Entire Pareto Front Research Articles

Extension of the Directed Search Domain algorithm for multi-objective optimization to higher dimensions

This paper addresses the problem of generating an evenly distributed set of Pareto solutions. It appears in real-life applications related to multi-objective optimization when it is important to represent the entire Pareto front with a minimal cost. There exist only a few algorithms which are able to tackle this problem in a general formulation. The Directed Search Domain (DSD) algorithm has proved to be efficient and quite universal. It has successfully been applied to different challengeable test cases. In this paper for the first time the DSD approach is systematically extended and applied to problems with higher dimensions. The modified algorithm does not have any formal limitation on the number of objective functions that is important for practical applications. The efficacy of the algorithm is demonstrated on a number of test cases.

A knee-oriented many-objective differential evolution with bi-strategy and Manhattan distance-domination range

Differential evolution multi-objective algorithms effectively address problems with two or three objectives. However, many-objective problems with more than three objectives are challenging because of inadequate selection pressure in environment selection, excessive variation in mutation, and difficulties in representing the Pareto front. This paper introduces a novel approach that combines differential evolution with a knee-oriented strategy to address the mentioned challenges. The approach allows the algorithm to focus on trade-off solutions for representing the surface instead of searching the entire Pareto front. Specifically, the algorithm consists of two stages: knee exploration stage and knee exploitation stage. Each stage utilizes different dominance relationships and mutation operators to enhance selection pressure in high-dimensional spaces. Moreover, a Manhattan distance-domination range is designed to update the reference vector by identifying multiple localized knee points in high-dimensional space. The experimental results demonstrate superior performance to other state-of-the-art algorithms on 50 knee-oriented benchmark problem test sets, including CKP, DO2DK, DEB2DK, DEB3DK, and PMOPs.

Research on decomposition-based multi-objective evolutionary algorithm with dynamic weight vector

In recent years, multi-objective evolutionary algorithm based on decomposition has gradually attracted people's interest. However, this algorithm has some problems. For example, the diversity of the algorithm is poor, and the convergence and diversity of the algorithm are unbalanced. In addition, users don't always care about the entire Pareto front. Sometimes they may only be interested in specific areas of entire Pareto front. Based on the above problems, this paper proposes a decomposition-based multi-objective evolutionary algorithm with dynamic weight vector (MOEA/D-DWV). Firstly, a weight vector generation model with uniform distribution or preference distribution is proposed. Users can decide which type of weight vector to generate according to their own wishes. Then, two combination evolution operators are proposed to better balance the convergence and diversity of the algorithm. Finally, a dynamic adjustment strategy of weight vector is proposed. This strategy can adjust the distribution of weight vector adaptively according to the distribution of solutions in the objective space, so that the population can be uniformly distributed in the objective space as much as possible. MOEA/D-DWV algorithm is compared with 9 advanced multi-objective evolutionary algorithms. The comparison results show that MOEA/D-DWV algorithm is more competitive. Data availabilityData will be made available on request.

Computational discovery of microstructured composites with optimal stiffness-toughness trade-offs.

The conflict between stiffness and toughness is a fundamental problem in engineering materials design. However, the systematic discovery of microstructured composites with optimal stiffness-toughness trade-offs has never been demonstrated, hindered by the discrepancies between simulation and reality and the lack of data-efficient exploration of the entire Pareto front. We introduce a generalizable pipeline that integrates physical experiments, numerical simulations, and artificial neural networks to address both challenges. Without any prescribed expert knowledge of material design, our approach implements a nested-loop proposal-validation workflow to bridge the simulation-to-reality gap and find microstructured composites that are stiff and tough with high sample efficiency. Further analysis of Pareto-optimal designs allows us to automatically identify existing toughness enhancement mechanisms, which were previously found through trial and error or biomimicry. On a broader scale, our method provides a blueprint for computational design in various research areas beyond solid mechanics, such as polymer chemistry, fluid dynamics, meteorology, and robotics.

