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Showing papers on "Heuristic (computer science) published in 2018"


Journal ArticleDOI
TL;DR: The results show that the OEMACS generally outperforms conventional heuristic and other evolutionary-based approaches, especially on VMP with bottleneck resource characteristics, and offers significant savings of energy and more efficient use of different resources.
Abstract: Virtual machine placement (VMP) and energy efficiency are significant topics in cloud computing research. In this paper, evolutionary computing is applied to VMP to minimize the number of active physical servers, so as to schedule underutilized servers to save energy. Inspired by the promising performance of the ant colony system (ACS) algorithm for combinatorial problems, an ACS-based approach is developed to achieve the VMP goal. Coupled with order exchange and migration (OEM) local search techniques, the resultant algorithm is termed an OEMACS. It effectively minimizes the number of active servers used for the assignment of virtual machines (VMs) from a global optimization perspective through a novel strategy for pheromone deposition which guides the artificial ants toward promising solutions that group candidate VMs together. The OEMACS is applied to a variety of VMP problems with differing VM sizes in cloud environments of homogenous and heterogeneous servers. The results show that the OEMACS generally outperforms conventional heuristic and other evolutionary-based approaches, especially on VMP with bottleneck resource characteristics, and offers significant savings of energy and more efficient use of different resources.

340 citations


Journal ArticleDOI
TL;DR: The Quantum Approximate Optimization Algorithm (QAOA) as discussed by the authors is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems and has been shown to achieve state-of-the-art performance on MaxCut problems.
Abstract: The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. Despite its promise for near-term quantum applications, not much is currently understood about QAOA's performance beyond its lowest-depth variant. An essential but missing ingredient for understanding and deploying QAOA is a constructive approach to carry out the outer-loop classical optimization. We provide an in-depth study of the performance of QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit non-adiabatic operations. Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasi-optimal $p$-level QAOA parameters in $O(\text{poly}(p))$ time, whereas the standard strategy of random initialization requires $2^{O(p)}$ optimization runs to achieve similar performance. We then benchmark QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps. The comparison reveals that QAOA can learn via optimization to utilize non-adiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps. Finally, we provide a realistic resource analysis on the experimental implementation of QAOA. When quantum fluctuations in measurements are accounted for, we illustrate that optimization will be important only for problem sizes beyond numerical simulations, but accessible on near-term devices. We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.

221 citations


Journal ArticleDOI
TL;DR: The most popular heuristic and meta-heuristic optimization algorithms are studied in this paper, and implementation of the optimization procedures for the solution of CHPED problem taking into account the objective functions and different constrains are discussed.
Abstract: Combined heat and power economic dispatch (CHPED) aims to minimize the operational cost of heat and power units satisfying several equality and inequality operational and power network constraints. The CHPED should be handled considering valve-point loading impact of the conventional thermal plants, power transmission losses of the system, generation capacity limits of the production units, and heat-power dependency constraints of the cogeneration units. Several conventional optimization algorithms have been firstly presented for providing the optimal production scheduling of power and heat generation units. Recently, experience-based algorithms, which are called heuristic and meta-heuristic optimization procedures, are introduced for solving the CHPED optimization problem. In this paper, a comprehensive review on application of heuristic optimization algorithms for the solution of CHPED problem is provided. In addition, the most popular heuristic and meta-heuristic optimization algorithms are studied in this paper, and implementation of the optimization procedures for the solution of CHPED problem taking into account the objective functions and different constrains are discussed. The main contributions of the reviewed papers are studied and discussed in details. Additionally, main considerations of equality and inequality constraints handled by different research studies are reported in this paper. Five test systems are considered for evaluating the performance of different optimization techniques. Optimal solutions obtained by employment of multiple heuristic and meta-heuristic optimization methods for test instances are demonstrated and the introduced methods are compared in terms of convergence speed, attained optimal solutions, and constrains. The best optimal solutions for five test systems are provided in terms of operational cost by employment of different optimization methods.

184 citations


Journal ArticleDOI
TL;DR: A simulated annealing algorithm with a mechanism of repeatedly cooling and rising the temperature is proposed to solve the four versions of this problem, with or without the LIFO constraint, and allowing rotation of goods or not.

