Weibull Analysis for Constant and Variant Stress Behavior Using the ALT Method for Single Stress and the Taguchi Method for Several Stress Variables

The article presents the advances on Weibull analysis which let practitioners to plan the data collection test, to fit the Weibull parameters directly from the applied stress values, to determine the capability indices cp and cpk, and to formulate the control charts to monitor the Weibull parameters. Thus, the theoretical background and formulation as well as numerical examples for constant and variant stress behavior are given. Also the references where the discussed methods were mathematically formulated and numerically applied are provided.


Introduction
Since the Weibull distribution is the best probability density function (pdf) used to model the behavior of a quadratic form [1,2], as they are any response surface model [3], the stress and strain matrix used in mechanical and structural analysis [4], the covariance matrix used in principal component analysis [5], and the branching process behavior [1]. Then an understanding of the Weibull distribution features is needed. Based on the above, this paper presents the advances on Weibull analysis to perform it from the planning data collection phase to the monitoring process phase. Therefore, the papers' structure is as follows. In section 2, the general new background of the Weibull distribution and the formulation to determine the sample size to perform a zero failure Weibull demonstration test plan are presented. In section 3, the formulation to determine the capability cp and cpk indices and the formulation of the control charts which let us to monitor the estimated Weibull parameters are given. In section 4, the formulation to fit the Weibull shape β and scale η parameters directly from the applied stress values is presented. In section 5 a numerical example for constant stress is given. In section 6 a numerical example for variant stress is given. Finally, in section 7 the conclusions are presented. The analysis is as follows:

Weibull distribution background and sample size
The two parameter Weibull distribution [6] is given by From Eq. (1), the Weibull reliability function is given by And since the Weibull distribution is generated by a nonhomogenous Poisson process (NHPP) [7] sec 4.3, then the Weibull risk function depends on the time also. Thus, the mean power function of the related NHPP in Weibull analysis is used as the cumulative hazard risk function ( ).
H t From Eq. (2), ( ). H t is given by In Weibull analysis Eq. (3) is of special interest because for the desired time t and known β value the effect that the stress variable s has on the estimated R(t) index is given by the linear relationship between η and s as In Eq. (4), α is a regression coefficient vector and Z is a nxp matrix which contains the k stress variables ( ) 1 2 , ,..., k s s s and p represents the k variables effects plus the possible effects generated among the k variables (e.g. interaction and quadratic effects). Also note from Eq. (2), that for known t and β values, the η value which corresponds to a desired R(t) index is completely addressed and it is given as On the other hand, because by setting H(t)=1 in Eq. (3) the sample size n which have to be tested to accurately estimate η , was derived in [8]. as Then, from Eq. (5 and 6) η in terms of n is given by Eq. (6) is too important in Weibull analysis because by using it in Eq. (7), theη value which corresponds to any feasible or R t values is completely determined without any experimentation or observed lifetime data. As an example suppose we desire to determine the η value which corresponds to R(t) =0 Hence, because the value estimated in Eq. (6) completely determine η , then this n value also represents the sample size which has to be tested without failures in order to accurately estimate the minimum η value for which the tested element will present at least the desired reliability index. As a simple example suppose we have to demonstrate a product fulfills with ( ) 0.95 Thus, because from Eq. (6) 19.4957, n = then 19 parts has to be tested by 1500hrs each and one part has to be tested by 0.4957(1500)=743.595hrs. It is to say, n in Eq. (6) is not a discrete value, instead it is continuous and it represents the times the desired time t has to be tested in order to demonstrate the tested product fulfills at least with the desired R(t) index. Now we know n in Eq. (6) represents the right sample size to design a zero failure Weibull demonstration test plan, let present the capability indices and the corresponding control charts.

