Island-based Crow Search Algorithm for solving optimal control problems

Journal Pre-proof Island-based Crow Search Algorithm for solving optimal control problems Mert Sinan Turgut, Oguz Emrah Turgut, Deniz Türsel Eliiyi

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S1568-4946(20)30110-1 https://doi.org/10.1016/j.asoc.2020.106170 ASOC 106170

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Applied Soft Computing Journal

Received date : 2 October 2019 Revised date : 8 February 2020 Accepted date : 12 February 2020 Please cite this article as: M.S. Turgut, O.E. Turgut and D.T. Eliiyi, Island-based Crow Search Algorithm for solving optimal control problems, Applied Soft Computing Journal (2020), doi: https://doi.org/10.1016/j.asoc.2020.106170. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Elsevier B.V. All rights reserved.

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Island-based Crow Search Algorithm for solving optimal control problems Mert Sinan Turguta, Oguz Emrah Turgutb, Deniz Türsel Eliiyic a

Department of Mechanical Engineering, Faculty of Engineering, Ege University, Bornova, Izmir, Turkey

b

Department of Mechanical Engineering, Faculty of Engineering and Architecture, Izmir Bakircay University, Menemen, Izmir, Turkey c

Department of Industrial Engineering, Faculty of Engineering and Architecture, Izmir Bakircay University, Menemen, Izmir, Turkey

Corresponding author: Mert Sinan Turgut

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E-mail: [email protected]

                     

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Declarations of interest: None

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Abstract

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Crow Search Algorithm (CROW) is one of the members of recently developed swarm-based metaheuristic algorithms. Literature includes different applications of this algorithm on engineering design problems. However, this optimization method suffers from some drawbacks such as premature convergence and trapping into local optima at the early phase of iterations. In order to conquer this algorithm specific inabilities, many research studies have been conducted in the literature dealing with the improvements and enhancements on the search mechanism of CROW. Structured population mechanism plays a vital role in preserving and controlling diversity, and thus increases the solution efficiency in evolutionary algorithms. Among the different types of methods used in structured algorithms, the island model is one of the widely applied solution strategies, in which the population individuals are subdivided into a predefined number of subpopulations. Migration mechanism is the key factor increasing population diversity, which takes place between independently running subpopulations during iterations to exchange valuable and useful solution information. This study embeds the fundamentals of the island model concepts into the Crow Search Algorithm to improve its probing capabilities of the search domain, by means of the periodically interacting subpopulations on the course of iterations. In addition, four different hierarchical migration topologies have been proposed, and their search effectiveness have been evaluated and compared over 45 optimization test functions. The optimization function test set includes classic benchmark optimization problems and CEC 2015 benchmark functions. Furthermore, each hierarchical island model is applied for solving six different optimal control problems in order to investigate their efficiencies on multi-dimensional real world optimization problems. The investigated optimal control problems are parallel reaction, continous stirred tank reactor, batch reactor consecutive reaction, nonlinear constrained mathematical system, nonlinear continous stirred tank reactor and nonlinear crane container problems. It is found out that the island model concepts improved the optimization performance of CROW. The proposed island models outperformed or showed similar performance compared to the six selected literature optimizers for 27 of 29 classic benchmark optimization problems. Moreover, incorporating the master sub-population to the island model improved the optimization capability of the algorithm further in most cases. The island models that employ the master sub-population came up with more favorable results compared to their non-master sub-population peers in all optimal control problems. The island model that includes the master sub-population and has the migration topology entitled “82” found the most desirable solutions for 4 of 6 optimal control problems. Keywords: Crow Search Algorithm, Island model, Hierarchical structured population, Optimal control.

1. Introduction

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Most of the real-world engineering problems intrinsically carry complex design constraints, involve interdependencies between the design variables and are highly nonlinear in their nature. These inherent complexities greatly affect the corresponding solution procedures to these problems, and leads to inefficiencies in their optimization [1]. Conventional optimization algorithms such as direct search and Newton-based methods experience some major difficulties during the solution process as they are highly dependent upon the initial solution. These algorithms also require derivative information of the search space and are prone to get stuck on sub-optimal points in the solution domain because of the mentioned difficulties [2]. It is also worth to mention that the utilization of such algorithms in parallel computing is highly inefficient. Metaheuristic algorithms can be a promising alternative for conventional optimization methods, as their application on optimization problems nearly eliminate most of the abovementioned drawbacks. Metaheuristics are iteration-based optimizers, controlling and guiding some subordinate heuristics and adopting some intelligent procedures from different environments for diversification and intensification

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purposes. Utmost care should be given on proper balance between diversification and intensification for developing an efficient metaheuristic algorithm [3]. Randomization also plays a vital role in the realization of possible solutions, which is the prominent core concept in stochastic-based metaheuristics. Randomization helps to provide accurate solution outcomes within a reasonable time without burdening heavy computational load. Another favorable benefit of using metaheuristics is that they are not problem specific and can be conveniently applied to any optimization problem. They are found to be very efficient in solving large scale NP-hard problems having nonlinear characteristics. The No Free Lunch Theorem [4] imposes that the optimization performance of the available metaheuristics is average for existing optimization problems. Therefore, plenty of metaheuristic-based solution procedures have been proposed over the years in order to obtain more reliable and robust results for various optimization problems. Some of the prevalent metaheuristics developed in the past decades are Genetic Algorithm [5], Differential Evolution [6], Ant Colony Optimization [7], and Simulated Annealing [8], Grey Wolf Optimizer [2], Bird Mating Optimizer [9], etc. Recent years have witnessed the development of some novel nature inspired metaheuristic algorithms. Algorithms developed by adopting some of the prominent natural phenomena into numerical simulation have demonstrated their efficiency in problem solving in terms of local minima avoidance, increased convergence speed and solution accuracy [10-14]. Relying on the mentioned advantages, nature inspired stochastic metaheuristic algorithms such as Grey Wolf Optimizer [2], Bird Mating Optimizer [9], Dolphin Echolocation Algorithm [15], Ray Optimization Algorithm [16], and Crow Search [1] have been successfully applied to real world optimization problems and very optimistic solution outcomes have been obtained. Among the various kinds of nature inspired algorithms, some of which are mentioned above, Crow Search Algorithm (CROW) takes a considerable part in the literature, and is utilized in many engineering applications, ranging from economic load dispatch [17] to DNA fragment assembly problem.

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Coined by Askerzadeh [1], the Crow Search Algorithm is a swarm-based nature-inspired algorithm, mimicking the intelligent foraging behaviors of crows. The algorithm has many advantages over its peer algorithms: It requires fewer mathematical equations and control parameters, its implementation is relatively easy, and the run-time is small compared to other nature-inspired optimization methods. It also has an in-built control mechanism that automatically shifts exploration (diversification) to exploitation (intensification) phases. However, the algorithm has a major shortcoming as it suffers from premature convergence resulting from the diversity loss during iterations. Therefore, many researchers proposed different type of improvements over the basic Crow Search Algorithm in order to overcome the said slow convergence and increase its optimization performance. Rizk-Allah [18] integrates chaotic sequences into manipulation equations of Crow Search, not only to increase global convergence speed but also to enhance diversification and intensification capabilities. Improved  form of the proposed method is applied for solving fractional optimization problems, and the superiority of the improved algorithm is endorsed by some powerful results. Allaoui et al. [19] hybridized Problem Aware Local Search (PALS)[20] with Crow Search algorithm for solving the DNA fragment assembly problem. Mohammadi and Abdi [17] proposed a different strategy on how a random crow chooses its target crow in the flock to discover its hiding place. In addition, they modified the valuing of the “flight length” control parameter and applied this improved version of the algorithm on electronic load dispatch problem. Horng and Lin [21] merged ordinary Crow Search algorithm with ordinal optimization with a view to solve NP-hard equality constrained simulation optimization problems. Solving real world optimization problems through metaheuristic algorithms has become more and more challenging due to the increased problem size and complexity resulting from the developments and needs of the industrial age. This is mainly due to the increased number of local optimum points in the search domain of the related optimization problem, and the computational burden resulting from an excessive number of function evaluations. Structured population-based evolutionary algorithms come into rescue when the traditional algorithms fail to cope with the abovementioned challenges. Mating of

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population individuals is mainly based on their corresponding fitness value along with their topological structure in these types of algorithms. The population members that are closer to each other have the chance to more freely interact and exchange information. This process not only properly and effectively spreads the high quality solutions over the search space, but also avoids the occurrence of premature convergence. Structured population models can be categorized into two distinct groups: fine-grained/cellular and coarse-grained/distributed models. Cellular models [22-24] distribute the population in a structurally coordinated grid, where the population members only interact within the neighborhood grid to exchange information. This type of interaction among the population entails slow diffusion of the available information through the neighboring grids and eliminates the diversity loss to some degree [25]. Distributed/coarse-grained model (a.k.a. the island model) [26-28] subdivides the population into a group of several sub-populations called demes. The separated populations evolve independently and concurrently by the commanding evolutionary algorithm, and the divided population members migrate across the islands based on the imposed migration policy. Independently evolving islands are able to explore different domains of the search space through the migrated individuals, and the diversity loss is tried to be avoided this way [29]. In a distributed topology, diversification is maintained through separated islands evolving independently, whereas intensification occurs during the interaction between each isolated island. This interaction scheme, composed of sequentially occurring migration and evolution processes, avoids inferiorities in solution efficiency and creates a feasible balance between exploration and exploitation.

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There are other options in developing structured population algorithms, such as combining the favorable merits of the island model with the cellular model. Hierarchical models [30] (a.k.a hybrid) are another alternative where the subdivided islands divide into smaller islands and individual exchange process between each island takes place at the same level [29]. There are also available models such as hierarchy [30], coevolution [31], pool [32], multi-agent systems [33], and master-slave models [34,35], which are all well-accepted methods to construct a successful distributed algorithm. The interested reader could find the details of these models in the study of Gong et al [36]. Among the abovementioned structured population models, island-based distributed algorithm is the most applied and popular method [28]. In the context of island-based models, panmictic individuals of the population are split into independent sub-populations called “islands”, each of which is executed by a unique processor. The information exchange between each subgroup is maintained by the periodically performed migration of some population individuals. Parallel processing of the sub-populations causes an increase in population diversity as well as the convergence capability of the algorithm.

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Many studies found their place in the literature about the research subject of island-based metaheuristic algorithms. Guo et al. [37] proposed variations of Bat algorithm that utilize island-based model to enhance the optimization capability of the conventional Bat algortihm. Afterwards, the authors applied the algorithms on 11 benchmark optimization problems compared the results with that of some other single-population variants of Bat algorithm. The authors found out that the multi-population algortihms performed better than their single-population counterparts. Liu et al. [38] developed a multi-island genetic algorithm and incorporated the algorithm into a multidisciplinary cooptimization method to accomplish the multi-objective optimization of crankshaft structure. The authors also utilized surrogate optimization techniques to carry out the optimization task and they found out that they can cut down the production development cycle and the costs significantly by applying the optimization algorithm. Abadlia et al. [39] proposed to integrate dynamic migration policy island models with Particle Swarm Optimization algorithm to improve the convergence and optimization performance of the algorithm. The authors compared the performance of the algorithm with that of other optimizers and found out that the island-based model improved the performance of the algorithm. Abed-alguni [40] incorporated the island model with variants of Cuckoo Search algorithm, Genetic algorithm and Harmony Search algorithm. The author assessed the performance of these algorithm by applying them on various

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benhcmark functions and concluded that island-based Cuckoo Search algorithm outperforms the other algorithms in terms of convergence and optimization capabilities. It is also observed that increasing the island size and decreasing the migration frequency and migration rate values improved the performance of the algorithm further. Gomez et al. [41] proposed an algorithm called S-PAMICRO to improve the optimization efficacy of the recently introduced SMS-EMOA algorithm. The idea behind S-PAMICRO was to divide the overall population into semi-independent subpopulations and perform the optimization task with all subpopulations communicating with each other. The authors compared the optimization performance of the proposed algorithm with some other island-based variants of literature optimizers and found out that S-PAMICRO is superior to other compared algorithms. Garcia-Hernandez et al. [42] introduced an island-based Coral Reefs Optimization (IMCRO) algorithm to solve the Unequal Area Facility Layout Problem (UA-FLP). The authors proposed two variants of island-based algorithms to solve the problem, namely basic and extended IMCRO. Then, they applied to these algorithms on various UA-FLPs in the literature and the results showed that both basic and extended IMCRO are robust and able to reach the most optimal solutions in most tested problems.This brief literature survey suggests that a very limited number of research studies investigate the influence of the migration topology on the solution accuracy and robustness of the optimization problem. In this study, different types of hierarchical migration models have been proposed to scrutinize the produced solution diversity in the interconnected sub-divided populations. Most of the past studies dealing with the applications of the island models on metaheuristic algorithms only consider a random or ring topology to enhance the population diversity and convergence accuracy. However, this type of application does not explicity reflect the optimization capability of the algorithm as some topologies perform well for some class of problems while some other population structures give better performance for some other class of problems. Therefore, it has not been clearly stated and mentioned yet in the literature studies as to which topological structure gives the best results for a given optimization problem. Main motivation in this study is to conquer this characteristic algorithmic drawback by proposing different hierarchical topologies. Different variants of topologies enable the interlinking population members to reach the unvisited paths of the search domain, which eliminates the premature convergence resulting from the quick information flow between the population individuals to some extent. This study does not consider the CPU cost burdened by the excessive function evaluations of the adopted manipulation schemes, which is the major disadvantage of the all structural population based algorithms, as evaluating the overall algorithm performance is the utmost concern in comparison between the developed schemes.

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This study adopts the essentials of the island model into the framework of the Crow Search algorithm with a view to enhance the solution efficiency and population diversity. Population individuals of Crow Search are grouped into several different islands, and each island population independently evolves using the basic manipulation schemes of Crow Search. The novelty of our study is that, it is the first application of island models on Crow Search. Another originality lies in the proposition of a novel manipulation scheme, whose equations are borrowed from the Opposition-based algorithm [43] and the Global Best Algorithm [44]. This scheme is applied on basic CROW to improve the solution efficiency and accuracy. Furthermore, incorporating a master sub-population into the proposed island models is also applied to enhance the solution efficacy. In addition, different types of migration topologies are firstly proposed and their corresponding solution outcomes are discussed, based on the numerical results of fourty-five benchmark optimization functions. It is seen that different island models with the novel manipulation scheme and master sub-population outperformed basic CROW and island models with basic CROW in most cases. Moreover, the proposed island models are applied to solve six optimal control problems having different difficulties. Once again, the island models with proposed manipulation scheme and master sub-population came up with more desirable solutions than that of previous studies in the literature, basic CROW and island models with basic CROW. The rest of this study is organized as follows, the Crow Search algortihm is briefly described in Section 2. The island models and their applications to the evolutionary algorithms along with a literature review are introduced in Section 3. Application of the different island topologies to the Crow Search algorithm

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is presented in Section 4. Optimization performances of the island models are compared with each other by applying the algorithms on unimodal, multimodal and CEC 2015 benchmark optimization functions in Section 5. Efficacy of the various island models are tested with six optimal control problems each having different difficulties in Section 6. Final remarks are given and the paper is concluded in Section 7. 2. The Crow Search Algorithm

A nature-inspired metaheuristic algorithm called Crow Search was proposed by Azkerzadeh [1]. This algorithm is conceptualized on the foraging behaviors of intelligent crows. Crows are fascinating birds having some notable characteristics during searching food resources. They have a huge memory to keep the location of the possible food stock places when compared to other types of birds. Crows follow other birds to find their food hiding places and steal the food stock of the owner as it leaves from the hiding place. As the crow pilfers the available food, it takes some extra measures such as moving the current location of the food stock to somewhere more desolate in order to keep it safe from other crows. In the context of CROW, each position represents the solution in the search space.

