Ranking of Pareto-optimal solutions and selecting the best solution in multi- and many-objective optimization problems using R-method

This paper presents a new multi-attribute decision-making (MADM) method, named as R-method, for ranking of Pareto-optimal solutions and selecting the best solution in multi- and many-objective optimization problems. The compromise among the optimization objectives is different for each Pareto-optimal solution and, hence, the solution that has the best compromise among the objectives can be considered as the best solution. The proposed R-method is used to identify such best compromise solution. The method ranks the objectives based on their importance for the given optimization problem and ranks the alternative solutions (i.e. Pareto-optimal solutions) based on their data corresponding to the objectives. The ranks assigned to the objectives and the ranks assigned to the alternative solutions with respect to each of the objectives are converted to appropriate weights and the final composite scores of the alternative solutions are computed using these weights. The final ranking of alternative solutions is done based on the composite scores. The steps of the proposed method are described along with a pseudocode. Three examples are considered to demonstrate and validate the proposed method. The first example contains 4-objectives and 50 alternative solutions, the second example contains 6-objectives and 30 alternative solutions, and the third example contains 3-objectives and 25 alternative solutions. The results of the proposed method are compared with those of the other widely used MADM methods for the three examples considered. Also, the proposed method is compared with four well-known ranking methods to demonstrate its rationality in assigning weights to the ranks of the objectives and the alternative solutions. The proposed method is comparatively easier, more logical, and can be used for choosing the best compromise solution in multi- and many-objective optimization problems.