# An Efficient Moth Flame Optimization (MFO) Algorithm for Solving Numerical Expressions

1. Introduction

Moth-flame optimization algorithm is a new metaheuristic optimization method, which is proposed by Seyedali Mirjalili in 2015 and based on the simulation of the behavior of moths for their special navigation methods in night [1]. They utilize a mechanism called transverse orientation for navigation. In this method, a moth flies by maintaining a fixed angle with respect to the moon, which is a very effective mechanism for travelling long distance in a straight path because the moon is far away from the moth. This mechanism guarantees that moths fly along straight line in night. However, we usually observe that moths fly spirally around the lights. In fact, moths are tricked by artificial lights and show such behaviors. Since such light is extremely close to the moon, hence, maintaining a similar angle to the light source causes a spiral fly path of moths. In MFO algorithm that Moths fly around flames in a Logarithmic spiral way and finally converges towards the flame [2]. Spiral way expresses the exploration area and it guarantees to exploit the optimum solution. Optimization refers to the process of finding the best possible solution(s) for a particular problem. As the complexity of problems increases, over the last few decades, the need for new optimization techniques becomes evident more than before [3]. Mathematical optimization techniques used to be the only tools for optimizing problems before the proposal of heuristic optimization techniques. Mathematical optimization methods are mostly deterministic that suffer from one major problem: local optima entrapment. Some of them such as gradient-based algorithm require derivation of the search space as well. This makes them highly inefficient in solving real problems [4].

2. Inspiration of Moth Flame Optimization Algorithm

In sufi literature one of the most loved metaphor is moth and flame [5]. The moth’s annihilation into the flame has been drawn again and again as an analogy for the seeker in the sufi path who seeks annihilation into Divine Essence. The sufistic term for the annihilation or passing away into Divine is Fana [6].

Like all insects, moths have a body with three main parts – head, thorax and abdomen. Moths have three pairs of jointed legs on the thorax [7]. Moths are also characterized by their two pairs of large, scale-covered wings and by mouthparts that form a long proboscis for sipping nectar. Moths have compound eyes and two antennae [8].

3. Moth flame Optimization Algorithm (MFO)

Moths are fancy insects, which are highly similar to the family of butterflies. Basically, there are over 160,000 various species of this insect in nature. They have two main milestones in their lifetime: larvae and adult. The larva is converted to moth by cocoons. The most interesting fact about moths is their special navigation methods in night [9]. They have been evolved to fly in night using the moon light. They utilized a mechanism called transverse orientation for navigation. In this method, a moth flies by maintaining a fixed angle with respect to the moon, a very effective mechanism for travelling long distances in a straight path. Since the moon is far away from the moth, this mechanism guarantees flying in straight line. The same navigation method can be done by humans [10]. Suppose that the moon is in the south side of the sky and a human wants to go the east. If he keeps moon of his left side when walking, he would be able to move toward the east on a straight line [11].

Despite the effectiveness of transverse orientation, we usually observe that moths fly spirally around the lights. In fact, moths are tricked by artificial lights and show such behaviors. This is due to the inefficiency of the transverse orientation, in which it is only helpful for moving in straight line when the light source is very far [12]. When moths see a human-made artificial light, they try to maintain a similar angle with the light to fly in straight line. Since such a light is extremely close compared to the moon, however, maintaining a similar angle to the light source causes a useless or deadly spiral fly path for moths.

3.1. Steps for MFO

• Parameter Setting
• Population Initialization
• Fitness function
• Iteration process
• Optimal selection

3.1.1. Parameter Setting

A parameter is a limit. In mathematics a parameter is a constant in an equation, but parameter isn’t just for math anymore: now any system can have parameters that define its operation. You can set parameters for your class debate [13].

3.1.2. Population Initialization

The MFO is one of the recent meta-heuristic optimization techniques. MFO algorithm imitates the navigation method of moths in the night. In this algorithm, the moths are the candidate solutions and the moths’ positions are the problem’s parameters. In this way, moths can fly in I-D, 2-D, 3-D, or hyper dimension space by exchanging their position vectors [14].

3.1.3. Fitness function

For the fitness function was a measure to determine the goodness or quality of a single solution in a population. At the end of each iteration, fitness value is calculated of each agent for evaluating quality search [15].

3.1.4. Iteration process

An iterative process, or on-going process, is systematic repetition of sequences or formulas that aims to achieve a given result. It is a process where different data is tested until the desired result is obtained [16].

3.1.5. Optimal selection

For each iteration, update the position and fitness of moths and flames according to. Moths update their positions in hyper spheres around the best solutions obtained so far. The sequence of flames is changed based on the best solutions in each iteration, and the moths are required to update their positions with respect to the updated flames. When the iteration criterion is satisfied, the best moth is returned as the best obtained approximation of the optimum [17].

3.2. Flow Chart of MFO

4. Numerical expression of MFO

Moth flame algorithm Numerical expressions are given [18],

5. Applications of MFO

• Training multi-layer preceptors[19]
• Optimal Power Flow
• Tomato diseases detection
• Terrorism detection
• Feature selection
• Multi area power system[21]
• Automatic Test Generation

• Population-based algorithms have high ability to avoid local optima since a set of solutions are involved during optimization.
• ILS is an improved hill climbing algorithm to decrease the probability of trapping in local optima [22].
• Hill climbing is also another local search and individual-based technique that starts optimization by a single solution.
• Although different improvements of individual-based algorithms promote local optima avoidance, the literature shows that population-based algorithms are better in handling this issue [23].
• A right balance between these two milestones can guarantee a very accurate approximation of the global optimum using population-based algorithms [24].