Assume an N-sized crow flock in a D-dimensional solution environment. Position of crow i is k k k k represented by a solution vector ci  [ci,1, ci,2,..., ci,D] (k=1,2,3,...,maxiter and i=1,2,3,...,N) where N is the size of the population and maxiter stands for the maximum number of iterations. Each crow keeps the position of the food hiding place in its memory. This is the best food location obtained by crow i so

k far and symbolized with mi . Crows continue their foraging process in the solution space to find more fruitful areas. k Assume that crow j wants to go to its hidden food source location denoted by mj at iteration k. In the

current iteration, a crow in the flock (say crow i) decides to chase crow j to reach the hiding place of crow j. At this situation, two different cases may occur:

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1- Crow j is not aware that crow i is chasing the hiding place of it. As a result, crow i stealthily arrives at the food stock of crow j. Position of the food location reached by crow i is mathematically formulated by the following equation:



cik 1  cik  rand ()  flik  mik  cik



(1) k

Where rand() is a uniformly distributed random number generated between 0 and 1; and fl i is the control parameter “flight length” responsible for specifying the flight distance of the crow i. Crow j is aware that crow i is in pursue and tries to confuse crow i in order to avoid being pilfered. For this purpose, crow j aims to fool crow i by directing crow to a different location in the search space. These two different scenarios can be merged into a simple mathematical equation by the following expression:

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2

k 1

ci

cik  rand1  flik   mik  cik  rand 2  APi k  k k k lbi  rand 3   ubi  lbi  otherwise

(2)

and in line with this position update, the memory update of crow i in the current iteration can be expressed by the below equation:

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 k 1 ci

if  cik 1  is better than f  cik 

m k  i

otherwise

mik 1  







(3)

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In equations (2) and (3), rand1,2,3 is a Gaussian random number defined in the range between 0 and 1;

lbi and ubi are respectively the predefined lower and upper bounds of the search space; and APik is the awareness probability of crow j at the current iteration. This control parameter plays a huge role in controlling the intensification and diversification phases of the algorithm. Smaller numerical values of this parameter lead to the exploitation of the promising regions of the search space obtained through the iterations. On the contrary, higher awareness probability rates will result in exploring unvisited valleys of the search region in a random manner and promote diversification. One of the benefits of CROW is that it requires fewer control parameters compared to some of the widely used swarm-based metaheuristics such as Particle Swarm Optimization [45] and Bat Algorithm [46]. It is well known that the optimization success of any metaheuristic depends on the proper tuning of these algorithm-specific parameters. The pseudo-code of CROW is given in Table 1. Table 1 Pseudo-code of Crow Search Algorithm

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Crow Search Algorithm Input: Problem objective – f(x), Problem dimension – D, Population size – N,              Define the maximum number of iteration (maxiter) Define the Awareness Probability (AP) and flight length (fl)             Define the upper (ub) and lower (lb) bounds of the search space                  Initialize the crow population within the prescribed boundaries Evaluate the position of each crow in the flock in terms of their corresponding fitness value Give the initial values to memory of each crow in the flock         While k < maxiter do for i = 1 to N (each crow in the flock) do Choose a random crow from the flock to follow ( crow j ) Generate uniformly distributed Gaussian random numbers (rand1 and rand2) if rand1 ≥ AP

cik1  cik  rand2  flik  mkj  cik  

else

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cik  generate a random position in the search space

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end end Apply the boundary check mechanism and eliminate infeasible solutions Assess the quality of each crow position Update the memory of each crow k=k+1              end Output: Best solution vector

3. Fundamentals of distributed evolutionary algorithms: The island model This section will briefly explain the fundamental principles of island models and their association with evolutionary algorithms. The island model is the most applied non-panmictic evolutionary algorithm framework, firstly proposed by Corcoran and Wainwright [47]. The total population is divided into several independently processed sub-populations (islands) under the island model concept. Information

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exchange among the partitioned islands are controlled by the periodically occurring migration process, through which the interaction between each separate island is maintained by sending and receiving a certain number of individuals. There are also some other decisive factors, including the migration rate and the migration frequency that are respectively responsible for deciding the number of individuals to be transferred from one subpopulation to another, and the migration interval of the selected population members during the iterations. Migration is a sporadic procedure, performed under the guidance of two different control mechanisms called the migration topology and the migration policy. Migration topology refers to specific and structured interaction links across the grouped islands. Different migration topologies have been investigated and utilized in literature [48-50], however two different topologies become prominent and promising outcomes have been obtained through their utilization. On the other hand, the migration topology is a decisive factor influencing the optimization performance of the island models. There are several topologies to be used for exchanging information between islands. The ring topology is the most common, in which the disposed islands form a neighboring ring structure to maintain exchanges between their sub-populations. Another alternative is the toroidal structure, which is composed of toroidally interconnected mesh of islands. Migration via this topology takes place between the nearest neighboring islands. Researchers classify the concept of migration topology into two distinct categories. Information exchange route and associated connection links between each neighboring island is predefined and remain constant during the population evolution in static topologies. Diversity loss may occur while applying a static topology on disposed islands as a result of the repetitive pattern of the information exchange procedure. However, island structures based on dynamical topology, in which destination sub-population changes randomly and dynamically in each algorithm run, eliminate the diversity stagnation to some extent [29]. Figure 1 visualizes the structural representation of the random-ring and static-ring topologies where the unidirectionally connected neighboring islands exchange their population members through a predefined path. As seen in the figure, the feasible path interlinking the islands is randomly established in random-ring topology, whereas the connecting route between the edges is prespecified before the information exchange and remain constant during the evolving process in static-ring topology. Several researchers have investigated the influences of different migration topologies over the probability of improving the optimization performance [5153].          

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Figure 1 Comparison between Random ring topology and Static ring topology

The migration policy determines the size of the copied population members from one island to another. The literature includes two common popular policies called the best-worst policy and the randomrandom policy. The best-worst policy considers the best individuals of a sub-population to replace with

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the worst population members of the selected sub-population. The random-random policy is based on selecting random population members of an island to be exchanged with random elements of the target island. According to Skolicki and De Jong [54], best-worst policy has a strong effect on the optimization performance of island models, whereas random-random policy has a negligible influence. The main motivation behind developing an island-based metaheuristic is to increase the population diversity as much as possible to maintain a plausible trade-off between intensification and diversification on the course of iterations. By doing so, the island model under the framework of a metaheuristic algorithm gains the capability to visit unexplored paths of the search domain, thereby providing promising sample solutions to be optimally evolved in the upcoming function evaluations. Unfit population individuals are eliminated and replaced with the fitter members during the evolution process, paving the way for obtaining the global best solution. 4. Hierarchical Island-based Crow Search Algorithm

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Firstly introduced under the framework of Genetic Algorithm and inspired by the Punctuated Equilibrium Theory [55], the island model is a coarse grained parallelization method, implemented on a variety of metaheuristic approaches ranging from Particle Swarm Optimization [56] to Differential Evolution [57]. Partitioned islands composed of members of the main population send and receive information via the injected individuals, through which exploration and exploitation mechanisms of the algorithm are greatly enhanced and diversity creation is excessively boosted. The island model is one of the reputed branches of distributed algorithms, which are predicated upon dividing the population into a certain amount of small-sized sub-populations. Each island evolves independently from the others by the same or different mutation schemes of the base algorithm. Structured island models using the same control parameters and mutation equations of the commanding algorithm are called homogenous methods. On the other hand, heterogeneous island models apply a different type of mutation scheme to each island individual. Island models have two important advantages promoting their utilization under the framework of various types of metaheuristic methods: 1- Semi-isolation resulting from the spatial separation of the population individuals leads to an increased performance in generating feasible offsprings in terms of fitness quality. 2- Concurrently evolving separated islands can be easily implemented into parallel hardware without the burden of excessive computational load on simultaneously working processors. These advantages render island models to get one step ahead of their contemporaries and widens their usage in metaheuristic community.

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The hierarchical model (a.k.a hybrid model) is mainly based on assembling two or more distributed algorithms into a hierarchically structured framework, to improve the problem-solving capability of the hybridized complex structure through utilizing the merits of each constituent model. Literature includes different types of hierarchically developed distributed algorithms, which can be categorized into three distinct subgroups according to Gong [36]: Island-master/slave model [58], island – cellular model [59], and island-island model [60]. Island-master/slave model considers several subdivided subpopulations dictated by different master processors and interact with each other in a predefined time interval. The master sends the available information to the associated slave individuals to further evaluate the refined fitness values of the related subpopulation. Then, the obtained fitness values are sent back to the ruling master for each generation. Island/cellular model involves several partitioned islands in the upper level while having cellular models in the lower level. Island-cellular model is run by multiple diversified island populations, each of which is processed by local search based cellular approaches. Island – island model considers island models for the upper and lower sections of the structured hierarchy. There are two types of migration strategies adopted in hybrid island-island model. Local migration takes place between the islands connected in the same layer. Global migration, however, occurs between the subpopulations interchanging the available information on different layers linking with each other. Cumulative effect of global and local migration strategies taking place between the connecting islands plays a dominant role in producing diversity in the population. Population based metaheuristic algorithms are known for their excellence in solving large scale optimization problems. However, their

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inherent panmictic structure may sometimes jeopardizes the ability to reach the peak valleys in the search domain and hampers the evolution process.

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In this study, a hierarchical island model based on CROW with different migration topologies is proposed. CROW suffer from diversity loss and premature convergence, resulting from the preservation of variation in the population. In order to circumvent the drawbacks of this method, island-island hierarchical hybrid model is embedded into basic CROW to effectively solve different type of real world engineering problems. We consider four different migration topologies incorporated into the hierarchical island-based CROW. In addition, a local search algorithm inspired by the mutation schemes of the Oppositional-learning algorithm [43] and Global Best Algorithm [44] is applied to ordinary CROW to upgrade the probing capabilities of the search agents. The proposed topologies consist of 16 subpopulations, running simultaneously at their corresponding level of hierarchy. Each hierarchy level has different number of sub-population based on the defined migration topology and the available information is exchanged among the subpopulations at the same level of hierarchy. Figure 2 to Figure 5 show the representation of the proposed migration topologies. Studied topologies will be briefly explained from the scratch and their characteristics will be discussed. The main aim to embed these proposed migration topologies into basic CROW is to perform the search conducted by the responsible agents for each hierarchy level. Therefore, independently evolving sub-populations in the lower level collaborate with the global search while the interacting islands in the higher level complete the local search. This hierarchical cooperation among islands upgrades the probing mechanism of the structurally enhanced distributed algorithm.

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Figure 2 Schematic representation of static ring topology for island Crow model

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Figure 3 Hierarchical topology composed of four island in the lower layer and four island in the upper layer

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Hierarchical topology consisting of two island in the lower layer and eight island in the upper layer

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Figure 4

Figure 5 Hierarchical topology constructed by the three communicating layers

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Figure 2 visualizes the ordinary ring topology, which is the most simple and applied topology used in the context of distributed algorithms. Sequentially organized 16 subpopulations (islands) communicate only with their neighboring islands. Interaction between islands can be either uni-directional or bidirectional in this type of migration topology. There are several literature applications of both variants of this topology, particularly embedded in Genetic Algorithm [61-63]. Figure 3 depicts the visual representation of hierarchical topology composed of 4 neighboring islands in the lower level and the 4 groups formed by the constituent 4 islands in the upper level. Figure 4 shows the proposed layered structure of the migration topology for distributed CROW. This architecture consists of 8 higher level structured populations, each of which involves 2 periodically communicating islands. A schematic illustration of the multi layered architecture, on which the subpopulations are executed in parallel, is shown in Figure 5. In this context, the evolution of 2 low-level subpopulations is accomplished. In the upper layer, the refined and updated information retained from the lower layer is circulated through 4 medium level organized structures comprising the lower level subpopulation individuals. The fittest individuals collected from the medium level subpopulations are exchanged in the distributed structure where the shared information is taken from the individuals of the lower level hierarchies.

Algorithm 1 shows the general representation of the developed migration topologies in pseudo-code. The proposed procedure consists of 8 distinctive algorithmic steps, each of which is responsible for establishing different duties to construct a reliable and robust distributed optimization algorithm. Step 1 deals with specifying the objective function ( f(x) ) and the problem dimension (D), initializing the

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associated algorithm parameters : Awareness Probability(AP) and flight length(fl) , defining the upper and lower bounds of the search space (ub and lb), and the maximum number of iterations (maxiter). Step 2 accomplishes initializing the random positions of the crow individuals in the search space and their associated memories according to their allocated positions. Step 3 subdivides the whole crow population into a predefined number of subpopulations, and the population members are randomly distributed into a set of islands. We consider 16 different islands ( I n  16 ) to be run concurrently on the course of iterations. Hence, the population size in each island ( I s ) becomes: I s  N / In . Only the highest performing individual of the subpopulation is considered to be migrated to the neighboring subpopulation within each iteration. The migration interval between the generation steps is noticeably frequent, i.e., it takes place in each iteration. As it is mentioned above, the selection strategy is based on replacing the fittest individual instead of a random individual with the worst individual of the connecting subpopulation neighborhood. Parallel evolution of the subdivided groups will take action more directly by copying the best individual to its neighboring subpopulation. However, this does not mean random selection is ineffective. This type of selection (random selection) creates a significant diversity in the population albeit convergence to the optimum solution will waste more computational effort compared to the case carried out by the best individual. Figure 6 depicts the main steps of the hierarchical Crow Search based island model. The algorithm starts by initializing the population and parameters and dividing the population into predetermined number of islands. Afterwards, an iterative procedure is initialized and each island sub-population explores the search space then sends their fittest individuals to the other islands which resides in upper layers than theirs in the hierarchy. This iterative procedure repeats until the algorithm termination criterion is met and the fittest individual in the highest rank of the hierarchy is selected as the global best answer.

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Algorithm 1: Hierarchical island based Crow Search Algorithm Step – 1 : Initialize the algorithm specific parameters Specify : the objective function – f(x) Define : the problem dimension – D and the population size – N Define : the maximum number of iterations (maxiter) Define : the prespecified upper (ub) and lower bounds (lb) of the search span Initialize : the algorithm parameters                  → Awareness Probability (AP) – flight length (fl) → Number of island (In) – Population size in each island (Is) Step – 2 : Initialize the crow population (C) and crow memory (m) for i = 1 to N for j = 1 to D Ci,j = lbj + (ubj-lbj) x rand1(0,1) mi,j = lbj + (ubj-lbj) x rand2(0,1) end end Step – 3 : Divide the whole population (C) into predefined number of islands (In = 16) Subdivide - C() → Ck k=1,2,3,...,In Step – 4 : Commence the function evaluations iter = 0 while (iter < maxiter) do for k = 1 to In do for i = 1 to Is do Pick a random crow from the flock to follow ( crow t ) t  1, Is 

Generate uniformly distributed Gaussian random numbers (rand3, rand4, rand5 ) if rand3(0,1) ≥ AP then Cik, j  Cik, j  rand 4 (0,1)  flik, j  mtk  Cik, j , where j  1, D else





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Cik, j  lbik, j 

ubik, j  lbik, j   rand  0,1 5

end k

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end end Apply boundary check mechanism Evaluate the fitness quality of each crow for each subpopulation Perform memory update of each crow for each subpopulation

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Obtain Cbest , which is the fittest individual of each subdivided population Perform the proposed local search strategy defined in Algorithm 1 on each individual to fine-tune the promising solutions (if necessary) Select the best member of the curent subpopulation and replace it with the worst member of the neighbouring subpopulation at same level of hierarchy Recieve the best member of the connected subpopulation at the same hierarchy level and replace it with the worst member of the current subpopulation Step – 5 : Send the fittest individuals of the current layer to upper layer Circulate the fittest members of the current hierarchy level among the upper hierarchy level to perform local search around the promising solutions Obtain the updated best solutions of each subpopulation acquired after performing the circulation Step – 6 : Repeat the procedure defined in Step – 5 until reaching to the most upper layer of hierarchy Step – 7 : Increment the iteration counter (iter = iter+1) end Step – 8 : Retain the global best answer of the optimization problem

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Figure 6 Main execution steps of the hierarchical Crow Search algorithm

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Step 4 realizes the function evaluations of each independently running island with a view to reposition the crow individuals in the decision space by means of the algorithm specific mutation schemes formulated by Eq. 1 to Eq. 3. Next, the boundary check mechanism is applied to restrict the violated solutions into the prescribed boundaries and quality assessment of the fitness values of each individual in each island is performed. Finally, position memory of the crows in the subdivided groups is updated based on the best solutions obtained so far. The fittest individuals of each subpopulation are retained in order to apply the local search-based mutation scheme. The proposed scheme randomly adopts the  manipulation equations of the Oppositional-based learning and Global Best Algorithm with an aim to avoid local optima and probe around promising regions of the search domain by using the merits of the best individual of the related subpopulation. Local search-based manipulation scheme is mathematically expressed by the below algorithm. Algorithm 2 : Local search strategy for i = 1 to N do for j = 1 to D do if ( rand1(0,1) < rand2 (0,1) ) k k k Cnew ,i, j  Cbest ,i  2.0  i, j  Ci, j

else

k  1,2,3,..., In

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k k k Cnew ,i, j  2.0  Cbest ,i  Ci, j

end end end

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In the algorithm,  is the chaotic random number defined in the range [0,1] generated by the equations th k of the Logistic map [64], and Cbest ,i is the best individual of the k subpopulation. Migration between

the partitioned subpopulations at the same level of hierarchy is maintained by replacing the best member of the current subpopulation with the worst member of the interlinked neighboring subpopulation. Step 5 is concerned with sending the promising population individuals of the current level to the upper hierarchy level to benefit their accumulated knowledge on the diversity information. This procedure can be considered as some kind of a local search as the upper level population individuals will have the opportunity to take advantage of utilizing the promising search agents transferred from the lower level hierarchy. The best search agents obtained by using this population update strategy are then exchanged with the worst member of each population in the higher level of hierarchy. The same procedure explained in Step - 5 repeats itself until the highest level of hierarchy is reached, which is the main issue in Step 6. Iteration counter is incremented, and the algorithmic process is completed for this iteration. Step 4 to Step 7 are repeated until the stopping condition is reached.

