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46 August 31, 2006 2:54 The Essence of Design for Six Sigma
A typical desired outcome of a reliability test is to claim that, with 100(1 − α)%
confidence, at most p % will fail by a certain time τ. Statistically, the largest value for
τ is the lower 100(1 − α) % confidence limit of the pth percentile of the time to failure.
For exponential failure time, this is given by
2r ˆ θ
TTF p,LCL = 2 (− ln(1 − p)) ,
χ
α,2r+2
where r is the number of failures in a time-censored test, ˆ θ is the estimate of θ
2
and χ α,2r+2 is the upper αth percentile of a chi-square statistic with 2r + 2 degrees of
freedom. If a reliability specification in terms of τ is given, then the lower confidence
limit obtained from the test must be more than τ. Since ˆ θ = TTT/r, where TTT is the
total time on test, we have
2
τχ α,2r+2
TTT ≥− .
2 ln(1 − p)
As a result, a high reliability requirement in terms of large τ or small p will require a
long test time and large sample size. For the same product tested under higher stress,
the same reliability requirement will translate into smaller τ and the test duration
can be shortened accordingly. In other words, a time compression can be achieved,
commonly refered to as the acceleration factor. This factor can be computed from
the model that relate the time to failure to the stress level. For example, the most
common model when the applied stress is temperature is the Arrhenius model, and
the acceleration factor is given by

E a 1 1
AF = exp − .
k T 0 T s
−1
where E a is the activation energy, k is the Boltzmann constant (8.617 × 10 −5 eV K ),
and T 0 and T s arethetemperatureinKelvinunderuseandstressconditionrespectively.
Putting these together, the total test time at higher stress level is then given by
2
τχ α,2r+2
TTT ≥− .
2AF ln(1 − p)
A simple reliability test plan that specifies the sample size and the test condition (say,
higher temperature) can then be derived from the above expression.
Subsequent to setting the tolerance, checking feasibility, and performing capability
and reliability assessment and the associated sensitivity studies, the result will be
verified against customers’ requirements. The final deliverables are a set of well-
conceived tolerance setting, the reliability and capability scores.

4.2.4 Validate
Themaintasksofthisphasearetoconductconfirmatoryrunswithappropriatesample
sizes and to prepare the design for production. The purpose of validation of the per-
formance of the selected design is to ensure that the key performance indices are
achievable under mass production. After validation, control plans will be put in place
to ensure that the KPOVs of the design are within acceptable limits. For example, to
validate a target process capability index (C pk ), an approximate sample size formula