Non-dominated Solutions Research Articles

Operative generative design using non-dominated sorting genetic algorithm II (NSGA-II)

Massing studies during the early stages of architectural design play an essential role in determining the final building’s performance across design objectives. This paper aims to answer the question: How can early-stage architectural design workflows be translated into a generative design process to create valuable massing solutions? In response, a new application of the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) using the Pymoo framework is proposed for the field of Operative Design. Nine experiments are discussed that test the algorithm’s geometry optimization capabilities based on objective functions reflecting common architectural design goals, including Floor Area Ratio (FAR), Non-Passive Zone (NPZ), Roofs and Best Oriented Surfaces (RBOS), and Usable Open Space (UOS). Selected cases are visualized among non-dominated solutions in each experiment demonstrating the trade-offs between different objectives while programmatically generating successful building designs. In the future, the proposed generative design workflow can be implemented to run optimizations independently from other software within immersive environments.

HV-Net: Hypervolume Approximation Based on DeepSets

In this letter, we propose HV-Net, a new method for hypervolume approximation in evolutionary multi-objective optimization. The basic idea of HV-Net is to use DeepSets, a deep neural network with permutation invariant property, to approximate the hypervolume of a non-dominated solution set. The input of HV-Net is a non-dominated solution set in the objective space, and the output is an approximated hypervolume value of this solution set. The performance of HV-Net is evaluated through computational experiments by comparing it with two commonly-used hypervolume approximation methods (i.e., point-based method and line-based method). Our experimental results show that HV-Net outperforms the other two methods in terms of both the approximation error and the runtime, which shows the potential of using deep learning techniques for hypervolume approximation.

Effects of Archive Size on Computation Time and Solution Quality for Multiobjective Optimization

An unbounded external archive has been used to store all nondominated solutions found by an evolutionary multi-objective optimization algorithm in some studies. It has been shown that a selected solution subset from the stored solutions is often better than the final population. However, the use of the unbounded archive is not always realistic. When the number of examined solutions is huge, we must pre-specify the archive size. In this study, we examine the effects of the archive size on three aspects: (i) the quality of the selected final solution set, (ii) the total computation time for the archive maintenance and the final solution set selection, and (iii) the required memory size. Unsurprisingly, the increase of the archive size improves the final solution set quality. Interestingly, the total computation time of a medium-size archive is much larger than that of a small-size archive and a huge-size archive (e.g., an unbounded archive). To decrease the computation time, we examine two ideas: periodical archive update and archiving only in later generations. Compared with updating the archive at every generation, the first idea can obtain almost the same final solution set quality using a much shorter computation time at the cost of a slight increase of the memory size. The second idea drastically decreases the computation time at the cost of a slight deterioration of the final solution set quality. Based on our experimental results, some suggestions are given about how to appropriately choose an archiving strategy and an archive size.

Evolutionary Multitasking for Large-Scale Multiobjective Optimization

Evolutionary transfer optimization (ETO) has been becoming a hot research topic in the field of evolutionary computation, which is based on the fact that knowledge learning and transfer across the related optimization exercises can improve the efficiency of others. However, rare studies employ ETO to solve large-scale multiobjective optimization problems (LMOPs). To fill this research gap, this paper proposes a new multitasking ETO algorithm via a powerful transfer learning model to simultaneously solve multiple LMOPs. In particular, inspired by adversarial domain adaptation in transfer learning, a discriminative reconstruction network (DRN) model (containing an encoder, a decoder, and a classifier) is created for each LMOP. At each generation, the DRN is trained by the currently obtained nondominated solutions for all LMOPs via backpropagation with gradient descent. With this well-trained DRN model, the proposed algorithm can transfer the solutions of source LMOPs directly to the target LMOP for assisting its optimization, can evaluate the correlation between the source and target LMOPs to control the transfer of solutions, and can learn a dimensional-reduced Pareto-optimal subspace of the target LMOP to improve the efficiency of transfer optimization in the large-scale search space. Moreover, we propose a real-world multitasking LMOP suite to simulate the training of deep neural networks on multiple different classification tasks. Finally, the effectiveness of the proposed algorithm has been validated on this real-world problem suite and other two synthetic problem suites.

