Pressure Vessel Design Problem Research Articles

FOX-TSA: Navigating Complex Search Spaces and Superior Performance in Benchmark and Real-World Optimization Problems

In the dynamic field of optimisation, hybrid algorithms have garnered significant attention for their ability to combine the strengths of multiple methods. This study presents the Hybrid FOX-TSA algorithm, a novel optimisation technique that merges the exploratory capabilities of the FOX algorithm with the exploitative power of the TSA algorithm. The primary objective is to evaluate the efficiency, robustness, and scalability of this hybrid approach across multiple CEC benchmark suites, including CEC2014, CEC2017, CEC2019, CEC2020, and CEC2022, alongside real-world engineering design problems. The results demonstrate that the Hybrid FOX-TSA algorithm consistently outperforms established optimisation techniques, such as PSO, GWO, and the original FOX and TSA algorithms, in terms of convergence speed, solution quality, and computational efficiency. Notably, the hybrid approach avoids premature convergence and navigating complex search spaces, producing optimal or near-optimal solutions in various test cases. For instance, the algorithm achieved superior performance in minimizing design costs in the Pressure Vessel and Welded Beam Design problems, as well as effectively handling the complex landscapes of the CEC2020 and CEC2022 benchmarks. These results affirm the Hybrid FOX-TSA algorithm as a powerful and adaptable tool for tackling complex optimization problems, particularly in high-dimensional and multimodal landscapes. The integration of statistical analyses, such as t-tests and Wilcoxon signed-rank tests, further supports the statistical significance of its performance improvements.

HSMAOA: An enhanced arithmetic optimization algorithm with an adaptive hierarchical structure for its solution analysis and application in optimization problems

The Arithmetic Optimization Algorithm (AOA) has recently gained significant attention as a novel meta-heuristic algorithm. However, it faces challenges such as premature convergence and entrapment in local optima when addressing complex optimization problems. To overcome these limitations, this paper proposes an enhanced AOA, termed the Self-Adaptive Hierarchical Arithmetic Optimization Algorithm (HSMAOA). The proposed method integrates three key strategies: Firstly, a spiral-guided random walk mechanism is introduced to improve global search ability. Secondly, a novel adaptive hierarchy leader and follower mechanism is proposed, which establishes a complete multi-branch tree hierarchy with decreasing branching degrees within the population, thereby increasing information exchange among population individuals to escape local optima. Finally, a differential mutation strategy based on ranked selection is introduced to enhance candidate solution quality. HSMAOAʼs performance was evaluated on the CEC2022 test suite against some state-of-the-art algorithms. Results, including optimization accuracy analysis, convergence curves, and various statistical tests, demonstrate HSMAOAʼs superior optimization capability and robustness. In addition, tests on eight engineering structure optimization problems, including the pressure vessel design problem, the multiple disk clutch brake design problem, and the step-cone pulley problem, and so forth, further validate its effectiveness. Thus, HSMAOA shows strong competitiveness in complex optimization tasks and potential for a wide range of applications, and is an advantageous and promising alternative solution for optimization problems.

Learning Harris Hawk Algorithm Based on Signal-to-Noise Ratio

In view of the problem of insufficient population learning and adaptability of the Harris Hawk optimization algorithm, this paper proposes a learning Harris Hawk algorithm based on signal-to-noise ratio, referred to as SLHHO. This algorithm introduces the concept of signal-to-noise ratio to determine the location information of individuals, and designs a coordinated learning strategy that can more reasonably update the location of individuals within the population, and then redesign the escape distance to improve the adaptation and optimization of the algorithm. ability. Using 12 benchmark functions as standards, the performance of this algorithm was tested with variants of the Harris Eagle algorithm and other algorithms, and comparative analysis was conducted in evaluation indicators such as time complexity, diversity, exploration and development, and the results show that SLHHO has strong competitiveness and feasibility. Finally, the practicality of SLHHO was verified in the pressure vessel design problem.

