**1. Introduction**

Although the truth of the algorithmic strategy for dealing with combinatorial optimization (CO) has been available for a long time, further application of evolutionary algorithms (EAs) to solve these problems provides a means to deal with largescale multi-objective optimization.

In this section, the current of my study, which is considered one of the most important studies in recent decades, has been dealt with, and we will explain in it: research objectives, research question, study significance, research breadth, and research limitations.

Often there is not one perfect solution in multi-objective function optimization, but rather a set of optimal Pareto options. Thus, cluster sampling is critical when the co-optimization of an algorithm to generate a comprehensive and varied approximation of the Pareto front (PF) is performed [1]. Using the rule of change of weights, a multi-objective bat algorithm (MOBAT) is introduced to determine the optimal Pareto array for multipurpose functions (MO).

The source [2] also presented bat for multi-objective problem-solving, as well as the multi-objective bat algorithm (MOBAT). To verify this, we will develop solutions against a subset of the multi-objective test functions first. We will now use it to address engineering design improvement challenges such as the total and partial steel beam.

MOBAT was used for this purpose, and it can be described as a successfully biologically inspired algorithm to address problem floor planning in VSLI design in a publication approach [3].

The author in [4] proposed a multi-purpose optimization problem (MOOP) to achieve both of the aforementioned goals. MOOP is solved using a new simple optimization algorithm called BAT Algorithm, which is based on weight addition method (WSM). Therefore, from the literature we can say here that there is no study before that combined many-objective bat algorithm with indicator convergence R2 (MaBAT/ R2). In addition, in another study, a comparison was made between the algorithms for feeding frontal neural networks (NN) and then the gradient descent (GD) algorithms (backpropagation and Levenberg–Marquardt), and three population-based statistical inference methods were used: the bat algorithm, the genetic algorithm (GA), and the particle swarm optimization (PSO) algorithm for the test. It has been shown that the BAT algorithm is superior to all other algorithms in training to feed-forward neural networks (NN) [5]. These results support the use of the best available techniques for further experiments, which greatly contributed to finding the optimal solution.

The advantage of using the bat algorithm is that it allows us to find solutions using population and local search techniques. This work introduced global diversity and rigorous local extraction, both of which are important for exploratory methods. As a result, the Bat algorithm was combined with PSO and local search, in addition to controlling the pulse rate and loudness [6].

MOBAT was used in many-objective optimization problems (MaOPs), which gave us a good balance between diversity and convergence, representing the main issue in MaOPs, by adapting the reference groups approach. Additionally, in 2021, a paper was published entitled using the multipurpose bat algorithm to solve the multipurpose nonlinear programming problem [7]. Moreover, in 2020 [8], a met heuristic hybrid method is proposed to solve multi-objective optimization problems.

We conclude from the above that the main objective of this study is to improve the performance of multi-objective algorithms by developing a new algorithm inspired by bats for multi-objective optimization problems that used a technique to achieve organization and to achieve goals and diversity. Therefore, we proposed a method of increment based on the R2 index distance algorithm to reduce processing efforts in the field of different objective challenges in this paper.
