An ensemble symbiosis organisms search algorithm and its application to real world problems

In this study, an ensemble algorithm has been proposed, called Quasi-Oppositional Symbiosis Organisms Search (QOSOS) algorithms, by incorporating the quasi-oppositional based learning (QOBL) strategy into the newly proposed Symbiosis Organisms Search (SOS) algorithm for solving unconstrained global optimization problems. The QOBL is incorporated into the basic SOS algorithm due to the balance of the exploration capability of QOBL and the exploitation potential of SOS algorithm. To validate the efficiency and robustness of the proposed Quasi-Oppositional Symbiosis Organisms Search (QOSOS) algorithms, it is applied to solve unconstrained global optimization problems. Also, the proposed QOSOS algorithm is applied to solve two real world global optimization problems. One is gas transmission compressor design optimization problem and another is optimal capacity of the gas production facilities optimization problem. The performance of the QOSOS algorithm is extensively evaluated and compares favorably with many progressive algorithms.


Introduction
In solving the non-linear problem, the optimization algorithm is most significant and sturdy.Since some of the problems have discontinuous and gradient based algorithms are not suitable for that sort of problem (Lee & Geem, 2004;Osman & Laporte, 1996), to overcome this difficulity, recently, several optimization algorithms have been introduced within the past decade.Some of them are given in the references (Holland, 1975;Kennedy & Eberhart, 1995;Storn & Price, 1997;Dorigo & Stützle, 2004;Geem et al., 2001;Karaboga & Basturk, 2007;Yang & Deb, 2009;Sadollah et al., 2012;Eskandar et al., 2012;Pham et al., 2006;Yang, 2009;Nama et al., 2015aNama et al., , 2016bNama et al., , 2017)).The idea of Oppositionbased learning (OBL) was initially developed by Tizhoosh (Tizhoosh, 2005).The inventor of OBL claims that a number's opposite is probably nearer than a random number to a solution.Integrating OBL in conventional evolutionary algorithms, one can increase the coverage of the solution space leading to increases accuracy and faster convergence.Some of the applications of OBL in the soft computing field include oppositional particle swarm optimization (OPSO) (Wang et al., 2011), oppositional biogeography based optimization (OBBO) (Roy and Mandal 2014), oppositional differential evolution (ODE) (Qingzheng et al. 2011).The idea of QOBL is employed in population initialization and generation jumping.A number of the application of QOBL within the soft computing field includes Quasi-oppositional TLBO (QOTLBO) (Mandal & Roy, 2013), quasi-oppositional BBO (QOBBO) (Roy & Mandal, 2011) and Multi-objective quasi-oppositional TLBO (MOQOTLBO) (Sultana & Roy, 2014).In this paper, quasi-opposition based learning (QOBL) strategy is introduced in the basic SOS algorithm which provides the additional likelihood of finding solutions nearer to the global optimum.In the proposed QOSOS algorithm, first, all the steps of SOS algorithm executed and then the concept of quasi-opposition based jumping is executed based on the jumping rate.
Therefore, the main contributions of this study are summarized as follows: i) A new ensemble algorithm (called QOSOS) is proposed by the combination of QOBL strategy and SOS algorithm.
ii) The proposed QOSOS algorithm have been applied to solve some well known benchmark functions.iii) Finally, the proposed QOSOS algorithm has been applied to solve two real world optimization problems.The remaining part of the paper is organized as follows: Section 2 discusses the overview of the basic Symbiosis Organisms Search algorithm.The Quasi-opposition-based learning (QOBL) strategy is presented in Section 3. Section 4 presents the details description of the newly proposed Quasi-Oppositional Symbiosis Organisms Search (QOSOS) algorithm.Section 5 and 6 demonstrate the efficiency and accuracy of the QOSOS algorithm for solving some well known unconstrained optimization problems and two real world optimization problems.Finally, Section 7 presents the summary of the contribution of this paper along with some future research directions.

