Solving constrained optimization problems with hybrid particle swarm optimization

Constrained optimization problems (COPs) are very important in that they frequently appear in the real world. A COP, in which both the function and constraints may be nonlinear, consists of the optimization of a function subject to constraints. Constraint handling is one of the major concerns when solving COPs with particle swarm optimization (PSO) combined with the Nelder–Mead simplex search method (NM-PSO). This article proposes embedded constraint handling methods, which include the gradient repair method and constraint fitness priority-based ranking method, as a special operator in NM-PSO for dealing with constraints. Experiments using 13 benchmark problems are explained and the NM-PSO results are compared with the best known solutions reported in the literature. Comparison with three different meta-heuristics demonstrates that NM-PSO with the embedded constraint operator is extremely effective and efficient at locating optimal solutions.     

Grey wolf optimizer and hybrid PSO-GWO for reliability optimization and redundancy allocation problem

Analysis and optimization of system reliability have very much importance for developing an optimal design for the system while using the available resources. Several studies are centered towards reliability optimization using metaheuristics. In this study, a recently developed metaheuristic optimization algorithm called hybrid PSO-GWO (HPSGWO) to solve the reliability-redundancy optimization problem has been proposed. The HPSGWO fuses the Particle Swarm Optimization's (PSO) exploitation ability with the grey wolf optimizer's (GWO) exploration ability. The comparison of results with prior best results of PSO and GWO for the four benchmarks of reliability redundancy allocation problem demonstrates the HPSGWO as a productive enhancement strategy since it got promising answers than other metaheuristic algorithms.     

Feasibility-Guided Constraint-Handling Techniques for Engineering Optimization Problems

The particle swarm optimization (PSO) algorithm is an established nature-inspired population-based meta-heuristic that replicates the synchronizing movements of birds and fish. PSO is essentially an unconstrained algorithm and requires constraint handling techniques (CHTs) to solve constrained optimization problems (COPs). For this purpose, we integrate two CHTs, the superiority of feasibility (SF) and the violation constraint-handling (VCH), with a PSO. These CHTs distinguish feasible solutions from infeasible ones. Moreover, in SF, the selection of infeasible solutions is based on their degree of constraint violations, whereas in VCH, the number of constraint violations by an infeasible solution is of more importance. Therefore, a PSO is adapted for constrained optimization, yielding two constrained variants, denoted SF-PSO and VCH-PSO. Both SF-PSO and VCH-PSO are evaluated with respect to five engineering problems: the Himmelblau’s nonlinear optimization, the welded beam design, the spring design, the pressure vessel design, and the three-bar truss design. The simulation results show that both algorithms are consistent in terms of their solutions to these problems, including their different available versions. Comparison of the SF-PSO and the VCH-PSO with other existing algorithms on the tested problems shows that the proposed algorithms have lower computational cost in terms of the number of function evaluations used. We also report our disagreement with some unjust comparisons made by other researchers regarding the tested problems and their different variants.     

Design of high-performing hybrid meta-heuristics for unrelated parallel machine scheduling with machine eligibility and precedence constraints

This study involves an unrelated parallel machine scheduling problem in which sequence-dependent set-up times, different release dates, machine eligibility and precedence constraints are considered to minimize total late works. A new mixed-integer programming model is presented and two efficient hybrid meta-heuristics, genetic algorithm and ant colony optimization, combined with the acceptance strategy of the simulated annealing algorithm (Metropolis acceptance rule), are proposed to solve this problem. Manifestly, the precedence constraints greatly increase the complexity of the scheduling problem to generate feasible solutions, especially in a parallel machine environment. In this research, a new corrective algorithm is proposed to obtain the feasibility in all stages of the algorithms. The performance of the proposed algorithms is evaluated in numerical examples. The results indicate that the suggested hybrid ant colony optimization statistically outperformed the proposed hybrid genetic algorithm in solving large-size test problems.     

A comparison of swarm intelligence algorithms for structural engineering optimization

This paper compares the performance of three swarm intelligence algorithms for the optimization of hard engineering problems. The algorithms tested were bacterial foraging optimization (BFO), particle swarm optimization (PSO), and artificial bee colony (ABC). Besides the regular BFO, two other variants reported in the literature were also included in the study: adaptive BFO and swarming BFO. Both PSO and ABC were tested using the regular algorithm and variants that include explosion (mass extinction). Three optimization problems of structural engineering were used: minimization of the cost of a welded beam, minimization of the construction cost of a pressure vessel, and minimization of the total weight of a 10‐bar plane truss. All problems are strongly constrained. The algorithms were evaluated using two criteria: quality of solutions and the number of function evaluations. The results show that PSO presented the best balance between these two criteria. For the optimization problems approached in this paper, we can also conclude that the explosion procedure resulted in no significant improvements. Copyright © 2012 John Wiley & Sons, Ltd.     

