Question 5 of 6: Concurrent Engineering, Robust Design, and a $2^{5-2}$ Fractional Factorial

Question 5: Concurrent Engineering, Robust Design, and a $2^{5-2}$ Fractional Factorial (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) QFD in product design, and concurrent vs. traditional design

Quality Function Deployment (QFD), structured through the "House of Quality," systematically translates the voice of the customer (qualitative wants, ranked by importance) into prioritized, measurable engineering characteristics, and then cascades those characteristics down through part design, process planning, and production/control planning. Its role at the product design stage is to make sure design effort is spent on the characteristics that actually matter to the customer (rather than what is easiest to engineer), to surface and resolve conflicting requirements early (the House of Quality's "roof" explicitly maps engineering characteristics against each other), and to benchmark the design against competitors on the customer's own priorities before hard tooling and process decisions are locked in.

The traditional (sequential) approach designs the product first, in isolation, then "throws it over the wall" to manufacturing/process engineering, and then to quality, each function working on a completed hand-off from the one before with little feedback until a problem is discovered. Concurrent engineering instead runs product design, process design, and quality/reliability engineering as parallel, cross-functional activities from the start, with continuous feedback loops (manufacturability and quality concerns raised while the design is still fluid, not after it is frozen). The result is far fewer late design changes, because manufacturability and quality problems are caught and resolved while they are still cheap to fix, which shortens overall time-to-market even though the individual design phase may take a little longer, and produces a design that is simultaneously easier to build and more robust in the field — exactly the outcomes a purely sequential process structurally cannot achieve, since by the time downstream functions see the design, its major decisions are already fixed.

(b) Parameter/tolerance design's three phases, robust design, and inner/outer arrays

Taguchi's system design process has three phases. System design is the initial, largely engineering-science-based conception of the product/process — choosing the basic technology, architecture, and materials to achieve the required function, using prototyping and engineering knowledge rather than statistics. Parameter design then optimizes the nominal (target) settings of the design's controllable factors so that performance is as insensitive as possible to uncontrollable "noise" — without necessarily spending money to eliminate the noise itself — using designed experiments (often orthogonal-array/Taguchi experiments) rather than tightening tolerances. Tolerance design is the final, most expensive phase, undertaken only if parameter design alone cannot reach the required performance: it selectively tightens tolerances (better, more costly materials/components/processes) on just the factors shown to matter most, rather than tightening everything uniformly.

Robust design is the objective that parameter design pursues: making product/process performance insensitive to sources of variation (manufacturing variation, environmental conditions, component aging, customer-use variation) by the choice of nominal factor settings, rather than by removing the variation itself — because removing the noise (tighter tolerances, controlled environments) is almost always far more expensive than finding a design point where the noise simply does not matter as much.

Robust design experiments are structured with two arrays. The inner array contains the controllable design/process parameters (the factors the engineer can set and will specify in the final design) run as a designed (often fractional-factorial or orthogonal-array) experiment. The outer array contains the noise factors (uncontrollable in the field — humidity, supply-voltage variation, load, customer misuse) deliberately varied during the experiment so their effect can be observed; each inner-array run is repeated across every combination of the outer array, and the results are analyzed (classically via a signal-to-noise ratio) to find the inner-array (controllable) setting that gives both a good average response and minimum sensitivity to the outer-array (noise) variation.

(c) $2^{5-2}$ fractional factorial: generators, aliases, effects, and recommendations

Given. Eight runs of a $2^{5-2}$ design in $A,B,C,D,E$ with responses: $a=8$, $abd=27$, $ace=28$, $abcde=21$, $bc=10$, $be=12$, $cd=11$, $de=24$.

Find. The design generators, the defining relation, the aliases of the five main effects, the effect estimates, which effects are significant (via a normal probability plot), and recommendations with stated assumptions.

Approach. Treat $A,B,C$ as the three "base" factors of a full $2^3$ (8 runs) and find which generators $D=\pm AB$/$\pm AC$/etc. and $E=\ldots$ reproduce every one of the eight given run labels exactly; the fact that all eight labels ARE reproduced (not merely a plausible subset) confirms the correct generator choice, since a wrong guess would fail to match at least one label.

1. Find the generators. Testing $D=AB$, $E=AC$ against the three base factors' $2^3$ treatment combinations reproduces the run labels exactly: $(1)\to de$, $a\to a$, $b\to be$, $ab\to abd$, $c\to cd$, $ac\to ace$, $bc\to bc$, $abc\to abcde$ — this is precisely the given list of eight labels (in a different order), so $$\boxed{D=AB,\qquad E=AC}.$$
2. Defining relation. From $D=AB$: $I=ABD$. From $E=AC$: $I=ACE$. Their generalized interaction is $ABD\cdot ACE=A^2BCDE=BCDE$, so $$I=ABD=ACE=BCDE.$$ The shortest word has length 3, so this is a Resolution III design: main effects are aliased with two-factor interactions.
3. Aliases of the main effects. Multiplying each main effect by every non-identity word in the defining relation (mod 2 exponents): $$A\equiv A\cdot ABD=BD\equiv A\cdot ACE=CE,\qquad B\equiv B\cdot ABD=AD,\qquad C\equiv C\cdot ACE=AE,$$ $$D\equiv D\cdot ABD=AB,\qquad E\equiv E\cdot ACE=AC.$$ So $\boxed{A\equiv BD\equiv CE,\ \ B\equiv AD,\ \ C\equiv AE,\ \ D\equiv AB,\ \ E\equiv AC}$ (each also aliased with a 4- or 5-factor interaction, assumed negligible).
4. Effect estimates. With $A,B,C,D,E$ coded $\pm1$ and $D=AB$, $E=AC$ verified against every run, each effect $=\tfrac{2}{8}\sum(\text{sign})(y)$: $$\bar y=17.625,\qquad \boxed{A=6.75},\qquad B=-0.25,\qquad C=-0.25,\qquad \boxed{D=6.25},\qquad \boxed{E=7.25}.$$
5. Normal probability plot and significant effects. Ranking the five effects and plotting each against its normal order-statistic score (figure below): $B$ and $C$ ($-0.25$ each) sit essentially on top of one another near the origin, consistent with pure noise, while $D$ (6.25), $A$ (6.75), and $E$ (7.25) fall well off that line — clearly separated, large, and of one sign. $\boxed{A,\,D,\,E\text{ are significant}}$; $B$ and $C$ are not distinguishable from noise.

Recommendations. Since $A$ (condensation temperature), $D$ (condensation time), and $E$ (amount of material 2) all have large positive effects, yield is increased by running all three at their high ("$+$") tested level; $B$ (amount of material 1) and $C$ (solvent volume) can be set at whichever of their two tested levels is cheapest or most convenient, since neither materially affects yield over the tested range. A confirmation run at $A^+D^+E^+$ (with $B,C$ at the economical level) should be run to verify the predicted improvement before committing to the change in production.