A framework for controllable Pareto front learning with completed scalarization functions and its applications

Pareto Front Learning (PFL) was recently introduced as an efficient method for approximating the entire Pareto front, the set of all optimal solutions to a Multi-Objective Optimization (MOO) problem. In the previous work, the mapping between a preference vector and a Pareto optimal solution is still ambiguous, rendering its results. This study demonstrates the convergence and completion aspects of solving MOO with pseudoconvex scalarization functions and combines them into Hypernetwork in order to offer a comprehensive framework for PFL, called Controllable Pareto Front Learning. Extensive experiments demonstrate that our approach is highly accurate and significantly less computationally expensive than prior methods in term of inference time.

Many-Objective Job-Shop Scheduling: A Multiple Populations for Multiple Objectives-Based Genetic Algorithm Approach.

The job-shop scheduling problem (JSSP) is a challenging scheduling and optimization problem in the industry and engineering, which relates to the work efficiency and operational costs of factories. The completion time of all jobs is the most commonly considered optimization objective in the existing work. However, factories focus on both time and cost objectives, including completion time, total tardiness, advance time, production cost, and machine loss. Therefore, this article first time proposes a many-objective JSSP that considers all these five objectives to make the model more practical to reflect the various demands of factories. To optimize these five objectives simultaneously, a novel multiple populations for multiple objectives (MPMO) framework-based genetic algorithm (GA) approach, called MPMOGA, is proposed. First, MPMOGA employs five populations to optimize the five objectives, respectively. Second, to avoid each population only focusing on its corresponding single objective, an archive sharing technique (AST) is proposed to store the elite solutions collected from the five populations so that the populations can obtain optimization information about the other objectives from the archive. This way, MPMOGA can approximate different parts of the entire Pareto front (PF). Third, an archive update strategy (AUS) is proposed to further improve the quality of the solutions in the archive. The test instances in the widely used test sets are adopted to evaluate the performance of MPMOGA. The experimental results show that MPMOGA outperforms the compared state-of-the-art algorithms on most of the test instances.

Five Facets of 6G: Research Challenges and Opportunities

While the fifth-generation systems are being rolled out across the globe, researchers have turned their attention to the exploration of radical next-generation solutions. At this early evolutionary stage, we survey five main research facets of this field, namelyFacet 1: next-generation architectures, spectrum, and services; Facet 2: next-generation networking; Facet 3: Internet of Things; Facet 4: wireless positioning and sensing; and Facet 5: applications of deep learning in 6G networks.In this article, we provide a critical appraisal of the literature of promising techniques ranging from the associated architectures, networking, and applications, as well as designs. We portray a plethora of heterogeneous architectures relying on cooperative hybrid networks supported by diverse access and transmission mechanisms. The vulnerabilities of these techniques are also addressed and carefully considered for highlighting the most of promising future research directions. Additionally, we list a rich suite of learning-driven optimization techniques. We conclude by observing the evolutionary paradigm shift that has taken place from pure single-component bandwidth efficiency, power efficiency, or delay optimization toward multi-component designs, as exemplified by the twin-component ultra-reliable low-latency mode of the fifth-generation system. We advocate a further evolutionary step toward multi-component Pareto optimization, which requires the exploration of the entire Pareto front of all optimal solutions, where none of the components of the objective function may be improved without degrading at least one of the other components.

Real Environment-Aware Multisource Data-Associated Cold Chain Logistics Scheduling: A Multiple Population-Based Multiobjective Ant Colony System Approach

Cold chain logistics (CCL) scheduling is important for smart cities as it directly affects the service quality and operating profits of logistics companies. However, traditional CCL models seldom reflect the real transportation environment, making the solutions hardly applicable to the real CCL scenes. Hence, this paper attempts to establish a multisource data-associated CCL model oriented to the real transportation environment. This environment is considered by employing the real-captured driving duration and distance between any two places. Three scheduling objectives (namely, quality losses, personnel and vehicle costs, and transportation costs) are taken into account. To efficiently solve the proposed multisource data-associated multiobjective CCL model, a multiple population-based multiobjective ant colony system (MPMOACS) approach is proposed. Based on the multiple populations for multiple objectives framework, the MPMOACS approach can optimize multiple objectives sufficiently, and thus obtain promising solutions distributed along the entire Pareto front. To further enhance the performance of the MPMOACS, a ranking-based local search strategy is also designed. Experiments are conducted on not only the existing benchmark instances but also a real environment-aware multisource dataset that is built based on real-captured transportation data of Guangzhou and Shenzhen, China. Compared with six state-of-the-art and very recent well-performing multiobjective optimization approaches, the proposed MPMOACS approach exhibits the overall best performance.