166 citations


Journal ArticleDOI
TL;DR: A hybrid multi-population genetic algorithm is proposed to solve a new city logistics problem arising in the last mile distribution of e-commerce, and the computational results obtained show the effectiveness of the different components of the algorithm.

164 citations


Journal ArticleDOI
TL;DR: A lightweight ECG noise analysis framework is highly demanded for real-time detection, localization, and classification of single and combined ECG noises within the context of wearable ECG monitoring devices which are often resource constrained.
Abstract: Electrocardiogram (ECG) signal quality assessment (SQA) plays a vital role in significantly improving the diagnostic accuracy and reliability of unsupervised ECG analysis systems. In practice, the ECG signal is often corrupted with different kinds of noises and artifacts. Therefore, numerous SQA methods were presented based on the ECG signal and/or noise features and the machine learning classifiers and/or heuristic decision rules. This paper presents an overview of current state-of-the-art SQA methods and highlights the practical limitations of the existing SQA methods. Based upon past and our studies, it is noticed that a lightweight ECG noise analysis framework is highly demanded for real-time detection, localization, and classification of single and combined ECG noises within the context of wearable ECG monitoring devices which are often resource constrained.

163 citations


Proceedings ArticleDOI
18 Jun 2018
TL;DR: This work proposes a novel hyperparameter optimization method that can find optimal hyperparameters for a given sequence using an action-prediction network leveraged on Continuous Deep Q-Learning.
Abstract: Hyperparameters are numerical presets whose values are assigned prior to the commencement of the learning process. Selecting appropriate hyperparameters is critical for the accuracy of tracking algorithms, yet it is difficult to determine their optimal values, in particular, adaptive ones for each specific video sequence. Most hyperparameter optimization algorithms depend on searching a generic range and they are imposed blindly on all sequences. Here, we propose a novel hyperparameter optimization method that can find optimal hyperparameters for a given sequence using an action-prediction network leveraged on Continuous Deep Q-Learning. Since the common state-spaces for object tracking tasks are significantly more complex than the ones in traditional control problems, existing Continuous Deep Q-Learning algorithms cannot be directly applied. To overcome this challenge, we introduce an efficient heuristic to accelerate the convergence behavior. We evaluate our method on several tracking benchmarks and demonstrate its superior performance1.

161 citations


Journal ArticleDOI
TL;DR: This survey reviews, classify and discusses several recent advances and results obtained for each variant, including theoretical complexity, exact solving algorithms, approximation schemes and heuristic approaches, and proves new complexity results and induce some solving algorithms through relationships established between different variants.

160 citations


Journal ArticleDOI
TL;DR: The present research on mobile robotics addresses the problems which are mainly on path planning algorithm and optimization in static as well as dynamic environments with a focus on meta-heuristic methods.

152 citations


Journal ArticleDOI
TL;DR: The simulation results demonstrate the superior performance of BGWO in solving UC problem for small, medium and large scale systems successfully compared to other well established heuristic and binary approaches.
Abstract: The unit commitment problem belongs to the class of complex large scale, hard bound and constrained optimization problem involving operational planning of power system generation assets. This paper presents a heuristic binary approach to solve unit commitment problem (UC). The proposed approach applies Binary Grey Wolf Optimizer (BGWO) to determine the commitment schedule of UC problem. The grey wolf optimizer belongs to the class of bio-inspired heuristic optimization approaches and mimics the hierarchical and hunting principles of grey wolves. The binarization of GWO is owing to the UC problem characteristic binary/discrete search space. The binary string representation of BGWO is analogous to the commitment and de-committed status of thermal units constrained by minimum up/down times. Two models of Binary Grey Wolf Optimizer are presented to solve UC problem. The first approach includes upfront binarization of wolf update process towards the global best solution (s) followed by crossover operation. While, the second approach estimates continuous valued update of wolves towards global best solution(s) followed by sigmoid transformation. The Lambda-Iteration method to solve the convex economic load dispatch (ELD) problem. The constraint handling is carried out using the heuristic adjustment procedure. The BGWO models are experimented extensively using various well known illustrations from literature. In addition, the numerical experiments are also carried out for different circumstances of power system operation. The solution quality of BGWO are compared to existing classical as well as heuristic approaches to solve UC problem. The simulation results demonstrate the superior performance of BGWO in solving UC problem for small, medium and large scale systems successfully compared to other well established heuristic and binary approaches.

