Monarch butterfly optimization
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Abstract
In nature, the eastern North American monarch population is known for its southward migration during the late summer/autumn from the northern USA and southern Canada to Mexico, covering thousands of miles. By simplifying and idealizing the migration of monarch butterflies, a new kind of nature-inspired metaheuristic algorithm, called monarch butterfly optimization (MBO), a first of its kind, is proposed in this paper. In MBO, all the monarch butterfly individuals are located in two distinct lands, viz. southern Canada and the northern USA (Land 1) and Mexico (Land 2). Accordingly, the positions of the monarch butterflies are updated in two ways. Firstly, the offsprings are generated (position updating) by migration operator, which can be adjusted by the migration ratio. It is followed by tuning the positions for other butterflies by means of butterfly adjusting operator. In order to keep the population unchanged and minimize fitness evaluations, the sum of the newly generated butterflies in these two ways remains equal to the original population. In order to demonstrate the superior performance of the MBO algorithm, a comparative study with five other metaheuristic algorithms through thirty-eight benchmark problems is carried out. The results clearly exhibit the capability of the MBO method toward finding the enhanced function values on most of the benchmark problems with respect to the other five algorithms. Note that the source codes of the proposed MBO algorithm are publicly available at GitHub (https://github.com/ggw0122/Monarch-Butterfly-Optimization, C++/MATLAB) and MATLAB Central (http://www.mathworks.com/matlabcentral/fileexchange/50828-monarch-butterfly-optimization, MATLAB).
Keywords
Evolutionary computation Monarch butterfly optimization Migration Butterfly adjusting operator Benchmark problems1 Introduction
In the areas of computer science, mathematics, control and decision making, a relatively new set of algorithms, called nature-inspired algorithms, has been proposed and used to address an array of complex optimization problems. Among various nature-inspired algorithms, swarm-based algorithms and evolutionary algorithms (EAs) are two of the most representative paradigms.
Swarm-based algorithms, also called swarm intelligence (SI) methods [1], are one of the most well-known paradigms in nature-inspired algorithms which have been widely used in various applications, such as scheduling, directing orbits of chaotic systems [2], wind generator optimization [3] and fault diagnosis [4]. Swarm intelligence (SI) concerns the collective, emerging behavior of multiple, interacting agents who follow some simple rules [5]. Two of widely used SI are particle swarm optimization (PSO) [6, 7, 8, 9, 10] and ant colony optimization (ACO) [11, 12]. The idea of PSO [6] originated from the social behavior of bird flocking when searching for the food. The ants in nature are well capable of keeping the past paths in mind by pheromone. Inspired by this phenomenon, the ACO algorithm [11] is proposed by Dorigo et al. Recently, more superior SI algorithms have been proposed, such as artificial bee colony (ABC) [13, 14], cuckoo search (CS) [15, 16, 17, 18, 19], bat algorithm (BA) [20, 21, 22], grey wolf optimizer (GWO) [23, 24], ant lion optimizer (ALO) [25], firefly algorithm (FA) [26, 27, 28, 29], chicken swarm optimization (CSO) [30] and krill herd (KH) [31, 32, 33]. These are inspired by the swarm behavior of honey bees, cuckoos, bats, grey wolves, chickens and krill, respectively.
By simplifying and idealizing the genetic evolution process, different kinds of EAs have been proposed and used in a wide range of applications. Genetic algorithm (GA) [34, 35], evolutionary programming (EP) [36, 37], genetic programming (GP) [38] and evolutionary strategy (ES) [39] are four of the most classical EAs among them. With the development of the evolutionary theory, some new methods have been proposed over the last decades that significantly improved the theory and search capacities of EAs. Differential evolution (DE) [40, 41] is a very efficient search algorithm that simulates the biological mechanisms of natural selection and mutation. The best-to-survive criteria are adopted in the above algorithms on a population of solutions. Stud genetic algorithm (SGA) [42, 43] is a special kind of GA that uses the best individual and the other randomly selected individuals at each generation for crossover operator. By incorporating the sole effect of predictor variable as well as the interactions between the variables into the GP, Gandomi and Alavi [44] proposed an improved version of GP algorithm, called multi-stage genetic programming (MSGP), for nonlinear system modeling. Recently, motivated by the natural biogeography, Simon has provided the mathematics of biogeography and accordingly proposed a new kind of EA: biogeography-based optimization (BBO) [45, 46, 47, 48, 49]. Inspired by the animal migration behavior, animal migration optimization (AMO) [50] is proposed and compared with other well-known heuristic search methods.
By simulating the migration behavior of the monarch butterflies in nature, a new kind of nature-inspired metaheuristic algorithm, called MBO, is proposed for continuous optimization problems in this paper. In MBO, all the monarch butterfly individuals are idealized and located in two lands only, viz. Southern Canada and the northern USA (Land 1) and Mexico (Land 2). Accordingly, the positions of the monarch butterflies are updated in two ways. At first, the offsprings are generated (position updating) by migration operator, which can be adjusted by the migration ratio. Subsequently, the positions of other butterflies are tuned by butterfly adjusting operator. In other words, the search direction of the monarch butterfly individuals in MBO algorithm is mainly determined by the migration operator and butterfly adjusting operator. Also, migration operator and butterfly adjusting operator can be implemented simultaneously. Therefore, the MBO method is ideally suited for parallel processing and well capable of making trade-off between intensification and diversification, a very important phenomenon in the field of metaheuristics. In order to demonstrate the performance of MBO method, it is compared with five other metaheuristic algorithms through thirty-eight benchmark problems. The results clearly show that the MBO method is able to find the better function values on most benchmark problems as compared to five other metaheuristic algorithms.
The goal of this paper is twofold. Firstly, the new optimization method called MBO is introduced. It is carried out by first studying the migration behavior of monarch butterflies and then generalizing it to formulate a general-purpose metaheuristic method. Secondly, a comparative study of the performance of MBO with respect to other population-based optimization methods is done. This has been addressed by looking at the commonalities and differences from an algorithmic point of view as well as by comparing their performances on an array of benchmark functions.
Section 2 reviews the migration behavior of monarch butterflies in nature, and Sect. 3 discusses how the migration behavior of monarch butterflies can be used to formulate a general-purpose search heuristic. Several simulation results comparing MBO with other optimization methods for general benchmark functions are presented in Sect. 4. Finally, Sect. 5 presents some concluding remarks along with scope for improvements and expansion of the present work.
2 Monarch butterfly and its migration behavior
As one of the most familiar North American butterflies, the monarch butterfly has an orange and black pattern that can be easily recognized [51]. It is a milkweed butterfly in the family Nymphalidae. Female and male monarchs have different wings that can be used to identify them.
The eastern North American monarch is known for its ability of migrating by flying thousands of miles from the USA and southern Canada to Mexico every summer. It involves which flying over west of the Rocky Mountains to California. In order to overwinter, they move thousands of miles to Mexico. Southward movements commence in August and end at the first frost. However, during the spring, opposite things happen. The female ones lay eggs for generating offspring during these movements [52]. Recent research shows some butterflies perform Lévy flight when they migrate or move [53].
3 Monarch butterfly optimization
- 1.
All the monarch butterflies are only located in Land 1 or Land 2. That is to say, monarch butterflies in Land 1 and Land 2 make up the whole monarch butterfly population.
- 2.
Each child monarch butterfly individual is generated by migration operator from monarch butterfly in Land 1 or in Land 2.
- 3.
In order to keep the population unchanged, an old monarch butterfly will pass away once a child is generated. In the MBO method, this can be performed by replacing its parent with newly generated one if it has better fitness as compared to its parent. On the other hand, the newly generated one is liable to be discarded if it does not exhibit better fitness with respect to its parent. Under this scenario, the parent is kept intact and undestroyed.
- 4.
The monarch butterfly individuals with the best fitness moves automatically to the next generation, and they cannot be changed by any operators. This can guarantee that the quality or the effectiveness of the monarch butterfly population will never deteriorate with the increment of generations.