Learning cooking algorithm for solving global optimization problems

In recent years, many researchers have made a continuous effort to develop new and efficient meta-heuristic algorithms to address complex problems. Hence, in this study, a novel human-based meta-heuristic algorithm, namely, the learning cooking algorithm (LCA), is proposed that mimics the cooking learning activity of humans in order to solve challenging problems. The LCA strategy is primarily motivated by observing how mothers and children prepare food. The fundamental idea of the LCA strategy is mathematically designed in two phases: (i) children learn from their mothers and (ii) children and mothers learn from a chef. The performance of the proposed LCA algorithm is evaluated on 51 different benchmark functions (which includes the first 23 functions of the CEC 2005 benchmark functions) and the CEC 2019 benchmark functions compared with state-of-the-art meta-heuristic algorithms. The simulation results and statistical analysis such as the t-test, Wilcoxon rank-sum test, and Friedman test reveal that LCA may effectively address optimization problems by maintaining a proper balance between exploitation and exploration. Furthermore, the LCA algorithm has been employed to solve seven real-world engineering problems, such as the tension/compression spring design, pressure vessel design problem, welded beam design problem, speed reducer design problem, gear train design problem, three-bar truss design, and cantilever beam problem. The results demonstrate the LCA’s superiority and capability over other algorithms in solving complex optimization problems.

Performance Assessment of Natural Survivor Method-Based Metaheuristic Optimizers in Global Optimization and Engineering Design Problems

This study presents the comparative performance analysis of Natural Survivor Method (NSM)-based algorithms in solving the IEEE CEC 2022 test suite benchmark problems and four real-world engineering design problems. Three different variants (Case1, Case2, Case3) of the NSM-TLABC, NSM-SFS and NSM-LSHADE-SPACMA algorithms were used in the study. The data obtained from the experimental studies were statistically analyzed using Friedman and Wilcoxon signed-rank tests. Based on the Friedman test results, NSM-LSHADE-SPACMA_Case2 showed the best performance with an average Friedman score of 3.96. The Wilcoxon signed-rank test showed that NSM-LSHADE-SPACMA_Case2 outperformed its competitors in 13 out of 16 experiments, achieving a success rate of 81.25%. NSM-LSHADE-SPACMA_Case2, which was found to be the most powerful of the NSM-based algorithms, is used to solve cantilever beam design, tension/compression spring design, pressure vessel design and gear train design problems. The optimization results are also compared with eight state-of-the-art metaheuristics, including Rime Optimization Algorithm (RIME), Nonlinear Marine Predator Algorithm (NMPA), Northern Goshawk Optimization (NGO), Kepler Optimization Algorithm (KOA), Honey Badger Algorithm (HBA), Artificial Gorilla Troops Optimizer (GTO), Exponential Distribution Optimization (EDO) and Hunger Games Search (HGS). Given that all results are together, it is seen that NSM-LSHADE-SPACMA_Case2 algorithm consistently produced the best results for the global and engineering design problems studied.

Gorilla optimization algorithm combining sine cosine and cauchy variations and its engineering applications

To address the issues of lacking ability, loss of population diversity, and tendency to fall into the local extreme value in the later stage of optimization searching, resulting in slow convergence and lack of exploration ability of the artificial gorilla troops optimizer algorithm (AGTO), this paper proposes a gorilla search algorithm that integrates the positive cosine and Cauchy's variance (SCAGTO). Firstly, the population is initialized using the refractive reverse learning mechanism to increase species diversity. A positive cosine strategy and nonlinearly decreasing search and weight factors are introduced into the finder position update to coordinate the global and local optimization ability of the algorithm. The follower position is updated by introducing Cauchy variation to perturb the optimal solution, thereby improving the algorithm's ability to obtain the global optimal solution. The SCAGTO algorithm is evaluated using 30 classical test functions of Test Functions 2018 in terms of convergence speed, convergence accuracy, average absolute error, and other indexes, and two engineering design optimization problems, namely, the pressure vessel optimization design problem and the welded beam design problem, are introduced for verification. The experimental results demonstrate that the improved gorilla search algorithm significantly enhances convergence speed and optimization accuracy, and exhibits good robustness. The SCAGTO algorithm demonstrates certain solution advantages in optimizing the pressure vessel design problem and welded beam design problem, verifying the superior optimization ability and engineering practicality of the SCAGTO algorithm.

Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution

The Gray Wolf Optimizer (GWO) is an established algorithm for addressing complex optimization tasks. Despite its effectiveness, enhancing its precision and circumventing premature convergence is crucial to extending its scope of application. In this context, our study presents the Cauchy Gray Wolf Optimizer (CGWO), a modified version of GWO that leverages Cauchy distributions for key algorithmic improvements. The innovation of CGWO lies in several areas: First, it adopts a Cauchy distribution-based strategy for initializing the population, thereby broadening the global search potential. Second, the algorithm integrates a dynamic inertia weight mechanism, modulated non-linearly in accordance with the Cauchy distribution, to ensure a balanced trade-off between exploration and exploitation throughout the search process. Third, it introduces a Cauchy mutation concept, using inertia weight as a probability determinant, to preserve diversity and bolster the capability for escaping local optima during later search phases. Furthermore, a greedy strategy is employed to incrementally enhance solution accuracy. The performance of CGWO was rigorously evaluated using 23 benchmark functions, demonstrating significant improvements in convergence rate, solution precision, and robustness when contrasted with conventional algorithms. The deployment of CGWO in solving the engineering challenge of pressure vessel design illustrated its superiority over traditional methods, highlighting its potential for widespread adoption in practical engineering contexts.

A novel chaotic transient search optimization algorithm for global optimization, real-world engineering problems and feature selection.

Metaheuristic optimization algorithms manage the search process to explore search domains efficiently and are used efficiently in large-scale, complex problems. Transient Search Algorithm (TSO) is a recently proposed physics-based metaheuristic method inspired by the transient behavior of switched electrical circuits containing storage elements such as inductance and capacitance. TSO is still a new metaheuristic method; it tends to get stuck with local optimal solutions and offers solutions with low precision and a sluggish convergence rate. In order to improve the performance of metaheuristic methods, different approaches can be integrated and methods can be hybridized to achieve faster convergence with high accuracy by balancing the exploitation and exploration stages. Chaotic maps are effectively used to improve the performance of metaheuristic methods by escaping the local optimum and increasing the convergence rate. In this study, chaotic maps are included in the TSO search process to improve performance and accelerate global convergence. In order to prevent the slow convergence rate and the classical TSO algorithm from getting stuck in local solutions, 10 different chaotic maps that generate chaotic values instead of random values in TSO processes are proposed for the first time. Thus, ergodicity and non-repeatability are improved, and convergence speed and accuracy are increased. The performance of Chaotic Transient Search Algorithm (CTSO) in global optimization was investigated using the IEEE Congress on Evolutionary Computation (CEC)'17 benchmarking functions. Its performance in real-world engineering problems was investigated for speed reducer, tension compression spring, welded beam design, pressure vessel, and three-bar truss design problems. In addition, the performance of CTSO as a feature selection method was evaluated on 10 different University of California, Irvine (UCI) standard datasets. The results of the simulation showed that Gaussian and Sinusoidal maps in most of the comparison functions, Sinusoidal map in most of the real-world engineering problems, and finally the generally proposed CTSOs in feature selection outperform standard TSO and other competitive metaheuristic methods. Real application results demonstrate that the suggested approach is more effective than standard TSO.

