# SSTNDS: A SET-Based Non-Dominated Sorting Algorithm for Multi-Objective Optimization Problems
## 1 Introduction  
In many practical living and production activities, we often need to make decisions through weighing the pros and cons of multiple objectives, but they always conflict with each other. Manufacturers need to select the optimal production program to make a tradeoff between the cost of production, time and quality and so on, which can be formalized as a multi-objective optimization problem (MOP) [1]. Evolutionary algorithms are more suitable to solve the MOPs than traditional optimization algorithms, as a set of optimal solutions can be found running at once [2]. Various kinds of multi-objective evolutionary algorithms (MOEAs) have been proposed over the past few decades to deal with MOPs; e.g., MOEA/D [3], SPEA 2 [4], NSGA II [2] and its improved version NSGA III [5], etc.  
As described in [2], non-dominated sorting, used to find pareto solutions or assign solutions to different fronts, is a key but time-consuming process in multi-objective evolutionary algorithms (MOEAs). It is very meaningful to improve the computational efficiency of non-dominated sorting to improve the whole efficiency of MOEAs. Studying existing non-dominated sorting algorithms, we come to the conclusion that there are two strains of thought to find the front each solution belongs to. A fast non-dominated sorting algorithm (FNDS) [2] is a representative of the first method. In thismethod, solutions which are not dominated by others are recognized as non-dominated solutions or a pareto set and they will be assigned to the first rank and deleted from the population. These solutions correspond to \(p6\), \(p7\) and \(p8\) in Figure 1 (an example of a bi-objective minimization problem with eight solutions). Then \(p1\) and \(p2\) are found as rank 2 through the same process. The process will be repeated until the population is empty. Another typical representative of the second method to assign solutions to the related rank is given in [6; 7]. In this method, every solution needs to compare with solutions whose ranks have been assigned. Suppose \(p2\) has the maximum rank in the dominant set \(\{p2,\)\(p6,\)\(p8\}\) who dominates \(p5\), and the rank of \(p2\) is 2, so the rank of \(p5\) is 2+1, i.e., 3. Above all, there are four fronts in Figure 1, set \(\{p6,\)\(p8,\)\(p7\}\) of rank 1, \(\{p1,\)\(p2\}\) of rank 2, \(\{p3,\)\(p5\}\) of rank 3 and \(\{p4\}\) of rank 4. In this paper, based on the analysis and study of the previous algorithms, a novel non-dominated sorting algorithm, combining the advantage of characteristics of the set, is proposed.  
The research contents of this paper are given below. Some general and recently studied nondominated sorting algorithms to find a pareto set or rank the solutions are introduced in Section 2. Section 3 presents the fundamental concepts and definitions as they will come up through the whole paper. Section 4 describes the proposed non-dominated algorithm and a detailed example explains how the proposed algorithm implemented is also given in this section. An algorithm complexity analysis is given in Section 4. Simulations and experimental results are discussed in Section 5. Finally, Section 6 makes a summary on the process of the work, and plans the next steps for following work.