MaOMFO: Many-objective moth flame optimizer using reference-point based non-dominated sorting mechanism for global optimization problems

Many-objective optimization (MaO) deals with a large number of conflicting objectives in optimization problems to acquire a reliable set of appropriate non-dominated solutions near the true Pareto front, and for the same, a unique mechanism is essential. Numerous papers have reported multi-objective evolutionary algorithms to explain the absence of convergence and diversity variety in many-objective optimization problems. One of the most encouraging methodologies utilizes many reference points to segregate the solutions and guide the search procedure. The above-said methodology is integrated into the basic version of the Moth Flame Optimization (MFO) algorithm for the first time in this paper. The proposed Many-Objective Moth Flame Optimization (MaOMFO) utilizes a set of reference points progressively decided by the hunt procedure of the moth flame. It permits the calculation to combine with the Pareto front yet synchronize the decent variety of the Pareto front. MaOMFO is employed to solve a wide range of unconstrained and constrained benchmark functions and compared with other competitive algorithms, such as non-dominated sorting genetic algorithm, multi-objective evolutionary algorithm based on dominance and decomposition, and novel multi-objective particle swarm optimization using different performance metrics. The results demonstrate the superiority of the algorithm as a new many-objective algorithm for complex many-objective optimization problems.


Introduction
Optimization problems can be found in a wide range of fields, in which the objective is to find optimal values for the unknowns of a given problem to minimize or maximize a set of objectives. This expectation is scientifically an advancement problem. There are many streamlining issues with different goals, and regular logical inconsistencies exist among these objectives (Abbasi et al., 2021;Brest et al., 2017). It is frequently challenging to locate the ideal arrangement that fulfils all the objectives simultaneously. In most real-time applications, multi-objective problems have many solutions as opposed to a single solution, and multi-objective enhancement calculations have pulled in an ever-increasing number of specialists' consideration. Generally, a problem with two to four objectives is referred to as a multi-objective problem (MOP), while problems with more than four objectives are called many-objective problems (MaOPs) (Behmanesh et al., 2021;Liu et al., more than one objective is no longer possible using relational operators. A new operator called Pareto optimality is applied in this circumstance. The following are the key principles in this regard: Def. 1. Pareto Optimality (Branke et al., 2001): Def. 2. Pareto Dominance (Branke et al., 2001): ∀ ∈ 1,2, … , : ⃗ ≤ ⃗ ∧ ∃ ∈ 1,2, … , : ⃗ ⃗ Def. 3. Pareto optimal set (Branke et al., 2001): Def. 4. Pareto optimal front (Branke et al., 2001): The parametric space and objective space are illustrated in Fig. 1. In Fig. 1, both search spaces are compared, in which the circle denotes the best solution than the rectangle as it dominates the rectangle considering all objectives.