An Improved Strength Pareto Evolutionary Algorithm 2 with Adaptive Crossover Operator for Bi-Objective Distributed Unmanned Aerial Vehicle Delivery

With the development of the e-commerce industry, using UAVs (unmanned aerial vehicles) to deliver goods has become more popular in transportation systems. This delivery method can reduce labor costs and improve the distribution efficiency, and UAVs can reach places that are difficult for humans to reach. Because some goods are perishable, the quality of the delivery will have an impact on the customer satisfaction. At the same time, the delivery time should also meet the needs of customers as much as possible. Therefore, this paper takes the distribution distance and customer satisfaction as the objective functions, establishes a bi-objective dynamic programming model, and proposes an improved SPEA2 (strength Pareto evolutionary algorithm 2). The improved algorithm introduces the local search strategy, on the basis of the original algorithm. It conducts a local search for the better non-dominated solutions obtained in each iteration. The new dominated solutions and non-dominated solutions are determined, and the crossover operator is improved, so that the local search ability is improved, on the basis of ensuring its global search ability. The numerical experiment results show that the improved algorithm achieves an excellent performance in three aspects: the Pareto front, generation distance, and spacing, and would have a high application value in UAV cargo delivery and other MOPs (multi-objective optimization problems). The average spacing value of the improved algorithm is more than 20% smaller than SPEA2 + SDE (strength Pareto evolution algorithm 2–shift-based density estimation), which is the second-best algorithm. In the comparison of the average generation distance value, this number reaches 30%.

Mission and Reliability Driven Fleet-Level Selective Maintenance Planning and Scheduling Two-Stage Method

This paper studies the problem of planning and scheduling in selective maintenance tasks of mission requirements and the health condition of the fleet. In order to deal with the problems of high maintenance cost and long time consumption in maintenance systems, a two-stage fleet maintenance optimization method is proposed. Firstly, a selective maintenance model of fleets based on age reduction is established to maximize the probability of completing the next mission and minimize the maintenance cost. Secondly, a multiobjective sparrow search algorithm is designed to solve the maintenance planning problem in the first stage, and a nondominated solution set of maintenance strategies satisfying the mission constraint is obtained. In the second stage, the simulated annealing algorithm is used to schedule the maintenance task and obtain the minimum maintenance hours required by the maintenance strategy. An example analysis of a vehicle fleet is launched to prove the effectiveness of this method. In a word, this method not only meets the mission requirements but also achieves the purpose of reducing maintenance cost and maintenance hours, which can provide reference for other types of equipment maintenance.

A multi-objective strategy for cost-effective microgrid solutions based on renewable energy sources

With the rapid urbanization and increasing energy demands, microgrids are recognizing renewable energy sources (RESs) as a valuable power generation option. However, efficiently managing the energy cost poses a significant challenge in integrating RESs with microgrids. To address this challenge, this study presents a novel approach utilizing a cost-effective multi-objective genetic algorithm (MOGA) to optimize power allocation among diverse generation units within the microgrid. The proposed MOGA algorithm aims to minimize generation costs by efficiently distributing the generated power from different sources in the microgrid vs the CO2 emission. By leveraging the genetic algorithm population, MOGA generates a diverse set of non-dominated solutions. Simulation results demonstrate the effectiveness of the proposed approach in reducing the cost of RESs in microgrids, surpassing the performance of other multi-objective optimization methods such as multi-objective particle swarm (MOPSO) and multi-objective wind-driven optimization (MOWDO).

Fabrication of Multi-Walled Structure through Parametric Study of Bead Geometries of GMAW-Based WAAM Process of SS309L.