A Novel Variant of the Salp Swarm Algorithm for Engineering Optimization

Abstract There are many design problems need to be optimized in various fields of engineering, and most of them belong to the NP-hard problem. The meta-heuristic algorithm is one kind of optimization method and provides an effective way to solve the NP-hard problem. Salp swarm algorithm (SSA) is a nature-inspired algorithm that mimics and mathematically models the behavior of slap swarm in nature. However, similar to most of the meta-heuristic algorithms, the traditional SSA has some shortcomings, such as entrapment in local optima. In this paper, the three main strategies are adopted to strengthen the basic SSA, including chaos theory, sine-cosine mechanism and the principle of quantum computation. Therefore, the SSA variant is proposed in this research, namely SCQ-SSA. The representative benchmark functions are employed to test the performances of the algorithms. The SCQ-SSA are compared with the seven algorithms in high-dimensional functions (1000 dimensions), seven SSA variants and six advanced variants on benchmark functions, the experiment reveals that the SCQ-SSA enhances resulting precision and alleviates local optimal problems. Besides, the SCQ-SSA is applied to resolve three classical engineering problems: tubular column design problem, tension/compression spring design problem and pressure vessel design problem. The design results indicate that these engineering problems are optimized with high accuracy and superiority by the improved SSA. The source code is available in the URL: https://github.com/ye-zero/SCQSSA/tree/main/SCQ-SSA.

Implementation of the Sine - Cosine Algorithm to the Pressure Vessel Design Problem

Implementation of the Sine - Cosine Algorithm to the Pressure Vessel Design Problem

An Improved Crystal Structure Algorithm for Engineering Optimization Problems

Crystal Structure Algorithm (CryStAl) is a new meta-heuristic algorithm, and it has been studied by many scholars because of its wide adaptability and the fact that there is no need to set parameters in advance. An improved crystal structure algorithm (GLCryStAl) based on golden sine operator and Levy flight is designed in this paper. The algorithm makes good use of the relationship between the golden sine operator and the unit circle to make the algorithm exploration space more comprehensive, and then gradually narrows the search space in the iterative process, which can effectively speed up the convergence rate of the algorithm. At the same time, a Levy operator is introduced to help the algorithm effectively get rid of the attraction of local optimal value. To evaluate the performance of GLCryStAl, 12 classic benchmark functions and eight CEC2017 test functions were selected to design a series of comparative experiments. In addition, the experimental data of these algorithms are analyzed using the Wilcoxon and Friedman tests. Through these two tests, it can be found that GLCryStAl has significant advantages over other algorithms. Finally, this paper further tests the optimization performance of GLCryStAl in engineering design. GLCryStAl was applied to optimize pressure vessel design problems and tension/compression spring design problems. The optimization results show that GLCryStAl is feasible and effective in optimizing engineering design.

Self-Adapting Spherical Search Algorithm with Differential Evolution for Global Optimization

The spherical search algorithm is an effective optimizer to solve bound-constrained non-linear global optimization problems. Nevertheless, it may fall into the local optima when handling combination optimization problems. This paper proposes an enhanced self-adapting spherical search algorithm with differential evolution (SSDE), which is characterized by an opposition-based learning strategy, a staged search mechanism, a non-linear self-adapting parameter, and a mutation-crossover approach. To demonstrate the outstanding performance of the SSDE, eight optimizers on the CEC2017 benchmark problems are compared. In addition, two practical constrained engineering problems (the welded beam design problem and the pressure vessel design problem) are solved by the SSDE. Experimental results show that the proposed algorithm is highly competitive compared with state-of-the-art algorithms.

An adaptive quadratic interpolation and rounding mechanism sine cosine algorithm with application to constrained engineering optimization problems

The sine cosine algorithm (SCA) is a well-known meta-heuristic optimization algorithm. SCA has received much attention in various optimization fields due to its simple structure and excellent optimization capabilities. However, the dimension of objective function also increases with the increasing complexity of optimization tasks. This makes the original SCA appear to have insufficient optimization capability and likely to fall into premature convergence. A multi-mechanism acting variant of SCA, called ARSCA, is proposed to address the above deficiencies. ARSCA is an enhanced SCA algorithm based on the adaptive quadratic interpolation mechanism (AQIM) and Rounding mechanism (RM). RM enables a more balanced state between exploration and exploitation of the ARSCA. AQIM enhances local exploitation capabilities. To verify the performance of ARSCA, we compared ARSCA with some advanced traditional optimization algorithms and variants of algorithms for 30 consecutive benchmark functions of IEEE CEC2014. In addition, ARSCA was applied to 6 constrained engineering optimization problems. These six algorithms include the tension–compression spring design problem, the welded beam design problem, the pressure vessel design problem, the I-beam design problem, the speed reducer design problem, and the three-bar design problem. Experimental results show that ARSCA outperforms its competitors in both the solution quality and the ability to jump out of the local optimum. The relevant codes for the paper are publicly available at https://github.com/YangXiao9799/paper_ARSCA.

