Single-objective Optimization Problem Research Articles

3D Path Planning of the Solar Powered UAV in the Urban-Mountainous Environment with Multi-Objective and Multi-Constraint Based on the Enhanced Sparrow Search Algorithm Incorporating the Levy Flight Strategy

In response to practical application challenges in utilizing solar-powered unmanned aerial vehicle (UAV) for remote sensing, this study presents a three-dimensional path planning method tailored for urban-mountainous environment. Taking into account constraints related to the solar-powered UAV, terrain, and mission objectives, a multi-objective trajectory optimization model is transferred into a single-objective optimization problem with weight factors and multi-constraint and is developed with a focus on three key indicators: minimizing trajectory length, maximizing energy flow efficiency, and minimizing regional risk levels. Additionally, an enhanced sparrow search algorithm incorporating the Levy flight strategy (SSA-Levy) is introduced to address trajectory planning challenges in such complex environments. Through simulation, the proposed algorithm is compared with particle swarm optimization (PSO) and the regular sparrow search algorithm (SSA) across 17 standard test functions and a simplified simulation of urban-mountainous environments. The results of the simulation demonstrate the superior effectiveness of the designed improved SSA based on the Levy flight strategy for solving the established single-objective trajectory optimization model.

Solving Convex Multi-Objective Optimization Problems via a Capable Neural Network Scheme

A neural network architecture is constructed to solve Convex Multi-objective Programming Problem (CMPP). By using the weighted sum method, a single-objective optimization problem related to the CMPP is formulated, in which the scalar objective is summation of weighted original objective functions. The Pareto Optimal Solution (POS) is given by using different values of weights. A neural network framework is then designed for solving the obtained convex problem. Based on employing Lyapunov theory, the proposed model is established to be stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the convex programming problem with different values of weights. The computer simulations shows the feasibility and the efficiency of the suggested model.

A short-term wind-hydrothermal operational framework in the presence of pumped-hydro storage

Short-term operation is one of the most significant aspects of power systems operation and planning and it mainly includes resource scheduling including renewable and non-renewable energy sources. The mentioned problem is modeled as an optimization problem with one or more objectives depending upon the priorities of the decision maker (system operator) and several equality and inequality constraints. In this respect, this paper proposes a short-term operational model for the hydrothermal power systems in the presence of wind power generation and pumped-hydro storage (PHS) units. The problem is formulated as a single-objective optimization problem that aims to minimize the total operating cost including the fuel cost of thermal units. The problem has been comprehensively investigated through five case studies where the first case study is used for the sake of validation. Then, the other case studies assessed the impacts of wind power generation, PHS unit, and uncertainty on the problem. The five case studies are as follows: Case 1: Short-term hydrothermal (STH) scheduling which aims to minimize the total cost. Case 2: Deterministic short-term hydrothermal-wind (STHTW) scheduling which aims to minimize the total cost. Case 3: Stochastic wind-hydro-thermal scheduling that seeks to minimize the total cost taking into consideration the uncertainty of the wind power generation. Case 4: Deterministic wind-hydro-thermal scheduling in the presence of the PHS unit with the total cost minimization as the objective function. Case 5: Stochastic wind-hydro-thermal scheduling in the presence of the PHS unit that is used to evaluate the impact of uncertainty of wind power generation with the total cost minimization as the objective function. It is noteworthy that the first three case studies have been formulated using a non-linear programming (NLP) model and solved using the CONOPT solver in GAMS. Case studies 4 and 5 have been presented using a mixed-integer non-linear programming (MINLP) model and solved using the DICOPT solver in GAMS. A sensitivity analysis has also been carried out to assess the system's performance with different loading conditions. In this respect, the total cost and PHS operation with different amounts of load demand were evaluated and the results have been presented and discussed.

