metaheuristics

Nature-Inspired Optimization Algorithms

This project implements two nature-inspired optimization algorithms: Moth Flame Optimization (MFO) and Honey Badger Optimization (HBO). Both algorithms are designed to solve complex optimization problems by mimicking behaviors observed in nature.

Moth Flame Optimization (MFO)

MFO is inspired by the navigation method of moths in nature, called transverse orientation. Moths fly at night by maintaining a fixed angle with respect to the moon, an effective technique for traveling in a straight line over long distances.

Contents:

• Introduction
  • Optimization Algorithms
  • Introduction of Moth-Flame algorithm
• Algorithm
  • What is MFO algorithm?
  • Moth-Flame Optimizer
  • Example
  • Explanation of MFO algorithm
  • Initial values
  • Algorithm iterations
  • Logarithmic spiral

Introduction

Optimization Algorithms

Optimization points to the process of finding the best solution(s) for a specific problem. Throughout the last few decades with complexity growth of problems, demand for new optimization tactics became more obvious.

Before the suggestion of Heuristic optimization tactics, mathematical optimization tactics were the only optimization tool for problems. Mathematical optimization tactics are mostly deterministic therefore they have local optima issues. On the other hand, meta-heuristic algorithms start from some initial solution and will get close to the solution through next iterations.

Some of the famous meta-heuristic algorithms are Genetic optimization, Grey Wolf optimization, Ant Colony optimization, and others.

Introduction of Moth-Flame algorithm

Moth-Flame optimization algorithm, also known as MFO or Moth-Flame algorithm, is one of the optimization and meta-heuristic algorithms that finds a solution for the problem from the behavior of moths around flame or fire.

This algorithm was proposed by Seyed Ali Mirjalili as an article named "Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm" in Knowledge-Based Systems journal back in 2015. This algorithm is a modern heuristic algorithm inspired by nature, moth's behavior, and their interest in flame or fire.

Algorithm

What is MFO algorithm?

Here we have a new optimization algorithm inspired by nature called Moth-Flame optimization algorithm or Moth and Flame.

The original inspiration of this optimization is moth's navigation method in nature called transverse orientation. Moths fly long distances through the night by maintaining a fixed angle with respect to the moon. However, these fancy insects are trapped in a useless/deadly spiral path around artificial lights.

Moth-Flame Optimizer

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 larvae is converted to moth in cocoons.

The most interesting fact about moths is their special navigation methods at night. They have evolved to fly at night using the moonlight. 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.

Example

A conceptual model of transverse orientation is shown in the picture above. 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. Suppose that the moon is in the south side of the sky and a human wants to go the east. If he keeps moon on his left side when walking, he would be able to move towards the east on a straight line.

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. 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. A conceptual model of this behavior is illustrated in the picture above. It may be observed that the moth eventually converges towards the light. This behavior is modeled mathematically to propose an optimizer called Moth-Flame Optimization (MFO) algorithm in the following subsection.

Explanation of MFO algorithm

In the proposed MFO algorithm, it is assumed that the candidate solutions are moths and the problem's variables are the position of moths in the space. Therefore, the moths can fly in 1-D, 2-D, 3-D, or hyper dimensional space with changing their position vectors.

Since the MFO algorithm is a population-based algorithm, the set of moths is represented in a matrix below, where n is the number of moths and d is the number of variables (dimension). For all the moths, we also assume that there is an array for storing the corresponding fitness values where again n is the number of moths.

Note that moths and flames are both solutions. The difference between them is their approach and update methods in each iteration. Moths are the actual search agents that move in the search space, while flames are moth's best calculated position so far. In other words, flames are flags/pins that are dropped by moths while searching the search space. Thus each moth searches around a flag (flame) and updates it in case of finding a better solution. With this mechanism, a moth never loses its best solution.

Initial values

Any random distribution can be used to generate initial solutions and calculate objective function values. This method is applied as default in which 'ub' is the upper bound and 'lb' is the lower bound for variable 'i'. Therefore for 'n' in dimension 'd' will be able to generate initial solutions. We can calculate the fitness for each solution after initialization, thus the fitness function value of matrix 'M' will be 'OM'.

Algorithm iterations

After the initialization, the 'P' function will iteratively run until the 'T' function returns true. The 'P' function is the main function that moves the moths around the search space. As mentioned above, the inspiration of this algorithm is the transverse orientation.

