A new harmony search algorithm for solving mixed–discrete engineering optimization problems

In general design optimization problems, it is usually assumed that the design variables are continuous. However, many practical problems in engineering design require considering the design variables as integer or discrete values. The presence of discrete and integer variables along with continuous variables adds to the complexity of the optimization problem. Very few of the existing methods can yield a globally optimal solution when the objective functions are non-convex and non-differentiable. This article presents a mixed–discrete harmony search approach for solving these nonlinear optimization problems which contain integer, discrete and continuous variables. Some engineering design examples are also presented to demonstrate the effectiveness of the proposed method.     

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.     

A fuzzy discrete harmony search algorithm applied to annual cost reduction in radial distribution systems

Similarly to other optimization algorithms, harmony search (HS) is quite sensitive to the tuning parameters. Several variants of the HS algorithm have been developed to decrease the parameter-dependency character of HS. This article proposes a novel version of the discrete harmony search (DHS) algorithm, namely fuzzy discrete harmony search (FDHS), for optimizing capacitor placement in distribution systems. In the FDHS, a fuzzy system is employed to dynamically adjust two parameter values, i.e. harmony memory considering rate and pitch adjusting rate, with respect to normalized mean fitness of the harmony memory. The key aspect of FDHS is that it needs substantially fewer iterations to reach convergence in comparison with classical discrete harmony search (CDHS). To the authors’ knowledge, this is the first application of DHS to specify appropriate capacitor locations and their best amounts in the distribution systems. Simulations are provided for 10-, 34-, 85- and 141-bus distribution systems using CDHS and FDHS. The results show the effectiveness of FDHS over previous related studies.     

Hybridizing the harmony search algorithm with a spreadsheet ‘Solver’ for solving continuous engineering optimization problems

In this article, a hybrid global–local optimization algorithm is proposed to solve continuous engineering optimization problems. In the proposed algorithm, the harmony search (HS) algorithm is used as a global-search method and hybridized with a spreadsheet ‘Solver’ to improve the results of the HS algorithm. With this purpose, the hybrid HS–Solver algorithm has been proposed. In order to test the performance of the proposed hybrid HS–Solver algorithm, several unconstrained, constrained, and structural-engineering optimization problems have been solved and their results are compared with other deterministic and stochastic solution methods. Also, an empirical study has been carried out to test the performance of the proposed hybrid HS–Solver algorithm for different sets of HS solution parameters. Identified results showed that the hybrid HS–Solver algorithm requires fewer iterations and gives more effective results than other deterministic and stochastic solution algorithms.     

A hybrid optimization method of harmony search and opposition-based learning

The harmony search (HS) method is an emerging meta-heuristic optimization algorithm. However, like most of the evolutionary computation techniques, it sometimes suffers from a rather slow search speed, and fails to find the global optimum in an efficient way. In this article, a hybrid optimization approach is proposed and studied, in which the HS is merged together with the opposition-based learning (OBL). The modified HS, namely HS-OBL, has an improved convergence property. Optimization of 24 typical benchmark functions and an optimal wind generator design case study demonstrate that the HS-OBL can indeed yield a superior optimization performance over the regular HS method.     

Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

Multipoint approximation method (MAM) focuses on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem within a trust region. To develop an optimization technique applicable to mixed integer-continuous design optimization problems in which the objective and constraint functions are computationally expensive and could be impossible to evaluate at some combinations of design variables, a simple and efficient algorithm, coordinate search, is implemented in the MAM. This discrete optimization capability is examined by the well established benchmark problem and its effectiveness is also evaluated as the discreteness interval for discrete design variables is increased from 0.2 to 1. Furthermore, an application to the optimization of a lattice composite fuselage structure where one of design variables (number of helical ribs) is integer is also presented to demonstrate the efficiency of this capability.     

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 local-best harmony search algorithm with dynamic subpopulations

This article presents a local-best harmony search algorithm with dynamic subpopulations (DLHS) for solving the bound-constrained continuous optimization problems. Unlike existing harmony search algorithms, the DLHS algorithm divides the whole harmony memory (HM) into many small-sized sub-HMs and the evolution is performed in each sub-HM independently. To maintain the diversity of the population and to improve the accuracy of the final solution, information exchange among the sub-HMs is achieved by using a periodic regrouping schedule. Furthermore, a novel harmony improvisation scheme is employed to benefit from good information captured in the local best harmony vector. In addition, an adaptive strategy is developed to adjust the parameters to suit the particular problems or the particular phases of search process. Extensive computational simulations and comparisons are carried out by employing a set of 16 benchmark problems from the literature. The computational results show that, overall, the proposed DLHS algorithm is more effective or at least competitive in finding near-optimal solutions compared with state-of-the-art harmony search variants.     

