Abstract

Introduction

     An engineering system consists of properly established activities and is put together to achieve a predefined goal. These activities include analysis, design, construction, research, and development. Designing and assembling structural systems, including buildings, bridges, highways, and other complex systems, have been developed over centuries. However, the evolution of these systems has been prolonged because the overall process is very costly and time-consuming, requiring primary human and material resources to be utilized. One of the options for overcoming these shortcomings is the use of metaheuristic algorithms as recently developed intelligent techniques. These algorithms can be utilized as upper-level search techniques for optimization procedures to achieve better results.

Objectives

     Shape and size optimization of truss structures are considered in this paper, utilizing the Chaos Game Optimization (CGO) as one of the recently developed metaheuristic algorithms. The principles of chaos theory and fractal configuration are considered inspirational concepts. For the numerical purpose, the 10-bar, 37-bar, 52-bar, 72-bar, and 120-bar truss structures as five of the benchmark problems in this field are considered as design examples in which the frequency constraints are considered as limits that have to be dealt with during the optimization procedure. Multiple optimization runs are also conducted for having a comprehensive statistical analysis, while a comparative investigation is also conducted with other algorithms in the literature.

Results

     Based on the results of the CGO and other approaches from the literature, the CGO can provide better and competitive results in dealing with the considered truss design problems.

Conclusion

     In summary, the CGO can provide better solutions in dealing with the considered real-size structural design problems with higher levels of complexity.

Introduction

     Over the past decades, human beings have put so much effort into maximizing the use of limited available resources. For example, one challenge is selecting design variables to consider design constraints in engineering designs and having the lowest constriction and material costs. In fact, the main goal is to properly meet the basic and advanced design standards by considering the project’s economic aspects. Recent advances in structural engineering reveal the need to consider greater accuracy, better performance, and higher construction speeds in the design of structural systems. Therefore, to address each of the above factors it is necessary to introduce new methods for design and optimization and implement them on complex and real-world systems. Optimization problems normally search for the minimum values ​​of a cost function to systematically select the values ​​for the variables that lead to the lowest cost. Metaheuristic algorithms are optimization methods that combine global and local search techniques to get the answers as close as possible to the optimal answer. Indeed, metaheuristic algorithms are types of approximate optimization algorithms capable of providing acceptable solutions and avoiding entrapment in local optimal points. Firefly Algorithm (FA) [1], Genetic Algorithm (GA) [2], Material Generation Algorithm (MGA) [3], Cuckoo Search Algorithm (CSA) [4], Chaos Game Optimization (CGO) [5], [6], Slime Mould Algorithm (SMA) [7], Atomic Orbital Search (AOS) [8], Particle Swarm Optimizer (PSO) [9], and Crystal Structure Algorithm (CSA) [10] are some of the recently developed metaheuristic algorithms. Nevertheless, the application of these algorithms alongside the improved or hybrid versions has been investigated in different fields. Investigation of Lévy flight distribution for engineering optimization [11], optimum design of engineering problems with dynamic differential annealed optimization [12], optimum design of reinforced concrete footings with metaheuristic algorithms [13], investigation of nature-inspired algorithms for getting of bridge scour information [14], performance assessments of an artificial bee colony in optimal design of steel skeletal structures [15], design optimization of reinforced concrete building structures with metaheuristics [16], and estimation of solar photovoltaic cell parameters with a new stochastic slime mould metaheuristic algorithm [17], are some of the recent researches in this field.