In the present study, an attempt is made to investigate and optimize the bead geometries of bead width (BW) and bead height (BH) of SS-309L using an SS316L substrate by employing a gas metal arc welding (GMAW)-based wire-arc additive manufacturing (WAAM) process. The Box-Behnken design approach was used to conduct the trials of single-layer depositions with input variables of travel speed (TS), voltage (V), and gas mixture ratio (GMR). The developed multi-variable regression models were tested for feasibility using ANOVA and residual plots. The data obtained indicated that V had the most significant impact on BW, followed by TS and GMR. For BH, TS had the most significant impact, followed by GMR and V. The results of single-response optimization using a passing vehicle search (PVS) algorithm showed a maximum BH of 9.48 mm and a minimum BW of 5.90 mm. To tackle the contradictory situation, a multi-objective PVS algorithm was employed, which produced non-dominated solutions. A multi-layered structure was successfully fabricated at the optimal parametric settings of TS at 20 mm/s, of voltage at 22 V, and of GMR at 3. For multi-layer structures, fusion among the layers was observed to be good, and they were found to be free from the disbonding of layers. This revealed the suitability of the PVS algorithm for generating suitable optimal WAAM variables. We consider the current work highly beneficial for users fabricating multi-layer structures.

Simultaneous optimization of design and maintenance for systems using multi-objective evolutionary algorithms and discrete simulation

When projecting and building new industrial facilities, getting integrated design alternatives and maintenance strategies are of critical importance to achieve the physical assets optimal performance, which is needed to be competitive in the actual global markets. Coupling Evolutionary Algorithms with Discrete Event Simulation has been explored both in relation to systems design and their maintenance strategy. However, it was not simultaneously considered when both the corrective and the preventive maintenance—consisting of achieving the optimum period of time to carry out a preventive maintenance activity—are taken into account before being considered by the authors of the present paper. This work couples Multi-objective Evolutionary Algorithms with Discrete Event Simulation in order to enhance the knowledge and efficiency of the methodology presented, which consists of exploring and optimizing simultaneously systems design alternatives and their preventive maintenance strategies. The aim consists of finding the best set of non-dominated solutions by using the system availability (first maximized objective function) with taking into consideration associated operational cost (second minimized objective function), while automatically selecting the system devices. Each solution proposed by the Multi-Objective Evolutionary Algorithm is analyzed by using Discrete Event Simulation in a procedure that looks at the effect of including periodic preventive maintenance activities all along the mission time. An industrial application case study is solved, and a comparison of the performance of five state-of-the-art and three more recently developed Multi-objective Evolutionary Algorithms is handled; moreover, the gap in the literature reviewed about the analysis regarding the effect of the discrete event simulation sampling size is faced with useful insights about the synergies of Multi-objective Evolutionary Algorithms and Discrete Event Simulation. Finally, the methodology is expanded to more complex systems which are successfully solved.

Optimization problems with evidential linear objective

We investigate a general optimization problem with a linear objective in which the coefficients are uncertain and the uncertainty is represented by a belief function. We consider five common criteria to compare solutions in this setting: generalized Hurwicz, strong dominance, weak dominance, maximality and E-admissibility. We provide characterizations for the non-dominated solutions with respect to these criteria when the focal sets of the belief function are Cartesian products of compact sets. These characterizations correspond to established concepts in optimization. They make it possible to find non-dominated solutions by solving known variants of the deterministic version of the optimization problem or even, in some cases, simply by solving the deterministic version.

Using an exact bi-objective decoder in a memetic algorithm for arc-routing (and other decoder-expressible) problems

We address the bi-objective Capacitated Arc Routing Problem (CARP) by considering two levels of solution interpretation: implicit and explicit solutions. An algorithm that translates implicit solutions into explicit solutions is called a decoder. In this work, the decoder takes as input a permutation of the required edges and generates a Pareto frontier of CARP solutions. While bi-objective CARP was our main focus and starting point, we could also use the proposed framework to solve a bi-objective version of the traveling salesman problem by plugging-in a different decoder. Recall that bi-objective CARP asks to service (the demands of) a set of required edges using a fleet of vehicles of limited capacity so as to minimize: (i) the total travelled distance and (ii) the length of the longest route. Any permutation s of the required edges constitutes an implicit CARP solution. The decoder constructs all non-dominated explicit solutions that service the edges in the order indicated by s, i.e., the decoder is an exact algorithm that returns the optimal Pareto frontier subject to the service order s. To achieve competitive CARP results it is also important to reinforce the decoder using a local search operator that acts on explicit routes (and that may change the service order s). For nine instances, the resulting algorithm was even able to find a new total-cost upper bound, improving upon the best solutions reported in the (considerably larger) mono-objective CARP literature. This shows that (some of) the proposed ideas can also be useful in single objective optimization: the second objective can be seen as a guide for the mono-objective search process.

