Showing papers in "Engineering Optimization in 2017"
TL;DR: A very recently proposed Jaya algorithm is presented that does not have any algorithm-specific control parameters and hence the burden of tuning the control parameters is minimized and is superior to or competitive with other optimization algorithms for the problems considered.
Abstract: This article presents the performance of a very recently proposed Jaya algorithm on a class of constrained design optimization problems. The distinct feature of this algorithm is that it does not have any algorithm-specific control parameters and hence the burden of tuning the control parameters is minimized. The performance of the proposed Jaya algorithm is tested on 21 benchmark problems related to constrained design optimization. In addition to the 21 benchmark problems, the performance of the algorithm is investigated on four constrained mechanical design problems, i.e. robot gripper, multiple disc clutch brake, hydrostatic thrust bearing and rolling element bearing. The computational results reveal that the Jaya algorithm is superior to or competitive with other optimization algorithms for the problems considered.
147 citations
TL;DR: In this paper, the authors further extended the Jaya algorithm for solving the unconstrained optimization problem, and proposed an extension of the algorithm to solve the constrained optimization problem with constraints.
Abstract: A Jaya algorithm was recently proposed for solving effectively both constrained and unconstrained optimization problems. In this article, the Jaya algorithm is further extended for solving the opti...
101 citations
TL;DR: In this paper, a new version of a biogeography-based optimization algorithm with Levy flight distribution (LFBBO) is introduced and used for the optimum design of reinforced concrete cantilever retaining walls under seismic loading.
Abstract: In this article, a new version of a biogeography-based optimization algorithm with Levy flight distribution (LFBBO) is introduced and used for the optimum design of reinforced concrete cantilever retaining walls under seismic loading. The cost of the wall is taken as an objective function, which is minimized under the constraints implemented by the American Concrete Institute (ACI 318-05) design code and geometric limitations. The influence of peak ground acceleration (PGA) on optimal cost is also investigated. The solution of the problem is attained by the LFBBO algorithm, which is developed by adding Levy flight distribution to the mutation part of the biogeography-based optimization (BBO) algorithm. Five design examples, of which two are used in literature studies, are optimized in the study. The results are compared to test the performance of the LFBBO and BBO algorithms, to determine the influence of the seismic load and PGA on the optimal cost of the wall.
62 citations
TL;DR: A new level-set based topology optimization method is developed for the computational design of multimaterial metamaterials with exotic thermomechanical properties that will produce material geometries with distinct interfaces and smoothed boundaries, which may facilitate the fabrication of the topologically optimized designs.
Abstract: Metamaterials are artificially engineered composites designed to have unusual properties. This article will develop a new level-set based topology optimization method for the computational design of multimaterial metamaterials with exotic thermomechanical properties. In order to generate metamaterials consisting of arrays of microstructures under periodicity, the numerical homogenization method is used to evaluate the effective properties of the microstructure, and a multiphase level-set model is used to evolve the boundaries of the multimaterial microstructure. The proposed method will produce material geometries with distinct interfaces and smoothed boundaries, which may facilitate the fabrication of the topologically optimized designs. Several numerical cases are used to demonstrate the effectiveness of the proposed method.
62 citations
TL;DR: Optimal sizing of hybrid renewable energy systems (HRES) to satisfy load requirements with the highest reliability and lowest cost is a crucial step in building HRESs to supply electricity to remot...
Abstract: Optimal sizing of hybrid renewable energy systems (HRES) to satisfy load requirements with the highest reliability and lowest cost is a crucial step in building HRESs to supply electricity to remot...
52 citations
TL;DR: This article suggests a parameter-free constraint handling approach called constrained non-dominated sorting (CNS), in which each solution in a population is assigned a constrainednon-dominated rank based on its constraint violation degree and Pareto rank.
Abstract: Constrained multi-objective optimization problems (cMOPs) are complex because the optimizer should balance not only between exploration and exploitation, but also between feasibility and optimality. This article suggests a parameter-free constraint handling approach called constrained non-dominated sorting (CNS). In CNS, each solution in a population is assigned a constrained non-dominated rank based on its constraint violation degree and Pareto rank. An improved hybrid multi-objective optimization algorithm called cMOEA/H for solving cMOPs is proposed. Additionally, a dynamic resource allocation mechanism is adopted by cMOEA/H to spare more computational efforts for those relatively hard sub-problems. cMOEA/H is first compared with the baseline algorithm using an existing constraint handling mechanism, verifying the advantages of the proposed constraint handling mechanism. Then cMOEA/H is compared with some classic constrained multi-objective optimizers, experimental results indicating that cMOEA...