A genetic algorithm approach to single and multiobjective structural optimization with discrete–continuous variables

A search procedure with a philosophical basis in molecular biology is adapted for solving single and multiobjective structural optimization problems. This procedure, known as a genetic algorithm (GA). utilizes a blending of the principles of natural genetics and natural selection. A lack of dependence on the gradient information makes GAs less susceptible to pitfalls of convergence to a local optimum. To model the multiple objective functions in the problem formulation, a co-operative game theoretic approach is proposed. Examples dealing with single and multiobjective geometrical design of structures with discrete–continuous design variables, and using artificial genetic search are presented. Simulation results indicate that GAs converge to optimum solutions by searching only a small fraction of the solution space. The optimum solutions obtained using GAs compare favourably with optimum solutions obtained using gradient-based search techniques. The results indicate that the efficiency and power of GAs can be effectively utilized to solve a broad spectrum of design optimization problems with discrete and continuous variables with similar efficiency.     

离散变量结构优化设计的拟满应力遗传算法

以力学准则法为基础,提出了一种求解离散变量结构优化设计的拟满应力方法;这种方法能直接求解具有应力约束和几何约束的离散变量结构优化设计问题。通过在遗传算法中定义拟满应力算子,建立了一种离散变量结构优化设计的混合遗传算法拟满应力遗传算法。算例表明:这种混合遗传算法对于离散变量结构优化设计问题具有较高的计算效率。     

Optimization of truss topology using tabu search

A design procedure for integrating topological considerations in the framework of structural optimization is presented. The proposed approach is capable of considering multiple load conditions, stress, displacement and local/global buckling constraints, and multiple objective functions in the problem formulation. Further, since the proposed method permits members to be added to or deleted from an existing topology and the topology is not defined by member areas, the difficulty of not being able to reach singular optima is also avoided. These objectives are accomplished using a discrete optimization procedure which uses 0–1 topological variables to optimize alternate designs. Since the topological variables are discrete in nature and the member cross-sections are assumed to be continuous, the topological optimization problem has mixed discrete-continuous variables. This non-linear programming problem is solved using a memory-based combinatorial optimization technique known as tabu search. Numerical results obtained using tabu search for single and multiobjective topological optimization of truss structures are presented. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. The results indicate that the optimum topologies obtained using tabu search compare favourably, and in some instances, outperform the results obtained using the ground–structure approach. However, this improvement occurs at the expense of a significant increase in computational burden owing to the fact that the proposed approach necessitates that the geometry of each trial topology be optimized.     

An improved harmony search algorithm with dynamically varying bandwidth

The present work demonstrates a new variant of the harmony search (HS) algorithm where bandwidth (BW) is one of the deciding factors for the time complexity and the performance of the algorithm. The BW needs to have both explorative and exploitative characteristics. The ideology is to use a large BW to search in the full domain and to adjust the BW dynamically closer to the optimal solution. After trying a series of approaches, a methodology inspired by the functioning of a low-pass filter showed satisfactory results. This approach was implemented in the self-adaptive improved harmony search (SIHS) algorithm and tested on several benchmark functions. Compared to the existing HS algorithm and its variants, SIHS showed better performance on most of the test functions. Thereafter, the algorithm was applied to geometric parameter optimization of a friction stir welding tool.     

Optimization of seismic isolation systems via harmony search

In this article, the optimization of isolation system parameters via the harmony search (HS) optimization method is proposed for seismically isolated buildings subjected to both near-fault and far-fault earthquakes. To obtain optimum values of isolation system parameters, an optimization program was developed in Matlab/Simulink employing the HS algorithm. The objective was to obtain a set of isolation system parameters within a defined range that minimizes the acceleration response of a seismically isolated structure subjected to various earthquakes without exceeding a peak isolation system displacement limit. Several cases were investigated for different isolation system damping ratios and peak displacement limitations of seismic isolation devices. Time history analyses were repeated for the neighbouring parameters of optimum values and the results proved that the parameters determined via HS were true optima. The performance of the optimum isolation system was tested under a second set of earthquakes that was different from the first set used in the optimization process. The proposed optimization approach is applicable to linear isolation systems. Isolation systems composed of isolation elements that are inherently nonlinear are the subject of a future study. Investigation of the optimum isolation system parameters has been considered in parametric studies. However, obtaining the best performance of a seismic isolation system requires a true optimization by taking the possibility of both near-fault and far-fault earthquakes into account. HS optimization is proposed here as a viable solution to this problem.     

An efficient and practical approach to obtain a better optimum solution for structural optimization

For many structural optimization problems, it is hard or even impossible to find the global optimum solution owing to unaffordable computational cost. An alternative and practical way of thinking is thus proposed in this research to obtain an optimum design which may not be global but is better than most local optimum solutions that can be found by gradient-based search methods. The way to reach this goal is to find a smaller search space for gradient-based search methods. It is found in this research that data mining can accomplish this goal easily. The activities of classification, association and clustering in data mining are employed to reduce the original design space. For unconstrained optimization problems, the data mining activities are used to find a smaller search region which contains the global or better local solutions. For constrained optimization problems, it is used to find the feasible region or the feasible region with better objective values. Numerical examples show that the optimum solutions found in the reduced design space by sequential quadratic programming (SQP) are indeed much better than those found by SQP in the original design space. The optimum solutions found in a reduced space by SQP sometimes are even better than the solution found using a hybrid global search method with approximate structural analyses.     

