Multiple heterogeneous agents are popular for executing pickup and delivery tasks for multiple pairs of customers. The scheduling solutions of agents are expected to complete each task within time windows, even under disturbances. Existing problem models tend to evaluate solutions through multiple simulations based on disturbances. This is time-consuming and implicit. In contrast, this article defines a robustness optimization objective based on the relationship between the agent's arrival time and the time windows for explicit evaluation. Taking robustness together with makespan and cost, the problem is modeled as a triobjective optimization problem. To solve the problem, this article proposes matrix-learning particle swarm optimization (MLPSO) to obtain diversified and high-quality solutions for decision-makers. In MLPSO, solutions are represented as an adjacency matrix of task sequences and an allocation matrix of agents to tasks. Corresponding to the matrix-based representation, solutions are constructed by planning the task order for execution and assigning agents to tasks. A matrix-distance-based learning (MDL) strategy is developed to select neighbors in the decision space for particle update. In this way, good task segments and allocation pairs can be extracted from learning exemplars and current positions to provide stable updating directions for generating high-quality solutions. To further enhance solution convergence and diversity, a dual-space local search (DSLS) is performed on elite and sparse nondominated solutions. Experimental results on 36 instances with various scales show that the proposed MLPSO is significantly better than state-of-the-art algorithms in terms of solution quality and diversity.

Set meal design (SMD) for online-to-offline (O2O) restaurant services presents a complex optimization problem, requiring the simultaneous satisfaction of diverse customer preferences, operational constraints, and profit maximization objective. To address this challenge, this article proposes a comprehensive mathematical formulation for the O2O-SMD problem. This formulation integrates complex operational requirements, such as dish variety, pricing, nutritional balance, and profitability, into a unified optimization problem with well-defined objective and constraints. To efficiently solve the O2O-SMD problem, we propose a tensor-based ant colony optimization (TACO) algorithm. Distinct from traditional ant colony optimization (ACO) variants, the core of TACO lies in reformulating the fundamental ACO operations into a tensor computational structure, enabling parallel optimization over O2O-SMD tasks at the algorithmic level. Furthermore, a dedicated local search strategy is integrated to refine solutions and accelerate convergence of the algorithm. The performance of TACO is evaluated on real-world restaurant data and benchmark instances. The experimental results show that TACO significantly outperforms a wide range of comparison algorithms in terms of solution quality, scalability, and computational efficiency, confirming its effectiveness and practical value for real-world O2O-SMD problems.

In dynamic environments, optimization algorithms tend to track the changing optima. However, frequent changes to decision solutions often cause high switching costs and system instability. Decision solutions are expected to be robust to changes. Such problems bring new challenges to existing algorithms, i.e., solution evaluation in future environments and solution selection. Though multiple surrogate models are developed for fitness evaluation, they mainly focus on predicting absolute fitness in static environments. Thus, this paper proposes conditional-surrogate-assisted particle swarm optimization (CSPSO), which adopts a conditional surrogate model to predict the relative fitness of solutions in future for robust solution selection. In CSPSO, a problem is decomposed into multiple subproblems, which are optimized by multiple swarms. Searching data of swarms is collected to learn network models for predicting the future fitness of solutions. In order to coordinate with the multi-modal, decoupled, and dynamic property of the problem, networks are conditioned on some extra information, i.e., time, subproblems, and peaks. Particularly, neural networks are first trained to predict new optimal solutions and fitness conditioned on time and subproblems, and a surrogate model is then trained to predict the relative fitness of solutions conditioned on the closest peak, corresponding subproblem, and time. By combining the current fitness with future ones, solutions with the highest fitness are selected for decision making. Experimental results show that the proposed algorithm outperforms state-of-the-art algorithms on problem instances up to 1000-D in terms of solution optimality. The proposed conditional surrogate model can well predict future fitness to assist decision making.

