Set Of Trade-off Solutions Research Articles

Evolutionary Multi-Objective Workflow Scheduling for Volatile Resources in the Cloud

The cloud has been widely used as a distributed computing platform for running scientific workflow applications. Most of the cloud providers encourage the use of their underutilized resources as spot instances for much cheaper prices compared with common resources as on-demand instances, however, the promise of lower costs for resources results in the volatility such that spot instances can be interrupted at any time by cloud providers. Many workflow scheduling algorithms have been proposed to deal with volatile resources. In this article, we consider the two most important features of the volatile resources namely fulfillment and interruption rates to fully model the instability of the cloud infrastructure. Subsequently, we propose a novel evolutionary multi-objective workflow scheduling approach to generate a set of trade-off solutions that outperform state-of-the-art algorithms in both makespan and economic costs. In addition, we explore the fluctuation of makespan and costs for our obtained schedules under different levels of fulfillment and interruption rates. Experimental results with the five well-known real-world workflows demonstrate that our evolutionary multi-objective workflow scheduling algorithm is competitive in terms of makespan and cost compared with state-of-the-art on-demand scheduling techniques.

Multi-Objective Optimization Design of a Modular Linear Permanent-Magnet Vernier Machine by Combined Approximation Models and Differential Evolution

In this article, a modular linear permanent-magnet (PM) vernier machine is presented and optimized for high-precision and safety-critical direct-drive applications. The novelty of this article is to not only propose a multi-objective optimization method, but also present a modular linear PM vernier machine. The presented machine has the merit of desired fault-tolerant capability and improved force performances by mainly adopting the modular mover structure. And, the design principle of modular mover structure is introduced in detail in this article. To realize low detent force, high thrust force, low force ripple, and good fault tolerance, modular mover structure, PM array with flux-intensifying effect and six-phase structure are adopted in machine design. Furthermore, a new multi-objective optimization method is proposed to obtain a set of tradeoff solutions among multiple optimization objectives in machine optimization. First, to decompose the high dimension design space, the design variables are stratified into nonsensitive and sensitive levels by utilizing a comprehensive sensitivity analysis. Second, combined approximation models are implemented to decrease the large amount of computations. Then, to enhance the efficiencies of two multi-objective optimization processes, multi-objective differential evolution algorithm with ranking-based mutation operator is employed. After the optimization by the proposed multi-objective optimization method, the desired comprehensive performances with high optimization efficiency and accuracy can be obtained. Finally, to verify the results of theoretical analysis, a prototype machine is fabricated.

Bi-objective optimization of biomass supply chains considering carbon pricing policies

Bi-objective optimization models considering carbon pricing policies are developed in this paper to obtain the trade-off between cost and emissions of biomass supply chain models, which is important for decision making. Solving bi-objective optimization models to obtain the set of trade-off solutions can be time consuming. To avoid the computational effort, in this paper, a new algorithm is developed to obtain the solutions for the bi-objective models with carbon pricing policies using the solutions of the bi-objective model without carbon pricing. The algorithm is based on mathematical properties of optimum solutions of bi-objective models with and without carbon pricing policies. These properties are proved mathematically. The developed algorithm is applied to a case study of a biomass-fed district heating system. Results indicate that the number of optimum solutions to the bi-objective models decrease when emissions are priced compared to when emissions are not priced. The increase in total cost for mitigating a given quantity of emission is more for the carbon offset model compared to the carbon tax and the carbon cap-and-trade models. Pair-wise comparison of the models indicates that the carbon tax model has more cost than the carbon cap-and-trade and the carbon offset models. The algorithm and results of this study are independent of the case study; therefore, they can be applied to other cases and industries.

