Achievement Scalarizing Function Research Articles

Preference-based multi-objective evolutionary algorithm with linear combination scalarizing function and reference point adjustment

In practice, the decision-maker (DM) may be only interested in a particular part of Pareto optimal front (PF). For this reason, many preference-based multi-objective evolutionary algorithms (MOEA) have been proposed to find a solution set that approximates the region of the interest (ROI). Most existing preference-based methods focus on selecting solutions in the ROI. Nevertheless, the enhancement of convergence in preference-based MOEAs has been neglected. Most decomposition-based approaches often employ the achievement scalarizing function (ASF) as their scalarizing function. However, it holds a weaker search ability than the weighted sum function (WSF) despite its capability to tackle problems with arbitrary PF geometries. In order to strengthen the selective pressure toward the PF, this paper proposes a new scalarizing function, LSF, which is a linear combination of the WSF and the ASF. Then, a simple adaptive penalty scheme is employed in LSF to balance the search ability and robustness. To focus the search on the ROI, we develop a reference point adjustment method that dynamically adjusts the position of the reference point according to the distance from the approximated target point. We apply the above two innovations to the MOEA/D framework and propose a new preference-based MOEA, namely RAMOEAD. Experimental results show that RAMOEAD is highly competitive when compared with five state-of-the-art preference-based MOEAs. Finally, the proposed algorithm is extended to double reference points for solving the problems in which the ROIs are defined by the reservation and aspiration points.

A new proximity metric based on optimality conditions for single and multi-objective optimization: Method and validation

With the widespread application of Evolutionary Algorithms (EAs), their performance needs to be evaluated using more than the usual performance metrics. In the EA literature, various metrics assess the convergence ability of these algorithms. However, many of them require prior knowledge of the Pareto-optimal front. Recently, two Karush–Kuhn–Tucker Proximity Metrics (KKTPMs) have been introduced to measure convergence without needing prior knowledge. One relies on the Augmented Achievement Scalarization Function (AASF) method (AASF-KKTPM), and the other on Benson’s method (B-KKTPM). However, both require specific parameters and reference points, making them computationally expensive. In this paper, we introduce a novel version of KKTPM applicable to single-, multi-, and many-objective optimization problems, utilizing the Penalty-based Boundary Intersection (PBI) method (PBI-KKTPM). Additionally, we introduce an approximate approach to reduce the computational burden of solving PBI-KKTPM optimization problems. Through extensive computational experiments across 23 case studies, our proposed metric demonstrates a significant reduction in computational cost, ranging from 20.68% to 60.03% compared to the computational overhead associated with the AASF-KKTPM metric, and from 16.48% to 61.15% compared to the computational overhead associated with the B-KKTPM metric. Noteworthy features of the proposed metric include its independence from knowledge of the true Pareto-optimal front and its applicability as a termination criterion for EAs. Another feature of the proposed metric is its ability to deal with black box problems very efficiently.

Interpretable self-organizing map assisted interactive multi-criteria decision-making following Pareto-Race

The problem-solving task of the multi-criteria decision-making (MCDM) approach involves decision makers’ (DMs’) interaction by incorporating their preferences to arrive at one or more preferred near Pareto-optimal solution(s). The Pareto-Race is an interactive MCDM approach that relies on finding preferred solutions as the DM navigates on the Pareto surface by selecting the preferred reference direction and uniform/varying speed using the concept of Achievement Scalarization Function (ASF) or its augmented version (AASF). Selecting a reference direction or speed is possible, algorithmically, but the effectiveness of the Pareto-Race concept relies on suitable visualization methods that present the series of past solutions and convey the current objective trade-off information to assist the DM’s in decision-making task. Traditional multi-dimensional Pareto-optimal front visualization methods like parallel coordinate plots, radial visualization, and heat maps are deemed insufficient for this purpose. Therefore, the paper proposes integrating the concept of the Pareto-Race MCDM approach with a recently developed interpretable self-organizing map (iSOM) based visualization method. The iSOM maps high-dimensional data into lower-dimensional space, facilitating visual convenience for the DM. The proposed method requires a finite-size representation of non-dominated solutions near the complete Pareto Front to generate iSOM plots of objectives. We also propose visualizing the metrics such as closeness to constraint boundaries, trade-off value, and robustness using iSOM to check the quality of the preferred solutions, which is rarely considered in existing MCDM approaches. iSOM plots of objective functions, closeness to constraint, trade-off, and robustness metrics assist the DM in effectively choosing new reference direction and step size in each iteration to arrive at the most preferred solution. The proposed iSOM-enabled Pareto-Race approach is demonstrated on benchmark analytical and real-world engineering examples with 3 to 10 objectives.