134 citations


Journal ArticleDOI
TL;DR: This paper addresses the virtual network function (VNF) placement problem in cloud datacenter considering users’ service function chain requests (SFCRs) and designs a Two-StAge heurisTic solution (T-SAT) designed to solve the ILP.
Abstract: Network function virtualization (NFV) brings great conveniences and benefits for the enterprises to outsource their network functions to the cloud datacenter. In this paper, we address the virtual network function (VNF) placement problem in cloud datacenter considering users’ service function chain requests (SFCRs). To optimize the resource utilization, we take two less-considered factors into consideration, which are the time-varying workloads, and the basic resource consumptions (BRCs) when instantiating VNFs in physical machines (PMs). Then the VNF placement problem is formulated as an integer linear programming (ILP) model with the aim of minimizing the number of used PMs. Afterwards, a Two-StAge heurisTic solution (T-SAT) is designed to solve the ILP. T-SAT consists of a correlation-based greedy algorithm for SFCR mapping (first stage) and a further adjustment algorithm for virtual network function requests (VNFRs) in each SFCR (second stage). Finally, we evaluate T-SAT with the artificial data we compose with Gaussian function and trace data derived from Google's datacenters. The simulation results demonstrate that the number of used PMs derived by T-SAT is near to the optimal results and much smaller than the benchmarks. Besides, it improves the network resource utilization significantly.

Journal ArticleDOI
TL;DR: In this study, a distance-constrained mobile hierarchical facility location problem is used in order to find the optimal number and locations of launch and recharge stations with the objective of minimizing the total costs of the system.
Abstract: In the last decade, aerial delivery system has been considered as a promising response to increasing traffic jams and incremental demand for transportation. In this study, a distance-constrained mobile hierarchical facility location problem is used in order to find the optimal number and locations of launch and recharge stations with the objective of minimizing the total costs of the system. System costs include establishment cost for launching and recharge stations, drone procurement, and drone usage costs. It is supposed that the demand occurs according to Poisson distribution, distributed uniformly along the network edges and is satisfied by the closest open facility. Since the flying duration of a drone is limited to its endurance, it may visit one or more recharge stations to reach to the demand point. This route is calculated by the shortest path algorithm, and the Euclidean distance is considered between nodes and facilities. It is proved that facility location problems are NP-hard on a general graph. Accordingly, heuristic algorithms are proposed as solution method. To illustrate the applicability of the algorithms, a case study is presented and the results are discussed.

Journal ArticleDOI
TL;DR: The computational experiments show that the proposed Hybrid Ant Colony algorithm provides better results relative to the other algorithms, compared to the Adaptive Learning Approach and Genetic Heuristic algorithm.

Journal ArticleDOI
TL;DR: A new approach based on ant colony optimization (ACO) to determine the trajectories of a fleet of unmanned air vehicles (UAVs) looking for a lost target in the minimum possible time is presented, including a new MTS heuristic that exploits the probability and spatial properties of the problem.

Journal ArticleDOI
12 Jan 2018-Energies
TL;DR: In this paper, the authors proposed an efficient home energy management controller (EHEMC) based on genetic harmony search algorithm (GHSA) to reduce electricity expense, peak to average ratio (PAR), and maximize user comfort.
Abstract: The traditional power grid is inadequate to overcome modern day challenges. As the modern era demands the traditional power grid to be more reliable, resilient, and cost-effective, the concept of smart grid evolves and various methods have been developed to overcome these demands which make the smart grid superior over the traditional power grid. One of the essential components of the smart grid, home energy management system (HEMS) enhances the energy efficiency of electricity infrastructure in a residential area. In this aspect, we propose an efficient home energy management controller (EHEMC) based on genetic harmony search algorithm (GHSA) to reduce electricity expense, peak to average ratio (PAR), and maximize user comfort. We consider EHEMC for a single home and multiple homes with real-time electricity pricing (RTEP) and critical peak pricing (CPP) tariffs. In particular, for multiple homes, we classify modes of operation for the appliances according to their energy consumption with varying operation time slots. The constrained optimization problem is solved using heuristic algorithms: wind-driven optimization (WDO), harmony search algorithm (HSA), genetic algorithm (GA), and proposed algorithm GHSA. The proposed algorithm GHSA shows higher search efficiency and dynamic capability to attain optimal solutions as compared to existing algorithms. Simulation results also show that the proposed algorithm GHSA outperforms the existing algorithms in terms of reduction in electricity cost, PAR, and maximize user comfort.