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In this study, we propose four different migration strategies, not only to evaluate the optimization performance of the crow search based island models running under different topologies, but also to improve the solution diversity in the population. The first proposed topology (CROWISL16) is simply based on the static ring structure composed of sixteen islands interlinked to their nearest neighbors to construct a feasible connection path, as shown in Figure 2. The second proposed migration topology consists of a two-level hierarchy. The lower hierarchy level is composed of four subpopulations (islands) connected to each other performing the local migrations. The upper hierarchy level is constructed by the best individuals adopted from the lower hierarchy level to realize the global migrations between the four neighboring sets of populations. The schematic representation of this migration topology, which is hereafter called CROWISL44 for simplicity, is illustrated in Figure 3. The third proposed migration topology is similar to second approach with small differences, as it has eight sets of subpopulations in the higher hierarchy level, each of which collects the available information from its associated 2 subpopulations belonging to the lower hierarchy level. Figure 4 depicts the schematic representation of the proposed migration policy, which is called CROWISL82 hereafter. The fourth and the last proposed migration topology is relatively the most complicated one as it consists of three levels of hierarchy. The upper hierarchy level gathers the best solution agents those obtained by the medium level performing both global and local migrations. The lower level hierarchy population individuals are responsible for accomplishing only local migrations. Figure 5 visualizes the basic structure of the fourth proposed migration topology, which namely called hereafter CROWISL242. To fine-tune the refined solutions obtained by the search agents of parallel processing islands, local search strategy defined in Algorithm 1 is adopted and applied on the most promising solutions. Crow search based island-based models utilizing the merits of local search strategy explained in Algorithm 1 will be respectively named after CROWMISL16, CROWMISL44, CROWMISL82, and CROWMISL242. Here “M” stands for “MASTER”, which denotes the master population composed of the most successful search agents. In each topology described above, the best individuals are replaced with the worst ones along the predetermined connection path within each completed iteration. The first layer (hierarchy level) corresponds to independent evolution of the structured populations (islands) frequently communicating with each other to facilitate the global search. The second and upper layers perform the local search nearby the promising regions of the search space by virtue of the refined search

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5. Computational Experiments

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agents adopted from the lower layer population individuals. Synchronous migration type is considered, which all migrant individuals migrate to their prespecified islands simultaneously. In this study, the computation cost burdened by the execution of the different migration topologies is not taken into account, as the main aim is to evaluate the overall optimization performance of the proposed algorithms having different migration topologies. Our main motivation behind developing these island-based algorithms is to obtain the most efficient optimization framework. A significant effort is also given to observe the influences of the number of hierarchical layers on the optimization performance of the proposed distributed algorithms.

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Each developed island model is executed in Java environment, and the experiments are conducted on a personal computer with Intel Core processor having 6.0 GB RAM at 2.50 GHz CPU. In order to assess the predictive performance of the island-based CROW running under different migration topologies, 45 well-known and highly reputed optimization test functions are employed, and their respective solution outcomes are compared. A total number of 80,000 function evaluations are performed on 30dimensional optimization test functions. 50 consecutive algorithm runs are made for each proposed method due to their complex stochastic nature. The benchmark functions can be categorized into different branches including unimodal, multimodal, and composite test functions, through which the optimization capability of many stochastic based optimization algorithms have been evaluated by researchers in the literature. Functions f1 to f15 represent multi-modal benchmark functions with considerable amount of local optimum points on the search space. These type of test functions have been generally utilized to evaluate the explorative capability of the developed algorithms, which is extremely desired to avoid local optima and explore the promising regions nearby the global optimum point. Optimization benchmark functions f16 to f29 are unimodal test functions, applied to the proposed methods to investigate their efficiency on exploitation of the promising search areas. On the other hand, the last 16 benchmark functions (f30-f45) are composed of multi-niche problems and composite test functions those were handled in CEC-2015 [65]. Test functions proposed in CEC-2015 are designed for assessing the convergence capabilities of the contestant optimization algorithms on single objective optimization problems. Detailed descriptions and characteristic features of these test problems are provided in [65]. Moreover, six selected literature optimizers, namely Bat Algorithm (BAT) [38], Cuckoo Search Algorithm (CUCKOO) [66], Big Bang Big-Crunch Algorithm (BB-BC) [67], Fruit Fly Optimization Algorithm (FRUITFLY) [68], Intelligent Tuned Harmony Search Algorithm [69] and Quantum Behaved Particle Swarm Optimizer (QPSO) [70], with the same number of function evaluations are also applied on the unimodal and multi-modal benchmark functions to determine the effectivity of the developed island models.

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Table 2 tabulates the results found by the island models, along with basic CROW in terms of best, worst mean and standard deviation values for multi modal test functions. As mentioned before, multi-modal test functions consist of many local optima, which are convenient test beds for scrutinizing the effectiveness of the algorithm in space exploration. Island models show quite impressive performance in exploring the unvisited sections of the search domain, as they find the global optimum solution of six out fifteen multi-modal test functions. Between the island models, CROWMISL242 shows much better performance compared to its counterparts for f1 to f15. CROWMISL242 gives the best minimum objective function values for all multi-modal benchmark functions except f5, f6, f14. Considering the mean values of the objective function, CROWMISL82 is the best performing algorithm, obtaining the minimum mean values for f1, f2, f3, f5, f6, f7, f9, f10, f12, f13, f15 test functions. CROWMISL242 is the second best for mean values with its prediction success on f4, f11 and f14. The consistency of CROWMISL82 is quite remarkable based on the standard deviation values. Figure 7 to Figure 10 show the convergence histories of the objective functions for island models for multi-modal test functions. It is evident that the

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island models reach their optimal solutions much quicker than basic CROW. From the statistical comparative results as well as the numerical solution comparison between each proposed method, it can be concluded that island models have remarkably superior capabilities on avoiding the local optimum points in the search space and exploring the unvisited paths of the search domain.

Best   8.00E-03 1.17E+00 9.98E-01 1.08E+00 3.86E-09 8.90E-02 1.79E-01 8.90E-02 3.86E-09

Std. Dev.   2.24E-02 3.11E+00 2.87E+00 2.59E+00 2.83E+00 2.36E-01 4.97E-01 2.93E+00 3.29E-01

Mean   3.75E-02 5.86E+00 5.72E+00 5.36E+00 5.38E+00 3.56E-01 5.56E-01 2.42E+00 4.31E-01

Worst   1.36E-01 1.47E+01 1.51E+01 1.31E+01 1.30E+01 1.08E+00 3.44E+00 1.37E+01 2.17E+00

2.91E-02 5.09E-14 2.79E-08 1.38E-13 2.19E-13 7.17E-16 1.17E-15 1.91E-15 7.24E-16

1.60E-01 8.96E-14 5.75E-09 1.49E-13 1.17E-13 4.14E-15 4.44E-15 6.80E-15 4.15E-15

2.32E-01 2.45E-13 1.69E-07 6.32E-13 1.82E-12 7.54E-15 7.54E-15 1.46E-14 7.54E-15

1.91E-04 1.20E-01 9.00E-03 8.00E-03 7.00E-03 0.00E+00 0.00E+00 3.44E-17 1.52E-17

4.71E-04 1.00E-02 8.00E-03 7.00E-03 5.00E-03 0.00E+00 0.00E+00 6.66E-18 2.13E-18

1.00E-03 6.60E-02 3.40E-02 4.20E-02 2.22E-02 0.00E+00 0.00E+00 2.22E-16 1.11E-16

1.37E+02 4.17E+01 3.58E+01 3.08E+01 3.28E+01 0.00E+00 0.00E+00 0.00E+00 0.00E+00

1.16E+01 1.78E+01 1.82E+01 1.85E+01 1.43E+01 8.24E+00 3.08E+00 2.20E+00 0.00E+00

1.72E+02 6.74E+01 7.44E+01 7.08E+01 6.62E+01 1.41E+01 5.86E-01 5.91E+01 0.00E+00

1.90E+02 1.14E+02 1.17E+02 1.13E+02 9.65E+01 5.96E+01 1.89E+00 1.10E+02 0.00E+00

5.11E+00 8.51E-07 2.04E-06 1.28E-06 1.28E-06 8.60E-160

1.13E+01 1.00E-03 2.02E-04 2.00E-03 3.93E-04 1.53E-96

8.44E+01 3.55E-04 1.21E-04 6.87E-04 2.04E-04 1.86E-97

1.04E+02 6.00E-03 1.00E-03 1.21E-03 1.00E-03 1.28E-95

1.04E-01 2.17E-14 7.14E-14 2.88E-14 2.17E-14 3.99E-15 3.99E-15 3.99E-15 3.99E-15

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1.59E-04 0.00E+00 2.22E-16 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

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  f1 - Levy CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242   f2 - Ackley CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242    f3 - Griewank CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242   f4 - Rastrigin CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242     f5 - Zakharov CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82

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Table 2 Statistical results of island models for multi-modal test functions

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3.75E-54 7.59E-09 2.46E-87

1.84E-52 7.31E-08 1.35E-85

2.30E-02 3.41E-13 7.89E-04 1.46E-12 1.23E-10 4.55E-16 2.23E-15 1.59E-13 3.97E-14

6.20E-02 8.42E-14 2.74E-04 3.35E-13 2.15E-11 1.06E-16 5.40E-16 3.14E-14 5.56E-15

1.43E-01 2.21E-12 3.00E-03 9.95E-12 8.79E-10 2.72E-15 1.49E-14 1.14E-12 2.89E-13

3.33E-06 3.51E-23 2.42E-02 2.50E-02 3.45E-23 3.61E-22 8.73E-23 1.50E-02 1.40E-02

1.09E-05 9.01E-11 6.00E-03 6.00E-03 9.01E-11 9.01E-11 9.01E-11 2.00E-03 2.00E-03

1.94E-05 9.01E-11 1.03E-01 1.03E-01 9.01E-11 9.01E-11 9.01E-11 1.03E-01 1.03E-01

5.71E-05 4.11E-02 4.13E-02 2.84E-02 2.56E-02 4.91E-02 3.37E-02 2.57E-02 3.62E-02

1.63E-04 1.79E-02 1.38E-02 9.21E-03 7.08E-03 2.22E-02 1.50E-02 7.00E-03 1.61E-02

2.92E-04 1.87E-01 2.22E-01 1.01E-01 1.11E-01 2.00E-01 9.05E-02 1.01E-01 1.03E-01

1.29E+01 2.20E-12 1.12E+00 8.49E-13 1.05E-08 4.60E-14 5.05E-07 7.48E-13 2.80E-13

3.70E+01 3.58E-13 2.19E-01 2.78E-13 1.33E-09 2.37E-14 6.63E-08 1.44E-13 9.88E-14

6.88E+01 1.60E-11 7.81E+00 4.74E-12 8.61E-08 2.99E-13 3.91E-06 5.30E-12 1.64E-12

5.00E-03 2.70E-07 4.52E-07 2.51E-07 3.37E-07 2.97E-63 7.21E-31 2.36E-16 7.65E-49

4.00E-03 1.78E-07 5.64E-07 2.38E-07 2.82E-07 4.29E-64 1.03E-31 3.49E-17 1.02E-49

3.30E-03 1.46E-06 2.03E-06 7.95E-07 1.13E-06 2.10E-62 5.40E-30 1.63E-15 5.82E-48

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2.60E-53 1.79E-08 1.81E-86

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CROWMISL44 2.89E-175 CROWMISL16 6.36E-13 CROWMISL242 2.95E-163    f6 - Alpine CROW 3.60E-02 CROWISL82 3.36E-15 CROWISL44 1.07E-14 CROWISL16 5.49E-15 CROWISL242 4.51E-15 CROWMISL82 6.27E-58 CROWMISL44 1.17E-85 CROWMISL16 1.05E-15 CROWMISL242 8.71E-66   f7 – Generalized Penalized1 CROW 4.50E-06 CROWISL82 9.01E-11 CROWISL44 9.01E-11 CROWISL16 9.01E-11 CROWISL242 9.01E-11 CROWMISL82 9.01E-11 CROWMISL44 9.01E-11 CROWMISL16 9.01E-11 CROWMISL242 9.01E-11    f8 – Generalized Penalized2 CROW 6.06E-05 CROWISL82 7.89E-11 CROWISL44 7.89E-11 CROWISL16 7.89E-11 CROWISL242 7.89E-11 CROWMISL82 7.89E-11 CROWMISL44 7.89E-11 CROWMISL16 7.89E-11 CROWMISL242 7.89E-11    f9 - Quintic CROW 1.86E+01 CROWISL82 0.00E+00 CROWISL44 3.02E-12 CROWISL16 4.44E-16 CROWISL242 4.44E-16 CROWMISL82 0.00E+00 CROWMISL44 3.99E-15 CROWMISL16 0.00E+00 CROWMISL242 0.00E+00   f10 - Csendes CROW 1.33E-04 CROWISL82 1.55E-12 CROWISL44 1.93E-10 CROWISL16 5.10E-16 CROWISL242 4.62E-10 CROWMISL82 8.77E-104 CROWMISL44 1.69E-85 CROWMISL16 1.77E-51 CROWMISL242 8.91E-104   

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1.70E-03 1.12E-03 1.00E-03 1.00E-03 1.12E-03 4.76E-04 2.21E-04 9.42E-04 5.43E-04

1.18E-02 6.00E-03 5.00E-03 6.12E-03 5.12E-03 1.12E-03 1.00E-03 2.00E-03 1.00E-03

1.63E-02 1.00E-02 1.00E-02 1.32E-02 9.83E-03 1.00E-03 1.45E-03 6.00E-03 1.00E-03

2.61E-01 8.00E-02 6.51E-02 7.41E-02 9.41E-02 1.81E-02 3.61E-02 5.81E-02 2.91E-02

3.08E-01 5.17E-01 5.33E-01 5.41E-01 5.01E-01 9.61E-02 1.15E-01 1.67E-01 9.90E-02

3.99E-01 6.99E-01 6.99E-01 6.99E-01 6.99E-01 9.91E-01 1.99E-01 2.99E-01 1.99E-01

1.41E-02 4.75E-01 5.94E-01 4.43E-01 5.48E-01 0.00E+00 0.00E+00 0.00E+00 0.00E+00

2.91E-02 1.41E+00 1.53E+00 1.33E+00 1.53E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

7.51E-02 2.50E+00 2.81E+00 2.35E+00 2.80E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

2.73E-01 1.73E-01 1.65E-01 1.49E-01 1.98E-01 1.41E-01 1.51E-01 1.59E-01 1.21E-01

3.34E+00 3.11E-01 3.59E-01 3.02E-01 3.30E-01 2.86E-01 2.91E-01 2.41E-01 2.21E-01

3.93E+00 8.01E-01 7.79E-01 7.04E-01 9.43E-01 5.28E-01 7.08E-01 5.34E-01 5.44E-01

1.96E-02 4.03E-02 3.72E-02 5.89E-02 4.14E-02 0.00E+00 0.00E+00 1.24E-01 0.00E+00

6.53E-01 3.16E-01 3.08E-01 2.98E-01 3.22E-01 0.00E+00 0.00E+00 6.97E-02 0.00E+00

6.96E-01 4.24E-01 3.91E-01 4.07E-01 4.05E-01 0.00E+00 0.00E+00 3.44E-01 0.00E+00

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 f11 - Schaffer CROW 7.00E-03 CROWISL82 3.00E-03 CROWISL44 3.12E-03 CROWISL16 3.00E-03 CROWISL242 3.11E-03 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 1.00E-03 CROWMISL242 0.00E+00   f12 - Salomon CROW 2.39E-01 CROWISL82 3.99E-01 CROWISL44 3.99E-01 CROWISL16 3.99E-01 CROWISL242 2.99E-01 CROWMISL82 3.06E-87 CROWMISL44 9.90E-02 CROWMISL16 9.91E-02 CROWMISL242 1.61E-107    f13 - Inverted cosine mixture function CROW 6.00E-03 CROWISL82 5.91E-01 CROWISL44 4.43E-01 CROWISL16 5.91E-01 CROWISL242 5.91E-01 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 0.00E+00 CROWMISL242 0.00E+00   f14 – Pathological CROW 2.66E+00 CROWISL82 5.41E-02 CROWISL44 6.84E-02 CROWISL16 6.37E-02 CROWISL242 8.57E-02 CROWMISL82 2.52E-02 CROWMISL44 4.61E-02 CROWMISL16 6.51E-03 CROWMISL242 5.43E-02    f15 – Wavy CROW 6.09E-01 CROWISL82 2.48E-01 CROWISL44 2.42E-01 CROWISL16 2.00E-01 CROWISL242 2.35E-01 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 0.00E+00 CROWMISL242 0.00E+00

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Figure 7 Evolution of objective functions with increasing number of function evaluations for Levy, Ackley, Griewank, and Rastrigin test functions

Figure 8 Comparison of convergence rates for Zakharov, Alpine, Penalized1, and Penalized2 test functions

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Figure 9 Convergence rate comparison for Quintic, Csendes, Schaffer and Salomon test functions

Figure 10 Evolution histories of Inverted cosine mixture function, Pathological function and Wavy function Table 3 depicts the results found by the selected literature optimizers for unimodal test functions. It can be seen from the table that proposed island models outperformed the literature optimizers for the most of the unimodal problems. However, is some cases literature optimizers showed a similar or better performance compared to the island models. Such as, CUCKOO showed a similar performance to island models for finding the optimal values of f7 , f8, f10 and FRUITFLY outperformed other algorithms at f12.