A constrained multi-objective evolutionary algorithm with Pareto estimation via neural network

The main challenge in addressing constrained multi-objective optimization problems (CMOPs) lies in achieving a balance among convergence, diversity, and feasibility. To address this issue, this paper proposes a constrained multi-objective evolutionary algorithm with Pareto estimation via neural network named CMOEA-PeNN. In order to exploit and explore the decision space, the proposed algorithm employs a dual-population mechanism, which is trained with a self-organizing map (SOM). Firstly, the population distribution structure in decision space is mapped to objective space while preserving neighborhood information, and then the neuron weight is utilized to estimate the Pareto front (PF). Secondly, a novel approach is devised to preserve the feasibility of the population and enhance the estimation of the Pareto front by SOM. The achievement scalarizing function (ASF) is employed to choose promising solutions. This strategy could guide the population toward the optimal solution while exploring the small feasible regions. Finally, the performance of CMOEA-PeNN is compared with five state-of-the-art constrained multi-objective evolutionary algorithms (CMOEAs) on three widely used benchmark problems and a real-world problem. The experimental results show that CMOEA-PeNN could archive competitive performance in solving CMOPs.

Kriging-Based Framework Applied to a Multi-Point, Multi-Objective Engine Air-Intake Duct Aerodynamic Optimization Problem

The forecasted growth in dynamic global air fleet size in the coming decades, together with the need to introduce disruptive technologies supporting net-zero emission air transport, demands more efficient design and optimization workflows. This research focuses on developing an aerodynamic optimization framework suited for multi-objective studies of small aircraft engine air-intake ducts in multiple flight conditions. In addition to the refinement of the duct’s performance criteria, the work aims to improve the economic efficiency of the process. The optimization scheme combines the advantages of Kriging-based Efficient Global Optimization (EGO) with the Radial Basis Functions (RBF)-based mesh morphing technique and the Chebyshev-type Achievement Scalarizing Function (ASF) for handling multiple objectives and design points. The proposed framework is applied to an aerodynamic optimization study of an I-31T aircraft turboprop engine intake system. The workflow successfully reduces the air-duct pressure losses and mitigates the flow distortion at the engine compressor’s front face in three considered flight phases. The results prove the framework’s potential for solving complex multi-point air-intake duct problems and the capacity of the ASF-based formulation to guide optimization toward the designer’s preferred objective targets.

A multi-objective approach to identify parameters of compartmental epidemiological models—Application to Ebola Virus Disease epidemics

In this work, we propose a novel methodology to identify parameters of compartmental epidemiological models. It is based on solving a multi-objective optimization problem that consists in fitting some of the model outputs to real observations. First, according to the available data of the considered epidemic, we define a multi-objective optimization problem where the model parameters are the optimization variables. Then, this problem is solved by considering a particular optimization algorithm called ParWASF-GA (Parallel Weighting Achievement Scalarizing Function Genetic Algorithm). Finally, the decision maker chooses, within the set of possible solutions, the values of parameters that better suit his/her preferences. In order to illustrate the benefit of using our approach, it is applied to estimate the parameters of a deterministic epidemiological model, called Be-CoDiS (Between-Countries Disease Spread), used to forecast the possible spread of human diseases within and between countries. We consider data from different Ebola outbreaks from 2014 up to 2019. In all cases, the proposed methodology helps to obtain reasonable predictions of the epidemic magnitudes with the considered model.