Interactive co-evolutionary multiple objective optimization algorithms for finding consensus solutions for a group of Decision Makers

We introduce interactive algorithms for identifying consensus solutions to multiple objective optimization problems. They learn the Decision Makers’ (DMs’) preferences from indirect judgments and represent the recognized aspirations using scalar optimization goals. Their role is to set guidelines for the evolution and discovery of consensuses. The novelty of our proposals lies in co-evolving two populations: primary and supportive. The former’s role is to discover solutions relevant to the committee. The latter approximates the entire Pareto front, revealing a variety of trade-offs between objectives. The method improves the potential of conducting more informative interactions with the DMs and prevents the evolution from stagnation. Furthermore, it improves the evolutionary pace by dynamically removing no longer worthwhile goals from the supportive population in favor of increasing the primary population size. We confirm the importance of using co-evolution and dynamic resource allocation in extensive experiments. Also, we prove the competitiveness of our proposals by comparing them with the state-of-the-art methods on the WFG benchmarks. Finally, we demonstrate their practical usefulness when applied to the real-world problem of designing an environmentally friendly supply chain.

Distributed Optimal Energy Management for Integrated Energy Systems

This article is concerned with the multiobjective energy management problem of integrated energy systems (IESs) in a distributed fashion. A multiobjective model is established where the operating costs and gas emissions are treated as competing objectives simultaneously. To achieve optimal energy management with reasonable operating costs and gas emissions, a primal-dual-based distributed algorithm with dynamic weights is proposed to assign the multiple energy sources. By this algorithm, each participant only needs to compute locally and share information among neighbors, and thus it offers better flexibility, reliability, and adaptivity, and lower communication burden compared with some centralized algorithms. Besides, the proposed algorithm can solve the multiobjective problem with no special initialization conditions and lower computational complexity different from existing distributed optimization algorithms for IESs. Furthermore, it is proved that each participant can converge to the global optimal solution asymptotically. Simulation studies demonstrate that the proposed algorithm can coordinate the two conflicting objectives of economy and environment simultaneously and obtain the entire Pareto front. The effectiveness and scalability of the proposed algorithm are also validated on a 39-32 power-heat IESs and the integrated energy unit system.

Surrogate-assisted automatic evolving of dispatching rules for multi-objective dynamic job shop scheduling using genetic programming

Dispatching rules are simple but efficient heuristics to solve multi-objective job shop scheduling problems, particularly useful to face the challenges of dynamic shop environments. A promising method to automatically evolve non-dominated rules represents multi-objective genetic programming based hyper-heuristic (MO-GP-HH). The aim of such methods is to approximate the Pareto front of non-dominated dispatching rules as good as possible in order to provide a sufficient set of efficient solutions from which the decision maker can select the most preferred one. However, one of the main drawbacks of existing approaches is the computational demanding simulation-based fitness evaluation of the evolving rules. To efficiently allocate the computational budget, surrogate models can be employed to approximate the fitness. Two possible ways, that estimate the fitness either based on a simplified problem or based on samples of fully evaluated individuals making use of machine learning techniques are investigated in this paper. Several representatives of both categories are first examined with regard to their selection accuracy and execution time. Furthermore, we developed a surrogate-assisted MO-GP-HH framework, incorporating a pre-selection task in the NSGA-II algorithm. The most promising candidates are consequently implemented in the framework. Using a dynamic job shop scenario, the two proposed algorithms are compared to the original one without using surrogates. With the aim to minimize the mean flowtime and maximum tardiness, experimental results demonstrate that the proposed algorithms outperform the former. Making use of surrogates leads to a reduction in computational costs of up to 70%. Another interesting finding shows that the enhanced ability to identify duplicates based on the phenotypic characterization of individuals is particularly helpful in increasing diversity within a population. This study illustrates the positive effect of this mechanism on the exploration of the entire Pareto front.

