key: cord-0879169-ac0rtdpj authors: Alfadhli, Jassim; Jaragh, Ali; Alfailakawi, Mohammad Gh.; Ahmad, Imtiaz title: FP-SMA: an adaptive, fluctuant population strategy for slime mould algorithm date: 2022-03-06 journal: Neural Comput Appl DOI: 10.1007/s00521-022-07034-6 sha: de0542259328f595f9a9561692e8bc17b4225d74 doc_id: 879169 cord_uid: ac0rtdpj In this paper, an adaptive Fluctuant Population size Slime Mould Algorithm (FP-SMA) is proposed. Unlike the original SMA where population size is fixed in every epoch, FP-SMA will adaptively change population size in order to effectively balance exploitation and exploration characteristics of SMA’s different phases. Experimental results on 13 standard and 30 IEEE CEC2014 benchmark functions have shown that FP-SMA can achieve significant reduction in run time while maintaining good solution quality when compared to the original SMA. Typical saving in terms of function evaluations for all benchmarks was between 20 and 30% on average with a maximum being as high as 60% in some cases. Therefore, with its higher computation efficiency, FP-SMA is much more favorable choice as compared to SMA in time stringent applications. Population-based meta-heuristics have been the dominant methods to find optimal or good solutions to many complex optimization problems in reasonable time [22] . The popularity of meta-heuristics has increased considerably due to their key role in learning and knowledge discovery within the emerging fields of big data and machine learning. These meta-heuristics derive their inspiration from mimicking intelligent processes arising in nature, and are commonly classified into two categories: evolutionary (EA) and swarm intelligence algorithms [13] . The most popular EA algorithms are genetic (GA) [17] and differential evolution (DE) [47] . As for the swarm intelligence category, this includes particle swarm (PSO) [47] , cuckoo search (CS) [58] , whale optimization [32] , monarch butterfly optimization (MBO) [53] , moth search (MSA) [52] , and Harris hawks optimization (HHO) [19] among others. Although the development of meta-heuristics has witnessed tremendous progress in recent years, there is still much room for improvement as no single algorithm can solve all problems efficiently as per the ''No Free Lunch'' theorem [55] . Recently, a new population-based metaheuristic called slime mould algorithm (SMA) has been proposed by Li et al. [27] . SMA is inspired by a unique slime mould, i.e., physarum polycephalum, which is an organism that can live freely as a single cell but can also form communicating aggregates when foraging food sources. In order to find food, slime mould starts the search process with a randomly distributed population. Once having identified food concentration during the random search, the slime mould will approach and wrap the food and secrete enzyme to digest it, while retaining certain exploration capability to search for better food sources. In order to imitate slime mould's exploration and exploitation behaviors, authors in [27] proposed a bio-oscillating policy that separates the population into two groups according to their fitness. The first group, called positive group, is the group of individuals from the population with the best fitness whereas the other one, labeled as negative group, is the one consisting of those with the lowest fitness. The better fitness group will be exploited to find the best & Mohammad Gh. Alfailakawi alfailakawi.m@ku.edu.kw possible solution whereas the lower fitness one will be used to explore outward regions. In addition, a vibration parameter based on the sigmoid function is introduced to simulate food-grabbing behavior of slime mould. Despite SMA being a recent algorithm, it has demonstrated excellent performance compared to state-of-the-art meta-heuristics in many fields. However, one of the key disadvantages of SMA lies in its relatively long run time and high computational cost. Being applied successfully in multiple fields, in this work we investigate the enhancement of the algorithm by augmenting it with a population size adaptation method that can reduce its prohibitively long run time. Population size plays a very important role in both run-time efficiency and optimization effectiveness of meta-heuristics and thus by balancing exploration and exploitation characteristics of the SMA algorithm, its performance and computational cost can be improved [27] . In order to balance exploration and exploitation phases of an algorithm, population size adaptation schemes can automatically adjust population size according to population diversity during the search process thus enhancing performance and reducing run time. Population size adaptation has been widely studied in genetic algorithms [5, 25] , differential evolution [40, 50] , artificial bee colony optimization [9] , swarm intelligence [7, 12, 41] and recently to sine cosine algorithm [3] . However, to the best of the author's knowledge, no such work has been reported for SMA. To fill this research gap, we propose herein an adaptive fluctuant population size SMA algorithm called FP-SMA. Unlike the original SMA where population size is fixed in every epoch, FP-SMA will adaptively change population size to enhance run time characteristics of both exploitation and exploration phases of the algorithm. Population adaptation concept used in the proposed algorithm is a clusterbased approach borrowed from K-mean clustering algorithm. The threshold used to start the adaptation process is a scaled sigmoid function that decreases smoothly initially, then dramatically midway, and again smoothly near the end of algorithm execution. Once population diversity is out of the threshold, population size increases or decreases using a sine wave pattern (for randomization). Therefore, key contributions of this study can be summarized as the following: -Propose an adaptive fluctuant population size slime mould algorithm (FP-SMA) that automatically adjust population size during the search process according to population diversity to effectively balance exploitation and exploration characteristics of conventional SMA algorithm. -FP-SMA performance is analyzed over several benchmark functions, including 13 standard and 30 IEEE CEC2014 benchmark functions. -Simulation results revealed that FP-SMA can achieve significant reductions in the number of function