Weibull capability indices and control charts
When the Weibull analysis is being performed for example in the quality field or in an improvement process, the related capability index cp and ability index cpk are of interest.
Fortunately by using the log-mean ( ) x µ and the log-standard ( ) x σ deviation parameters of the observed (or expected) failure times, as was demonstrated in [9] and [10], the Weibull cp and cpk indices can both be estimated. In particular, they are formulated based on the direct relationships between the Weibull β η parameters with the log x µ and parameters. These relationships are In Eq. (8) In Eq. (9) y µ and in Eq. (8) y σ are the mean and the standard deviation of the response vector given by In Eq.
Therefore, based on the y σ parameters given in Eqs. (8 and 9), the corresponding cp and cpk indices are In Eq. (10 and 11), USL and LSL are the upper and lower product's specifications limits measured in time units. And if they are unknown, then the minimum and maximum expected lifetimes of the Weibull analysis can be used to estimate them. These maximum and minimum lifetimes are estimated from Eq. (10) by using the n Y maximum element to estimate the maximum lifetime and by using the i Y minimum element to estimate the minimum lifetime as On the other hand, based on the facts that 2) The β and η are determined by the ( ) x σ [Eqs. 8 and 9].
Then in [12], the Weibull control charts to monitor β and η were formulated by setting ( ) x σ as the upper control limit to monitoring. And by setting ( ) x µ as the lower control limit to monitoring η. Here it is too important to highlight because represent the central parameters in logarithm scale, then in the case where we are analyzing several variables can be estimated by using the Taguchi method as it is made in [11]. And of course, once they had been already estimated, they can be used in Eqs. (8 and 9) to estimate the corresponding β and η parameters, as well as in Eqs. (12 and 13) to determine the corresponding capability indices; for details see [12]. Now let present how to estimate β and η directly from the applied stresses values.

Estimation of β and η directly from the stresses values
Since in Weibull analysis β and η completely determine the estimated ( ) R t index, and because β represents the aging of the analyzed system, then their accurate estimation is needed. Fortunately, as demonstrated in [4] both β and η can be directly estimated from the maximum and minimum applied stresses values. The estimation is made by using the eigen values of the analyzed quadratic form as the maximum and minimum applied stress values. As examples of quadratic forms suppose we are analyzing a biological phenomenon by using a response surface model given by Thus, its quadratic form is 11  λ of the quadratic form Q, the β value is given as

Numerical example for constant stress behavior
In this section two numerical example with constant stress behavior are presented. In the case of single stress variable, the temperature (T) is used as the stress variable. And for several stress variables the Taguchi method is used.

Analysis with a single stress variable
Therefore, from Eq. (9),

9.7136675
x µ = and from Eq.

Analysis with several stress variables
In this section, the analysis is based on [11]. The used data given in Table 2 was first published in [15]. The analyzed factors were the aluminum-alloy content in manganese (Mn) and magnesium (Mg) as well as hot mill pass counts (HMPC) and cold mill reduction rate (CMR). Mn, Mg, and CMR were measured in percentage units until dome yielded 1mm on a constantly applied external pressure of 500psi. From the Taguchi analysis ( [11] sec.3.3.2) the robust level is Mn=1.6, Mg=1.8, HMPC=35, and CMR=30. The found robust's log-parameters are 3.5442  On the other hand, because in [11] ( ) 0.90 R t = was used, then in this case µ y =-0.52311 and 1.115.34. represents the lower allowed value to be monitored. For details on how a Weibull analysis with several variables can be performed see [16]. Now let present the analysis for nonconstant stress variable.

Numerical example for variable stress behavior
In this section two numerical example with variable stress behavior are presented. In the first case data published in [4] is used to show how a product can be designed. In the second case data published in [17] is use to show how the ALT analysis can be used to analyze variable stress behavior. The analysis is as follows

Variable stress analysis focused on product's design
In this section the focus is on product's design. The analysis is performed following [4]. The objective consists on determining the ( ) . mpa η = Therefore the addressed Weibull family is W(1.984693, 96.7367). However, because in variable stress analysis, the stress variable instead of be a single value it is a range of stress values, then the addressed Weibull family only represents the stress distribution which models the stress range behavior as shown in Table 2 in [4]. As a consequence, in order to determine the ( ) R t index of the analyzed product, the Weibull distribution which models the strength of the product to overcome the applied stress has to be determined. In [4]   is ( ) 95.58.