Application of Cuckoo Search Algorithm in Cost Estimation of Building Energy Engineering

In order to solve the problem that the construction project cost estimation is at a relatively advanced stage, many information about the project cannot be determined, and many unforeseen things affecting the project cost will occur in the process of project construction, which makes the preparation of project investment estimation very difficult. An optimized firefly algorithm is proposed. By introducing cuckoo algorithm, the initial population of fireflies is optimized, which greatly improves the quality of the initial population and speeds up the convergence of fireflies to the optimal solution; Secondly, the performance of CSFA algorithm is tested by six standard test functions; Finally, the algorithm is applied to solve the pressure vessel design problem. The effectiveness of the experiment is verified.

A novel version of slime mould algorithm for global optimization and real world engineering problems: Enhanced slime mould algorithm

The slime mould algorithm is a stochastic optimization algorithm based on the oscillation mode of nature’s slime mould, and it has effective convergence. On the other hand, it gets stuck at the local optimum and struggles to find the global optimum. Location updates of slime moulds are very important in terms of convergence to optimum. In this study, the position updates of the sine cosine algorithm are combined with the slime mould algorithm. In these updates, besides the existing sine cosine algorithm, different types of sine cosine algorithmic transformations are used and the oscillation processes of the slime moulds are also modified. In the mathematical model of the slime mould algorithm, the arctanh function that stacks two random slime moulds in a certain interval has been replaced by a novel modified sigmoid function. The proposed function is presented with its theoretical derivations based on Schwarz lemma. According to experimental results, it has been observed that the exploration and exploitation capabilities of the proposed algorithm are highly effective. In the study, sine cosine trigonometric functions have been used while updating the position in slime mould algorithm. The performance of the presented algorithm has been considered for fifty benchmark functions and has also been tested on cantilever beam design, pressure vessel design, 3-bar truss and speed reducer real world problems. Accordingly, it is possible to conclude that the proposed hybrid algorithm has better ability to escape from local optima with faster convergence than standard sine cosine and slime mould algorithms.

Gaussian Perturbation Specular Reflection Learning and Golden-Sine-Mechanism-Based Elephant Herding Optimization for Global Optimization Problems.

Elephant herding optimization (EHO) has received widespread attention due to its few control parameters and simple operation but still suffers from slow convergence and low solution accuracy. In this paper, an improved algorithm to solve the above shortcomings, called Gaussian perturbation specular reflection learning and golden-sine-mechanism-based EHO (SRGS-EHO), is proposed. First, specular reflection learning is introduced into the algorithm to enhance the diversity and ergodicity of the initial population and improve the convergence speed. Meanwhile, Gaussian perturbation is used to further increase the diversity of the initial population. Second, the golden sine mechanism is introduced to improve the way of updating the position of the patriarch in each clan, which can make the best-positioned individual in each generation move toward the global optimum and enhance the global exploration and local exploitation ability of the algorithm. To evaluate the effectiveness of the proposed algorithm, tests are performed on 23 benchmark functions. In addition, Wilcoxon rank-sum tests and Friedman tests with 5% are invoked to compare it with other eight metaheuristic algorithms. In addition, sensitivity analysis to parameters and experiments of the different modifications are set up. To further validate the effectiveness of the enhanced algorithm, SRGS-EHO is also applied to solve two classic engineering problems with a constrained search space (pressure-vessel design problem and tension-/compression-string design problem). The results show that the algorithm can be applied to solve the problems encountered in real production.

