Question 5 of 6: Taguchi's Three Design Stages, the Taguchi Toolkit, and a Two-Factor ANOVA on CO Emissions

Question 5: Taguchi's Three Design Stages, the Taguchi Toolkit, and a Two-Factor ANOVA on CO 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) The three product-design stages, and where QFD fits

System design is the initial, creative engineering stage: choosing the basic technology, architecture, materials, and components that will accomplish the required function, drawing on scientific/engineering knowledge and prior experience — it decides WHAT the product fundamentally is. Parameter design chooses the NOMINAL SETTINGS of the design's controllable parameters so that performance is as insensitive as possible to uncontrollable noise (manufacturing variation, environmental conditions, usage patterns, component aging) — robustness achieved purely through clever parameter choice, without eliminating the noise itself and without yet tightening any tolerances. Tolerance design is the final, most expensive stage, used only when parameter design alone cannot meet the performance target: it tightens tolerances on the most sensitive parameters (identified during parameter design) and may specify higher-grade materials or components, trading increased production cost for reduced output variability.

The typical OBJECTIVES pursued during parameter design are: minimizing the mean-squared deviation of the response from its target (equivalently maximizing the signal-to-noise ratio) as the primary robustness goal; reducing how strongly noise-factor variation transmits into the response; and, once variability is minimized, using an available "adjustment factor" (a parameter that shifts the mean without materially affecting variability) to move the mean exactly onto target.

Quality Function Deployment (QFD) translates customer requirements ("voice of the customer") into specific, prioritized engineering characteristics via the House of Quality matrix, then cascades those characteristics down into part characteristics, process plans, and production requirements across a chain of related matrices. QFD is applied primarily during, and just before, the system-design stage: it establishes WHAT the design must achieve and which characteristics matter most before system-level configuration and technology choices are finalized, ensuring the concept selected in system design is already aimed at the attributes customers actually value, before parameter and tolerance design begin refining HOW well those characteristics are delivered.

(b) The Taguchi toolkit and its limitations

The Taguchi Methods are a structured, experimentally-driven approach to achieving robust parameter design: rather than eliminating sources of variation (expensive), the methods systematically search for controllable-parameter settings that make the response insensitive to UNCONTROLLABLE noise, using a compact, highly fractionated experimental plan and a single combined performance statistic to guide the search. They are used at the parameter-design stage of new product/process development to select robust nominal operating settings before tolerances are finalized.

The signal-to-noise (S/N) ratio is a single statistic computed from the replicated response at each experimental run that combines mean level and variability into one number to be maximized (e.g., for nominal-the-best, $S/N=10\log_{10}(\bar y^2/s^2)$; larger-the-better and smaller-the-better use their own forms). The inner array lists the settings of the CONTROLLABLE design parameters being tested (an orthogonal array such as an $L_8$ or $L_9$); the outer array lists settings of the UNCONTROLLABLE noise factors, replicated under every inner-array run so each design combination is exercised across the full range of noise conditions the product will actually see in the field. A linear graph is a diagram (dots for factors, connecting edges for interactions) used to assign factors and interactions to the columns of a standard orthogonal array without introducing unwanted confounding — a graphical column-assignment aid. A graph of marginal (row) averages plots the average S/N ratio (or average response) at each level of each controllable factor, letting the analyst read off, factor by factor, which level maximizes robustness — this is how the optimal parameter-design settings are ultimately selected.

The main limitations are: the S/N ratio conflates mean and variance into a single number, which can be statistically inefficient and can obscure which one actually moved (analyzing mean and $\log(s^2)$ separately is often more diagnostic); the highly fractionated orthogonal arrays frequently confound interactions heavily with main effects, so an apparently significant "main effect" may actually be a disguised interaction; and the associated analysis (marginal averages, ANOVA on S/N ratios) is less statistically rigorous/efficient than a standard factorial-design analysis of the same raw data.