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Table 3 Statistical results of selected literature optimizers for multi-modal test functions Mean   2.23E+01 3.86E-09 3.63E+01 3.11E+00 1.39E-02 5.07E+00

Worst   7.09E+01 3.86E-09 5.20E+01 3.25E+00 6.30E-02 1.30E+01

1.01E+00 2.50E-15 1.82E-03 2.58E-08 4.09E-02 3.94E-01

4.72E-01 9.85E-15 2.05E-02 6.20E-05 6.15E-02 4.04E-01

4.33E+00 1.46E-14 2.48E-02 6.21E-05 1.62E-01 1.94E+00

1.32E-01 2.51E-08 2.77E-02 4.92E-13 7.60E-03 5.11E-02

5.77E-02 4.26E-09 1.71E-02 4.32E-10 3.43E-03 3.58E-02

7.91E-01 1.53E-07 1.26E-01 4.34E-10 4.22E-02 2.30E-01

1.30E+02 3.16E+01 1.15E+02 1.45E-09 2.37E+01 2.05E+01

3.15E+02 1.16E+02 5.18E+02 1.43E-06 3.43E+01 8.86E+01

7.92E+02 1.64E+02 8.47E+02 1.43E-06 7.27E+01 1.37E+02

1.03E+04 2.36E+02 2.09E+02 1.29E-08 3.52E+00 1.23E+01

3.64E+03 4.26E+02 4.21E+02 1.10E-05 2.04E+00 1.63E+01

4.55E+04 1.32E+03 8.53E+02 1.10E-05 1.70E+01 4.99E+01

3.84E+00 8.65E-03 3.26E+00 2.27E-08 1.59E-01 2.63E-01

9.06E+00 1.22E-02 9.22E+00 4.65E-05 7.28E-02 1.40E-01

2.02E+01 3.58E-02 1.80E+01 4.66E-05 7.62E-01 1.40E+00

1.69E+00 1.35E-23 1.69E+00 4.13E-08 8.45E-04 2.39E-01

1.42E+00 9.01E-11 3.80E+00 1.66E+00 4.34E-04 1.41E-01

8.42E+00 9.01E-11 9.93E+00 1.66E+00 4.90E-03 1.23E+00

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Std. Dev.   1.38E+01 6.24E-22 8.11E+00 7.93E-02 1.70E-02 3.18E+00

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Best   f1 - Levy   BAT 6.34E+00 CUCKOO 3.86E-09 BB-BC 2.08E+01 FRUITFLY 2.90E+00 ITHS 1.56E-06 QPSO 3.58E-01   f2 - Ackley BAT 2.50E-03 CUCKOO 7.54E-15 BB-BC 1.58E-02 FRUITFLY 6.20E-05 ITHS 4.00E-04 QPSO 4.77E-02    f3 - Griewank BAT 7.87E-07 CUCKOO 0.00E+00 BB-BC 3.84E-05 FRUITFLY 4.31E-10 ITHS 1.93E-06 QPSO 8.49E-05   f4 - Rastrigin BAT 9.94E+01 CUCKOO 2.59E+01 BB-BC 3.37E+02 FRUITFLY 1.42E-06 ITHS 1.07E-03 QPSO 4.99E+01     f5 - Zakharov BAT 1.04E+01 CUCKOO 1.38E+02 BB-BC 1.23E+02 FRUITFLY 1.09E-05 ITHS 3.03E-03 QPSO 1.16E+00    f6 - Alpine BAT 2.42E+00 CUCKOO 3.65E-04 BB-BC 3.03E+00 FRUITFLY 4.65E-05 ITHS 3.04E-04 QPSO 2.40E-03   f7 – Generalized Penalized1 BAT 2.97E-07 CUCKOO 9.01E-11 BB-BC 1.28E+00 FRUITFLY 1.66E+00 ITHS 7.18E-07 QPSO 6.42E-05   

Journal Pre-proof

1.95E+01 1.78E-23 1.72E+01 4.59E-02 7.57E-03 1.84E+00

4.09E+01 7.89E-11 6.19E+01 2.97E+00 5.45E-03 8.46E-01

9.38E+01 7.89E-11 9.10E+01 3.26E+00 3.51E-02 1.11E+01

1.42E+02 4.12E-01 1.28E+01 2.45E-06 1.48E+01 2.15E+01

5.94E+01 1.14E-01 2.65E+01 1.16E+02 9.00E+00 9.53E+00

1.05E+03 2.38E+00 6.81E+01 1.16E+02 7.01E+01 1.43E+02

3.17E+02 2.89E-50 2.28E+00 2.31E-20 5.24E-05 8.41E+00

5.88E+01 9.52E-51 4.06E-01 2.41E-20 1.70E-05 4.06E+00

2.14E+03 1.22E-49 1.44E+01 1.19E-19 3.10E-04 3.75E+01

4.16E-02 8.98E-04 3.03E-02 5.56E-13 3.78E-02 3.30E-02

1.90E-01 4.12E-03 2.34E-01 3.79E-12 4.58E-02 1.36E-01

2.55E-01 6.23E-03 3.13E-01 6.43E-12 1.78E-01 1.78E-01

3.57E-01 1.53E-02 3.33E-01 1.99E-01 1.31E-01 1.13E-01

2.12E+00 1.97E-01 2.68E+00 4.69E-01 2.61E-01 6.99E-01

2.79E+00 1.99E-01 3.29E+00 8.15E-01 6.99E-01 8.99E-01

3.61E+00 0.00E+00 1.54E+00 1.04E-10 7.18E-01 3.94E+00

1.48E+01 0.00E+00 5.49E+00 9.63E-08 3.80E-01 3.28E+00

3.49E+01 0.00E+00 1.02E+01 9.66E-08 3.06E+00 2.03E+01

8.65E-01 2.96E-01 6.23E-01 4.00E-17 3.94E-01 5.25E-01

3.33E+00 5.29E+00 3.71E+00 1.70E-14 1.19E+00 3.37E+00

5.76E+00 5.77E+00 4.77E+00 1.71E-14 2.13E+00 4.15E+00

6.63E-02 4.99E-02

8.14E-01 7.51E-01

9.20E-01 8.46E-01

Jou

rna

lP repro of

f8 – Generalized Penalized2 BAT 1.29E+01 CUCKOO 7.89E-11 BB-BC 2.82E+01 FRUITFLY 2.96E+00 ITHS 3.47E-05 QPSO 5.57E-04    f9 - Quintic BAT 1.34E+01 CUCKOO 3.70E-07 BB-BC 5.59E+00 FRUITFLY 1.16E+02 ITHS 3.70E-05 QPSO 1.32E-01   f10 - Csendes BAT 2.69E-06 CUCKOO 2.00E-56 BB-BC 6.20E-12 FRUITFLY 4.86E-21 ITHS 6.89E-14 QPSO 2.12E-03     f11 - Schaffer BAT 1.07E-01 CUCKOO 2.48E-03 BB-BC 1.79E-01 FRUITFLY 3.61E-12 ITHS 9.71E-03 QPSO 7.81E-02   f12 - Salomon BAT 1.49E+00 CUCKOO 1.00E-01 BB-BC 1.99E+00 FRUITFLY 1.50E-01 ITHS 9.98E-02 QPSO 4.99E-01    f13 - Inverted cosine mixture function BAT 1.14E+01 CUCKOO 0.00E+00 BB-BC 2.81E+00 FRUITFLY 9.60E-08 ITHS 3.56E-04 QPSO 1.11E+00   f14 – Pathological BAT 1.65E+00 CUCKOO 4.33E+00 BB-BC 2.15E+00 FRUITFLY 1.69E-14 ITHS 4.60E-01 QPSO 1.51E+00    f15 – Wavy BAT 6.12E-01 CUCKOO 6.39E-01

Journal Pre-proof

BB-BC FRUITFLY ITHS QPSO

6.84E-01 1.21E-08 1.52E-01 2.37E-01

6.72E-02 1.22E-11 5.58E-02 6.28E-02

8.41E-01 1.21E-08 3.02E-01 3.56E-01

9.55E-01 1.22E-08 4.15E-01 4.84E-01

lP repro of

Table 4 reports the optimal solutions found by island models for unimodal test functions. Although none of the compared island models find the global optimum solution of any benchmark function given in Table 4, satisfactory statistical results are obtained with regards to best objective function value, as each compared model obtains very close numerical values to the global optimum point of each test function. Among the proposed algorithms, CROWMISL44 shows much better optimization performance in terms of both best and mean values. CROWMISL44 finds the minimum optimal values for f16, f19, f21, f22, f25, f26, f27, f28, f29 unimodal test functions. In addition, mean values for the test functions of f16, f18, f21, f22, f23, f25, f26, f27, f28, f29 are much superior compared to the other compared algorithms. It is also noteworthy to mention that CROWMISL44 is the most consistent and robust island model among them. In terms of best and mean values, CROWMISL242 becomes the second-best algorithm after CROWMISL44. Figure 11 to Figure 14 show the convergence curves of the island models for unimodal test functions. For most of the unimodal test functions, island models trap into stagnation period at the early phases of the iterations then converge into their optimal points at the end. Table 4 Statistical results for unimodal test functions Best 4.83E-04 7.84E-32 7.34E-29 3.37E-30 1.03E-32 1.99E-69 4.03E-82 4.57E-41 3.48E-73

f17 - Rosenbrock CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

Std.dev.

Mean

Worst

5.38E-04 1.12E-28 5.03E-14 5.14E-27 1.84E-28 4.38E-61 1.32E-66 1.71E-37 1.98E-63

1.12E-03 2.91E-29 7.42E-15 1.56E-27 3.68E-29 6.45E-62 2.07E-67 4.34E-38 3.39E-64

2.54E-03 8.07E-28 3.66E-13 3.34E-26 1.40E-27 3.53E-60 9.94E-66 1.22E-36 1.38E-62

2.26E-01 1.08E+01 8.31E+00 1.12E+01 1.20E+01 5.85E-01 6.13E+00 8.17E+00 9.54E-01

2.81E+01 7.69E+01 8.01E+01 7.66E+01 7.76E+01 2.30E+01 7.11E+01 8.06E+01 2.30E+01

rna

f16 - Sphere CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

2.75E+01 2.55E+01 2.53E+01 2.55E+01 2.59E+01 2.19E+01 2.28E+01 2.27E+01 2.19E+01

f18 - Schwefel 2.22 CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

3.05E-02 2.01E-18 5.27E-11 3.07E-17 4.20E-18 2.68E-59 1.52E-83 1.34E-25 1.87E-64

8.18E-02 5.46E-15 6.72E-02 1.18E-13 1.53E-10 7.62E-50 2.40E-65 6.45E-24 6.75E-55

5.19E-02 1.67E-15 1.11E-02 2.48E-14 2.20E-11 1.29E-50 3.82E-66 5.58E-24 1.23E-55

6.75E-02 3.00E-14 4.89E-01 8.49E-13 1.13E-09 5.54E-49 2.02E-64 2.70E-23 4.87E-54

f19 - Schwefel 2.23 CROW

3.33E-09

4.01E-06

2.54E-06

2.05E-05

Jou

2.68E+01 1.73E+01 1.89E+01 9.69E+00 1.03E+01 2.00E+01 1.94E+01 1.41E+01 1.69E+01

Journal Pre-proof

5.12E-69 8.33E-66 1.94E-64 2.89E-70 1.00E-109 5.77E-97 4.04E-82 6.95E-121

7.74E-56 9.30E-44 4.56E-50 3.56E-57 5.81E-79 1.53E-65 7.42E-63 4.08E-75

1.17E-56 1.38E-44 6.56E-51 8.50E-58 9.07E-80 2.29E-66 1.12E-63 5.89E-76

5.63E-55 6.38E-43 3.32E-49 2.22E-56 4.10E-78 1.05E-64 5.03E-62 2.88E-74

f20 – Schwefel 2.25 CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

2.85E-01 1.85E-11 4.06E-07 4.66E-09 5.86E-09 3.18E-11 2.45E-09 1.12E-13 1.43E-11

1.54E-01 1.29E-06 4.71E-02 8.41E-06 1.61E-05 9.43E-09 4.98E-06 3.33E-09 6.45E-08

5.21E-01 5.27E-07 2.01E-02 1.94E-06 4.84E-06 9.87E-09 1.97E-06 1.29E-09 2.41E-08

9.89E-01 8.08E-06 2.52E-01 6.10E-05 1.32E-04 4.31E-08 2.80E-05 2.17E-08 4.26E-07

f21 - Brown CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

3.81E-02 1.21E-30 8.92E-29 4.07E-29 3.45E-31 5.53E-70 4.31E-85 1.39E-38 1.41E-74

1.02E-01 3.24E-27 2.34E-13 3.88E-26 3.36E-26 2.33E-60 1.07E-65 4.10E-36 2.46E-63

1.73E-01 1.29E-27 4.92E-14 2.30E-26 1.18E-26 6.22E-61 1.73E-66 1.62E-36 4.36E-64

6.59E-01 1.40E-26 1.39E-12 1.86E-25 1.58E-25 1.13E-59 8.51E-65 1.98E-35 1.67E-62

1.07E+00 6.15E+00 7.82E+00 7.30E+00 7.63E+00 1.61E-34 9.25E-47 2.81E-06 7.40E-39

9.28E+00 3.21E+01 3.43E+01 3.44E+01 3.31E+01 5.22E-35 1.33E-47 3.38E-07 2.73E-39

1.16E+01 5.01E+01 5.69E+01 4.95E+01 5.21E+01 9.53E-34 6.80E-46 2.37E-05 3.61E-38

f23 – Powell Singular CROW 6.65E-01 CROWISL82 1.31E-13 CROWISL44 9.07E-12 CROWISL16 1.44E-12 CROWISL242 4.33E-13 CROWMISL82 2.90E-226 CROWMISL44 3.22E-217 CROWMISL16 1.88E-74 CROWMISL242 4.55E-206

1.02E+00 1.09E-09 1.49E-07 4.07E-09 3.30E-10 3.49E-65 6.38E-81 9.48E-13 3.33E-79

2.29E+00 3.52E-10 2.63E-08 9.03E-10 2.44E-10 5.22E-66 7.74E-82 1.98E-13 5.30E-80

5.77E+00 7.40E-09 1.17E-06 3.12E-08 1.47E-09 2.39E-64 5.34E-80 4.93E-12 2.28E-78

f24 – Sum of Different Powers CROW 9.38E-07 CROWISL82 1.17E-54 CROWISL44 4.29E-50 CROWISL16 1.21E-51 CROWISL242 4.55E-55 CROWMISL82 7.70E-135