Machine Learning Enabled Design Automation and Multi-Objective Optimization for Electric Transportation Power Systems

This article presents an automated design and optimization framework for electric transportation power systems (ETPS) enabled by machine learning (ML). The use of physical models, simulations, and optimization methods can greatly aid the engineering design process. However, when considering the optimal co-design of multiple interdependent subsystems that span multiple physical domains, such model-based simulations can be computationally expensive, and traditional metaheuristic optimization methods can be unreliable. Bayesian optimization (BO), an ML framework, paves one feasible pathway to realize an efficient design process practically. However, current state-of-the-art BO algorithms are non-compatible or perform poorly when applied to system-level ETPS design with multiple objectives and constraints. This article proposes a novel BO algorithm referred to as max-value entropy search for multiobjective optimization with constraints (MESMOC) to solve multiobjective optimization (MOO) problems with black-box constraints that can only be evaluated through design simulations. After a full presentation of the algorithm, MESMOC is applied to a realistic ETPS design case using a heavy-duty electric vertical-takeoff-landing (eVTOL) urban aerial vehicle (UAV) power system. Two MOO experimental trials show a drastic reduction in the number of design simulations to discover a high-quality Pareto front. In Trial 1, MESMOC uncovered the entire Pareto front while only requiring to explore ~4% of the design space. With expanded design parameters and larger design space in Trial 2, a near complete but high-quality Pareto front was uncovered. Both trials compared MESMOC to the popular genetic algorithm NSGA-II and another BO algorithm predictive entropy search for multi-objective Bayesian optimization with constraints (PESMOC), showing superior performance.

Uniform mixture design via evolutionary multi‐objective optimization

Design of experiments is a branch of statistics that has been employed in different areas of knowledge. A particular case of experimental designs is uniform mixture design. A uniform mixture design method aims to spread points (mixtures) uniformly distributed in the experimental region. Each mixture should meet the constraint that the sum of its components must be equal to one. In this paper, we propose a new method to approximate uniform mixture designs via evolutionary multi-objective optimization. For this task, we formulate three M -objective optimization problems whose Pareto optimal fronts correspond to a mixture design of M components (or dimensions). In order to obtain a uniform mixture design, we consider six well-known algorithms used in the area of evolutionary multi-objective optimization to solve M -objective optimization problems. Thus, a set of solutions approximates the entire Pareto front of each M -objective problem, while it implicitly approximates a uniform mixture design. We evaluate our proposed methodology by generating mixture designs in two, three, and up to eight dimensions, and we compare the results obtained concerning those produced by different methods available in the specialized literature. Our results indicate that the proposed strategy is a promising alternative to approximate uniform mixture designs. Unlike most of the existing approaches, it obtains mixture designs for an arbitrary number of points. Moreover, the generated design points are properly distributed in the experimental region.

A Region Preference-based Optimization Algorithm for Satellite Mission Planning Problem

For the earth observation satellite mission planning problem, objectives such as observed target quantity, observation profit, energy consumption, image quality should be considered simultaneously, which is a many-objective optimization problem. Classical optimization-based mission planning algorithms obtain a set of non-dominated solutions in the entire search space, while only a single satisfy final plan is desired by decision maker. In this paper, a five-objective optimization model for satellite mission planning problem is constructed, then a region preference-based evolutionary algorithm, HMOEA-T, is applied to obtain the desired solutions. The decision makers describe the preference on each objective in target region form, then the algorithm guides a more detailed search within the preference region rather than the entire Pareto front. Comparative studies with preference-based methods (T-NSGA-III) and classical methods (NSGA-III) are conducted. We have exemplified the proposed method manage to obtain the solutions satisfying the mission planning preference and achieve better performance in convergence and diversity.

