An algebraic and combinatorial approach to the construction of experimental designs

Experimental design is a well-known and broadly applied area of statistics. The expansion of this field to the areas of industrial processes and engineered systems has meant interest in an optimal set of experimental tests. This is achieved through the use of combinatorial and algebraic approaches. As such, the present study states the theoretical basis to construct and enumerate experimental designs using non-isomorphic mathematical structures in the form of matrix arrangements called orthogonal arrays (OAs). These entities are characterized by their number of rows, columns, entries (symbols), and strength. Thus, each different column could represent some measurable feature of interest (temperature, pressure, speed). The runs, expressed through OA rows, define the number of different combinations of a particular design. Similarly, the symbols allocated in OAs' entries could be the distinct levels of the phenomenon under study. During the OA construction process, we used group theory to deal with permutation groups, and combinatorics to create the actual OAs following a particular design. The enumeration process involved the use of algebraic-based algorithms to list all possible combinations of arrays according to their isomorphic equivalent. To test isomorphism, we used graph theory to convert the arrays into their corresponding canonical graph. The outcomes for this study are, firstly, a powerful computational technique to construct OAs from 8 to 80 runs; and secondly, additions in the published list of orbit sizes and number of non-isomorphic arrays given in [1] for 64, 72, and 80 runs.