A smooth proximity measure for optimality in multi-objective optimization using Benson’s method

Multi-objective optimization problems give rise to a set of trade-off Pareto-optimal solutions. To evaluate a set-based multi-objective optimization algorithm, such as an evolutionary multi-objective optimization (EMO) algorithm, for its convergence and diversity attainment, more than one performance metrics are required. To measure the convergence aspect, a new Karush-Kuhn-Tucker proximity measure (KKTPM) was recently proposed based on the extent of satisfaction of KKT optimality conditions on the augmented achievement scalarization function (AASF) formulation. However, the Pareto-optimality of a point depends on a parameter needed to be used in the AASF formulation. In this paper, we use Benson’s method as a scalarized version of the multi-objective optimization problem, mainly because it is parameter-less and is a popularly used in the multi-criterion decision-making (MCDM) literature. The proposed Benson’s method based metric (B-KKTPM) is applied to optimized solutions of popular EMO algorithms on standard two to 10-objective test problems and to a few engineering design problems. B-KKTPM is able to determine relative closeness of a set of trade-off solutions from the strictly efficient solutions without any prior knowledge of them. To reduce the computational cost of solving an optimization problem to compute B-KKTPM, we also propose a direct, but approximate, method. The obtained results from our extensive study indicates that (i) the proposed optimization based and direct B-KKTPMs can be used for a termination check for any optimization algorithm, and (ii) the direct B-KKTPM method can be used as a replacement of the optimization-based version for a good trade-off between computational time and accuracy.

Inference of gene regulatory networks with multi-objective cellular genetic algorithm

Reverse engineering of biochemical networks remains an important open challenge in computational systems biology. The goal of model inference is to, based on time-series gene expression data, obtain the sparse topological structure and parameters that quantitatively understand and reproduce the dynamics of biological systems. In this paper, we propose a multi-objective approach for the inference of S-System structures for Gene Regulatory Networks (GRNs) based on Pareto dominance and Pareto optimality theoretical concepts instead of the conventional single-objective evaluation of Mean Squared Error (MSE). Our motivation is that, using a multi-objective formulation for the GRN, it is possible to optimize the sparse topology of a given GRN as well as the kinetic order and rate constant parameters in a decoupled S-System, yet avoiding the use of additional penalty weights. A flexible and robust Multi-Objective Cellular Evolutionary Algorithm is adapted to perform the tasks of parameter learning and network topology inference for the proposed approach. The resulting software, called MONET, is evaluated on real-based academic and synthetic time-series of gene expression taken from the DREAM3 challenge and the IRMA in vivo datasets. The ability to reproduce biological behavior and robustness to noise is assessed and compared. The results obtained are competitive and indicate that the proposed approach offers advantages over previously used methods. In addition, MONET is able to provide experts with a set of trade-off solutions involving GRNs with different typologies and MSEs.

Evolutionary Machine Learning for Multi-Objective Class Solutions in Medical Deformable Image Registration

Current state-of-the-art medical deformable image registration (DIR) methods optimize a weighted sum of key objectives of interest. Having a pre-determined weight combination that leads to high-quality results for any instance of a specific DIR problem (i.e., a class solution) would facilitate clinical application of DIR. However, such a combination can vary widely for each instance and is currently often manually determined. A multi-objective optimization approach for DIR removes the need for manual tuning, providing a set of high-quality trade-off solutions. Here, we investigate machine learning for a multi-objective class solution, i.e., not a single weight combination, but a set thereof, that, when used on any instance of a specific DIR problem, approximates such a set of trade-off solutions. To this end, we employed a multi-objective evolutionary algorithm to learn sets of weight combinations for three breast DIR problems of increasing difficulty: 10 prone-prone cases, 4 prone-supine cases with limited deformations and 6 prone-supine cases with larger deformations and image artefacts. Clinically-acceptable results were obtained for the first two problems. Therefore, for DIR problems with limited deformations, a multi-objective class solution can be machine learned and used to compute straightforwardly multiple high-quality DIR outcomes, potentially leading to more efficient use of DIR in clinical practice.

