Planning Experiments Orthogonal Arrays

An orthogonal array is an xm matrix ( = trials or experiments, m = factors) with s different levels (elements) in each factor so that any pair of columns has all the possible pairs of elements with the same frequency. Taguchi represented these matrices by the L (/ ) nomenclature. For instance, an L8(2 ) matrix describes 8 runs and has 7 columns. It is orthogonal because for any pair of columns the four combinations (1,1), (1,2), (2,1) and (2,2) appear with the same frequency. 

These designs are of general use because they are reasonably small and easily adaptable to different problems. Some of Taguchi s orthogonal matrices consider the case where not all factors have the same number of levels and they are denoted L (/ x z ), where 5 and t denote the number of levels decided for some of the factors. Hence we would say that m factors have s different levels, whereas u factors have t levels. Taguchi described 18 orthogonal matrices, 12 for those cases where all factors have the same number of levels and 6 for the opposite situation. These matrices are as follows  

If we analyse Table 2.7, we can observe that columns 1, 2 and 4 are those of a 2 factorial design (replace symbols [1,2] by [—1, +1] and see columns A, B and C in Table 2.1) and the remaining columns are those of the respective interactions in the algorithm. This is because the Taguchi matrix is in 

In addition to the L8(2 ) and L f,(2 ) orthogonal matrices, matrices L27(3 h and L32(2 ) are often used successfully. The L32 matrix is applied when the factors have two levels whereas the Lg and L27 matrices work with factors at three levels. 

Li8(2 X 3 ) is also a useful orthogonal matrix. It allows for the study of seven factors at three levels and a factor at two levels, and also the interaction of the latter with the factor situated in column 2, as displayed in Table 2.10. 

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