Velella Swarm Optimization Algorithm (VSO)

Some algorithms have been developed in heuristic optimization area inspired by swarm behaviour in nature which are the most popular optimization problems. In this paper, a theory for subgroups of Velella are introduced and a new biologically-inspired optimization algorithm is proposed based on subgroups behaviour of the individual Velella in colonies. Our proposed method is compared with some of the state-of-the-art heuristic optimization algorithms. Finally, performance and results of the proposed method is prepared for solving various benchmark functions. Obtained results prove that our method can solve same problems significantly better than the others.


Introduction
Many collective behaviors are exhibited by social insects and animals as well as swarm's phenomena in nature which demonstrates their intelligent characteristics. Swarm Systems benefit from swarm intelligence (SI) to solve existed challenges [1]. An SI system is typically consist a population of individuals inspired by natural phenomena. Interaction rules between individuals or agents show especial patterns and behavior as SI systems. Birds flying in flocks, ants' pathplanning behavior and honey bee waggle dances which are all performed to find food sources and interactions of human society to learn from each other are a few biological examples that exhibit multiple necessary behavior for success of colony in SI systems. [2], [3].
Many phenomena can be observed in nature and velella animal is one of these remarkable one getting advantage of SI system that emerge subgroup behavior. 2 The velella are floating on the surface of Open Ocean. They have a purple color with a flat elliptical transparent float and an erect sail projecting a vertical at an angle to the axis of its body. They feed on small prey and fish by their venom that can be caught below the surface of the water. The windbased projection of the sail, water flow and swimming by its tentacles can take the best advantage of movement at any given moment.
The velella feeding is limited to the surface of water because it is not big enough to cover a large surface of water by venom and its tentacles do not sense far distances. Venom establish trap in a radius for bait and velella sense the food caught in trap by its tentacles. Their mouth is located in the middle of the underside of body and feed the bait by shaking their tentacles [4].
Older zoological opinions assumed velella was a colony of specialized individuals. Some taxonomists classified velella as Siphonophore because their view did not consider velella to be a single animal but they are organisms which were linked together with their interactions. (A colony of specialized polyps). However, more recent studies, have classified the velella as a highly modified individual hydroid polyps, but not a colonial hydrozoan. Because it is believed that velella is a single highly modified polyp. For example, Research playwright, Nick Darke describe the movement of this species as an action of wind upon each subgroup which can push them in different directions. The interplay of the wind and the animal's physical structure results in a simple behavior that (presumably) is beneficial for the species [5] [6]. However, behavior of the velella is not so important if one refers to the old definition or the new one in the evaluation of velella. The effect of the sail is so that the velella can take the useful advantages of the wind and water force, at any given moment [4].
In this article, we proposed a hypothesis that logically prove velella subgroups are a colony and have interactions together. As we know that velella feed from small prey and fish below the surface of water and also, they use traps with venom to hunt its bait [6]. Thus, according to the evidence of velella's research behavior, when velellas are together on the surface of water, the probability of catching food is increased. In fact, the more rate of venom increases the probability prey in the water surface and for this reason, they prefer to make subgroup formation. Subgroup state and staying together makes much of water area surface covered by venom traps (see Fig. 1). Velellas do not organize subgroups formation without reason although air and water are effective factor on their movements. Subgroup formation on water surface need to be intelligent to make velella resist against the wind force and water flow or to use them efficiently. In subgroups behavior, tentacles of velella sense surroundings to hunt the bait. It's feasible that tentacles of velella sense the nearest neighbors in sense radius and move toward it by useful water flow, wind force on sailing and even with swimming by tentacles. After staying on next neighbors, velella try to keep balance due to changes of subgroups. If weather conditions are being suitable, (without storm) subgroups become in forms of swarms or colonies that try to feed on surface of water [7] [8]. Research on Swarm intelligence (SI) behavior for solving optimization problem have been widely developed in artificial intelligence sciences. An SI system ability refers to artificial intelligence techniques in swarm system to behave in a complex pattern and self-organized way without any specific individual control. In solving optimization problem by a heuristic algorithm, each single individual of swarms has an intelligence interaction in problem space which represents a potential solution. In problem space, local interaction between simple agent's exhibit swarm intelligence for local search and then exchanged processing information emerge complex patterns as global search by lapse of time. Thus, interaction between individuals is very important to develop a new heuristic optimization algorithm based on a biologically-inspired behaviour. This paper is organized in seven section. In the next section originality of this work is introduced and after that motivation and theoretical approach is discussed. Methodology is in the four section and section five is about comparing methods and experimental results are shown in section six. Finally conclusion section is at the end of this article.