Coevolutionary Multiobjective Evolutionary Algorithms: Survey of the State-of-the-Art

In the last 20 years, evolutionary algorithms (EAs) have shown to be an effective method to solve multiobjective optimization problems (MOPs). Due to their population-based nature, multiobjective EAs (MOEAs) are able to generate a set of tradeoff solutions (called nondominated solutions) in a single algorithmic execution instead of having to perform a series of independent executions, as normally done with mathematical programming techniques. Additionally, MOEAs can be successfully applied to problems with difficult features such as multifrontality, discontinuity and disjoint feasible regions, among others. On the other hand, coevolutionary algorithms (CAs) are extensions of traditional EAs which have become subject of numerous studies in the last few years, particularly for dealing with large-scale global optimization problems. CAs have also been applied to the solution of MOPs, motivating the development of new algorithmic and analytical formulations that have advanced the state-of-the-art in CAs research, while simultaneously opening a new research path within MOEAs. This paper presents a critical review of the most representative coevolutionary MOEAs (CMOEAs) that have been reported in the specialized literature. This survey includes a taxonomy of approaches together with a brief description of their main features. In the final part of this paper, we also identify what we believe to be promising areas of future research in the field of CMOEAs.

Simplex-based navigation tool for a posteriori selection of the preferred deformable image registration outcome from a set of trade-off solutions obtained with multiobjective optimization for the case of breast MRI.

Multiobjective optimization approaches for deformable image registration (DIR) remove the need for manual adjustment of key parameters and provide a set of solutions that represent high-quality trade-offs between objectives of interest. Choosing a desired outcome a posteriori is potentially far more insightful as differences between solutions can be immediately visualized. The purpose of this work is to investigate whether such an approach allows clinical experts to intuitively select their preferred DIR outcome. To this end, we developed a simplex-based tool for solution navigation and asked 10 clinical experts to use it to choose their preferred DIR outcome from sets of trade-off solutions obtained for 10 breast magnetic resonance DIR cases of low (prone-prone DIR; ) and high (prone-supine DIR; ) difficulty, of patients and volunteers, respectively. The usability of the software is subsequently evaluated by the observers using the system usability scale. Further, the quality of the selected DIR outcomes is evaluated using the mean target registration error. Results show that the users are able to identify and select high-quality DIR outcomes, and attested to high learnability and usability of our software, supporting the validity of the presumed added value of taking a multiobjective perspective on DIR in clinical practice.

Multi-layer interaction preference based multi-objective evolutionary algorithm through decomposition

Many problems in real world have not only one objective to be met. In the majority of cases, a set of trade-off solutions which spread evenly along the entire Pareto optimal front are generated by multi-objective evolutionary algorithms (MOEAs). Taking the preference of decision maker (DM) into consideration, some specified solutions can be obtained, which is of great interest in practical applications. In this paper, a novel multi-layer interaction preference based multi-objective evolutionary algorithm through decomposition (denoted as MLIP-MOEA/D) is proposed. In MLIP-MOEA/D, a multi-layer interactive strategy is developed during evolution, in the first-layer interaction, the DM will provide a reference vector and an initial radius to determine a preference range, then all solutions in this range will be updated. The algorithm will stop if the DM is satisfied with the first output result, otherwise it will go on to the second-layer interaction. In this step, the most preferred solution generated from the first-layer interaction will be chosen as the new preference direction, and the weight vector is redefined by the angle-based method, and the range of preferred region is reduced gradually, until the closest solution that meet the DM’s need is found. The algorithm is tested on a set of benchmark problems including DTLZ problems with more than three objectives, the experimental studies show that the proposed algorithm can effectively search the preferred solutions with the preference information and successfully deal with many-objective optimization problems.

Enhanced directed search: a continuation method for mixed-integer multi-objective optimization problems

Multi-objective optimization problems (MOPs) commonly arise in various applications of engineering and management fields. Many real-world MOPs are mixed-integer multi-objective optimization problems (MMOP), where the solution space consists of real and integer decision variables. The research regarding MMOPs is still scarce due to the mixture nature of the solution space and difficulty of finding the set of trade-off solutions. In this work we propose a continuation based method that efficiently solves MMOP problems. Our method, called Enhanced Directed Search (EDS), is capable of steering the search along a predefined direction along the Pareto front in the objective function space. EDS traces the Pareto front by following closest predictor and corrector solutions in the course of optimization. By searching around the objective function boundary, EDS can solve problems with $$k > 2$$ objectives. With five example problems widely studied in the literature, we demonstrate that EDS outperforms the recently developed Direct Zig Zag algorithm and the popular NSGA-II method.

