Improved symbiotic organisms search algorithm for solving unconstrained function optimization

Recently, Symbiotic Organisms Search (SOS) algorithm is being used for solving complex problems of optimization. This paper proposes an Improved Symbiotic Organisms Search (I-SOS) algorithm for solving different complex unconstrained global optimization problems. In the improved algorithm, a random weighted reflective parameter and predation phase are suggested to enhance the performance of the algorithm. The performances of this algorithm are compared with the other state-of-the-art algorithms. The parametric study of the common control parameter has also been performed.


Introduction
In the real life scenario, the optimization problems are not as simple that they can be solved by deterministic search techniques.There may have more than one global optimal solution and one may be interested to find many global solutions for various reasons depending on the need of the problem.In addition, some functions may have discontinuities, and thus the derivative information is not easy to obtain for those functions.This may pose a strong challenge to many traditional methods of optimization.To overcome this difficulty, efficient optimization algorithms are needed which can deal with this kind of challenges.There are many optimization algorithms which can be classified in many ways, depending on the focus and characteristics.Some of the well-known algorithms found by the literature survey are Genetic Algorithm (GA) (Holland, 1975), Differential Evolution (DE) (Storn & Price, 1997), Particle Swarm Optimization (PSO) (Kennedy & Eberhart, 1995;Shi & Eberhart 1998), Artificial Bee Colony (ABC) (Dorigo et al., 1991), Harmony Search (HS) (Geem et al., 2001), Ant Colony Optimization (ACO) (Blum, 2005), Teaching Learning Based Optimization (TLBO) (Rao et al. 2011), Water Cycle Algorithm (WCA) (Eskandar et al., 2012) etc. GA is a search and optimization techniques that mimics the natural law of evolution and chromosomal processing in genetics.DE is another evolutionary optimization algorithm imitating the Darwin theory of evolution.PSO is based on the behavior of bird flocking or fish schooling.ABC uses the foraging behavior of a honey bee, ACO implemented based on the searching behavior of ant from the source to the destination, and HS is related to the improvisation process of a musician.Some of the applications of these algorithms are given in references (Chander et al., 2011;Wang et al., 2012;Canelas et al., 2013;Hecker et al., 2014;Ghasemi et al., 2014;Li & He, 2014;Baykasoğlu et al., 2014;Nama et al., 2015;Rao, 2016;Bolañosa et al., 2015;Mohammadia et al., 2015;Bhunia et al., 2015;Hosseini et al., 2014;Aulady, 2013;Nama et al., 2016;Rao & Patel, 2014;Barati et al., 2016;Eshraghi, 2016;Rout et al., 2016;Mir & Rezaeian, 2016;Abido, 2016;Gen et al., 2016;Aickelin & Dowsland, 2014;Hecker et al., 2013).
Recently, a new metaheuristic optimization algorithm Symbiotic Organisms Search (SOS), proposed by Min-Yuan and Doddy (2014), is based on the interactions relationship between two organisms in ecosystems.Some of the application of this algorithm has been found in literature which are given in references (Prasad and Mukherjee 2015;Abdullahi and Ngadi 2016;Cheng et al. 2015;Kavousi-Fard et al. 2015;Eki et al. 2015;Verma et al. 2015).In ecosystem predation is a biological interaction where a predator (an organism that is hunting) feeds on its prey (the organism that is attacked).Predators may or may not kill their prey prior to feeding on them, but the act of predation often results in the death of its prey and the ultimate absorption of the prey's tissue through consumption.
Therefore, in the present work, the authors suggest the predation phase to improve the performance of the algorithm and introduce the random weighted reflection vector to enhance the search ability of the SOS algorithm influenced by Satapathy and Naik (2012), which used the random weighted difference vector to improve the performance of TLBO.
The remaining portion of this paper is organized as follows: Section 2 presented the reviews of the original concept of SOS.The new improve SOS (ISOS) is presented in Section 3. Section 4 and 5 presented the result and discussion on solving unconstrained and real world optimization problem.Finally, section 6 summarizes the conclusion of the paper on the whole study.

