Question 5 of 6: Robust Design Stages and QFD, Taguchi Methods, and Two-Way ANOVA on Emissions Data

Question 5: Robust Design Stages and QFD, Taguchi Methods, and Two-Way ANOVA on Emissions Data (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

System design is the first, creative engineering stage: applying scientific and engineering knowledge and prior experience to choose the basic technology, architecture, and components that will accomplish the product's required function — it decides WHAT the product fundamentally is, at a prototype/feasibility level, before any optimisation of settings or tolerances begins. Parameter design follows: given the chosen system, it selects the NOMINAL SETTINGS of the design's controllable factors so that performance is made as insensitive as possible to noise (manufacturing variation, environmental conditions, component aging, usage variation), achieving robustness purely through a clever choice of parameter LEVELS, without eliminating any noise source and without yet tightening tolerances — this is the stage where Taguchi's designed experiments (orthogonal arrays, signal-to-noise ratios) are applied, and is the most cost-effective of the three stages because it buys robustness essentially for free. Tolerance design is the final, most expensive stage, invoked only when parameter design alone cannot meet the performance target: it tightens the manufacturing tolerances (and/or upgrades to higher-grade, lower-variability components) on the specific factors shown to matter most, trading additional manufacturing cost for reduced variability.

Quality Function Deployment (QFD) is the structured methodology (the "House of Quality") that translates the VOICE OF THE CUSTOMER (qualitative wants and needs) into prioritized, measurable engineering characteristics, and cascades those characteristics down through product design, component/part specifications, process planning, and production requirements, so nothing gets lost in translation between what the customer wants and what the shop floor actually controls. Its role sits at the very front of, and feeds directly into, the system design stage: QFD is what determines WHICH functions and characteristics the system design must deliver and how heavily each should be prioritized (via its relationship matrices and competitive-benchmarking weights), before parameter design ever begins optimizing the settings of a system that QFD has already defined.

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

The Taguchi Methods are a structured, statistically-efficient experimental approach for parameter design: rather than a full factorial experiment over every controllable-factor combination, they use small, balanced fractional-factorial orthogonal arrays to estimate the main effects of many design factors economically, with the explicit GOAL of finding factor settings that minimize sensitivity to noise while hitting the performance target (robust design), rather than simply optimizing the mean response in isolation.

The signal-to-noise (S/N) ratio is Taguchi's single combined performance-and-robustness metric for each trial: rather than judging a trial on mean response alone, S/N folds mean and variance together (e.g. for "nominal-is-best," $S/N=10\log_{10}(\bar y^2/s^2)$; for "smaller-is-better" and "larger-is-better," analogous log-mean-square forms), so that the factor-level combination selected is the one that simultaneously performs well AND stays consistent under noise, not merely the one with the best average outcome.

The inner array is the orthogonal array of CONTROLLABLE design factors (the ones the engineer sets and can hold at chosen levels in production); the outer array is a separate, smaller orthogonal array of NOISE factors (the uncontrollable-in-production conditions — temperature, humidity, load, component tolerance stack-up) that are deliberately, artificially varied DURING the experiment so their effect can be measured. Every inner-array (control-factor) trial is run against every outer-array (noise) combination, so an S/N ratio can be computed for each control-factor trial across the full spread of noise conditions it was exposed to — this crossed inner/outer structure is precisely what lets the experiment optimize for robustness rather than just a single-noise-condition mean.

A linear graph is Taguchi's visual tool for assigning factors (and, where relevant, their interactions) to the columns of a chosen orthogonal array without violating the array's confounding structure — each array has a small set of published linear graphs showing which column pairs are already interaction-confounded, so factors needing to be estimated free of a particular interaction are assigned to non-confounded columns. The graph of marginal averages (main-effects plot) is the standard result-interpretation tool: for each factor, the mean response (or mean S/N ratio) is averaged across all trials at each of its levels and plotted level-by-level, so the level giving the best mean/S/N for each factor can be read off directly and the best overall combination selected level-by-level (rather than only from among the trials actually run).