An ensemble weighted average conservative multi-fidelity surrogate modeling method for engineering optimization

Multi-fidelity (MF) surrogate models have been widely used in engineering optimization problems to reduce the design cost by replacing computat ional expensive simulations. Ignoring the prediction uncertainty of the MF model that is caused by a limited number of samples may result in infeasible solutions. Conservative MF surrogate model, which can effectively improve the feasibility of the constraints, has been a promising way to address this issue. In this paper, an ensemble weighted average (EWA) conservative multi-fidelity modeling method that integrates the performance of different error metrics is proposed. In the proposed method, the bootstrap method and mean-square-error method are reasonably weighted to calculate the safety margin of the MF surrogate model. The weights for the two metrics are determined through an optimization problem, which considers the performance of the two metrics in different subsets of the sample points. The effectiveness of the proposed method is illustrated through several numerical examples and a pressure vessel design problem. Results show that the proposed method constructs a more accurate conservative MF surrogate model than other methods in different problems. Furthermore, applying the constructed conservative MF surrogate model into optimization problems obtains more accurate optimal solutions while ensuring the feasibility of it.

Application of Particle Swarm Optimization for Classical Engineering Problems

In the design of mechanical elements, designers usually consider certain objectives that are related with cost, time, quality and reliability of product, depending on the requirements. In this paper, parametric optimization of spring design problem, pressure vessel design problem, cantilever beam design problem, cone coupling design problem and welded beam design problem has been carried out using Particle Swarm Optimization (PSO for short). The pseudo code for this algorithm was written using Matlab R2018a software suite. Results of the PSO algorithm will be compared to results obtained by the Differential Evolution (DE), Modified Ant Colony Algorithm, (MACA), Grasshopper Optimization Algorithm (GOA), Water Cycle Algorithm (WCA), Cucko Search (CS) , Genetic Algorithm (GA), Ant Lion Optimization (ALO), Firefly Algorithm (FA) and Method of Moving Asymptotes (MMA), depending of solutions found in literature. The source code of Particle Swarm Optimization (PSO) algorithm is publicly available at https://seyedalimirjalili.com.

Global Optimization for Mixed–Discrete Structural Design

Several structural design problems that involve continuous and discrete variables are very challenging because of the combinatorial and non-convex characteristics of the problems. Although the deterministic optimization approach theoretically guarantees to find the global optimum, it usually leads to a significant burden in computational time. This article studies the deterministic approach for globally solving mixed–discrete structural optimization problems. An improved method that symmetrically reduces the number of constraints for linearly expressing signomial terms with pure discrete variables is applied to significantly enhance the computational efficiency of obtaining the exact global optimum of the mixed–discrete structural design problem. Numerical experiments of solving the stepped cantilever beam design problem and the pressure vessel design problem are conducted to show the efficiency and effectiveness of the presented approach. Compared with existing methods, this study introduces fewer convex terms and constraints for transforming the mixed–discrete structural problem and uses much less computational time for solving the reformulated problem to global optimality.

Hybrid metaheuristic algorithm using butterfly and flower pollination base on mutualism mechanism for global optimization problems

The butterfly optimization algorithm (BOA) is a new metaheuristic algorithm that is inspired from food foraging behavior of the butterflies. Because of its simplicity and effectiveness, the algorithm has been proved to be effective in solving global optimization problems and applied to practical problems. However, BOA is prone to local optimality and may lose its diversity, thus suffering losses of premature convergence. In this work, a hybrid metaheuristic algorithm using butterfly and flower pollination base on mutualism mechanism called MBFPA was proposed. Firstly, the flower pollination algorithm has good exploration ability and the hybrid butterfly optimization algorithm and the flower pollination algorithms greatly improve the exploration ability of the algorithm; secondly, the symbiosis organisms search has a strong exploitation capability in the mutualism phase. By introducing the mutualism phase, the algorithm's exploitation capability is effectively increased and the algorithm's convergence speed is accelerated. Finally, the adaptive switching probability is increased to increase the algorithm's balance in exploration and exploitation capabilities. In order to evaluate the effectiveness of the algorithm, in the 49 standard test functions, the proposed algorithm was compared with six basic metaheuristic algorithms and five hybrid metaheuristic algorithms. MBFPA has also been used to solve five classic engineering problems (three-bar truss design problem; multi-plate disc clutch brake design; welded beam design; pressure vessel design problem; and speed reducer design). The results show that the proposed method is feasible and has good application prospect and competitiveness.