Integrating fitness predator optimizer with multi-objective PSO for dynamic partitional clustering

Clustering is an unsupervised classification task in data mining, which partitions data set into certain number of clusters. Though many metaheuristics have been suggested to optimize clustering results, most of them are marred by the three key issues. First, clustering performance is devised on a single validity measure; hence, it produces a single best solution biased toward the criterion; it is unacceptable as clustering has multiple conflicting criteria. Second, they find difficulty in avoiding local optima owing to lack of balancing in exploration and exploitation in the search space. Third, they require number of clusters a priori which is difficult to decide in the real-world problems. To deal with the first issue, we follow Pareto-based approach to obtain diverse trade-off solutions by optimizing conceptually contradicting validity measures sum of squared error and connectedness. The well-known swarm intelligence-based metaheuristic particle swarm optimization (PSO) has a severe drawback that it converges prematurely owing to single directional information sharing mechanism among particles in the swarm. Therefore, for the second issue, we introduce fitness predator optimizer (FPO) to enhance diversity in the PSO in multi-objective optimization scenario; it is named as FPO-MOPSO. To address the third issue, we use dynamic clustering method to decide an optimum number of clusters (k) in the range of 2 to \(\sqrt{n}\), where n is number of objects in the data set. We use eight real-world and two artificial data sets for comparison of the FPO-MOPSO results with three well-known competitive methods MOPSO, MABC, and NSGA-II using four quality measures convergence, diversity, coverage, and ONVG. The obtained results show superiority of the FPO-MOPSO over its competitors. Two external validity indices of clustering results F-measure and Rand index also establish the above finding. As the decision maker picks only a single solution from the set of trade-off solutions, we employ an additional measure silhouette index to select a final single solution from the archive and compare it with known number of clusters in the data sets.

Comparative Study of Knee-Based Algorithms for Many-Objective Optimization Problems

Nowadays, most real-world optimization problems consist of many and often conflicting objectives to be optimized simultaneously. Although, many current Multi-Objective optimization algorithms can efficiently solve problems with 3 or less objectives, their performance deteriorates proportionally with the increasing of the objectives number. Furthermore, in many situations the decision maker (DM) is not interested in all trade-off solutions obtained but rather interested in a single optimum solution or a small set of those trade-offs. Therefore, determining an optimum solution or a small set of trade-off solutions is a difficult task. However, an interesting method for finding such solutions is identifying solutions in the Knee region. Solutions in the Knee region can be considered the best obtained solution in the obtained trade-off set especially if there is no preference or equally important objectives. In this paper, a pruning strategy was used to find solutions in the Knee region of Pareto optimal fronts for some benchmark problems obtained by NSGA-II, MOEA/D-DE and a promising new Multi-Objective optimization algorithm NSGA-III. Lastly, those knee solutions found were compared and evaluated using a generational distance performance metric, computation time and a statistical one-way ANOVA test.

Multi-objective sizing and topology optimization of truss structures using genetic programming based on a new adaptive mutant operator

Most real-world engineering problems deal with multiple conflicting objectives simultaneously. In order to address this issue in truss optimization, this paper presents a multi-objective genetic programming approach for sizing and topology optimization of trusses. It aims to find the optimal cross-sectional areas and connectivities between the nodes to achieve a set of trade-off solutions to satisfy all the optimization objective functions subjected to some constraints such as kinematic stability, maximum allowable stress in members and nodal deflections. It also uses the variable-length representation of potential solutions in the shape of computer programs and evolves to the potential final set of solutions. This approach also employs an adaptive mutant factor besides the classical genetic operators to improve the exploring capabilities of Genetic Programming in structural optimization. The intrinsic features of genetic programming help to identify redundant truss members and nodes in the design space, while no violation of constraints occurs. Our approach applied to some numerical examples and found a better non-dominated solution set in the most cases in comparison with the competent methods available in the literature.