Conclusion

     Shape and size optimization of different large-scale truss structures are considered in this paper using the Chaos Game Optimization (CGO) as one of the recently proposed metaheuristic optimization algorithms. In this algorithm, the principles of chaos theory and the configuration of fractals are utilized as inspirational concepts. For the numerical purpose, the 10-bar, 37-bar, 52-bar, 72-bar and 120-bar truss structures as five of the benchmark problems in this field are considered design examples, in which the frequency constraints are considered as limits to be dealt with during the optimization procedure. Multiple optimization runs are also conducted for having a comprehensive statistical analysis, while a comparative investigation is also performed with other algorithms in the literature. Based on the results, the CGO can reach 524.4545 kg in dealing with the 10-bar truss problem, which is better than the previously calculated weights. The CGO can provide 524.5099 kg as the mean of 30 independent runs with 524.7488 kg as the worst run, which are the best statistical results among other approaches for this structure. The CGO provides a best optimum weight of 359.7893 kg for the 37-bar truss structure, while the other attempts in this case, such as IDE, calculate 359.8194 kg, which demonstrates the capability of the CGO. The CGO can provide 359.8842 kg as the mean of 30 independent runs with 360.0873 kg as the worst run, which are the best statistical results among other approaches. Based on the results of other algorithms from the literature for the 52-bar truss problem, CGO can reach 193.1876 kg which is the best among other approaches, while the IDE with 193.2085 kg is the next competitive result. The mean, worst and standard deviation of the conducted runs demonstrate that CGO provides very stable results with mean of 195.4586 kg and standard deviation of 3.8183. The CGO can reach 324.197 kg for the 72-bar truss structure, which is better than the results of other methods.

1- Introduction

2- Related work

3- Optimization by continuation

4- Results and discussion

5- Conclusions and recommendations

References

Abstract

Artificial Neural Networks research field is among the areas of major activity in Artificial Intelligence. Training a neural network is an NP-hard optimization problem that presents several theoretical and computational limitations. In optimization, continuation refers to an homotopy transformation of the fitness function that is used to obtain simpler versions of such fitness function and improve convergence. In this paper we propose an approach for Artificial Neural Network training based on optimization by continuation and meta-heuristic algorithms. The goal is to reduce overall execution time of training without causing negative effects in accuracy. We use continuation together with Particle Swarm Optimization, Firefly Algorithm and Cuckoo Search for training neural networks on public benchmark datasets. The continuation variations of the studied meta-heuristic algorithms reduce execution time required to complete training in about 5–30% without statistically significant loss of accuracy when compared with standard variations of the meta-heuristics.

Introduction

In the latest years, we have seen the efforts in optimization field to solve very complex and challenging tasks. At the same time, the success in this area has led researchers and practitioners to address much larger instances and more difficult classes of problems, specially in meta-heuristic area [1,2]. Training a neural network is an NP-hard optimization problem [3] that usually involves thousands and millions of parameters. Additionally, the fitness function of Artificial Neural Networks (ANNs) is highly non-convex [4] and presents a poor correspondence between its local and global structure [5]. Also, the number of local minima increases exponentially respect to the number of parameters of the network. In fact, such local minima appears to be saddle points instead of true minima [6] reducing the convergence speed of optimization, specially for gradient based training algorithms. In general, second-order methods have not obtained better results than first-order methods in this area [5]. These facts become even more problematic as we start using modern deep learning architectures. ANNs are widely used in several applications of image recognition [7], signal processing [8], speech [9] and several other fields Nevertheless, neural networks are playing a major role in the field of pattern recognition and learning representations from data. The most popular algorithm for training is Stochastic Gradient Descent (SGD) which is a gradient based first order1 method that has been proven to converge into local minima in the parameter space. In general, first and second order2 optimization methods require the calculation of derivatives and Hessians for which Automatic Reverse Mode Differentiation (ARMD) have to be used [10]. The parallel implementation of such methods faces important challenges due to the inherent sequential nature of ARMD. Recently, several meta-heuristic algorithms for optimization have been actively used in ANN training. Meta-heuristic algorithms have features to escape local minima and increase the probability of global convergence. In the literature there are reports of natureinspired meta-heuristics such as Cuckoo Search [11,12], Firefly Algorithm [13,14], Wolf Search Optimization [15,16], Particle Swarm Optimization (PSO) [17,18] and others.

