The Optimize button invokes an algorithm that is conceptually similar to a class of covering array optimizers sometimes referred to as post-construction randomized optimizers (Nayeri et al., 2013). However, JMP’s algorithm differs from most in that it also addresses designs with constraints. In particular, it optimizes constrained covering arrays as well as unsatisfiable, constrained covering arrays.
The algorithm assumes that the design to be optimized is a covering array of the specified strength. For a K factor design of strength t, the algorithm iteratively examines all KCt factor projections to determine whether runs can be eliminated or merged. Consequently, as K or t increases, the run time of the algorithm quickly escalates. To improve performance, the JMP implementation is threaded to use as many CPU cores as are available on your workstation.
uCv is the number of combinations of u things taken v at a time
t is the strength of the design
K is the number of factors
M =  KCt
i = 1, 2, ..., M is an index that orders all combinations, or projections, of t factors
vik is the number of levels for the kth factor
ni is the number of distinct t tuples in the design for the ith projection
pi is the product of the vik for the factors in the ith projection
r is the number of runs in the design
In a constrained design, certain t tuples are not allowed. This can result in missing values for some t tuples. For some combinations of t factors, there might be no valid t tuples whatsoever. Coverage and diversity must be defined in terms of the possible valid combinations. For this reason, the formulas for constrained designs require additional notation:
ai is the number of invalid t tuples arising from factors in the ith projection
m is the number of projections where there are no valid t tuples
qi is the number of runs in the design with missing values for any factor in the ith projection
ri = r - qi
M’ = M - m
If there are no invalid t tuples (M’ = M) and if there are no missing values (ri - r, for all i), then the definitions for coverage and diversity for constrained designs reduce to the definitions for unconstrained designs. For details, see Morgan (2014).