The next subsections will present a snapshot of the migration operator and butterfly adjusting operator.
3.1 Migration operator
Through the above analyses, it can be seen that the MBO method can balance the direction of migration operator by adjusting the ratio p. If p is big, more elements from monarch butterflies in Land 1 will be selected. This indicates that the Subpopulation 1 plays a more important role in newly generated monarch butterfly. If p is small, more elements from monarch butterflies in Land 2 will be selected. This indicates Subpopulation 2 plays a more important role in newly generated monarch butterfly. In the current work, p is set to 5/12 as per migration period. Accordingly, the migration operator can be represented in Algorithm 1. Open image in new window
3.2 Butterfly adjusting operator
3.3 Schematic presentation of MBO algorithm
According to Algorithm 3, firstly, all the parameters are initialized followed by the generation of initial population and evaluation of the same by means of its fitness function. Subsequently, the positions of all monarch butterflies are updated step by step until certain conditions are satisfied. It should be mentioned that, in order to make the population fixed and reduce fitness evaluations, the number of monarch butterflies, generated by migration operator and butterfly adjusting operator, are NP_{1} and NP_{2}, respectively. Open image in new window
4 Simulation results
Benchmark functions
No. | Name | No. | Name |
---|---|---|---|
F01 | Ackley | F20 | Schwefel 2.21 |
F02 | Alpine | F21 | Sphere |
F03 | Brown | F22 | Step |
F04 | Dixon and price | F23 | Sum function |
F05 | Fletcher-Powell | F24 | Zakharov |
F06 | Griewank | F25 | Wavy1 |
F07 | Holzman 2 function | F26 | Beale |
F08 | Levy | F27 | Bohachevsky #1 |
F09 | Pathological function | F28 | Bohachevsky #2 |
F10 | Penalty #1 | F29 | Bohachevsky #3 |
F11 | Penalty #2 | F30 | Booth |
F12 | Perm | F31 | Branin |
F13 | Powel | F32 | Easom |
F14 | Quartic with noise | F33 | Foxholes |
F15 | Rastrigin | F34 | Freudenstein-Roth |
F16 | Rosenbrock | F35 | Goldstein-price |
F17 | Schwefel 2.26 | F36 | Hump |
F18 | Schwefel 1.2 | F37 | Matyas |
F19 | Schwefel 2.22 | F38 | Shubert |
In order to obtain fair results, all the implementations are conducted under the same conditions as shown in [56].
The same parameters for MBO method are set as follows: max step S_{max} = 1.0, butterfly adjusting rate BAR = 5/12, migration period peri = 1.2, and the migration ratio p = 5/12. Note that, for high-dimensional functions (F01–F25), both population size NP and maximum generation MaxGen are set to 50; while for low-dimensional functions (F26–F38), less population size NP and maximum generation MaxGen are used in our experiments, and both of them are set to 30. Accordingly, for F01–F25, NP_{1} and NP_{2} are 21 and 29, respectively; while for F26–F38, NP_{1} and NP_{2} are 13 and 17, respectively.
Metaheuristic algorithms are based on certain distribution, and hence, 200 independent trials have been made with the aim of decreasing the influence of the randomness. In the following experiments, the optimal solution for each test problem is highlighted. For all the tables, the total number of functions in which the optimization method has the best performance is provided in the last row. It is worth mentioning that the scales which are used to normalize values in the tables are fully different with each other and hence values from different tables are not comparative. The detailed normalization process can be defined as follows:
Firstly, the real function values are presented by matrix \( {\mathbf{A}}_{m \times n} = [A_{1} , \, A_{2} , \ldots , \, A_{i} , \ldots , \, A_{m} ]^{T} \). A_{i} means the ith row, and a_{ij} is an item in A, 1 ≤ i ≤ m, 1 ≤ j ≤ n. Here, m = 38 and n = 6 denote the number of benchmark functions and algorithms used in this paper, respectively.
4.1 Comparisons of the MBO method with other methods
Mean, best and worst function values obtained by six methods
ABC | ACO | BBO | DE | MBO | SGA | |
---|---|---|---|---|---|---|
F01 | ||||||
Mean | 2.16 | 2.45 | 1.33 | 1.95 | 1.00 | 1.42 |
Best | 1.2E5 | 1.4E5 | 6.2E4 | 1.2E5 | 1.00 | 7.2E4 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F02 | ||||||
Mean | 11.80 | 19.87 | 4.07 | 23.27 | 1.00 | 5.16 |
Best | 2.8E5 | 5.2E5 | 8.7E4 | 7.1E5 | 1.00 | 8.0E4 |
Worst | 1.00 | 3.09 | 3.09 | 3.09 | 3.09 | 3.09 |
F03 | ||||||
Mean | 8.80 | 12.55 | 1.39 | 2.16 | 2.42 | 1.00 |
Best | 3.4E7 | 2.4E8 | 1.6E7 | 3.5E7 | 1.00 | 1.3E7 |
Worst | 1.10 | 1.00 | 1.59 | 1.59 | 1.59 | 1.59 |
F04 | ||||||
Mean | 31.96 | 53.35 | 2.25 | 10.18 | 16.04 | 1.00 |
Best | 1.5E4 | 3.0E4 | 783.47 | 6.3E3 | 1.00 | 272.34 |
Worst | 371.34 | 618.56 | 10.60 | 266.90 | 1.00 | 83.07 |
F05 | ||||||
Mean | 2.61 | 10.04 | 1.00 | 3.97 | 5.80 | 1.20 |
Best | 2.51 | 21.50 | 1.00 | 7.73 | 9.32 | 1.34 |
Worst | 5.39 | 23.31 | 1.00 | 8.60 | 8.86 | 7.36 |
F06 | ||||||
Mean | 4.70 | 1.57 | 1.04 | 2.54 | 1.68 | 1.00 |
Best | 15.77 | 4.98 | 3.17 | 13.62 | 1.00 | 3.44 |
Worst | 27.98 | 12.97 | 4.16 | 19.64 | 1.00 | 12.93 |
F07 | ||||||
Mean | 33.21 | 56.60 | 2.94 | 12.93 | 16.01 | 1.00 |
Best | 2.9E16 | 1.8E17 | 1.7E15 | 7.6E16 | 1.00 | 9.5E14 |
Worst | 1.8E3 | 2.7E3 | 21.74 | 112.65 | 1.00 | 150.45 |
F08 | ||||||
Mean | 6.29 | 10.61 | 1.41 | 7.67 | 1.28 | 1.00 |
Best | 1.6E8 | 5.1E8 | 5.7E7 | 4.7E8 | 1.00 | 4.8E7 |
