Generally, response surface methodology (RSM) assumes that the response follows normal distribution with independent and homoscedastic errors (Box and Hunter 1957; Box and Draper 2007). RSM is frequently used in lifetime improvement (or quality engineering) experiments for locating the optimal level combinations to attain the specific target (Nair
The article considers four lifetime distributions such as lognormal, gamma, exponential, Weibull and to be added. Suppose the lifetime (
Suppose
where
where
If
For gamma lifetime (
where Γ(
where
For the lifetime (
where
In the following section, it is shown that for a real lifetime data set, the lifetime distribution is important for locating the optimal level combinations from its fitted second-order model with constant variance. The article is organized as follows. The following section presents a motivating example, while Sections 3 and 4 present correlated second-order model and rotatability conditions, respectively. Section 5 presents robust second-order rotatable designs, which is then followed by the conclusion.
Firth (1988) has shown that the regression parameter estimates from the log-normal, or the gamma model have almost identical results with constant variance. In this section, a real lifetime experimental data set is presented which has different regression parameter estimates for these two models with constant variance. For non-constant variance, regression parameter estimates for these two models may be different which has been shown by Das and Lee (2009).
Watkins
The lognormal and gamma model analyses of the above data set have been derived by Das and Park (2012), where the estimates of the two models are not similar, and also the model selection criteria (based on Akaike information criterion (AIC)) are not same. For reference, both analyses results are reproduced in Table 1. From Table 1, it is clear that the regression parameter estimates are not the same in both the models. In addition, the lognormal model gives a better fit than the gamma based on AIC. It is clear that
First-order models are often used if the experimenters are remote from the optimum operating conditions of the process. The direction of steepest ascent indicates the direction in which the estimated response (
For a lifetime
where
where
The model (
considering
For the deriving model (
As in (
For the gamma distribution as in (
For the lognormal lifetime distribution as in (
Note that for the considered four lifetime distributions, the second-order response surface model is same as in (
For the second-order correlated lifetime model (
where
Note that
Following Das (1997), the necessary and sufficient conditions of second-order rotatability for the model (
(I)
(i)
(ii)
(iii) (1)
(2)
(3)
(4)
(II)
(i)
(ii)
(iii)
(III)
(i)
(ii)
(IV)
The non-singularity condition of correlated second-order rotatable design is
(V)
where
Following Das (1997), the variance of the estimated response at
which is a function of
and
Note that a design ‘
The present section derives the necessary and sufficient conditions for second-order rotatability following Das (1997) for lognormal, gamma, exponential, andWeibull lifetime distributions under intraclass and inter-class error variance-covariance structure
The simplest dispersion matrix, or errors variance-covariance structure, is known as ‘intra-class’ or ‘uniform’ correlation structure. This dispersion structure occurs if any two observational errors have the identical correlation coefficient (
where
For the second-order model (
The necessary and sufficient conditions of a robust second-order rotatable design (for all the four considered lifetime distributions) under the intra-class error structure (
(I)
(II) (i)
(III)
(IV)
(V)
The above conditions (I)–(V) as in (
Inter-class error dispersion matrix, or errors correlation structure is often observed in practice. For instance, products from different furnaces, or in batch products, where it is noted that observations are grouped into some batches such that within each batch there is the same intra-class correlation and between batches there is no correlation. Let ‘
where ⊗ denotes Kronecker product,
For all the considered four lifetime distributions with second-order model (
(I) (i)
(ii)
(iii) (1)
(2)
(3)
(4)
(II) (i)
(ii)
(iii)
(III) (i)
(ii)
(IV)
(V)
where
For all the considered four lifetime distributions with model (
It can be readily examined that (
For a non-singularity condition, the following design parameters can be noted.
Using these design parameters, the non-singularity condition
which is the non-singularity condition for each set of observations of ‘
For all the considered four lifetime distributions with model (
Let us consider a matrix
Suppose
Obviously, in this design construction method, a factor point (obtained either from the appropriate fraction of a 2
The RSORDs obtained by the Method II satisfy condition (I) of (
and they are the same for
Note that
and they are the same for 1 ≤
Note that ‘
or, 3
Thus, with
Using the above design parameters and noting
Thus, the non-singularity condition (V) in (
Therefore, the designs described by Method II satisfy the non-singularity condition (
Note that the above Method II for constructing RSORDs is quite powerful as the original design
Using the BIB design for Method II, let us consider an experimental design with seven factors, eight independent groups, each of fifteen observations i.e.,
Group totals for
Note that the design points within each group do not satisfy the usual SORD conditions, but the overall design forms a RSORD under the inter-class structure (
The present article considers the univariate response lifetime random variable having distributions such as gamma, lognormal, Weibull and exponential, which are often observed in practice. Regressor variables are considered in lifetime models for handling heterogeneity. First-order lifetime location-scale regression models with constant variance are extended to second-order correlated models. Mixed linear logarithms design of lifetime correlated models are derived from two random components, whereas one component is associated with the response lifetime distribution and the other is the error term. Second-order rotatable designs are derived under these four lifetime distributions with errors having intra and inter-class structures. For the intra-class error structure, usual SORDs are robust SORDs and invariant for these four lifetime distributions. For inter-class structure, two design construction methods have been suggested which give invariant robust SORDs.
