Engineering Design Optimization Problems Research Articles

An optimization method: hummingbirds optimization algorithm

This paper introduces an optimization algorithm, the hummingbirds optimization algorithm (HOA), which is inspired by the foraging process of hummingbirds. The proposed algorithm includes two phases: a self-searching phase and a guide-searching phase. With these two phases, the exploration and exploitation abilities of the algorithm can be balanced. Both the constrained and unconstrained benchmark functions are employed to test the performance of HOA. Ten classic benchmark functions are considered as unconstrained benchmark functions. Meanwhile, two engineering design optimization problems are employed as constrained benchmark functions. The results of these experiments demonstrate HOA is efficient and capable of global optimization.

A multi-objective robust optimization approach for engineering design under interval uncertainty

PurposeEngineering system design and optimization problems are usually multi-objective and constrained and have uncertainties in the inputs. These uncertainties might significantly degrade the overall performance of engineering systems and change the feasibility of the obtained solutions. This paper aims to propose a multi-objective robust optimization approach based on Kriging metamodel (K-MORO) to obtain the robust Pareto set under the interval uncertainty.Design/methodology/approachIn K-MORO, the nested optimization structure is reduced into a single loop optimization structure to ease the computational burden. Considering the interpolation uncertainty from the Kriging metamodel may affect the robustness of the Pareto optima, an objective switching and sequential updating strategy is introduced in K-MORO to determine (1) whether the robust analysis or the Kriging metamodel should be used to evaluate the robustness of design alternatives, and (2) which design alternatives are selected to improve the prediction accuracy of the Kriging metamodel during the robust optimization process.FindingsFive numerical and engineering cases are used to demonstrate the applicability of the proposed approach. The results illustrate that K-MORO is able to obtain robust Pareto frontier, while significantly reducing computational cost.Practical implicationsThe proposed approach exhibits great capability for practical engineering design optimization problems that are multi-objective and constrained and have uncertainties.Originality/valueA K-MORO approach is proposed, which can obtain the robust Pareto set under the interval uncertainty and ease the computational burden of the robust optimization process.

A Two-stage State Transition Algorithm for Constrained Engineering Optimization Problems

In this study, a state transition algorithm (STA) is investigated into constrained engineering design optimization problems. After an analysis of the advantages and disadvantages of two well-known constraint-handling techniques, penalty function method and feasibility preference method, a two-stage strategy is incorporated into STA, in which, the feasibility preference method is adopted in the early stage of an iteration process whilst it is changed to the penalty function method in the later stage. Then, the proposed STA is used to solve three benchmark problems in engineering design and an optimization problem in power-dispatching control system for the electrochemical process of zinc. The experimental results have shown that the optimal solutions obtained by the proposed method are all superior to those by typical approaches in the literature in terms of both convergency and precision.

A hybrid global optimization method based on multiple metamodels

PurposeThis paper aims to propose hybrid global optimization based on multiple metamodels for improving the efficiency of global optimization.Design/methodology/approachThe method has fully utilized the information provided by different metamodels in the optimization process. It not only imparts the expected improvement criterion of kriging into other metamodels but also intelligently selects appropriate metamodeling techniques to guide the search direction, thus making the search process very efficient. Besides, the corresponding local search strategies are also put forward to further improve the optimizing efficiency.FindingsTo validate the method, it is tested by several numerical benchmark problems and applied in two engineering design optimization problems. Moreover, an overall comparison between the proposed method and several other typical global optimization methods has been made. Results show that the global optimization efficiency of the proposed method is higher than that of the other methods for most situations.Originality/valueThe proposed method sufficiently utilizes multiple metamodels in the optimizing process. Thus, good optimizing results are obtained, showing great applicability in engineering design optimization problems which involve costly simulations.