The SOS algorithm
Symbiotic Organisms Search (SOS) is a comparatively new algorithm which simulates the interactive behaviour of the organisms in nature.The mostly common symbiotic relations between the organisms in ecosystem are mutualism, commensalism, and parasitism.Mutualism is a symbiosis relationship in which both organisms benefit.Commensalism is symbiosis in which one organism benefits and the other is not harmed or helped.If in a interaction one organism benefits and other organism harmed, the relation is called Parasitism.Based on the principle of biological interaction in ecosystem mutualism phase, commensalism phase, and parasitism phase are developed.
In SOS, the initial population called the ecosystem and in ecosystem a group of organisms is generated randomly within the search space.The candidate solution of the problem is the organism of the ecosystem and the fitness value of each organism, reflects the degree of adaptation to the desired objective.In SOS, the new solution is generated by executing the mutualism phase, commensalism phase, and parasitism phase.The descriptions of these three phases are given below.

Mutualism phase:
In this phase, an organism Xi to interact an organism Xj which select randomly from the ecosystem.Both the organisms try to increases of their mutual survival advantage in the ecosystem.The new candidate solutions Xi new and Xj new for the organism Xi and Xj, is calculated by Eqn.(1) and Eqn.(2).
where _ 2 (3) Here, the value of benefit factors BF1 and BF2 is either 1 or 2 which represent level of benefit to each organisms, i.e., whether an organism is benefitted partially or fully from the interaction.The ''Mutual_Vector'' signifies the relationship characteristic between organism Xi and Xj.And Xbest represents an organism with best objective function value in the ecosystem.
Commensalism phase: Since in this phase Xi attempts to increase the benefits from Xj, new candidate solution Xi new is calculated by the commensal symbiosis relationship between organisms Xi and Xj, according to the Eq. ( 4).

1,1 * (4)
Parasitism phase: In SOS, organism Xi is given a role similar to the anopheles mosquito through the creation of an artificial parasite called ''Parasite_Vector''.By modifying the randomly selected dimensions of organism Xi, Parasite_Vector is created.Organism Xj is chosen randomly from the ecosystem and serves as a host to the parasite vector.Parasite_Vector attempts to replace Xj in the ecosystem.Both organisms are then evaluated to measure their fitness.If Parasite_Vector has a better fitness value, it will kill organism Xj and assume its position in the ecosystem.If the fitness value of Xj is better, Xj will have immunity from the parasite and the Parasite_ Vector will no longer be able to live in that ecosystem.
The implementation steps of SOS algorithm are as follows: Step 1: Initialized each organism with uniform random generation and evaluate the objective function value in the ecosystem.
Step2: Generation Step 2.1: Evaluate new organisms by mutualism phase and update in the ecosystem which are calculated by Eqn.( 1) and (2).
Step 2.2: Evaluate new organisms by commensalism phase and update in the ecosystem which are calculated by using Eqn.(4).
Step 2.3: Evaluate the new organisms by parasitism phase (given in section 2) and update them in the ecosystem.
Step 3: Check the termination criterion, if it is fails, go to Step 2; otherwise the best objective function value is considered as the required solution.

I-SOS
In this section, the proposed Predation phase and Random weighed reflection vector are discussed briefly.