Worst | 1.00 | 1.32 | 1.10 | 1.11 | 1.10 | 1.10 |
F09 | ||||||
Mean | 3.47 | 3.62 | 2.92 | 1.90 | 1.00 | 2.62 |
Best | 195.13 | 216.54 | 164.82 | 75.47 | 1.00 | 137.17 |
Worst | 1.13 | 1.00 | 2.66 | 2.66 | 2.66 | 2.66 |
F10 | ||||||
Mean | 6.0E3 | 1.8E5 | 81.64 | 707.18 | 79.71 | 1.00 |
Best | 1.0E17 | 1.00 | 2.4E16 | 8.7E16 | 5.9E6 | 1.1E16 |
Worst | 3.2E5 | 1.00 | 77.01 | 7.3E4 | 2.36 | 2.36 |
F11 | ||||||
Mean | 383.00 | 6.7E3 | 14.05 | 95.25 | 48.23 | 1.00 |
Best | 2.8E20 | 1.00 | 1.5E17 | 1.2E21 | 8.8E7 | 4.7E16 |
Worst | 7.9E5 | 3.8E7 | 4.0E3 | 8.9E4 | 1.00 | 1.2E3 |
F12 | ||||||
Mean | 7.3E4 | 5.2E4 | 4.1E5 | 1.00 | 1.4E5 | 1.1E4 |
Best | 2.8E4 | 5.9E8 | 5.9E8 | 1.00 | 6.0E3 | 5.9E8 |
Worst | 1.6E6 | 7.8E5 | 1.0E8 | 1.00 | 7.8E5 | 7.8E5 |
F13 | ||||||
Mean | 5.39 | 30.11 | 1.98 | 14.09 | 3.27 | 1.00 |
Best | 1.5E6 | 5.4E6 | 5.2E5 | 6.8E6 | 1.00 | 2.0E5 |
Worst | 11.91 | 16.14 | 1.51 | 21.06 | 1.00 | 1.33 |
F14 | ||||||
Mean | 32.44 | 33.33 | 2.75 | 13.53 | 16.31 | 1.00 |
Best | 7.0E13 | 4.4E14 | 2.5E13 | 3.6E14 | 1.00 | 6.2E12 |
Worst | 1.00 | 18.16 | 18.16 | 18.16 | 18.16 | 18.16 |
F15 | ||||||
Mean | 2.58 | 4.76 | 1.03 | 4.19 | 1.00 | 1.37 |
Best | 3.6E6 | 9.9E6 | 1.5E6 | 9.0E6 | 1.00 | 2.1E6 |
Worst | 3.97 | 7.89 | 1.00 | 5.94 | 3.10 | 1.77 |
F16 | ||||||
Mean | 6.24 | 35.89 | 2.14 | 5.56 | 1.00 | 2.21 |
Best | 2.3E5 | 1.1E6 | 7.1E4 | 2.4E5 | 1.00 | 8.3E4 |
Worst | 25.84 | 81.22 | 9.43 | 21.26 | 1.00 | 5.26 |
F17 | ||||||
Mean | 3.07 | 2.07 | 1.00 | 3.97 | 1.72 | 1.15 |
Best | 7.4E6 | 2.4E6 | 1.2E6 | 1.0E7 | 1.00 | 1.2E6 |
Worst | 4.73 | 3.36 | 1.00 | 5.66 | 3.71 | 1.61 |
F18 | ||||||
Mean | 1.82 | 1.69 | 1.00 | 2.27 | 1.44 | 1.47 |
Best | 1.9E8 | 1.5E8 | 7.6E7 | 2.3E8 | 1.00 | 1.1E8 |
Worst | 1.51 | 2.06 | 1.00 | 1.85 | 1.17 | 1.62 |
F19 | ||||||
Mean | 2.57 | 7.06 | 1.00 | 3.07 | 1.19 | 1.42 |
Best | 4.6E4 | 8.5E4 | 1.8E4 | 6.6E4 | 1.00 | 3.0E4 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F20 | ||||||
Mean | 3.66 | 2.32 | 2.58 | 2.97 | 1.00 | 2.17 |
Best | 1.5E3 | 872.07 | 847.55 | 1.3E3 | 1.00 | 693.45 |
Worst | 1.00 | 160.40 | 160.40 | 160.40 | 160.40 | 160.40 |
F21 | ||||||
Mean | 5.11 | 13.22 | 1.00 | 2.58 | 1.27 | 1.08 |
Best | 3.0E7 | 7.0E7 | 3.2E6 | 2.0E7 | 1.00 | 4.1E6 |
Worst | 1.00 | 5.44 | 21.21 | 21.21 | 21.21 | 21.21 |
F22 | ||||||
Mean | 5.76 | 2.66 | 1.17 | 2.94 | 2.66 | 1.00 |
Best | 3.9E18 | 2.3E18 | 1.5E18 | 3.9E18 | 1.00 | 8.6E17 |
Worst | 3.03 | 1.23 | 1.00 | 2.40 | 2.40 | 2.40 |
F23 | ||||||
Mean | 4.98 | 10.92 | 1.06 | 2.31 | 2.22 | 1.00 |
Best | 7.4E6 | 1.3E7 | 1.3E6 | 4.2E6 | 1.00 | 1.2E6 |
Worst | 5.45 | 17.28 | 1.00 | 5.18 | 2.24 | 1.39 |
F24 | ||||||
Mean | 1.72 | 1.3E4 | 1.00 | 2.09 | 1.99 | 1.57 |
Best | 4.0E5 | 3.1E5 | 1.7E5 | 5.9E5 | 1.00 | 2.6E5 |
Worst | 2.82 | 1.15 | 1.59 | 2.91 | 1.00 | 17.13 |
F25 | ||||||
Mean | 2.33 | 1.82 | 1.00 | 2.49 | 1.18 | 1.05 |
Best | 1.7E5 | 1.1E5 | 6.6E4 | 1.9E5 | 1.00 | 5.5E4 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F26 | ||||||
Mean | 1.01 | 1.03 | 1.07 | 1.00 | 1.44 | 1.14 |
Best | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.80 | 1.00 |
F27 | ||||||
Mean | 1.00 | 2.27 | 5.91 | 1.13 | 69.49 | 2.01 |
Best | 1.00 | 1.13 | 1.30 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 1.63 | 3.55 | 1.15 | 293.62 | 2.05 |
F28 | ||||||
Mean | 1.00 | 1.99 | 6.25 | 1.10 | 58.76 | 1.45 |
Best | 1.00 | 1.00 | 1.24 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 3.20 | 8.77 | 1.11 | 1.46 | 1.42 |
F29 | ||||||
Mean | 1.00 | 1.93 | 7.06 | 1.07 | 83.07 | 1.54 |
Best | 1.00 | 1.04 | 1.04 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 3.44 | 3.78 | 1.09 | 4.43 | 1.40 |
F30 | ||||||
Mean | 1.03 | 1.03 | 1.23 | 1.00 | 2.08 | 1.03 |
Best | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Worst | 1.02 | 1.05 | 1.36 | 1.00 | 1.98 | 1.98 |
F31 | ||||||
Mean | 1.00 | 1.04 | 1.09 | 1.06 | 1.46 | 1.05 |
Best | 1.00 | 1.01 | 1.01 | 1.00 | 1.00 | 1.01 |
Worst | 1.00 | 1.13 | 1.02 | 1.07 | 1.23 | 2.77 |
F32 | ||||||
Mean | 1.00 | 1.07 | 1.13 | 1.09 | 1.04 | 1.05 |
Best | 1.5E3 | 2.0E3 | 1.2E4 | 3.2E3 | 1.00 | 91.68 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F33 | ||||||
Mean | 1.00 | 1.00 | 1.01 | 1.00 | 5.80 | 1.00 |
Best | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F34 | ||||||
Mean | 2.82 | 2.50 | 9.35 | 2.30 | 15.43 | 1.00 |
Best | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 1.00 | 3.55 | 66.68 | 3.55 | 3.55 |
F35 | ||||||
Mean | 1.43 | 1.09 | 1.58 | 1.00 | 4.74 | 2.33 |
Best | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 1.00 | 1.97 | 1.02 | 6.13 | 1.00 |
F36 | ||||||
Mean | 1.00 | 1.01 | 1.02 | 1.00 | 1.19 | 1.01 |
Best | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 303.21 | 303.21 |
F37 | ||||||
Mean | 1.01 | 1.01 | 1.02 | 1.00 | 1.02 | 1.01 |
Best | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Worst | 1.01 | 1.00 | 1.00 | 1.00 | 1.08 | 2.05 |
F38 | ||||||
Mean | 1.00 | 63.93 | 34.42 | 31.31 | 82.54 | 2.76 |
Best | 2.54 | 452.25 | 452.25 | 237.31 | 1.00 | 452.25 |
Worst | 1.00 | 21.89 | 1.00 | 2.81 | 1.00 | 1.00 |
Total | ||||||
Mean | 8 | 1 | 7 | 7 | 6 | 13 |
Best | 11 | 10 | 8 | 12 | 34 | 10 |
Worst | 19 | 13 | 15 | 10 | 13 | 8 |
From Table 2, it can be seen that, on average, MBO has the fifth performance on six out of thirty-eight benchmarks. SGA, ABC, BBO and DE have better performance than MBO method, and they rank 1, 2, 3 and 4, respectively.