7.98E-05 6.71E-46 9.93E-35 1.77E-42 2.75E-44 5.52E-57

4.96E-05 1.15E-46 1.46E-35 3.29E-43 4.66E-45 7.67E-58

4.31E-04 4.98E-45 6.88E-34 1.27E-41 2.05E-43 3.66E-56

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f22 – Streched V Sine Wave CROW 6.56E+00 CROWISL82 2.02E+01 CROWISL44 2.15E+01 CROWISL16 1.99E+01 CROWISL242 1.97E+01 CROWMISL82 0.00E+00 CROWMISL44 4.10E-55 CROWMISL16 1.66E-12 CROWMISL242 1.92E-46

lP repro of

CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

Journal Pre-proof

0.00E+00 1.16E-62 4.25E-214

1.30E-53 2.00E-53 3.21E-62

1.65E-54 5.19E-54 4.86E-63

1.07E-52 1.12E-52 2.18E-61

f25 – High Conditioned Elliptic CROW 1.51E-01 CROWISL82 4.18E-29 CROWISL44 6.85E-27 CROWISL16 3.46E-27 CROWISL242 2.53E-29 CROWMISL82 2.90E-68 CROWMISL44 1.96E-78 CROWMISL16 5.23E-38 CROWMISL242 3.89E-72

9.91E-02 6.38E-26 7.36E-13 1.11E-24 2.68E-25 1.75E-58 1.46E-62 1.97E-34 1.15E-59

3.25E-01 2.46E-26 1.51E-13 5.88E-25 6.87E-26 4.21E-59 1.98E-63 6.19E-35 2.31E-60

5.81E-01 3.05E-25 4.95E-12 5.75E-24 1.45E-24 1.08E-57 1.11E-61 1.38E-33 7.08E-59

5.15E-03 1.86E-27 1.91E-16 1.02E-26 7.43E-27 3.69E-60 1.67E-64 6.59E-36 3.98E-63

1.23E-02 4.92E-28 3.13E-17 3.95E-27 1.76E-27 7.74E-61 3.41E-65 1.05E-36 7.30E-64

2.71E-02 1.28E-26 1.40E-15 7.17E-26 4.38E-26 2.55E-59 1.05E-63 5.15E-35 2.91E-62

1.50E+00 5.96E-28 5.35E-16 7.73E-27 9.30E-28 2.13E-58 5.41E-64 3.15E-37 2.61E-64

9.72E-01 2.62E-28 1.38E-16 3.31E-27 2.88E-28 2.93E-59 9.42E-65 1.43E-37 7.01E-65

3.97E+00 3.44E-27 2.51E-15 4.37E-26 4.69E-27 1.62E-57 3.19E-63 1.68E-36 1.08E-63

2.19E-03 7.83E-31 1.77E-31 3.16E-29 1.80E-31 3.48E-69 4.00E-80 9.56E-40 7.36E-73

f27 – Hyperellipsoid CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

2.13E-02 4.29E-31 3.01E-27 1.53E-29 6.04E-32 3.10E-69 2.45E-82 3.63E-40 4.19E-75 3.04E+02 1.85E-26 8.00E-25 1.14E-24 2.66E-26 7.55E-67 5.38E-74 4.76E-36 1.90E-68

2.10E+02 2.42E-23 5.01E-10 4.52E-22 1.13E-21 1.22E-54 1.70E-57 5.95E-32 2.33E-60

6.61E+02 1.32E-23 7.91E-11 2.91E-22 2.51E-22 2.66E-55 3.65E-58 2.23E-32 6.40E-61

1.16E+03 1.09E-22 3.28E-09 1.55E-21 5.80E-21 6.90E-54 8.36E-57 2.85E-31 9.71E-60

6.10E-04 2.50E-31 9.74E-30 1.04E-30 4.17E-32 4.96E-69 6.30E-79 1.97E-41 4.87E-75

7.67E-04 1.80E-28 6.18E-18 9.38E-27 2.59E-29 1.03E-60 1.59E-65 1.90E-37 4.29E-63

1.73E-03 5.66E-29 1.10E-18 2.25E-27 1.28E-29 2.65E-61 3.03E-66 1.10E-37 8.12E-64

5.72E-03 1.04E-27 3.61E-17 8.58E-26 1.08E-28 4.42E-60 9.57E-65 8.37E-37 2.63E-62

Jou

f28 - Bent Cigar CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

rna

f26 – Sum squares CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

f29 - Discus CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

lP repro of

CROWMISL44 CROWMISL16 CROWMISL242

lP repro of

Journal Pre-proof

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Figure 11 Comparison of the convergence capabilities of the island models for Sphere, Rosenbrock, Schwefel 2.22, and Schwefel 2.23 test functions

Figure 12 Convergence curves for Schwefel 2.25 function, Brown function, Streched V Sine Wave, Powell Singular functions

lP repro of

Journal Pre-proof

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Figure 13 Comparison of the convergence curves for Sum of Different Powers function, High Conditioned Elliptic function, Sumsquares function, and Hyperellipsoid function

Figure 14 Convergence curves for Bent Cigar function and Discus function

Table 5 reports the results of multi-modal test functions found by the literature optimizers. The proposed island models outperformed the literature optimizers for almost all of the unimodal test functions except f17. Figure 15 to Figure 19 depict the convergence charts of literature optimizers for the unimodal and multi-modal test functions.

Journal Pre-proof

Table 5 Statistical results of selected literature optimizers for unimodal test functions Std.dev.

Mean

Worst

1.40E-05 8.10E-33 5.95E-04 7.21E-09 7.06E-06 5.02E-03

1.19E+01 1.02E-31 1.16E-04 7.11E-12 3.31E-02 8.91E-01

4.04E+00 8.02E-32 8.43E-04 7.22E-09 2.86E-02 4.64E-01

6.21E+01 4.02E-31 1.06E-03 7.24E-09 1.33E-01 5.32E+00

f17 - Rosenbrock BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

2.22E+01 8.82E+00 2.32E+01 2.86E+01 1.21E-01 1.03E+02

9.70E+02 2.54E+00 1.37E+02 1.49E-02 5.19E+01 5.32E+02

4.10E+02 1.90E+01 1.03E+02 2.87E+01 6.04E+01 5.01E+02

4.76E+03 3.00E+01 6.57E+02 2.87E+01 1.80E+02 3.38E+03

f18 - Schwefel 2.22 BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

1.56E+01 1.47E-15 1.74E+01 4.65E-04 1.81E-03 1.97E-02

2.13E+06 3.37E-14 3.42E+09 2.53E-07 3.58E-01 1.35E+00

4.77E+05 1.92E-14 5.58E+08 4.65E-04 4.27E-01 1.06E+00

1.32E+07 1.52E-13 2.24E+10 4.66E-04 1.75E+00 6.93E+00

f19 - Schwefel 2.23 BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

1.24E-28 7.75E-85 2.78E-19 2.41E-47 4.97E-29 3.14E-04

1.45E+04 1.38E-73 4.16E+00 1.36E-49 3.39E-07 3.88E+02

2.36E+03 3.25E-74 9.06E-01 2.44E-47 5.79E-08 8.74E+01

9.35E+04 7.66E-73 2.56E+01 2.46E-47 2.09E-06 2.43E+03

f20 – Schwefel 2.25 BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

1.41E-04 6.13E-16 4.57E-03 2.30E+01 7.81E-02 7.64E-01

1.81E+03 4.33E-14 3.36E+01 3.34E-01 3.43E+00 6.80E+01

4.12E+02 2.83E-14 3.45E+01 2.37E+01 2.25E+00 9.62E+01

1.09E+04 1.83E-13 1.13E+02 2.45E+01 2.38E+01 2.50E+02

2.69E-05 7.50E-31 9.49E+00 1.39E-08 2.94E-06 1.20E-03

1.24E+02 1.26E-27 3.32E+09 1.30E-11 9.29E-02 9.57E+00

7.50E+01 2.60E-28 5.19E+08 1.39E-08 5.42E-02 5.26E+00

6.53E+02 7.28E-27 2.17E+10 1.39E-08 4.11E-01 4.14E+01

f22 – Streched V Sine Wave BAT 4.85E+01 CUCKOO 1.72E-02 BB-BC 5.87E+01 FRUITFLY 4.67E+00 ITHS 5.23E-01 QPSO 1.40E+01

6.34E+00 2.83E-02 6.60E+00 1.19E+00 6.59E+00 7.06E+00

5.91E+01 7.25E-02 7.04E+01 6.10E+00 9.88E+00 2.82E+01

7.68E+01 1.11E-01 8.82E+01 7.62E+00 2.71E+01 4.36E+01

rna

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f21 - Brown BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

lP repro of

Best f16 - Sphere BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

Journal Pre-proof

f23 – Powell Singular BAT 1.07E-02 CUCKOO 3.74E-10 BB-BC 2.14E-01 FRUITFLY 7.75E-07 ITHS 1.29E-03 QPSO 1.62E+00

2.31E+03 1.44E-09 1.10E+02 7.77E-07 6.23E+00 1.14E+03

5.11E+04 3.16E-09 1.87E+03 7.79E-07 2.58E+01 1.22E+04

1.09E+09 3.49E-50 3.82E+06 1.43E-08 3.41E-02 1.41E+06

1.82E+09 7.76E-51 1.02E+06 1.02E-05 1.48E-02 2.65E+06

6.93E+09 2.00E-49 2.47E+07 1.02E-05 1.54E-01 9.05E+07

8.51E+04 9.97E-28 3.64E+04 8.79E-07 1.32E+03 4.80E+03

6.10E+04 8.44E-28 6.95E+05 4.65E-04 4.88E+02 2.27E+03

3.29E+05 5.43E-27 1.40E+05 4.67E-04 8.39E+03 2.87E+04

1.43E+02 2.16E-30 2.02E+01 1.09E-10 5.68E-01 1.53E+01

5.71E+01 1.17E-30 1.00E+01 1.08E-07 3.63E-01 6.69E+00

8.42E+02 1.22E-29 1.15E+02 1.09E-07 2.90E+00 9.08E+01

2.63E+03 8.63E-31 3.86E+02 2.45E-09 1.43E+01 3.70E+02

1.15E+03 9.51E-31 3.36E+02 1.91E-06 8.01E+00 1.32E+02

1.26E+04 4.08E-30 2.14E+03 1.92E-06 7.51E+01 2.28E+03

1.23E+01 3.06E-27 5.60E+02 6.93E-03 7.66E+00 3.26E+02

4.95E+06 6.87E-26 1.08E+02 8.04E-06 2.51E+04 1.02E+06

1.60E+06 5.97E-26 7.88E+02 6.95E-03 2.01E+04 4.26E+06

2.63E+07 3.62E-25 1.08E+03 6.97E-03 1.23E+05 6.43E+07

3.62E+02 2.46E-31 5.12E+02 1.04E-04 7.50E-04 1.67E-02

4.99E+02 8.28E-30 2.93E+02 6.46E-07 1.15E+01 8.07E+00

1.06E+03 4.63E-30 1.02E+03 1.05E-04 2.52E+00 4.17E+00

2.28E+03 4.49E-29 1.88E+03 1.08E-04 7.44E+01 3.43E+01

f24 – Sum of Different Powers BAT 5.28E-03 CUCKOO 4.89E-56 BB-BC 1.07E+01 FRUITFLY 1.02E-05 ITHS 1.29E-05 QPSO 1.69E-02 f25 – High Conditioned Elliptic BAT 1.05E+03 CUCKOO 2.72E-29 BB-BC 1.09E+04 FRUITFLY 4.64E-04 ITHS 4.68E+00 QPSO 8.94E+00 7.12E-04 7.28E-32 2.13E-02 1.08E-07 2.99E-04 1.02E-02

f27 – Hyperellipsoid BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

8.01E-01 1.06E-31 1.52E+01 1.91E-06 1.64E-04 6.21E-02

Jou

f28 - Bent Cigar BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

rna

f26 – Sum squares BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

f29 - Discus BAT CUCKOO BB-BC FRUITFLY ITHS QPSO

lP repro of

7.80E+03 6.77E-10 3.31E+02 7.31E-10 6.03E+00 2.27E+03

lP repro of

Journal Pre-proof

rna

Figure 15 Convergence charts of the selected literature optimizers for Levy, Ackley, Griewank, Rastrigin, Zakharov and Alpine test functions

Jou

Figure 16 Convergence curves of the selected literature optimizers for the test functions Penalized1, Penalized2, Quintic, Csendes, Schaffer and Salomon

lP repro of

Journal Pre-proof

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Figure 17 Comparison of the convergence capabilities of the selected literature optimizers for Inverted cosine mixture, Pathological, Wavy, Sphere function, Rosenbrock and Schwefel 2.22 functions

Jou

Figure 18 Evolution histories of the selected literature optimizers for Schwefel 2.23, Schwefel 2.25, Brown, Stretched V Sine Wave, Powell Singular and Sum of Different Powers functions

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Figure 19 Convergence curves of the selected literature optimizers for High Conditioned Elliptic, Sumsquares, Hyperellipsoid, Bent Cigar and Discus functions

Jou

rna

Figure 20 and Figure 21 display the relation between the iteration count and execution time of developed island models for eight selected benchmark optimization problems. As can be seen from the figures, execution time of the algorithms extend with respect to increasing number of iterations.

Figure 20 Relation between the run time and number of iterations for different island models

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Figure 21 Run time of the test functions with increasing function evaluations for different island models

rna

Table 6 reports the statistical analysis results of island models obtained for CEC 2015 benchmark function. These functions have plenty of global and local optimum points on their search domain hampering the ability to acquire the global optimum of the problem by the proposed algorithms. For f30 function, all algorithms failed to converge to the optimal solution, nevertheless CROWISL16 performs well with respect to the best solution values while CROWMISL44 having the worst predictions. Similar behavior is also observed for f31. While the solution found by each island model variant is far away from the global optimum point, CROWISL16 again provides the best predictions. All island variants obtain the same best value of 3.81E-04 for f32 function. CROWMISL242 retains the most accurate results between the compared algorithms for the composite test functions of f33, f35, f36, and f37, which proves that the exploration capacity and local minima avoidance of the best performing island model is quite satisfactory as these composite functions consist of many local optimum points. Each of the master assisted island model reaches the optimal solution of f34, f39, f41, f42, f45 test functions. Table 6 Optimal results of island models for CEC 2015 benchmark functions Best

Std. Dev.

