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.     

Evolutionary computation of unconstrained and constrained problems using a novel momentum-type particle swarm optimization

This study proposes a novel momentum-type particle swarm optimization (PSO) method, which will find good solutions of unconstrained and constrained problems using a delta momentum rule to update the particle velocity. The algorithm modifies Shi and Eberhart's PSO to enhance the computational efficiency and solution accuracy. This study also presents a continuous non-stationary penalty function, to force design variables to satisfy all constrained functions. Several well-known and widely used benchmark problems were employed to compare the performance of the proposed PSO with Kennedy and Eberhart's PSO and Shi and Eberhart's modified PSO. Additionally, an engineering optimization task for designing a pressure vessel was applied to test the three PSO algorithms. The optimal solutions are presented and compared with the data from other works using different evolutionary algorithms. To show that the proposed momentum-type PSO algorithm is robust, its convergence rate, solution accuracy, mean absolute error, standard deviation, and CPU time were compared with those of both the other PSO algorithms. The experimental results reveal that the proposed momentum-type PSO algorithm can efficiently solve unconstrained and constrained engineering optimization problems.     

Multi-objective hierarchical genetic algorithms for multilevel redundancy allocation optimization

Multilevel redundancy allocation optimization problems (MRAOPs) occur frequently when attempting to maximize the system reliability of a hierarchical system, and almost all complex engineering systems are hierarchical. Despite their practical significance, limited research has been done concerning the solving of simple MRAOPs. These problems are not only NP hard but also involve hierarchical design variables. Genetic algorithms (GAs) have been applied in solving MRAOPs, since they are computationally efficient in solving such problems, unlike exact methods, but their applications has been confined to single-objective formulation of MRAOPs. This paper proposes a multi-objective formulation of MRAOPs and a methodology for solving such problems. In this methodology, a hierarchical GA framework for multi-objective optimization is proposed by introducing hierarchical genotype encoding for design variables. In addition, we implement the proposed approach by integrating the hierarchical genotype encoding scheme with two popular multi-objective genetic algorithms (MOGAs)—the strength Pareto evolutionary genetic algorithm (SPEA2) and the non-dominated sorting genetic algorithm (NSGA-II). In the provided numerical examples, the proposed multi-objective hierarchical approach is applied to solve two hierarchical MRAOPs, a 4- and a 3-level problems. The proposed method is compared with a single-objective optimization method that uses a hierarchical genetic algorithm (HGA), also applied to solve the 3- and 4-level problems. The results show that a multi-objective hierarchical GA (MOHGA) that includes elitism and mechanism for diversity preserving performed better than a single-objective GA that only uses elitism, when solving large-scale MRAOPs. Additionally, the experimental results show that the proposed method with NSGA-II outperformed the proposed method with SPEA2 in finding useful Pareto optimal solution sets.     

An uncertain multidisciplinary design optimization method using interval convex models

This article proposes an uncertain multi-objective multidisciplinary design optimization methodology, which employs the interval model to represent the uncertainties of uncertain-but-bounded parameters. The interval number programming method is applied to transform each uncertain objective function into two deterministic objective functions, and a satisfaction degree of intervals is used to convert both the uncertain inequality and equality constraints to deterministic inequality constraints. In doing so, an unconstrained deterministic optimization problem will be constructed in association with the penalty function method. The design will be finally formulated as a nested three-loop optimization, a class of highly challenging problems in the area of engineering design optimization. An advanced hierarchical optimization scheme is developed to solve the proposed optimization problem based on the multidisciplinary feasible strategy, which is a well-studied method able to reduce the dimensions of multidisciplinary design optimization problems by using the design variables as independent optimization variables. In the hierarchical optimization system, the non-dominated sorting genetic algorithm II, sequential quadratic programming method and Gauss–Seidel iterative approach are applied to the outer, middle and inner loops of the optimization problem, respectively. Typical numerical examples are used to demonstrate the effectiveness of the proposed methodology.     

Multi-objective optimization of a flexible slider-crank mechanism synthesis,based on dynamic responses

This work deals with a multi-body system synthesis. A flexible slider crank mechanism has been investigated as an illustrative application. The main interest is focused on the mechanism design variables’ identification based on its dynamic responses. Three responses have been involved such as the slider velocity, the slider acceleration and the mid-point transversal deflection of the flexible connecting rod. Each of these responses has been embroiled separately in a mono-objective optimization. Subsequently, the multi-objective optimization subsuming these responses has been established. Two different optimization methods have been studied namely the genetic algorithm (GA) and the particle swarm optimization (PSO) technique. It has been proved that the multi-objective optimization presents more accurate results beside the mono-objective optimization. Compared to the GA, the PSO is more powerful and is able to identify the mechanism design variable with better accuracy, in spite of the affordable computational time allowed with the GA optimization.     

