A Note on the Construction of Near-Orthogonal Arrays With Mixed Levels and Economic Run Size

Consider a screening experiment for factors that may influence the properties of pulps prepared from woodchips by a variant of the kraft pulping process. The 10 factors in this experiment are (1) impregnation temperature (30”, 80”, or 12O”C), (2) chip steaming time (0 or 10 minutes), (3) initial concentration of alkali (6% or 12%) and (4) sulfide (2% or 10%) in the impregnation step, (5) impregnation pressure (30 or 120 psi) and (6) time (10 or 40 minutes), (7) ratio of pulping liquor to wood (3.5:1 or 6:1), (8) addition of anthraquinone (none or .OS%), (9) temperature of the main cook stage (160”or 170°C) and (10) final quenching of the cook with water (yes/no). The smallest orthogonal array (OA) found for 9 two-level factors and 1 three-level factor requires 24 runs (Wang and Wu 1991). The scientist wants to reduce the cost of this screening experiment and wishes to know which array to use when many fewer than 24 runs can be performed. The concept of near-orthogonal arrays (near-OA’s) (see Taguchi 1959; Tukey 1959; Wang and Wu 1992, hereafter called WW) provides a genuine answer to the preceding question. An OA (of strength 2) of size n with Ici si-level columns (‘i = 1,. . , T), denoted by L,(sf’, . . , s,k,.), is an rb x k: matrix (Ic = C ki) in which all possible combinations of levels in any two columns appear the same number of times (Rao 1947). In a near-OA L:, (.sf’, . . , s,“?), to reduce the run size the orthogonality of some pairs of columns is necessarily sacrificed. A near-OA in 12 runs suitable for the previously mentioned experiment was given in figure 3 of WW. The usefulness of arrays of this type is flexibility in the choice of factor levels and small run size. Perhaps, the best-known use of a near-OA is the one for the integrated circuit (IC) fabrication experiment performed by Phadke and his coworkers in collaboration with Taguchi at AT&T Bell Laboratories (Phadke, Kackar, Speeney, and Greico 1983). This article details a method for constructing near-OA’s in which the number of runs is divisible by the number of levels of each factor.