1- Introduction

2- Existing algorithms for the OP

3- Clustering and the OP

4- An ALNS-based heuristic for the OP

5- Results

6- Conclusions

References

Abstract

We propose a new heuristic algorithm for the well-known Orienteering Problem, which aims at maximising the prize collected at vertices of a graph, visiting them through a simple closed tour whose length (also known as travel time) is limited by an upper bound. The algorithm is based on the Adaptive Large Neighbourhood Search metaheuristic paradigm, and uses a clustering of the graph to operate on groups of nearby vertices. Extensive computational results show that the proposed algorithm is able to find very good solutions (it produced 27 new best known solutions on a set of benchmark instances) and that it fares favourably compared with other state-of-the-art heuristics when tested with both long and short computation time budgets.

Introduction

The Orienteering Problem (OP) was introduced by Golden et al. (1987) and combines the selection of a set of customers to visit with the determination of the shortest tour to visit them. Imagine a tour operator who has to design a walking tour of a city. He has a vast set of possible interest points and associates a score to each of them, based on their desirability. Because the tour time is limited, he has to operate a selection of points to visit and decide the visit order, with the objective of maximising the scores of the visited points. Besides applications in tourist trip design (Borràs et al., 2014; Souffriau et al., 2008; Vansteenwegen and Van Oudheusden, 2007; Wang et al., 2008), the OP has been considered as a generalisation of the classical Travelling Salesman Problem when the salesman does not have enough time to visit all the cities (Tsiligirides, 1984). In this case, the prizes reflect the expected profit that the salesman can earn in each city. Tsiligirides (1984) also mentions an application in sporting events, which actually gave the name to the OP: an orienteering event is a competition usually held in forests where participants start from and come back to a basecamp within a time limit; during their tour they visit intermediate locations where they are awarded points. The participant who returns to the basecamp with the highest number of points wins. The survey by Vansteenwegen et al. (2011) also mentions applications to the home fuel delivery problem (Golden et al., 1987), where customers need to be resupplied with fuel. The lower is the level of fuel at the customer, the higher the urgency of visiting them during the current day; we can then model this problem as an OP where the prizes are inversely proportional to the fuel level. Another application arises in telecommunication network design (Thomadsen and Stidsen, 2003). The ring topology is often used to build wired networks, since it is resistent to single-failure disruptions. The length of the ring is constrained since noise is proportional to the total length and because a ring too long would increase the probability of double-failure disruptions. If the operator assigns a level of importance to each node, for example proportional to the amount of traffic it is expected to route, the problem of linking the most important nodes with a ring network can be modelled as an OP. A general history of the problem, a description of its many variants, and an overview of exact and heuristic algorithms developed to solve them, can be found in the two surveys by Vansteenwegen et al. (2011) and Gunawan et al. (2016).

1- Introduction

2- Whale optimization algorithm

3- Differential evolution

4- The proposed approach

5- Simulation results and discussion

6- Conclusion

References

Abstract

The whale optimization algorithm (WOA) is a new bio-inspired meta-heuristic algorithm which is presented based on the social hunting behavior of humpback whales. WOA suffers premature convergence that causes it to trap in local optima. In order to overcome this limitation of WOA, in this paper WOA is hybridized with differential evolution (DE) which has good exploration ability for function optimization problems. The proposed method is named Improved WOA (IWOA). The proposed method, combines exploitation of WOA with exploration of DE and therefore provides a promising candidate solution. In addition, IWOA+ is presented in this paper which is an extended form of IWOA. IWOA+ utilizes re-initialization and adaptive parameter which controls the whole search process to obtain better solutions. IWOA and IWOA+ are validated on a set of 25 benchmark functions, and they are compared with PSO, DE, BBO, DE/BBO, PSO/GSA, SCA, MFO and WOA. Furthermore, the effects of dimensionality and population size on the performance of our proposed algorithms are studied. The results demonstrate that IWOA and IWOA+ outperform the other algorithms in terms of quality of the final solution and convergence rate.