An on-line variable fidelity metamodel assisted Multi-objective Genetic Algorithm for engineering design optimization

Population-based algorithms, which require a large number of fitness evaluations, can become computationally intractable when applied in engineering design optimization problems involving computational expensive simulations. To address this challenge, this paper proposes an on-line variable-fidelity metamodel assisted Multi-Objective Genetic Algorithm (OLVFM-MOGA) approach. In OLVFM-MOGA, the variable-fidelity metamodel (VFM) is constructed to replace the expensive simulation models to ease the computational burden. Besides, a novel model updating strategy is developed to address the issues of 1) which sample points should be sent for simulation analysis to improve the optimization quality, and 2) whether the low-fidelity (LF) model or the high-fidelity (HF) model should be selected to run for a selected sample point. Six numerical examples and an engineering case with different degrees of complexity are used to demonstrate the applicability and efficiency of the proposed approach. Results illustrate that the proposed OLVFM-MOGA is able to obtain comparable convergence and diversity of the Pareto frontier as to that obtained by MOGA with HF model, while at the same time significantly reducing the computational cost.

Ensemble of surrogate based global optimization methods using hierarchical design space reduction

For computationally expensive black-box problems, surrogate models are widely employed to reduce the needed computation time and efforts during the search of the global optimum. However, the construction of an effective surrogate model over a large design space remains a challenge in many cases. In this work, a new global optimization method using an ensemble of surrogates and hierarchical design space reduction is proposed to deal with the optimization problems with computation-intensive, black-box objective functions. During the search, an ensemble of three representative surrogate techniques with optimized weight factors is used for selecting promising sample points, narrowing down space exploration and identifying the global optimum. The design space is classified into: Original Global Space (OGS), Promising Joint Space (PJS), Important Local Space (ILS), using the newly proposed hierarchical design space reduction (HSR). Tested using eighteen representative benchmark and two engineering design optimization problems, the newly proposed global optimization method shows improved capability in identifying promising search area and reducing design space, and superior search efficiency and robustness in identifying the global optimum.

Locally weighted regression models for surrogate-assisted design optimization

We consider engineering design optimization problems where the objective and/or constraint functions are evaluated by means of computationally expensive blackboxes. Our practical optimization strategy consists of solving surrogate optimization problems in the search step of the mesh adaptive direct search algorithm. In this paper, we consider locally weighted regression models to build the necessary surrogates, and present three ideas for appropriate and effective use of locally weighted scatterplot smoothing (LOWESS) models for surrogate optimization. First, a method is proposed to reduce the computational cost of LOWESS models. Second, a local scaling coefficient is introduced to adapt LOWESS models to the density of neighboring points while retaining smoothness. Finally, an appropriate order error metric is used to select the optimal shape coefficient of the LOWESS model. Our surrogate-assisted optimization approach utilizes LOWESS models to both generate and rank promising candidates found in the search and poll steps. The “real” blackbox functions that govern the original optimization problem are then evaluated at these ranked candidates with an opportunistic strategy, reducing CPU time significantly. Computational results are reported for four engineering design problems with up to six variables and six constraints. The results demonstrate the effectiveness of the LOWESS models as well as the order error metric for surrogate optimization.

Volleyball Premier League Algorithm

This article proposes a novel metaheuristic algorithm called Volleyball Premier League (VPL) inspired by the competition and interaction among volleyball teams during a season. It also mimics the coaching process during a volleyball match. To solve global optimization problems using the volleyball metaphor, there are terms such as substitution, coaching, and learning, which are captured in the VPL algorithm. The proposed algorithm is benchmarked on 23 well-known test functions, which are categorized into three groups, namely unimodal, multimodal and fixed-dimension multimodal functions. The solutions obtained using the VPL have been compared with other metaheuristic algorithms including Particle Swarm Optimization (PSO), Differential Evolution (DE), Genetic Algorithm (GA), Artificial Bee Colony (ABC), Firefly Algorithm (FA), Harmony Search (HS), Sin Cosine Algorithm (SCA), Soccer League Competition (SLC), and League Championship Algorithm (LCA). In addition, VPL has been used to solve three classical engineering design optimization problems. Results show that VPL algorithm possesses a strong capability to produce superior performance over the other well-known metaheuristic algorithms. The results of the experiments also show that the VPL is effectively applicable to solve problems with complex search space.