A Mating Selection Based on Modified Strengthened Dominance Relation for NSGA-III

In multi/many-objective evolutionary algorithms (MOEAs), to alleviate the degraded convergence pressure of Pareto dominance with the increase in the number of objectives, numerous modified dominance relationships were proposed. Recently, the strengthened dominance relation (SDR) has been proposed, where the dominance area of a solution is determined by convergence degree and niche size (θ¯). Later, in controlled SDR (CSDR), θ¯ and an additional parameter (k) associated with the convergence degree are dynamically adjusted depending on the iteration count. Depending on the problem characteristics and the distribution of the current population, different situations require different values of k, rendering the linear reduction of k based on the generation count ineffective. This is because a particular value of k is expected to bias the dominance relationship towards a particular region on the Pareto front (PF). In addition, due to the same reason, using SDR or CSDR in the environmental selection cannot preserve the diversity of solutions required to cover the entire PF. Therefore, we propose an MOEA, referred to as NSGA-III*, where (1) a modified SDR (MSDR)-based mating selection with an adaptive ensemble of parameter k would prioritize parents from specific sections of the PF depending on k, and (2) the traditional weight vector and non-dominated sorting-based environmental selection of NSGA-III would protect the solutions corresponding to the entire PF. The performance of NSGA-III* is favourably compared with state-of-the-art MOEAs on DTLZ and WFG test suites with up to 10 objectives.

Tuned-mass-damper-inerter optimal design and performance assessment for multi-storey hysteretic buildings under seismic excitation

Inerter-based vibration absorbers (IVAs), such as the tuned-mass-damper-inerter (TMDI), have become popular in recent years for the earthquake protection of building structures. Previous studies using linear structural models have shown that IVAs can achieve enhanced vibration suppression, but at the expense of increased control forces exerted from the IVA to the host building structure. The authors recently developed a bi-objective IVA design framework for linearly behaving buildings to balance between structural performance (drift/acceleration suppression) and IVA forces. This paper extends the framework to multi-storey hysteretic/yielding structures under seismic excitation. Though the proposed design framework can accommodate any type of IVA, the focus is herein on TMDI applications, with tuned-mass-damper (TMD) and tuned-inerter-damper (TID) treated as special cases of the TMDI. Earthquake hazard is modeled through representative, design-level acceleration time-histories and response of the IVA-equipped structure is evaluated through nonlinear response-history analysis. A high-fidelity finite element model (FEM) is established to accurately describe hysteretic structural behavior. To reduce the computational burden, a reduced order model (ROM) is based on the original FEM, using the framework proposed recently by the first and second authors. The ROM maintains the accuracy of the original FEM while enabling for a computationally efficient solution to the optimization problem. As an illustrative example, the bi-objective design for different IVA placements along the height of a non-linear benchmark 9-storey steel frame structure is examined. The accuracy of the ROM-based design is evaluated by comparing performance to the FEM-based response predictions across the entire Pareto front resulting from the bi-objective optimization. Then, the designs and associated performance predicted by using a linear or a nonlinear structural model are compared to evaluate how the explicit consideration of nonlinearities, as well as the degree of nonlinear behavior, impact the IVA design and efficiency.

Locating Pareto optimal designs of antenna through parallel multiobjective evolutionary algorithm

Population-based multiobjective metaheuristics are potential in microwave component designs because all electric properties could be optimized simultaneously to find several tradeoff designs in a single run. However, they often suffer extremely-expensive computational time caused by too many full-time objective function evaluations. In this paper, firstly, an efficient parallel implementation of MOEA/D (multiobjective evolutionary algorithm based on decomposition) is proposed to reduce the overall computational time. Secondly, optimization speed, approximation performance to the entire Pareto front, and algorithm robustness of the proposed parallel MOEA/D are investigated through extensive standard test instances. Then, a complete design procedure based on the proposed optimization algorithm is presented to locate multiple Pareto optimal designs of antenna. Finally, a compact broadband dual-polarized antenna operating at 3.3-5.0 GHz is designed for 5G base station applications. Comparisons are made to verify the great performance of the proposed method.

