sheet in the excel-file "2^(9-4) randomised". If "Noise" is a hard-to-change factor (like in a split-plot design) and (really!) can't be changed for each experiment, the runs could be partly sorted (see second sheet in the excel-file "2^(9-4) partly randomised". If any kind of non-random structure is present in a design, the analysis has to be done with respect to this changes (e.g. a split-plot design is a nested design).

Ich hoffe wir haben Dich jetzt nicht komplett abgehängt, Hannes

Viele Grüße

Barbara
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File Type: pdf Taguchi Design V2 2011 08 10.pdf (2.10 MB, 112 views)
File Type: xls Alternative Designs 2011 08 10.xls (29.0 KB, 135 views)
Thanks to Barbara B for your informative Post and/or Attachment!
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