A review of nature-inspired algorithms on single-objective optimization problems from 2019 to 2023

The field of nature inspired algorithm (NIA) is a vital area of research that consistently aids in solving optimization problems. One of the metaheuristic algorithm classifications that has drawn attention from researchers in recent decades is NIA. It makes a significant contribution by addressing numerous large-scale problems and achieving the best results. This research aims to identify the optimal NIA for solving single-objective optimization problems. The NIA discovered between 2019 and 2023 is presented in this study with a brief description. About 83 distinct NIAs have been studied in this study in order to address the optimization issues. In order to accomplish this goal, we have taken into consideration eight real-world single-objective optimization problems: the 3-bar truss design problem, the rolling element bearing, the pressure vessel, the cantilever beam, the I beam, the design of a welded beam, and the design of a spring. Based on a comparative study and bibliographic analysis, we have determined that two algorithms—the flow direction algorithm, and prairie dog optimization—give us the best results and optimal solutions for all eight of the engineering problems listed. Lastly, some perspectives on the limitations, difficulties, and future course are provided. In addition to providing future research guidelines, this will assist the novice and emerging researcher in providing a more comprehensive perspective on advanced NIA.

Power allocation scheme for sum rate and fairness trade-off in downlink NOMA networks

Non-orthogonal multiple access (NOMA) is an essential enabler technology that is expected to help satisfy the key requirements of increased system throughput in future wireless networks. However, another equally important aspect that should go hand-in-hand with system throughput is user fairness for any network. But, to the best of our knowledge, even though there have been works that look at system throughput or user fairness maximization for NOMA-based networks, they looked at these as a single objective optimization problem, where one is the objective and the other is one of the constraints. However, quite often, joint optimization of both system throughput and user fairness is required to make optimized decisions in the face of trade-offs between these two equally important but conflicting objectives. In this regard, this paper formulates a multi-objective optimization problem to jointly maximize the sum rate and user fairness in a downlink transmission NOMA system, through optimize power allocation (PA), under system-imposed constraints. A weighted sum approach is used to turn the multi-objective optimization problem into a single-objective optimization problem to make it analytically tractable. The optimized PA is then obtained using Lagrange dual decomposition method and Karush–Kuhn–Tucker (KKT) conditions. Using our derived expressions, we propose an iterative PA algorithm that converges fast enough to be employed in practical NOMA networks. We also present simulation results to highlight the effectiveness of the proposed solution. Further, the performance of the proposed method is compared with that of the benchmark methods.

Augmenting bi-objective branch and bound by scalarization-based information

AbstractWhile branch and bound based algorithms are a standard approach to solve single-objective (mixed-)integer optimization problems, multi-objective branch and bound methods are only rarely applied compared to the predominant objective space methods. In this paper we propose modifications to increase the performance of multi-objective branch and bound algorithms by utilizing scalarization-based information. We use the hypervolume indicator as a measure for the gap between lower and upper bound set to implement a multi-objective best-first strategy. By adaptively solving scalarizations in the root node to integer optimality we improve both, upper and lower bound set. The obtained lower bound can then be integrated into the lower bounds of all active nodes, while the determined solution is added to the upper bound set. Numerical experiments show that the number of investigated nodes can be significantly reduced by up to 83% and the total computation time can be reduced by up to 80%.

The Multiobjective Control Based on Tolerance Optimization in a Multienergy System

To address the issue of multiobjective control in multienergy systems with diverse operational objectives, a two-stage optimization framework based on expected point tolerance has been proposed in this paper. In the first stage, a single objective function is used for optimization control to obtain the expected point of the multiobjective optimization problem. Then, in the second stage, by defining the allowable deviation between each optimization objective and the expected point, the original multiobjective optimization problem is transformed into a single objective optimization problem solution with tolerance measurement. Finally, in the simulation scene of a multienergy system, it is demonstrated that compared with the optimal results under each single objective method, the proposed method increases power line loss, maximum voltage deviation, new energy consumption, and economy by 2.22, 2.30, 1.02, and 2.45 times, respectively. Compared with the suboptimal results, the proposed method reduces power line loss by 22.26, 1.74, 1.09, and 0.97 times, respectively. Combining the shape of the Pareto frontier, it is demonstrated that the proposed method can comprehensively consider the needs of multiple power optimization objectives for forming a more reasonable and effective system optimization scheduling and also provide a new approach for solving multiobjective optimization problems.