Unmanned aerial vehicles (UAVs) are increasingly used for firefighting due to security. Multiple UAVs depart from a site and cooperate to execute tasks with time-varying demands in different locations. To better execute firefighting tasks, the reasonable scheduling of UAVs is crucial. Although multiple methods have been developed to solve the multi-UAV scheduling problem, they mainly focus on instances with sufficient UAV resources. They face challenges on instances with limited resources and multiple objectives in terms of solution diversity and convergence, especially when the number of tasks is larger than that of UAVs. This paper models the problem as a bi-objective optimization problem, aiming to minimize two important objectives: the makespan and the total travel distance of UAVs. A dual-encoding-based genetic algorithm (DEGA) is developed to solve the problem. In DEGA, a dual-encoding scheme is adopted for solution representation, in which the execution order of all tasks is represented as a sequence and the task assignment for UAVs is represented as a binary matrix. Correspondingly, solutions can be constructed in two steps: task sequence generation first and then task assignment. Crossover and mutation operations are specifically designed to explore the search space for enhancing solution diversity. In addition, a local search is employed to improve solution quality. Experimental results on instances with various scales demonstrate that DEGA outperforms state-of-the-art algorithms on most instances in terms of solution optimality and diversity.

Although knowledge transfer methods are developed for many-task optimization problems, they tend to utilize solutions from a single task for knowledge transfer. Indeed, there are usually multiple relevant source tasks with commonality. Multisource data fusion can capture complementary knowledge of distinct source tasks to better assist the optimization of target tasks. However, biases potentially flow with the interaction between tasks during multisource fusion, resulting in performance degeneration. Thus, how to select multiple relevant source tasks and perform multisource knowledge transfer is challenging. To address these issues, this article proposes a multisource knowledge fusion (MKF) method based on graph attention networks. In MKF, tasks are structured using a relational graph, in which each vertex represents a task and each directed edge from vertex u to v represents that u is a source task of v. Particularly, for each task, multiple source tasks are selected based on the distribution similarity and evolutionary performance. In the graph, local message is passed from source tasks to target tasks using graph attention networks, which automatically learn the adjacency weight of each directed edge and aggregate solutions from multiple source tasks to obtain fused representations for target tasks. These fused representations are adopted to generate new solutions through mutation. In this way, multisource knowledge is fused and transferred according to their importance to the target task. Integrating MKF into differential evolution, a new algorithm named MKF-DE is put forward. Experimental results on GECCO2020MaTOP and CEC2022MaTOP show that MKF-DE outperforms state-of-the-art algorithms on most instances.

In many real-world scenarios (e.g., product manufacturing), multiple heterogeneous robots cooperate to complete complex tasks with precedence constraints. In these heterogeneous multirobot systems, the multirobot task allocation problem is important and has attracted increasing attention. The problem usually involves multiple optimization objectives for decision making. However, existing approaches meet challenges on multi-objective problems with large-scale tasks and precedence constraints in terms of solution diversity and convergence. Therefore, this paper formulates a tri-objective model and proposes a cooperative ant colony system (CACS) to optimize three objectives, i.e., minimizing the makespan, average robot traveling time, and average task waiting time. In CACS, three ant colonies are created to simultaneously optimize the three objectives. To coordinate with the precedence constraints of the problem, solutions are encoded as a task-alliance sequence. A new solution construction method is developed to generate feasible solutions using dynamic heuristic information and two pheromone matrices. Particularly, one matrix deposits pheromone between tasks for task selection and the other between tasks and robots for alliance building. To further improve solution diversity and convergence, a fusion-based local search is adopted to generate high-quality solutions by combining information from multiple colonies. Thirty instances are constructed with different numbers of tasks and robots under complex precedence constrains. Experimental results show that CACS outperforms state-of-the-art methods in terms of the inverted generational distance and hypervolume metrics.

In a capacitated vehicle routing problem (CVRP), multiple vehicles are planned to travel for serving customers so as to reduce transportation costs in logistics. Taking each CVRP as a task, multiple CVRPs can form a multi-task optimization problem. Based on the similarity between tasks, knowledge transfer methods are developed to improve optimization performance by utilizing the search experience of related tasks. However, existing methods tend to use one task only for knowledge transfer. Indeed, in a target task, the different parts of customer distributions are similar to that of multiple related tasks. The information of multiple source tasks can be fused to assist the optimization of target tasks. Thus, this paper proposes a genetic algorithm with fragment-based knowledge transfer (FKT-GA), which fuses route fragments from multiple related tasks to assist the optimization of target tasks. In FKT-GA, multiple source tasks are selected for each target task based on distribution features that are invariant to rotation, shift, and scaling. Solutions of source tasks are aligned to target spaces for sampling route fragments, which are integrated to construct high-quality solutions for target tasks. In addition, mutation and crossover operators are developed to enhance solution diversity. Experimental results on fifteen 6-task instances show that FKT-GA outperforms state-of-the-art algorithms in terms of solution optimality. The proposed FKT can improve algorithm performance.