Mean

Worst

4.31E-01 2.79E-01 2.91E-01 1.50E-01 1.61E-01 3.95E-01 4.32E-01 2.96E-01 3.78E-01

7.35E-02 1.49E-01 1.46E-01 1.61E-01 1.44E-01 1.39E-01 1.62E-01 1.76E-01 1.51E-01

6.09E-01 5.15E-01 5.64E-01 5.51E-01 5.25E-01 6.84E-01 7.22E-01 5.97E-01 6.75E-01

7.58E-01 8.68E-01 9.51E-01 9.09E-01 1.06E+00 1.11E+00 1.09E+00 1.15E+00 9.17E-01

2.85E-01 1.98E-01 2.56E-01 1.82E-01 2.64E-01 2.74E-01 2.76E-01

1.83E-01 3.27E-01 3.03E-01 2.58E-01 2.64E-01 1.02E-01 1.72E-01

5.30E-01 6.29E-01 6.20E-01 5.49E-01 5.59E-01 4.56E-01 5.09E-01

1.08E+00 1.26E+00 1.11E+00 1.06E+00 1.07E+00 7.23E-01 9.92E-01

Jou

f30 Happy Cat CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242 f31 HGBat CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44

Journal Pre-proof

2.08E-01 1.77E-01

5.08E-01 5.00E-01

9.13E-01 1.09E+00

f32 Modified Schwefel’s Function CROW 2.07E-02 CROWISL82 3.81E-04 CROWISL44 3.81E-04 CROWISL16 3.81E-04 CROWISL242 3.81E-04 CROWMISL82 3.81E-04 CROWMISL44 3.81E-04 CROWMISL16 3.81E-04 CROWMISL242 3.81E-04

1.95E-02 6.29E-12 1.25E-09 9.22E-12 8.78E-11 0.00E+00 8.33E-13 8.04E-13 5.71E-13

4.82E-02 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04

1.26E-01 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04 3.81E-04

f33 Expanded Griewank’s plus Rosenbrock’s Function CROW 2.73E+03 1.09E+06 CROWISL82 2.58E-01 2.21E-01 CROWISL44 5.00E-01 3.16E-01 CROWISL16 3.00E-05 2.24E-01 CROWISL242 4.83E-01 1.95E-01 CROWMISL82 3.23E-01 1.53E-01 CROWMISL44 3.29E-01 1.61E-01 CROWMISL16 3.38E-01 2.06E-01 CROWMISL242 6.07E-01 1.12E-01

6.60E+05 6.13E-01 9.42E-01 7.66E-01 8.48E-01 7.89E-01 8.01E-01 7.12E-01 8.18E-01

9.51E+06 9.75E-01 1.73E+00 1.36E+00 1.32E+00 1.18E+00 1.20E+00 1.18E+00 1.05E+00

f34 Expanded Schaffer’s Function CROW 1.18E+01 CROWISL82 7.58E+00 CROWISL44 9.10E+00 CROWISL16 7.35E+00 CROWISL242 9.02E+00 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 0.00E+00 CROWMISL242 0.00E+00

1.16E+01 9.77E+00 9.98E+00 9.83E+00 1.03E+01 4.75E-01 1.30E+00 7.02E+00 9.94E-01

1.19E+01 1.14E+01 1.10E+01 1.11E+01 1.16E+01 8.06E+00 9.06E+00 1.03E+01 8.16E+00

f35 Expanded High Conditioned Elliptic plus Bent Cigar Function CROW 6.39E+10 1.79E+11 CROWISL82 7.87E-29 4.52E-17 CROWISL44 2.74E-26 1.06E-16 CROWISL16 2.34E-26 2.21E-22 CROWISL242 9.72E-30 1.71E-25 CROWMISL82 3.57E-75 5.74E-59 CROWMISL44 2.41E-70 2.59E-55 CROWMISL16 7.26E-45 2.48E-35 CROWMISL242 1.63E-78 1.18E-62

2.83E+11 9.65E-18 3.37E-17 9.71E-23 8.96E-26 1.58E-59 7.82E-56 9.49E-36 3.34E-63

7.45E+11 2.22E-16 4.01E-16 7.19E-22 5.84E-25 2.44E-58 9.39E-55 1.03E-34 5.38E-62

f36 Expanded Rastrigin plus Rosenbrock Function CROW 3.95E+07 6.36E+09 CROWISL82 3.71E+02 7.01E+02 CROWISL44 1.65E+03 1.41E+03 CROWISL16 1.03E+03 7.93E+02 CROWISL242 7.28E+02 1.08E+03 CROWMISL82 2.98E+01 1.08E+03 CROWMISL44 2.98E+01 1.13E+03 CROWMISL16 2.98E+01 8.11E+02 CROWMISL242 2.98E+01 9.93E+02

4.52E+09 1.91E+03 3.73E+03 2.04E+03 2.15E+03 9.98E+02 1.50E+03 1.19E+03 9.50E+02

2.48E+10 3.05E+03 6.39E+03 4.37E+03 6.58E+03 3.07E+03 4.03E+03 2.81E+03 2.96E+03

lP repro of

2.94E-01 2.76E-01

2.20E-01 1.03E+00 5.57E-01 7.29E-01 7.86E-01 1.63E+00 2.71E+00 3.01E+00 2.45E+00

Jou

rna

CROWMISL16 CROWMISL242

f37 Expanded Rastrigin plus Ackley Function

Journal Pre-proof

5.38E-02 5.74E-02 1.35E-01 6.17E-02 4.83E-02 9.45E+00 8.18E+00 7.12E-02 6.82E+00

2.10E+01 2.05E+01 2.09E+01 2.00E+01 2.00E+01 1.33E+01 1.58E+01 2.00E+01 1.73E+01

2.10E+01 2.02E+01 2.07E+01 2.02E+01 2.01E+01 2.01E+01 2.01E+01 2.02E+01 2.02E+01

f38 Expanded Five-Uneven-Peak Trap Function CROW 1.00E+03 1.55E+02 CROWISL82 1.23E-02 8.22E+01 CROWISL44 6.00E+01 7.95E+01 CROWISL16 8.32E+01 1.08E+02 CROWISL242 4.00E+01 1.12E+02 CROWMISL82 0.00E+00 1.29E+02 CROWMISL44 0.00E+00 1.29E+02 CROWMISL16 8.00E+01 8.97E+01 CROWMISL242 0.00E+00 1.19E+02

1.33E+03 2.04E+02 2.16E+02 2.37E+02 2.25E+02 1.31E+02 1.86E+02 2.47E+02 8.74E+01

1.62E+03 4.00E+02 3.40E+02 4.80E+02 4.80E+02 4.20E+02 4.40E+02 3.80E+02 4.40E+02

f39 Expanded Equal Minima Function CROW 1.51E+01 CROWISL82 0.00E+00 CROWISL44 1.59E-13 CROWISL16 0.00E+00 CROWISL242 3.55E-15 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 0.00E+00 CROWMISL242 0.00E+00

1.05E+00 1.44E-13 5.09E-01 1.07E-11 2.70E-14 5.22E-15 5.52E-15 1.70E-14 1.17E-14

1.72E+01 4.24E-14 2.19E-01 1.45E-12 2.29E-14 1.77E-15 2.56E-15 1.42E-14 3.90E-15

1.93E+01 6.71E-13 2.02E+00 8.19E-11 1.17E-13 2.13E-14 2.13E-14 5.32E-14 3.90E-14

f40 Expanded Two Peak Trap Function CROW 7.32E+02 CROWISL82 3.20E+02 CROWISL44 2.80E+02 CROWISL16 2.40E+02 CROWISL242 3.60E+02 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 2.80E+02 CROWMISL242 0.00E+00

1.78E+02 9.54E+01 1.15E+02 1.01E+02 1.01E+02 2.26E+02 2.48E+02 1.10E+02 1.94E+02

1.09E+03 5.58E+02 5.80E+02 5.49E+02 5.54E+02 1.30E+02 1.73E+02 5.21E+02 9.98E+01

1.43E+03 8.00E+02 8.80E+02 7.60E+02 7.20E+02 6.00E+02 6.80E+02 7.20E+02 5.60E+02

f41 Expanded Decreasing Minima Function CROW 1.63E+01 CROWISL82 2.41E+00 CROWISL44 2.95E+00 CROWISL16 2.53E+00 CROWISL242 2.86E+00 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 0.00E+00 CROWMISL242 0.00E+00

9.67E-01 1.11E+00 1.46E+00 1.22E+00 1.40E+00 0.00E+00 0.00E+00 2.22E+00 0.00E+00

1.89E+01 4.96E+00 4.97E+00 5.18E+00 5.65E+00 0.00E+00 0.00E+00 3.31E+00 0.00E+00

2.08E+01 7.07E+00 8.23E+00 8.94E+00 8.52E+00 0.00E+00 0.00E+00 6.98E+00 0.00E+00

f42 Expanded Uneven Minima Function CROW 1.55E+01 CROWISL82 0.00E+00 CROWISL44 7.10E-15 CROWISL16 0.00E+00 CROWISL242 0.00E+00

9.62E-01 4.11E-01 5.95E-01 3.59E-01 2.33E-01

1.73E+01 1.02E-01 2.98E-01 1.87E-01 6.73E-02

1.92E+01 1.75E+00 2.19E+00 8.75E-01 8.75E-01

lP repro of

2.08E+01 2.00E+01 2.00E+01 2.00E+01 2.00E+01 4.44E-16 4.44E-16 2.00E+01 4.44E-16

Jou

rna

CROW CROWISL82 CROWISL44 CROWISL16 CROWISL242 CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

Journal Pre-proof

3.21E-01 2.13E-15 3.20E-01 2.25E-01

1.41E-01 7.10E-16 1.40E-01 6.25E-02

8.75E-01 7.63E-15 8.75E-01 8.75E-01

f43 Expanded Himmelblau’s Function CROW 3.58E+02 CROWISL82 5.67E-26 CROWISL44 2.27E-24 CROWISL16 6.21E-25 CROWISL242 4.42E-27 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 7.42E-26 CROWMISL242 0.00E+00

1.11E+02 8.06E-23 8.25E-12 1.50E-21 3.00E-22 1.27E-24 1.00E-18 1.81E-23 3.78E-25

6.42E+02 2.82E-23 2.41E-12 7.95E-22 1.20E-22 4.65E-25 2.69E-19 8.35E-24 1.23E-25

8.34E+02 3.67E-23 3.81E-12 6.00E-21 1.31E-21 4.86E-24 4.04E-18 8.39E-23 2.11E-24

f44 Expanded Six-Hump Camel Back Function CROW 9.26E+01 CROWISL82 9.79E+00 CROWISL44 9.79E+00 CROWISL16 9.79E+00 CROWISL242 3.26E+00 CROWMISL82 2.79E-06 CROWMISL44 2.79E-06 CROWMISL16 2.79E-06 CROWMISL242 2.79E-06

1.31E+02 6.66E+00 1.13E+01 1.12E+01 9.05E+00 8.44E+00 9.68E+00 1.10E+01 7.00E+00

2.15E+02 1.86E+01 2.63E+01 2.18E+01 2.14E+01 5.44E+00 7.74E+00 1.34E+01 3.53E+00

6.36E+02 3.48E+01 5.39E+01 5.74E+01 4.19E+01 2.61E+01 2.88E+01 3.86E+01 2.28E+01

f45 Modified Vincent Function CROW 3.97E-01 CROWISL82 0.00E+00 CROWISL44 1.11E-16 CROWISL16 0.00E+00 CROWISL242 0.00E+00 CROWMISL82 0.00E+00 CROWMISL44 0.00E+00 CROWMISL16 0.00E+00 CROWMISL242 0.00E+00

4.17E-02 5.81E-03 5.84E-03 3.10E-16 6.16E-03 4.50E-16 4.84E-03 4.50E-16 5.38E-03

4.74E-01 1.45E-03 1.49E-03 1.99E-16 1.64E-03 4.25E-16 9.88E-04 4.08E-16 1.23E-03

5.48E-01 2.47E-03 2.47E-02 1.22E-15 2.47E-02 1.88E-15 2.47E-02 1.88E-15 2.47E-02

lP repro of

0.00E+00 0.00E+00 0.00E+00 0.00E+00

rna

CROWMISL82 CROWMISL44 CROWMISL16 CROWMISL242

Jou

Table 7 reports the variations of the objective function values of the some multimodal test functions investigated in this study with different flight length (fl) and Awareness Probability (AP) values. It is seen that smaller AP and fl values increase the solution quality and robustness for Levy test function. Flight length higher than 1.0 along with relatively smaller AP values provide more better results compared to the other cases for Pathological test funtion. Smaller AP and fl values give more accurate results for Alpine test function. Variations of AP and fl values do not have any significant influence on the optimization results for Salomon test function.

Table 8 gives the variations of the mean and statistical results of some of the unimodal test functions discussed in this experimental study section with different Flight length and Awareness Probability values. It seems that parameter settings of AP=0.4 and fl=2.5 obtain the best solution accuracy for Rosenbrock test function. It is observed that randomized numerical values generated for AP and fl jeopardize the solution quality obtained for Brown, High conditioned elliptic, and Bent cigar optimization bencmark functions.

Salomon

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Alpine

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Pathological

fl  2.5

Mean - Std.dev

Mean - Std.dev 2.201E-08 ± 5.827E-08 8.448E-10 ± 1.972E-09 3.682E-09 ± 4.367E-09 1.929E-134 ± 1.350E-133 6.975E-09 ± 7.451E-09 4.301E-117 ± 2.821E-116 1.889E-08 ± 5.148E-08 2.347E-123 ± 1.862E-122

Mean - Std. dev 5.442E-01 ± 4.782E-01 5.326E-01 ± 5.332E-01 7.083E-01 ± 8.271E-01 6.136E-01 ± 8.224E-01 6.326E-01 ± 7.732E-01 3.761E-01 ± 3.886E-01 5.097E-01 ± 6.798E-01 6.645E-01 ± 8.753E-01

rna

Mean - Std. dev. 7.998E-01 ± 5.387E-01 2.193E-01 ± 1.953E-01 6.253E-01 ± 4.679E-01 2.685E-01 ± 1.702E-01 8.167E-01 ± 6.033E-01 1.387E-01 ± 1.311E-01 9.306E-01 ± 4.443E-01 2.142E-01 ± 1.662E-01

AP  0.4

Jou

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Levy

fl  0.5

fl  1.0

Mean - Std.dev 1.078E+00 ± 2.554E-01 3.446E-01 ± 1.531E-01 1.141E+00 ± 1.931E-01 6.035E-01 ± 2.571E-01 1.024E+00 ± 1.787E-01 3.201E-01 ± 1.941E-01 1.153E+00 ± 2.654E-01 3.671E-01 ± 1.543E-01

AP  0.5

fl   0,5

Mean - Std.dev 3.465E+00 ± 4.515E+00 1.010E+00 ± 8.431E-01 5.721E+00 ± 4.387E+00 1.024E+00 ± 8.157E-01 4.873E+00 ± 3.288E+00 9.412E-01 ± 7.972E-01 3.692E+00 ± 4.181E+00 9.542E-01 ± 7.909E-01

AP   0,1

Mean - Std.dev

Mean - Std.dev 2.102E+00 ± 1.258E+00 5.763E-94 ± 3.451E-93 2.351E+00 ± 1.532E+00 5.338E-140 ± 4.237E-139 2.221E+00 ± 1.1498E+00 8.044E-140 ± 3.631E-139 2.098E+00 ± 1.191E+00 1.073E-148 ± 6.487E-148

Mean - Std. dev 3.536E+00 ± 3.065E-01 3.496E+00 ± 4.072E-01 3.507E+00 ± 2.377E-01 3.249E+00 ± 4.072E+00 3.406E+00 ± 3.367E-01 2.345E+00 ± 1.575E+00 2.927E+00 ± 3.157E-01 2.728E+00 ± 1.409E+00

Mean - Std.dev

Mean - Std.dev 2.281E+00 ± 1.246E+00 1.441E-03 ± 1.939E-03 3.419E+00 ± 1.295E+00 1.042E-04 ± 6.674E-04 2.234E+00 ± 1.076E+00 4.685E-05 ± 2.657E-04 3.074E+00 ± 1.833E+00 1.083E-05 ± 7.631E-05

Mean - Std. dev 1.078E+00 ± 5.219E-01 5.129E-01 ± 2.494E-01 1.003E+00 ± 4.567E-01 7.739E-01 ± 4.076E-01 1.001E+00 ± 4.763E-01 4.548E-01 ± 2.556E-01 1.001E+00 ± 0 .814E-01 5.156E-01 ± 4.942E-01

Mean - Std.dev

Mean - Std.dev 4.917E+00 ± 6.157E+00 3.276E-03 ± 2.388E-02 1.052E-06 ± 8.040E-06 5.957E-11 ± 3.741E-10 5.719E+00 ± 6.702E+00 6.848E-06 ± 4.842E-05 4.386E+00 ± 5.843E+00 4.650E-08 ± 3.103E-07

Mean - Std. dev 2.678E+00 ± 1.267E+00 2.410E+00 ± 1.405E+00 2.918E+00 ± 2.272E+00 1.907E+00 ± 1.417E+00 2.201E+00 ± 1.433E+00 1.788E+00 ± 1.567E+00 2.202E+00 ± 1.551E+00 2.092E+00 ± 1.538E+00

lP repro of

Mean - Std.dev 1.141E+00 ± 2.439E-01 3.997E-01 ± 1.829E-01 1.122E+00 ± 2.166E-01 6.846E-01 ± 1.885E-01 1.125E+00 ± 1.702E-01 5.297E-01 ± 1.888E-01 1.124E+00 ± 1.715E-01 5.634E-01 ± 1.671E-01

AP  0.8

Table 7 Variations of the numerical results obtained by the proposed island topologies for multimodal test functions with different Flight length and Awareness probability values

Journal Pre-proof

4.826E-01 1.254E-01 5.053E-01 7.621E-02 4.892E-01 6.653E-02 5.186E-01 6.773E-02

± ± ± ± ± ± ± ±

7.905E-02 4.365E-02 8.381E-02 4.291E-02 7.682E-02 4.704E-02 8.795E-02 4.664E-02

± ± ± ± ± ± ± ±

3.462E-02 4.985E-02 3.853E-02 3.479E-02 4.807E-02 2.709E-02 4.731E-02 3.997E-02

3.288E-01 1.797E-01 3.417E-01 1.145E-01 3.332E-01 9.988E-02 3.271E-01 1.010E-01

± ± ± ± ± ± ± ±

5.576E-02 6.602E-02 5.628E-02 3.625E-02 5.067E-02 3.331E-02 4.479E-02 3.308E-02

5.837E-01 1.097E-01 6.097E-01 5.302E-02 4.583E-01 3.376E-02 5.905E-01 3.992E-02

± ± ± ± ± ± ± ±

1.862E-01 7.398E-02 2.278E-01 5.464E-02 2.693E-01 5.287E-02 2.111E-01 1.998E-01

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2.913E-01 5.362E-02 2.859E-01 1.368E-02 2.845E-01 7.989E-03 2.837E-01 1.991E-02