Preference-Inspired Co-Evolutionary Algorithms With Local PCA Oriented Goal Vectors for Many-Objective Optimization

It remains a challenge to identify a satisfactory set of tradeoff solutions for many-objective optimization problems that have more than three objectives. Coevolving the solutions with preference is becoming increasingly popular due to the enhanced local search capability, which makes it suitable for solving many-objective optimization problems. The framework of preference-inspired co-evolutionary algorithms (PICEAs) is suitable for obtaining promising performance for such problems, and the PICEA with goal vectors (PICEA-g) has achieved good performance in many applications. In this paper, an improved PICEA-g is proposed to further resolve this long-standing problem. The local principal component analysis operator is used as a controller to further expand the ability of the PICEA-g algorithm and enhance the convergence of PICEA-g. The proposed algorithm was evaluated using several widely used benchmark test suites that had 3–15 objectives and made a systematic comparison with five state-of-the-art multi-objective evolutionary algorithms. The resulting substantial amount of experimental results revealed that the algorithm we proposed could have good performance on most of the test suites assessed in our research, and it performs very well compared with other many-objective optimization algorithms. In addition, a sensitivity test was carried out to explore the impact of a key parameter in the algorithm we proposed in this study.

Gravitational search algorithm and K-means for simultaneous feature selection and data clustering: a multi-objective approach

Clustering is an unsupervised classification method used to group the objects of an unlabeled data set. The high dimensional data sets generally comprise of irrelevant and redundant features also along with the relevant features which deteriorate the clustering result. Therefore, feature selection is necessary to select a subset of relevant features as it improves discrimination ability of the original set of features which helps in improving the clustering result. Though many metaheuristics have been suggested to select subset of the relevant features in wrapper framework based on some criteria, most of them are marred by the three key issues. First, they require objects class information a priori which is unknown in unsupervised feature selection. Second, feature subset selection is devised on a single validity measure; hence, it produces a single best solution biased toward the cardinality of the feature subset. Third, they find difficulty in avoiding local optima owing to lack of balancing in exploration and exploitation in the feature search space. To deal with the first issue, we use unsupervised feature selection method where no class information is required. To address the second issue, we follow pareto-based approach to obtain diverse trade-off solutions by optimizing conceptually contradicting validity measures silhouette index (Sil) and feature cardinality (d). For the third issue, we introduce genetic crossover operator to improve diversity in a recent Newtonian law of gravity-based metaheuristic binary gravitational search algorithm (BGSA) in multi-objective optimization scenario; it is named as improved multi-objective BGSA for feature selection (IMBGSAFS). We use ten real-world data sets for comparison of the IMBGSAFS results with three multi-objective methods MBGSA, MOPSO, and NSGA-II in wrapper framework and the Pearson’s linear correlation coefficient (FM-CC) as a multi-objective filter method. We employ four multi-objective quality measures convergence, diversity, coverage and ONVG. The obtained results show superiority of the IMBGSAFS over its competitors. An external clustering validity index F-measure also establish the above finding. As the decision maker picks only a single solution from the set of trade-off solutions, we employee the F-measure to select a final single solution from the external archive. The quality of final solution achieved by IMBGSAFS is superior over competitors in terms of clustering accuracy and/or smaller subset size.

Multi‐objective generation dispatch considering the trade‐off between economy and security

This study presents a multi-objective generation dispatch model based on security-constrained optimal power flow, in order to handle the contradictory economic and security objectives. In addition to the deterministic N−1 security criterion, a risk index is proposed to quantify the system security degree on a probabilistic basis. The risk index is defined as the summation of the products of the contingency probability and the associated severity function. Moreover, the cost of interruptible load is added into the generation cost, formulating the economic objective. In the meantime, wind power outputs and load variations are considered as uncertainties in the dispatch process. Furthermore, a multi-objective evolutionary algorithm is employed and developed to provide a set of trade-off solutions. The proposed approach is verified by the 4-bus and IEEE 24-bus systems. Numerical results show that the proposed approach can effectively investigate the impacts of demand response on system economy and security. The obtained Pareto optimal solutions are less likely to incur insecurity risk and are robust to the studied stochastic variations.