evaluations as compared to conventional SMA without impacting solution quality. The remainder of the paper is organized as follows. Section 2 highlights SMA algorithm, literature review, and population diversity adaptation techniques. Section 3 provides details of the proposed algorithm. Experimental results are reported and analyzed in Sect. 4. Finally, conclusions and some future directions are presented in Sect. 5. In this section, we introduce SMA algorithm's mathematical models followed by the short literature review of SMA, and finally brief discussion of population diversity-based adaptation techniques [56] that motivated this work. In [27] , a new stochastic optimizer called slime mould algorithm (SMA) was proposed. The algorithm models the morphological changes of slime mould, namely Physarum polycephalum, when foraging. Slime mould is eukaryote where its organic matter utilizes a process called Plasmodium as its main process to seek food. Once a food source is identified, the slime mould surrounds it and secrete enzymes to digest it. The foraging process of the Slime mould is divided into three phases, where the first process is finding food source, followed by wrapping, and then food grabble. The mathematical model for the various processes of the slime mould is described as follows [27] : where T represents maximum number of iterations. Parameter W ! is a vector representing the slime mould weight and is described mathematically as [27] : SmellIndex ¼ sortðSÞ ð 4Þ where F b /F w represents best/worst fitness value at the current iteration and r being a random number 2 [0,1]. Moreover, S(i) represents the individual's fitness, condition indicates the rank of S(i) in the first half of the population (i.e., the positive group). In Eq. (4), the term SmellIndex denotes the result of sorting S in an ascending order. Parameter p in Eq. (1) is calculated using [27] : where DF is the overall global best fitness and S(i) represents the individual's fitness. A flowchart for SMA algorithm is depicted in Fig. 1 where it starts with initializing parameters D, P, T, LB, UB, z, and a random population X i ! ðt ¼ 0Þ. In each iteration, SMA will calculate individuals' fitness and find the best one in the current iteration. The next population is then updated according to Eq. (1) and conditional weighting parameter W. This iterative process is repeated until maximum number of iteration is reached where the best fitness and the solution are stored as the final result. In Eq. (1), the exploration capability is guaranteed with a probability of at least z while exploitation is performed with a probability of at least 1 À p. When rand is less than z, SMA will take a random walk within the boundaries defined by LB and UB. However, when another random number r is larger than p, SMA will perform exploitation and search in the neighbourhood of the current location. When r is less than p, SMA will wrap around the current best position, X b ! ðtÞ, with wrapping direction and radius depending on the current position's fitness. Such behaviors can be much more evident when plugging the definition of W in Eq. (3) back to Eq. (1) when r\p resulting in [27] : The SMA algorithm will wrap the food in two directions depending on the fitness of the current position's with the radius depending on the amplitude of vb ! and population variance. In Eq. (1), vb ! and vc ! are two tuning parameters oscillating towards 0 with iterations imitating food-grabbing behaviour and hence exploitation around the best food source. SMA and its variants were successfully applied to many problems such as image segmentation [34, 61] , breast cancer detection [29, 36] , COVID-19 early detection [4, 46] , parameters estimation of photovoltaic cells [14, 31, 33] , medical data classification [54] , feature selection [23] , proportional-integral-derivative (PID) motor speed controllers [11, 43] , power systems [38, 45] , bearings defect identification [51] , travelling salesman problem [30] , and machine learning models parameter tuning for support vector machine [8] and artificial neural network [63] to name a few. Initialize D, P, T , LB, U B, z However, SMA may suffer from some shortcomings such as slow convergence rate because of being trapped in local minima and having an unbalanced exploitation and exploration phases. Therefore, to further improve SMA performance, researchers have reported a variety of specific stochastic operators such as Levy distribution [4, 10] , cosine function for controlling parameters [16, 18] , quantum rotation gate [59] , opposition-based learning [35, 45, 54] , and chaotic operator [31] . In addition, SMA has been hybridized with other metaheuristics such as Harris hawk optimization [60] , whale optimization [1], particle swarm [15] , differential evolution [20, 29] , and arithmetic optimization algorithm [62] to efficiently solve specific optimization problems. Furthermore, variants of SMA to solve discrete binary [2, 26] and multi-objective optimization problems [21, 24, 44] have been proposed. However, none of these works have considered population size adaptation to enhance the performance of SMA. Population size adaption has become prevalent in many population-based metaheuristic algorithms (MAs). The choice of a proper population size can substantially enhance the efficiency of many meta-heuristic algorithms including SMA. In a typical linear population size reduction algorithm, there is a large number of individuals in the population initially to enhance exploration capability. During population evolution, population size is decreased linearly until reaching smallest population size at the end of algorithm execution in order to allow for proper exploitation. However, for more complex objective functions, it is also possible to increase population size towards the end of the search process to avoid premature convergence or stagnation. A common criteria to control population size direction is to use population diversity as a metric (i.e., population distribution). For example, in [6, 39, 42, 56, 57] , the authors proposed to use population diversity to start and stop population adaption process in differential evolution. The following is a review of population diversity adaptation technique based on the work presented in [56] . Parameters and variables associated with this technique are listed in Table 1 .