Weibull and lognormal Taguchi analysis using multiple linear regression

Based on the additive property of the normal distribution, the normal distribution has been widely used to designing, monitoring, controlling and optimizing processes when the aging and wearing factors are not significant. Examples of tools based on the normal distribution are experimental design of the K-series and the Taguchi method (both are based on the ANOVA), six sigma and response surface methodology, the reliability factor estimated in mechanical design (K_R=1-0.08Z_a), the control charts (mean and standard deviation), measurement system and capability and stress-strength analysis. However, when the aging and wearing factors are significant, then the Weibull (damage cumulates in additive form/Non-homogeneous Poison process) and the lognormal (damage cumulates in multiplicative form/Geometric Brownian motion) have to be used. Unfortunately, because the Weibull distribution does not has the additive property (for different stress variables or variables combination, the Weibull shape parameter is not constant), the above mentioned tools cannot be directly applied to the Weibull distribution.

Thus the Significance of this paper can be seen in almost three points of view.

1. Scientific significance: It is given by the fact that with the addressed direct relation between the Weibull parameters with the observed mean and standard deviation given in eq.(41 and 46), we can pass from the Weibull distribution to the normal one (in logarithm scale) or vice versa; and as a consequence all the normal tools, as the mentioned above, could be generalized to the Weibull analysis. And in the fact that eq.(45) estimates depending only on the desired reliability index R(t), the right sample size n (not lower nor large) to estimate in accurate form the Weibull family of each row of the used experiment design. And on the formulas given in eq.(48a-48f) which let perform the anova analysis of the desired Weibull family (as could be the optimum or robust family) based only on both the desired R(t) index and the expected Weibull parameters (or its expected failures times). And on the fact that the Taguchi’s prediction polynomial can be used as a life-stress model in the reliability analysis, for either the normal and/or ALT analysis. Finally, in the fact that by using the proposed method of section 3.3.1, we determine, based on the observed Weibull families of the used orthogonal array (OA) (into the OA, each row presents its own Weibull family), the Weibull family that represents either the robust, optimal or desired variables setting. (note that this last Weibull family is determined regardless the Weibull distribution does not have the additive property). 

2. Practice significance: With the paper content, researchers and practitioners can perform accurately the Taguchi analysis for any Weibull variables. They will estimate the Weibull parameters based only on the observed mean and standard deviation of the expected or observed data. They can determine in accurate form the anova analysis only based on the Weibull parameters (or expected failure data). They can perform the normal or accelerated analysis for Weibull processes subjected to several variables by using the Taguchi method, and by using the Taguchi´s prediction polynomial as a life-stress model. They can perform the robust analysis, the tolerance design and or the parameter design for any Weibull process, and used the mean and standard deviation of the observed data (or derived from the expected Weibull parameters) to designing, monitoring and controlling the Weibull process. And since the estimated n in eq.(45) is unique and it only depends on the desired reliability index (R(t)), n could be used to formulate the standards to perform a Weibull demonstration test plans for either the normal operational or accelerated level with zero failures or with a specific percentage of failures. 

3. Generation of knowledge significance: From the paper content, since the addressed relationships between the Weibull parameters with the mean and standard deviation of the observed data always hold, these relations can be used by researches to determine methods to generalize the normal tools (as the above mentioned) to the Weibull analysis, and to generalize the findings of this paper to the extreme value distributions which are closed related to the Weibull distribution. Under the stochastic point of view, we believe that for experts in this field, the fact that the addressed n depends only on the reliability index R(t), which in turns depends only on the cumulative risk function H(t), and since the Weibull is determined by a non-homogeneous Poisson process, it will be possible to use this n, maybe as the number of shocks, to formulate a Weibull cumulative damage model, where the critical threshold can be the cumulated damage that determines R(t). In mechanical design where the strength is generally Weibull (Bearings) and the stress is lognormal, the relationships between the Weibull and lognormal distribution given in eq.(41 and 46) can be used to determine a Weibull/lognormal reliability factor to determine the final endurance limit (fatigue limit) that fulfills with a desired reliability index. Finally, since now is possible to perform the stress-strength analysis for normal/normal and lognormal/lognormal distributions, then since eq.(41 and 46) let us pass from the Weibull to the lognormal and vice versa, then it seem be possible, based on these relations, create a methodology for the Weibull/Weibull, Weibull/lognormal and lognormal/Weibull stress-strength methodology.