Predation phase
In ecology, predation is a biological interaction where a predator (an organism that's searching) feeds on its prey (the organism that is attacked).Predators may or would possibly not kill their prey previous to feeding on them, but the act of predation regularly outcome in the death of its prey and the eventual absorption of the prey's tissue by means of consumption.Predation and parasitism, in both cases, one organism is harmed and the other is benefited.On the other hand, predators almost always kill and eat their prey, while not all parasites kill their hosts.In this phase, one predator generated by Organism Xi, called the Predation_Vector.The Predation_Vector is generated by Eq. ( 5).
where and are the maximum and minimum value of dimension of organism .Then the worst organisms in the ecosystem are replaced by the Predation_Vector.In this work the predator size (i.e. the number of predator in the ecosystem) is chosen as 4. 3.2 Random weighted reflection vector Satapathy and Naik (2012) used the random weighted differential vector (Satapathy and Naik, 2012) in a random manner in the range (0.5, 1) by using the Eqn.( 6 where rand (0, 1) is a uniformly distributed random number within the range [0, 1].Therefore, the mean value of this weighted differential scale factor is 0.75.In this work, the authors suggested the random weighted reflection vector (RWRV) to enhance the search ability of the algorithm which is calculated by Eq. ( 7).
So the new sets of Mutualism phase & Commensalism phase which are formulated by Eq. (1), Eq. ( 2), and Eq. ( 4) can be modified by using the following equation: If the organism violates the boundary constraint, violating organism is reflected back from the violated boundary using the following rule: 0.5 * rand 0,1 * ub 0.5 * rand 0,1 * ub The Pseudo code of the I-SOS algorithm for solving benchmark functions are shown in Fig. 1 and the algorithm steps can be summarized in the following way: Step 1: Randomly initialize the common control parameter i.e.Eco-size (number of organisms in the ecosystem); function evaluation; the ecosystem organisms and evaluate the fitness of each organism.
Step 2: Calculate the new organisms and update by mutualism phase using Eq. ( 8) and Eq. ( 9) and repair the infeasible organisms of the ecosystem to be feasible using En.(11).
Step 3: Calculate the new organisms and update by commensalism phase using Eq. ( 4) and repair the infeasible organisms of the ecosystem to be feasible using Eq.(10).
Step 4: Calculate the new organisms and update by parasitism phase which is given in Section 2.
Step 5: Update organisms by predation phase in Eq. ( 5) and repair the infeasible organisms of the ecosystem to the organisms to be feasible using Eq.(11).
Step 6: Replace the worst organism by Predation_Vector.
Step 7: If the stopping criterion is not satisfied go to Step 2, until the best fitness value is obtained.

Experimental Result and Discussion
Three experiments are considered for the investigation of the validity of the proposed algorithms.All the benchmark function for these experiments is given in Table 1, Table 2 and Table 3 respectively.

Effect of the Eco-size
Experiment l: The effects of Eco-size (number of organism in the ecosystem) of the algorithm are discussed in this section by taking different values of Eco-size.Here the values of Eco-size are considered as 10, 20, 30, 40, and 50 respectively.Twenty five independent runs are carried out for each eco-size in each problem.The best, mean and standard deviation of the function error values * among 25 runs are recorded on each problem, where the best solution is found by the algorithm in a run and * is the global optimum of the test problem.A run is successful if its function error value is not larger than the target error accuracy level ε, which is set to 10 ( Suganthan et al. 2005).We record the number of successful runs and report the success rate, the percentage of the successful runs among 25 runs.The effect of Eco-size, justified on 12 different benchmark functions which are given in Table 1.Table 4, 5 and 6 show the effect of Eco-size of these 12 benchmark functions for dimension (D) 3, 5, and 7 respectively with D*10000 function evaluations.In these tables, the boldface represents the best result found after reaching the maximum number of function evaluation.From Table 4, it is seen that strategy with Eco-size 10 produce the best result for function F1, F8, F10, F11, and F12; strategy with Eco-size 20 produce the best result for function F5 and F9 but for function F2 strategy with Eco-size 20 and 30 produce the identical result; for function F6 strategy with Eco-size 40 produce the best result; for function F4 and F7 strategy with Eco-size 50 produce the best result but for function F3 produce the identical result and hence there is no effect of Eco-size for this function.From Table 6 it is observed that for function F1, F8, F11 and F12 strategy with Eco size 10 produce the best result but for function F3 produce the identical result; for function F2 and F6 strategy with Eco size 20 produce the best result but for function F5 produce the identical result with the strategy of Ecosize 30; for function F7 and F10 strategy with Eco size 30 produce the best result; for function F4 and F9 strategy with Eco size 40 produce the identical result with the strategy of Eco-size 50.Table 7, 8, and 9 present the number of fitness evolution and number of successful run to obtain the accuracy level ε (1e-08) (Suganthan et al., 2005).In this table the boldface represents the minimum function evaluation require to obtain the accuracy level ε.From Table 7, it is observed that strategy with Eco-size 10 requires minimum function evaluation on one function; strategy with Eco-size 20 requires minimum function evaluation on six functions; strategy with Eco-size 30 requires minimum function evaluation on four functions.Here "Mean (SD)" represents the statistical result of fitness evolution requires, "SR" represents the success rate.
From Table 8, it is observed that strategy with Eco-size 10 requires minimum function evaluation on four functions; strategy with Eco-size 20 requires minimum function evaluation on two functions; strategy with Eco-size 30 requires minimum function evaluation on one function; strategy with Ecosize 40 requires minimum function evaluation on two functions.From