Journal ArticleDOI
01 Mar 2018
TL;DR: The experimental results show that PACO-3Opt is more efficient and reliable than the other algorithms and can reach the global optimum.
Abstract: This article presented a parallel cooperative hybrid algorithm for solving traveling salesman problem. Although heuristic approaches and hybrid methods obtain good results in solving the TSP, they cannot successfully avoid getting stuck to local optima. Furthermore, their processing duration unluckily takes a long time. To overcome these deficiencies, we propose the parallel cooperative hybrid algorithm (PACO-3Opt) based on ant colony optimization. This method uses the 3-Opt algorithm to avoid local minima. PACO-3Opt has multiple colonies and a master---slave paradigm. Each colony runs ACO to generate the solutions. After a predefined number of iterations, each colony primarily runs 3-Opt to improve the solutions and then shares the best tour with other colonies. This process continues until the termination criterion meets. Thus, it can reach the global optimum. PACO-3Opt was compared with previous algorithms in the literature. The experimental results show that PACO-3Opt is more efficient and reliable than the other algorithms.

Journal ArticleDOI
TL;DR: This work designs and implements a policy network based on reinforcement learning to make node mapping decisions and uses policy gradient to achieve optimization automatically by training the policy network with the historical data based on virtual network requests, the first to utilize historical requests data to optimize network embedding automatically.

Journal ArticleDOI
TL;DR: Effective heuristic methods, including multi-round linear weight optimization and enhanced multi-objective particle swarm optimization algorithms are proposed to achieve adequate Pareto-optimal allocation in heterogeneous spatial crowdsourcing.
Abstract: With the rapid development of mobile networks and the proliferation of mobile devices, spatial crowdsourcing, which refers to recruiting mobile workers to perform location-based tasks, has gained emerging interest from both research communities and industries. In this paper, we consider a spatial crowdsourcing scenario: in addition to specific spatial constraints, each task has a valid duration, operation complexity, budget limitation, and the number of required workers. Each volunteer worker completes assigned tasks while conducting his/her routine tasks. The system has a desired task probability coverage and budget constraint. Under this scenario, we investigate an important problem, namely heterogeneous spatial crowdsourcing task allocation (HSC-TA), which strives to search a set of representative Pareto-optimal allocation solutions for the multi-objective optimization problem, such that the assigned task coverage is maximized and incentive cost is minimized simultaneously. To accommodate the multi-constraints in heterogeneous spatial crowdsourcing, we build a worker mobility behavior prediction model to align with allocation process. We prove that the HSC-TA problem is NP-hard. We propose effective heuristic methods, including multi-round linear weight optimization and enhanced multi-objective particle swarm optimization algorithms to achieve adequate Pareto-optimal allocation. Comprehensive experiments on both real-world and synthetic data sets clearly validate the effectiveness and efficiency of our proposed approaches.

Journal ArticleDOI
TL;DR: The VNE-NTANRC algorithm adopts a novel node-ranking approach to rank all substrate and virtual nodes before embedding each given VN, and Simulation results reveal that V NE-NTAnRC algorithm outperforms typical and latest heuristic algorithms, only considering single network topology attribute and local resources.
Abstract: Network virtualization (NV) is a promising approach to remove the ossification of current Internet. Virtual network embedding (VNE) is the key issue in NV which efficiently and effectively maps various of virtual networks (VNs), with different node and link resource requests, onto the shared substrate network(s) with finite underlying resources. Previous VNE algorithms in the literature are mostly heuristic. Single network topology attribute and each node’s local resources are assisted to rank nodes in most heuristic algorithms, leading to inefficient resource utilization of substrate network in the long run. To deal with this issue, we propose the network topology attribute and network resource-considered algorithm (VNE-NTANRC). The VNE-NTANRC algorithm adopts a novel node-ranking approach to rank all substrate and virtual nodes before embedding each given VN. The novel node-ranking approach has two subapproaches and considers five important network topology attributes and global network resources altogether. One subapproach is able to calculate all node values (NoV) directly. The other subapproach, stimulating from the Google PageRank website algorithm, enables to calculate NoVs in a stable state. Simulation results reveal that VNE-NTANRC algorithm outperforms typical and latest heuristic algorithms, only considering single network topology attribute and local resources.

