Question 5 of 6: Product/Process Design Stages, Taguchi Methods, and a Fractional Factorial Experiment

Question 5: Product/Process Design Stages, Taguchi Methods, and a Fractional Factorial Experiment (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) System, parameter, and tolerance design; the role of QFD

Taguchi's three-stage framework separates product/process design into a deliberate sequence of increasingly expensive decisions. System design is the conceptual/engineering stage: using scientific and engineering knowledge to choose the basic technology, architecture, and materials that will meet the functional requirement — it selects what the product or process fundamentally is. Parameter design chooses the nominal (target) settings of the design's controllable factors so that performance is as insensitive as possible to uncontrollable noise (environmental variation, manufacturing variation, customer usage variation) — the central Taguchi idea that robustness can often be achieved by clever choice of nominal levels alone, at essentially no material cost. Only after parameter design has reduced sensitivity to noise as far as it economically can does tolerance design tighten the manufacturing tolerances (and, where needed, use higher-grade materials or components) on the factors that remain sensitive — the most expensive lever, because tighter tolerances directly raise unit cost, so it is deliberately used last and sparingly.

Quality function deployment (QFD) is the structured methodology (built around the "House of Quality" matrix) that translates the voice of the customer into prioritized engineering characteristics, and cascades those characteristics down through part characteristics, process parameters and production requirements. Its role in product design is to make sure system design begins from a rigorously captured, prioritized set of customer requirements rather than engineering assumption — it is applied primarily at the system design stage (and cascaded again as system design's outputs feed the more detailed part/process House of Quality matrices), ensuring that what gets carried into parameter and tolerance design is the right set of characteristics to make robust in the first place.

(b) Taguchi Methods: description, S/N ratio, arrays, linear graphs, marginal averages, limitations, robust-design examples

The Taguchi Methods are a structured, statistically-based approach to parameter design: use a designed experiment (often an orthogonal array) that varies both the product/process's controllable factors and a representative sample of the noise it will actually face, then choose the controllable-factor settings that maximize robustness (minimum sensitivity to noise) before, separately, adjusting a factor with negligible noise-interaction to bring the mean onto target. It is used whenever performance depends on factors that cannot be fully controlled in the field or on the shop floor, and the goal is to find design settings that make the product perform consistently despite that uncontrolled variation, rather than trying to eliminate the noise itself.

The signal-to-noise (S/N) ratio is a single log-scale performance statistic, computed for each experimental run from its noise-array replicates, that combines mean level and variability into one number to be maximized regardless of whether the ideal response is a target value ("nominal-is-best," $S/N=10\log_{10}(\bar y^2/s^2)$), a minimum ("smaller-is-better," e.g. a particle-contamination or defect-count response), or a maximum ("larger-is-better"); maximizing S/N is equivalent to minimizing the Taguchi quadratic loss, which is why it is the analysis target rather than the raw mean. The inner array is the orthogonal array of controllable (design) factors under the engineer's control; the outer array is a second, typically smaller orthogonal array of noise factors (temperature, humidity, supply variation, usage pattern) that is run in full for every single inner-array row, so each design combination is tested across the realistic range of noise it will face and an S/N ratio can be computed for that row. A linear graph is a graphical aid (a set of dots for factors and connecting lines for interactions) used to assign factors and their interactions to the columns of a standard orthogonal array without violating the array's confounding structure — it tells the experimenter which column to use for a given interaction once two factors have been placed. The graph of marginal averages (main-effects plot) shows, for each factor, the average response (or average S/N ratio) at each of its levels, connected by a line; its slope and direction show which factors most affect the response/S/N and which level of each is preferred, and is the standard visual tool for selecting the final parameter-design settings.

Limitations of the Taguchi Methods include: the S/N ratio, being a single aggregated statistic, can mask information available from the individual mean and variance separately, and its "nominal-is-best" formula is not statistically appropriate under all noise models; the recommended orthogonal arrays are typically saturated, resolution-III designs, so main effects are aliased with two-factor interactions and an apparently significant main effect may actually be (or be confounded with) an interaction, which can mislead the parameter choice if interactions are actually important; the method's two-step (robustness-then-mean-adjustment) procedure assumes an "adjustment factor" with negligible interaction with noise actually exists, which is not always true; and the philosophy's engineering statistics (S/N transformations, marginal-average analysis) have been criticized by mainstream statisticians as often less efficient, and sometimes less rigorous, than a properly designed and analyzed classical factorial/response-surface experiment that explicitly separates mean and variance effects and estimates interactions.