Introduction

Nature-inspired algorithms are very popular for solving different problems in various fields such as engineering (Hadavandi, Mostafayi, & Soltani, 2018; Lucas, Nasiri-Gheidari, & Tootoonchian, 2010), bioinformatics (Das, Abraham, & Konar, 2008), economy (Hafezi, Shahrabi, & Hadavandi, 2015) and medicine (Lin, Mimori, & Chen, 2012), for they don’t require prior knowledge about the problems (Ghasemi, Ghavidel, Rahmani, Roosta, & Falah, 2014). Swarm intelligence (SI) is one of the main categories of nature-inspired algorithms which is developed by simulating the social behavior of some simple organisms. Some of the most popular swarm intelligence algorithms are Particle Swarm Optimization (PSO) (Clerc & Kennedy, 2002; Eberhart & Shi, 2004), Krill-Herd (KH) (Gandomi & Alavi, 2012), Moth-flame Optimization (Mirjalili, 2015), Cuckoo Search (CS) (Yang & Deb, 2009), Biogeography-Based Optimization (BBO) (Simon & Member, 2008) and Grey Wolf Optimization (GWO) (Mirjalili, Mirjalili, & Lewis, 2014). The Whale Optimization Algorithm (WOA) (Mirjalili & Lewis, 2016) is a new swarm intelligence algorithm developed by Mirjalili and Lewis and it is inspired from social behavior of humpback whales. WOA searches the global optimum through encircling prey, searching for prey and attacking the prey. The performance of WOA was tested on 29 optimization benchmark functions and some 6 structural design problems. Results indicates the effectiveness of WOA in comparison with some state-of-the-art natureinspired algorithms (Mirjalili & Lewis, 2016). Several studies of WOA are presented which can be divided into two categories, i.e., (1) improving the WOA’s performance and (2) Appling the WOA to solve some optimization problems. Some of WOA’s improvements are (Kaur & Arora, 2017; Ling, Zhou, & Luo, 2017; Sun, Wang, Chen, & Liu, 2018). Kaur and Arora proposed chaotic WOA (CWOA) (Kaur & Arora, 2017) in which the chaos theory is used to tuning the main parameters of WOA to enhance the convergence speed of it. The experimental results on 20 optimization benchmark functions demonstrated that CWOA can improve the performance of WOA. In Ling et al. (2017) Lévy flight trajectory-based WOA (LWOA) is introduced. The LWOA employed Lévy flight trajectory to increase the diversity of population. The results indicated that LWOA outperformed the original WOA. Sun et al. (2018) proposed modified WOA (MWOA) for solving largescale global optimization problems. In MWOA the Lévy flight strategy was employed to improve WOA’s exploration ability.

1- Introduction

2- Related work

3- The proposed model

4- Results and discussion

5- Conclusions

References

Abstract

The Artificial Bee Colony (ABC) algorithm is a swarm intelligence approach which has initially been proposed to solve optimisation of mathematical test functions with a unique neighbourhood search mechanism. This neighbourhood search mechanism could not be directly applied to combinatorial discrete optimisation problems. In order to tackle combinatorial discrete optimisation problems, the employed and onlooker bees need to be equipped with problem-specific perturbative heuristics. However, a large variety of problem-specific heuristics are available, and it is not an easy task to select an appropriate heuristic for a specific problem. In this paper, a hyper-heuristic method, namely a Modified Choice Function (MCF), is applied such that it can regulate the selection of the neighbourhood search heuristics adopted by the employed and onlooker bees automatically. The Lin-Kernighan (LK) local search strategy is integrated to improve the performance of the proposed model. To demonstrate the effectiveness of the proposed model, 64 Traveling Salesman Problem (TSP) instances available in TSPLIB are evaluated. On average, the proposed model solves the 64 instances to 0.055% from the known optimum within approximately 2.7 min. A performance comparison with other state-of-the-art algorithms further indicates the effectiveness of the proposed model.

