# A Novel Multi-Objective Harmony Search Algorithm with Pitch Adjustment by Genotype
## 1 Introduction  
Most real engineering problems are multi-objective in nature, since commonly they present several objective functions to be optimized that are compromised with each other, that is, the improvement of one produces the deterioration of another. Therefore, without a function priority, a unique solution cannot be determined as in a single objective optimization. Instead, multi-objective optimization seeks to obtain the best compromises. The best compromised solutions are known as non-dominated and form the Pareto-optimal front [1]. Harmony Search (HS) is a metaheuristic algorithm developed by [PERSON] et al. [2], emulating musical improvisation. It was proposed as a mono-objective algorithm and has been successfully applied in practical and scientific problems [3, 4, 5, 6]. HS integrates three resources for the quantitative optimization process: use of a harmonic memory (HM), pitch adjustment, and randomization [7]. The implementation of multi-objective HS is a growing trend in scientific research, and its use for solving highly complex problems is considered a future challenge [8]. The first multi-objective cases solved by HS [9, 10, 11] were engineering problems treated with the weighting function method. In [12, 11], the authors began to handle the Pareto front term, without explicitly establishing a multi-objective HS algorithm. In [13], [PERSON] et al. developed for the first time a multi-objective HS to generate a Pareto-optimal front of five points for a robotics model. [PERSON] et al. [14] established two formal multi-objective proposals known as MOHS1 and MOHS2. Both of them are very similar to the single objective form of HS but with a fundamental difference in the ranking of the harmonies. [PERSON] et al. [15] presented a proposal similar to MOHS2 (denominated as MOHS3 in this paper) that incorporates the crowding distancing and dynamic pitch adjustment proposed by [PERSON] et al. [16]. [PERSON] et al. [17] developed amulti-objective memetic algorithm by combining HS and PSO. However, it greatly differs from the original structure of HS. In its original structure, HS is very effective at identifying promising areas in the solution space (exploration) but presents difficulties in carrying out refined searches (exploitation). For this reason, it is required to improve the balance between exploration and exploitation. There are two trends in this sense [18]. The first one is based on the synergistic combination of HS with other metaheuristic algorithms. In [19; 20; 21], PSO is applied to give a social component to the pitch adjustment. In [22; 23], the authors proposed hybrid algorithms to improve the exploitation in HS by combining it with Artificial Bee Colony (ABC) and Stochastic Hill Climbing. However, these alternatives are often disruptive to the conceptual essence of HS. The second stream focuses on improving the pitch adjustment [24; 25]. In [26], pitch adjustments with fixed parameters are proposed, using bandwidths that are proportional to the range of each variable. On the other hand, in [7; 16; 27], the authors propose the use of dynamic parameters to intensify exploitation as the algorithm execution advances. [PERSON] [28] proposed an HS variant using non-lineal dynamic parameters with a Gaussian distribution for the Harmony Memory Accepting Rate (HMCR) and bandwidth with a social component. In the case of multi-objective HS, the conventional mono-objective pitch and the dynamic pitch adjustments of [PERSON] et al. [16] have barely been implemented, both based on their effectiveness for single-objective problems. However, Pareto fronts can be generated for different dispersion degrees of the search space, also known as decision space [29]. Currently, most multi-objective HS proposals present a lack of analysis of the behavior of harmonies in the search space during the formation of Pareto fronts. In this paper, the Multi-Objective HS algorithm MOHSg is proposed.