An analysis of selection methods in memory consideration for harmony search
Introduction
Evolutionary algorithms (EA) are a class of optimization methods starting with a set of provisional solutions that are normally generated randomly. They iterate toward the global optimal solution using specific operators controlled by certain parameters [1]. In principle, they imitate the Darwinian’s natural selection of the ‘survival of the fittest’. Iteratively, they explore a wide range of problem search space regions,while, at the same time, they exploit the existing solutions utilizing learning strategies to manage information that will result in an efficient solution [2], [3]. Naturally, EA deals with optimization problems that can be generally modeled as an n-dimensional minimization problem as follows:where is the objective function; is the set of decision variables. is the possible value range for each decision variable, where , where and are the lower and upper bounds for the decision variable respectively and N is the number of decision variables.
The Harmony search (HS) algorithm is a recent EA proposed by Geem et al. [4] which mimics the musical process of improvising a pleasing Harmony. It has several advantages over other EAs: it has a novel stochastic derivative [5], it necessitates fewer mathematical requirements which generate a new solution after considering all the existing solutions at each iteration [6]. In other words, such advantages are related to simplicity, flexibility, adaptability, generality, and scalability [7], [8].
Therefore, HS has been adopted for a wide variety of optimization problems such as engineering optimization problems, scheduling problems, bio-informatics and biomedical problems, Clustering, [9], [10], [11], [12] any many others [13], [14], [15], [16], [17], [18], [19], [20], [21]. To keep with the combinatorial nature of some optimization problems, the performance of HS algorithm has been improved by hybridizing, modifying, replacing, and adopting the HS operators [6], [22], [23], [19]. Furthermore, the parameter tuning of HS is also studied leading to a parameter-less HS [24]. However, very few investigation directed toward the analysis of the search power of the HS algorithm to reveal why it succeed [25], [7].
HS terms and strategies can be connected to the Harmonized tuning that musicians endeavour to achieve in a musical rehearsal. HS algorithm begins with a population of individuals randomly generated and stored in the Harmony memory (HM). A new individual is evolutionary generated using three operators: (i) memory consideration, (ii) random consideration, and (iii) pitch adjustment. The new individual then substitutes the worst individual in HM, if better. This process is evolved until a termination criterion is met.
Memory consideration (MC) operator of HS is the main concern of this paper. In MC, the HS selects a value of a decision variable from the individuals stored in HM through the process of generating a new one. Initially, the MC randomly selects any individual from HM, then the value of each decision variable is taken from that individual. To avoid the random selection process in MC, Al-Betar et al. [7] initially explored some selection schemes adapted from EA methods to employ in the MC process for the purpose of imitating the natural selection of the ‘survival of the fittest’ principle. These were Global-best, proportional, tournament, linear rank, and exponential rank selection methods. In their study, Al-Betar et al. [7] maintain that these selection methods are proposed and adapted to be workable with MC in the HS algorithm.
The main objective of this study is to analyze some selection methods proposed in [7] in terms of their effect in the selection pressure concepts (i.e., the amount of bias that selection expresses toward the better individuals in the population during search) and the effect of their parameters on the behavior of HS algorithm. Furthermore, the proportional selection method is scaled in this study to overcome its shortcoming in dealing with problems of negative fitness values.
The rest of this paper is organized as follows: basic acquired knowledge for HS algorithm is presented in Section 2. The selection methods are expressed in Section 3. The analysis of the results obtained by selection schemes in MC is discussed in Section 4. Finally, Section 5 provides a conclusion to the findings and suggests possible future guidelines.
Section snippets
Fundamentals to the harmony search algorithm
The recent evolutionary algorithm that mimics the musical improvisation process is the HS algorithm developed by Geem et al. [4]. The musical improvisation process in the musical context consists of a set of musicians each playing on his own instrument. The process is launched aiming for a pleasing Harmony resulting from musicians’ practices. Each pitch is tuned at each practice in line with the following rules: (i) considering one pitch stored in memory; (ii) adjusting a pitch in the memory;
Selection methods in memory consideration
In evolutionary algorithms (EAs), it is the selection that decides the survivability of the individuals during the search, and drives the search toward the global minima [27]. Overtime, several selection methods were proposed to improve the performance of EAs. Proportional [28], tournament [29], linear rank [30], [31], and exponential rank [32] are examples on the selection methods utilizing the ‘survival of the fittest’ principle in their work to ensure an efficient performance.
In HS
Experiments and results
To evaluate the effect of the selection methods with different parameter settings on the performance of HS algorithm, extensive experiments have been conducted using global minimization benchmark functions described in Table 1. The experiments studied HS algorithm with each memory consideration variation are presented in Section 3. The effect of the selection method parameters is studied individually and a comparative evaluation between the HS with memory consideration variations is conducted.