Introduction

A computational optimisation methodology involves finding feasible solutions from a finite set of solutions, and identifying only the optimal solution(s). Swarm intelligence algorithms constitute a sub-class of computational optimisation methodology [1]. Swarm intelligence algorithms are developed based on emergence of collective behaviours pertaining to a population of interacting individuals in adapting to the local and/or global environments. Examples of swarm intelligence algorithms include Particle Swarm Optimisation (PSO) [2], Ant Colony Optimisation (ACO) [3], Bat Algorithm (BA) [4], Firefly Algorithm (FA) [5], Cuckoo Search Algorithm (CSA) [6], and bee-inspired algorithms [7–9]. Bees are highly organised social insects. Their survival relies on assigning an important task to each bee in a cooperative mode. The tasks include reproduction, foraging, and constructing hive. Within these tasks, foraging is one of the most important tasks, because the bee colony must ensure an undisrupted supply of food to survive. The food foraging behaviours of bees can be computationally realised as algorithmic tools to solve various optimisation problems. The Artificial Bee Colony (ABC) algorithm is one of the popular beeinspired algorithms. Proposed by Karaboga [7], it is inspired by the foraging behaviours of honey bees in a colony. In the ABC algorithm, a food source represents a possible solution to the optimisation problem in the search space, and the nectar amount of the food source represents the fitness of that solution. The ABC algorithm defines three types of bees: employed bees, onlooker bees, and scout bees. An employed bee looks for new food sources around the neighbourhood of the food source that it has previously visited. An onlooker bee observes dances and selects a food source to visit. It tends to select good food sources from those found by the employed bees. A scout bee searches for new food sources randomly. The mechanism of the ABC algorithm is as follows. The employed bees first perform a neighbourhood search nearby the food source in their memory (i.e. solution). Then, they go back to the hive and perform dances. The dances inform the onlooker bees about the fitness of each solution. Each onlooker bee observes and selects a food source to perform another neighbourhood search based on a probability proportional to the food source fitness (i.e. a roulette wheel selection). The onlooker bees tend to select good food sources from those found by the employed bees. The employed and onlooker bees perform neighbourhood search by perturbing an existing solution to produce a new solution.

Scheduling allows optimal allocation of resources among given tasks in a finite time to achieve desired quality of service. Formally, scheduling problem involves tasks that must be scheduled on resources subject to some constraints to optimize some objective function. The aim is to build a schedule that specifies when and on which resource each task will be executed [1]. It has remained a topic of research in various fields for decades, may it be scheduling of processes or threads in an operating system, job shop, flow shop or open shop scheduling in production environment, printed circuit board assembly scheduling or scheduling of tasks in distributed computing systems such as cluster, grid or cloud. In recent years, distributed computing paradigm has gained much attention due to high scalability, reliability, information sharing and low-cost than single processor machines. Cloud computing has emerged as the most popular distributed computing paradigm out of all others in the current scenario. It provides on-demand access to shared pool of resources in a self-service, dynamically scalable and metered manner with guaranteed Quality of service to users. To provide guaranteed Quality of Service (QoS) to users, it is necessary that jobs should be efficiently mapped to given resources. If the desired performance is not achieved, the users will hesitate to pay. Therefore scheduling is considered as a central theme in cloud computing systems. In general, the problem of mapping tasks on apparently unlimited computing resources in cloud computing belongs to a category of problems known as NP-hard problems. There are no algorithms which may produce optimal solution within polynomial time for such kind of problems. Solutions based on exhaustive search are not feasible as the operating cost of generating schedules is very high [2]. Metaheuristic based techniques [3] deal with these problems by providing near optimal solutions within reasonable time. Metaheuristics have gained huge popularity in the past years due to its efficiency and effectiveness to solve large and complex problems. In this paper, we present an extensive review of various scheduling algorithms based on five metaheuristic techniques namely Ant Colony Optimization (ACO), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), League Championship Algorithm (LCA) and BAT algorithm. Fig. 1 demonstrates a general framework of the paper.