Fuzzy correlation entropy-based NSGA-II for energy-efficient hybrid flow-shop scheduling problem

Green scheduling of manufacturing industry with energy saving as the core has been paid more and more attention in academia and industry. As a classic scheduling problem, hybrid flow-shop scheduling problem (HFSP) has been receiving increasing research attention. However, most studies only focus on time-related metrics while neglecting energy consumption. In this article, we studied energy-efficient HFSP and assumed that machines can operate at different speeds. This is an assumption that has been rarely explored but can make the problem more relevant to real-world production. Moreover, the energy-efficient HFSP at a variable machine speed (EHFSP-VMS) was formulated as a multiobjective mathematical optimization model aiming to optimize make-span and total energy consumption simultaneously. As a landmark achievement in the field of multiobjective optimization, non-dominated sorting genetic algorithm-II (NSGA-II) is adopted and improved as the solver, which is called fuzzy correlation entropy (FCE)-based NSGA-II (FCENSGA-II). Firstly, FCE, a fusion of fuzzy mathematics and information theory, is used to describe the difference and the FCE-based crowding distance is proposed for the first time. Its time complexity is lower than the original crowding distance. In addition, a machine learning strategy, namely opposition-based learning (OBL), is used to learn from opposite regions of the search space and increase the exploratory ability of the algorithm and the diversity of solutions. Finally, a critical path knowledge-based energy saving strategy (ESC) is adopted to discover non-dominant solutions by reducing the speed of machines on non-critical paths. A large number of experiments are conducted to test the performance of FCENSGA-II. The results show that in all test instances, the average, best and worst values of the solution obtained by FCENSGA-II are better than the compared state-of-the-art algorithms, and even the worst values obtained by FCENSGA-II in 75% of test instances are better than the best values of compared algorithms, which strongly confirms that FCENSGA-II outperforms the compared state-of-the-art algorithms for solving EHFSP-VMS.

A Multi–Objective Gaining–Sharing Knowledge-Based Optimization Algorithm for Solving Engineering Problems

Metaheuristics in recent years has proven its effectiveness; however, robust algorithms that can solve real-world problems are always needed. In this paper, we suggest the first extended version of the recently introduced gaining–sharing knowledge optimization (GSK) algorithm, named multiobjective gaining–sharing knowledge optimization (MOGSK), to deal with multiobjective optimization problems (MOPs). MOGSK employs an external archive population to store the nondominated solutions generated thus far, with the aim of guiding the solutions during the exploration process. Furthermore, fast nondominated sorting with crowding distance was incorporated to sustain the diversity of the solutions and ensure the convergence towards the Pareto optimal set, while the ϵ-dominance relation was used to update the archive population solutions. ϵ-dominance helps provide a good boost to diversity, coverage, and convergence overall. The validation of the proposed MOGSK was conducted using five biobjective (ZDT) and seven three-objective test functions (DTLZ) problems, along with the recently introduced CEC 2021, with fifty-five test problems in total, including power electronics, process design and synthesis, mechanical design, chemical engineering, and power system optimization. The proposed MOGSK was compared with seven existing optimization algorithms, including MOEAD, eMOEA, MOPSO, NSGAII, SPEA2, KnEA, and GrEA. The experimental findings show the good behavior of our proposed MOGSK against the comparative algorithms in particular real-world optimization problems.