For the best solutions, Table 2 shows that MBO is well capable of finding the optimal solutions on thirty-four out of thirty-eight benchmarks. DE and ABC can search for the best solutions on twelve and eleven out of thirty-eight benchmarks. Generally, ABC, ACO, DE and SGA have identical performance with respect to each other and this is more evident for ACO and SGA. By carefully looking at the Table 2, we found that, for twenty-five high-dimensional complicated functions, MBO has absolute advantage over the other five methods. For thirteen low-dimensional functions, all the methods can solve them well under the given conditions.
For the worst function values shown in Table 2, the first two algorithms are ABC and BBO, and they perform the best on nineteen and fifteen out of thirty-eight benchmarks, respectively. MBO and ACO are well capable of finding the best solutions on thirteen functions, which are nothing but inferior to the above two methods.
From Table 2, for low-dimensional benchmarks (D = 2), the performance of the six methods has little difference. In particular, ABC and DE are the two best methods when dealing with 2D functions. For high-dimensional benchmarks (D = 20), especially for the best average performance, MBO is the best method at searching for the optimal function values.
From Fig. 2, there is no ambiguity in concluding that even though all the methods commence the process of optimization at the beginning of same fitness, MBO can find the satisfactory function values faster and the function values found by MBO are always smaller than other five methods during the whole optimization process.
For this test problem, it is obvious that, though BBO and SGA perform well and have the similar performance, both of them are outperformed by MBO from the beginning of the optimization process. Looking carefully at generations 1–5 from Fig. 3, MBO has a faster convergence than other methods.
As shown in Fig. 4, it is obvious that the convergent trajectory of MBO is far away from others. This indicates that MBO significantly outperforms other five methods for Pathological function. Furthermore, DE and SGA rank 2 and 3 among six methods.
For this case, though MBO and SGA have similar final fitness value, the convergent trajectory of MBO is always below SGA. This implies that the fitness found by MBO is better than SGA during the convergent process.
For this case as shown in Fig. 6, SGA and BBO have similar convergent trajectories and final fitness values, and both of them are only inferior to MBO method. MBO method has better fitness from generation 1 to generation 50.
For this case, the convergent trajectory of MBO is far below other methods, and this indicates MBO has found much better function value than other five comparative methods. Furthermore, for other five methods, SGA and ACO are only worse than MBO and rank 2 and 3, respectively.
From the above analyses about the Figs. 2, 3, 4, 5, 6 and 7, we can arrive at a conclusion that MBO algorithm significantly outperforms the other five comparative algorithms. On most of the occasions, SGA and BBO have better performance as compared to the other methods, but worse only than that of MBO.
4.2 Comparisons with other optimization methods by using t test
Comparisons between MBO and other methods at α = 0.05 on a two-tailed t tests
ABC | ACO | BBO | DE | SGA | |
---|---|---|---|---|---|
F01 | 14.67 | 18.19 | 4.18 | 12.13 | 5.34 |
F02 | 31.91 | 42.91 | 10.61 | 58.56 | 13.55 |
F03 | 3.30 | 5.22 | −0.54 | −0.13 | −0.75 |
F04 | 2.88 | 6.42 | −2.68 | −1.14 | −2.93 |
F05 | −18.14 | 16.35 | −28.89 | −10.51 | −27.46 |
F06 | 12.87 | −0.51 | −3.00 | 4.05 | −3.22 |
F07 | 3.25 | 6.72 | −2.67 | −0.63 | −3.07 |
F08 | 12.73 | 18.77 | 0.34 | 16.31 | −0.80 |
F09 | 39.70 | 42.85 | 30.18 | 13.44 | 24.70 |
F10 | 9.52 | 7.01 | 0.04 | 8.27 | −1.87 |
F11 | 8.28 | 8.24 | −1.27 | 1.72 | −1.77 |
F12 | −2.23 | −3.10 | 4.60 | −5.16 | −4.75 |
F13 | 2.85 | 21.29 | −1.76 | 13.25 | −3.15 |
F14 | 2.92 | 3.21 | −2.72 | −0.55 | −3.07 |
F15 | 13.87 | 31.34 | 0.27 | 27.77 | 3.31 |
F16 | 13.59 | 42.65 | 3.23 | 12.44 | 3.43 |
F17 | 8.96 | 2.25 | −4.83 | 14.98 | −3.76 |
F18 | 2.92 | 1.90 | −3.36 | 6.34 | 0.24 |
F19 | 7.85 | 29.35 | −1.14 | 10.70 | 1.30 |
F20 | 31.38 | 15.85 | 17.71 | 24.92 | 12.26 |
F21 | 9.63 | 23.83 | −0.73 | 3.44 | −0.52 |
F22 | 6.88 | 0.02 | −3.50 | 0.66 | −3.89 |
F23 | 6.50 | 16.69 | −2.85 | 0.23 | −2.99 |
F24 | −1.27 | 1.87 | −4.67 | 0.50 | −1.95 |
F25 | 9.06 | 5.00 | −1.41 | 10.31 | −0.99 |
F26 | −5.72 | −5.30 | −4.72 | −5.79 | −3.46 |
F27 | −4.36 | −4.27 | −4.03 | −4.35 | −4.29 |
F28 | −5.04 | −4.95 | −4.57 | −5.03 | −5.00 |
F29 | −3.74 | −3.69 | −3.45 | −3.73 | −3.71 |
F30 | −4.29 | −4.29 | −3.41 | −4.40 | −4.28 |
F31 | −4.38 | −3.97 | −3.41 | −3.73 | −3.93 |
F32 | −0.64 | 0.65 | 2.14 | 0.99 | 0.15 |
F33 | −1.55 | −1.55 | −1.55 | −1.55 | −1.55 |
F34 | −3.86 | −4.24 | −1.70 | −4.03 | −4.82 |
F35 | −4.54 | −5.27 | −4.48 | −5.42 | −2.95 |
F36 | −3.18 | −3.07 | −2.83 | −3.17 | −3.05 |
F37 | −1.61 | −2.62 | −0.73 | −3.52 | −1.85 |
F38 | −5.99 | −1.07 | −3.19 | −3.58 | −5.85 |
Better | 22 | 20 | 7 | 15 | 6 |
Equal | 4 | 7 | 12 | 10 | 12 |
Worse | 12 | 11 | 19 | 13 | 20 |
4.3 The study of the number of fitness evaluations
In order to further investigate the performance of MBO method, fitness evaluations are also studied from the following two respects.
