Taguchi Robust Design

The underlying motive in Taguchi Robust Design is to improve the quality of a product or process by not only striving to achieve performance targets but also by minimizing performance variation.

In Taguchi Robust Design, parameters are classified using the following terminology:

  • Control Factors (x). These parameters can be specified freely by a designer. These are equivalent to design variables in optimization.

  • Noise Factors (z). These parameters are uncertain. The parameters are either not under a designer’s control, or their settings are difficult or expensive to control. Noise factors cause the response, y, to vary and lead to quality loss (performance variation).

    Examples include system wear, variations in the operating environment, and economic uncertainties.

  • Responses (y). These parameters are dependent performance characteristics. Responses are the system outputs and are functions of the control and noise factors.

The focus in robust design is to reduce the variation of system performance responses caused by uncertainty of noise factor values or to reduce system sensitivity. Solutions, which are system designs represented through settings of the control factors, are sought that minimize response variation and achieve performance targets (mean, μy, on target and minimized variance, σy2).

Taguchi’s parameter design, one implementation of robust design, is built on the foundation of statistically designed experiments (DOE). In this approach the evaluation of mean performance and performance variation is accomplished through a product array experimental design, constructed by “crossing” two arrays: a “control” array, designed in the control factors, and a “noise” array, designed in the noise factors.

An example product array is shown in the following figure:

Each array in the product array can be any designed experiment. In the previous figure, a two-level factorial design is depicted for both the control and noise array, as indicated by the “+” and “–” levels for each control factor (x1,x2,x3,) and each noise factor (z1,z2,). For each row of the control array (indicating one set of values of the control factors), response values are generated for each noise factor combination (each row of the noise array). For example, control array row 1 (run with noise row 1) leads to the response value y11, control row 1 (run with noise row 2) leads to response value y12, and so on. This experimentation strategy then leads to multiple response values for each set of control factor settings, from which a response mean, μyi, and variance or standard deviation, σyi, can be computed, as shown in the previous figure.

Given the mean and variance data for each control experiment, the experiments can be compared to determine which set of control settings best achieves “mean on target” and “minimized variation” performance goals. Alternatively, performance characteristics or metrics that combine the effects of mean performance and performance variation can be calculated and used to compare the set of designs represented by the control factor experiments. The performance characteristics used by Taguchi are the signal-to-noise ratio (shown as S/Nyi for each control experiment in the previous figure) and quality loss (measured using a loss function, Lyi in the previous figure).

The S/N ratio calculation depends on the particular response being investigated:

  • A response for which a specified target value is desired is categorized as a Nominal is Best response type.

    While Nominal is Best can support positive and negative response values, the analysis for this response type is traditionally and ideally intended for responses that take only positive values and when a positive, nonzero target is appropriate.

    Nominal is Best is also appropriate for cases in which the variance is proportional to the mean (the variance increases if the mean increases).

  • A response that can take positive and negative values, with a target value of zero, is categorized as a Zero Nominal is Best response type.

    Zero Nominal is Best is intended specifically for cases in which a response takes both positive and negative values and when the variance is independent of the mean (the variance does not increase if the mean increases).

  • A response for which low values are desired is categorized as a Lower is Better response type.
  • A response for which high values are desired is categorized as a Higher is Better response type.

The S/N ratios for these response types are typically formulated as described in the following table:

Response Type

S/N Ratio

 

Nominal is best:

101 og 10 μ 2 σ 2

0y

Zero nominal is best:

10log10(1Ve)

y

Lower is better:

10log10(1ni=1nyi2)

0y

Higher is better:

10log10(1ni=1n1yi2)

0y

In the S/N Ratio formulation for the Zero Nominal is Best case, Ve is the error variance, calculated as follows:

Total sum of squares:

ST=1nyi2

Sum of squares of mean:

Sm=n(y¯)2=(1nyi)2n

Sum of squares of error:

S e = S T S m

Error variance:

V e = S e n1

where n is the number of noise experiments.

The S/N ratio equations are formulated such that high values are desired (the control experiment with the highest S/N ratio value is considered the best set of factor settings of those included in the experiment), when the S/N ratio is the performance characteristic chosen for comparison. This formulation is consistent with the name of this performance characteristic (signal-to-noise ratio) because the effects of noise are reduced by increasing the ratio.