48 citations
TL;DR: In this paper, a beam search based heuristic algorithm is proposed to minimize the sum of cycle times over all models of a mixed-model assembly line balancing problem, and the proposed heuristic is tested on benchmark problems and compared with the optimal solutions.
Abstract: In recent years, there has been an increasing trend towards using robots in production systems. Robots are used in different areas such as packaging, transportation, loading/unloading and especially assembly lines. One important step in taking advantage of robots on the assembly line is considering them while balancing the line. On the other hand, market conditions have increased the importance of mixed-model assembly lines. Therefore, in this article, the robotic mixed-model assembly line balancing problem is studied. The aim of this study is to develop a new efficient heuristic algorithm based on beam search in order to minimize the sum of cycle times over all models. In addition, mathematical models of the problem are presented for comparison. The proposed heuristic is tested on benchmark problems and compared with the optimal solutions. The results show that the algorithm is very competitive and is a promising tool for further research.
43 citations
TL;DR: The results demonstrate that the particle categorization mechanism greatly reduces the computational requirements of the PSO-based approaches while maintaining the original search capability of the algorithms in solving optimization problems with computationally cheap objective function and expensive constraints.
Abstract: This article presents an enhanced particle swarm optimization (EPSO) algorithm for size and shape optimization of truss structures. The proposed EPSO introduces a particle categorization mechanism into the particle swarm optimization (PSO) to eliminate unnecessary structural analyses during the optimization process and improve the computational efficiency of the PSO-based structural optimization. The numerical investigation, including three benchmark truss optimization problems, examines the efficiency of the EPSO. The results demonstrate that the particle categorization mechanism greatly reduces the computational requirements of the PSO-based approaches while maintaining the original search capability of the algorithms in solving optimization problems with computationally cheap objective function and expensive constraints.
42 citations
TL;DR: This article presents the first method to simultaneously balance and sequence robotic mixed-model assembly lines (RMALB/S), which involves three sub-problems: task assignment, model sequencing and robot allocation.
Abstract: This article presents the first method to simultaneously balance and sequence robotic mixed-model assembly lines (RMALB/S), which involves three sub-problems: task assignment, model sequencing and robot allocation. A new mixed-integer programming model is developed to minimize makespan and, using CPLEX solver, small-size problems are solved for optimality. Two metaheuristics, the restarted simulated annealing algorithm and co-evolutionary algorithm, are developed and improved to address this NP-hard problem. The restarted simulated annealing method replaces the current temperature with a new temperature to restart the search process. The co-evolutionary method uses a restart mechanism to generate a new population by modifying several vectors simultaneously. The proposed algorithms are tested on a set of benchmark problems and compared with five other high-performing metaheuristics. The proposed algorithms outperform their original editions and the benchmarked methods. The proposed algorithms are a...
41 citations
TL;DR: An efficient improved hybrid Jaya algorithm based on time-varying acceleration coefficients (TVACs) and the learning phase introduced in teaching–learning-based optimization (TLBO), named the LJaya-TVAC algorithm, for solving various types of nonlinear mixed-integer reliability–redundancy allocation problems (RRAPs)
Abstract: This article proposes an efficient improved hybrid Jaya algorithm based on time-varying acceleration coefficients (TVACs) and the learning phase introduced in teaching–learning-based optimization (TLBO), named the LJaya-TVAC algorithm, for solving various types of nonlinear mixed-integer reliability–redundancy allocation problems (RRAPs) and standard real-parameter test functions. RRAPs include series, series–parallel, complex (bridge) and overspeed protection systems. The search power of the proposed LJaya-TVAC algorithm for finding the optimal solutions is first tested on the standard real-parameter unimodal and multi-modal functions with dimensions of 30–100, and then tested on various types of nonlinear mixed-integer RRAPs. The results are compared with the original Jaya algorithm and the best results reported in the recent literature. The optimal results obtained with the proposed LJaya-TVAC algorithm provide evidence for its better and acceptable optimization performance compared to the orig...
40 citations
TL;DR: Comparing the results of the proposed HGFA with other approaches using the well-known PSPLIB benchmark sets validates the effectiveness of the propose algorithm to solve the MRCPSP.