Comparison with other algorithms
Experiment-ll: In this experiment performance of the results of I-SOS algorithm compared to others PSO, DE and SOS algorithm.For this experiment the details of the parameter setting of all algorithms are given in the Table 10, the details of the function formulation are reported in Table 2, and the entire algorithms run 30 times.The performances results of this experiment are reported in Table11.The bold of the Table 11 are represented the best performance of the algorithm.From this table it is observed that I-SOS gives best value on eight problems, identical result on ten problems and the inferior on other problems.Based on the above discussion it is conclude that the proposed I-SOS better than other algorithms.Experiment-lll: For this experiment a set of benchmark function which are given in Table 3 is taken and the experimental result are compared to CoDE (Wang et al. 2011) and EPSDE (Mallipeddi et al. 2010).The algorithms were run 25 times with Eco-size 50 and 200,000 function evolution.The performance results are reported in Table 12.Here the bold value represented the best performance of the algorithms.From this table it is observed that the proposed I-SOS algorithms give us good performed on nine functions out of eleven functions.

Real world application
The proposed I-SOS is applied to two real world problems.The problems are taken from (Li et al. 2011).The formulation of these problems is given below: The experimental results of this problem are given in Table 13.In Table 13, results except I-SOS are taken from (Li et al. 2011).The bold face represents the best result than other algorithms.From this table it is observed that the performance results show of I-SOS algorithm possesses superior with other algorithms.

Conclusion
In SOS method, proposed by Min-Yuan and Prayogo, a new phase called predation phase is suggested to enhance the performance of the algorithm.Also a random weighted reflection vector is suggested to enhance the search ability of the algorithm.The modified algorithm thus obtained, called improved SOS (I-SOS) Algorithm, is presented in this paper to solve global numerical optimization problems.
For the validity of this algorithm, the performance of the proposed algorithm is testified on a set of benchmark functions and reported in Table 1, 2 and 3.The effect of the common control parameters i.e.Eco-size and number of function evaluation are investigated by varying these two parameters.Also the performance results are compared with other basic algorithms and state of the art DE variants.The performance of proposed I-SOS is also applied on two real world problems.From the above discussion of the performance results it is conclude that I-SOS algorithm is superior to the other algorithms.Future research may be carried out by employing I-SOS in solving complex real problems even for constrained and multi-objective optimization problems.

Table 4
Effect of the eco-size of 12 different benchmark function (given in Table1) with dimension 3 and function evaluation D×10000

Table 5
Effect of the eco-size of 12 different benchmark function given in Table1) with dimension 5 and function evaluation D×10000From Table5, it is seen that strategy with Eco-size 10 produce the best result for function F1, F8, F11 and F12 but for function F3 produce the identical result and hence there is no effect of Eco-size for this function; for function F7 and F9 strategy with Eco-size 20 produce the best result; for function F2 strategy with Eco-size 30 produce the best result but for function F5 produce the identical result with the strategy of Eco-size 40; for function F6 and F10 strategy with Ecosize 50 produce the best result.

Table 7
No of fitness function evolution required to obtain the accuracy level of 12 benchmark function (given in Table1) with dimension 3 and function evaluation D×10000 SD)" represents the statistical result of fitness evolution requires, "SR" represents the success rate.

Table 8
No of fitness function evolution required to obtain the accuracy level of 12 benchmark function (given in Table1) with dimension 5 and function evaluation D×10000 SD)" represents the statistical result of fitness evolution requires, "SR" represents the success rate.

Table 9
No of fitness function evolution required to obtain the accuracy level of 12 benchmark function (given in Table1) with dimension 7 and function evaluation D×10000 Table 9, it is observed that strategy with Eco-size 10 requires minimum function evaluation on five functions; strategy with Eco-size 20 requires minimum function evaluation on two functions; strategy with Eco-size 30 requires minimum function evaluation on two functions; strategy with Ecosize 40 requires minimum function evaluation on one function; strategy with Eco-size 50 requires minimum function evaluation on one function.

Table 10
Parameter setting for Experiment-ll (MAX_FEs: maximum function evaluation)

Table 11
Performance Comparison of I-SOS of 25 benchmark function (given inTable 2)

Table 11
Performance Comparison of I-SOS of 25 benchmark function (given in Table 2) (Continued)