Lifetime data sets can be homogeneous, or heterogeneous with lognormal, or gamma distributed (Das and Lee, 2009; Das and Park, 2012; Das, 2013; Das, Kim and Park, 2015). In industrial design of experiments, experimental units can not be treated as independent, as they produce correlation among observations via a repeated measures scenario as in split plot design (Myers
Group 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | 1 | 1 | −1 | −1 | 1 | 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | −1 | −1 | 1 | 1 | 1 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | |
1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | |
Group2 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | −1 | −1 | 1 | 1 | 1 | 1 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | |
1 | −1 | −1 | 1 | 1 | −1 | −1 | 1 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | |
1 | 1 | −1 | −1 | −1 | −1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | |
Group3 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | |
1 | −1 | −1 | 1 | 1 | −1 | −1 | 1 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | |
1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | |
−1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | |
Group4 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
−1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −2 | 0 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | |
1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | |
1 | 1 | −1 | −1 | −1 | −1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | |
Group5 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | 1 | 1 | −1 | −1 | 1 | 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | |
1 | 1 | −1 | −1 | −1 | −1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | |
−1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | |
Group6 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
−1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −2 | 0 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | −1 | −1 | 1 | 1 | 1 | 1 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | |
−1 | 1 | 1 | −1 | 1 | −1 | −1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | |
Group7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
−1 | 1 | −1 | 1 | −1 | 1 | −1 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | 1 | 1 | −1 | −1 | 1 | 1 | 0 | −2 | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | |
−1 | −1 | −1 | −1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | |
Group8 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | 1 | 1 | −1 | −1 | 1 | 1 | 0 | −2 | 0 | 0 | 0 | 0 | 0 | |
−1 | −1 | −1 | −1 | 1 | 1 | 1 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 | 0 | 0 | 0 | |
1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −2 |
The developed designs in the article can be used to predict the optimal operating conditions that achieves the target mean value, while reducing the variance. Dual response surface (DRS) approach was introduced by Myers and Carter (1973) for this purpose, whereas second-order response surface designs are used for separate modeling of mean and variance. For this purpose, the joint generalized linear models (JGLMs) have been proposed by Nelder and Lee (1991). Therefore, for both the DRS and JGLMs approaches, the present developed designs can be used to attain the specific target mean value, and simultaneously reduce the process variability.
Group | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
2 | −2 | 2 | −2 | 2 | −2 | 2 | −2 | |
2 | 2 | −2 | −2 | 2 | 2 | −2 | −2 | |
2 | −2 | −2 | 2 | 2 | −2 | −2 | 2 | |
2 | 2 | 2 | 2 | −2 | −2 | −2 | −2 | |
2 | −2 | 2 | −2 | −2 | 2 | −2 | 2 | |
2 | 2 | −2 | −2 | −2 | −2 | 2 | 2 | |
2 | −2 | −2 | 2 | −2 | 2 | 2 | −2 |
Results for mean models of threading machine lifetime data from log-normal and gamma fits
Log-normal model | Gamma model | ||||||||
---|---|---|---|---|---|---|---|---|---|
Covar. | est. | s.e. | est. | s.e. | |||||
Mean | const. | 4.10 | 0.11 | 37.46 | 4.37 | 0.11 | 39.73 | ||
−0.84 | 0.11 | −7.66 | −0.84 | 0.11 | −7.55 | ||||
0.48 | 0.11 | 4.32 | 0.47 | 0.11 | 4.24 | ||||
0.44 | 0.11 | 4.01 | 0.42 | 0.11 | 3.83 | ||||
−0.18 | 0.11 | −1.66 | 0.10 | −0.14 | 0.11 | −1.28 | 0.21 | ||
Model | 0.04 | 0.11 | 0.38 | 0.71 | 0.07 | 0.11 | 0.59 | 0.55 | |
−0.36 | 0.11 | −3.30 | −0.31 | 0.11 | −2.76 | 0.01 | |||
0.22 | 0.11 | 1.96 | 0.05 | 0.18 | 0.11 | 1.65 | 0.10 | ||
−0.19 | 0.11 | −1.71 | 0.09 | −0.17 | 0.11 | −1.58 | 0.12 | ||
Var model | const. | −0.44 | 0.20 | −2.19 | 0.03 | −0.54 | 0.20 | −2.64 | 0.01 |
AIC | 599.0 | 604.4 |