On the estimation of pareto front and dimensional similarity in many-objective evolutionary algorithm

Evolutionary algorithms have been proven to be effective in solving multi-objective optimization problems. However, their performance deteriorates progressively in handling many-objective optimization problems due to the sensitivity upon the curvature of Pareto front, as well as the implicit evaluation on similarity in high dimensionality. This paper proposes an on-line Pareto front curvature estimator for an adaptive selection, in which the achievement scalarizing function is used to identify the pivotal solution to extrapolate the geometric information. Then an adaptive scalarizing function based fitness assessment, which guarantees the Pareto optimality, is presented. The diversity of the Pareto optimal solutions is also ensured by introducing a novel similarity metric. Finally, an extensive experimental analysis is presented to corroborate the analytical result by evaluating problems with various types of Pareto fronts. The experimental results substantiate the efficacy of the results with competitive performance.

Reference point based evolutionary multi-objective optimization algorithms with convergence properties using KKTPM and ASF metrics

In a preference-based multi-objective optimization task, the goal is to find a subset of the Pareto-optimal set close to a supplied set of aspiration points. The reference point based non-dominated sorting genetic algorithm (R-NSGA-II) was proposed for such problem-solving tasks. R-NSGA-II aims to finding Pareto-optimal points close, in the sense of Euclidean distance in the objective space, to the supplied aspiration points, instead of finding the entire Pareto-optimal set. In this paper, R-NSGA-II method is modified using recently proposed Karush–Kuhn–Tucker proximity measure (KKTPM) and achievement scalarization function (ASF) metrics, instead of Euclidean distance metric. While a distance measure may not produce desired solutions, KKTPM-based distance measure allows a theoretically-convergent local or global Pareto solutions satisfying KKT optimality conditions and the ASF measure allows Pareto-compliant solutions to be found. A new technique for calculating KKTPM measure of a solution in the presence of an aspiration point is developed in this paper. The proposed modified R-NSGA-II methods are able to solve as many as 10-objective problems as effectively or better than the existing R-NSGA-II algorithm.

Achievement scalarizing function sorting for strength Pareto evolutionary algorithm in many-objective optimization

Multi-objective evolutionary algorithms (MOEAs) have proven their effectiveness in solving two or three objective problems. However, recent research shows that Pareto-based MOEAs encounter selection difficulties facing many similar non-dominated solutions in dealing with many-objective problems. In order to reduce the selection pressure and improve the diversity, we propose achievement scalarizing function sorting strategy to make strength Pareto evolutionary algorithm suitable for many-objective optimization. In the proposed algorithm, we adopt density estimation strategy to redefine a new fitness value of a solution, which can select solution with good convergence and distribution. In addition, a clustering method is used to classify the non-dominated solutions, and then, an achievement scalarizing function ranking method is designed to layer different frontiers and eliminate redundant solutions in the environment selection stage, thus ensuring the convergence and diversity of non-dominant solutions. The performance of the proposed algorithm is validated and compared with some state-of-the-art algorithms on a number of test problems with 3, 5, 8, 10 objectives. Experimental studies demonstrate that the proposed algorithm shows very competitive performance.