Motivation and originality
Providing a simple mathematical model to describe the phenomena of the real world has a significant effect on perception of heuristic algorithms. Hence, in last decades, a wide range of approaches for heuristics algorithms were presented such as Simulated Annealing(SA) [9], Genetic Algorithm(GA) [10], Particle Swarm Optimization (PSO) [11] and Ant Colony Optimization (ACO) [12], which are the most representative samples in heuristic approaches that use different phenomena to solve complex computational and optimization problems [10]. These algorithms well suited to solve different problems such as finding objective functions [6], pattern recognition [13], image processing [4], swarm robotics [14], etc. However, each heuristic algorithm has a various performance in a specific problem to find the best solution [15].

Sailing physics
Due to details study of velella simulation, movement behavior on the water surface with water flow and wind power is a problematic task. Normally, in the real world, whether conditions and environment vary and they have impact on velella sail. Water and wind force variation with diverse direction are effective samples of velella movement. However, using motion Physics can propose a simple and comprehensible equations for velella movement. A simple sailing theory is practically attractive and can describe performance of those animals because physical structure of velella is similar to sailboats. Hence, in this research, sailboat physics is used to describe performance of the velella sail.
In sailboats physic study, a mathematical equation is presented for movements of sailboats which is relevant and similar to the aerodynamic performance of the velella sail explained in [17].
In equation (5), () F wind is the wind force with a direction which enters to the sail of boat, D C is a constant (usually equals to 4/3), A is coefficient of sail cross-section,  is density of the air and V is the velocity of wind which hits the sail. Proposed equation can be expressed in equation (7) which is derived from the above equations: In artificial system of velella that is proposed here, a movement is introduced based on sailing of sailboat physics. Therefore, according to the physics movements of sailboat we are describe a new simple equation for artificial system model which's expressed in equation (8) as follows: In the equation above, the number of coefficients for the artificial model is reduced. In equation (8) α, is parameter that controls the coefficient of wind power (coefficient cross-section of sail) and β, is a parameter that controls the coefficient of water force (coefficient of influence of water force to body or tentacle of velella). In next section, our proposed method is expressed based on the above methodology.