Journal ArticleDOI
TL;DR: Simulation results prove that proposed congestion control algorithm based on the multi-objective optimization algorithm named PSOGSA for rate optimization and regulating arrival rate of data from every child node to the parent node has better results than existing approaches.

Journal ArticleDOI
TL;DR: Carbon tax policies are introduced to analyze the impact of carbon tax on the total costs and carbon emissions, which proves that carbon tax policy can effectively reduce carbon dioxide emissions in cold chain logistics network.
Abstract: In order to solve the optimization problem of logistics distribution system for fresh food, this paper provides a low-carbon and environmental protection point of view, based on the characteristics of perishable products, and combines with the overall optimization idea of cold chain logistics distribution network, where the green and low-carbon location-routing problem (LRP) model in cold chain logistics is developed with the minimum total costs as the objective function, which includes carbon emission costs. A hybrid genetic algorithm with heuristic rules is designed to solve the model, and an example is used to verify the effectiveness of the algorithm. Furthermore, the simulation results obtained by a practical numerical example show the applicability of the model while provide green and environmentally friendly location-distribution schemes for the cold chain logistics enterprise. Finally, carbon tax policies are introduced to analyze the impact of carbon tax on the total costs and carbon emissions, which proves that carbon tax policy can effectively reduce carbon dioxide emissions in cold chain logistics network.

Journal ArticleDOI
TL;DR: This work proposes a hybrid of two methods which has the advantage of providing a good learning from fewer examples and a fair selection of features from a really large set, all these while ensuring a high standard classification accuracy of the data.
Abstract: Selection of a representative set of features is still a crucial and challenging problem in machine learning. The complexity of the problem increases when any of the following situations occur: a very large number of attributes (large dimensionality); a very small number of instances or time points (small-instance set). The first situation poses problems for machine learning algorithm as the search space for selecting a combination of relevant features becomes impossible to explore in a reasonable time and with reasonable computational resources. The second aspect poses the problem of having insufficient data to learn from (insufficient examples). In this work, we approach both these issues at the same time. The methods we proposed are heuristics inspired by nature (in particular, by biology). We propose a hybrid of two methods which has the advantage of providing a good learning from fewer examples and a fair selection of features from a really large set, all these while ensuring a high standard classification accuracy of the data. The methods used are antlion optimization (ALO), grey wolf optimization (GWO), and a combination of the two (ALO-GWO). We test their performance on datasets having almost 50,000 features and less than 200 instances. The results look promising while compared with other methods such as genetic algorithms (GA) and particle swarm optimization (PSO).

Journal ArticleDOI
TL;DR: A new hybrid algorithm ABC-TRR is proposed to improve the parameter extraction of PV models that combines the global exploration capability of the ABC and the local exploitation of the TRR, which achieves a good tradeoff among accuracy, convergence and reliability.

Journal ArticleDOI
TL;DR: The proposed hybrid genetic algorithm (HGA) to solve the QoS-aware cloud service composition problem outperforms the simple genetic algorithm, simple fruit fly optimization algorithm, and another recently proposed algorithm (DGABC) in terms of optimality, computation time, convergence speed and feasibility rate.
Abstract: This paper addresses the QoS-aware cloud service composition problem, which is known as a NP-hard problem, and proposes a hybrid genetic algorithm (HGA) to solve it. The proposed algorithm combines two phases to perform the evolutionary process search, including genetic algorithm phase and fruit fly optimization phase. In genetic algorithm phase, a novel roulette wheel selection operator is proposed to enhance the efficiency and the exploration search. To reduce the computation time and to maintain a balance between the exploration and exploitation abilities of the proposed HGA, the fruit fly optimization phase is incorporated as a local search strategy. In order to speed-up the convergence of the proposed algorithm, the initial population of HGA is created on the basis of a heuristic local selection method, and the elitism strategy is applied in each generation to prevent the loss of the best solutions during the evolutionary process. The parameter settings of our HGA were tuned and calibrated using the taguchi method of design of experiment, and we suggested the optimal values of these parameters. The experimental results show that the proposed algorithm outperforms the simple genetic algorithm, simple fruit fly optimization algorithm, and another recently proposed algorithm (DGABC) in terms of optimality, computation time, convergence speed and feasibility rate.