An effective hybrid cuckoo search and genetic algorithm for constrained engineering design optimization

This article presents an effective hybrid cuckoo search and genetic algorithm (HCSGA) for solving engineering design optimization problems involving problem-specific constraints and mixed variables such as integer, discrete and continuous variables. The proposed algorithm, HCSGA, is first applied to 13 standard benchmark constrained optimization functions and subsequently used to solve three well-known design problems reported in the literature. The numerical results obtained by HCSGA show competitive performance with respect to recent algorithms for constrained design optimization problems.     

A dynamic programming-based particle swarm optimization algorithm for an inventory management problem under uncertainty

This article presents a dynamic programming-based particle swarm optimization (DP-based PSO) algorithm for solving an inventory management problem for large-scale construction projects under a fuzzy random environment. By taking into account the purchasing behaviour and strategy under rules of international bidding, a multi-objective fuzzy random dynamic programming model is constructed. To deal with the uncertainties, a hybrid crisp approach is used to transform fuzzy random parameters into fuzzy variables that are subsequently defuzzified by using an expected value operator with optimistic–pessimistic index. The iterative nature of the authors’ model motivates them to develop a DP-based PSO algorithm. More specifically, their approach treats the state variables as hidden parameters. This in turn eliminates many redundant feasibility checks during initialization and particle updates at each iteration. Results and sensitivity analysis are presented to highlight the performance of the authors’ optimization method, which is very effective as compared to the standard PSO algorithm.     

Drilling path optimization by the particle swarm optimization algorithm with global convergence characteristics

Drilling path optimization is one of the key problems in holes-machining. This paper presents a new approach to solve the drilling path optimization problem belonging to discrete space, based on the particle swarm optimization (PSO) algorithm. Since the standard PSO algorithm is not guaranteed to be global convergent or local convergent, based on the mathematical model, the algorithm is improved by adopting the method to generate the stop evolution particle once again to obtain the ability of convergence on the global optimization solution. Also, the operators are proposed by establishing the Order Exchange Unit (OEU) and the Order Exchange List (OEL) to satisfy the need of integer coding in drilling path optimization. The experimentations indicate that the improved algorithm has the characteristics of easy realization, fast convergence speed, and better global convergence capability. Hence the new PSO can play a role in solving the problem of drilling path optimization.     

An optimization technique using the characteristics of genetic algorithm

Optimization problems could happen often in discrete or discontinuous search space. Therefore, the traditional gradient‐based methods are not able to apply to this kind of problems. The discrete design variables are considered reasonably and the heuristic techniques are generally adopted to solve this problem, and the genetic algorithm based on stochastic search technique is one of these. The genetic algorithm method with discrete variables can be applied to structural optimization problems, such as composite laminated structures or trusses. However, the discrete optimization adopted in genetic algorithm gives rise to a troublesome task that is a mapping between each strings and discrete variables. And also, its solution quality could be restricted in some cases. In this study, a technique using the genetic algorithm characteristics is developed to utilize continuous design variables instead of discrete design variables in discontinuous solution spaces. Additionally, the proposed algorithm, which is manipulating a fitness function artificially, is applied to example problems and its results are compared with the general discrete genetic algorithm. The example problems are to optimize support positions of an unstable structure with discontinuous solution spaces.     

Time-response-based evolutionary optimization

Solutions to engineering problems are often evaluated by considering their time responses; thus, each solution is associated with a function. To avoid optimizing the functions, such optimization is usually carried out by setting auxiliary objectives (e.g. minimal overshoot). Therefore, in order to find different optimal solutions, alternative auxiliary optimization objectives may have to be defined prior to optimization. In the current study, a new approach is suggested that avoids the need to define auxiliary objectives. An algorithm is suggested that enables the optimization of solutions according to their transient behaviours. For this optimization, the functions are sampled and the problem is posed as a multi-objective problem. The recently introduced algorithm NSGA-II-PSA is adopted and tailored to solve it. Mathematical as well as engineering problems are utilized to explain and demonstrate the approach and its applicability to real life problems. The results highlight the advantages of avoiding the definition of artificial objectives.     