4.3.1 Fixed fitness
The number of fitness evaluations for different methods
ABC | ACO | BBO | DE | MBO | SGA | |
---|---|---|---|---|---|---|
F01 | 47,232 | 50,000 | 36,145 | 18,090 | 19,085 | 33,135 |
F02 | 25,480 | 50,000 | 5585 | 27,450 | 1680 | 4500 |
F03 | 16,922 | 50,000 | 3315 | 6880 | 1365 | 2740 |
F04 | 50,000 | 50,000 | 48,835 | 27,460 | 42,860 | 47,350 |
F05 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 |
F06 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 |
F07 | 21,787 | 50,000 | 11,130 | 13,990 | 7235 | 6055 |
F08 | 17,710 | 50,000 | 4405 | 12,540 | 1135 | 3850 |
F09 | 50,000 | 50,000 | 50,000 | 13,600 | 3235 | 50,000 |
F10 | 21,455 | 45,005 | 8630 | 17,490 | 35,340 | 5435 |
F11 | 27,020 | 50,000 | 16,700 | 18,890 | 35,515 | 9440 |
F12 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 |
F13 | 46,777 | 50,000 | 30,730 | 46,900 | 36,310 | 22,670 |
F14 | 7,875 | 2485 | 830 | 4230 | 860 | 940 |
F15 | 50,000 | 50,000 | 34,125 | 50,000 | 26,420 | 35,035 |
F16 | 50,000 | 50,000 | 50,000 | 50,000 | 40,595 | 50,000 |
F17 | 50,000 | 50,000 | 50,000 | 50,000 | 45,135 | 44,690 |
F18 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 | 50,000 |
F19 | 26,565 | 50,000 | 16,245 | 15,590 | 2420 | 18,125 |
F20 | 50,000 | 50,000 | 50,000 | 46,810 | 50,000 | 50,000 |
F21 | 16,152 | 50,000 | 3640 | 8550 | 1520 | 4015 |
F22 | 37,205 | 50,000 | 48,995 | 19,920 | 31,260 | 26,400 |
F23 | 27,230 | 50,000 | 22,345 | 14,830 | 18,560 | 14,020 |
F24 | 50,000 | 50,000 | 50,000 | 50,000 | 45,230 | 50,000 |
F25 | 48,440 | 50,000 | 50,000 | 29,150 | 28,325 | 50,000 |
Total | 4 | 4 | 5 | 8 | 14 | 10 |
4.3.2 Fixed fitness evaluations
Mean function values with the fixed evaluations
ABC | ACO | BBO | DE | MBO | SGA | |
---|---|---|---|---|---|---|
F01 | ||||||
Mean | 7.02 | 5.35 | 2.11 | 4.84 | 1.00 | 2.04 |
Best | 1.3E4 | 9.3E3 | 3.5E3 | 8.8E3 | 1.00 | 3.1E3 |
F02 | ||||||
Mean | 1.1E3 | 732.21 | 76.42 | 1.2E3 | 1.00 | 72.60 |
Best | 1.4E5 | 8.6E4 | 5.9E3 | 1.3E5 | 1.00 | 4.5E3 |
F03 | ||||||
Mean | 1.7E3 | 566.72 | 20.46 | 116.53 | 1.00 | 10.42 |
Best | 8.4E6 | 5.4E6 | 9.5E4 | 9.1E5 | 1.00 | 4.4E4 |
F04 | ||||||
Mean | 3.1E3 | 233.30 | 3.48 | 114.24 | 978.41 | 1.00 |
Best | 9.8E4 | 6.5E3 | 54.42 | 4.6E3 | 1.00 | 32.11 |
F05 | ||||||
Mean | 7.34 | 16.02 | 1.00 | 6.17 | 2.60 | 1.19 |
Best | 17.45 | 40.36 | 1.00 | 20.76 | 3.07 | 2.49 |
F06 | ||||||
Mean | 50.64 | 3.16 | 1.50 | 8.48 | 31.10 | 1.00 |
Best | 50.57 | 2.40 | 1.45 | 8.53 | 1.00 | 1.39 |
F07 | ||||||
Mean | 1.2E4 | 802.84 | 7.85 | 599.97 | 811.66 | 1.00 |
Best | 4.0E13 | 3.6E12 | 3.0E10 | 4.2E12 | 1.00 | 1.6E9 |
F08 | ||||||
Mean | 52.55 | 20.08 | 1.52 | 23.65 | 4.54 | 1.00 |
Best | 1.6E8 | 4.8E7 | 3.8E6 | 6.7E7 | 1.00 | 3.3E6 |
F09 | ||||||
Mean | 18.80 | 17.50 | 11.08 | 7.95 | 1.00 | 10.16 |
Best | 270.96 | 242.87 | 145.71 | 65.85 | 1.00 | 128.36 |
F10 | ||||||
Mean | 2.0E7 | 1.4E8 | 3.26 | 1.1E5 | 5.8E5 | 1.00 |
Best | 1.8E14 | 1.0E8 | 4.1E8 | 4.6E10 | 1.00 | 1.3E8 |
F11 | ||||||
Mean | 1.2E7 | 4.2E7 | 21.31 | 2.0E5 | 1.8E4 | 1.00 |
Best | 6.4E14 | 6.8E7 | 2.4E8 | 1.5E12 | 1.00 | 6.5E7 |
F12 | ||||||
Mean | 1.2E6 | 1.6E5 | 4.5E6 | 1.00 | 1.9E6 | 7.3E4 |
Best | 6.5E5 | 2.7E9 | 2.7E9 | 1.00 | 1.0E6 | 2.7E9 |
F13 | ||||||
Mean | 70.58 | 100.47 | 2.04 | 60.12 | 28.39 | 1.00 |
Best | 8.3E5 | 1.6E6 | 2.2E4 | 1.2E6 | 1.00 | 1.0E4 |
F14 | ||||||
Mean | 7.6E3 | 294.50 | 6.56 | 389.58 | 367.33 | 1.00 |
Best | 6.3E12 | 2.7E11 | 3.6E9 | 7.4E11 | 1.00 | 3.4E8 |
F15 | ||||||
Mean | 10.93 | 11.90 | 1.50 | 12.06 | 1.00 | 2.10 |
Best | 2.3E6 | 2.4E6 | 2.7E5 | 2.8E6 | 1.00 | 3.7E5 |
F16 | ||||||
Mean | 10.86 | 22.38 | 1.06 | 3.48 | 1.00 | 1.03 |
Best | 2.2E5 | 4.2E5 | 1.5E4 | 6.3E4 | 1.00 | 1.3E4 |
F17 | ||||||
Mean | 13.17 | 3.34 | 1.34 | 13.01 | 7.82 | 1.00 |
Best | 1.1E7 | 1.4E6 | 5.7E5 | 1.0E7 | 1.00 | 3.5E5 |
F18 | ||||||
Mean | 2.96 | 1.69 | 1.00 | 3.06 | 2.28 | 1.57 |
Best | 1.9E3 | 609.84 | 570.97 | 2.1E3 | 1.00 | 738.67 |
F19 | ||||||
Mean | 10.41 | 14.27 | 1.12 | 6.08 | 1.00 | 1.42 |
Best | 2.0E4 | 2.4E4 | 1.9E3 | 1.3E4 | 1.00 | 2.5E3 |
F20 | ||||||
Mean | 2.53 | 1.26 | 1.32 | 1.70 | 1.08 | 1.00 |
Best | 108.74 | 33.67 | 39.73 | 69.79 | 1.00 | 23.21 |
F21 | ||||||
Mean | 3.0E6 | 1.7E6 | 5.3E4 | 4.2E5 | 1.00 | 3.6E4 |
Best | 1.2E8 | 6.1E7 | 1.3E6 | 1.6E7 | 1.00 | 8.5E5 |
F22 | ||||||
Mean | 124.97 | 8.69 | 2.58 | 19.84 | 27.72 | 1.00 |
Best | 2.0E19 | 1.6E18 | 2.9E17 | 3.5E18 | 1.00 | 1.6E17 |
F23 | ||||||
Mean | 86.30 | 59.00 | 1.74 | 11.94 | 6.44 | 1.00 |
Best | 3.0E7 | 1.7E7 | 2.0E5 | 4.8E6 | 1.00 | 1.9E5 |
F24 | ||||||
Mean | 3.03 | 4.1E4 | 1.28 | 3.50 | 1.00 | 1.76 |
Best | 6.9E4 | 3.2E4 | 1.9E4 | 8.6E4 | 1.00 | 2.2E4 |
F25 | ||||||
Mean | 8.23 | 2.61 | 1.10 | 5.00 | 2.74 | 1.00 |
Best | 6.3E4 | 1.9E4 | 7.1E3 | 4.3E4 | 1.00 | 5.1E3 |
Total | ||||||
Mean | 0 | 0 | 2 | 1 | 9 | 13 |
Best | 0 | 0 | 1 | 1 | 23 | 0 |
From Fig. 8, it can be seen that, BBO and SGA have almost the same fitness when using the same fitness evaluations. MBO has better fitness than BBO and SGA when using the same fitness evaluations during the optimization process.
For this test problem, though MBO has the similar performance with BBO and SGA that perform well too at the beginning of the search, MBO outperforms them after 100 fitness evaluations. Finally, MBO has far better fitness than BBO and SGA.
As shown in Fig. 10, similarly with Figs. 8 and 9, though SGA has better fitness than BBO finally, both of them have the similar convergent trend with the same fitness evaluations. MBO has better fitness than other five methods, including BBO and SGA.
