Algorithms for Constructing Ideal Gradings for Nonhoming Selectors (An Application of the Theory of Block Designs)

Algorithms are presented for designing high-efficiency gradings for nonhoming selectors and for expanding an existing grading by the addition of trunks or selectors with minimal rearrangement of the grading while maintaining its efficiency. The grading design problem is formulated as a problem in constructing block designs. The block design problem is shown to be equivalent to a nonlinear programming problem. The solution to the problem is a (partially) balanced incomplete block design that is a representation of a cyclic homogeneous grading. Flowcharts of algorithms and examples are given for solutions of variants of the programming problem corresponding to 1) the expansion of a grading by the addition of a selector, 2) the expan sion of a grading by the addition of a trunk, and 3) the optimization of a homogeneous grading.