Multiple response surface optimization with correlated data

Abstract

Setting of process variables to meet the required specification of quality characteristics is an important problem in the process quality control. There are often several conflicts in quality characteristics, which should be simultaneously satisfied. These types of problems are called “Multiple Response Optimization” (MRO). When quality characteristics are correlated, MRO problems may become increasingly difficult. In design of experiments, identifying covariates effects could reduce error and uncovered variances as well as give more insight about the process. This study aims to identify process variables to consider correlated covariates and correlated quality characteristics. It also accommodates dispersion effects and specification limits as well as location effects in a unified framework based on desirability functions. The features of the proposed method are investigated and the results are compared with some existing techniques by applying two numerical examples.

Appendix: Principal Component Analysis

Principal component analysis was first introduced by Pearson [26] and then developed by Hotelling [27]. PCA is a multivariate technique for forming new uncorrelated variables which are linear composites of the original variables. The maximum number of new variables that can be formed is equal to the number of original variables. Based on Kaiser’s study [3] the components with an eigen-value greater than one are chosen to replace the original variables for further analysis. Assume that there are P original variables; PCA generates P uncorrelated linear combinations as follow:

$$ p{{c}_1} = {{w}_{{11}}}{{x}_1} + {{w}_{{12}}}{{x}_2} + \cdots {{w}_{{1p}}}{{x}_p} $$

(44)

$$ p{{c}_2} = {{w}_{{21}}}{{x}_1} + {{w}_{{22}}}{{x}_2} + \cdots + {{w}_{{2p}}}{{x}_p} $$

(45)

$$ \vdots $$

$$ p{{c}_p} = {{w}_{{p1}}}{{x}_1} + {{w}_{{p2}}}{{x}_2} + \cdots + {{w}_{{pp}}}{{x}_p} $$

(46)

where pc₁, pc₂,···, pc _p are the p principal components and w _ij is the weight of the jth variable for the ith principal component. The weights, w _ij , are estimated such that:

• 1. The principal components are created in order to decreasing variance, and therefore the first principal component accounts for most variance in the data, second new variable accounts for the maximum of the variance that is not accounted for by first new variable and so on.

• 2.
  $$ \matrix{ {w_{{i1}}^2 + w_{{i2}}^2 + \cdots + w_{{ip}}^2 = 1} &{i = 1, \ldots, p} \\ }<!end array> $$

  (47)
• 3.
  $$ \matrix{ {{{w}_{{i1}}}{{w}_{{j1}}} + {{w}_{{i2}}}{{w}_{{j2}}} + \cdots {{w}_{{ip}}}{{w}_{{jp}}} = 0} &{for\,all\,i \ne j} \\ }<!end array> $$

  (48)

The Condition (2) is used to fix the scale of the new variables and is necessary because it is possible to increase the variance of a linear combination by changing the scale of the weights. The condition (3) ensures that $W = {{\left( {{{w}_{{ij}}}} \right)}_{{p \times p}}}$ is an orthogonal matrix (Sharma [28]).

More than one correlated quality characteristic is usually considered in a manufactured product. PCA is an effective method for determining a small number of uncorrelated variables which account for the main sources of variation in such a set of correlated quality characteristics.