For this case, MBO has far better fitness than other methods with the same fitness evaluations. For other methods, DE, SGA and BBO rank 2, 3, and 4 among six methods, respectively. In addition, ACO and ABC have the similar convergent curves that indicate their fitness decrease little with the increment of fitness evaluations.
For this case as shown in Fig. 12, at first glance, the six methods can easily be divided into two groups: MBO, BBO, SGA and ABC, ACO, DE. It is quite clear that the first group has far better fitness than the second one with the same fitness evaluations. In more detail, MBO and ABC have the best fitness among three methods in group one and group two, respectively.
Similar to Rastrigin function as shown in Fig. 12, the six methods can be divided into two groups: MBO, BBO, SGA and DE, ABC, ACO. It is quite clear that the first group has far better fitness than the second one with the same fitness evaluations. Furthermore, MBO and DE have the best fitness among three methods in group one and group two, respectively.
From above analyses about the Figs. 8, 9, 10, 11, 12 and 13, we can infer that MBO algorithm is well capable of finding far better function values with the same fitness evaluations. For the other methods, in most cases, SGA and BBO have better fitness as compared to the rest with the same fitness evaluations though they are worse as compared to that of MBO.
4.4 Comparisons of the MBO method with other methods on higher dimensions
In order to further investigate the performance of MBO method, it is further investigated on twenty-five higher-dimensional functions. Population size and maximum generations are set to 100. For other parameters used in this section, they are the same as the Sect. 4.1, except the dimension of test functions.
4.4.1 D = 60
Mean, best and worst function values obtained by six methods when D = 60
ABC | ACO | BBO | DE | MBO | SGA | |
---|---|---|---|---|---|---|
F01 | ||||||
Mean | 4.18 | 4.80 | 2.09 | 4.43 | 1.00 | 2.66 |
Best | 6.8E4 | 8.0E4 | 3.0E4 | 7.5E4 | 1.00 | 3.9E4 |
Worst | 1.00 | 1.10 | 1.00 | 1.02 | 1.00 | 1.00 |
F02 | ||||||
Mean | 84.47 | 158.42 | 14.21 | 158.89 | 1.00 | 30.83 |
Best | 1.1E6 | 2.5E6 | 1.9E5 | 2.6E6 | 1.00 | 3.8E5 |
Worst | 2.56 | 4.27 | 1.00 | 4.66 | 1.00 | 1.00 |
F03 | ||||||
Mean | 1.3E5 | 1.0E10 | 3.3E3 | 2.1E4 | 1.00 | 2.4E3 |
Best | 6.4E17 | 1.00 | 2.8E16 | 2.2E17 | 2.3E9 | 2.6E16 |
Worst | 20.30 | 2.28 | 1.00 | 5.13 | 1.00 | 1.00 |
F04 | ||||||
Mean | 188.93 | 614.44 | 2.04 | 127.94 | 1.00 | 2.43 |
Best | 3.8E6 | 1.8E7 | 2.9E4 | 3.3E6 | 1.00 | 3.0E4 |
Worst | 1.2E5 | 3.0E5 | 768.16 | 5.0E4 | 1.00 | 1.4E3 |
F05 | ||||||
Mean | 5.05 | 16.32 | 1.00 | 7.42 | 11.24 | 1.24 |
Best | 4.69 | 17.75 | 1.00 | 9.00 | 11.47 | 1.05 |
Worst | 6.64 | 18.58 | 1.00 | 9.16 | 15.36 | 2.06 |
F06 | ||||||
Mean | 15.82 | 2.85 | 1.04 | 14.27 | 1.00 | 1.87 |
Best | 250.71 | 47.38 | 15.12 | 286.12 | 1.00 | 27.79 |
Worst | 20.67 | 2.54 | 1.00 | 15.98 | 3.19 | 2.37 |
F07 | ||||||
Mean | 661.78 | 2.3E3 | 9.38 | 523.02 | 1.00 | 10.92 |
Best | 4.9E18 | 2.0E19 | 3.4E16 | 4.1E18 | 1.00 | 4.9E16 |
Worst | 6.8E3 | 2.9E4 | 54.63 | 3.9E3 | 1.00 | 110.50 |
F08 | ||||||
Mean | 14.23 | 35.30 | 1.00 | 24.00 | 1.36 | 1.67 |
Best | 4.1E8 | 9.6E8 | 2.4E7 | 7.0E8 | 1.00 | 4.0E7 |
Worst | 1.00 | 2.74 | 1.00 | 2.14 | 1.00 | 1.00 |
F09 | ||||||
Mean | 8.28 | 7.95 | 6.29 | 6.70 | 1.00 | 5.66 |
Best | 1.8E3 | 1.7E3 | 1.3E3 | 1.4E3 | 1.00 | 1.1E3 |
Worst | 1.19 | 1.14 | 1.00 | 1.00 | 1.00 | 1.00 |
F10 | ||||||
Mean | 1.7E7 | 2.8E7 | 1.6E4 | 1.5E7 | 1.00 | 978.27 |
Best | 4.7E17 | 1.8E8 | 4.5E11 | 3.4E17 | 1.00 | 1.8E11 |
Worst | 2.3E6 | 2.5E6 | 380.40 | 2.2E6 | 1.00 | 1.71 |
F11 | ||||||
Mean | 7.2E3 | 1.4E4 | 35.00 | 6.1E3 | 1.00 | 11.93 |
Best | 3.3E16 | 1.0E9 | 1.9E13 | 2.8E16 | 1.00 | 6.9E12 |
Worst | 5.0E7 | 1.00 | 5.6E4 | 4.5E7 | 2.7E3 | 6.7E4 |
F12 | ||||||
Mean | 7.4E10 | 2.2E10 | 3.2E11 | 1.00 | 7.9E10 | 4.0E9 |
Best | 1.2E13 | 7.0E14 | 7.0E14 | 1.00 | 1.0E13 | 7.0E14 |
Worst | 1.0E11 | 1.0E11 | 4.2E12 | 1.00 | 5.6E11 | 1.1E10 |
F13 | ||||||
Mean | 148.91 | 165.11 | 14.29 | 390.71 | 1.00 | 10.28 |
Best | 2.5E9 | 3.6E9 | 9.2E7 | 7.2E9 | 1.00 | 1.4E8 |
Worst | 17.44 | 16.18 | 1.14 | 46.21 | 1.00 | 1.06 |
F14 | ||||||
Mean | 105.27 | 306.90 | 1.00 | 79.78 | 4.33 | 1.51 |
Best | 2.1E17 | 7.6E17 | 2.2E15 | 2.2E17 | 1.00 | 9.8E14 |
Worst | 2.49 | 6.51 | 1.00 | 2.04 | 1.00 | 1.00 |
F15 | ||||||
Mean | 7.07 | 14.69 | 1.74 | 11.12 | 1.00 | 3.81 |
Best | 4.1E7 | 8.9E7 | 7.4E6 | 6.4E7 | 1.00 | 1.9E7 |
Worst | 4.00 | 9.45 | 1.00 | 6.26 | 2.05 | 2.39 |
F16 | ||||||
Mean | 10.71 | 48.30 | 1.03 | 12.17 | 1.00 | 1.27 |
Best | 3.2E5 | 1.6E6 | 3.2E4 | 3.8E5 | 1.00 | 3.3E4 |
Worst | 37.63 | 224.41 | 5.41 | 43.88 | 1.00 | 4.23 |
F17 | ||||||
Mean | 7.06 | 4.06 | 1.71 | 9.47 | 1.00 | 2.72 |
Best | 1.2E7 | 5.4E6 | 2.1E6 | 1.8E7 | 1.00 | 3.5E6 |
Worst | 84.64 | 48.52 | 26.79 | 108.91 | 1.00 | 37.36 |
F18 | ||||||
Mean | 1.77 | 2.85 | 1.00 | 2.67 | 2.21 | 1.13 |
Best | 378.19 | 607.80 | 174.11 | 611.18 | 1.00 | 157.11 |
Worst | 485.06 | 661.79 | 249.21 | 783.07 | 1.00 | 300.79 |
F19 | ||||||
Mean | 2.3E3 | 4.6E3 | 411.10 | 3.1E3 | 1.00 | 993.53 |
Best | 1.2E5 | 2.5E5 | 1.9E4 | 1.6E5 | 1.00 | 4.7E4 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F20 | ||||||
Mean | 5.06 | 4.07 | 3.43 | 5.02 | 1.00 | 3.28 |
Best | 3.2E4 | 2.2E4 | 1.8E4 | 3.2E4 | 1.00 | 1.7E4 |
Worst | 1.00 | 5.6E4 | 5.6E4 | 5.6E4 | 5.6E4 | 5.6E4 |
F21 | ||||||
Mean | 7.1E6 | 2.1E7 | 4.3E5 | 5.7E6 | 1.00 | 1.1E6 |
Best | 2.7E8 | 9.4E8 | 1.5E7 | 2.3E8 | 1.00 | 3.7E7 |
Worst | 1.00 | 2.93 | 106.52 | 106.52 | 106.52 | 106.52 |
F22 | ||||||
Mean | 37.92 | 20.20 | 2.49 | 31.40 | 1.00 | 4.36 |
