Multi-objective feasibility enhanced particle swarm optimization

This article introduces a new method entitled multi-objective feasibility enhanced partical swarm optimization (MOFEPSO), to handle highly-constrained multi-objective optimization problems. MOFEPSO, which is based on the particle swarm optimization technique, employs repositories of non-dominated and feasible positions (or solutions) to guide feasible particle flight. Unlike its counterparts, MOFEPSO does not require any feasible solutions in the initialized swarm. Additionally, objective functions are not assessed for infeasible particles. Such particles can only fly along sensitive directions, and particles are not allowed to move to a position where any previously satisfied constraints become violated. These unique features help MOFEPSO gradually increase the overall feasibility of the swarm and to finally attain the optimal solution. In this study, multi-objective versions of a classical gear-train optimization problem are also described. For the given problems, the article comparatively evaluates the performance of MOFEPSO against several popular optimization algorithms found in the literature.     

Non-dominated archiving multi-colony ant algorithm for multi-objective optimization: Application to multi-purpose reservoir operation

Multi-objective optimization using heuristic methods has been established as a subdiscipline that combines the fields of heuristic computation and classical multiple criteria decision making. This article presents the Non-dominated Archiving Ant Colony Optimization (NA-ACO), which benefits from the concept of a multi-colony ant algorithm and incorporates a new information-exchange policy. In the proposed information-exchange policy, after a given number of iterations, different colonies exchange information on the assigned objective, resulting in a set of non-dominated solutions. The non-dominated solutions are moved into an offline archive for further pheromone updating. Performance of the NA-ACO is tested employing two well-known mathematical multi-objective benchmark problems. The results are promising and compare well with those of well-known NSGA-II algorithms used in real-world multi-objective-optimization problems. In addition, the optimization of reservoir operating policy with multiple objectives (i.e. flood control, hydropower generation and irrigation water supply) is considered and the associated Pareto front generated.     

Multi-objective scheduling problems: Determination of pruned Pareto sets

There are often multiple competing objectives for industrial scheduling and production planning problems. Two practical methods are presented to efficiently identify promising solutions from among a Pareto optimal set for multi-objective scheduling problems. Generally, multi-objective optimization problems can be solved by combining the objectives into a single objective using equivalent cost conversions, utility theory, etc., or by determination of a Pareto optimal set. Pareto optimal sets or representative subsets can be found by using a multi-objective genetic algorithm or by other means. Then, in practice, the decision maker ultimately has to select one solution from this set for system implementation. However, the Pareto optimal set is often large and cumbersome, making the post-Pareto analysis phase potentially difficult, especially as the number of objectives increase. Our research involves the post Pareto analysis phase, and two methods are presented to filter the Pareto optimal set to determine a subset of promising or desirable solutions. The first method is pruning using non-numerical objective function ranking preferences. The second approach involves pruning by using data clustering. The k-means algorithm is used to find clusters of similar solutions in the Pareto optimal set. The clustered data allows the decision maker to have just k general solutions from which to choose. These methods are general, and they are demonstrated using two multi-objective problems involving the scheduling of the bottleneck operation of a printed wiring board manufacturing line and a more general scheduling problem.     

A novel evolutionary meta-heuristic for the multi-objective optimization of real-world water distribution networks

Genetic algorithms are currently one of the state-of-the-art meta-heuristic techniques for the optimization of large engineering systems such as the design and rehabilitation of water distribution networks. They are capable of finding near-optimal cost solutions to these problems given certain cost and hydraulic parameters. Recently, multi-objective genetic algorithms have become prevalent in the water industry due to the conflicting nature of these hydraulic and cost objectives. The Pareto-front of solutions can aid decision makers in the water industry as it provides a set of design solutions which can be examined by experienced engineers. However, multi-objective genetic algorithms tend to require a large number of objective function evaluations to arrive at an acceptable Pareto-front. This article investigates a novel hybrid cellular automaton and genetic approach to multi-objective optimization (known as CAMOGA). The proposed method is applied to two large, real-world networks taken from the UK water industry. The results show that the proposed cellular automaton approach can provide a good approximation of the Pareto-front with very few network simulations, and that CAMOGA outperforms the standard multi-objective genetic algorithm in terms of efficiency in discovering similar Pareto-fronts.     