Velella algorithm
Now, according to the physics of boat sailing movement and similarity of velella physics sailing, it is possible to design and implement a new optimization algorithm inspired by the behavior of this animal in nature. In this section, we proposed an artificial model based on biological behavior and theory performance of the velella sail. According to subgroups behavior, each agent on the water surface has interaction with the other agents by tentacles. In the proposed method in accordance with a case study, competency is required to create subgroups formation behavior. For this propose, the venom amount of agent is set as objective function. In other words, the position of each agent has a rate of venom and radius neighborhoods which exchange processing information to each other as local search. Deaths is considered as global search due to storms and directing to the shore by the water of flow. In fact, the power of wind always guides agents toward the ocean shore which is a common property of agents.
In the proposed algorithm, water surface is set as problem space in which velellas are agents and their competency are measured by rate of their venom. We have assumed as a strategy to determine the global search which an agent to have more competency comparing with other agents is nearest agent to the shore. In our proposed method navigate agents toward the shore considered as the strategy of global search. In fact, an agent with the most rate of venom is defined as ocean shore.
Where, N presents the number of agents and vector x describes the position of agents for each i in the dth dimension space. In addition, each agent with vector x represents the quality of fitness (rate of venom). The value of venom is determined by given agent's position on the objective function. In other words, the position of each agent represents quality of venom or concentration rate of venom which deposited by the velella and determined as ( ( )) .
In maximization and minimization problems the fitness value is calculated by the following equations respectively: ( 1) ( ) It should be noted that in our proposed method, the radius of the neighborhood does not mean the step movement length of the agent and only determines the range of vision for that agent.
Therefore, in the problem space wind and water force determine movement step length of the agent. In this algorithm, the status of each agent is updated by velocity according to the global and local optimization strategy. Therefore, for global search strategy, agent with best fitness is defined as the edge of shore and global optimization in problem space. In the other word global optimization strategy is a common feature between agents which wind force navigated them toward the edge of shore. Thus, the way of velella's death in nature is considered to be as global optimization strategies as illustrated in  Thus, in the artificial models, a simple equation is defined to update global search strategy of agent movement from x(t) state to x(t+1)as follows: Above equation determine wind impact on agent movement on the water surface toward the global optimization. α is sail cross-section coefficient, V is velocity of wind which push sail of artificial agent and it is considered to be 2. rand is a random number in range of [0,1] with a uniform distribution. The value of Wind force impact coefficient (α) considered to be 0.9. G_best is the best position of an agent with high competency which is considered as the global search (ocean shore).
Local search strategies are also defined based on subgroup behavior formation of velella.
Subgroups behavior movement of agents to target is in Sin form. In local search strategies, there are three possible occurrences that describes in the following respectively.

First state:
If there is a best neighborhood ( ℎ _ ) for the sense radius of neighborhood agent x(t) in search space, agent x(t) moves one step towards its neighbor which its force is defined in equation 13: In equation (13) β is impact coefficient of water force on agent movement and U is velocity of water pushed to the agent body (water force) and consider constant (U=2). Sin, is sinus shape of water, and rand is a random number in the range of [0,1] with a uniform distribution. Impact  Second state: If there was no agent for the best neighbor in the sense radius, then it would act according to the best memory of its own as a step, which is calculated as follows:  In general, total forces entered into agents is calculated as follows: With regard to the equation (16), the wind force is common feature of agent's move. In the other side, each of the three state provides resistance to create subgroups behavior preventing them from navigation to the shore. Logically, coefficients of impact forces determine the value of forces push to the agents. Because acceleration is a proportional to speed, acceleration is replaced with velocity movement in the proposed method.
Regard to methodology of this optimization algorithm, the wind force coefficient considered as exploration approach and water force coefficient describes exploitation capability of the algorithm.
According to the coefficient of wind and water force they can be increases and decreases in exploration and exploitation approaches at the end of each step of the algorithm.
In the above equation, swimming (swim) parameter considered to be 0.04 and relation of impact coefficient of water and wind leads to increase rate of water velocity. Thus, the best neighbour in each of those three states (local optimal) is selected with respect to the fitness of neighbourhoods. The fitness level is adjustable for the best neighbourhoods regarding the type of optimization problem (maximization and minimization).
Pseudo code of the proposed algorithm is described below:  Fig.8 and its flowchart is illustrated in Fig.9.