Logarithmic spiral

A logarithmic spiral is chosen as the main update mechanism of moths in this paper. However, any types of spiral can be applied here subject to the following conditions:

• Spiral's initial point should start from the moth.
• Spiral's final point should be the position of the flame.
• Fluctuation of the range of spiral should not exceed from the search space.

Considering these points, a logarithmic spiral is defined for the MFO algorithm as follows:

where 'D(i)' indicates the distance of the 'i-th' moth for the 'j-th' flame (which its formula is down below), 'b' is a constant for defining the shape of the logarithmic spiral, and 't' is a random number in [-1, 1].

Logarithmic spiral equation is where the spiral flying path of moths is simulated. As may be seen in this equation, the next position of a moth is defined with respect to a flame. The 't' parameter in the spiral equation defines how much the next position of the moth should be close to the flame ('t' = '-1' is the closest position to the flame, while 't' = '1' shows the farthest). Therefore, a hyper ellipse can be assumed around the flame in all directions and the next position of the moth would be within this space.

The spiral movement is the main component of the proposed method because it dictates how the moths update their positions around flames. The spiral equation allows a moth to fly "around" a flame and not necessarily in the space between them. Therefore, the exploration and exploitation of the search space can be guaranteed. The logarithmic spiral, space around the flame, and the position considering different 't' on the curve are illustrated in the picture below.

A conceptual model of position updating of a moth around a flame is shown in the picture below. Note that the vertical axis shows only one dimension (1 variable/parameter of a given problem), but the method can be applied for changing all the variables of the problem. The possible positions (dashed black lines) that can be chosen as the next position of the moth (blue horizontal line) around the flame (green horizontal line) clearly show that a moth can explore and exploit the search space around the flame in one dimension.

Exploration happens when the next position is outside the space between the moth and flame as can be seen in the arrows labelled by 1, 3, and 4. Exploitation occurs when the next position is inside the space between the moth and flame as can be observed in the arrow labelled by 2.

Key Features of MFO:

1. Population: Consists of moths and flames.
2. Movement: Moths update their positions with respect to flames using a spiral flying path.
3. Adaptive Flame Number: The number of flames decreases over iterations, intensifying the exploitation phase.
4. Spiral Update: Uses a logarithmic spiral to define the moth's flying path around flames.

Honey Badger Optimization (HBO)

HBO is inspired by the foraging behavior and aggressive nature of honey badgers. These animals are known for their fearlessness and ability to adapt to various environments.

Key Features of HBO:

1. Population: Consists of honey badgers.
2. Behavior: Alternates between foraging and aggressive behaviors.
3. Movement: Updates positions based on the best solution found (foraging) or a random member of the population (aggressive).
4. Simplicity: Uses a straightforward update rule without complex mathematical functions.

Comparison between MFO and HBO

1. Inspiration Source:

   • MFO: Based on moth navigation using moonlight.
   • HBO: Based on honey badger foraging and aggressive behaviors.

2. Population Structure:

   • MFO: Uses two sets of solutions (moths and flames).
   • HBO: Uses a single set of solutions (honey badgers).

3. Update Mechanism:

   • MFO: Uses a logarithmic spiral path for updates.
   • HBO: Uses linear position updates based on best solution or random population member.

4. Adaptive Parameters:

   • MFO: Adapts the number of flames over iterations.
   • HBO: Does not have adaptive parameters.

5. Complexity:

   • MFO: More complex due to spiral paths and flame number adaptation.
   • HBO: Simpler implementation with straightforward update rules.

6. Exploration vs Exploitation:

   • MFO: Balances exploration and exploitation through flame number reduction.
   • HBO: Balances through alternating between foraging (exploitation) and aggressive (exploration) behaviors.

Usage

Both algorithms are implemented to work with various fitness functions. To use:

1. Ensure you have numpy and matplotlib installed.
2. Place fitness_functions.py and report_generator.py in the same directory as the main scripts.
3. Run either moth_flame_optimization.py or honey_badger_optimization.py.
4. The algorithms will generate reports for each fitness function.
5. You'll be prompted if you want to see visualizations of the optimization process.

Conclusion

Both MFO and HBO offer unique approaches to optimization inspired by nature. MFO provides a more structured approach with its spiral paths and adaptive flame number, potentially making it suitable for problems requiring a balance of exploration and exploitation. HBO, with its simpler implementation, might be more adaptable to a wider range of problems and easier to tune.