Symbiosis Organism Search Algorithm
The Symbiosis Organisms Search algorithm, proposed by Cheng and Prayogo (2014), simulates the interactive behavior seen among organisms in nature.Symbiosis is derived from the Greek word for ''living together''.De Bary first used the term in 1878 to explain the cohabitation behavior of unlike organisms (Sapp 1994).Today, symbiosis is used to describe a relationship between any two distinct species.Symbiotic relationships may be either obligate, meaning the two organisms depend on each other for survival or facultative, i.e., the two organisms choose to cohabitate in a mutually beneficial but non-essential relationship.
In an ecosystem, the most common symbiotic relationships are mutualism, commensalism, and parasitism.Mutualism relationship may be between two or more organisms and in this symbiosis relationship, both organisms get benefit from each other.In Commensalism relationships, one organism is benefited from the interaction and the other is not harmed or helped.A parasitism relationship is a symbiosis relationship in which one organism gets the benefit and the other is harmed but not always to its death.SOS begins with an initial population referred to as the ecosystem.In the initial ecosystem, a bunch of organisms is generated randomly within the search area.Each organism represents one candidate solution to the corresponding problem.Every organism within the ecosystem is associated with an explicit fitness value that reflects the degree of adaptation to the required objective.In SOS, new solution generation is governed by imitating the biological interaction between two organisms in the ecosystem.The detailed description of SOS algorithm can be seen in Cheng and Prayogo (2014).Also some applications and improve vesion of SOS algorithm are presented in the literature (Nama et al., 20016c, 2016d;Prasad & Mukherjee, 2015;Abdullahi & Ngadi, 2016;Cheng et al., 2015;Verma et al., 2015).

Quasi-opposition-based learning (QOBL)
We recall some definitions related to Quasi-opposition-based learning (QOBL) for the convenience of the proposed algorithm.(Tizhoosh, 2005) Let

Definition
,  be a real number.The opposite number of x is defined by .where c is given by: .(Roy & Mandal, 2011) Let x be any real number between (a, b).Then quasi-opposite point is defined as , where , ∈ , ; ∈ 1, 2, 3, … … . ., .

Quasi-opposition based optimization
The quasi-oppositional based optimization is based on quasi-opposition based initialization and quasiopposition based generation jumping which are described below:

Quasi-opposition initialization
By utilizing opposite points, better initial candidate solutions, namely opposite population (OP) may be obtained even when there is no prior knowledge about the solutions.Initialization of quasi-opposite population (QOP) may be described as follows (Roy & Mandal, 2011): For i = 1: eco-size (Number of organism in the ecosystem) For j = 1: D (Dimension of the variable) , , ; ; , , , ; End End