A modification of tree-seed algorithm using Deb’s rules for constrained optimization

This study focuses on the modification of Tree-Seed Algorithm (TSA) to solve constrained optimization problem. TSA, which is one of the population-based iterative search algorithms, has been developed by inspiration of the relations between trees and seeds grown on a land, and the basic version of TSA has been first used to solve unconstrained optimization problems. In this study, the basic algorithmic process of TSA is modified by using Deb’s rules to solve constrained optimization problems. Deb’s rules are based on the objective function and violation of constraints and it is used to select the trees and seeds that will survive in next iterations. The performance of the algorithm is analyzed under different conditions of control parameters of the proposed algorithm, CTSA for short, and well-known 13 constrained maximization or minimization standard benchmark functions and engineering design optimization problems are employed. The results obtained by the CTSA are compared with the results of particle swarm optimization (PSO), artificial bee colony algorithm (ABC), genetic algorithm (GA) and differential evolution (DE) algorithm on the standard benchmark problems. The results of state-of-art methods are also compared with the proposed algorithm on engineering design optimization problems. The experimental analysis and results show that the proposed method produces promising and comparable results for the constrained optimization benchmark set in terms of solution quality and robustness.

DSLC-FOA : Improved fruit fly optimization algorithm for application to structural engineering design optimization problems

• A new algorithm named DSLC-FOA is developed. • The novel algorithm is tested on four different well-known benchmark problems. • It is compared with the GA, PSO, FOA, LGMS-FOA, and CFOA algorithms. • Experiments show the superiority of the DSLC-FOA in terms of obtained statistical results. In this study, we propose an improved fruit fly optimization algorithm (FOA) based on linear diminishing step and logistic chaos mapping (named DSLC-FOA) for solving benchmark function unconstrained optimization problems and constrained structural engineering design optimization problems. Based on comparisons with genetic algorithm, particle swarm optimization, FOA, LGMS -FOA, and chaotic FOA methods, we demonstrated that DSLC-FOA performed better at searching for the optimal solutions of four typical benchmark functions. The approximate optimal results were obtained using DSLC-FOA for three structural engineering design optimization problems as examples of applications. The numerical results demonstrated that the proposed DSLC-FOA algorithm is superior to the basic FOA and other metaheuristic or deterministic methods.

An on-line Kriging metamodel assisted robust optimization approach under interval uncertainty

PurposeUncertainty is inevitable in real-world engineering optimization. With an outer-inner optimization structure, most previous robust optimization (RO) approaches under interval uncertainty can become computationally intractable because the inner level must perform robust evaluation for each design alternative delivered from the outer level. This paper aims to propose an on-line Kriging metamodel-assisted variable adjustment robust optimization (OLK-VARO) to ease the computational burden of previous VARO approach.Design/methodology/approachIn OLK-VARO, Kriging metamodels are constructed for replacing robust evaluations of the design alternative delivered from the outer level, reducing the nested optimization structure of previous VARO approach into a single loop optimization structure. An on-line updating mechanism is introduced in OLK-VARO to exploit the obtained data from previous iterations.FindingsOne nonlinear numerical example and two engineering cases have been used to demonstrate the applicability and efficiency of the proposed OLK-VARO approach. Results illustrate that OLK-VARO is able to obtain comparable robust optimums as to that obtained by previous VARO, while at the same time significantly reducing computational cost.Practical implicationsThe proposed approach exhibits great capability for practical engineering design optimization problems under interval uncertainty.Originality/valueThe main contribution of this paper lies in the following: an OLK-VARO approach under interval uncertainty is proposed, which can significantly ease the computational burden of previous VARO approach.

A rank-based constraint handling technique for engineering design optimization problems solved by genetic algorithms

This work presents a constraint handling technique (CHT) for the solution of real-world engineering optimization problems by evolutionary algorithms. Referred to as the Multiple Constraint Ranking (MCR), it extends the rank-based approach from many CHTs, by building multiple separate queues based on the values of the objective function and the violation of each constraint. This way, it overcomes difficulties found by other techniques when faced with complex problems characterized by several constraints with different orders of magnitude and/or different units.The MCR follows an “uncoupled” approach where the CHT is not embedded into the optimization algorithm. Extensive studies are performed to assess its accuracy and robustness, compared to six other up-to-date CHTs, all implemented into the same canonical Genetic Algorithm to allow a neutral and unbiased evaluation. The numerical experiments comprise benchmark functions from the IEEE-CEC competitions on constrained optimization, and also classical structural engineering problems. The performance of the CHTs is compared using efficiency measures in terms of nonparametric statistical tests. The results indicate that the MCR is remarkably more accurate and robust for the subset of problems presenting different-magnitude constraints, while remaining very competitive and one of the top-performers for all other benchmark problems comprising the case studies.