平流层飞艇的多目标优化与决策

该文建立了以平流层飞艇阻力最小、自重最轻、极限承载力最大及刚度最大为优化目标的多目标优化模型;采用强度Pareto进化算法(SPEA)进行了多目标优化设计;基于优化所得的Pareto解集,采用基于信噪比的决策方法选择满足实际需要的最终方案。结果表明:采用的SPEA算法是合理有效的,可以得到非劣解分布较均匀的Pareto曲面;通过基于信噪比的决策方法,可从非劣解集中获得满足实际要求的最稳健设计方案。     

A particle swarm optimizer for constrained multi-objective engineering design problems

This article presents a particle swarm optimizer (PSO) capable of handling constrained multi-objective optimization problems. The latter occur frequently in engineering design, especially when cost and performance are simultaneously optimized. The proposed algorithm combines the swarm intelligence fundamentals with elements from bio-inspired algorithms. A distinctive feature of the algorithm is the utilization of an arithmetic recombination operator, which allows interaction between non-dominated particles. Furthermore, there is no utilization of an external archive to store optimal solutions. The PSO algorithm is applied to multi-objective optimization benchmark problems and also to constrained multi-objective engineering design problems. The algorithmic effectiveness is demonstrated through comparisons of the PSO results with those obtained from other evolutionary optimization algorithms. The proposed particle swarm optimizer was able to perform in a very satisfactory manner in problems with multiple constraints and/or high dimensionality. Promising results were also obtained for a multi-objective engineering design problem with mixed variables.     

MONSS: A multi-objective nonlinear simplex search approach

This article presents a novel methodology for dealing with continuous box-constrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple elements of the Pareto optimal set. The search is directed by a well-distributed set of weight vectors, each of which defines a scalarization problem that is solved by deforming a simplex according to the movements described by Nelder and Mead's method. Considering an MOP with n decision variables, the simplex is constructed using n+1 solutions which minimize different scalarization problems defined by n+1 neighbor weight vectors. All solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a proper representation of the Pareto optimal front. In this article, it is shown that a well-designed strategy using just mathematical programming techniques can be competitive with respect to the state-of-the-art multi-objective evolutionary algorithms against which it was compared.     

An improved imperialist competitive algorithm for multi-objective optimization

This article proposes an improved imperialistic competitive algorithm to solve multi-objective optimization problems. The proposed multi-objective imperialistic competitive algorithm (MOICA) uses the elitist strategy, based on the mutation and crossover as in genetic algorithms, and the Pareto concept to store simultaneously optimal solutions of multiple conflicting functions. Three performance metrics are used to evaluate the performance of the new algorithm: convergence to the true Pareto-optimal set, solution diversity and robustness, characterized by the variance over 10 runs. To validate the efficiency of the proposed algorithm, several multi-objective standard test functions with true solutions are used. The obtained results show that the MOICA outperforms most of the methods available in the literature. The proposed algorithm can also handle multi-objective engineering design problems with high dimensions.     

An enhanced preference-inspired co-evolutionary algorithm using orthogonal design and an ε-dominance archiving strategy

The concept of co-evolution of preferences and candidate solutions has proven effective for many-objective optimization. One realization of this concept, namely preference-inspired co-evolutionary algorithms using goal vectors (PICEA-g), is found to outperform many state-of-the-art multi-objective evolutionary algorithms for many-objective problems. However, PICEA-g is susceptible to unevenness in the solution distribution. This study seeks to tackle this issue and to improve the performance of PICEA-g further. Two established strategies are incorporated into PICEA-g: (i) an adaptive ε-dominance archiving strategy which is applied to obtain a set of well spread solutions online; and (ii) the orthogonal design method which is used to initialize candidate solutions. The improved algorithm, denoted as aε-ODPICEA-g, shows a better performance than PICEA-g on both 2- and 7-objective benchmark problems as well as a real-world problem.     

