Analyzing Dominance Move (MIP-DoM) Indicator for Multiobjective and Many-Objective Optimization

Dominance move (DoM) is a binary quality indicator that can be used in multiobjective and many-objective optimization to compare two solution sets obtained from different simulations. The DoM indicator can differentiate the sets for certain important features, such as
convergence
,
spread
,
uniformity
, and
cardinality
. DoM does not require any reference point or any representative Pareto solution set, and it has an intuitive and physical meaning, similar to the
$\epsilon $
-indicator. It calculates the minimum total move of members of one set so that all elements in another set are to be dominated or identical to at least one member of the first set. Despite the aforementioned desired properties, DoM is hard to calculate, particularly for higher dimensions. There is an efficient and exact method to calculate it in biobjective problems. This work proposes a novel approach to calculate DoM using a mixed-integer programming (MIP) approach, which can handle two sets with two or more objectives and is shown to overcome the issue of information loss associated with the
$\epsilon $
-indicator. Experiments in the biobjective space are done to verify the model’s correctness. Furthermore, other experiments, using 3-, 5-, 10-, 15-, 20-, 25-, and 30-objective problems, are performed to show how the model behaves in higher dimensional cases. Algorithms, such as IBEA, MOEA/D, NSGA-III, NSGA-II, and SPEA2, are used to generate the solution sets; however, any other algorithm can also be used with the proposed MIP-DoM indicator. Further extensions are discussed to handle certain idiosyncrasies with some solution sets and improve the quality indicator and its use for other scenarios.