Optimal Constrained Groove Pressing Process Parameters Applying Modified Taguchi Technique and Multi-Objective Optimization

Most engineering problems are complicated, and developing mathematical models for such problems requires understanding the phenomena through experiments. It is well known that as processing parameters with assigned levels increase, so does the number of experiments. By minimizing the number of experiments, Taguchi’s method of experimental design will help to furnish the idea of full factorial experimental design. Taguchi’s method is more appropriate for single-objective optimization problems and needs modifications while dealing with multi-objective optimization problems. Aluminum alloys are in great demand in today’s automotive and aerospace sectors due to their low density, good corrosion resistance, and excellent machinability. The material is subjected to a constrained groove pressing (CGP) process to obtain microstructural grain refinement with enhanced mechanical behavior. This paper considers AA6061 material having major alloys such as silicon and magnesium. For this work, 3 CGP process parameters (viz., displacement rate, plate thickness and number of passes) are assigned 3 levels to each parameter, acquired the test data, viz., grain size (gs), micro hardness (hs), and tensile strength (ult) based on L9 orthogonal array of Taguchi. Using a modified version of Taguchi’s methodology, it is possible to estimate the range of grain size (gs), micro hardness (hs), and tensile strength (σult) for effective combinations of the CGP processing parameters and validate the results with existing test data. A more dependable and simpler multi-objective optimization procedure is used to choose the optimal CGP processing parameters.

A nondominated sorting genetic model for co-clustering

Co-clustering aims to cluster the rows and columns of data simultaneously and can be often formulated as a two-objective optimization problem (one objective for rows and the other for columns) and the solution is a Pareto-optimal solution set in principle. Existing methods usually convert the co-clustering problem into a single-objective optimization problem by setting a hyper-parameter λ between the two objectives. However, finding a good value of λ is not easy because of its large parameter space; also, there may exist many equally good λs. In this paper, a nondominated sorting genetic model (NSGC) is proposed to tackle the co-clustering problem, totally bypassing the trade-off parameter λ and returning to the original two-objective problem. The core of our model is to group a row objective function and a column objective function and integrate them into a genetic algorithm as the fitness functions. After this reformulation, we follow a standard genetic algorithm procedure to iteratively find the Pareto-optimal solutions. Finally, to fish out a single best solution we further design a sorting criterion according to which the Pareto-optimal solution set can be totally ordered. Extensive experiments with 16 public datasets are conducted, and the results demonstrate the superiority of our approach.

A Lagrangian bounding and heuristic principle for bi-objective discrete optimization

Lagrangian relaxation is a common and often successful way to approach computationally challenging single-objective discrete optimization problems with complicating side constraints. Its aim is often twofold; first, it provides bounds for the optimal value, and, second, it can be used to heuristically find near-optimal feasible solutions, the quality of which can be assessed by the bounds. We consider bi-objective discrete optimization problems with complicating side constraints and extend this Lagrangian bounding and heuristic principle to such problems. The Lagrangian heuristic here produces non-dominated candidates for points on the Pareto frontier, while the bounding forms a polyhedral outer approximation of the Pareto frontier, which can be used to assess the quality of the candidate points. As an illustration example we consider a facility location problem in which both CO2 emission and cost should be minimized. The computational results are very encouraging, both with respect to bounding and the heuristically found non-dominated solutions. In particular, the Lagrangian bounding is much stronger than the outer approximation given by the Pareto frontier of the problem’s linear programming relaxation.

An Improved Grey Wolf Optimizer with Multi-Strategies Coverage in Wireless Sensor Networks