Parallel simple cell mapping for multi-objective optimization

In this article the Parallel Simple Cell Mapping (pSCM) is presented, a novel method for the numerical treatment of multi-objective optimization problems. The method is a parallel version of the simple cell mapping (SCM) method which also integrates elements from subdivision techniques. The classical SCM method exhibits nice properties for parallelization, which is used to speed up computations significantly. These statements are underlined on some classical benchmark problems with up to 10 decision variables and up to 5 objectives and provide comparisons to sequential SCM. Further, the method is applied on illustrative examples for which the method is also able to find the set of local optimal solutions efficiently, which is interesting in multi-objective multi-modal optimization, as well as the set of approximate solutions. The latter is of potential interest for the decision maker since it comprises an extended set of possible realizations of the given problem.     

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.     

Solving initial and boundary value problems using learning automata particle swarm optimization

In this article, the particle swarm optimization (PSO) algorithm is modified to use the learning automata (LA) technique for solving initial and boundary value problems. A constrained problem is converted into an unconstrained problem using a penalty method to define an appropriate fitness function, which is optimized using the LA-PSO method. This method analyses a large number of candidate solutions of the unconstrained problem with the LA-PSO algorithm to minimize an error measure, which quantifies how well a candidate solution satisfies the governing ordinary differential equations (ODEs) or partial differential equations (PDEs) and the boundary conditions. This approach is very capable of solving linear and nonlinear ODEs, systems of ordinary differential equations, and linear and nonlinear PDEs. The computational efficiency and accuracy of the PSO algorithm combined with the LA technique for solving initial and boundary value problems were improved. Numerical results demonstrate the high accuracy and efficiency of the proposed method.     

Reliability Analysis Using Chaotic Particle Swarm Optimization

Reliability analysis is an important and practical way of considering uncertainty in an engineering system. In practical engineering, the limit state functions are usually implicit in terms of random variables. Traditional reliability analysis methods are time‐consuming and require derivative computing. In this paper, particle swarm optimization (PSO) is applied to reliability analysis by combining it with the first‐order reliability method. To improve the global search performance, chaotic PSO (CPSO) was proposed by combining PSO with a chaotic system. CPSO‐based reliability analysis is described and applied to four classic examples. The results show that implementation of CPSO‐based reliability analysis is easy and can yield a reliability index and design point with good accuracy. The proposed method was applied to analyze the reliability of a circular tunnel in different cases. The results show that the CPSO algorithm is very efficient at solving global optimization problems and is a good approach for reliability analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

基于改进粒子群算法的制造单元设施布局问题研究

为避免单元系统布局和单元内设施布局分开孤立研究所导致的问题解空间损失,利用并行工程的思想对单元布局的两个环节集成考虑,对单元系统布局、单元内设施布置、设施摆放方向进行同时描述,并建立多目标集成优化模型。针对模型的复杂性,设计了改进粒子群算法,算法吸收了遗传算法中的交叉操作算子,具有跳出局部最优解的能力。最后通过求解单元设施布置实例,验证模型和算法的有效性。     

Probabilistic optimal design of laminates using improved particle swarm optimization

A new approach to the particle swarm optimization (PSO) is proposed for the solution of non-linear optimization problems with constraints, and is applied to the reliability-based optimum design of laminated composites. Special mutation-interference operators are introduced to increase swarm variety and improve the convergence performance of the algorithm. The reliability-based optimum design of laminated composites is modelled and solved using the improved PSO. The maximization of structural reliability and the minimization of total weight of laminates are analysed. The stacking sequence optimization is implemented in the improved PSO by using a special coding technique. Examples show that the improved PSO has high convergence and good stability and is efficient in dealing with the probabilistic optimal design of composite structures.     

A highly accurate differential evolution–particle swarm optimization algorithm for the construction of initial value problem solvers

In this work a new evolutionary computation technique is introduced for the construction of initial value solvers based on Runge–Kutta (RK) pairs. The derivation of RK pairs corresponds to solving a nonlinear optimization problem with a multimodal objective function in a high dimensional search space; additional difficulty stems from the fact that only solutions with accuracy at least equal to machine epsilon are acceptable. The proposed approach involves hybridizing a Differential Evolution (DE) strategy with elements from Particle Swarm Optimization (PSO) in order to produce a method for solving optimization problems with high accuracy. The resulting methodology is applied to two different problems of RK pair derivation of orders 5 and 4 and compared with standard DE techniques. Numerical experiments show that the proposed hybrid DE-PSO satisfies the strict accuracy requirements imposed by the particular problem, while outperforming its rivals.     

