Question 5 of 6: Product/Process Design Stages, Taguchi Methods, and a Randomized-Block ANOVA for Vehicle Emissions

Question 5: Product/Process Design Stages, Taguchi Methods, and a Randomized-Block ANOVA for Vehicle Emissions (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; where QFD fits

System design is the stage where engineering and scientific knowledge is applied to develop the basic technology, configuration and prototype that will do the job — largely a matter of engineering judgment about which concept to pursue. Parameter design determines the best levels of the design's controllable factors so that performance is made as insensitive as possible to uncontrollable "noise" (environmental variation, manufacturing variation, degradation) — the stage where designed experiments (orthogonal arrays) are used, and, crucially, one that can improve robustness without increasing unit cost, by exploiting nonlinearities in the system rather than tightening tolerances. Tolerance design is used only if parameter design alone cannot deliver the required performance: it selectively tightens the tolerance on whichever parameters the system is most sensitive to, which generally does raise cost (better components, tighter processes) and is therefore treated as a last resort rather than a default.

Quality function deployment (the "house of quality") translates the customer's stated and unstated requirements ("voice of the customer") into prioritized engineering characteristics and specifications. It is applied primarily during the system design stage — at the concept/product-planning level — so that the basic technology and configuration chosen are already targeted at what customers actually value, before parameter and tolerance design refine how well that concept is realized.

(b) Taguchi methods

The Taguchi methods use designed experiments to find the combination of controllable-factor levels that makes product or process performance robust to noise, rather than trying to eliminate or control the noise itself. The signal-to-noise (S/N) ratio is a single performance statistic, in decibels, that folds both "hits the target" and "stays consistent" into one number to be maximized — e.g. nominal-the-best $S/N=10\log_{10}(\bar y^2/s^2)$, smaller-the-better $S/N=-10\log_{10}\big(\overline{y^2}\big)$, larger-the-better $S/N=-10\log_{10}\big(\overline{1/y^2}\big)$. The inner array is an orthogonal array of the controllable design factors whose levels the engineer is choosing between. The outer array is a (typically smaller) orthogonal array of noise factors, crossed with every inner-array run, so each candidate factor-level combination is evaluated under several different noise conditions and reduced to one S/N value. The linear graph is a graphical tool (dots for main-effect columns, connecting lines for interaction columns) used to assign factors, without violating confounding rules, to the columns of a standard orthogonal array (e.g. $L_8$, $L_9$). The graph of marginal (average) S/N plots the average S/N at each level of each factor and is the basis for selecting the final parameter set — picking, for each factor independently, the level with the highest average S/N.

Limitations: the crossed inner/outer-array approach implicitly assumes factor effects are mostly additive and does not explicitly test for interactions, which it can therefore miss; the single S/N number can obscure whether an apparent improvement came from a shifted mean or a reduced variance, and the choice of which S/N formula to use is itself somewhat subjective; the standard Taguchi orthogonal arrays are often highly confounded (low resolution) compared to modern response-surface or D-optimal designs, and large crossed inner$\times$outer arrays can require many experimental runs. Robust-design examples: for a product, choosing a power-supply circuit topology and component-value set whose output voltage transfer function has low sensitivity to component-tolerance drift and input-voltage fluctuation, rather than specifying expensive tight-tolerance parts; for a process, selecting injection-molding parameters (melt temperature, injection speed, packing pressure) so that a critical part dimension stays within spec despite the normal batch-to-batch variation in incoming resin.

(c) Randomized complete block design: speed effect and temperature block effect on CO emissions

Given. A single observation per (speed, temperature) cell — speed (15, 20, 25 mph) is the treatment of interest, external temperature (20, 40, 60°F) is the blocking factor:

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

Approach. This is a randomized complete block design (RCBD) with $a=3$ speed levels and $b=3$ temperature blocks, one observation per cell (no replication), so the total sum of squares splits three ways: $SS_{total}=SS_{speed}+SS_{block}+SS_E$, with $SS_E$ obtained by subtraction (the interaction and error are confounded when there is no replication, so $SS_E$ is used as the error term for both $F$-tests).

1. Sums of squares. Grand total $=677$, grand mean $\bar{\bar y}=677/9=75.222$. $$SS_{total}=\sum(y_{ij}-\bar{\bar y})^2=\boxed{3263.56},$$ $$SS_{speed}=b\sum_i(\bar y_{i\cdot}-\bar{\bar y})^2=3\Big[\big(\tfrac{283}{3}-75.222\big)^2+\big(\tfrac{220}{3}-75.222\big)^2+\big(\tfrac{174}{3}-75.222\big)^2\Big]=\boxed{1996.22},$$ $$SS_{block}=a\sum_j(\bar y_{\cdot j}-\bar{\bar y})^2=3\Big[\big(\tfrac{272}{3}-75.222\big)^2+\big(\tfrac{217}{3}-75.222\big)^2+\big(\tfrac{188}{3}-75.222\big)^2\Big]=\boxed{1213.56},$$ $$SS_E=SS_{total}-SS_{speed}-SS_{block}=3263.56-1996.22-1213.56=\boxed{53.78}.$$
2. Degrees of freedom and mean squares. $df_{speed}=a-1=2$, $df_{block}=b-1=2$, $df_E=(a-1)(b-1)=4$, $df_{total}=8$. $$MS_{speed}=\frac{1996.22}{2}=998.11,\qquad MS_{block}=\frac{1213.56}{2}=606.78,\qquad MS_E=\frac{53.78}{4}=13.44.$$
3. $F$ statistics and hypothesis tests, $\alpha=0.05$. $$F_{speed}=\frac{MS_{speed}}{MS_E}=\frac{998.11}{13.44}=\boxed{74.24},\qquad F_{block}=\frac{MS_{block}}{MS_E}=\frac{606.78}{13.44}=\boxed{45.13}.$$ Both are compared against $F_{0.05,2,4}=6.94$ (Appendix table, $v_1=2$, $v_2=4$). Since $74.24\gg6.94$, reject $H_0:$ "speed has no effect on emissions" — speed has a highly significant effect. Since $45.13\gg6.94$ as well, reject $H_0:$ "temperature has no effect" — the block (temperature) effect is also highly significant, confirming that blocking on temperature was the right design choice (it removed a real, large source of variation from the error term rather than a negligible one).