Abstract: In this article, the genetic algorithm (GA) and fully informed particle swarm (FIPS) are hybridized for solving the multi-mode resource-constrained project scheduling problem (MRCPSP) with minimization of project makespan as the objective subject to resource and precedence constraints. In the proposed hybrid genetic algorithm–fully informed particle swarm algorithm (HGFA), FIPS is a popular variant of the particle swarm optimization algorithm. A random key and the related mode list representation schemes are used as encoding schemes, and the multi-mode serial schedule generation scheme (MSSGS) is considered as the decoding procedure. Furthermore, the existing mode improvement procedure in the literature is modified. The results show that the proposed mode improvement procedure remarkably improves the project makespan. Comparing the results of the proposed HGFA with other approaches using the well-known PSPLIB benchmark sets validates the effectiveness of the proposed algorithm to solve the MRCPSP.
TL;DR: In this paper, a robust algorithm is developed in the context of minimizing the expectation of the structural compliance, while conforming to a material volume constraint, to guarantee optimal solutions, sufficient cloud drops are used, which in turn leads to low efficiency.
Abstract: Few researchers have paid attention to designing structures in consideration of uncertainties in the loading locations, which may significantly influence the structural performance. In this work, cloud models are employed to depict the uncertainties in the loading locations. A robust algorithm is developed in the context of minimizing the expectation of the structural compliance, while conforming to a material volume constraint. To guarantee optimal solutions, sufficient cloud drops are used, which in turn leads to low efficiency. An innovative strategy is then implemented to enormously improve the computational efficiency. A modified soft-kill bi-directional evolutionary structural optimization method using derived sensitivity numbers is used to output the robust novel configurations. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed algorithm.
TL;DR: In this article, a multi-objective mathematical model is developed to minimize total time and cost while maximizing the production rate and surface finish quality in the grinding process, which aims to determine optimal values of the decision variables considering process constraints.
Abstract: In this article a multi-objective mathematical model is developed to minimize total time and cost while maximizing the production rate and surface finish quality in the grinding process. The model aims to determine optimal values of the decision variables considering process constraints. A lexicographic weighted Tchebycheff approach is developed to obtain efficient Pareto-optimal solutions of the problem in both rough and finished conditions. Utilizing a polyhedral branch-and-cut algorithm, the lexicographic weighted Tchebycheff model of the proposed multi-objective model is solved using GAMS software. The Pareto-optimal solutions provide a proper trade-off between conflicting objective functions which helps the decision maker to select the best values for the decision variables. Sensitivity analyses are performed to determine the effect of change in the grain size, grinding ratio, feed rate, labour cost per hour, length of workpiece, wheel diameter and downfeed of grinding parameters on each valu...
TL;DR: In this article, a variant of the well-known CVRP called the capacitated vehicle routing problem with order available time (CVRPOAT) is considered, which is observed in the operations of the current e-commerce industry.
Abstract: In this article, a variant of the well-known capacitated vehicle routing problem (CVRP) called the capacitated vehicle routing problem with order available time (CVRPOAT) is considered, which is observed in the operations of the current e-commerce industry. In this problem, the orders are not available for delivery at the beginning of the planning period. CVRPOAT takes all the assumptions of CVRP, except the order available time, which is determined by the precedent order picking and packing stage in the warehouse of the online grocer. The objective is to minimize the sum of vehicle completion times. An efficient tabu search algorithm is presented to tackle the problem. Moreover, a Lagrangian relaxation algorithm is developed to obtain the lower bounds of reasonably sized problems. Based on the test instances derived from benchmark data, the proposed tabu search algorithm is compared with a published related genetic algorithm, as well as the derived lower bounds. Also, the tabu search algorithm is...
TL;DR: In this article, a new multi-objective model for facility location problem with congestion and pricing policies is presented, which considers situations in which immobile service facilities are congested by a stochastic demand following M/M/m/k queues.
Abstract: This article presents a new multi-objective model for a facility location problem with congestion and pricing policies. This model considers situations in which immobile service facilities are congested by a stochastic demand following M/M/m/k queues. The presented model belongs to the class of mixed-integer nonlinear programming models and NP-hard problems. To solve such a hard model, a new multi-objective optimization algorithm based on a vibration theory, namely multi-objective vibration damping optimization (MOVDO), is developed. In order to tune the algorithms parameters, the Taguchi approach using a response metric is implemented. The computational results are compared with those of the non-dominated ranking genetic algorithm and non-dominated sorting genetic algorithm. The outputs demonstrate the robustness of the proposed MOVDO in large-sized problems.