Related Works
In recent years, multi-objective evolutionary algorithms (MOEAs) have improved significantly since Schaffer first effectively utilized evolutionary algorithms to solve MOPs A. Zhou et al., 2011). Researchers are unhappy with the in-depth analysis of Multi-Objective Optimization (MOO) but pay even more importance to MaOPs. More many-objective evolutionary algorithms (MaOEAs) have been investigated in recent years to address practical manyobjective problems (He & Yen, 2014;Liu et al., 2017). However, almost all of these methods only concentrate on multi/many-objective small-scale problems, but few concentrate on large-scale optimisation problems. In order to solve MOPs, Schaffer suggested a vector-assessed genetic algorithm (VEGA) that integrated GA with MOO for the first time.
Since more and more researchers have introduced GAs to address MOPs (B. Li et al., 2015). Goldberg integrates the Pareto dominance method with EA to address MOPs for the first time. The principle of Goldberg and Schaffer influences many classical MOEAs, such as Niched Pareto Genetic Algorithm (NPGA) (Horn et al., 1994), NPGA2 (Erickson et al., 2001), NSGA (H. Li & Zhang, 2009), and NSGA-II (H. Li & Zhang, 2009).
Multi/Many objective optimizations first effectively utilized transformative calculation to multiple-objective problems by Schaffer; however, they give increasingly more consideration to many-objective problems. Lately, several MaOEAs have been developed and used to tackle many-objective problems and common-sense issues. In any case, the vast majority of these calculations centred around little scope multi/many-objective issues, and not many concentrated on huge scope improvement issues. Here we will quickly present some related EAs: NSGA-III (Vesikar et al., 2019), MOEA/D (Q. Zhang & Li, 2007), and Vector Genetic Algorithm (VEGA) Schaffer, 1984 .
NSGA-II is the most well-regarded algorithm for tackling MOPs, and numerous MOEAs depend on the concept of nondominated sorting integrated into this algorithm. This algorithm sort individual in each population based on their domination Objective Space Parametric Space 1 x 2 x 2 f 1 f Pareto Optimal Front level. Like GA, the solutions are then selected using inversely proportional to the domination level to undergo crossover and then mutation. Molina et al. (Luo et al. , 2019) used data from the reference point, and afterwards, they joined the gpredominance with NSGA-II calculation to manage a few MOPs with muddled PFs better. To improve the decent variety of NSGA-II, Vachhani et al. (Vachhani et al. , 2016) introduced an enhanced version of NSGA-II, in which another assorted variety of strategies followed agglomerative various levelled grouping techniques and outrageous arrangements protection to supplant the swarming separation technique. The test results indicated that the proposed technique improved the decent variety of unique NSGA-II on two-objective test cases. Dissimilar to the decay technique, a MOP can be broken into problems with multiple objectives and solved by MOEA/D. Qi et al. (Zheng et al. , 2018) set forward a versatile weight vector modification way to deal with improving the presentation of MOEA/D just utilized another approach to introducing weight vector and a procedure which can adaptively re-arrange sub-problems and use an outer best population data to help include new sub-problems into the genuine inadequate area of the best PF. These calculations referenced above are extremely powerful in settling MOPs; however, a considerable number of them have terrible showing in managing MaOPs. For instance, NSGA-II is serious about illuminating MOPs, but it does not perform well when managing MaOPs. To solve MaOPs and handle having a large number of non-dominated solutions in each iteration, Deb et al. recommended an evolutionary algorithm (EA) in light of the reference point and NSGA-II system (NSGA-III) (Vesikar et al., 2019). He utilized the reference point to choose parents for crossover. This method substantially improved the decent variety of populations and the capacity to illuminate MaOPs. In this work, MaoMFO proposes a similar manner in NSGA-III.

Many-Objective Moth-Flame Optimizer (MaOMFO)
This section briefly presents the basic notions of the original moth flame optimizer and comprehensively discusses the formulation procedure of the proposed Many-Objective Moth Flame Optimizer (MaOMFO) algorithm.

Moth-Flame Optimizer
The Moth-Flame Optimizer (Mirjalili, 2015) (MFO) was proposed by Mirjalili in 2015. This algorithm mimics the phototactic phenomenon in moths and other insects, in which they move towards a light source. Moths use traverse orientation by keeping the moon as their main light source at night for navigation. These insects maintain a fixed angle with the moon, which allows them to travel in a straight line due to the long distance to the moon. When replacing the moon with artificial light, moths get trapped in a spiral movement, which disrupts their navigation but converges them towards a single point. This behaviour has been mathematically modelled in the MFO algorithm as follows: where i M represent the i th moth, Fj represents the j th flame, and where i D expresses the path length of the i th moth for the j th flame. The authors are encouraged to read the base paper for more details about the MFO algorithm.

Many-Objective Moth-Flame Optimizer (MaOMFO)
The proposed MaoMFO algorithm uses diversity preservation and an elitist non-dominated sorting with a well-distributed Pareto front reference point mechanism .
The following measures are included in non-dominated sorting: • Determine the non-dominated solution • Apply non-dominated sorting (NDS) mechanism • Find the non-dominated ranking (NDR) of all non-dominated solutions • Apply reference point mechanism The NDR procedure, with two fronts presented, is shown in Fig. 2. Because any other solutions do not dominate them, the solutions in the first front have an index of 0, but at minimum, one of the solutions in the second front dominates the solutions in the first front. The number of solutions that exceed such solutions is equal to their NDR.
ℎ 2 is the overall framework of the MaoMFO algorithm. There are several important functions in ℎ 2, including MFO(), Normalize(), Associate(), Reference point mechanism(). The respective pseudocodes are shown in ℎ 1, 3, 4, and 5, respectively. get all individuals associated with : Pi 3: sort Pi by ascending order of vertical distance from individual to 4: for j = 1: length(Pi) 5: set Pij's Reference-value = j + dj 6: end for 7: end for The complete procedure of the proposed MaOMFO algorithm is presented in ℎ 6. Step by Step presentation of the Many-objective moth flame optimizer (MaoMFO) is given below: Step 1: Generating reference points using the uniform point function, a utility function for generating about N uniformly distributed points with M objectives on the unit hyperplane. Z is the set of reference points, and the Moth Position Size N is reset to the same as the number of reference points in Z.
Step 2: An initial random population is generated using the initialization function.
Step 3: Find the minimum objective value using a random Moth Position.
Step 4: After that, the termination criteria are invoked to check whether the number of evaluated fitness exceeds the maximum number of function evaluations, and Moth Position is passed to the function to be the final output.
Step 5: Afterwards, the mating pool selection using Tournament Selection. Returns the indices of N solutions by twotournament selection based on their fitness values. In each selection, the candidate having the minimum value can be selected.
Step 7: Find the minimum and maximum objective value generated via Moth Flame Optimizer; then combine both values using the union operator.
Step 8: Then apply the non-dominated sorting approach using objective value, constraint violation, and Moth Position size until it cannot be reached at the maximum front number or Select part of the solutions in the last front.
Step 9: After applying the normalization approach, detect the extreme points and calculate the intercepts of the hyperplane constructed by the extreme points on the axes.
Step 10: After that, calculate the distance of each solution to each reference vector that associates each solution with its nearest reference point and calculates the number of associated solutions except for the last front of each reference point.
Step 11: Afterwards, select K-remaining solutions one by one and find the least crowded reference point. Then, select one solution associated with this reference point Step 12: Afterwards, get Moth Position for the next generation