In dynamic time-linkage optimization, current decisions influence the future state of environments. To make good decisions that have a positive impact on future states, existing methods usually build a model to predict the future rewards of solutions for decision making. However, these prediction models present low accuracy since decision data are not enough to train such a complex model. To address this issue, this article proposes a contrastive-learning-based decision making (CLDM) method, which builds a contrastive model to learn the relationship between solutions but not absolute rewards and adopts a quick decision strategy to select solutions. In CLDM, a clustering-based time-linkage detection (CD) strategy is developed to measure the intensity of the time linkage, which determines whether to make decisions based on future rewards. To represent the relative relationship between solutions, a large number of contrastive samples are constructed using the limited historical decisions. A contrastive model is trained for solution comparison in terms of the combination of current fitness and future rewards. Candidate solutions are clustered into multiple groups to filter poor ones, and a few solutions are preserved to rank using the contrastive model. The winner is taken as the decision solution. Integrating CLDM into particle swarm optimization (PSO), a new algorithm named contrastive-learning-based PSO (CL-PSO) is put forward. Experimental results on multiple dynamic time-linkage optimization instances demonstrate that CL-PSO outperforms state-of-the-art algorithms in terms of solution quality. CL-PSO can also well solve the mobile robot path planning problem.

Cooperative coevolutionary algorithms are popular to solve large-scale dynamic optimization problems via divide-and-conquer mechanisms. Their performance depends on how decision variables are grouped and how changing optima are tracked. However, existing decomposition methods are computationally expensive, resulting in limitations under dynamic variable interactions. Quick online decomposition is still a challenging issue, along with solution reconstruction for new subproblems. This paper proposes transfer-based particle swarm optimization, which adopts a dynamic differential grouping for online decomposition and a solution transfer strategy in response to environmental changes. Particularly, once an environmental change occurs, the dynamic differential grouping readjusts historical groupings based on the change severity of variable interactions. In addition, according to the similarity between subproblems in successive environments, the solution transfer strategy constructs new solutions from historical ones through dimension mapping. Multiple swarms are created to explore subareas of subproblems. Experimental results show that the proposed algorithm outperforms state-of-the-art algorithms on problem instances up to 1000-D in terms of solution optimality. The dynamic differential grouping obtains accurate groupings using less function evaluations.

Surrogate-assisted evolutionary algorithms (SAE-As), mainly including regression-based SAEAs and classification-based SAEAs, are promising for solving expensive multi-objective optimization problems (EMOPs). Regression-based SAEAs usually use complex regression models to approximate the fitness evaluation, which will suffer from high training costs to obtain a fine-accuracy surrogate. In contrast, classification-based SAEAs can achieve solution selection via coarse binary relations predicted by classifiers, thus avoiding high requirements in prediction accuracy and training costs. However, most of the binary relations in existing classification-based SAEAs mainly only involve convergence comparison whereas diversity maintenance is neglected. Considering the capacity of the grid technique in maintaining both convergence and diversity, we propose a new classification method called grid classification to discretize the objective space into grids and train a lightweight grid classification-based surrogate (GCS), for which low training costs are needed. The GCS can evaluate the solution performance in terms of both convergence and diversity simultaneously according to the predicted grid locations, which opens up a new field for follow-up research on classification-based SAEAs. Following this, a GCS-assisted particle swarm optimization algorithm is proposed for tackling EMOPs. Experimental results on widely-used benchmark problems (including high-dimensional EMOPs) and a 222-high-dimensional real-world application problem show its competitiveness in terms of both optimization performance and computational cost.