TL;DR: In this paper, a multi-fidelity optimization technique by an efficient global optimization process using a hybrid surrogate model is investigated for solving real-world design problems, where the model constructs the local deviation using the kriging method and the global model using a radial basis function.
Abstract: A multi-fidelity optimization technique by an efficient global optimization process using a hybrid surrogate model is investigated for solving real-world design problems The model constructs the local deviation using the kriging method and the global model using a radial basis function The expected improvement is computed to decide additional samples that can improve the model The approach was first investigated by solving mathematical test problems The results were compared with optimization results from an ordinary kriging method and a co-kriging method, and the proposed method produced the best solution The proposed method was also applied to aerodynamic design optimization of helicopter blades to obtain the maximum blade efficiency The optimal shape obtained by the proposed method achieved performance almost equivalent to that obtained using the high-fidelity, evaluation-based single-fidelity optimization Comparing all three methods, the proposed method required the lowest total number
TL;DR: In this article, an energy absorption model for predicting the effect of loading rates, concrete compressive strength, shear span-to-depth ratio, and longitudinal and transverse reinforcement ratio of reinforced concrete (RC) beams using the particle swarm optimization (PSO) technique.
Abstract: This study proposes an energy absorption model for predicting the effect of loading rates, concrete compressive strength, shear span-to-depth ratio, and longitudinal and transverse reinforcement ratio of reinforced concrete (RC) beams using the particle swarm optimization (PSO) technique. This technique avoids the exhaustive traditional trial-and-error procedure for obtaining the coefficient of the proposed model. Fifty-six RC slender and deep beams are collected from the literature and used to build the proposed model. Three performance measures, namely, mean absolute error, mean absolute percentage error and root mean square error, are investigated in the proposed model to increase its accuracy. The design procedure and accuracy of the proposed model are illustrated and analysed via simulation tests in a MATLAB/Simulink environment. The results indicate the minimal effect of swarm size on the convergence of the PSO algorithm, as well as the ability of PSO to search for an optimum set of coeffici...
TL;DR: In this paper, the authors presented an effective estimation of distribution algorithm, named P-EDA, to solve the blocking flow-shop scheduling problem (BFSP) with the makespan criterion.
Abstract: This article presents an effective estimation of distribution algorithm, named P-EDA, to solve the blocking flow-shop scheduling problem (BFSP) with the makespan criterion. In the P-EDA, a Nawaz–Enscore–Ham (NEH)-based heuristic and the random method are combined to generate the initial population. Based on several superior individuals provided by a modified linear rank selection, a probabilistic model is constructed to describe the probabilistic distribution of the promising solution space. The path relinking technique is incorporated into EDA to avoid blindness of the search and improve the convergence property. A modified referenced local search is designed to enhance the local exploitation. Moreover, a diversity-maintaining scheme is introduced into EDA to avoid deterioration of the population. Finally, the parameters of the proposed P-EDA are calibrated using a design of experiments approach. Simulation results and comparisons with some well-performing algorithms demonstrate the effectiveness...
TL;DR: A continuous-time mixed-integer linear programming model is proposed, which improves an existing discrete-time model in the literature and can run hundreds of times faster than the existing ones for large-size instances.
Abstract: This article addresses bi-objective single-machine batch scheduling under time-of-use electricity prices to minimize the total energy cost and the makespan. The lower and upper bounds on the number of formed batches are first derived and a continuous-time mixed-integer linear programming model is proposed, which improves an existing discrete-time model in the literature. Two improved heuristics are proposed based on the improved model. Computational experiments demonstrate that the improved model and heuristics can run hundreds of times faster than the existing ones for large-size instances.
TL;DR: A two-stage multi-objective buffer allocation approach is applied for robust project scheduling using the meta-heuristic non-dominated sorting genetic algorithm (NSGA-II) based on the decisions made in the buffer allocation step for Pareto-optimal robust schedules.