Optimal design of a parallel robotic gripper using enhanced multi-objective ant lion optimizer with a sensitivity analysis approach

PurposeFrom the past few decades, parallel grippers are used successfully in the automation industries for performing various pick and place jobs due to their simple design, reliable nature and its economic feasibility. So, the purpose of this paperis to design a suitable gripper with appropriate design parameters for better performance in the robotic production systems.Design/methodology/approachIn this paper, an enhanced multi-objective ant lion algorithm is introduced to find the optimal geometric and design variables of a parallel gripper. The considered robotic gripper systems are evaluated by considering three objective functions while satisfying eight constraint equations. The beta distribution function is introduced for generating the initial random number at the initialization phase of the proposed algorithm as a replacement of uniform distribution function. A local search algorithm, namely, achievement scalarizing function with multi-criteria decision-making technique and beta distribution are used to enhance the existing optimizer to evaluate the optimal gripper design problem. In this study, the newly proposed enhanced optimizer to obtain the optimum design condition of the design variables is called enhanced multi-objective ant lion optimizer.FindingsThis study aims to obtain optimal design parameters of the parallel gripper with the help of the developed algorithms. The acquired results are investigated with the past research paper conducted in that field for comparison. It is observed that the suggested method to get the best gripper arrangement and variables of the parallel gripper mechanism outperform its counterparts. The effects of the design variables are needed to be studied for a better design approach concerning the objective functions, which is achieved by sensitivity analysis.Practical implicationsThe developed gripper is feasible to use in the assembly operation, as well as in other pick and place operations in different industries.Originality/valueIn this study, the problem to find the optimum design parameter (i.e. geometric parameters such as length of the link and parallel gripper joint angles) is addressed as a multi-objective optimization. The obtained results from the execution of the algorithm are evaluated using the performance indicator algorithm and a sensitivity analysis is introduced to validate the effects of the design variables. The obtained optimal parameters are used to develop a gripper prototype, which will be used for the assembly process.

Multiobjective Mixed Integer Nonlinear Model to Plan the Schedule for the Final Disposal of the Spent Nuclear Fuel in Finland

The safe disposal of the spent nuclear fuel is the important part of the nuclear power production. In this paper, we model the geological disposal in Finland covering objectives related to the interim storage, the encapsulation facility, the disposal facility, and the costs. A notable fact is that all the fuel types used in Finland are taken into account. The resulting optimization model is of a multiobjective nonlinear mixed integer type having eight objectives. The model is solved with the interactive method utilizing the special type of the achievement scalarizing functions. From this, we obtain a disposal schedule giving amounts of canisters to encapsulate in each time period. The results obtained are analyzed from the practical point of view.

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.

Investigating the equivalence between PBI and AASF scalarization for multi-objective optimization

Abstract Scalarization refers to a generic class of methods to combine multiple conflicting objectives into one in order to find a Pareto optimal solution to the original problem. Augmented achievement scalarizing function (AASF) is one such method used popularly in the multi-criterion decision-making (MCDM) field. In evolutionary multi-objective optimization (EMO) literature, scalarization methods such as penalty boundary intersection (PBI) are commonly used to compare similar solutions within a population. Both AASF and PBI methods require a reference point and a reference direction for their calculation. In this paper, we aim to analytically derive and understand the commonalities between these two metrics and gain insights into the limitations of their standard parametric forms. We show that it is possible to find an equivalent modified AASF formulation for a given PBI parameter and vice versa for bi-objective problems. Numerical experiments are presented to validate the theory developed. We further discuss the challenges in extending this to higher objectives and show that it is still possible to achieve limited equivalence along symmetric reference vectors. The study connects the two philosophies of solving multi-objective optimization problems, provides a means to gain a deeper understanding of both these measures, and expands their parametric range to provide more flexibility of controlling the search behavior of the EMO algorithms.

