Butterfly optimization algorithm

In the Linnaean Animal Kingdom system, butterflies are in the class Lepidoptera. There are more than 18,000 species of butterflies in the world. The reason for their survival for millions of years lies in their senses (Saccheri et al. 1998). Butterflies use their senses of smell, sight, taste, touch, and hearing to find food and mating partners. These senses also help in migrating from one place to another, escaping from the predator, and laying eggs in suitable places. Among all these senses, smell is the most important sense that helps the butterfly find food even from long distances (Blair and Launer 1997). Butterflies use sensory receptors used for smelling to find the source of nectar, and these receptors are distributed over body parts such as the antennae, legs, and fingers of the butterfly. These receptors are nerve cells on the body surface of the butterfly and are called chemoreceptors. These chemoreceptors guide the butterfly to find the best mating partner to maintain a strong genetic line. A male butterfly can identify a female by means of pheromones which are scent secretions that the female butterfly emits to cause certain reactions (Arora and Singh 2019). Based on scientific observations, it has been found that butterflies have a very accurate perception of the source of the odor (Raguso 2008). They can also distinguish different odors and sense their intensity (Wyatt 2003).

The butterfly optimization algorithm (BOA) which is based on swarm intelligence was developed by Arora and Singh (2019) inspired by nature to solve global optimization problems. BOA is mainly based on the moving strategy of butterflies which uses their sense of smell to locate nectar or mating mates. Butterflies detect and analyze odor with their sense sensor as noted above to determine the potential direction of a nectar/mating mate. The BOA mimics this behavior to find the optimum in the search space. The BOA is an algorithm designed by modeling butterflies to find food and mating mates using their senses of smell, sight, taste, touch, and hearing. Among all these senses, smell is the most important which helps the butterfly find food, usually nectar, even from long distances. Butterflies are search agents for optimization in the BOA algorithm. A butterfly will produce scent with an intensity associated with its fitness. Namely, as a butterfly moves from one location to another its fitness will change. The scent spreads over the distance, other butterflies can sense it, and butterflies can share their personal information with other butterflies and create a collective social information network. When a butterfly can detect the scent of another butterfly, the butterfly will move toward it, and this step is called global search in the proposed algorithm (Arora and Singh 2019).

The BOA was developed by taking the following three features as role models. (i) All butterflies are expected to emit a scent that makes the butterflies attract each other. (ii) Each butterfly will move randomly or toward the best butterfly that emits more scents. (iii) The stimulus intensity of a butterfly is affected or determined by the value of its objective function.

In the BOA which is based on the behavior of butterflies, the three stages can be explained as follows: (1) Initialization Phase: Parameters are determined, and an initial population is generated for the algorithm. When creating the initial population, the position of the butterflies is randomly assigned by calculating the odor values. (2) Iteration Phase: This is the part where the main processes are carried out and each butterfly tries to reach the best result with parameters specific to the BOA. At each iteration, all butterflies in the search space are moved to new positions, and then their fitness values are evaluated. (3) The last stage: It is the part where the stopping criterion is met and the optimum or closest to the optimum result is reported.

Understanding the BOA modality relies on three key concepts. These concepts are sensory method (c), stimulus intensity (I), and power exponent (α). The sensory method refers to the raw input used by the sensors to measure the sensory energy form and process it in similar ways. The stimulus intensity parameter I is limited to an exponential value. According to previous studies by scientists, this is because as the stimulus gets stronger, the insects go intensely to the stimulus and eventually become less sensitive to it. The parameter α is used to correct this situation. The parameter α is the power base that is dependent on modality (smell in BOA). If α = 1, it means that there is no odor absorption. Namely, the amount of scent emitted by a particular butterfly is perceived by other butterflies with the same capacity. This brings us closer to a single solution, usually the optimum. If α = 0, it means that the scent emitted by any butterfly cannot be perceived by other butterflies. This provides the local search. α and c represent a random number between [0, 1], f represents the perceived magnitude of the odor, and I represents the stimulus intensity. There are two important stages in the algorithm: local search and global search. The global search is shown in Eq. (5) and the local search is shown in Eq. (6). In the local search, the butterfly x_i^t does not move toward the global best (g∗), but instead exhibits a random walk in the search space.

where g∗ represents the best available solution among all the solutions in the current iteration, X_i^t and X_k^t represent the butterflies in the search spaces, r represents the parameter that provides randomness in the range [0, 1], and f_i represents the perceived scent of the butterfly.

where, unlike the global search, the butterfly x_i^t does not move toward the global best (g∗), but instead exhibits a random walk in the search space.

Searching for food and mating partners with butterflies can occur on both the local and global scales. Thus, a switching probability p is used in the BOA to switch between the global search and the local search. The switching probability decides whether a butterfly will move to the best butterfly or randomly. The flowchart of the BOA algorithm is shown in Fig. 3.

Flowchart of the BOA algorithm