Abstract: Unpredictable uncertainties cause delays and additional costs for projects. Often, when using traditional approaches, the optimizing procedure of the baseline project plan fails and leads to delays. In this study, a two-stage multi-objective buffer allocation approach is applied for robust project scheduling. In the first stage, some decisions are made on buffer sizes and allocation to the project activities. A set of Pareto-optimal robust schedules is designed using the meta-heuristic non-dominated sorting genetic algorithm (NSGA-II) based on the decisions made in the buffer allocation step. In the second stage, the Pareto solutions are evaluated in terms of the deviation from the initial start time and due dates. The proposed approach was implemented on a real dam construction project. The outcomes indicated that the obtained buffered schedule reduces the cost of disruptions by 17.7% compared with the baseline plan, with an increase of about 0.3% in the project completion time.
TL;DR: In this paper, an evolutionary structural topology optimization method for the design of completely submerged buoyant modules with design-dependent fluid pressure loading is presented, which is used to support offshore rig installation and pipeline transportation at all water depths.
Abstract: This paper presents an evolutionary structural topology optimization method for the design of completely submerged buoyant modules with design-dependent fluid pressure loading. This type of structure is used to support offshore rig installation and pipeline transportation at all water depths. The proposed optimization method seeks to identify the buoy design that has the highest stiffness, allowing it to withstand deepwater pressure, uses the least material and has a minimum prescribed buoyancy. Laplace's equation is used to simulate underwater fluid pressure, and a polymer buoyancy module is considered to be linearly elastic. Both domains are solved with the finite element method. Using an extended bi-directional evolutionary structural optimization (BESO) method, the design-dependent pressure loads are modelled in a straightforward manner without any need for pressure surface parametrization. A new buoyancy inequality constraint sets a minimum required buoyancy effect, measured by the joint volume of the structure and its interior voids. Solid elements with low strain energy are iteratively removed from the initial design domain until a certain prescribed volume fraction. A test case is described to validate the optimization problem, and a buoy design problem is used to explore the features of the proposed method.
TL;DR: An improved multi-objective system reliability-based design optimization (MOSRBDO) method is developed and used to explore the lightweight and high-performance design of a concept car body under uncertainty, and the optimized body-in-white structure signifies a noticeable improvement from the baseline model.
Abstract: This article investigates multi-objective optimization under reliability constraints with applications in vehicle structural design. To improve computational efficiency, an improved multi-objective system reliability-based design optimization (MOSRBDO) method is developed, and used to explore the lightweight and high-performance design of a concept car body under uncertainty. A parametric model knowledge base is established, followed by the construction of a fully parametric concept car body of a multi-purpose vehicle (FPCCB-MPV) based on the knowledge base. The structural shape, gauge and topology optimization are then designed on the basis of FPCCB-MPV. The numerical implementation of MOSRBDO employs the double-loop method with design optimization in the outer loop and system reliability analysis in the inner loop. Multi-objective particle swarm optimization is used as the outer loop optimization solver. An improved multi-modal radial-based importance sampling (MRBIS) method is utilized as the s...
TL;DR: A newly developed metaheuristic method called the cyclical parthenogenesis algorithm (CPA) is used for layout optimization of truss structures subjected to frequency constraints, which imitates the reproductive and social behaviour of some animal species which alternate between sexual and asexual reproduction.
Abstract: Structural optimization with frequency constraints is seen as a challenging problem because it is associated with highly nonlinear, discontinuous and non-convex search spaces consisting of several local optima. Therefore, competent optimization algorithms are essential for addressing these problems. In this article, a newly developed metaheuristic method called the cyclical parthenogenesis algorithm (CPA) is used for layout optimization of truss structures subjected to frequency constraints. CPA is a nature-inspired, population-based metaheuristic algorithm, which imitates the reproductive and social behaviour of some animal species such as aphids, which alternate between sexual and asexual reproduction. The efficiency of the CPA is validated using four numerical examples.
TL;DR: A novel dynamic programming method with a reduced state space algorithm (RSS-DP) is proposed, which decomposing the amount of SBPG into the reference and subsequent allocation, reduces the state space of the SDP model significantly such that the computation time is significantly reduced.
Abstract: Surplus by-product gas (SBPG) in a steel plant is the difference between gas production and consumption. Dynamic programming (DP) has been observed to be a useful method for SBPG dynamic al...
TL;DR: A heuristic algorithm based on a greedy criterion and the linear relaxation algorithm are designed to solve the model and a fuzzy expected value model is presented.
Abstract: A generalized interval fuzzy mixed integer programming model is proposed for the multimodal freight transportation problem under uncertainty, in which the optimal mode of transport and the optimal ...