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.     

A new optimization algorithm based on a combination of particle swarm optimization,convergence and divergence operators for single-objective and multi-objective problems

Particle swarm optimization (PSO) is a randomized and population-based optimization method that was inspired by the flocking behaviour of birds and human social interactions. In this work, multi-objective PSO is modified in two stages. In the first stage, PSO is combined with convergence and divergence operators. Here, this method is named CDPSO. In the second stage, to produce a set of Pareto optimal solutions which has good convergence, diversity and distribution, two mechanisms are used. In the first mechanism, a new leader selection method is defined, which uses the periodic iteration and the concept of the particle's neighbour number. This method is named periodic multi-objective algorithm. In the second mechanism, an adaptive elimination method is employed to limit the number of non-dominated solutions in the archive, which has influences on computational time, convergence and diversity of solution. Single-objective results show that CDPSO performs very well on the complex test functions in terms of solution accuracy and convergence speed. Furthermore, some benchmark functions are used to evaluate the performance of periodic multi-objective CDPSO. This analysis demonstrates that the proposed algorithm operates better in three metrics through comparison with three well-known elitist multi-objective evolutionary algorithms. Finally, the algorithm is used for Pareto optimal design of a two-degree of freedom vehicle vibration model. The conflicting objective functions are sprung mass acceleration and relative displacement between sprung mass and tyre. The feasibility and efficiency of periodic multi-objective CDPSO are assessed in comparison with multi-objective modified NSGAII.     

An engineering method for complex structural optimization involving both size and topology design variables

This work presents an engineering method for optimizing structures made of bars, beams, plates, or a combination of those components. Corresponding problems involve both continuous (size) and discrete (topology) variables. Using a branched multipoint approximate function, which involves such mixed variables, a series of sequential approximate problems are constructed to make the primal problem explicit. To solve the approximate problems, genetic algorithm (GA) is utilized to optimize discrete variables, and when calculating individual fitness values in GA, a second-level approximate problem only involving retained continuous variables is built to optimize continuous variables. The solution to the second-level approximate problem can be easily obtained with dual methods. Structural analyses are only needed before improving the branched approximate functions in the iteration cycles. The method aims at optimal design of discrete structures consisting of bars, beams, plates, or other components. Numerical examples are given to illustrate its effectiveness, including frame topology optimization, layout optimization of stiffeners modeled with beams or shells, concurrent layout optimization of beam and shell components, and an application in a microsatellite structure. Optimization results show that the number of structural analyses is dramatically decreased when compared with pure GA while even comparable to pure sizing optimization.     

Dynamic optimization of chemical engineering problems using a control vector parameterization method with an iterative genetic algorithm

An approach that combines genetic algorithm (GA) and control vector parameterization (CVP) is proposed to solve the dynamic optimization problems of chemical processes using numerical methods. In the new CVP method, control variables are approximated with polynomials based on state variables and time in the entire time interval. The iterative method, which reduces redundant expense and improves computing efficiency, is used with GA to reduce the width of the search region. Constrained dynamic optimization problems are even more difficult. A new method that embeds the information of infeasible chromosomes into the evaluation function is introduced in this study to solve dynamic optimization problems with or without constraint. The results demonstrated the feasibility and robustness of the proposed methods. The proposed algorithm can be regarded as a useful optimization tool, especially when gradient information is not available.     

A surrogate assisted parallel multiobjective evolutionary algorithm for robust engineering design

A number of multi-objective evolutionary algorithms have been proposed in recent years and many of them have been used to solve engineering design optimization problems. However, designs need to be robust for real-life implementation, i.e. performance should not degrade substantially under expected variations in the variable values or operating conditions. Solutions of constrained robust design optimization problems should not be too close to the constraint boundaries so that they remain feasible under expected variations. A robust design optimization problem is far more computationally expensive than a design optimization problem as neighbourhood assessments of every solution are required to compute the performance variance and to ensure neighbourhood feasibility. A framework for robust design optimization using a surrogate model for neighbourhood assessments is introduced in this article. The robust design optimization problem is modelled as a multi-objective optimization problem with the aim of simultaneously maximizing performance and minimizing performance variance. A modified constraint-handling scheme is implemented to deal with neighbourhood feasibility. A radial basis function (RBF) network is used as a surrogate model and the accuracy of this model is maintained via periodic retraining. In addition to using surrogates to reduce computational time, the algorithm has been implemented on multiple processors using a master–slave topology. The preliminary results of two constrained robust design optimization problems indicate that substantial savings in the actual number of function evaluations are possible while maintaining an acceptable level of solution quality.     