Reference

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[2] Mirjalili, S. (2015). Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowledge-Based Systems, 89, pp.228-249.

[3] Aziz, M., Ewees, A. and Hassanien, A. (2017). Whale Optimization Algorithm and Moth-Flame Optimization for multilevel thresholding image segmentation. Expert Systems with Applications, 83, pp.242-256.

[4] Allam, D., Yousri, D. and Eteiba, M. (2016). Parameters extraction of the three diode model for the multi-crystalline solar cell/module using Moth-Flame Optimization Algorithm. Energy Conversion and Management, 123, pp.535-548.

[5] Li, C., Li, S. and Liu, Y. (2016). A least squares support vector machine model optimized by moth-flame optimization algorithm for annual power load forecasting. Applied Intelligence, 45(4), pp.1166-1178.

[6] Xi, Y. and Peng, H. (2012). Training Multi-Layer Perceptrons with the Unscented Kalman Particle Filter. Advanced Materials Research, 542-543, pp.745-748.

[7] Mirjalili, S. (2015). Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowledge-Based Systems, 89, pp.228-249.

[8] Khalilpourazari, S. and Khalilpourazary, S. (2017). An efficient hybrid algorithm based on Water Cycle and Moth-Flame Optimization algorithms for solving numerical and constrained engineering optimization problems. Soft Computing, 23(5), pp.1699-1722.

[9] Hassanien, A., Gaber, T., Mokhtar, U. and Hefny, H. (2017). An improved moth flame optimization algorithm based on rough sets for tomato diseases detection. Computers and Electronics in Agriculture, 136, pp.86-96.

[10] Khalilpourazari, S. and Khalilpourazary, S. (2017). An efficient hybrid algorithm based on Water Cycle and Moth-Flame Optimization algorithms for solving numerical and constrained engineering optimization problems. Soft Computing, 23(5), pp.1699-1722.

[11] Singh, P. and Prakash, S. (2017). Optical network unit placement in Fiber-Wireless (FiWi) access network by Moth-Flame optimization algorithm. Optical Fiber Technology, 36, pp.403-411.

[12] Ng Shin Mei, R., Sulaiman, M., Mustaffa, Z. and Daniyal, H. (2017). Optimal reactive power dispatch solution by loss minimization using moth-flame optimization technique. Applied Soft Computing, 59, pp.210-222.

[13] K, S., Panwar, L., Panigrahi, B. and Kumar, R. (2018). Solution to unit commitment in power system operation planning using binary coded modified moth flame optimization algorithm (BMMFOA): A flame selection based computational technique. Journal of Computational Science, 25, pp.298-317.

[14] Wang, M., Chen, H., Yang, B., Zhao, X., Hu, L., Cai, Z., Huang, H. and Tong, C. (2017). Toward an optimal kernel extreme learning machine using a chaotic moth-flame optimization strategy with applications in medical diagnoses. Neurocomputing, 267, pp.69-84.

[15] Khairuzzaman, A. and Chaudhury, S. (2017). Moth-Flame Optimization Algorithm Based Multilevel Thresholding for Image Segmentation. International Journal of Applied Metaheuristic Computing, 8(4), pp.58-83.

[16] Sayed, G. and Hassanien, A. (2017). Moth-flame swarm optimization with neutrosophic sets for automatic mitosis detection in breast cancer histology images. Applied Intelligence, 47(2), pp.397-408.

[17] Savsani, V. and Tawhid, M. (2017). Non-dominated sorting moth flame optimization (NS-MFO) for multi-objective problems. Engineering Applications of Artificial Intelligence, 63, pp.20-32.

[18] Zhang, L., Mistry, K., Neoh, S. and Lim, C. (2016). Intelligent facial emotion recognition using moth-firefly optimization. Knowledge-Based Systems, 111, pp.248-267.

[19] Dash, K., Puhan, N. and Panda, G. (2015). Handwritten numeral recognition using non-redundant Stockwell transform and bio-inspired optimal zoning. IET Image Processing, 9(10), pp.874-882.

[20] Özyön, S. and Aydin, D. (2013). Incremental artificial bee colony with local search to economic dispatch problem with ramp rate limits and prohibited operating zones. Energy Conversion and Management, 65, pp.397-407.

[21] Mohanty, B. (2018). Performance analysis of moth flame optimization algorithm for AGC system. International Journal of Modelling and Simulation, 39(2), pp.73-87.

[22] Ebrahim, M., Becherif, M. and Abdelaziz, A. (2018). Dynamic performance enhancement for wind energy conversion system using Moth-Flame Optimization based blade pitch controller. Sustainable Energy Technologies and Assessments, 27, pp.206-212.

[23] Bhattacharjee, K. and Patel, N. (2018). A Comparative Study of Economic Load Dispatch using Sine Cosine Algorithm. Scientia Iranica, 0(0), pp.0-0.

[24] Zhang, P., Sun, P. and Li, G. (2014). Opposition-Based Learning Harmony Search Algorithm Solving Unconstrained Optimization Problems. Advanced Materials Research, 1006-1007, pp.1035-1038.