Effective multi-objective optimization with the coral reefs optimization algorithm

In this article a new algorithm for multi-objective optimization is presented, the Multi-Objective Coral Reefs Optimization (MO-CRO) algorithm. The algorithm is based on the simulation of processes in coral reefs, such as corals' reproduction and fight for space in the reef. The adaptation to multi-objective problems is a process based on domination or non-domination during the process of fight for space in the reef. The final MO-CRO is an easily-implemented and fast algorithm, simple and robust, since it is able to keep diversity in the population of corals (solutions) in a natural way. The experimental evaluation of this new approach for multi-objective optimization problems is carried out on different multi-objective benchmark problems, where the MO-CRO has shown excellent performance in cases with limited computational resources, and in a real-world problem of wind speed prediction, where the MO-CRO algorithm is used to find the best set of features to predict the wind speed, taking into account two objective functions related to the performance of the prediction and the computation time of the regressor.     

A multi-objective approach for multi-material topology and shape optimization

An automated multi-material approach that integrates multi-objective Topology Optimization (TO) and multi-objective shape optimization is presented. A new ant colony optimization algorithm is presented and applied to solving the TO problem, estimating a trade-off set of initial topologies or distributions of material. The solutions found usually present irregular boundaries, which are not desirable in applications. Thus, shape parameterization of the internal boundaries of the design region, and subsequent shape optimization, is performed to improve the quality of the estimated Pareto-optimal solutions. The selection of solutions for shape optimization is done by using the PROMETHEE II decision-making method. The parameterization process involves identifying the boundaries of different materials and describing these boundaries by non-uniform rational B-spline curves. The proposed approach is applied to the optimization of a C-core magnetic actuator, with two objectives: the maximization of the attractive force on the armature and the minimization of the volume of permanent magnet material.     

Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.     

A new multi-objective particle swarm optimizer using empirical movement and diversified search strategies

Most real-world optimization problems involve the optimization task of more than a single objective function and, therefore, require a great amount of computational effort as the solution procedure is designed to anchor multiple compromised optimal solutions. Abundant multi-objective evolutionary algorithms (MOEAs) for multi-objective optimization have appeared in the literature over the past two decades. In this article, a new proposal by means of particle swarm optimization is addressed for solving multi-objective optimization problems. The proposed algorithm is constructed based on the concept of Pareto dominance, taking both the diversified search and empirical movement strategies into account. The proposed particle swarm MOEA with these two strategies is thus dubbed the empirical-movement diversified-search multi-objective particle swarm optimizer (EMDS-MOPSO). Its performance is assessed in terms of a suite of standard benchmark functions taken from the literature and compared to other four state-of-the-art MOEAs. The computational results demonstrate that the proposed algorithm shows great promise in solving multi-objective optimization problems.     

A Hopfield Neural Network Approach to the Dual Response Problem

The application of neural networks to optimization problems has been an active research area since the early 1980s. Unconstrained optimization, constrained optimization and combinatorial optimization problems have been solved using neural networks. This study presents a new approach using Hopfield neural networks (HNNs) for solving the dual response system (DRS) problems. The major aim of the proposed method is to produce a string of solutions, rather than a ‘one‐shot’ optimum solution, to make the trade‐offs available between the mean and standard deviation responses. This gives more flexibility to the decision‐maker in exploring alternative solutions. The proposed method has been tested on two examples. The HNN results are very close to those obtained by using the NIMBUS (Nondifferentiable Interactive Multiobjective Bundle‐based Optimization System) algorithm. Choosing an appropriate solution method for a certain multi‐objective optimization problem is not easy, as has been made abundantly clear. Unlike the NIMBUS method, the HNN approach does not set any specific assumptions on the behaviour or the preference structure of the decision maker. As a result, the proposed method will still work and generate alternative solutions whether or not the decision maker has enough time and capabilities for co‐operation. Copyright © 2005 John Wiley & Sons, Ltd.     

