Optimal additions to and deletions from two-level orthogonal arrays

Consider the problem of selecting a two-level factorial design. It is well known that two-level orthogonal arrays of strength 4 or more with "e" extra runs have various optimality properties including generalized Cheng (type 1) optimality when "e"=1, restricted Cheng (type 1) optimality when "e"=2 and "E"-optimality when 3⩽"e"⩽5. More general Schur optimality results are derived for more general values of "e" within the more restricted class of augmented two-level orthogonal arrays. Similar results are derived for the class of orthogonal arrays with deletions. Examples are used to illustrate the results and in many cases the designs are confirmed to be optimal across all two-level designs. Copyright 2007 Royal Statistical Society.