A multi-point sampling method based on kriging for global optimization

In general, sampling strategy plays a very important role in metamodel based design optimization, especially when computationally expensive simulations are involved in the optimization process. The research on new optimization methods with less sampling points and higher convergence speed receives great attention in recent years. In this paper, a multi-point sampling method based on kriging (MPSK) is proposed for improving the efficiency of global optimization. The sampling strategy of this method is based on a probabilistic distribution function converted from the expected improvement (EI) function. It can intelligently draw appropriate new samples in an area with certain probability according to corresponding EI values. Besides, three strategies are also proposed to speed up the sequential sampling process and the corresponding convergence criterions are put forward to stop the searching process reasonably. In order to validate the efficiency of this method, it is tested by several numerical benchmark problems and applied in two engineering design optimization problems. Moreover, an overall comparison between the proposed method and several other typical global optimization methods has been made. Results show that the higher global optimization efficiency of this method makes it particularly suitable for design optimization problems involving computationally expensive simulations.

A Hybrid Lightning Search Algorithm-Simplex Method for Global Optimization

In this paper, a novel hybrid lightning search algorithm-simplex method (LSA-SM) is proposed to solve the shortcomings of lightning search algorithm (LSA) premature convergence and low computational accuracy and it is applied to function optimization and constrained engineering design optimization problems. The improvement adds two major optimization strategies. Simplex method (SM) iteratively optimizes the current worst step leaders to avoid the population searching at the edge, thus improving the convergence accuracy and rate of the algorithm. Elite opposition-based learning (EOBL) increases the diversity of population to avoid the algorithm falling into local optimum. LSA-SM is tested by 18 benchmark functions and five constrained engineering design problems. The results show that LSA-SM has higher computational accuracy, faster convergence rate, and stronger stability than other algorithms and can effectively solve the problem of constrained nonlinear optimization in reality.

Global optimization method using adaptive and parallel ensemble of surrogates for engineering design optimization

As a robust and efficient technique for global optimization, surrogate-based optimization method has been widely used in dealing with the complicated and computation-intensive engineering design optimization problems. It’s hard to select an appropriate surrogate model without knowing the behaviour of the real system a priori in most cases. To overcome this difficulty, a global optimization method using an adaptive and parallel ensemble of surrogates combining three representative surrogate models with optimized weight factors has been proposed. The selection of weight factors is treated as an optimization problem with the desired solution being one that minimizes the generalized mean square cross-validation error. The proposed optimization method is tested by considering several well-known numerical examples and one industrial problem compared with other optimization methods. The results show that the proposed optimization method can be a robust and efficient approach in surrogate-based optimization for locating the global optimum.

Sequential Radial Basis Function Using Support Vector Machine for Expensive Design Optimization

To improve the efficiency and quality of solving expensive engineering design optimization problems, recently metamodel-based design and optimization technologies have been widely employed. In this paper, a novel adaptive metamodel-based optimization method called sequential radial basis function using support vector machine is proposed to solve the practical engineering optimization problems involving computationally expensive objective and constraints. The proposed metamodel-based optimization method uses the radial basis function metamodel to approximate the expensive simulations in the interesting sampling region, which is a subregion of the design space. To identify the interesting sampling region, a novel feature space fuzzy -means clustering method is proposed to obtain the cluster center of superior cheap points classified by support vector machine. The procedure of sequential radial basis function using support vector machine is presented first, and the interesting sampling region and feature space fuzzy -means clustering methods are discussed in detail. Several numerical benchmark problems are used to compare the proposed method with other well-known metamodel-based design and optimization methods including efficient global optimization, multiple surrogate efficient global optimization methodology, mode-pursuing sampling, constraint importance mode pursuing sampling, and constrained optimization by radial basis function interpolation. The comparison results show that the proposed method generally outperforms the competitive methods in terms of efficiency, convergence, and robustness. Finally, sequential radial basis function using support vector machine is applied in an Earth observation satellite multidisciplinary design optimization problem. The results illustrate that compared with genetic algorithm and sequential quadratic programming, the proposed method can find a better solution with fewer function evaluations, which demonstrates the practicality and effectiveness of the proposed method in solving real-world engineering design optimization problems.

