In recent years, the use of meta-heuristic algorithms has significantly expanded in solving complex computational problems, and they also have many applications in various branches of science.

Unlike exact optimization methods, these approaches look for points that are as close as possible to the global optimum and are called inexact methods [1].

The Grey Wolf Optimization (GWO) algorithm was introduced by Mirjalili et al. [2], which is based on the hunting behavior of wolves and their hierarchy. In this algorithm, the location of the optimal point is calculated based on the location of the three points (alpha, beta, and delta) (the best points in each iteration). Simon proposed the Biogeography-Based Optimization (BBO) algorithm, which is based on the mathematical models of biogeography. In the principle of this algorithm, there are two main operators called migration and mutation before optimization. Each individual in the population is evaluated and then goes through migration and mutation steps to reach the global minimum [3].

The algorithm of bees was proposed by Pham et al. [4], which is inspired by the natural search behavior of bees to find the best flower beds (optimal solution).

Leading honey bees communicate the direction of the food source, the distance of the food source, and the richness of the food source to other bees by their body movements. Choosing the best sites and sending more bees to those sites is the method of implementing the bees algorithm (inspired by the behavior of honey bees) [5].

The Chicken Swarm Optimization (CSO) algorithm was presented by Meng et al. [6], which is inspired by the behavior of roosters, chickens and chicks and is proposed to solve practical problems.

To compare the performance of the wolf algorithm with the three mentioned algorithms, first, their mathematical models are coded by MATLAB 2014 and then implemented for 9 criteria mathematical functions. In order to familiarize ourselves with these new algorithms, the steps (pseudo-code) of optimization algorithms for wolves and chicken swarms are described in Figs. 1 and 2.

It seems that in order to compare different optimization algorithms, their behaviors should be examined when dealing with solving different mathematical functions.

In this research, 9 criteria functions (Sphere, Schwefel2.22, Schwefel2.21, Rosenbrock, Rastrigin, Ackley, Griewank, Penalty#1 and Penalty#2) were selected [7].

Each of the algorithms has been investigated in different conditions: changing the number of generations (200, 1000, 1500) while keeping the number of the population at 30 and changing the number of the population (20, 30, 50) while keeping the number of generations at 1000.

It should be noted that for each of the specific conditions, 20 executions were performed, and the final answer was obtained from the average of 20 executions in order to reduce the errors.

From the 10 prepared tables, 4 examples of these tables are given below.

I. Table 1: fitness functions in repetitions 200 and population number 30.

II. Table 2: fitness functions in repetitions 1500 and population number 30.

III. Table 3: fitness functions in repetitions 1000 and population number 20.

IV. Table 4: fitness functions in repetitions 1000 and population number 50).

According to the results of the comparison of the time to reach the optimal point in 4 algorithms, the GWO algorithm reaches the solution faster than the other stated algorithms in all conditions. However, regarding the fitness value, the results are different as the number of iterations and population size change.

The calculation results showed in the case of the Sphere function, which is a simple function without local optimal points; if we have many or few repetitions or if we have a large or small population, the Wolf algorithm still gives us a better answer than other algorithms (Fig. 3).

The same rule exists for the Schwefel2.21 function because it is a simple function and does not have local optimal points. As a result, in the case of simple functions that do not have local points, the best algorithm for solving is the wolf algorithm (Fig. 4).

In the case of the Schwefel2.22 function, in almost all cases, the chicken swarm algorithm produces a better answer than the rest of the algorithms. This function is a bit more complicated than the Sphere function because it contains both addition and multiplication operations.

On the other hand, this function also has local optimal points. Therefore, the chicken swarm algorithm can be used to solve slightly complex problems with local optimal points (Fig. 5).

The Rosenbrock function is relatively complex, but it does not have local optimal points, and it has no variable multiplication operation. In this function, when we have a population limit or a repetition limit, the gray wolf algorithm still gives us better answers. But when there is no limitation of population and repetitions, the bees algorithm gives us a better answer. The result is that for relatively complex functions where there are no local optimal points, in order to give us a better answer, we need more iterations and more population in the bees algorithm.

In Rastrigin and Ackley functions, in almost all cases, the chicken swarm algorithm gives us a better answer. These functions have many local optimal points, and like the Schwefel2.22 function, which has local optimal points, they get their best solutions from the chicken swarm algorithm (Fig. 6).

In the Griewank function, which is a relatively complex function, since it does not have local optimal points, the gray wolf algorithm still gives us the best answers in situations of high or low amounts of population and repetitions.

Penalty#1 and Penalty#2 functions are very complex functions and have local optimal points.

In the Penalty #1 function, the best answers are created by the biogeography algorithm, and in the Penalty #2 function, the best answers are created by the bees algorithm (Fig. 7).

The result is that the gray wolf algorithm is usually caught in the trap of local optimal points, so it is not suitable for problems that have local optimal points. However, in other problems, due to the high speed of convergence of this algorithm, it is a suitable option for solving.

Unlike the wolf algorithm, the chicken swarm algorithm does not fall into the trap of local optima, and it seems that using the CSA advantages is useful in modifying the wolf algorithm. For complex problems, bees' algorithms and biogeography have the ability to reach optimal points, and they are suitable for such issues.