There aren't any tidy equations or mathematical expressions describing these design techniques. They are based on heuristic computer programmed algorithms. For example, for Complete Enumeration, the algorithm is as follows:
1. Draw an initial random product concept (that doesn't violate any attribute prohibitions). This becomes the first concept in the first task for the first questionnaire version (block).
2. Examine all possible product concepts that can be used in the next concept position. For each possible product concept, tally up the resulting 1-way and 2-way level balance (including this and all previous concepts taken into the questionnaire at this point). Select the one product concept that results in the best 1-way and 2-way balance AND that doesn't violate prohibitions AND will tend to strongly favor no level repeating (level overlap) within the same task. There is a loss function we compute to differentially weight these goals (except violating an attribute prohibition is a hard stop constraint) and identify the best candidate concept.
Repeat step 2 until as many concepts, tasks, and versions (blocks) have been generated that the researcher requests.
Balanced Overlap does exactly the same thing as Complete Enumeration, except that the amount of penalty for having levels repeat within the same task (level overlap) is greatly reduced in the loss function. This allows a modest amount of level overlap within tasks (this modest overlap has been shown through simulated data to greatly improve the precision of interaction effects at only a modest loss in precision for main effects).
You will find that Complete Enumeration and Balanced Overlap designs are 1) near perfectly level balanced, 2) near perfectly orthogonal, and are near optimally D-efficient.
If the researcher has in mind a specific set of main effects and targeted subset of interaction effects, then a more targeted design that might be found in a catalogue or generated by a D-optimality search routine will probably have a slightly better D-efficiency than resulting from Sawtooth Software's designs. But, Sawtooth Software's designs have the benefit of supporting all potential main effects and first-order interactions (not just those hypothesized by the researcher prior to collecting the data).