Improving Multi-Objective Optimization Methods of Water Distribution Networks

Water distribution network design is a complex multi-objective optimization problem and multi-objective evolutionary algorithms (MOEAs) such as NSGA II have been widely used to solve this optimization problem. However, as networks get larger, NSGA II struggles to find the diverse and uniform solutions that are critical in multi-objective optimization. This research proposes an improved version of NSGA II that uses three new-generation methods to target different regions of the Pareto front and thus increase the number of solutions in critical regions. These methods include saving an archive, local search around extreme and uncrowded Pareto front, and local search around the knee area of the Pareto front. The improved NSGA II is tested on benchmark networks of different sizes and compared to the best-known Pareto front of the networks determined by MOEAs. The results show that the proposed algorithm outperforms the original NSGA II in terms of broadening the Pareto front solution range, increasing solution density, and discovering more non-dominated solutions. The improved NSGA II can find solutions that cover all parts of the Pareto front using a single algorithm without increasing computational effort.

MOIMPA: multi-objective improved marine predators algorithm for solving multi-objective optimization problems

This paper introduces a multi-objective variant of the marine predators algorithm (MPA) called the multi-objective improved marine predators algorithm (MOIMPA), which incorporates concepts from Quantum theory. By leveraging Quantum theory, the MOIMPA aims to enhance the MPA’s ability to balance between exploration and exploitation and find optimal solutions. The algorithm utilizes a concept inspired by the Schrödinger wave function to determine the position of particles in the search space. This modification improves both exploration and exploitation, resulting in enhanced performance. Additionally, the proposed MOIMPA incorporates the Pareto dominance mechanism. It stores non-dominated Pareto optimal solutions in a repository and employs a roulette wheel strategy to select solutions from the repository, considering their coverage. To evaluate the effectiveness and efficiency of MOIMPA, tests are conducted on various benchmark functions, including ZDT and DTLZ, as well as using the evolutionary computation 2009 (CEC’09) test suite. The algorithm is also evaluated on engineering design problems. A comparison is made between the proposed multi-objective approach and other well-known evolutionary optimization methods, such as MOMPA, multi-objective ant lion optimizer, and multi-objective multi-verse optimization. The statistical results demonstrate the robustness of the MOIMPA approach, as measured by metrics like inverted generational distance, generalized distance, spacing, and delta. Furthermore, qualitative experimental results confirm that MOIMPA provides highly accurate approximations of the true Pareto fronts.

Multi-objective medical supplies distribution open vehicle routing problem with fairness and timeliness under major public health emergencies

Fair and timely delivery of supplies plays a critical role under major public health emergencies. In this paper, aiming at fairness and timeliness, an optimization model of open vehicle routing problem for medical supplies distribution is established considering the urgency of the demand. We adopt a differential evolutionary algorithm with fast non-dominated solution sorting to solve the proposed model, obtaining an approximate Pareto optimal solution set. Through the comparison of algorithms, the results showed that the differential evolutionary algorithm with non-dominated sorting is superior with a shorter runtime and more diverse solutions, while the epsilon constraint method has more accurate solutions. In the case verification, the quality of the solutions of both algorithms was within the acceptable range, but the runtime of the epsilon constraint method was too long to be applicable. The results can provide theoretical suggestions and practical guidance for decision-makers in emergency supplies distribution.

An improved multi-objective framework for the Rich arc routing problem

This article introduces a Rich Arc Routing Problem (RARP). Required arcs are partitioned into given clusters, and each cluster shall be serviced for prescribed times during the planning horizon. There is a time limit that restricts the visits that can be performed in a single day. Each vehicle has to stop at one of intermediate facilities before the end of the specific hours per day. Accordingly, several days may be necessary to complete all tasks, and paths traveled by different vehicles on the same day shall not overlap. The RARP consists of finding a set of routes with (i) the minimum total distance, (ii) the minimum number of vehicles, (iii) the minimum pure working days, and (iv) the temporal service time interval consistency of each cluster. Recently, a Local-Ideal-Points based Autonomous Space Decomposition (LIP-ASD) algorithm has been demonstrated to be a competitive multi-objective framework. However, randomly generating an initial population in each decomposed space decreases the convergence speed of LIP-ASD. In addition, retaining only one individual preserved in each sub-space limits population diversity. In response to these issues, an Improved LIP-ASD (I-LIP-ASD) for RARP is developed. The two improvements of the proposed algorithm are as follows: (i) the initial population in the current sub-space is generated by combining existing nondominated individuals, and (ii) all nondominated solutions will be output from the evolved population in each decomposed sub-space. Finally, we conduct a series of experiments over converted capacitated arc routing problem instances, as well as analyze them under a real-world case to evaluate and demonstrate the effectiveness of the proposed solution algorithm.

