SC QUALITY GUILDTaguchi loss function and signal-to-noise ratio
SCQUALITYGUILD.COM/TOOLS/TAGUCHI-LOSS-FUNCTION.HTML
Put a cost on being off target. Enter the tolerance and what a unit costs at its edge; get the loss constant k, the loss for any unit, the average loss of your process split into variation and off-target parts, and signal-to-noise ratios to compare settings.
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Q-BotClick for an explanation of this tool.For educational use. This page is a study aid. The method behind it has far more depth and detail than one page and one tool can cover, and what you see here is not everything the method offers or everything this site offers. For full understanding and application, use the governing standards and texts, and seek further education, training or qualified advice. Results depend on the data you enter, so check them before relying on them.
The idea
The traditional view of quality is the goalpost: a part inside the specification is good and costs nothing, a part outside is bad. Genichi Taguchi argued that this is wrong. A part just inside the limit performs almost exactly like a part just outside it, and both perform worse than a part on target. Loss to the customer and to society starts as soon as a characteristic leaves its target and grows the further it goes.
He modeled the loss as a parabola, L(y) = k(y − T)2. The constant k comes from one known point: if a unit at the edge of the tolerance, T ± Δ, costs A to scrap, rework or replace, then k = A / Δ2.
Three kinds of characteristic
- Nominal is best (a diameter, a voltage): L = k(y − T)2, k = A / Δ2.
- Smaller is better (wear, noise, impurity; the ideal is zero): L = k y2, k = A / Δ2, where Δ is the value at which the loss reaches A.
- Larger is better (strength, life): L = k / y2, k = AΔ2.
Average loss of a process
Averaged over production, the nominal-is-best loss becomes k[σ2 + (μ − T)2]. The two terms are the two ways to reduce it: cut the variation, or move the mean onto target. Centering is usually cheap; reducing variation usually needs a more robust design or process. The tool shows how much of the loss comes from each. (Texts differ on whether σ2 is estimated with n or n − 1; with n, the formula equals the exact average of the per-unit losses.) For smaller is better it is k(σ2 + μ2).
Signal-to-noise ratios
In a Taguchi experiment, each run (a combination of control-factor settings) is measured several times under the noise factors the product will meet in use. The S/N ratio, in decibels, rolls the mean and the spread into one figure, and larger is always better:
- Nominal is best: S/N = 10 log10(ȳ2 / s2)
- Smaller is better: S/N = −10 log10(mean of y2)
- Larger is better: S/N = −10 log10(mean of 1/y2)
For nominal is best the optimization has two steps: pick the settings with the highest S/N, which makes the process insensitive to noise, then use a factor that moves the mean without changing S/N to put it on target. That is robust design: making the product or process work well despite variation it cannot control, instead of trying to control everything.
On the exam
The CQPA Body of Knowledge asks you to identify and describe Taguchi concepts: the quality loss function, robustness, controllable and uncontrollable factors, and the signal-to-noise ratio (III.F.4). The CSSBB covers robust design and tolerance design (IX.C). Expect to calculate k from a cost at the tolerance limit, the loss for one unit, or the average loss of a process from its mean and standard deviation, and to say which S/N ratio fits a characteristic.