Examples of robust design: for a PRODUCT, an automotive suspension bushing's rubber-compound hardness and geometric dimensions can be chosen so ride quality stays consistent across the wide temperature range (noise factor) the vehicle experiences, rather than tightening the rubber's manufacturing tolerance; for a MANUFACTURING PROCESS, a wave-soldering process's conveyor speed, preheat temperature, and flux density can be chosen so solder-joint quality stays high despite normal, uncontrolled variation in incoming PCB thickness and component lead spacing (noise factors), rather than imposing tighter incoming-material tolerances on every supplier.

(c) Two-factor ANOVA: speed and temperature on CO emissions

Given. A $3\times3$ factorial (Speed $\in\{15,20,25\}$ mph $\times$ Temperature $\in\{20,40,60\}\,{}^{\circ}\text{F}$), $r=2$ replicates per cell, response = CO emissions (g/vehicle-mile).

Find. Test $H_0$: no Speed effect, no Temperature effect, no Speed$\times$Temperature interaction, each at $\alpha=0.05$; identify which effects are significant.

Approach. Standard fixed-effects two-way ANOVA: partition the total sum of squares into Speed, Temperature, Interaction, and Error components, form the corresponding mean squares, and compare each $F$ ratio to the tabulated $F_{0.05}$ critical value (Appendix V, this paper).

1. Grand mean and total sum of squares. $N=18$ observations, $\bar{\bar y}=75.33$ g/mi. $$SS_T=\sum(y_{ijk}-\bar{\bar y})^2=\boxed{3884.0}.$$
2. Row (Speed) sum of squares ($b=3$ temp. levels, $r=2$ reps, $a=3$ speed levels): $$SS_{\text{Speed}}=br\sum_i(\bar y_{i\cdot\cdot}-\bar{\bar y})^2=\boxed{2250.33},\quad df=a-1=2.$$
3. Column (Temperature) sum of squares: $$SS_{\text{Temp}}=ar\sum_j(\bar y_{\cdot j\cdot}-\bar{\bar y})^2=\boxed{1361.33},\quad df=b-1=2.$$
4. Interaction and error sums of squares. Cell-means SS $=r\sum_{ij}(\bar y_{ij\cdot}-\bar{\bar y})^2=3695.67$; interaction is the remainder after removing the two main effects, and error is the remainder within cells: $$SS_{\text{Speed}\times\text{Temp}}=3695.67-2250.33-1361.33=\boxed{84.33},\quad df=(a-1)(b-1)=4;$$ $$SS_E=SS_T-SS_{\text{cells}}=3884.0-3695.67=\boxed{188.0},\quad df=ab(r-1)=9.$$ (Check: $2250.33+1361.33+84.33+188.0=3884.0=SS_T$. ✓)
5. Mean squares and $F$ ratios. $$MS_{\text{Speed}}=2250.33/2=1125.17,\quad MS_{\text{Temp}}=1361.33/2=680.67,\quad MS_{\text{Int}}=84.33/4=21.08,\quad MS_E=188.0/9=20.89.$$ $$F_{\text{Speed}}=\frac{1125.17}{20.89}=\boxed{53.86},\qquad F_{\text{Temp}}=\frac{680.67}{20.89}=\boxed{32.59},\qquad F_{\text{Int}}=\frac{21.08}{20.89}=\boxed{1.01}.$$
6. Compare to $F_{0.05}$ critical values (Appendix V). Main effects: $F_{0.05,2,9}=4.26$. Interaction: $F_{0.05,4,9}=3.63$. $$F_{\text{Speed}}=53.86\gg4.26\ (\text{reject }H_0,\ \text{significant}),\qquad F_{\text{Temp}}=32.59\gg4.26\ (\text{reject }H_0,\ \text{significant}),$$ $$F_{\text{Int}}=1.01<3.63\ (\text{fail to reject }H_0,\ \text{not significant}).$$

Conclusion. Both speed and external temperature have a statistically significant effect on CO emissions (each $p<0.001$; emissions fall as either speed or temperature increases, from a maximum of $104$ g/mi at 15 mph/20°F to a minimum of $50$ g/mi at 25 mph/60°F), and the two factors act ADDITIVELY — there is no evidence ($p=0.45$) that the effect of speed depends on temperature or vice versa, so the two main effects can be interpreted and reported independently of one another.