Automatic preference based multi-objective evolutionary algorithm on vehicle fleet maintenance scheduling optimization

A preference based multi-objective evolutionary algorithm is proposed for generating solutions in an automatically detected knee point region. It is named Automatic Preference based DI-MOEA (AP-DI-MOEA) where DI-MOEA stands for Diversity-Indicator based Multi-Objective Evolutionary Algorithm). AP-DI-MOEA has two main characteristics: firstly, it generates the preference region automatically during the optimization; secondly, it concentrates the solution set in this preference region. Moreover, the real-world vehicle fleet maintenance scheduling optimization (VFMSO) problem is formulated, and a customized multi-objective evolutionary algorithm (MOEA) is proposed to optimize maintenance schedules of vehicle fleets based on the predicted failure distribution of the components of cars. Furthermore, the customized MOEA for VFMSO is combined with AP-DI-MOEA to find maintenance schedules in the automatically generated preference region. Experimental results on multi-objective benchmark problems and our three-objective real-world application problems show that the newly proposed algorithm can generate the preference region accurately and that it can obtain better solutions in the preference region. Especially, in many cases, under the same budget, the Pareto optimal solutions obtained by AP-DI-MOEA dominate solutions obtained by MOEAs that pursue the entire Pareto front.

Sequential Multi-objective Genetic Algorithm

Many of the real-world issues have multiple conflicting objectives that the optimization between contradictory objectives is very difficult. In recent years, the Multi-objective Evolutionary Algorithms (MOEAs) have shown great performance to optimize such problems. So, the development of MOEAs will always lead to the advancement of science. The Non-dominated Sorting Genetic Algorithm II (NSGAII) is considered as one of the most used evolutionary algorithms, and many MOEAs have emerged to resolve NSGAII problems, such as the Sequential Multi-Objective Algorithm (SEQ-MOGA). SEQ-MOGA presents a new survival selection that arranges individuals systematically, and the chromosomes can cover the entire Pareto Front region. In this study, the Archive Sequential Multi-Objective Algorithm (ASMOGA) is proposed to develop and improve SEQ-MOGA. ASMOGA uses the archive technique to save the history of the search procedure, so that the maintenance of the diversity in the decision space is satisfied adequately. To demonstrate the performance of ASMOGA, it is used and compared with several state-of-the-art MOEAs for optimizing benchmark functions and designing the I-Beam problem. The optimization results are evaluated by Performance Metrics such as hypervolume, Generational Distance, Spacing, and the t-test (a statistical test); based on the results, the superiority of the proposed algorithm is identified clearly.

A SHADE-based multimodal multi-objective evolutionary algorithm with fitness sharing

In the multimodal multi-objective optimization problems (MMOPs), at least two equivalent Pareto optimal solutions in decision space with an identical objective value are desired. The challenge for solving MMOPs is locating equivalent Pareto optimal solutions in decision space, and maintaining a fine balance between diversity and convergence of Pareto optimal solutions in both decision space and objective space, simultaneously. To address this issue, a success-history based parameter adaptation for multimodal multi-objective differential evolution algorithm using fitness sharing (MMOSHADE) is proposed in this paper. A success-history based parameter adaptation for differential evolution (SHADE) is integrated into MMOSHADE to find elite individuals and locate Pareto optimal solutions in decision space. Subsequently, a modified selection operation in differential evolution (DE) is introduced into MMOSHADE to explore outstanding convergence solutions. Furthermore, a double fitness sharing method is available for maintaining the diversity of Pareto optimal solutions in both decision space and objective space, simultaneously. The proposed MMOSHADE is performed on three categories of problems to test the performance of MMOSHADE. The comparison between MMOSHADE and six competing algorithms demonstrates the superiority of the proposed MMOSHADE in solving MMOPs and large-scale polygon-based MMOPs. MMOSHADE is also capable of finding the entire Pareto front in most cases when it is used to address multi-objective optimization problems. Additionally, the effectiveness of several strategies is validated by the designed experiments, and the parameters involved in MMOSHADE are discussed.