  We introduce systematic methods to create optimal designs for
order-of-addition (OofA) experiments, those that study the order in which $m$
components are applied---for example, the order in which chemicals are added to
a reaction or layers are added to a film. Full designs require $m!$ runs, so we
investigate design fractions. Balance criteria for creating such designs employ
an extension of orthogonal arrays (OA's) to OofA-OA's. A connection is made
between $D$-efficient and OofA-OA designs. Necessary conditions are found for
the number of runs needed to create OofA-OA's of strengths 2 and 3. We create a
number of new, optimal, designs: 12-run OofA-OA's in 4 and 5 components, 24-run
OofA-OA's in 5 and 6 components, and near OofA-OA's in 7 components. We extend
these designs to include (a) process factors, and (b) the common case in which
component orderings are restricted. We also suggest how such designs may be
analyzed.