Adaptive Global WASF-GA to handle many-objective optimization problems

Abstract In this paper, a new version of the aggregation-based evolutionary algorithm Global WASF-GA (GWASF-GA) for many-objective optimization is proposed, called Adaptive Global WASF-GA (A-GWASF-GA). The fitness function of GWASF-GA is defined by an achievement scalarizing function (ASF) based on the Tchebychev distance, which considers two reference points (the nadir and utopian points) and a set of weight vectors. Despite of the benefits of using these two reference points simultaneously and a well-distributed set of weight vectors, it is necessary to go a step further to get better approximations in problems with complicated Pareto optimal fronts. For this, in A-GWASF-GA, some of the weight vectors are re-calculated during the optimization process based on the sparsity of the solutions found so far, and taking into account some theoretical results demonstrated in this paper regarding the ASF considered. Different strategies are carried out to accelerate the convergence and to maintain the diversity. The computational results, carried out in comparison with RVEA, NSGA-III, and different versions of MOEA/D, show the potential of A-GWASF-GA in well-known but also in novel many-objective optimization benchmark problems.

Planning the Schedule for the Disposal of the Spent Nuclear Fuel with Interactive Multiobjective Optimization

Several countries utilize nuclear power and face the problem of what to do with the spent nuclear fuel. One possibility, which is under the scope in this paper, is to dispose of the fuel assemblies in the disposal facility. Before the assemblies can be disposed of, they must cool down their decay heat power in the interim storage. Next, they are loaded into canisters in the encapsulation facility, and finally, the canisters are placed in the disposal facility. In this paper, we model this process as a nonsmooth multiobjective mixed-integer nonlinear optimization problem with the minimization of nine objectives: the maximum number of assemblies in the storage, maximum storage time, average storage time, total number of canisters, end time of the encapsulation, operation time of the encapsulation facility, the lengths of disposal and central tunnels, and total costs. As a result, we obtain the disposal schedule i.e., amount of canisters disposed of periodically. We introduce the interactive multiobjective optimization method using the two-slope parameterized achievement scalarizing functions which enables us to obtain systematically several different Pareto optimal solutions from the same preference information. Finally, a case study adapting the disposal in Finland is given. The results obtained are analyzed in terms of the objective values and disposal schedules.

Preference-based cone contraction algorithms for interactive evolutionary multiple objective optimization

We introduce a family of interactive evolutionary algorithms for Multiple Objective Optimization (MOO). In the phase of preference elicitation, a Decision Maker (DM) is asked to compare some pairs of solutions from the current population. Such holistic information is represented by a set of compatible instances of achievement scalarizing or quasi-convex preference functions, which contribute to the construction of preference cones in the objective space. These cones are systematically contracted during the evolutionary search, because an incremental specification of the DM's pairwise comparisons is progressively reducing the DM's most preferred region in the objective space. An inclusion of evolved solutions in this region is used along with the dominance relation to revise an elitism principle of the employed optimization algorithm. In this way, the evolutionary search is focused on a subset of Pareto optimal solutions that are particularly interesting to the DM. We investigate, moreover, how the convergence is influenced by the use of some pre-defined and newly proposed self-adjusting (dynamic) interaction patterns. We also propose a new way for visualizing the progress of an evolutionary search. It supports understanding the differences between effects of a selection pressure imposed by various optimization algorithms. • We propose interactive evolutionary algorithms for multiple objective optimization. • Interactive cone contraction is used to bias the evolutionary search. • The proposed methods outperform selected state-of-the-art algorithms. • We introduce a novel approach for visualizing a progress in the evolutionary search. • Novel interaction patterns are used to control the questioning of a decision maker.

Optimal trajectory planning of industrial robot for improving positional accuracy

