Accepted Manuscript

In this study, we focus on the modification of the artificial algae algorithm (AAA), proposed for solving continuous optimization problems, for binary optimization problems by using exclusive or (xor) logic operators and stigmergic behavior. In the binary version of the algorithm, three of them have been modified to overcome the structure of binary optimization problems. The binary version of the PSO algorithm called BPSO [33], also explored by Kennedy and Eberhart in 1997.

AAA is one of the swarm intelligence algorithms, inspired by the foraging behavior of algae colonies. In the initialization of the basic AAA, all the cells are generated using Eq.1. The size of the algae colonies are important in the AAA algorithm because there is an evolutionary process that is performed in this algorithm.

After all colonies have been energized, the colony sizes are recalculated using Eq. In the solution update mechanism, three dimensions are updated as in the basic version of the AAA algorithm. After the spiraling phase, the fitness value of the candidate solution is compared with the old one.

In the adaptation phase of the algorithm, another modification is performed to work with the binary decision variables.

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Update Mechanism 2: Stigmergic Behavior

The concept of stigmergy was first introduced by Grassé[57] in 1959 in the field of entomology. One of the goals of his work is to discover how simple individual termites are able to create large termite mounds. The actions of termites are not coordinated from start to finish with any deliberate plan.

This behavior is observed not only in termites but also in ants, bees and many others. In this approach, information about the actions performed in the past (𝐶01 and 𝐶10 counters) is used to guide the moves to be performed in the future. Briefly, the information obtained from the processed solution to the problem controls the behavior of algae colonies in the proposed algorithm.

In the basic version of the AAA algorithm, three dimensions are updated to find new food sources to realize helical motion. In parallel with this approach, up to three dimensions can be updated to produce candidate solutions. For this purpose, method-specific dimension selection probability (DSP) control parameters are added.

If the random number produced for a dimension in the range [0,1] is less than the DSP parameter, the shift is performed on that dimension. Therefore, 𝑉𝐴(𝑎) denotes a random dimension (cell) that has a value of 1 and 𝑉𝐵(𝑏) denotes a random dimension that has a value of 0. If the value of 𝑟1 is less than the probability value of 𝑝10, the decision variable in the candidate solution that has a value of 1 ( 𝑉𝐴(𝑎)) is set to 0.

If the value of 𝑟1 is equal to or greater than the probability value of 𝑝10, the random decision variable in the proposed solution that has the value 0 (𝑉𝐵(𝑏)) is set to 1. Ultimately, for minimization problems, if the value of the objective function is the candidate solution is smaller than the selected individual in the population, the candidate solution is copied to that individual and the equation is given as follows:.

Experiments

Experimental Material
Comparison on UFLPs
Comparison on Numeric Benchmark Problems

The proposed method is coded in the MATLAB R2015a environment, and the results of other algorithms are taken directly from [49]. In order to make a fair comparison of the algorithms, the common control parameters of the algorithms are chosen to be equal to each other. The trend change of the control parameters of the curve (T) is chosen as 1.5 and 2 for the logistic functions Tanh(x) and Sig(x), respectively.

The upper limit of particle velocities (𝑉𝑚𝑎𝑥) is taken as 6, the lower limit of particle velocities (𝑉𝑚𝑖𝑛) is taken as -6 and positive acceleration constants (𝑐1, 𝑐2) are taken as 2 for the BPSO algorithm. When the dimensionality of the problems is increased, the proposed algorithm is superior to the compared algorithms in terms of solution quality. To overcome this problem, the mutation probability must be tuned according to the characteristics and dimensionality of the optimization problem.

Our algorithm outperforms GA in almost all cases because both exploration and exploitation provide the binarization process and the origin of the algorithm. In the first part of the UFLP experiment, BPSO, GA, BAAA variants and the proposed algorithms are compared to show the performance of the algorithms. The corrected rank is also used to rank algorithms based on performance.

For these problems, the proposed algorithm is run 30 times with random seeds and the results obtained are compared with the results of state-of-the-art algorithms [49]. The population size is selected to 50 and the maximum number of fitness evaluations is used for the algorithm's termination condition and is set to 100,000. For the particular control parameters of the proposed method, the energy loss is set to 0.3 and the rate of adaptation to 0.5, as used in the previous experiment and basic version of AAA[38].

Since the artificial colonies of AAA operate on binary structured solution space, the use of the shear force parameter of the basic algorithm is not required in the proposed binary version of the algorithm. Under these conditions, obtained results are reported as mean and standard deviation of the runs in Table 6. On the rest of the functions, the proposed algorithm is better than comparative algorithms in terms of solution quality and robustness based on the standard deviations shown in the comparison table.

Conclusion and Future Works

However, when we compare the optimal solution and the obtained solutions of these test functions, we see that the binary optimization algorithms require much more effort to achieve optimal or near-optimal results. In the near future, we will apply this algorithm to solve various binary optimization problems, especially knapsack problems that have been extensively studied, and we will also use the stigmergic behavior proposed in the present study in the other swarm intelligence algorithms to solve the binary optimization problems to solve. Cunha, an ABC heuristic for optimizing the location of a movable ambulance station and vehicle repositioning for the city of Sao Paulo.

Meenambal, Velocity Bounded Boolean Particle Swarm Optimization for Improved Feature Selection in the Diagnosis of Liver and Kidney Disease. Liang, J., et al., Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization. Suganthan, P.N., et al., Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization.

Suganthan, Problem Definitions and Evaluation Criteria for CEC 2011 Competition on Testing Evolutionary Algorithms on Real World Optimization Problems. Chen, Q., et al., Problem Definition and Evaluation Criteria for CEC 2015 Special Session and Competition on Bound Constrained Single Objective Computational Duration Numerical Optimization. If the random number is less than the adaptation rate parameter, find the most hungry colony and apply Eq.11 with the largest colony.

Fig. 1. The pseudo-code of the AAA algorithm

Comparison of the proposed method with the binary structured optimization algorithms for the CapA, CapB and CapC problems. The comparison of the proposed algorithm with the most modern binary versions of ABC, DE and PSO by using gap and rank problems. The comparison of the proposed algorithm with the most modern methods in the CEC2015 test suite.