For wireless sensor network (WSN) coverage problems, since the sensing range of sensor nodes is a circular area with symmetry, taking symmetry into account when deploying nodes will help simplify problem solving. In addition, in view of two specific problems of high node deployment costs and insufficient effective coverage in WSNs, this paper proposes a WSN coverage optimization method based on the improved grey wolf optimizer with multi-strategies (IGWO-MS). As far as IGWO-MS is concerned, first of all, it uses Sobol sequences to initialize the population so that the initial values of the population are evenly distributed in the search space, ensuring high ergodicity and diversity. Secondly, it introduces a search space strategy to increase the search range of the population, avoid premature convergence, and improve search accuracy. And then, it combines reverse learning and mirror mapping to expand the population richness. Finally, it adds Levy flight to increase the disturbance and improve the probability of the algorithm jumping out of the local optimum. To verify the performance of IGWO-MS in WSN coverage optimization, this paper rasterizes the coverage area of the WSN into multiple grids of the same size and symmetry with each other, thereby transforming the node coverage problem into a single-objective optimization problem. In the simulation experiment, not only was IGWO-MS selected, but four other algorithms were also selected for comparison, namely particle swarm optimization (PSO), grey wolf optimizer (GWO), grey wolf optimization based on drunk walk (DGWO), and grey wolf optimization led by two-headed wolves (GWO-THW). The experimental results demonstrate that when the number of nodes for WSN coverage optimization is 20 and 30, the optimal coverage rate and average coverage rate using IGWO-MS are both improved compared to the other four comparison algorithms. To make this clear, in the case of 20 nodes, the optimal coverage rate of IGWO-MS is increased by 13.19%, 1.68%, 4.92%, and 3.62%, respectively, compared with PSO, GWO, DGWO, and GWO-THW; while IGWO-MS performs even better in terms of average coverage rate, which is 16.45%, 3.13%, 11.25%, and 6.19% higher than that of PSO, GWO, DGWO, and GWO-THW, respectively. Similarly, in the case of 30 nodes, compared with PSO, GWO, DGWO, and GWO-THW, the optimal coverage rate of the IGWO-MS is increased by 15.23%, 1.36%, 5.55%, and 3.66%; the average coverage rate is increased by 16.78%, 1.56%, 10.91%, and 8.55%. Therefore, it can be concluded that IGWO-MS has certain advantages in solving WSN coverage problems, which is reflected in that not only can it effectively improve the coverage quality of network nodes, but it also has good stability.

OPTIMAL SIZING OF SOLAR-WIND HYBRID MICROGRID USING IMPROVED GREY WOLF OPTIMIZATION ALGORITHM A CASE STUDY OF KADUNA - NIGERIA

This paper presents an improved grey wolf optimization algorithm (IGWOA) for optimal sizing of an isolated photovoltaic (PV), wind turbine (WT), and battery energy storage (BES) hybrid microgrid. To demonstrate the effectiveness of the proposed approach, atmospheric data sets comprising of wind, solar, and temperature of Kaduna International Airport were collected from Nigerian Meteorological Agency while the load demand data was collected from Kaduna International Airport Electricity Distribution Center. The microgrid optimal sizing was formulated as a constrained single objective optimization problem. Constraints including, loss of power supply probability (LPSP), power balance, generation limits and battery state of charge (SOC) were imposed. Three simulation scenarios were considered. Firstly, the target allowable maximum LPSP was fixed at 25% and the algorithm was able to determine the optimal sizing of the hybrid microgrid components and minimize the initial cost from 169,880.00 USD to 112,356.40 USD per annum resulting in 34% savings in cost. Secondly, the effect of the target allowable maximum LPSP variation was investigated, and it was found that the total installed capacity of the system decreases with increase in LPSP thereby decreasing the total cost. Additionally, a novel electricity price index (EPI) was introduced in order to quantify the degree of optimality of the solution. The EPI was found to increase exponentially with increase in LPSP, resulting in an EPI of < 0.05USD/kWh at 20% LPSP. Lastly, to validate the proposed approach, a comparative analysis between the IGWOA and other algorithms was carried out, and the proposed IGWOA proved applicable.

Nested Markov chain hyper-heuristic (NMHH): a hybrid hyper-heuristic framework for single-objective continuous problems.

This article introduces a new hybrid hyper-heuristic framework that deals with single-objective continuous optimization problems. This approach employs a nested Markov chain on the base level in the search for the best-performing operators and their sequences and simulated annealing on the hyperlevel, which evolves the chain and the operator parameters. The novelty of the approach consists of the upper level of the Markov chain expressing the hybridization of global and local search operators and the lower level automatically selecting the best-performing operator sequences for the problem. Numerical experiments conducted on well-known benchmark functions and the comparison with another hyper-heuristic framework and six state-of-the-art metaheuristics show the effectiveness of the proposed approach.