Purpose The purpose of this paper is to improve the positional accuracy, smoothness on motion and productivity of industrial robot through the proposed optimal joint trajectory planning method. Also a new improved algorithm, i.e. non-dominated sorting genetic algorithm-II (NSGA-II) with achievement scalarizing function (ASF) has been proposed to obtain better optimal results compared to previously used optimization methods. Design/methodology/approach The end effector positional errors can be reduced by limiting the uncertainties of dynamic parameter variations like torque rate of joints. The jerk induced in robot joints due to acceleration variations are need to be minimized which otherwise induces vibrations in the manipulator that causes deviation in the encoders. But these lead to a vast increase in total travel time which affects the cost function of trajectory planning. Therefore, these three objectives need to be minimized individually so that an optimal trajectory path can be achieved with minimum positional error. Findings The simulation results have been obtained by running the proposed hybrid NSGA-II with ASF in MATLAB R2017a software. The optimal time intervals have been used to calculate jerk, acceleration and torque values for consecutive points on the trajectory path. From the simulation and experimental results, it can be concluded that the optimization technique could be used effectively for the trajectory planning of six-axis industrial manipulator in the joint space on the basis of minimum time-jerk-torque rate criteria. Originality/value In this paper, a new approach based on hybrid multi-objective optimization technique by combining NSGA-II with ASF has been applied to find the minimal time-jerk- torque rate joint trajectory of a six-axis industrial robot for obtaining higher positional accuracy. The results obtained from the execution of algorithm have been validated through experimentation using Kawasaki RS06L industrial robot for a particular defined path.

An Adaptative Reference Vector Based Evolutionary Algorithm for Many-Objective Optimization

Recent studies have shown difficulties in balancing convergence and diversity for many-objective optimization problems with various types of Pareto fronts. This paper proposes an adaptive reference vector based evolutionary algorithm for many-objective optimization, termed as ARVEA. The ARVEA develops a reference vector adaptation method, which can adapt different types of Pareto fronts by adjusting the distribution of reference vectors. Besides, this algorithm adopts Pareto dominance as the first selection criterion, and the achievement scalarizing function (ASF) is introduced as the secondary selection criterion. The empirical results demonstrate that the proposed ARVEA has good performance for solving problems with various types of Pareto fronts, surpassing several state-of-the-art evolutionary algorithms designed for many-objective optimization.

A Multi-Objective Particle Swarm Optimization Algorithm Based on Enhanced Selection

Most multi-objective particle swarm optimization algorithms, which have demonstrated their good performance on various practical problems involving two or three objectives, face significant challenges in complex problems. For overcoming this challenges, a multi-objective particle swarm optimization algorithm based on enhanced selection(ESMOPSO) is proposed. In order to increase the ability of exploration and exploitation, enhanced selection strategy is designed to update personal optimal particles, and objective function weighting is used to update global optimal particle adaptively. In addition, R2 indicator is incorporated into the achievement scalarizing function to layer particles in archive, which promotes the archive update. Besides, Gaussian mutation strategy is designed to avoid particles falling into local optimum, and polynomial mutation is applied in archive to increase the diversity of elite solutions. The performance of the proposed algorithm is validated and compared with some state-of-the-art algorithms on a number of test problems. Experimental results demonstrate that ESMOPSO algorithm shows very competitive performance when dealing with complex MOPs.

Statistical Learning of Discrete States in Time Series.

Time series obtained from time-dependent experiments contain rich information on kinetics and dynamics of the system under investigation. This work describes an unsupervised learning framework, along with the derivation of the necessary analytical expressions, for the analysis of Gaussian-distributed time series that exhibit discrete states. After the time series has been partitioned into segments in a model-free manner using the previously developed change-point (CP) method, this protocol starts with an agglomerative hierarchical clustering algorithm to classify the detected segments into possible states. The initial state clustering is further refined using an expectation-maximization (EM) procedure, and the number of states is determined by a Bayesian information criterion (BIC). Also introduced here is an achievement scalarization function, usually seen in artificial intelligence literature, for quantitatively assessing the performance of state determination. The statistical learning framework, which is comprised of three stages, detection of signal change, clustering, and number-of-state determination, was thoroughly characterized using simulated trajectories with random intensity segments that have no underlying kinetics, and its performance was critically evaluated. The application to experimental data is also demonstrated. The results suggested that this general framework, the implementation of which is based on firm theoretical foundations and does not require the imposition of any kinetics model, is powerful in determining the number of states, the parameters contained in each state, as well as the associated statistical significance.