Multiple autonomous underwater vehicles (AUVs) are popular for executing submarine missions, which involve multiple targets distributed in a large and complex underwater environment. The path planning of multiple AUVs is a significant and challenging problem, which determines the location of surface points for AUV launch and plans the paths of AUVs for target traveling. Most existing works model the problem as a single-objective static optimization problem. However, the target missions may change over time, and multiple optimization objectives are usually expected for decision making. Thus, this article models the problem as a dynamic multiobjective optimization problem and proposes a cooperative evolutionary computation algorithm to provide diverse and high-quality solutions for decision makers. In the proposed method, solutions are represented using a bilayer encode scheme, in which the first layer indicates the surface location points and the second layer represents the traveling sequences of target missions. Multiple populations for multiple objectives framework is adopted to efficiently solve the dynamic multiobjective AUV optimization problem. In addition, a recombination-based sampling strategy is developed to improve convergence by fusing the information of multiple populations. Once a change occurs, an incremental response strategy is adopted to generate high-quality solutions for population evolution. Based on the dataset of New Zealand bathymetry, six complex underwater scenarios are constructed with a size of 50 km × 50 km× 10 km and 400 target missions for tests. Experimental results show that the proposed method outperforms the state-of-the-art algorithms in terms of solution diversity and optimality.

Although cooperative coevolutionary algorithms are developed for large-scale dynamic optimization via subspace decomposition, they still face difficulties in reacting to environmental changes, in the presence of multiple peaks in the fitness functions and unevenness of subproblems. The resource allocation mechanisms among subproblems in the existing algorithms rely mainly on the fitness improvements already made but not potential ones. On the one hand, there is a lack of sufficient computing resources to achieve potential fitness improvements for some hard subproblems. On the other hand, the existing algorithms waste computing resources aiming to find most of the local optima of problems. In this article, we propose a cooperative particle swarm optimization algorithm to address these issues by introducing a bilevel balanceable resource allocation mechanism. A search strategy in the lower level is introduced to select some promising solutions from an archive based on solution diversity and quality to identify new peaks in every subproblem. A resource allocation strategy in the upper level is introduced to balance the coevolution of multiple subproblems by referring to their historical improvements and more computing resources are allocated for solving the subproblems that perform poorly but are expected to make great fitness improvements. Experimental results demonstrate that the proposed algorithm is competitive with the state-of-the-art algorithms in terms of objective function values and response efficiency with respect to environmental changes.

Dynamic multiobjective optimization poses great challenges to evolutionary algorithms due to the change of optimal solutions or Pareto front with time. Learning-based methods are popular to extract the changing pattern of optimal solutions for predicting new solutions. They tend to use all variables as features (i.e., inputs) to build prediction models. However, there are usually some irrelevant and redundant variables, which increase training difficulty and decrease prediction accuracy. This article proposes a new interaction-based prediction (IP) method, which captures the correlation of variables with prediction targets and selects the most relevant variables to build prediction models using neural networks. In particular, the interaction between variables is detected to remove redundant variables. In addition, a correction procedure is developed to further improve predicted solutions according to the prediction error in past environments. The predicted solutions are used to update the population according to a specifically designed update strategy. Integrating the IP method into the framework of multiobjective evolutionary algorithm based on decomposition (MOEA/D), a new algorithm named IP-DMOEA is put forward. Experimental results on a typical dynamic multiobjective test suite demonstrate the better performance of the proposed IP-DMOEA than state-of-the-art algorithms in terms of convergence speed and solution quality. The proposed IP-DMOEA is also successfully applied to the multirobot task scheduling problem.

Cooperative heterogeneous multirobot systems have attracted increasing attention in recent years. They use multiple heterogeneous robots to execute complex tasks in a coordinated way. The allocation of heterogeneous robots to cooperative tasks is a significant and challenging optimization problem. However, little work has gone into scheduling large-scale cooperative tasks with precedence constraints and multiple conflicting optimization objectives. Existing methods are insufficient to address the issue. We propose a multiobjective model and develop strength learning particle swarm optimization (SLPSO) to optimize multiple objectives. In this article, the problem is converted into a two-step problem of task permutation construction and robot subset selection. In order to coordinate with the time-extended property of the problem, SLPSO utilizes a hybrid encode scheme: an element-based representation for task permutations and a binary representation for robot coalitions. A strength learning strategy with heuristic information guides particles to enhance their best-performing objectives for improving swarm convergence. In addition, an estimation-based local search is developed to improve spare solutions for enhancing swarm diversity, which determines the search direction by estimating fitness improvements. Experimental results on thirty problem instances are elaborated to demonstrate that the proposed SLPSO significantly outperforms the state-of-the-art algorithms in terms of inverted generational distance and hypervolume metrics. The proposed SLPSO can obtain a set of high-quality and diversified solutions.