MOEA/D-SQA: a multi-objective memetic algorithm based on decomposition

A multi-objective memetic algorithm based on decomposition is proposed in this article, in which a simplified quadratic approximation (SQA) is employed as a local search operator for enhancing the performance of a multi-objective evolutionary algorithm based on decomposition (MOEA/D). The SQA is used for a fast local search and the MOEA/D is used as the global optimizer. The multi-objective memetic algorithm based on decomposition, i.e. a hybrid of the MOEA/D with the SQA (MOEA/D-SQA), is designed to balance local versus global search strategies so as to obtain a set of diverse non-dominated solutions as quickly as possible. The emphasis of this article is placed on demonstrating how this local search scheme can improve the performance of MOEA/D for multi-objective optimization. MOEA/D-SQA has been tested on a wide set of benchmark problems with complicated Pareto set shapes. Experimental results indicate that the proposed approach performs better than MOEA/D. In addition, the results obtained are very competitive when comparing MOEA/D-SQA with other state-of-the-art techniques.     

基于分组策略的多目标三维装箱算法

目的 针对现有三维装箱算法优化目标单一、优化效率低的问题,提出适用于求解大规模货物装载问题的多目标装箱算法,以提高装箱规划效率,确保货物运输安全。方法 考虑5种现实约束条件,以体积利用率和装载垛型重心偏移量为优化目标,建立多目标货物装载优化模型。采用拟人式装箱对货物进行预分组,减小决策空间,然后结合分组信息与装箱算法生成初始解;引入数据驱动的装箱交叉算子提高算法收敛性;设计多策略变异算子提高算法结果的多样性。结果 以公共数据集和真实航空货物数据作为实验数据进行实验。实验结果表明,在满足多种约束条件下,集装箱装载强异构货物平均体积利用率达到92.0%,重心位置空间偏移从20 cm减少到7.5 cm,并且算法运行时间减少了73.5%。结论 本文所提算法应用于求解大规模多目标三维装箱问题,提高了装箱质量和效率,可为三维装箱算法的工程应用提供参考。     

Pareto Tracer: a predictor–corrector method for multi-objective optimization problems

This article proposes a novel predictor–corrector (PC) method for the numerical treatment of multi-objective optimization problems (MOPs). The algorithm, Pareto Tracer (PT), is capable of performing a continuation along the set of (local) solutions of a given MOP with k objectives, and can cope with equality and box constraints. Additionally, the first steps towards a method that manages general inequality constraints are also introduced. The properties of PT are first discussed theoretically and later numerically on several examples.     

Multi-objective robust design optimization of fatigue life for a welded box girder

To reduce the scatter of fatigue life for welded structures, a robust optimization method is presented in this study based on a dual surrogate modelling and multi-objective particle swam optimization algorithm. Considering the perturbations of material parameters and environment variables, the mean and standard deviation of fatigue life are fitted using dual surrogate modelling and selected as the objective function to be minimized. As an example, a welded box girder is presented to reduce the standard deviation of fatigue life. A set of non-dominated solutions is produced through a multi-objective particle swam optimization algorithm. A cognitive approach is used to select the optimum solution from the Pareto sets. As a comparative study, traditional single objective optimizations are also presented in this study. The results reduced the standard deviation of the fatigue life by about 16.5%, which indicated that the procedure improved the robustness of the fatigue life.     

Stability analysis and regularization of uncertain linear multi-objective integer optimization problems

This article presents a systematic approach to analysing linear integer multi-objective optimization problems with uncertainty in the input data. The goal of this approach is to provide decision makers with meaningful information to facilitate the selection of a solution that meets performance expectations and is robust to input parameter uncertainty. Standard regularization techniques often deal with global stability concepts. The concept presented here is based on local quasi-stability and includes a local regularization technique that may be used to increase the robustness of any given efficient solution or to transform efficient solutions that are not robust (i.e. unstable), into robust solutions. An application to a multi-objective problem drawn from the mining industry is also presented.