Conclusion and future work
This paper studies the effect of the selection methods in memory consideration in terms of the selection pressure power. The parameter of the selection methods directly affects the balance between exploration and exploitation during the search. The parameters of four common selection methods were carefully analyzed. These included the tournament size t in harmony search with tournament memory consideration (HS with TMC), the parameter c in harmony search with scaled proportional memory
Acknowledgments
The authors sincerely appreciate the helpful and insightful comments from the anonymous referees, that have greatly improved the clarity of the paper. The first author is grateful to be awarded a Postdoctoral Fellowship from the School of Computer Sciences (USM).
References (42)
Novel derivative of harmony search algorithm for discrete design variables
Applied Mathematics and Computation
(2008)- et al.
An improved harmony search algorithm for solving optimization problems
Applied Mathematics and Computation
(2007) - et al.
Novel selection schemes for harmony search
Applied Mathematics and Computation
(2012) - et al.
Dynamic multi-swarm particle swarm optimizer with harmony search
Expert Systems with Applications
(2011) - et al.
Damage detection under ambient vibration by harmony search algorithm
Expert Systems with Applications
(2012) - et al.
Broadcast scheduling in packet radio networks using harmony search algorithm
Expert Systems with Applications
(2012) - et al.
Optimal synthesis of linear antenna arrays using a harmony search algorithm
Expert Systems with Applications
(2011) - et al.
A novel grouping harmony search algorithm for the multiple-type access node location problem
Expert Systems with Applications
(2012) - et al.
Discrete harmony search based expert model for epileptic seizure detection in electroencephalography
Expert Systems with Applications
(2012) - et al.
A coevolutionary differential evolution with harmony search for reliability-redundancy optimization
Expert Systems with Applications
(2012)
Global-best harmony search
Applied Mathematics and Computation
Modified harmony search optimization for constrained design problems
Expert Systems with Applications
Parameter-setting-free harmony search algorithm
Applied Mathematics and Computation
A self-adaptive global best harmony search algorithm for continuous optimization problems
Applied Mathematics and Computation
Novel global harmony search algorithm for unconstrained problems
Neurocomputing
Introduction to Evolutionary Computing
Metaheuristics: a bibliography
Annals of Operations Research
Metaheuristics in combinatorial optimization
Annals of Operations Research
A new heuristic optimization algorithm: harmony search
Simulation
University course timetabling using a hybrid harmony search metaheuristic algorithm
IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews
A harmony search algorithm for university course timetabling
Annals OR
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2017, Expert Systems with ApplicationsCitation Excerpt :The HS algorithms of Section 3.1 convey the selection in a random manner because they don’t utilize the fitness of the solutions in the improvisation step (Al-Betar, Doush, Khader, & Awadallah, 2012). To equip the HS algorithms with Natural Selection characteristic, some recent works have proposed to append a selection scheme into the algorithm (Al-Betar et al., 2012; Al-Betar et al., 2013; Castelli, Silva, Manzoni, & Vanneschi, 2014; Karimi, Askarzadeh, & Rezazadeh, 2012). They have also analyzed the effect of this modification on the performance of the algorithm.
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2016, Applied Soft Computing JournalTournament-based harmony search algorithm for non-convex economic load dispatch problem
2016, Applied Soft Computing JournalCitation Excerpt :One of the main shortcomings of the HS algorithm raised is in the area of memory consideration selection process where the natural selection of survival-of-the-fittest principle is omitted [3]. Thus, the investigations of novel selection methods in the memory consideration for the HS algorithm are proposed in [3,13] and analyzed in [6], where five selection schemes were investigated: proportional, tournament, global best, linear rank, and exponential rank. Interestingly, the tournament-based HS algorithm achieved the best performance for the global optimization problems and adopted by other researchers as an efficient variant of the HS algorithm [25,75,49].
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2015, Digital Signal Processing: A Review JournalCitation Excerpt :It has received much attention in comparative to conventional mathematical optimization algorithms in that it is not limited by requiring substantial gradient information and not sensitive to initialization [1]. As a typical swarm intelligent algorithm and characterized by simplicity, utilizing real-number encoding and fewer mathematical requirements and so forth, harmony search (HS) and its variants [2–15,51–54], mimicking the process of improvising a musical harmony, have been found to be potential in solving optimization problems. They have been applied to many fields of science and engineering successfully (e.g., pipe network design optimization problems [16], structural optimization problems [17,18], nurse rostering problems [19], economic load dispatch problems [20–22], PID controller optimization problems [23], location of wireless sensor networks [24], trajectory planning for robots [25], vehicle routing optimization problems [26,27], reliability problems [28], 0–1 knapsack problems [29], feature selection [30,31] and so on [32–46]).