Optimization of a Two-stage Bevel Helical Gearbox using Multiple Objectives to Increase Efficiency and Reduce Gearbox Bottom Area

This study aims to look at multi-target optimization of two-stage bevel helical gearboxes to determine the best major design factors for reducing gearbox bottom area (GBA) and increasing gearbox efficiency (GE). Grey relation analysis (GRA) and the Taguchi technique were used to address the problem in two steps. Prioritizing the closure of the variable level gap, the single-objective optimization problem was addressed, followed by the multi-objective optimization problem, which identified the ideal primary design variables. Additionally, the first-stage gear ratio, allowable contact stresses (ACS), and first and second-stage coefficients of wheel face width (CWFW) were calculated. The outcomes of the study were used to determine the best values for five essential design features of a two-stage bevel helical gearbox (BHG).

Investigation on the effect of wavelet mutation and opposition learning on slime mould algorithm

The study focuses on a hypothesis-based critical analysis of proposed modifications to the slime mould algorithm. The modifications being investigated are opposition learning and wavelet mutation. These modifications help the slime mould algorithm to avoid local optima for better exploration of the search space. The goal is to assess the effectiveness of these modifications in improving the performance of slime mould algorithm. Both, the basic slime mould algorithm, and the proposed variant eventually achieve the desired convergence. However, to compare the efficiency of the two algorithms, the study introduces a measurement index called the ‘swiftness’ of the algorithm. This index quantifies the speed at which an algorithm reaches convergence. It is calculated as the area under the convergence curve of each algorithm. The rationale behind this approach is that visual inspection alone may not be sufficient to discriminate between the algorithms based on the convergence curve. The hybrid approach, incorporating opposition learning and wavelet mutation, is evaluated statistically using the CEC 2008 benchmark function. Additionally, the study examines small and medium single-objective power dispatch optimization problems that adhere to non-convex system limitations. By assessing the performance of the modified slime mould algorithm on these different problem domains, the researchers aim to provide a comprehensive analysis of its effectiveness.

Ship pipe route design based on NSGA-III and multi-population parallel evolution

Ship pipe route design (SPRD) aims to generate pipe layouts while considering various objectives and constraints in a confined 3D space, leading to extremely high complexity. To solve the problem of SPRD, traditional methods often adopt weighted summation of sub-objectives, introduce penalty functions to convert it to a single-objective optimization problem. However, these methods tend to yield a single optimal pipe route design pattern, and some existing multi-objective pipe routing algorithms do not have high search performance or sufficient treatment of constraints. Based on the grid-space decomposition model, this paper establishes a multi-objective pipe routing optimization model with several sub-objectives including path length, bends number, path energy, number of air-pockets, and number of violated bending distance, which also considers pipe routing for different diameters and interface direction requirements. A ship pipe route design framework based on the NSGA-III (Non-dominated Sorting Genetic algorithm III) is proposed, and its embedded pathfinding algorithm can explore pipe route using heuristic information and failure retry strategy, which greatly reduces the algorithm complexity compared to the classical algorithms such as A* and LEE and also facilitates parallel implementation. By employing OpenMP technology to achieve parallel evolution of multiple populations, more Pareto optimal solutions can be obtained in approximately the same timeframe as single population evolution. Finally, through comparative experiments using simulated cases, the feasibility and advancement of the proposed method are verified.

MOEA/D with customized replacement neighborhood and dynamic resource allocation for solving 3L-SDHVRP

Split delivery heterogeneous vehicle routing problem with three-dimensional loading (3L-SDHVRP) is a critical issue in manufacturing logistics. Current mainstream research formulates this problem as a single objective optimization problem, which fails to reveal the relationship among multiple conflicting objectives and cannot provide various trade-off solutions for decision-makers in real-world logistics scenarios. This paper concentrates on a multi-objective 3L-SDHVRP and proposes a multi-objective evolutionary algorithm based on decomposition with customized replacement neighborhood and dynamic resource allocation (called MOEA/D-RD) to deal with it. The numerical experiments on the dataset from the 11th International Conference on Evolutionary Multi-Criterion Optimization (EMO 2021) show the superior performance of the proposed method over some state-of-the-art ones. The source code of MOEA/D-RD is available at https://github.com/CIAM-Group/EvolutionaryAlgorithm_Codes/tree/main/MOEAD-RD.