A product-robustness example: designing an electronic circuit's resistor network so the output voltage is insensitive to the normal manufacturing tolerance and temperature drift of the individual resistors (choosing nominal resistor values/topology so small resistor-to-resistor variation barely moves the output) rather than specifying every resistor to an expensive tight tolerance. A process-robustness example: choosing a wave-soldering process's conveyor speed, preheat temperature and flux density so that solder-joint quality stays acceptable across the day-to-day variation in incoming board thickness and ambient humidity, instead of trying to control board thickness and humidity themselves.

(c) Completing and analyzing the fractional factorial contamination experiment

Given. A 6-factor (A–F), 8-run experiment; the design matrix and response $Y$ (particles/million) as legibly printed for six of the eight runs, with runs $i$ and $v$ having every factor setting unreadable:

Find. The missing settings for runs $i$ and $v$; the seven independent contrasts (main effects A–F plus one unused/error contrast) obtainable from the 8 runs; and which effects are significant.

1. Completed table. Applying $D=BC$, $E=ABC$, $F=AB$ to the two missing corners:

   Run	A	B	C	D	E	F	Y
   i	−	+	+	+	−	−	16
   v	+	+	+	+	+	+	21

2. Effect estimates. With 8 runs and $\pm1$ coding, each factor's effect is $\text{Effect}=\dfrac{1}{4}\sum(\text{sign}_i\times Y_i)$. The design uses only 6 of the 7 available orthogonal contrasts (A, B, C, $D{=}BC$, $E{=}ABC$, $F{=}AB$); the seventh, unused contrast ($AC$) is computed the same way and serves as a pseudo-error/noise reference: $$\text{Effect}_A=-5.75,\quad \text{Effect}_B=+0.75,\quad \text{Effect}_C=+1.25,\quad \text{Effect}_D=-1.75,$$ $$\text{Effect}_E=+5.75,\quad \text{Effect}_F=+4.25,\quad \text{Effect}_{AC(\text{unused})}=+0.75,\qquad \bar Y=\boxed{18.375}.$$
3. Normal probability plot of the seven contrasts. Ordering the seven effects and plotting each against its normal order-statistic median score:
   Normal probability plot of the seven contrasts. A, E and F (red) sit well off the line formed by B, C, D and the unused AC contrast (blue), which cluster near zero as expected of noise.
   Lenth's pseudo-standard-error from the same seven contrasts is $PSE=1.5\times\text{median}|effect|_{trimmed}=2.625$; A, E and F each exceed $1.5\,PSE=3.94$ by a wide margin, while B, C, D and the unused $AC$ contrast are all $\leq1.75$ — comfortably inside noise. A, E and F are the significant effects; B, C and D are not distinguishable from experimental noise at this sample size.
4. Interpretation and aliasing. Because this is a resolution-III design, each main effect is aliased with 2-factor interactions of the other factors: $A\equiv A+BF+DE$, $E\equiv E+AD+CF$, $F\equiv F+AB+CE$. Since B, C and D's own main effects are all small, the standard effect-heredity argument (a large 2-factor interaction is rare without at least one sizeable parent main effect) supports attributing the three large contrasts to the main effects of A, E and F themselves, not to their aliased interactions. The signs give a direct recommendation: $\text{Effect}_A<0$ (raising A from $-$ to $+$ reduces particle count), while $\text{Effect}_E,\text{Effect}_F>0$ (raising E or F increases particle count) — so the setting that minimizes contamination is $A^+,E^-,F^-$. This exactly matches run $viii$ ($A{+}{,}E{-}{,}F{-}$, the observed minimum, $Y=12$) and is exactly contradicted by run $vii$ ($A{-}{,}E{+}{,}F{+}$, the observed maximum, $Y=27$), reinforcing the significance call.