A joint reliability-redundancy optimization approach for multi-state series-parallel systems

In this paper, we present a practical approach for the joint reliability-redundancy optimization of multi-state series-parallel systems. In addition to determining the optimal redundancy level for each parallel subsystem, this approach also aims at finding the optimal values for the variables that affect the component state distributions in each subsystem. The key point is that technical and organizational actions can affect the state transition rates of a multi-state component, and thus affect the state distribution of the component and the availability of the system. Taking this into consideration, we present an approach for determining the optimal versions and numbers of components and the optimal set of technical and organizational actions for each subsystem of a multi-state series-parallel system, so as to minimize the system cost while satisfying the system availability constraint. The approach might be considered to be the multi-state version of the joint system reliability-redundancy optimization methods.     

Stability assessment of earth slope using modified particle swarm optimization

This paper has proposed an effective method to determine the minimum factor of safety (FS) and associated critical failure surface in slope stability analysis. The search for the minimum FS based on limit equilibrium methods is a complex optimization problem as the objective function is non-smooth and non-convex. Recently, particle swarm optimization (PSO) as a meta-heuristic optimization algorithm has been developed with success in treating various types of problems. In the current study, a new approach of PSO is proposed to calculate the safety factor of earth slopes. The safety factors of the general slip surfaces are calculated using Spencer method of slices, and each new slip surface is randomly generated by straight line technique. The reliability and efficiency of the proposed algorithm are examined by considering a number of published cases. The results indicate that the new method can predict a more critical failure mechanism with a lower FS and can outperform the other methods in the literature as well as standard PSO. Finally, the proposed method will be validated by considering an existing slope failure in Ulu Klang, Malaysia.     

Fuzzy multiobjective system reliability optimization by genetic algorithms and clustering analysis

System reliability optimization is a key element for a competitive and safe industrial plant. This paper addresses the multiobjective system reliability optimization in the presence of fuzzy data. A framework solution approach is proposed and based on four steps: defuzzify the data into crisp values by the ranking function procedure, the defuzzified problems are solved by the non-sorting genetic algorithms II and III (NSGA-II and NSGA-III), the Pareto fronts are compared by the spacing method for selecting the best one, and then the best Pareto front is reduced by the clustering analysis for helping the decision maker. A case study presented in the literature as a mono-objective redundancy allocation problem with fuzzy data is investigated in the present paper as multiobjective redundancy allocation and reliability-redundancy allocation problems show the applicability of the approach.     

A new relaxed flexible bay structure representation and particle swarm optimization for the unequal area facility layout problem

The facility layout problem (FLP) with unequal area departments is a very hard problem to be optimally solved. In this article, a hybrid particle swarm optimization (PSO) and local search approach is proposed to solve the FLP with unequal area departments. The flexible bay structure (FBS), which is a very common layout in manufacturing and retail facilities, is used. Furthermore, the FBS is relaxed by allowing empty spaces in bays, which results in more flexibility while assigning departments in bays. The proposed PSO approach is used to solve the FLP instances from the literature with varying sizes. The comparative results show that the PSO approach is very promising and able to find the previously known-optimal solutions in very short CPU times. In addition, new best solutions have been found for some test problems. Improvements have been achieved by allowing partially filled bays.     

Guided particle swarm optimization method to solve general nonlinear optimization problems

The development of hybrid algorithms is becoming an important topic in the global optimization research area. This article proposes a new technique in hybridizing the particle swarm optimization (PSO) algorithm and the Nelder–Mead (NM) simplex search algorithm to solve general nonlinear unconstrained optimization problems. Unlike traditional hybrid methods, the proposed method hybridizes the NM algorithm inside the PSO to improve the velocities and positions of the particles iteratively. The new hybridization considers the PSO algorithm and NM algorithm as one heuristic, not in a sequential or hierarchical manner. The NM algorithm is applied to improve the initial random solution of the PSO algorithm and iteratively in every step to improve the overall performance of the method. The performance of the proposed method was tested over 20 optimization test functions with varying dimensions. Comprehensive comparisons with other methods in the literature indicate that the proposed solution method is promising and competitive.     

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。     

Biogeography-based particle swarm optimization with fuzzy elitism and its applications to constrained engineering problems

In evolutionary algorithms, elites are crucial to maintain good features in solutions. However, too many elites can make the evolutionary process stagnate and cannot enhance the performance. This article employs particle swarm optimization (PSO) and biogeography-based optimization (BBO) to propose a hybrid algorithm termed biogeography-based particle swarm optimization (BPSO) which could make a large number of elites effective in searching optima. In this algorithm, the whole population is split into several subgroups; BBO is employed to search within each subgroup and PSO for the global search. Since not all the population is used in PSO, this structure overcomes the premature convergence in the original PSO. Time complexity analysis shows that the novel algorithm does not increase the time consumption. Fourteen numerical benchmarks and four engineering problems with constraints are used to test the BPSO. To better deal with constraints, a fuzzy strategy for the number of elites is investigated. The simulation results validate the feasibility and effectiveness of the proposed algorithm.