Limitations of the Taguchi Methods include: orthogonal arrays are built to estimate MAIN EFFECTS efficiently but deliberately confound many two-factor interactions, so a design dominated by strong interactions can be badly misread by a main-effects-only analysis; the S/N ratio, by combining mean and variance into one number, can obscure which of the two is actually driving a result, and its statistical properties (distribution, appropriate significance testing) are less rigorously established than classical ANOVA; the method also requires the experimenter to correctly identify and include the relevant noise factors in the outer array up front — a noise source omitted from the experiment is never optimized against; and because the array is fractional, the number of runs saved comes at the cost of resolving fewer effects than a full factorial would.

Robust-design examples: a PRODUCT example is an automotive suspension bushing whose rubber compound formulation and geometry (controllable design parameters) are chosen via parameter design so that ride stiffness stays consistent across the temperature range and rubber-aging variation the vehicle will see in service (noise factors), rather than specifying a narrower, more expensive rubber-hardness tolerance to hold stiffness constant. A MANUFACTURING-PROCESS example is a wave-soldering process whose conveyor speed, preheat temperature, and solder-pot temperature (controllable process parameters) are optimized via a designed experiment so that solder-joint quality stays acceptable despite normal variation in incoming board thickness and ambient humidity (noise factors), instead of tightening incoming-board or solder-alloy purchase specifications to compensate.

(c) Two-way ANOVA (randomized complete block design): speed effect and temperature-block effect

Given. A single observation per speed×temperature cell (no replication) — speed (15, 20, 25 mph) is the treatment of interest, external temperature (20°F, 40°F, 60°F) is the block:

Find. Whether speed (treatment) has a significant effect on emissions, and whether temperature (block) has a significant effect, both at $\alpha=0.05$.

Approach. With one replicate per cell, run a two-factor ANOVA WITHOUT replication (randomized complete block design): partition $SS_{total}$ into $SS_{speed}$ (row/treatment), $SS_{temp}$ (column/block), and $SS_{error}=SS_{total}-SS_{speed}-SS_{temp}$ (the interaction term is confounded with error and cannot be separated with only one replicate, so it is assumed negligible), then form $F$-ratios against $F_{crit}(2,4,0.05)$.

1. Hypotheses. $$H_0^{(speed)}: \mu_{15}=\mu_{20}=\mu_{25}\ \ \text{vs.}\ \ H_1^{(speed)}:\text{ at least one speed mean differs};$$ $$H_0^{(temp)}: \mu_{20^\circ F}=\mu_{40^\circ F}=\mu_{60^\circ F}\ \ \text{vs.}\ \ H_1^{(temp)}:\text{ at least one temperature-block mean differs}.$$
2. Sums of squares. Grand mean $\bar{\bar y}=76.11$ over the 9 observations; row (speed) means $91.33,\ 73.33,\ 58.67$; column (temperature) means $87.33,\ 72.67,\ 63.33$. $$SS_{total}=\sum(y_{ij}-\bar{\bar y})^2=2510.22,\qquad SS_{speed}=b\sum(\bar y_{i\cdot}-\bar{\bar y})^2=3(\ldots)=1606.22,$$ $$SS_{temp}=a\sum(\bar y_{\cdot j}-\bar{\bar y})^2=3(\ldots)=878.22,\qquad SS_{error}=SS_{total}-SS_{speed}-SS_{temp}=2510.22-1606.22-878.22=25.78.$$
3. ANOVA table and $F$-tests ($df_{speed}=df_{temp}=2$, $df_{error}=(3{-}1)(3{-}1)=4$). $$MS_{speed}=\frac{1606.22}{2}=803.11,\quad MS_{temp}=\frac{878.22}{2}=439.11,\quad MS_{error}=\frac{25.78}{4}=6.44.$$ $$F_{speed}=\frac{803.11}{6.44}=\boxed{124.6},\qquad F_{temp}=\frac{439.11}{6.44}=\boxed{68.1},\qquad F_{crit}(2,4,0.05)=6.94.$$
4. Decision. Both $F_{speed}=124.6$ and $F_{temp}=68.1$ vastly exceed $F_{crit}=6.94$ (equivalently, $p<0.001$ for both), so BOTH null hypotheses are rejected: speed has a highly significant effect on CO emission, and the blocking variable (external temperature) also has a highly significant effect — confirming blocking on temperature was the right design choice, since it was itself a real source of variation the experiment needed to control for.