Aggregation of multi-objective fuzzy symmetry-based clustering techniques for improving gene and cancer classification

The current work reports about the application of a cluster ensemble approach in combining results produced by some multiobjective-based clustering techniques. Firstly, some multiobjective-based fuzzy clustering techniques are developed using the search capabilities of differential evolution and particle swarm optimization. Both these clustering techniques utilize a recently developed point symmetry-based distance for allocation of points to different clusters. The appropriate partitioning from a data set is identified by optimizing simultaneously two cluster quality measures, namely Xie–Beni index and FSym-index. First objective function uses Euclidean distance as a similarity measure, and the second objective function uses point symmetry-based distance in its computation. A set of trade-off solutions are produced by each of these clustering techniques on the final Pareto optimal front. Finally, this set of solutions are combined using a link-based cluster ensemble technique. The effectiveness of ensemble techniques is illustrated on partitioning some real-life gene expression and cancer data sets where automatic identification of set of genes or set of cancer tissues is a pressing issue. The potency of the ensemble techniques applied on both the multi-objective DE- and PSO-based clustering approaches is shown in comparison with several state-of-the-art techniques.

Bridging the Gap: Many-Objective Optimization and Informed Decision-Making

The field of many-objective optimization has grown out of infancy and a number of contemporary algorithms can deliver well converged and diverse sets of solutions close to the Pareto optimal front. Concurrently, the studies in cognitive science have highlighted the pitfalls of imprecise decision-making in presence of a large number of alternatives. Thus, for effective decision-making, it is important to devise methods to identify a handful (7 ± 2) of solutions from a potentially large set of tradeoff solutions. Existing measures such as reflex/bend angle, expected marginal utility (EMU), maximum convex bulge/distance from hyperplane, hypervolume contribution, and local curvature are inadequate for the purpose as: 1) they may not create complete ordering of the solutions; 2) they cannot deal with large number of objectives and/or solutions; and 3) they typically do not provide any insight on the nature of selected solutions (internal, peripheral, and extremal). In this letter, we introduce a scheme to identify solutions of interest based on recursive use of the EMU measure. The nature of the solutions (internal or peripheral) is then characterized using reference directions generated via systematic sampling and the top ${K}$ solutions with the largest relative EMU measure are presented to the decision maker. The performance of the approach is illustrated using a number of benchmarks and engineering problems. In our opinion, the development of such methods is necessary to bridge the gap between theoretical development and real-world adoption of many-objective optimization algorithms.

Solving multi-objective water management problems using evolutionary computation

Water as a resource is becoming increasingly more valuable given the changes in global climate. In an agricultural sense, the role of water is vital to ensuring food security. Therefore the management of it has become a subject of increasing attention and the development of effective tools to support participative decision-making in water management will be a valuable contribution. In this paper, evolutionary computation techniques and Pareto optimisation are incorporated in a model-based system for water management. An illustrative test case modelling optimal crop selection across dry, average and wet years based on data from the Murrumbidgee Irrigation Area in Australia is presented. It is shown that sets of trade-off solutions that provide large net revenues, or minimise environmental flow deficits can be produced rapidly, easily and automatically. The system is capable of providing detailed information on optimal solutions to achieve desired outcomes, responding to a variety of factors including climate conditions and economics.

Optimal Placement and Sizing of Renewable Distributed Generations and Capacitor Banks into Radial Distribution Systems

In recent years, renewable types of distributed generation in the distribution system have been much appreciated due to their enormous technical and environmental advantages. This paper proposes a methodology for optimal placement and sizing of renewable distributed generation(s) (i.e., wind, solar and biomass) and capacitor banks into a radial distribution system. The intermittency of wind speed and solar irradiance are handled with multi-state modeling using suitable probability distribution functions. The three objective functions, i.e., power loss reduction, voltage stability improvement, and voltage deviation minimization are optimized using advanced Pareto-front non-dominated sorting multi-objective particle swarm optimization method. First a set of non-dominated Pareto-front data are called from the algorithm. Later, a fuzzy decision technique is applied to extract the trade-off solution set. The effectiveness of the proposed methodology is tested on the standard IEEE 33 test system. The overall results reveal that combination of renewable distributed generations and capacitor banks are dominant in power loss reduction, voltage stability and voltage profile improvement.