Quasi-opposition generation jumping
By applying a similar approach to the current population, the evolutionary process can be forced to jump to a new solution candidate, which may be better than the current one.Based on a jumping rate Jr, after generating new population by using all the steps of SOS, the opposite population is generated.Quasi-Opposite population jumping based on jumping rate is described as follows: In the development of heuristic global optimization algorithm, the balance between exploration and exploitation capability plays a major role (Crepinšek et al., 2013).According to Crepinšek et al. ( 2013) "Exploration is the process of visiting entirely new regions of a search space, whilst exploitation is the process of visiting those regions of a search space within the neighborhood of previously visited points".As discussed above the QOP method may be used for better exploration when executing the optimization process, on the other hand, Cheng and Prayoga (2014) elaborately discussed the better exploitation ability of SOS algorithm for global optimization.By incorporating the exploration capability of QOP into the SOS algorithm, the quasi oppositional symbiosis organisms search (QOSOS) algorithm has been proposed.The QOSOS is able to explore the new search region with the SOS algorithm and to exploit the population information with the QOP.This ensemble method can increase the robustness of the algorithm as well as searching capability of the algorithm for attaining the global optimum.During the execution of the optimization process of the proposed QOSOS, two variants of organisms are evaluated with the same size.One is randomly initialized and another is quasioppositional based initialize organisms.From these, eco-size number organisms have been selected as initial organisms.If an organism violates boundary constraints, the organism is reflected back inside the boundary using the following rule (Gong et al., 2006): where are the lower and upper bound of the i th organism.
The flowchart of the proposed QOSOS algorithm is given in Fig. 1 and the procedure for implementing the QOSOS algorithm can be summarized in the following steps: Step 1: Initialize the algorithm parameter.
Step 2: Randomly generate the organisms and evaluate the fitness value for each organism.
Step 3: Generate quasi-oppositional population i.e. organism (QOP) set and evaluates the fitness value using the procedure given in Section 3.2.1.
Step 4: Select eco-size number of fittest organisms from {Org, QOP} as initial organisms set.Org is the set of the organism.Eco-size is the number of the organism in the ecosystem.
Step 5: Main Loop Step 5.1: Update organisms by mutualism phase: In this phase, an organism Orgj is selected randomly from the ecosystem to interact with organism Orgi, the i th member in the ecosystem.Both organisms engage in a mutualistic relationship with the goal of increasing mutual survival advantage in the ecosystem.The new organism in the ecosystem for Orgi and Orgj are calculated based on the mutualistic symbiosis between organism Orgi and Orgj, by Eqs. (2-3). where, Here, benefit factors (BF1 and BF2) are determined randomly as either 1 or 2. These factors represent the level of benefit to each organism, i.e., whether an organism partially or fully benefits from the interaction.Eq. ( 4) shows a vector called ''Mutual_Vector'' which represents the relationship characteristic between organism Orgi and Orgj.Org is the best organism in the ecosystem.
Step 5.2: Update organisms by commensalism phase: In the commensalism phase, organism Orgi benefits by organism Orgj and organism Orgi increased the beneficial advantage in the ecosystem to the higher degree of adaption.New organism Orgi is calculated by Eq.( 5) and updated in the ecosystem only if its new fitness is better than its pre-interaction fitness.

Org
Org rand 1,1 * Org Org (5) Step 5.3: Update organisms by parasitism phase: In SOS, by duplicating organism Org i an artificial parasite called ''Parasite_Vector'' is created in the search space.Select another organism Orgj randomly from the ecosystem and serves as a host to the parasite vector.If Parasite_Vector has a better fitness value, it will kill organism Orgj and assume its position in the ecosystem.If the fitness value of Orgj is better, Orgj will have immunity from the parasite and the Parasite_Vector will no longer be able to live in that ecosystem.
Step 5.4: Update Organisms by Quasi-opposition generation jumping: Based on a jumping rate Jr (i.e.Jumping probability), QOP is generated and fitness value of the QOP is calculated as given in Section 3.3.2.Select require numbers of the fittest organisms from {Org, QOP} as current Organisms.
Step 6: If the stopping criterion is not satisfied, go to Step 5, else return the organism with the best fitness value as the optimum solution.

Performance Evaluation of Well Known Test Functions
To validate a new optimization technique, it is a common practice to compare different algorithms using different benchmark problems.In this work, also, the convergence graph, three experiments, and two real life problems are considered for verifying the performance of the proposed algorithm.The statistical measures are obtained with different independent runs.Details of benchmark functions are given in the Appendices.The proposed algorithm coded in MATLAB7.10.0 (R2010a).

Performance Evaluation on Twenty Six Well Known Benchmark Functions
Table 1 shows the statistical measures of 26 benchmark functions which are given in Appendix-1.The performance results are compared with another algorithm like GA, DE, PSO, BA, PBA and SOS (Cheng & Prayogo, 2014).To compare the performance results of the proposed algorithm on twenty six benchmark functions, the algorithm runs 30 times with 500000 function evaluations.In Table 1, results except QOSOS are taken from Cheng and Prayogo (2014).From Table 1, it can be seen that the performance of proposed QOSOS algorithm is better than other compared algorithm on 22 benchmark functions out of 26 benchmark functions.The convergence rate of the QOSOS algorithm is also examined.The comparison of QOSOS is performed with SOS algorithm., (e) F20, (f) F21, (g) F22, (h) F23, (i) F24 and (j) F26).