In dynamic multiobjective optimization, the Pareto front (PF) or Pareto set varies over time as the problem environment changes. In such scenarios, optimization algorithms are required to efficiently find and continuously track a set of Pareto-optimal and diverse solutions. However, existing algorithms often result in the imbalanced approximation of PFs since some objectives are usually harder to optimize than others. In addition, the prediction strategies of the existing algorithms usually entail additional parameters (e.g., reference points, weights, and clustering parameters) to match available Pareto-optimal solutions for prediction in dynamically changing environments. This article presents a cooperative differential evolution algorithm with an attention-based prediction strategy. Multiple populations are adopted to optimize multiple objectives in search of subparts of PFs. Every population adopts a new fusion-based mutation strategy for coevolution. In addition, an expanding procedure is proposed on archived solutions to further expand the objective space covered by the populations to the entire PF. Specifically, once an environment change is detected, the populations are updated by using a new attention-based prediction strategy according to the historical variation of the objective functions. In this way, every population is adapted to the change of its attentive objective in the dynamic environment. Experimental results on a recent test suite of scalable dynamic multiobjective optimization problems are elaborated to demonstrate the superiority of the proposed method to state-of-the-art algorithms.

The neural-network (NN)-based control method is a new emerging promising technique for controller design in a power electronic circuit (PEC). However, the optimization of NN-based controllers (NNCs) has significant challenges in two aspects. The first challenge is that the search space of the NNC optimization problem is such complex that the global optimization ability of the existing algorithms still needs to be improved. The second challenge is that the training process of the NNC parameters is very computationally expensive and requires a long execution time. Thus, in this article, we develop a powerful evolutionary computation-based algorithm to find a high-quality solution and reduce computational time. First, the differential evolution (DE) algorithm is adopted because it is a powerful global optimizer in solving a complex optimization problem. This can help to overcome the premature convergence in local optima to train the NNC parameters well. Second, to reduce the computational time, the DE is extended to distribute DE (DDE) by dispatching all the individuals to different distributed computing resources for parallel computing. Moreover, a resource-aware strategy (RAS) is designed to further efficiently utilize the resources by adaptively dispatching individuals to resources according to the real-time performance of the resources, which can simultaneously concern the computing ability and load state of each resource. Experimental results show that, compared with some other typical evolutionary algorithms, the proposed algorithm can get significantly better solutions within a shorter computational time.

In dynamic optimization problems (DOPs), as the environment changes through time, the optima also dynamically change. How to adapt to the dynamic environment and quickly find the optima in all environments is a challenging issue in solving DOPs. Usually, a new environment is strongly relevant to its previous environment. If we know how it changes from the previous environment to the new one, then we can transfer the information of the previous environment, e.g., past solutions, to get new promising information of the new environment, e.g., new high-quality solutions. Thus, in this paper, we propose a neural network (NN)-based information transfer method, named NNIT, to learn the transfer model of environment changes by NN and then use the learned model to reuse the past solutions. When the environment changes, NNIT first collects the solutions from both the previous environment and the new environment and then uses an NN to learn the transfer model from these solutions. After that, the NN is used to transfer the past solutions to new promising solutions for assisting the optimization in the new environment. The proposed NNIT can be incorporated into population-based evolutionary algorithms (EAs) to solve DOPs. Several typical state-of-the-art EAs for DOPs are selected for comprehensive study and evaluated using the widely used moving peaks benchmark. The experimental results show that the proposed NNIT is promising and can accelerate algorithm convergence.

The application of multiobjective evolutionary algorithms to many-objective optimization problems often faces challenges in terms of diversity and convergence. On the one hand, with a limited population size, it is difficult for an algorithm to cover different parts of the whole Pareto front (PF) in a large objective space. The algorithm tends to concentrate only on limited areas. On the other hand, as the number of objectives increases, solutions easily have poor values on some objectives, which can be regarded as poor bottleneck objectives that restrict solutions' convergence to the PF. Thus, we propose a coevolutionary particle swarm optimization with a bottleneck objective learning (BOL) strategy for many-objective optimization. In the proposed algorithm, multiple swarms coevolve in distributed fashion to maintain diversity for approximating different parts of the whole PF, and a novel BOL strategy is developed to improve convergence on all objectives. In addition, we develop a solution reproduction procedure with both an elitist learning strategy (ELS) and a juncture learning strategy (JLS) to improve the quality of archived solutions. The ELS helps the algorithm to jump out of local PFs, and the JLS helps to reach out to the missing areas of the PF that are easily missed by the swarms. The performance of the proposed algorithm is evaluated using two widely used test suites with different numbers of objectives. Experimental results show that the proposed algorithm compares favorably with six other state-of-the-art algorithms on many-objective optimization.