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82

Mean - Std.dev 6.967E-14 ± 1.212E-16 8.158E-33 ± 3.867E-32 1.768E-13 ± 7.567E-13 2.021E-245 ± 0.000E+00 1.427E-13 ± 2.324E-13 5.234E-182 ± 0.000E+00

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Brown

Mean - Std.dev 4.071E+01 ± 2.815E+01 2.948E+01 ± 1.894E+01 3.994E+01 ± 2.647E+01 2.410E+01 ± 2.589E-01 4.216E+01 ± 2.905E+01 2.407E+01 ± 2.514E-01 4.525E+01 ± 2.984E+01 2.410E+01 ± 2.936E-01

fl  2.5

AP  0.4

Rosenbrock

fl  0.5

Mean - Std.dev 3.187E+00 ± 1.051E+00 2.291E-157 ± 9.963E-157 2.864E+00 ± 1.031E+00 3.918E-253 ± 0.000E+00 3.138E+00 ± 1.004E+00 1.471E-220 ± 0.000E+00

Mean - Std.dev 2.095E+02 ± 6.392E+01 2.491E+01 ± 7.858E-01 2.062E+02 ± 5.482E+01 2.645E+01 ± 4.102E-01 2.064E+02 ± 5.975E+01 2.482E+01 ± 7.464E-01 1.958E+02 ± 5.825E+01 2.472E+01 ± 6.362E-01

AP  0.8

fl  1.0

Mean - Std.dev 3.352E+00 ± 1.393E+00 2.225E-43 ± 7.188E-43 5.170E+00 ± 1.969E+00 8.861E-230 ± 0.000E+00 3.940E+00 ± 1.551E+00 4.131E-167 ± 0.000E+00

Mean - Std.dev 2.212E+02 ± 6.437E+01 2.453E+01 ± 1.243E+01 3.539E+02 ± 1.362E+02 2.491E+01 ± 7.819E-01 2.382E+02 ± 7.721E+01 2.319E+01 ± 8.733E+00 3.082E+02 ± 1.219E+02 2.196E+01 ± 1.273E+00

AP  0.5

fl   0,5

Mean - Std.dev 1.453E+05 ± 1.171E+06 5.356E-34 ± 3.654E-33 5.537E+01 ± 9.758E+01 1.297E-62 ± 8.310E-62 5.892E+03 ± 2.933E+03 3.347E-62 ± 1.706E-61

Mean - Std.dev 5.608E+03 ± 1.086E+04 3.089E+01 ± 1.525E+01 4.471E+03 ± 1.011E+04 2.592E+01 ± 2.063E+00 5.075E+03 ± 9.454E+03 2.545E+01 ± 2.745E+00 3.782E+03 ± 7.702E+03 2.497E+01 ± 2.761E+00

AP   0,1

Table 8 Comparison of the objective function results for unimodal test functions with varying Flight length and Awareness Probability values

rna

Jou

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

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CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Bent cigar

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

High conditioned elliptic

Mean - Std.dev 7.281E-09 ± 9.635E-09 4.991E-26 ± 3.034E-25 1.011E-08 ± 1.621E-08 5.12E-234 ± 0.000E+00 2.114E-08 ± 5.939E-08 1.017E-187 ± 0.000E+00 1.249E-08 ± 2.387E-08 8.405E-196 ± 0.000E+00

Mean - Std.dev 1.419E-11 ± 3.499E-11 6.575E-30 ± 4.024E-29 1.355E-11 ± 2.858E-11 2.068E-238 ± 0.000E+00 1.716E-11 ± 2.859E-11 1.963E-179 ± 0.000E+00 1.791E-11 ± 3.170E-11 1.007E-194 ± 0.000E+00

rna

4.052E-13 ± 1.072E-12 2.143E-201 ± 0.000E+00

Jou

CROWISL242 CROWMISL242 Mean - Std.dev 1.276E+04 ± 7.153E+03 2.363E-39 ± 1.213E-38 1.688E+04 ± 8.843E+03 1.111E-81 ± 6.812E-81 1.417E+04 ± 7.491E+03 2.661E-166 ± 0.000E+00 1.608E+04 ± 8.507E+03 2.043E-189 ± 0.000E+00

4.463E+00 ± 1.481E+00 8.698E-178 ± 0.000E+00 Mean - Std.dev 2.433E+04 ± 7.013E+04 4.172E+00 ± 3.231E-04 5.725E+04 ± 2.091E+05 3.962E-66 ± 3.120E-65 3.751E+04 ± 1.929E+05 4.559E-36 ± 3.156E-35 4.764E+04 ± 1.465E+05 6.025E-42 ± 5.076E-41

1.529E+05 ± 9.613E+05 5.794E-49 ± 5.187E-48

Mean - Std.dev 1.329E+06 ± 4.071E+05 1.927E-145 ± 1.282E-144 1.244E+06 ± 3.662E+05 2.122E-243 ± 0.000E+00 1.270E+06 ± 4.164E+05 5.633E-219 ± 0.000E+00 1.316E+06 ± 4.335E+05 6.651E-222 ± 0.000E+00

Mean - Std.dev 1.219E+06 ± 4.778E+05 1.566E-37 ± 5.187E-37 2.100E+06 ± 6.986E+05 2.455E-221 ± 0.000E+00 1.514E+06 ± 5.651E+05 7.842E-169 ± 0.000E+00 1.775E+06 ± 6.486E+05 1.898E-168 ± 0.000E+00

Mean - Std.dev 5.756E+06 ± 1.129E+07 2.538E-26 ± 2.303E-26 5.014E+06 ± 1.086E+07 6.424E-65 ± 4.157E-64 1.015E+07 ± 1.9504E+07 4.465E-31 ± 4.443E-30 1.101E+07 ± 1.824E+07 1.121E-04 ± 1.043E-03

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Mean - Std.dev 1.954E+04 ± 8.818E+03 1.035E-154 ± 7.633E-154 1.786E+04 ± 8.872E+03 5.008E-243 ± 0.000E+00 1.984E+04 ± 9.386E+03 4.291E-225 ± 1.915E-223 1.889E+04 ± 9.715E+03 3.229E-213 ± 0.000E+00

3.139E+00 ± 1.077E+00 9.638E-230 ± 0.000E+00

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6. Solving optimal control problems through island models

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This section presents an alternative numerical approach based on an island-based optimization strategy to maintain optimal control of nonlinear systems. Design problems that will be discussed in this section are highly nonlinear with equality and inequality constraints. Optimal control problems deal with obtaining the correct sequence of the control inputs that both satisfies the state equations and system constraints and ensures the minimization of the problem specific predefined performance index. There are several methods in the literature to successfully solve optimal control problems, which can be categorized under the branch of dynamic optimization. Mathematical models devoted to solving these type of problems are built upon numerical solution procedures, most of which have significantly evolved over the past two decades thanks to the rapid evolution of computer technology [71]. Traditional solution procedures are called “indirect methods” that aim to solve two-point boundary value problem defined by the Euler-Lagrange Equation, which also should satisfy the Hamilton-JacobiBellman equation. However, these type of methods experience the following drawbacks that make them inefficient and impractical during the solution stage: (1) Correct solution of the Hamilton-JacobiBellman equation is a very tedious process, (2) The decision maker should have a deep knowledge on the mathematical and physical characteristics of the related optimal control problem, (3) Accurate solution of the defined problem is highly dependent upon the initial conditions [72]. Therefore, researchers have proposed numerical indirect methods in order to avoid these drawbacks. In the context of direct solution methods, optimal solution is attained by minimization of the defined objective function subject to problem specific constraints. The optimal control problem is converted into a finite dimensional nonlinear programming problem in order to achieve the correct sequence of the controller inputs. However, utilization of the direct methods may not always provide the best result, as this type of methods produces semi-optimal results that are less accurate than those found by conventional indirect methods. Moreover, discretized solutions of the optimal control problem may often get stuck in local optimum points, which jeopardizes the quality and accuracy of the obtained sequence of the control signals [73].

Jou

rna

As mentioned above, this section proposes the application of Crow Search based island models as a favorable alternative to indirect methods. Proposed solution procedure aims to overcome the abovementioned limitations and disadvantages of the indirect problem solvers. Metaheuristic algorithms require a little information on the specific problem to be dealt with. In addition, they are able to provide solution outcomes with an acceptable accuracy without violating the prescribed constraints. However, it should be admitted that the computational load burdened by the variants of island-based Crow search algorithm proposed in this study is much higher as compared to conventional indirect methods for solving the optimal control problem. In order to gain insightful inference on the performance of the various type of island-based optimization models over optimal control problems, six different dynamic optimization problems have been solved by the proposed metaheuristic optimizers and their corresponding numerical outcomes have been evaluated and compared via statistical analysis. Island based models have been run 30 times with a maximum number of 5000 iterations.

6.1 Case 1: Parallel Reaction Problem This problem, which was previously solved by several researchers [74,75], is based on a chemical reaction A  B , A  C taking place in a batch reactor. Considered design objective is to maximize the yield concentration of B when the reaction is completed. Below dimensionless mathematical model describes the dynamical behavior of the reactants and product concentrations during chemical reaction:

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 dz1

s.t

dt dz2 dt

z  0   [1 0]T





  u  0.5u 2 z1                                                                                                       (4)

 uz1

0u5

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Max J  z2  t f

t f  1.0

Where z1  C A / C A and z2  C B / C A and the numerical value of the control parameter u is bounded 0

0

between 0.0 and 5.0 during the course of the occurring chemical reaction.

Literature studies dealing with this optimal control problem obtained different objective function values. Biegler [76] found J=0.57349 with using Control Vector Iteration (CVI) method. Logdson and Biegler [77] obtained slightly better value of J=0.57353 with using a modified collocation-based NLP formulation. Dadebo and Mcauley [78] also found the same value obtained by the Logdson and Biegler [77] by means of Iterative Dynamic Programming (IDP) method. Rajesh et al. [79] used a well reputed metaheuristic algorithm of ant colony optimizer to reach the optimal solution of this problem. They obtained an objective function value of J=0.57284, which is the worst solution compared to abovereported studies. Table 9 provides the comparison of the objective function values retained by the proposed island-based Crow Search models for the parallel reaction problem. Table 9  Statistical results for parallel reaction problem  Best 0.59759042 0.59790271 0.59809412  0.59837006  0.59790017  0.59790862  0.59792493  0.59789002

rna

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Standard dev 4.5152878E-4 4.8701712E-4 2.0395218E-4 2.5642123E-4 3.3082951E-4 3.0586242E-4 3.0182694E-4 2.9800375E-4

Mean 0.59696705 0.59719455 0.59774351 0.59772138 0.59731339 0.59742626 0.59747731 0.59755294

Worst 0.5954575 0.5948054 0.5969967 0.5968127 0.5959694 0.5965380 0.5964902 0.5959593

Jou

Superiority of the island-based models is evident for this case, as the worst solution found by these methods is much better than those found by the literature studies discussed above. Between the island models, CROW44ISLMASTER is the best performing optimizer with the corresponding fitness value of J = 0.59837006. One can also easily notice that, the consistency of the solutions obtained from the island models is significant as each compared algorithm in Table 9 has a standard deviation rate no more than 5.0E-4. Figure 22 shows the optimal control policy and the corresponding state trajectories obtained by CROW44ISLMASTER for parallel reaction problem.

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Figure 22 State variables and optimal control profile for parallel reaction problem

6.2 Case 2: Continuous Stirred Tank Reactor Problem

Min J  

0.78

0

s.t

z

2 1

rna

This problem is a typical representation of a nonlinear continuous stirred tank reactor problem comprising two local minimum points [80]. In addition, this optimal control problem takes place in the Handbook of Test Problems in Local and Global Optimization [81], in which various kind of control problems are defined and formulated. Concurrently occurring heat and mass transfer in a chemical reaction takes place in a continuous stirred reactor and is formulated by the below given set of differential equations.



 z22  0.1u 2 dt dz1 dt

 25.0 z1    z1  2.0 

   u  2  z1  0.25    z2  0.5  exp 

Jou

 25.0 z1   0.5  z2   z2  0.5  exp   dt  z1  2.0 

(5)

dz2

z  0    0.09 0.09

T

0u5

Where z1 symbolizes the deviation from the dimensionless state temperature; z2 denotes the deviation from a dimensionless steady-state concentration and the control parameter u manipulates mass flow rate of the cooling fluid. The main aim of the control problem is to determine the optimal trajectory of the control parameter u that minimizes the performance index defined in Eq. (10). Mathematical explanation of the performance measure can be given as to keep state variables as close as to their steady-state values without making an excessive amount of effort in adjusting the defined control parameter.

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Ali et al [80] applied eight different stochastic optimization algorithms to solve this optimal control problem whose corresponding performance index rates vary between J=0.135 and J=0.245. Lopez Cruz et al [82] applied well-known metaheuristic algorithms, i.e. Differential Evolution and Genetic Algorithm along with the analytic methods of First-order Gradient Algorithm and Iterative Dynamic Programming (IDP) Algorithm to this control problem. Results revealed that optimization success of the first-order gradient algorithm is highly dependent upon initial conditions of the control parameter. Gradient algorithm obtained J=0.2444 as a local optimum if static initial conditions are applied to the problem. When initial state of the control parameter was taken below a predetermined value, say it 1.8, algorithm converges into its global optimum of J=0.1330. IDP also converges to local optimum of J=0.2444 when initial guess of the control parameter is too small. Under other circumstances, IDP found its global optimum of J=0.1365 for this optimal control problem. Optimal objective function values found by the different variants of Differential Evolution and Genetic algorithm are in the range between J=0.1355970 and J=0.1449189. Modares and Sistani [83] hybridized improved Particle Swarm Optimization (IPSO) and Successive Quadratic Programming (SQP) for solving for this optimal control problem. Optimal results found by IPSO-SQP algorithm along with PSO variants used for benchmarking in the corresponding study vary between J=0.13549 and J=0.13849. Table 10 reports the optimal performance index values found by the island-based model proposed in this study. It is clear that all island models reported in Table 10 have better solution accuracy than those found in the literature studies discussed above. Among the Crow search variants, CROWISL82MASTER provides the best optimization performance with a performance index value of J=0.11697667, which is relatively much better than the contender methods. Figure 23 shows the optimal trajectories of the control input and state variables.

Table 10 Comparison of the accuracy of the solutions found by the island models for Stirred Tank Reactor Problem

rna

Best 0.11827787 0.11810994 0.11826336  0.11801464  0.11816971  0.11697667  0.11779097  0.11748503

Jou

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Standard dev. 2.712711E-3 2.526206E-3 3.017562E-3 2.087034E-3 2.495079E-3 1.160762E-2 3.135965E-3 1.321749E-2

Mean 0.12139902 0.12098024 0.12042803 0.12044025 0.12124387 0.12272896 0.12114492 0.12401462

Continuous

Worst 0.130661293 0.128124844 0.131948954 0.126971539 0.126247034 0.178694243 0.130355538 0.188805473

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Figure 23 State trajectories and optimal control profile for Continuous Stirred Tank Reactor Problem

6.3 Case 3: Batch Reactor Consecutive Reaction Problem

In this case, a consecutive chemical reaction taking place in a batch reactor: A  B  C is considered [74]. Optimization objective is to retain the optimal sequence of reactor temperature that maximizes the yield of product B at the end of the reaction. Problem dynamics are formulated by the below given equations. Max J  CB  t f

dC A dt

 2500  2  CA  T 

rna

s.t



 4000 exp  

 2500  2  5000   4000 exp    C A  620000 exp    CB dt  T   T 

dCB

C A (0)  1.0 C B (0)  0.0

298  T  398

                                        (6)

where t f  1.0

Jou

Where CA and CB respectively represent the concentration of A and B. Previously, this problem was solved by Renfro et al [84] by means of the global spine collocation method. They reported an optimal objective function value of J=0.610. Logdson and Biegler [77] used a collocation-based nonlinear programming method to solve this problem. It takes 88 iterations to reach a Kuhn-Tucker tolerance of 10-7 in order to obtain the optimal fitness value of J=0.610767. By means of the reduced space SQP method, Logdson and Biegler [85] retained the optimal value of J=0.610775 which is slightly better than that found by their former study reported in [77]. Applying Iterative Dynamic Programming (IDP) method, optimal solution J=0.610775 was obtained after 16 iterations with considering P=80 stages by Dadebo and Mcauley [78]. They also concluded that quality and accuracy of the optimal solutions are strongly affected by the number of stages considered for the related problem.