R t =
It is to say a product designed with sy=400mpa performing in a variant stress range of [42.28, 127.72]mpa will present a ( ) 95.58.
On the other hand, although in variable stress analysis the cp and cpk indices defined in Eqs. (12 and 13) are not defined for the Weibull/Weibull stress/strength function yet, the related control charts can be used to monitor the stress and the strength Weibull parameters. Thus, should be used as the maximum value to monitor the common β value, and 4.29707726 should be used as the maximum value to monitor the stress η value. Also observe because here represents the stress distribution it was set as the maximum allowed value. On the other hand, when it represents the strength distribution, it should be set as the minimal allowed value. Finally, the value to monitor the strength S η value is determined by using The estimated value to be monitored as the minimal allowed stress value is 5.845891.
x µ = Now let present the stress variable case using ALT data.

Variable stress analysis by using ALT data
In this section the focus is to show how the ALT analysis can be used to analyze variable stress behavior. The used data was published in [17]. Data represents a set of 65 ball bearings tested at loads of 3500, 3800 and 4500 pounds. (Observe because the stress variable behavior by itself represents the normal stress behavior, then in the ALT analysis no extrapolation is needed). In the analysis the lognormal distribution with log parameters  Here it is too important to observe the addressed Weibull family only represents the behavior for fixed t=30000, if other t value is desired the above process has to be repeated [17]. Finally, by using the lognormal stress distribution and the Weibull strength distribution in the lognormal/Weibull stress/strength analysis [18], the corresponding ( ) R t index was determined. The found On the other hand, although the cp and cpk indices are not defined for the lognormal/Weibull stress/strength analysis yet, the monitoring process of the addressed stress and strength parameters can be performed. This can be done because both the lognormal and the Weibull distribution are based on the log parameters. Therefore the lognormal parameter 7.6 x µ = can be monitored by set 7.6 x µ = as the maximum allowed stress level in a control chart. Note it was set as the maximum because it represents the stress distribution. It is to say, because In the stress/strength analysis the higher the stress value the lower the R(t) index then it has to be set as the maximum allowed value. Similarly, the log normal parameter can be monitored by using 0.4 x σ = as the maximum allowed value also [18].
On the other hand, because in the monitoring process the variables which determine the values are the process variables which have to be monitored. And because in order to correctly monitor the contribution that each variable has on the observe values have to be determined, then their contribution have to be determined also. Fortunately because µ x is directly given from the determinant of the Q matrix as it is made in [4], then the decomposition method given in [19] can be used to determine the mentioned variable contributions. Also it is important to mention that the mathematical formulation to determine the value and the variables which determine its value, can be performed by partitioning the Q matrix as it is made in [20]. Finally the value to monitor 1.847127 β = is determined by using R t = was used. This is made in this way because in the stress/strength analysis, the stress distribution is independent of the strength distribution. Therefore, the generated stress/strength function only represents the effect that all possible stress values have over the all possible strength values [18].

Conclusion
In this paper the advances on the theoretical interpretation of the features of the Weibull distribution which let practitioners to perform an integral analysis from the test planning phase to the monitor phase are presented. This is made based on the fact the Weibull distribution efficiently model a quadratic form.  [4] that because this n value also represent the base lifetime of any Weibull analysis, then Eq. (6) can be used to determine the expected lifetimes after any reliability improvement process.
The mean of the expected logarithm of the lifetimes is completely determined by the logarithm of the square root of the determinant of the analyzed quadratic form Q.
Although the Weibull cp and cpk indices are estimated by using the logarithm parameters because they are dimensionless they are efficient.
By monitoring the log-parameters the β and η parameters are completely controlled.