An effective multi-objective bald eagle search algorithm for solving engineering design problems

In this paper, a multi-objective bald eagle search algorithm (MOBES) is proposed. The MOBES introduces an archive mechanism to store the non-dominated solutions obtained by the algorithm. When the archive overflows, remove the most crowded solutions by using the roulette method. The MOBES also adds elite selection strategy to guide other individuals to optimize by selecting elite individuals in the population. The efficiency of MOBES is validated on CEC 2020 benchmark functions, and the results demonstrate that the proposed algorithm is more efficient than its competitors in terms of convergence, diversity and distribution of solutions. The MOBES is also applied to two-objective, tri-objective and four-objective engineering design problems in real world. The results show its superiority in handling challenging multi-objective optimization problems with unknown true Pareto optimal solutions and fronts, and it is more competitive than other algorithms.

Solving distributed low carbon scheduling problem for large complex equipment manufacturing using an improved hybrid artificial bee colony algorithm

Multi-agent collaborative manufacturing, high energy consumption and pollution, and frequent operation outsourcing are the three main characteristics of large complex equipment manufacturing enterprises. Therefore, the production scheduling problem of large complex equipment to be studied is a distributed flexible job shop scheduling problem involving operation outsourcing (Oos-DFJSP). Besides, the influences of each machine on carbon emission and job scheduling at different processing speeds are also involved in this research. Thus the Oos-DFJSP of large complex equipment consists of the following four sub-problems: determining the sequence of operations, assigning jobs to manufactories, assigning operations to machines and determining the processing speed of each machine. In the Oos-DFJSP, if a job is assigned to a manufactory of a group manufacturing enterprise, and the manufactory cannot complete some operations of the workpiece, then these operations will be assigned to other manufactories with related processing capabilities. Aiming at solving the problem, a multi-objective mathematical model including costs, makespan and carbon emission was established, in which energy consumption, power generation of waste heat and treatment capacity of pollutants were considered in the calculation of carbon emission. Then, a multi-objective improved hybrid genetic artificial bee colony algorithm was developed to address the above model. Finally, 45 groups of random comparison experiments were presented. Results indicate that the developed algorithm performs better than other multi-objective algorithms involved in the comparison experiments not only on quality of non-dominated solutions but also on Inverse Generational Distance and Error Ratio. That is, the proposed mathematical model and algorithm were proved to be an excellent method for solving the multi-objective Oos-DFJSP.

A Many-Objective Evolutionary Algorithm Based on Dual Selection Strategy.

In high-dimensional space, most multi-objective optimization algorithms encounter difficulties in solving many-objective optimization problems because they cannot balance convergence and diversity. As the number of objectives increases, the non-dominated solutions become difficult to distinguish while challenging the assessment of diversity in high-dimensional objective space. To reduce selection pressure and improve diversity, this article proposes a many-objective evolutionary algorithm based on dual selection strategy (MaOEA/DS). First, a new distance function is designed as an effective distance metric. Then, based distance function, a point crowding-degree (PC) strategy, is proposed to further enhance the algorithm's ability to distinguish superior solutions in population. Finally, a dual selection strategy is proposed. In the first selection, the individuals with the best convergence are selected from the top few individuals with good diversity in the population, focusing on population convergence. In the second selection, the PC strategy is used to further select individuals with larger crowding distance values, emphasizing population diversity. To extensively evaluate the performance of the algorithm, this paper compares the proposed algorithm with several state-of-the-art algorithms. The experimental results show that MaOEA/DS outperforms other comparison algorithms in overall performance, indicating the effectiveness of the proposed algorithm.