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Table 11 gives the best fitness values obtained by island-based models after 30 consecutive iterations for this optimal control problem. CROWISL82MASTER provides the best prediction accuracy and solution robustness with an optimal value of J=0.64760654 with corresponding standard deviation rate of 3.297921E-4. Figure 24 depicts the optimal control policy and evolution of the state trajectories obtained by CROWISL82MASTER for this problem. Table 11 Comparison of the statistical results obtained by island models for Batch Reactor Consecutive Reaction Problem Best 0.64707256 0.64683681 0.64717105  0.64711927  0.64696071  0.64760654  0.64705056 0.64700566

Standard dev. 4.084732E-4 3.474941E-4 1.868372E-4 2.523956E-4 3.278544E-4 3.297921E-4 2.935948E-4 2.617601E-4

Mean 0.64622571 0.64618352 0.64758192 0.64657144 0.64635454 0.64639768 0.64647272 0.64652761

Worst 0.64390045 0.64506512 0.64626182 0.64578341 0.64542392 0.64548242 0.64540812 0.64556801

Jou

rna

CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Figure 24 Variation of state variables and optimal control profile for Batch Reactor Consecutive Reaction Problem

6.4 Case 4: Nonlinear constrained mathematical system A mathematical system consisting of four state variables, where the fourth variable is to be minimized, is considered in this case to assess the optimization accuracy of the proposed Crow search based island

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models. Numerical formulation of the set of differential equations representing the mathematical model can be found in [86].

s.t.

 dz1 dt dz2 dt dz3 dt dz4 dt

 z2

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Min J  z4  t f

  z3u  16.0t  8.0

                                                         (7)

u



 z12  z22  0.005 z2  16t  8.0  0.1z3u 2

z (0)  [0 -1 - 5 0]T

 4.0  u  10.0



2

where t f  1.0

Jou

rna

This complex nonlinear control problem has been solved many researchers up to now. Rajesh et al [79] obtained the optimal value of J = 0.1290 by means of the ant colony framework. Zhang and Sanderson [87] applied the improved version of Differential Evolution algorithm (JADE) on this problem and obtained the optimal value of J = 0.1235. Asgari and Pishvaie [88] reported an optimal value of J=0.1220 and J=0.1599 respectively obtained by a region reduction strategy and CVP based on ant colony optimization. Zhu et al. [89] used a modified Differential Evolution algorithm to solve this problem and found the optimal values of J=0.1203 and J=0.1196 with using different parameter settings. Optimal value attained by Xu et al [86] J=0.1195 through the upgraded version of Differential Evolution method is much better than those obtained in literature studies, however, optimal solutions found by the proposed island models are more accurate and robust than all abovementioned reported literature solutions. CROW82ISLMASTER is the best performing optimizer with the optimal value J = 0.0707283 for this case among the compared algorithms given in Table 12. Figure 25 shows the optimal control profile along with the corresponding state trajectories for this optimal control problem.

Figure 25 Optimal control and state trajectories for nonlinear constrained mathematical system

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Table 12 Comparison of the optimal results retained by the island models for Nonlinear constrained mathematical system Best Standard dev. Mean Worst CROWISL16 0.0796312 1.735243E-3 0.08261035 0.0878187 CROWM16ISL 0.0765287 1.121291E-3 0.07903369 0.0803356 CROWISL44 0.0790028 8.254504E-4 0.08089324 0.0826514 CROWMISL44 0.0771499 1.769598E-3 0.07983691 0.0827149 CROWISL82 0.0792481  1.717831E-3 0.08246832 0.0870235 CROWMISL82 0.0707283  1.339811E-3 0.07800982 0.0804585 CROWISL242 0.0797261 1.074051E-3 0.08145734 0.0834270 CROWMISL242 0.0767469 1.405843E-3 0.07866831 0.0826607 6.5 Case 5 : Non-linear Continuous Stirred Tank Reactor

Firstly introduced to the literature by Jensen [90], this problem aims to determine the optimal values of the four control parameters with a view to maximum economic gain. Simultaneous chemical reactions taking place in an isothermal continuous stirred tank reactor is mathematically formulated by the following set of differential equations. Max J  z8  t f dt dz2 dt dz3 dt dz4 dt dz5 dt dz6 dt dz7 dt dz8 dt

 

 u4   u1  u2  u4  z1  17.6 z1 z2  23.6 z1 z6u3  u1   u1  u2  u4  z2  17.6 z1 z2  146 z2 z3  u2   u1  u2  u4  z3  73.0 z2 z3

   u1  u2  u4  z4  35.2 z1 z2  51.3 z 4 z5

   u1  u2  u4  z5  219.0 z2 z3  51.3 z4 z5

rna

s.t

dz1



   u1  u2  u4  z6  102.6 z 4 z5  23.0 z1 z6 u3

       

                                          (8)      

   u1  u2  u4  z7  46.0 z1 z6 u3

 

 5.8  u1  u2  u4  z1  u4   3.7u1  4.1u2

 

Jou

+  u1  u2  u4  23 z4  11z5  28 z6  35 z7   5.0u32  0.099

z(0)  [0.1883 0.2507 0.0467 0.0899 0.1804 0.1394 0.1046 0.000]T 0.0  u1  20.0 where t f  0.2

0.0  u2  6.0

0.0  u3  4.0 0.0  u4  20.0

       

Balsa-Canto et al. [91] reported the optimal value of J = 21.757 when considered discretization level is 11 and obtained J = 21.807 when the discretization level is 80 by using restricted second order information. Xu et al [86] got the best optimal value of J =21.757 by conducting improved version Differential Evolution algorithm which is slightly worse than the optimal value J = 21.807 found by Balsa-Canto [91]. They attributed this fact to the discretization of the time space into 20 segments, which is relatively a lower number to discretize the whole time span to achieve the optimum solution.

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In this study, we have discretized the entire time span into 50 different segments to run Runge-Kutta 4/5 differential equation solver and apply island-based models to obtain the optimal trajectories of the four control variables. CROW242ISLMASTER retains the best value of J = 23.2463786, which is much better than not only the results found by the literature algorithms but also those obtained by the compared island-based algorithms reported in Table 13. It is also interesting to see that the worst solution J = 21.8577781 found by island-based model CROW82ISL is better than those acquired by the literature methods. Figure 26 exhibits optimal control profiles and state trajectories obtained by the best performing optimizer for this problem.

Table 13 Optimum objective function values for optimal control of Non-linear Continuous Stirred Tank Reactor Best 21.8794717 22.0716633 22.0452161  22.0443771  21.8577781  22.1446484  21.9313924 23.2463786

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CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Standard dev. 8.729013E-2 9.870211E-2 1.165435E-2 7.779968E-2 9.852244E-2 1.086432E-1 1.018909E-1 1.142872E-1

Mean 21.7217692 21.8535974 21.7936142 21.9391541 21.6779174 21.9415559 21.7542032 21.9434122

Worst 21.5634555 21.7101901 21.5890709 21.7564722 21.4386461 21.6760768 21.5419343 21.7452074

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Figure 26 Optimal control profile along with state trajectories for Nonlinear Continuous Stirred Tank Reactor 6.6 Case 6: Nonlinear Crane Container Problem

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This complex mechanical engineering design problem deals with determining an optimal path of a crane that transfers containers from a ship to cargo truck at Kobe port. Considered design objective is to minimize the swing of the containers during the transportation while they move along their optimized trajectory as well as to minimize the total amount of swing at the end of the container transfer process. Set of differential equations describing the characteristics of the system was modified by Teo et al [92] and takes the final form as given below

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1

9  z  t   z  t  dt 2 1

0

2 3

2 6

dz1  t 

s.t

dt dz2  t  dt dz3  t 

  

dt dz4  t  dt dz5  t  dt dz6  t  dt

 9.0 z4  t   9.0 z5  t   9.0 z6  t 

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Min J 

                                            (9)

 9.0 u1  t   17.25656 z3  t   9.0u2  t  

9.0 u1  t   27.0756 z3  t   2.0 z5  t  z6  t 

 

z2  t 

With the defined initial state values:

z  0    0.0 22.0 0.0 0.0 -1.0 0.0

T

(10)

Control variables of torque of the hoist motor and trolley drive motor are restricted into below defined bounds

u1  2.834

(11)

0.809  u2  0.713

z 4  2.5 z5  1.0

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State constraints imposed on the velocities of the hoist and trolley are defined as

(12)

And terminal state constraints are

z 1  10.0 14.0 0.0 2.5 0.0 0.0

T

(13)

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Literature researchers have faced severe challenges while obtaining the accurate solution of this optimal control problem due to the complex nature of the mathematical model. For instance, Sakawa and Shindo [93] proposed a novel algorithm to maintain optimal control of the transported containers by hybridizing convergence control parameters (CCP) method firstly coined by Jarmark [94] with the method of multipliers introduced by Hestenes [95] and Nahra [96]. They obtained the optimal trajectories of the two control parameters; however, they did not report the optimal value of the defined performance index. Goh and Teo [97] revisited this optimizable control problem by performing a unified solution approach based on control parametrization technique and retained the best performance index value of J=0.00540. Teo et al [92] also found an optimum value of J=0.004684 for this problem after applying some kind of relaxation method to problem constraints. Dadebo and Mcauley [78] used Iterative Dynamic Programming (IDP) method to solve this optimal control problem to circumvent the extravagant usage of the computational power. They proposed the

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application of absolute error penalty function to pose the problem as a nonsmooth dynamic optimization problem. They obtained the optimal performance index value of J=0.004157 with a reasonable degree of violation on design constraints. Island models introduced in this study is applied to this optimal control problem and their corresponding statistical results reveal that CROW82ISLMASTER gives the minimum objective function value of J=0.00446933 as reported in Table 14. Although this best performance index value is inferior to the best solution found by Dadebo and Mcauley [78], no clear constraint violation is observed for the island models reported in Table 14, which is an undeniable superiority of these proposed algorithms over the literature approaches. Figure 27 shows the optimal control policy and the evolution of state trajectories for crane container problem.

Table 14 Comparison of the optimal results in terms of statistical analysis for Nonlinear Crane Container Problem Best 0.00493094 0.00487019 0.00495234  0.00477517  0.00468691  0.00446933  0.00470213 0.00485293

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CROWISL16 CROWM16ISL CROWISL44 CROWMISL44 CROWISL82 CROWMISL82 CROWISL242 CROWMISL242

Standard dev. 5.7376719E-4 3.1282057E-4 3.9220433E-4 3.3599123E-4 6.2723977E-4 4.7503005E-4 5.2118537E-4 6.3029861E-4

Mean 0.00562574 0.00528334 0.00541283 0.00530446 0.00535524 0.00524308 0.00532505 0.00547153

Worst 0.007056728 0.005903305 0.006415882 0.005970044 0.007669823 0.006942612 0.007089701 0.007273062

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Figure 27 Optimal control policy for Nonlinear Crane Container Problem

7. Conclusion

This research study adopts the basics of island model concepts into the Crow Search Algorithm with an aim to improve and upgrade its search mechanism. The island-based Crow Search algorithm proposed in this study divides the population into predefined number of subpopulations called “islands” and each island runs algorithm specific manipulation schemes simultaneously but independently. After a predefined number of iterations, the algorithm applies “best-worst” migration policy in order to exchange the useful population information among the interlinking islands. Furthermore, four different hierarchical island models have been proposed and their probing capabilities have been investigated and compared on 45 optimization test functions. Numerical results obtained from the benchmark functions

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comprised of unimodal, multimodal and CEC 2015 test problems reveal that structured island models can provide more efficient predictive performance compared to basic Crow Search Algorithm.

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Moreover, hierarchical models have been utilized in solving optimal control problems those were previously solved by the literature researchers. Comparison between the optimal solutions found by island models and past solutions from the literature studies shows that incorporating the basics of island models on Crow Search Algorithm are helpful and useful for improving the search capacity. An interesting and beneficial future research study on hierarchical island models would be to propose different migration topologies working on different hierarchical layers and implement these structured search mechanisms on different metaheuristic optimization algorithms.

References

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Table Captions

Table 1 Pseudo-code of Crow Search Algorithm

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Table 2 Statistical results of island models for multi-modal test functions Table 3 Statistical results of selected literature optimizers for multi-modal test functions Table 4 Statistical results for unimodal test functions Table 5 Statistical results of selected literature optimizers for unimodal test functions Table 6 Optimal results of island models for CEC 2015 benchmark functions Table 7 Variations of the numerical results obtained by the proposed island topologies for multimodal test functions with different Flight length and Awareness probability values Table 8 Comparison of the objective function results for unimodal test functions with varying Flight length and Awareness Probability values

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Table 9  Statistical results for parallel reaction problem  Table 10 Comparison of the accuracy of the solutions found by the island models for Stirred Tank Reactor Problem

Continuous

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Table 11 Comparison of the statistical results obtained by island models for Batch Reactor Consecutive Reaction Problem Table 12 Comparison of the optimal results retained by the island models for Nonlinear constrained mathematical system Table 13 Optimum objective function values for optimal control of Non-linear Continuous Stirred Tank Reactor

Table 14 Comparison of the optimal results in terms of statistical analysis for Nonlinear Crane Container Problem Figure Captions

Figure 1 Comparison between Random ring topology and Static ring topology

Figure 2 Schematic representation of static ring topology for island Crow model

Figure 3 Hierarchical topology composed of four island in the lower layer and four island in the upper layer Figure 4 Hierarchical topology consisting of two island in the lower layer and eight island in the upper layer Figure 5 Hierarchical topology constructed by the three communicating layers Figure 6 Main execution steps of the hierarchical Crow Search algorithm

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Figure 7 Evolution of objective functions with increasing number of function evaluations for Levy, Ackley, Griewank, and Rastrigin test functions Figure 8 Comparison of convergence rates for Zakharov, Alpine, Penalized1, and Penalized2 test functions Figure 9 Convergence rate comparison for Quintic, Csendes, Schaffer and Salomon test functions Figure 10 Evolution histories of Inverted cosine mixture function, Pathological function and Wavy function

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Figure 11 Comparison of the convergence capabilities of the island models for Sphere, Rosenbrock, Schwefel 2.22, and Schwefel 2.23 test functions Figure 12 Convergence curves for Schwefel 2.25 function, Brown function, Streched V Sine Wave, Powell Singular functions Figure 13 Comparison of the convergence curves for Sum of Different Powers function, High Conditioned Elliptic function, Sumsquares function, and Hyperellipsoid function Figure 14 Convergence curves for Bent Cigar function and Discus function Figure 15 Convergence charts of the selected literature optimizers for Levy, Ackley, Griewank, Rastrigin, Zakharov and Alpine test functions

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Figure 16 Convergence curves of the selected literature optimizers for the test functions Penalized1, Penalized2, Quintic, Csendes, Schaffer and Salomon Figure 17 Comparison of the convergence capabilities of the selected literature optimizers for Inverted cosine mixture, Pathological, Wavy, Sphere function, Rosenbrock and Schwefel 2.22 functions

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Figure 18 Evolution histories of the selected literature optimizers for Schwefel 2.23, Schwefel 2.25, Brown, Stretched V Sine Wave, Powell Singular and Sum of Different Powers functions Figure 19 Convergence curves of the selected literature optimizers for High Conditioned Elliptic, Sumsquares, Hyperellipsoid, Bent Cigar and Discus functions Figure 20 Relation between the run time and number of iterations for different island models

Figure 21 Run time of the test functions with increasing function evaluations for different island models Figure 22 State variables and optimal control profile for parallel reaction problem

Figure 23 State trajectories and optimal control profile for Continuous Stirred Tank Reactor Problem Figure 24 Variation of state variables and optimal control profile for Batch Reactor Consecutive Reaction Problem Figure 25 Optimal control and state trajectories for nonlinear constrained mathematical system

Figure 26 Optimal control profile along with state trajectories for Nonlinear Continuous Stirred Tank Reactor Figure 27 Optimal control policy for Nonlinear Crane Container Problem

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Highlights

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Four different hierarchical island models are proposed Proposed models are implemented into Crow Search Algorithm Island models are applied into 45 well-known optimization test problems Optimal control problems are solved by the proposed island models

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   

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*Declaration of Interest Statement

Declaration of interests

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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*Credit Author Statement

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Mert Sinan Turgut: Conceptualization, Metholodology, Software, Validation, Writing- Original Draft, Writing- Review & Editing Oğuz Emrah Turgut: Conceptualization, Software, Writing-Original Draft Deniz Türsel Eliiyi: Writing- Original Draft