New Approach to Solving Fuzzy Multiobjective Linear Fractional Optimization Problems

In this paper, an iterative approach based on the use of fuzzy parametric functions is proposed to find the best preferred optimal solution to a fuzzy multiobjective linear fractional optimization problem. From this approach, the decision-maker imposes tolerance values or termination conditions for each parametric objective function. Indeed, the fuzzy parametric values are computed iteratively, and each fuzzy fractional objective is transformed into a fuzzy non-fractional parametric function using these values of parameters. The core value of fuzzy numbers is used to transform the fuzzy multiobjective non-fractional problem into a deterministic multiobjective non-fractional problem, and the ε-constraint approach is employed to obtain a linear single objective optimization problem. Finally, by setting the value of parameter ε, the Dangtzig simplex method is used to obtain an optimal solution. Therefore, the number of solutions is equal to the number of used values, and the optimal solution is chosen according to the preference of the decision-maker. We have provided a didactic example to highlight the step of our approach and its numerical performances.

A Cooperative Evolutionary Computation Algorithm for Dynamic Multiobjective Multi-AUV Path Planning

Multiple autonomous underwater vehicles (AUVs) are popular for executing submarine missions, which involve multiple targets distributed in a large and complex underwater environment. The path planning of multiple AUVs is a significant and challenging problem, which determines the location of surface points for AUV launch and plans the paths of AUVs for target traveling. Most existing works model the problem as a single-objective static optimization problem. However, the target missions may change over time, and multiple optimization objectives are usually expected for decision making. Thus, this paper models the problem as a dynamic multiobjective optimization problem and proposes a cooperative evolutionary computation algorithm to provide diverse and high-quality solutions for decision makers. In the proposed method, solutions are represented using a bi-layer encode scheme, in which the first layer indicates the surface location points and the second layer represents the traveling sequences of target missions. Multiple populations for multiple objectives (MPMO) framework is adopted to efficiently solve the dynamic multiobjective AUV optimization problem. In addition, a recombination-based sampling strategy is developed to improve convergence by fusing the information of multiple populations. Once a change occurs, an incremental response strategy is adopted to generate high-quality solutions for population evolution. Based on the dataset of New Zealand bathymetry, six complex underwater scenarios are constructed with a size of 50 km×50 km×10 km and 400 target missions for tests. Experimental results show that the proposed method outperforms the state-of-the-art algorithms in terms of solution diversity and optimality.

Cam Profile Design of Low-Speed Marine Diesel Engine Based on the Grey Theory

The camshaft profile is one of the important parameters of the engine, the reasonableness or otherwise of its design directly influences the engine’s power output, fuel consumption, emissions, noise and reliability. However, most existing cam profile optimization designs are based on the basic software operation, profile parameter selection and tuning lack a scientific theoretical basis. In order to address this issue, this study establishes a single valve kinematic model of the gas delivery mechanism of diesel engines by using the AVL-EXCITE Timing Drive, and performs a model-based multinomial kinematic acceleration cam profile design. Subsequently, an orthogonal design method is employed to obtain the parameter combinations that can satisfy the dynamic requirements. The research investigates the influence of cam profile design parameters (C 4, p, q, r, and s) on valve fullness, maximum jerk, cam contact stress, and valve seating force through a systematic study. Lastly, based on the grey correlation theory, a thorough analysis of the performance indices of cams with different profile parameter combinations is conducted. The optimization problem of the profile parameters with multiple performance indicators is transformed into a single-objective grey relational optimization problem, resulting in the optimal cam profile parameter combination of C4 =0.1, p=14, q=20, r=28 and s=36. The optimal method proposed in the paper is straightforward to implement and is applicable to the optimum design of cam profiles of the engine, which is of great importance for the optimal design of cam fabrication processes.