The Construction of Optimal Design for Order-of-Addition Experiment via Threshold Accepting

The objective of the order-of-addition (OofA) experiment is to find the optimal addition order by comparing all responses with different orders. Assuming that the OofA experiment involves $$m (\ge 2)$$ components, there are m! different orders of adding sequence. When m is large, it is infeasible to compare all m! possible solutions (for example, $$10!\approx 3.6$$ millions). Two potential construction methods are systematic combinatorial construction and computer algorithmic search. Computer search methods presented in the literature for constructing optimal fractional designs of OofA experiments appear rather simplistic. In this paper, based on the pairwise-order (PWO) model and the tapered PWO model, the threshold accepting algorithm is applied to construct the optimal design (D-efficiency for the present application) with subsets of size n among all possible size m!. In practical, the designs obtained by threshold accepting algorithm for $$4 \le m\le 30$$ with $$n=m(m-1)/2+1, m(m-1)+1, 3m(m-1)/2+1$$ respectively are provided for practical uses. This is apparently the most complete list of order-of-addition (OofA) designs via computer search for $$4 \le m\le 30$$ in the literature. Their efficiencies are illustrated by a scheduling problem.