As the mutation strategy and algorithmic parameters in differential evolution (DE) are sensitive to the problems being solved, a hot research topic is to adaptively control the strategy and parameters according to the requirements of the problem. In the literature, most adaptive DE use either historical experiences of the population or heuristic information of the individuals to promote adaptation. In this paper, we develop a novel variant of adaptive DE, utilizing both the historical experience and heuristic information for the adaptation. In this novel historical and heuristic DE (HHDE), each individual dynamically adjusts its mutation strategy and associated parameters not only by learning from previous successful experience of the whole population, but also according to heuristic information related with its own current state. These help the algorithm select a more suitable mutation strategy and determinate better parameters for each individual in different evolutionary stages. The performance of the proposed HHDE is extensively evaluated on 30 benchmark functions with different dimensions. Experimental results confirm the competitiveness of the proposed algorithm to a number of DE variants.

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.

Existing differential evolution (DE) algorithms often face two challenges. The first is that the optimization performance is significantly affected by the ad hoc configurations of operators and parameters for different problems. The second is the long runtime for real-world problems whose fitness evaluations are often expensive. Aiming at solving these two problems, this paper develops a novel double-layered heterogeneous DE algorithm and realizes it in cloud computing distributed environment. In the first layer, different populations with various parameters and/or operators run concurrently and adaptively migrate to deliver robust solutions by making the best use of performance differences among multiple populations. In the second layer, a set of cloud virtual machines run in parallel to evaluate fitness of corresponding populations, reducing computational costs as offered by cloud. Experimental results on a set of benchmark problems with different search requirements and a case study with expensive design evaluations have shown that the proposed algorithm offers generally improved performance and reduced computational time, compared with not only conventional and a number of state-of-the-art DE variants, but also a number of other distributed DE and high-performing evolutionary algorithms. The speedup is significant especially on expensive problems, offering high potential in a broad range of real-world applications.

Differential evolution (DE) is an efficient and powerful population-based stochastic evolutionary algorithm, which evolves according to the differential between individuals. The success of DE in obtaining the optima of a specific problem depends greatly on the choice of mutation strategies and control parameter values. Good parameters lead the individuals towards optima successfully. The increasing of the success rate (the ratio of entering the next generation successfully) of population can speed up the searching. Adaptive DE incorporates success-history or population-state based parameter adaptation. However, sometimes poor parameters may improve individual with small probability and are regarded as successful parameters. The poor parameters may mislead the parameter control. So, in this paper, we propose a novel approach to distinguish between good and poor parameters in successful parameters. In order to speed up the convergence of algorithm and find more "good" parameters, we propose a dichotomy adaptive DE (DADE), in which the successful parameters are divided into two parts and only the part with higher success rate is used for parameter adaptation control. Simulation results show that DADE is competitive to other classic or adaptive DE algorithms on a set of benchmark problem and IEEE CEC 2014 test suite.

Cloud computing provides resources as services in pay-as-you-go mode to customers by using virtualization technology. As virtual machine (VM) is hosted on physical server, great energy is consumed by maintaining the servers in data center. More physical servers means more energy consumption and more money cost. Therefore, the VM placement (VMP) problem is significant in cloud computing. This paper proposes an approach based on ant colony optimization (ACO) to solve the VMP problem, named as ACO-VMP, so as to effectively use the physical resources and to reduce the number of running physical servers. The number of physical servers is the same as the number of the VMs at the beginning. Then the ACO approach tries to reduce the physical server one by one. We evaluate the performance of the proposed ACO-VMP approach in solving VMP with the number of VMs being up to 600. Experimental results compared with the ones obtained by the first-fit decreasing (FFD) algorithm show that ACO-VMP can solve VMP more efficiently to reduce the number of physical servers significantly, especially when the number of VMs is large.