10.4 Tolerance Design and Statistical Tolerancing

Where tolerance design sits

Taguchi's three-stage design sequence puts tolerance design last, and the order is deliberate:

Tighten tolerances last. Tightening a tolerance adds cost to every unit ever produced -- better material, closer machining, more inspection, more scrap. Parameter design achieves robustness for free by choosing nominal values at which the response is flat with respect to the noise. A team that reaches for tighter tolerances before exhausting parameter design is buying with recurring cost what could have been had with design effort.

Worst-case tolerance stacking

For a linear stack, the assembly tolerance is the arithmetic sum of the component tolerances:

$$T_{assembly} = \sum_{i=1}^{n} |T_i|$$

Worked example. Four components stacked, each $\pm 0.10$ mm:

$$T_{assembly} = 4 \times 0.10 = \pm 0.40 \text{ mm}$$

Worst case guarantees that every assembly built from in-tolerance components will conform. Its cost is that it assumes all four components are simultaneously at the same extreme, which for independent components each centred with a Cpk of 1.33 is an event of vanishingly small probability.

Use worst-case when: the number of components is small, non-conformance is catastrophic or a safety issue, production volume is low, or component distributions are unknown or known to be non-normal and non-centred.

Statistical tolerancing: root sum of squares

Because variances add rather than standard deviations, independent variation combines by the root sum of squares:

$$T_{assembly} = \sqrt{\sum_{i=1}^{n} T_i^2}$$

Same example, four components at $\pm 0.10$ mm:

$$T_{assembly} = \sqrt{4 \times 0.10^2} = \sqrt{0.04} = \pm 0.20 \text{ mm}$$

Half the worst-case stack. Turned around, if the assembly requirement is $\pm 0.20$ mm, RSS permits each component to hold $\pm 0.10$ mm where worst-case would demand $\pm 0.05$ mm -- a doubling of the permissible component tolerance, which is a very large manufacturing cost difference.

Allocating a known assembly tolerance

With $n$ equal components and an assembly requirement $T_A$:

$$T_i = \frac{T_A}{\sqrt{n}}$$

For $T_A = \pm 0.30$ mm across six components: $T_i = 0.30/\sqrt{6} = \pm 0.1225$ mm each. Worst-case allocation would give $0.30/6 = \pm 0.05$ mm.

Unequal allocation is usually better in practice: give the loosest tolerances to the components that are expensive to control tightly, and the tightest to those that are cheap, subject to the RSS total.

Assumptions -- and what happens when they fail

The centring assumption is the one that fails most often in practice, and it is why statistical tolerancing must be paired with SPC on the component characteristics. RSS is a promise about variation that only holds if the components are actually controlled at nominal.

Where components are not centred, a modified RSS or a mean-shift model applies a correction factor, typically inflating the RSS result by 1.4 to 1.6, or an explicit shift allowance is added.

Six sigma tolerancing

If each of $n$ components is held to a capability $C_p$, the assembly capability under RSS is approximately

$$C_{p,\,assembly} \approx C_{p,\,component}$$

when tolerances are allocated by RSS -- which is the point. RSS allocation preserves the component capability at the assembly level, whereas worst-case allocation buys a far higher assembly capability than required and pays for it in every component.

Non-linear stacks and simulation

Real assemblies are frequently non-linear -- angular relationships, geometric dimensioning and tolerancing, kinematic constraints -- so a linear stack understates or overstates the result. Two approaches:

• Sensitivity (linearized) analysis: weight each component by its partial derivative, so $T_A = \sqrt{\sum (\partial f/\partial x_i)^2 T_i^2}$. This identifies which components actually matter, which is often surprising.
• Monte Carlo simulation: sample each component from its actual measured distribution, compute the assembly response many thousands of times, and read the resulting distribution directly. This handles non-linearity, non-normality, and correlation, and is the modern default when the model is available.

The Taguchi loss function

Tolerance design also rests on a different philosophy of what a tolerance means. The conventional "goalpost" view treats everything inside the limits as equally good. Taguchi's quadratic loss function treats any deviation from target as a loss to society that grows with the square of the deviation:

$$L(y) = k(y - T)^2, \qquad k = \frac{A_0}{\Delta_0^2}$$

where $A_0$ is the cost incurred when the deviation reaches $\Delta_0$, typically the specification limit.

Worked example. A characteristic has a target of 50.0 with limits at $\pm 2.0$, and a unit at the limit costs $80 to rectify. Then $k = 80/2^2 = 20$ per unit squared.

• A unit at 51.0 (half the tolerance away): $L = 20 \times 1.0^2 = \text{\textdollar}20$.
• A unit at 50.2: $L = 20 \times 0.04 = \text{\textdollar}0.80$.

Average loss over a population with mean $\mu$ and standard deviation $\sigma$:

$$\bar{L} = k\left[\sigma^2 + (\mu - T)^2\right]$$

Two consequences follow directly, and both are exam-relevant. Centring matters even inside the tolerance, because the $(\mu - T)^2$ term is a real cost. And reducing variation has value even when nothing is out of specification, because the $\sigma^2$ term falls. This is the formal justification for continuing to improve a process that is already producing zero defects, and it is the philosophical bridge between tolerance design and the rest of Six Sigma.