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Use of polynomial indicator functions to enumerate fractional factorialdesigns with given properties is first introduced by Fontana, Pistone andRogantin (2000) for two-level factors, and generalized by Aoki (2019) formulti-level factors. In this paper, we apply this theory to enumeratecross-array designs. For the experiments of several control factors and noisefactors, use of the cross-array designs with direct product structure iswidespread as an effective robust strategy in Taguchi method. In this paper, werelax this direct product structure to reduce the size of the designs. Weobtain 24-runs cross-array designs without direct product structure with somedesirable properties for 6 control factors and 3 noise factors, each withtwo-levels, instead of 32-runs design that is widely used.

Use of indicator functions to enumerate cross-array designs without direct product structure