Journal ArticleDOI
TL;DR: An integrated berth allocation and quay crane assignment and scheduling problem motivated by a real case where a heterogeneous set of cranes is considered and a new model is introduced to avoid the big-M constraints included in the RPF.

Journal ArticleDOI
TL;DR: In this article, a model predictive control (MPC) strategy was proposed to maximize the photovoltaic utilization and minimize battery degradation in a local residential area, while satisfying all grid constraints.
Abstract: In this paper, we present a novel methodology for leveraging Receding Horizon Control, also known as Model Predictive Control (MPC) strategies for distributed battery storage in a planning problem using a Benders decomposition technique. Longer prediction horizons lead to better storage placement strategies but also higher computational complexity that can quickly become computationally prohibitive. The MPC strategy proposed here in conjunction with a Benders decomposition technique effectively reduces the computational complexity to a manageable level. We use the CIGRE low voltage benchmark grid as a case study for solving an optimal placement and sizing problem for different control strategies with different MPC prediction horizons. The objective of the MPC strategy is to maximize the photovoltaic utilization and minimize battery degradation in a local residential area, while satisfying all grid constraints. For this case study, we show that the economic value of battery storage is higher when using MPC-based storage control strategies than when using heuristic storage control strategies, because MPC strategies explicitly exploit the value of forecast information. The economic merit of this approach can be further increased by explicitly incorporating a battery degradation model in the MPC strategy.

Journal ArticleDOI
TL;DR: A meta-heuristic based algorithm for workflow scheduling that considers minimization of makespan and cost and introduces a new factor called cost time equivalence to make the bi-objective optimization more realistic.

Journal ArticleDOI
01 Sep 2018
TL;DR: The main optimization methods used in laminated composite structure design, including the gradient-based methods, heuristic methods, and hybrid methods, are presented in this article, where the advantages and shortcomings of each method are discussed in detail.
Abstract: The advantages of laminated composite structures make them attractive in industrial applications. To achieve the maximum advantages of the composite structures, the design optimization is of great importance. This paper classifies and compares various optimization problems and methods in laminated composite structure design. Three kinds of problems are illustrated in this paper: constant stiffness design, variable stiffness design, and topology optimization. The main optimization methods used in laminated composite structure design, including the gradient-based methods, heuristic methods, and hybrid methods, are presented. The advantages and shortcomings of each method are discussed in detail. Finally, constant and variable stiffness design and topology optimization of laminated composite structures which have been solved in literature are reviewed.

Journal ArticleDOI
Yifan Gu1, He Chen1, Yonghui Li1, Ying-Chang Liang1, Branka Vucetic1 
01 Mar 2018
TL;DR: It is shown that the proposed heuristic approach can achieve a near-optimal system performance and the ETMRS scheme outperforms the existing single-relay selection scheme and common energy threshold scheme.
Abstract: This paper investigates a wireless-powered cooperative network (WPCN) consisting of one source-destination pair and multiple decode-and-forward relays. We develop an energy threshold based multi-relay selection (ETMRS) scheme for the considered WPCN. The proposed ETMRS scheme can be implemented in a fully distributed manner as the relays only need local information to switch between energy harvesting and information forwarding modes. By modeling the charging/discharging behaviors of the finite-capacity battery at each relay as a finite-state Markov chain, we derive an analytical expression for the system outage probability of the proposed ETMRS scheme over mixed Nakagami- ${m}$ and Rayleigh fading channels. Based on the derived expression, the optimal energy thresholds for all the relays corresponding to the minimum system outage probability can be obtained via an exhaustive search. However, this approach becomes computationally prohibitive when the number of relays and the associated number of battery energy levels are large. To resolve this issue, we propose a heuristic approach to optimize the energy threshold for each relay. To gain some useful insights for practical relay design, we also derive the upper bound for system outage probability corresponding to the case that all relays are equipped with infinite-capacity batteries. Numerical results validate our theoretical analysis. It is shown that the proposed heuristic approach can achieve a near-optimal system performance and our ETMRS scheme outperforms the existing single-relay selection scheme and common energy threshold scheme.