TL;DR: In this article, the makespan and the total completion time minimization problems in the scheduling of jobs with non-decreasing rates of job processing time on a single machine are considered.
Abstract: There is a situation found in many manufacturing systems, such as steel rolling mills, fire fighting or single-server cycle-queues, where a job that is processed later consumes more time than that same job when processed earlier. The research finds that machine maintenance can improve the worsening of processing conditions. After maintenance activity, the machine will be restored. The maintenance duration is a positive and non-decreasing differentiable convex function of the total processing times of the jobs between maintenance activities. Motivated by this observation, the makespan and the total completion time minimization problems in the scheduling of jobs with non-decreasing rates of job processing time on a single machine are considered in this article. It is shown that both the makespan and the total completion time minimization problems are NP-hard in the strong sense when the number of maintenance activities is arbitrary, while the makespan minimization problem is NP-hard in the ordinary ...
TL;DR: It is proved that the problem of minimizing the makespan subject to limited resource availability can be solved in polynomial time under the condition that the setup times of groups are independent.
Abstract: In this article, single-machine group scheduling with learning effects and convex resource allocation is studied. The goal is to find the optimal job schedule, the optimal group schedule, and resource allocations of jobs and groups. For the problem of minimizing the makespan subject to limited resource availability, it is proved that the problem can be solved in polynomial time under the condition that the setup times of groups are independent. For the general setup times of groups, a heuristic algorithm and a branch-and-bound algorithm are proposed, respectively. Computational experiments show that the performance of the heuristic algorithm is fairly accurate in obtaining near-optimal solutions.
TL;DR: The symbiotic organisms search (SOS) algorithm is presented, a novel metaheuristic optimization technique for solving TEP problems in power systems, inspired by the interactions among organisms in an ecosystem.
Abstract: Transmission expansion planning (TEP) has become a complex problem in restructured electricity markets. This article presents the symbiotic organisms search (SOS) algorithm, a novel metaheuristic optimization technique for solving TEP problems in power systems. The SOS algorithm is inspired by the interactions among organisms in an ecosystem. The TEP problem is formulated here as an optimization problem to determine the cost-effective expansion planning of electrical power systems. Several constraints, such as power flow of the lines, right-of-way validity and maximum line addition, are taken into consideration. First, the SOS algorithm is tested with several benchmark functions. Then, it is applied on three standard power system networks (IEEE 24-bus system, Brazilian 46-bus system and Brazilian 87-bus system) in a TEP study to demonstrate the optimization capability of the proposed SOS algorithm. The results are compared with those produced by other state-of-the-art algorithms.
TL;DR: It is rigorously proved that a surrogate model providing only correct sign prediction of the performance function can meet the accuracy requirement of evidence-theory-based reliability analysis.
Abstract: Structural reliability analysis under evidence theory is investigated. It is rigorously proved that a surrogate model providing only correct sign prediction of the performance function can meet the accuracy requirement of evidence-theory-based reliability analysis. Accordingly, a method based on the active learning kriging model which only correctly predicts the sign of the performance function is proposed. Interval Monte Carlo simulation and a modified optimization method based on Karush–Kuhn–Tucker conditions are introduced to make the method more efficient in estimating the bounds of failure probability based on the kriging model. Four examples are investigated to demonstrate the efficiency and accuracy of the proposed method.
TL;DR: In this paper, a finite element (FE) model for tailor-rolled blank thin-walled (TRB-TH) structures is established and validated by performing a dynamic axial crash test.
Abstract: Tailor-rolled blank thin-walled (TRB-TH) structures have become important vehicle components owing to their advantages of light weight and crashworthiness. The purpose of this article is to provide an efficient lightweight design for improving the energy-absorbing capability of TRB-TH structures under dynamic loading. A finite element (FE) model for TRB-TH structures is established and validated by performing a dynamic axial crash test. Different material properties for individual parts with different thicknesses are considered in the FE model. Then, a multi-objective crashworthiness design of the TRB-TH structure is constructed based on the -support vector regression (-SVR) technique and non-dominated sorting genetic algorithm-II. The key parameters (C, and σ) are optimized to further improve the predictive accuracy of -SVR under limited sample points. Finally, the technique for order preference by similarity to the ideal solution method is used to rank the solutions in Pareto-optimal frontiers a...