Discussions
The ZDT, DTLZ & IMOP test suite is considered as one of the most crucial testing benchmarks sets in writing and incorporates exceptionally unimodal, multimodal, rotated, hybrid, and composite test features. The optimal Pareto front of such features is of various shapes and coherence. This subsection discusses the MaOMFO findings of these test suites and contrasts the outcomes with NSGA-III, NMPSO, and MOEA/DD. The outcomes are given in Table 1, The MaOMFO is based on the NSGA-III and adds the history data of individuals from earlier iterations to the production of offspring. The individuals chosen in this method may be excellent or undesirable since they are selected arbitrarily or in a predetermined manner rather than the best individuals in the population. This would somewhat slow down the algorithm's convergence rate and prevent the algorithm from reaching a local optimum. Regardless, MaOMFO can also be used to solve problems with three or more objective functions. Due to the Pareto dominated-based solution in MaOMFO turns out to be less successful for the higher objective problems. This is because of the way that in problems with many objectives, an enormous number of solutions is non-dominated, so the archive becomes full. In this way, the MaOMFO performance is reasonable for handling problems with more than four objectives. The outcomes demonstrated that MaOMFO could be extremely successful for multi/many-objective problems. A better combination of MaOMFO is expected than the moth's position update around the best non-dominated solutions with reference point mechanism. The high coverage of MaOMFO is a direct result of the archive maintenance and solution update. When the archive is full, non-dominated solutions are discarded by MaOMFO, which improves the solution diversity along the complete front. The proposed MaOMFO has the features such as high search accuracy, neighbourhood solution distribution, exploitation, and quick convergence. The MaOMFO can keep away from nearby fronts and coverage towards the best Pareto front shown in Figure 3, Figure 4, and Figure 5. The obtained results and above-all discussions demonstrate that the MaOMFO algorithm is reliable and simple too to handle the multi/many-objective optimization problems with different search spaces.

Conclusion
This paper proposes a simple and new many-objective algorithm called Many-Objective Moth Flame Optimizer (MaOMFO) based on the original version of the MFO, reference point strategy, and non-dominated sorting mechanism. This study presents a unique MaOMFO for processing MaOPs to enhance the overall performance in terms of convergence and diversity. An MFO algorithm with outstanding convergence ability is used as the MaOMFO's optimization procedure.
Both strong convergence speed and better diversity are characteristics of the MaOMFO algorithm. Experimental studies contrasting the MaOMFO with the other three most reputed algorithms while running them on the DTLZ, IMOP, and ZDT test cases demonstrate the MaOMFO-efficiency. The findings of the experiments reveal that the suggested MaOMFO algorithm succeeds well in most of the DTLZ, IMOP, and ZDT test cases that we investigated, and the generated solution set shows good convergence and diversity. The statistical metrics, such as GD, IGD, HV, and MS, demonstrate that the proposed MaOMFO algorithm performs better than other state-of-the-art many-objective optimizers. However, the result also demonstrates that none of the algorithms can surpass any other algorithms in any instance. This highlights the significance of making an informed decision regarding which algorithms to use while attempting to solve the MaOPs.
In addition, to further validate MaOMFO's efficacy, it is suggested to extend it to be able to tackle constrained manyobjective problems by different constraint handling mechanisms. This will allow MaOMFO to handle several objectives of real-world many-objective engineering design problems simultaneously.