Principles of VSO algorithm is shown in
To see how the proposed algorithm is efficient some remarks are noted: -In VSO algorithm an agent with more venom rate considered as global search strategy.
-Global search strategy is equivalent to the oceans which is determined as a common feature between searcher agents.
-Agents with wind force moves toward the global search in the search space. The value of coefficient determines impact of wind force to all agents.
-Exploration capability is strengthened with wind force in VSO. If the value of is high then wind force increases effect of agents.
-Subgroups behavior formation on search space is described with three state as local search strategies. -Size of radius sense neighbourhood is adjustable in a large area due to the steps of the agents move defined with water and wind forces. But limited radius sense reduces the performance of search algorithm.
-In local search strategies agents move by water force which is similar to water flow described as Sin form. The value of coefficient determines impact of water force to all the agents.
-Best neighbourhood evaluation, personal memory and non-neighbourhood agents in radius sense is the component of local search strategies.
-Local search strategies guarantee the exploitation capability with water force. If the value of is high enough then water force increases effectiveness of the agents.
-It is possible to increase or decrease coefficients of wind and water force in final stages according to the type of optimization problem.
By comparing theoretical differences, it can be concluded that, both PSO and VSO inspired from biological behavior in nature but updating equation movement of agent in VSO is different with PSO. Although VSO determines global best similar to PSO strategies, PSO uses a pbest and gbest to update the velocity while, VSO uses best local neighborhood and personal memories to obtain global best. VSO agents limited to radius neighborhood sense in search space, but, PSO uses direct interaction between agents to exchange information in problem space.
-Genetic Algorithms GAs inspired from biological organisms which is used to solve optimization problems. In GA each individual represents a possible solution which is assigned by a fitness according to problem types.
High fitness of chromosomes produces new generation with mutation and crossover operator.
Mating of the best fitness individuals in the population leads to converge to an optimal solution.
In the next section experimental results of those algorithms are compared with proposed method.

-Gravity Search Algorithms
Gravitational search algorithm is inspired by the law of gravity and the mass concept. Searching agents in this algorithm are a set of objects. According to the laws of physics all materials attract each other by gravity and heavier objects have a greater impact than light objects. In this algorithm, the gravitational force is considered as an interface for data transfer and communicate between objects. Therefore heaviest object is the optimal point of interest in ideal conditions. Objects follow by gravity and moving law in GSA. In the gravity law, any object, absorb other objects by a force directly proportional to their masses and inversely proportional to the square of the distance between them [16].

Experimental results
In this section, proposed method is applied to 26 minimization CEC 2014 standard benchmark functions and its results is compared with the results of GAs and PSO optimization algorithms.
The characteristics of GA, PSO algorithms are described in table 3. In all cases of testing phases, population size on search space is 50 agents (N=50) and maximum number of iterations for each run is set to 500. Obtained results is averaged over 30 independent runs. Benchmark functions conducted on 26 well-known classic functions which their related images are illustrated in the following forms in Fig 10.a and Fig 10.b.

Test 1. Unimodal and Separable benchmarks Functions
Converging to optimal solution is very important in unimodal and separable benchmark functions with high-dimension that can be seen in table 1.

Test 2. Unimodal and non-separable benchmark functions
Probability of being caught in local optimums is high in optimization algorithms limited to radius neighbourhood using unimodal and non-separable with convex function.             Total best of Average mean results can be seen in Table 13 and this results show the proposed algorithm knockout the other two algorithm both in Multimodal and Unimodal (separable or nonseparable) benchmark functions. Since VSO is limited to neighbourhood sense radius it can fall into the local optimum in separable multimodal functions because of their multi-picks and convex structures. This problem is solved with exploration and exploitation parameters so being cough in a local optimum is prevented.
Exploitation capability of VSO is configurable by water impact coefficients. In contrast exploration capability is configurable by wind impact coefficients. According to the principal of searching objective function, if objective function is more methodical then it is recommended to 38 increase exploitation and for less methodical functions (more disturbed changes), it is better to increase exploitation. Concerning Table 13 it can be concluded that VSO has a better performance than GA and PSO regarding to the results.

Conclusion
In recent decades, various biological heuristic optimization algorithm has been developed. In this Population size is considered 50 and for higher values our results will be better compare to classical algorithms.
Funding: There is no funding for this paper.

Conflict of interests:
There is no conflict of interests between authors of this article.

Availability of data and material:
There is no data for this paper.

Availability of code:
Code is available by request.