Best | 1.6E20 | 8.6E19 | 8.1E18 | 1.3E20 | 1.00 | 1.3E19 |
Worst | 14.09 | 7.56 | 1.00 | 310.54 | 310.54 | 310.54 |
F23 | ||||||
Mean | 43.45 | 107.82 | 2.81 | 28.86 | 1.00 | 5.77 |
Best | 5.9E8 | 1.6E9 | 3.2E7 | 4.0E8 | 1.00 | 7.1E7 |
Worst | 813.93 | 1.8E3 | 36.08 | 484.62 | 1.00 | 101.38 |
F24 | ||||||
Mean | 1.33 | 9.5E8 | 1.00 | 1.78 | 109.21 | 1.24 |
Best | 798.75 | 347.30 | 537.91 | 1.1E3 | 1.00 | 598.94 |
Worst | 8.32 | 5.8E9 | 6.32 | 10.29 | 1.00 | 261.56 |
F25 | ||||||
Mean | 9.67 | 8.67 | 2.09 | 12.20 | 1.00 | 3.64 |
Best | 4.9E5 | 4.1E5 | :.0E4 | 6.3E5 | 1.00 | 1.6E5 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Total | ||||||
Mean | 0 | 0 | 5 | 1 | 19 | 0 |
Best | 0 | 1 | 1 | 1 | 22 | 0 |
Worst | 6 | 3 | 12 | 4 | 17 | 8 |
From Table 6, it can be seen that, on average, MBO is well capable of finding the optimal solutions on twenty-two out of twenty-five benchmarks. ACO, BBO and DE can search for the best solutions on only one out of twenty-five benchmarks. Generally, ACO, BBO and DE have the similar performance each other.
For the best solutions, Table 6 indicates that MBO has the best performance on nineteen out of twenty-five benchmarks. BBO and DE have better performance than ABC, ACO and SGA methods, and they rank 2 and 3, respectively.
For the worst function values shown in Table 6, the first two algorithms are MBO and BBO, and they perform the best on seventeen and twelve out of twenty-five benchmarks, respectively. SGA is well capable of finding the best solutions on eight functions, which are only inferior to the above two methods.
4.4.2 D = 100
Mean, best and worst function values obtained by six methods when D = 100
ABC | ACO | BBO | DE | MBO | SGA | |
---|---|---|---|---|---|---|
F01 | ||||||
Mean | 9.68 | 9.94 | 5.63 | 10.13 | 1.00 | 7.63 |
Best | 2.9E4 | 2.9E4 | 1.6E4 | 3.1E4 | 1.00 | 2.2E4 |
Worst | 1.00 | 1.00 | 1.00 | 1.03 | 1.00 | 1.00 |
F02 | ||||||
Mean | 8.5E3 | 1.4E4 | 1.5E3 | 1.3E4 | 1.00 | 3.8E3 |
Best | 2.8E5 | 4.6E5 | 4.9E4 | 4.2E5 | 1.00 | 1.2E5 |
Worst | 6.77 | 11.05 | 1.20 | 9.79 | 1.00 | 2.91 |
F03 | ||||||
Mean | 1.6E8 | 3.1E13 | 174.23 | 1.4E4 | 1.00 | 81.69 |
Best | 9.0E10 | 1.6E5 | 5.0E6 | 1.5E8 | 1.00 | 2.8E6 |
Worst | 4.0E6 | 1.21 | 3.69 | 283.69 | 1.00 | 1.97 |
F04 | ||||||
Mean | 58.62 | 51.96 | 1.00 | 48.15 | 1.17 | 3.36 |
Best | 2.6E7 | 2.5E7 | 3.3E5 | 1.9E7 | 1.00 | 1.2E6 |
Worst | 99.81 | 84.28 | 1.00 | 84.78 | 20.40 | 4.87 |
F05 | ||||||
Mean | 5.11 | 12.09 | 1.00 | 6.82 | 8.24 | 1.26 |
Best | 6.05 | 13.98 | 1.00 | 7.88 | 8.58 | 1.18 |
Worst | 5.43 | 11.59 | 1.00 | 7.14 | 8.16 | 1.64 |
F06 | ||||||
Mean | 16.83 | 2.40 | 1.36 | 17.16 | 1.00 | 2.87 |
Best | 1.1E3 | 124.95 | 72.58 | 1.0E3 | 1.00 | 152.28 |
Worst | 67.01 | 9.27 | 5.34 | 71.93 | 1.00 | 9.54 |
F07 | ||||||
Mean | 715.21 | 581.76 | 10.31 | 605.13 | 1.00 | 32.33 |
Best | 6.9E17 | 3.5E17 | 7.6E15 | 5.6E17 | 1.00 | 2.0E16 |
Worst | 2.5E4 | 2.2E4 | 397.86 | 1.6E4 | 1.00 | 1.0E3 |
F08 | ||||||
Mean | 204.85 | 263.04 | 15.50 | 326.96 | 1.00 | 38.53 |
Best | 9.6E8 | 9.3E8 | 6.5E7 | 1.5E9 | 1.00 | 1.7E8 |
Worst | 1.71 | 2.24 | 1.00 | 2.88 | 1.00 | 1.00 |
F09 | ||||||
Mean | 6.89 | 6.32 | 5.46 | 6.03 | 1.00 | 4.91 |
Best | 49.17 | 44.17 | 36.34 | 41.71 | 1.00 | 32.61 |
Worst | 1.83 | 1.67 | 1.43 | 1.59 | 1.00 | 1.40 |
F10 | ||||||
Mean | 2.0E7 | 3.2E7 | 6.3E4 | 2.9E7 | 1.00 | 8.7E4 |
Best | 2.4E17 | 4.0E9 | 1.7E14 | 3.6E17 | 1.00 | 4.8E12 |
Worst | 2.8E6 | 4.9E5 | 1.6E3 | 4.1E6 | 1.00 | 1.9E4 |
F11 | ||||||
Mean | 4.8E7 | 5.8E7 | 4.7E5 | 5.9E7 | 1.00 | 7.2E5 |
Best | 1.6E16 | 1.8E10 | 7.7E13 | 2.2E16 | 1.00 | 8.4E13 |
Worst | 9.8E7 | 2.7E4 | 4.0E5 | 1.1E8 | 1.00 | 7.5E5 |
F12 | ||||||
Mean | – | – | – | – | – | – |
Best | – | – | – | – | – | – |
Worst | – | – | – | – | – | – |
F13 | ||||||
Mean | 2.4E3 | 1.8E3 | 232.31 | 4.2E3 | 1.00 | 249.88 |
Best | 1.4E8 | 1.4E8 | 1.4E7 | 3.3E8 | 1.00 | 1.7E7 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F14 | ||||||
Mean | 54.98 | 100.66 | 1.00 | 52.55 | 1.42 | 2.62 |
Best | 4.5E17 | 1.0E18 | 7.3E15 | 5.2E17 | 1.00 | 1.7E16 |
Worst | 25.02 | 54.13 | 1.00 | 30.85 | 1.00 | 1.57 |
F15 | ||||||
Mean | 7.21 | 11.78 | 1.86 | 9.61 | 1.00 | 4.65 |
Best | 1.6E7 | 2.6E7 | 3.8E6 | 2.3E7 | 1.00 | 1.0E7 |
Worst | 9.11 | 16.91 | 2.83 | 12.93 | 1.00 | 6.35 |
F16 | ||||||
Mean | 96.51 | 245.71 | 7.24 | 113.11 | 1.00 | 13.92 |
Best | 4.9E4 | 1.5E5 | 3.4E3 | 5.3E4 | 1.00 | 6.9E3 |
Worst | 14.38 | 32.90 | 1.00 | 17.60 | 5.26 | 5.26 |
F17 | ||||||
Mean | 3.56 | 2.35 | 1.00 | 4.35 | 1.27 | 1.78 |
Best | 1.7E7 | 8.8E6 | 4.2E6 | 2.2E7 | 1.00 | 6.1E6 |
Worst | 3.39 | 2.40 | 1.00 | 4.11 | 1.72 | 1.30 |
F18 | ||||||
Mean | 1.86 | 3.50 | 1.02 | 2.82 | 3.00 | 1.00 |
Best | 103.86 | 171.96 | 47.83 | 161.13 | 1.00 | 56.85 |
Worst | 2.26 | 3.37 | 1.45 | 3.96 | 4.24 | 1.00 |
F19 | ||||||
Mean | 134.12 | 200.41 | 26.65 | 154.27 | 1.00 | 68.72 |
Best | 9.5E4 | 1.5E5 | 1.6E4 | 1.1E5 | 1.00 | 4.5E4 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
F20 | ||||||
Mean | 2.58 | 2.41 | 1.98 | 2.57 | 1.00 | 1.83 |
Best | 20.99 | 16.73 | 15.30 | 21.26 | 1.00 | 13.77 |
Worst | 1.00 | 9.6E4 | 9.6E4 | 9.6E4 | 9.6E4 | 9.6E4 |
F21 | ||||||
Mean | 18.32 | 36.28 | 1.45 | 17.64 | 1.00 | 5.06 |
Best | 3.3E8 | 5.8E8 | 2.1E7 | 2.6E8 | 1.00 | 6.2E7 |
Worst | 1.00 | 2.02 | 300.50 | 300.50 | 300.50 | 300.50 |
F22 | ||||||
Mean | 39.77 | 25.78 | 3.14 | 40.29 | 1.00 | 6.94 |
Best | 4.9E20 | 2.5E20 | 3.7E19 | 5.0E20 | 1.00 | 6.4E19 |
Worst | 13.07 | 10.39 | 1.00 | 926.54 | 926.54 | 926.54 |
F23 | ||||||
Mean | 69.36 | 75.70 | 5.21 | 54.62 | 1.00 | 16.70 |
Best | 2.2E9 | 2.1E9 | 1.3E8 | 1.7E9 | 1.00 | 4.2E8 |
Worst | 19.64 | 19.08 | 1.00 | 13.62 | 597.00 | 597.00 |