Performance Evaluation on Eight Well Known Benchmark Functions
Considering the problem given in Appendix-2, the performance results and maximum function evaluation of all eight well known test functions are reported in Table 2 and Table 3 along with the results as quoted in (Gao et al., 2012)

Performance Results on Two Real World Problems
Table 5 and Table 6 show the performance results on two real life optimization problems.The mathematical models of these two problems are given in Appendix-3.The minimum objective function values for RP1 and RP2 are reported in Tables 5 and Table 6, respectively, along with results quoted in (Das & Singh, 2014).The performance results are compared with DE, GSA, DE-DSA, DFO, DFO (QA) and Beightler and Phillips (1976).In Table 5 and Table 6, values in the bold print show best values for QOSOS that are superior to DE, GSA, and Beightler and Phillips (25).From these two tables, it is seen that QOSOS gives better performance than the other algorithms.show that the proposed QOSOS method has higher efficiency in terms of the numerical results than the other reported optimization techniques to solve the benchmark problems.Also, the proposed algorithm has been applied to solve two real world optimization problems and the results obtained by proposed algorithm were compared with other algorithms available in the literature.The comparative study shows the satisfactory performance of the proposed QOSOS algorithm.
Fig.2 shows the convergence graph of ten different benchmark functions which are given in Appendix-1 ( Fig 2.(a) for Step function, (b) Sphere function, (c) for Sum Squares function, (d) for Quartic function, (e) for Schwefel2.22,(f) for Schwefel1.2,(g) for Rosenbrock, (h) for Dixon-Price, (i) for Rastrigin function, and (j) for Ackley function).From Fig. 2 it can be easily seen that QOSOS converges faster than SOS algorithm.Above all, these experimental results demonstrate that the proposed QOSOS algorithm performs better than other compared algorithms.FEs) Objective fu nctio n v alu e (a).F16: Convergence graph of step function QOSOS FEs) O b jectiv e fu n ctio n v alu e (b).F17:Convergence graph of sphere function SOS FEs) Objective function value (c).F18: Convergence graph of sum squares function QOSOS SOS
Optimal capacity of gas production facilities (Beightler and Phillips, 1976):

Table 1
Performance results of GA, DE, PSO, BA, PBA, SOS and QOSOS after reaching 500000 FEs of twenty six well known test functions over 30 runs.''Mean '' and ''SD'' show the average and standard deviation of the test function respectively.Boldface represents the best result among the compared algorithms Gao et al. (2012)200 and 300 dimensions, respectively with 20 independent runs.The authors inGao et al. (2012)compares CSPSO with several state of the art algorithms, namely SDE, SPSO, MPSO-TVSE, DPSO with 5000×D (D = dimension of the problem) function evaluations and 40 eco-size.In Table2, the bold face results represent the best performance results among the compared algorithms.From the results in Table2, it is observed that QOSOS performs better than all other algorithms for solving lower to high dimensional problems with less number of function evaluations.

Table 5
Performance results of DE, GSA, DE-GSA, DFO and QOSOS on gas transmission compressor design optimization problem.Boldface represents the best result among the compared algorithms

Table 6
Performance results of DE, GSA, DE-GSA, DFO and QOSOS on Optimal Capacity of the Gas production facilities optimization problem.Boldface represents the best result among the compared algorithms This paper has developed an ensemble algorithm combining quasi-oppositional based learning with Symbiosis Organisms Search algorithm, called quasi-oppositional Symbiosis Organisms Search (QOSOS) algorithm.The proposed algorithm solved several unconstrained benchmark function to enrich the searching behavior and to avoid being trapped into a local optimum.The comparison results ), Cuckoo search via levy flights.In: Proceedings of the world congress