The Nominal is Best, Lower is Better, and Higher is Better S/N ratio equations given above are valid only for positive response values (0y), When response values are negative and positive, the Zero Nominal is Best case is recommended. If the Nominal is Best, Lower is Better, or Higher is Better response types are selected and negative values of a response are encountered in Isight, the following S/N ratio formulations are substituted for the standard formulations:

Response Type

S/N

 

Nominal is Best:

10log10[1n1i=1n(yiT)2]
y

Lower is Better:

10log10[1ni=1neyi/max|y|]
y

Higher is Better:

10log10[1ni=1neyi/max|y|]
y

For the Lower is Better and Higher is Better functions, when negative values are encountered the above formulations represent a modification of the standard S/N ratio formulation. No standard S/N ratio formulation is available for this case (again, the Zero Nominal is Best formulation is recommended) and the modified exponential formulation supports responses with both negative and positive values. In these modified formulations, the response values are normalized by the maximum absolute response value as shown (max|y|). This normalization is performed to prevent the exponential values from getting excessively large.

For the Nominal is Best and Zero Nominal is Best cases, the S/N ratio is typically used to identify factors that can help address the “minimize variation” goal of Taguchi Robust Design. Control factors that drive the S/N ratio are used to reduce the effect of noise on performance variation. To achieve the “mean on target” goal for the Nominal is Best and Zero Nominal is Best cases, another metric, called sensitivity, is used. Sensitivity is calculated as follows:

Response Type

Sensitivity

 

Nominal is Best:

10 log 10 ( 1 n ( S m V e ) )
y

Zero Nominal is Best:

μ y
y

Since Sm=(y)n2 and E(Ve)=σ2, sensitivity for the Nominal is Best case can also be written as

10log10(μy2σy2n)

Another performance characteristic used by Taguchi Robust Design, the loss function, is used to measure the “loss of quality” associated with deviating from a targeted performance value. The basic concept is shown in the following figure, where it is compared to the conventional specification-limit quality assessment approach.



With the conventional approach, lower and upper specification limits (USL and LSL in the previous figure) are used to define the acceptable quality range. All values within the limits are assumed to have no quality loss, all values outside the limits are defined as having 100% quality loss, and related parts are reworked or scrapped. The best example of this concept is dimensioning and tolerancing. If a part is to be manufactured to a length of 10 in, with a tolerance of ±0.1 in, parts inspected to be between 9.9 and 10.1 in will be accepted (no quality loss) and parts less than 9.9 or greater than 10.1 in will be rejected (100% quality loss). According to Taguchi Robust Design, however, loss of quality occurs gradually when moving in either direction from the target value, rather than as a sharp cutoff with the conventional approach. Therefore, quality loss is measured by the deviation from the target.

The standard form of the loss function L(y) is given as follows:

L(y)=k(yT)2

In this equation y is the quality characteristic, such as a dimension or performance parameter, T is the target value for the quality characteristic, and k is the loss constant. This loss function is derived from a Taylor series expansion with higher-order terms neglected. The loss constant is dependent upon the cost structure of a manufacturing process or an organization. Therefore, the quality loss is proportional to the deviation from a desired ideal target yT; it is zero at the target and increases quadratically with deviations in either direction from the target. The loss constant is a problem-specific, user-defined parameter. It is used to convert the squared deviation from the desired target to the appropriate measure of quality loss. For example, if possible, the loss constant can be defined as $/in2, and the quality loss is then measured as an actual monetary value (for the case of a manufactured dimension). Therefore, the loss constant is a scaling factor. Individual values for loss, for a given design, are generally meaningless without a meaningfully defined loss constant. However, relative values, between designs, are still meaningful. If no meaningful loss constant can be defined, k should be set to 1.0 (default). The loss values calculated for different designs (different cases defined by the control array, in the first figure) are compared, and the design with the lowest loss value is chosen as the best design (minimum quality loss, for the control points evaluated).

With the presence of noise factors, the loss can be calculated for each control/noise factor combination in the crossed product array, as shown in the first figure. The loss values must then be averaged across the noise factor cases, for each control factor case, to compare designs (control cases) and select the case with the minimum loss. The quality loss can also be calculated for the Lower is Better and the Higher is Better response types, as with the S/N ratio (the traditional loss equation given above obviously applies to the Nominal is Best case because a target value is desired).

The typical loss function formulations, when noise is present, are then given as follows:

Response Type

Loss Function

 

Nominal is Best:

1n1=1nk(yiT)2
y

Lower is Better:

1ni=1nkyi2
0y

Higher is Better:

1ni=1nk1yi2
0y

As with the traditional S/N ratio equations, the loss function equations given above for the Lower is Better and Higher is Better response types are valid only for positive response values (0y).

If negative values of a response are encountered in Isight, the following loss function formulations are substituted for the standard formulations, with the response values again normalized by the maximum absolute response value (as with S/N ratios):

Response Type

Loss Function

 

Lower is Better:

1ni=1nkeyi/max|y|
y

Higher is Better:

1ni=1nkeyi/max|y|
y

For all loss function formulations, low values are desired (loss of quality is always to be reduced where possible). The control experiment case with the lowest loss value is considered the best set of factor settings of those included in the experiment, when the loss function is the performance characteristic chosen for comparison.