Parallel global optimization with the particle swarm algorithm

Present day engineering optimization problems often impose large computational demands, resulting in long solution times even on a modern high-end processor. To obtain enhanced computational throughput and global search capability, we detail the coarse-grained parallelization of an increasingly popular global search method, the particle swarm optimization (PSO) algorithm. Parallel PSO performance was evaluated using two categories of optimization problems possessing multiple local minima-large-scale analytical test problems with computationally cheap function evaluations and medium-scale biomechanical system identification problems with computationally expensive function evaluations. For load-balanced analytical test problems formulated using 128 design variables, speedup was close to ideal and parallel efficiency above 95% for up to 32 nodes on a Beowulf cluster. In contrast, for load-imbalanced biomechanical system identification problems with 12 design variables, speedup plateaued and parallel efficiency decreased almost linearly with increasing number of nodes. The primary factor affecting parallel performance was the synchronization requirement of the parallel algorithm, which dictated that each iteration must wait for completion of the slowest fitness evaluation. When the analytical problems were solved using a fixed number of swarm iterations, a single population of 128 particles produced a better convergence rate than did multiple independent runs performed using sub-populations (8 runs with 16 particles, 4 runs with 32 particles, or 2 runs with 64 particles). These results suggest that (1) parallel PSO exhibits excellent parallel performance under load-balanced conditions, (2) an asynchronous implementation would be valuable for real-life problems subject to load imbalance, and (3) larger population sizes should be considered when multiple processors are available.     

混流装配线调度问题的离散粒子群优化解

混流装配线调度问题是JIT生产中的一个重要问题。借鉴二进制遗传算法中的交叉操作过程，对传统的连续型粒子群算法进行改进，使其适用于离散问题的优化处理。然后以丰田公司的汽车组装调度函数作为目标函数，利用改进的离散粒子群算法进行求解。对比分析表明：新算法所得结果优于常用的目标追随法、遗传算法、模拟退火等方法。     

Engineering optimization using a real-parameter genetic-algorithm-based hybrid method

A hybrid optimization algorithm which combines the respective merits of the genetic algorithm and the simulated annealing algorithm is proposed. The proposed algorithm incorporates adaptive mechanisms designed to adjust the probabilities of the cross-over and mutation operators such that its hill-climbing ability towards the optimum solution is improved. The algorithm is used to optimize the weight of four planar or space truss structures and the results are compared with those obtained using other well-known optimization schemes. The evaluation trials investigate the performance of the algorithm in optimizing over discrete sizing variables only and over both discrete sizing variables and continuous configuration variables. The results show that the proposed algorithm consistently outperforms the other optimization methods in terms of its weight-saving capabilities. It is also shown that the global searching ability and convergence speed of the proposed algorithm are significantly improved by the inclusion of adaptive mechanisms to adjust the values of the genetic operators. Hence the hybrid algorithm provides an efficient and robust technique for solving engineering design optimization problems.     

结构拓扑优化中敏度过滤技术研究

为了抑制连续体结构拓扑优化结果中的棋盘格和灰度单元问题,借鉴粒子群优化算法中粒子状态的更新方法,提出一种改进的敏度更新技术.以结构的柔度最小为优化目标,构建了基于固体各项同性微惩罚结构的结构拓扑优化模型,根据结构的力学响应分析,采用优化准则法进行设计变量更新,进行载荷作用下二维连续体结构的拓扑优化设计,得到了材料在设计域内的最优分布.通过与已有敏度过滤技术的对比分析,验证了文中方法的正确性和有效性．     

The harmony search heuristic algorithm for discrete structural optimization

Many methods have been developed and are in use for structural size optimization problems, in which the cross-sectional areas or sizing variables are usually assumed to be continuous. In most practical structural engineering design problems, however, the design variables are discrete. This paper proposes an efficient optimization method for structures with discrete-sized variables based on the harmony search (HS) heuristic algorithm. The recently developed HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. In this article, a discrete search strategy using the HS algorithm is presented in detail and its effectiveness and robustness, as compared to current discrete optimization methods, are demonstrated through several standard truss examples. The numerical results reveal that the proposed method is a powerful search and design optimization tool for structures with discrete-sized members, and may yield better solutions than those obtained using current methods.