Optimal cost design of water distribution networks using harmony search

This study presents a cost minimization model for the design of water distribution networks. The model uses a recently developed harmony search optimization algorithm while satisfying all the design constraints. The harmony search algorithm mimics a jazz improvisation process in order to find better design solutions, in this case pipe diameters in a water distribution network. The model also interfaces with a popular hydraulic simulator, EPANET, to check the hydraulic constraints. If the design solution vector violates the hydraulic constraints, the amount of violation is considered in the cost function as a penalty. The model was applied to five water distribution networks, and obtained designs that were either the same or cost 0.28–10.26% less than those of competitive meta-heuristic algorithms, such as the genetic algorithm, simulated annealing and tabu search under similar or less favorable conditions. The results show that the harmony search-based model is suitable for water network design.     

离散复合形法的改进及应用研究

在工程设计中,复合形法是求解约束非线性规划的一种直接方法.它计算简单、适用性强,是解决工程技术问题的常用方法.通常情况下,复合形法多用于连续变量优化设计问题的求解,但在工程实际中还存在大量的离散变量优化问题.为此,我们在连续变量复合形算法的基础上,采用平均值舍入法获取离散点,并改进了复合形迭代过程和停机准则,使之成为一种能直接求解离散变量优化解的方法.几个典型工程离散变量优化设计算例的计算结果表明,我们提出的方法十分有效.     

Evolutionary Algorithm Integrating Stress Heuristics for Truss Optimization

The optimal truss design using problem-oriented evolutionary algorithm is presented in the paper. The minimum weight structures subjected to stress and displacement constraints are searched. The discrete design variables are areas of members, selected from catalogues of available sections. The integration of the problem specific knowledge into the optimization procedure is proposed. The heuristic rules based on the concept of fully stressed design are introduced through special genetic operators, which use the information concerning the stress distribution of structural members. Moreover, approximated solutions obtained by deterministic, sequential discrete optimization methods are inserted into the initial population. The obtained hybrid evolutionary algorithm is specialized for truss design. Benchmark problems are calculated in numerical examples. The knowledge about the problem integrated into the evolutionary algorithm can enhance considerably the effectiveness of the approach and improve significantly the convergence rate and the quality of the results. The advantages and drawbacks of the proposed method are discussed.     

Self-adaptive multi-objective harmony search for optimal design of water distribution networks

In multi-objective optimization computing, it is important to assign suitable parameters to each optimization problem to obtain better solutions. In this study, a self-adaptive multi-objective harmony search (SaMOHS) algorithm is developed to apply the parameter-setting-free technique, which is an example of a self-adaptive methodology. The SaMOHS algorithm attempts to remove some of the inconvenience from parameter setting and selects the most adaptive parameters during the iterative solution search process. To verify the proposed algorithm, an optimal least cost water distribution network design problem is applied to three different target networks. The results are compared with other well-known algorithms such as multi-objective harmony search and the non-dominated sorting genetic algorithm-II. The efficiency of the proposed algorithm is quantified by suitable performance indices. The results indicate that SaMOHS can be efficiently applied to the search for Pareto-optimal solutions in a multi-objective solution space.     

Discrete harmony search algorithm for scheduling and rescheduling the reprocessing problems in remanufacturing: a case study

In this article, scheduling and rescheduling problems with increasing processing time and new job insertion are studied for reprocessing problems in the remanufacturing process. To handle the unpredictability of reprocessing time, an experience-based strategy is used. Rescheduling strategies are applied for considering the effect of increasing reprocessing time and the new subassembly insertion. To optimize the scheduling and rescheduling objective, a discrete harmony search (DHS) algorithm is proposed. To speed up the convergence rate, a local search method is designed. The DHS is applied to two real-life cases for minimizing the maximum completion time and the mean of earliness and tardiness (E/T). These two objectives are also considered together as a bi-objective problem. Computational optimization results and comparisons show that the proposed DHS is able to solve the scheduling and rescheduling problems effectively and productively. Using the proposed approach, satisfactory optimization results can be achieved for scheduling and rescheduling on a real-life shop floor.     

A hybrid cellular automaton–bi-directional evolutionary optimization algorithm for topological optimization of crashworthiness

This article proposes a new algorithm for topological optimization under dynamic loading which combines cellular automata with bi-directional evolutionary structural optimization (BESO). The local rules of cellular automata are used to update the design variables, which avoids the difficulty of obtaining gradient information under nonlinear collision conditions. The intermediate-density design problem of hybrid cellular automata is solved using the BESO concept of 0–1 binary discrete variables. Some improvement strategies are also proposed for the hybrid algorithm to solve certain problems in nonlinear topological optimization, e.g. numerical oscillation. Some typical examples of crashworthiness problems are provided to illustrate the efficiency of the proposed method and its ability to find the final optimal solution. Finally, numerical results obtained using the proposed algorithms are compared with reference examples taken from the literature. The results show that the hybrid method is computationally efficient and stable.