Set-based many-objective optimization guided by a preferred region

Set-based evolutionary optimization based on performance indicators is one of effective methods to solve many-objective optimization problems. However, preference information of a high-dimensional objective space has not yet been fully used to guide the evolution of a population. In this paper, we propose a set-based many-objective evolutionary algorithm guided by a preferred region. In the set-based evolution, the preferred region of a high-dimensional objective space is dynamically determined, a selection strategy on sets by combining the Pareto dominance on sets with the above preferred region is designed, and the crossover operators on sets guided by the above preferred region are developed to produce a Pareto front with superior performances. The proposed method is applied to four benchmark many-objective optimization problems and a real-world engineering design optimization problem, and the experimental results empirically demonstrate its effectiveness.

Improved global-best-guided particle swarm optimization with learning operation for global optimization problems

In this paper, an improved global-best-guided particle swarm optimization with learning operation (IGPSO) is proposed for solving global optimization problems. The particle population is divided into current population, historical best population and global best population, and each population is assigned a corresponding searching strategy. For the current population, the global neighborhood exploration strategy is employed to enhance the global exploration capability. A local learning mechanism is used to improve local exploitation ability in the historical best population. Furthermore, stochastic learning and opposition based learning operations are employed to the global best population for accelerating convergence speed and improving optimization accuracy. The effects of the relevant parameters on the performance of IGPSO are assessed. Numerical experiments on some well-known benchmark test functions reveal that IGPSO algorithm outperforms other state-of-the-art intelligent algorithms in terms of accuracy, convergence speed, and nonparametric statistical significance. Moreover, IGPSO performs better for engineering design optimization problems.

Global optimization method using ensemble of metamodels based on fuzzy clustering for design space reduction

For most engineering design optimization problems, it is difficult or even impossible to find the global optimum due to the unaffordable computational cost. To overcome this difficulty, a global optimization method that integrates the ensemble of metamodels and fuzzy clustering is proposed to deal with the optimization problems involving the computation-intensive, black-box computer analysis and simulation. The ensemble of metamodels combining three representative metamodeling techniques with optimized weight factors is used to decrease the computational expense during the optimization procedure and the fuzzy clustering technique is applied to obtain the reduced design space. In this way, the efficiency and capability of capturing the global optimum will be improved in the reduced design space. To demonstrate the superior performance of the proposed global optimization method over existing methods, it is examined using various benchmark optimization problems and applied to solve an engineering design optimization problem. The results show that the proposed global optimization method is robust and efficient in capturing the global optimum.

Parallel chaotic local search enhanced harmony search algorithm for engineering design optimization

In this paper, we present a parallel chaotic local search enhanced harmony search algorithm (MHS–PCLS) for solving engineering design optimization problems. The concept of chaos has been previously successfully applied in metaheuristics. However, chaos sequences are sensitive to their initial conditions and cause unstable performance in algorithms. The proposed parallel chaotic local search method searches from several different initial points and diminishes the sensitivity of the initial condition, thereby increasing the robustness of the harmony search method. Numerical benchmark problems are tested to validate the effectiveness of MHS–PCLS. The simulation results confirm that MHS–PCLS obtains superior results for mathematical examples compared to other harmony search variants. Several well-known constrained engineering design problems are also tested using the new approach. The computational results demonstrate that the proposed MHS–PCLS algorithm requires a smaller number of function evaluations and in the majority of cases delivers improved and more robust results compare to other algorithms.