F24 | ||||||
Mean | 1.24 | 6.2E10 | 1.00 | 1.47 | 2.9E3 | 1.07 |
Best | 9.98 | 1.00 | 8.06 | 11.04 | 1.37 | 7.70 |
Worst | 1.22 | 1.1E11 | 1.00 | 1.22 | 2.19 | 393.64 |
F25 | ||||||
Mean | 8.08 | 7.61 | 1.89 | 9.94 | 1.00 | 3.86 |
Best | 3.2E5 | 2.7E5 | 6.9E4 | 3.9E5 | 1.00 | 1.3E5 |
Worst | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Total | ||||||
Mean | 0 | 0 | 5 | 0 | 18 | 1 |
Best | 0 | 1 | 1 | 0 | 22 | 0 |
Worst | 6 | 4 | 13 | 3 | 15 | 6 |
From Table 7, for average values, MBO is well capable of finding the optimal solutions on eighteen out of twenty-five benchmarks. BBO and SGA can search for the best solutions on five and one out of twenty-five benchmarks, respectively.
For the best solutions, Table 7 indicates that MBO is well capable of finding the optimal solutions on twenty-two out of twenty-five benchmarks. ACO and BBO can search for the best solutions on only one out of twenty-five benchmarks. Hence, it is safe to conclude that MBO has the absolute advantage over other five methods at this respect.
For the worst function values shown in Table 7, MBO is a little better than BBO that significantly outperforms the other four methods. ABC and SGA have the similar performance, and they can find the best solutions on six functions, which are only inferior to the above two methods.
Its function values are related to dimension, and it is too large to be represented by MATLAB. In our experiment, it can be up to 10E4000. Therefore, the six methods cannot solve this function in our experiment.
From Tables 6 and 7, it can be seen, for high-dimensional functions, MBO has the absolute advantage over other five methods. This indicates that MBO can solve more complicated problem more efficiently and effectively than the other five comparative algorithms.
5 Discussion and conclusion
By simulating the migration behavior of the monarch butterflies in nature, a new kind of nature-inspired metaheuristic algorithm, called MBO, is presented for continuous optimization problems in this paper. In MBO, all the monarch butterfly individuals are idealized and only located in two lands: southern Canada and the northern USA (Land 1) and Mexico (Land 2). Accordingly, the positions of the monarch butterflies are updated in two ways. Firstly, the offsprings are generated (position updating) by migration operator, which can be adjusted by the migration ratio. And then, for other butterflies, their positions are tuned by means of butterfly adjusting operator. In other words, the search direction of the monarch butterfly individuals in MBO algorithm are mainly determined by the migration operator and butterfly adjusting operator. With the aim of showing the performance of MBO method, it is compared with five other metaheuristic algorithms through thirty-eight benchmark problems. The results show that the MBO method is able to find the better function values on most benchmark problems than five other metaheuristic algorithms.
In addition, MBO algorithm is simple and has no complicated calculation and operators. This makes the implementation of MBO algorithm easy and fast.
Despite various advantages of the MBO method, the following points should be clarified and focused on in the future research.
Firstly, it is well known that the parameters used in a metaheuristic method have great influence on its performance. In the present work, we do little effort to fine-tune the parameters used in MBO method. The best parameter settings will be selected through theoretical analyses or empirical experiments.
Secondly, computational requirements are of vital importance for any metaheuristic method. It is imperative to improve the search speed by analyzing the MBO method.
Thirdly, we use only thirty-eight benchmark functions to test our proposed MBO method. In future, more benchmark problems, especially real-world applications, should be used for effective implementation of the MBO method, such as image segmentation, constrained optimization, knapsack problem, scheduling, dynamic optimization, antenna and microwave design problems, and water, geotechnical and transport engineering.
Fourthly, in the current work, the characteristics of the migration behavior (essentially migration operator and butterfly adjusting operator) are idealized to form the MBO method. In future, more characteristics, such as swarm, defense against predators, and human interactions, can be idealized and simplified to be added to the MBO method.
Fifthly, as discussed in Sect. 4, MBO method has the absolute advantage over other five methods on best performance. However, for the average performance, MBO is not the best one among six methods. Furthermore, bad average performance must lead to bad standard deviation (SD). Efforts should be made to improve the average performance by updating the search process.
Sixthly, as shown in Sect. 4, MBO method has the absolute advantage over other five methods when dealing with high-dimensional functions. However, for the low-dimensional functions, MBO performs equally to or worse than other five methods. We plan to investigate this and endeavor to find out the reasons and thereby address this disadvantage in our future research.
And last, in the current work, the performance of MBO method is experimentally tested only using benchmark problems. The convergence of MBO method will be analyzed theoretically by dynamic systems and Markov chain. This can ensure stable implementation of MBO method.
Notes
Acknowledgments
This work was supported by Research Fund for the Doctoral Program of Jiangsu Normal University (No. 13XLR041).
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