Quantifying the Shape of Pareto Fronts During Multi-Objective Trade Space Exploration

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Mehmet Unal ◽  
Gordon P. Warn ◽  
Timothy W. Simpson
Keyword(s):  
Pareto Front ◽  
Space Exploration ◽  
Problem Formulation ◽  
Shape Index ◽  
Multi Objective ◽  
Conflicting Objectives ◽  
Complex Engineering ◽  
Trade Space Exploration ◽  
Pareto Fronts ◽  
Future Work

Recent advances in simulation and computation capabilities have enabled designers to model increasingly complex engineering problems, taking into account many dimensions, or objectives, in the problem formulation. Increasing the dimensionality often results in a large trade space, where decision-makers (DM) must identify and negotiate conflicting objectives to select the best designs. Trade space exploration often involves the projection of nondominated solutions, that is, the Pareto front, onto two-objective trade spaces to help identify and negotiate tradeoffs between conflicting objectives. However, as the number of objectives increases, an exhaustive exploration of all of the two-dimensional (2D) Pareto fronts can be inefficient due to a combinatorial increase in objective pairs. Recently, an index was introduced to quantify the shape of a Pareto front without having to visualize the solution set. In this paper, a formal derivation of the Pareto Shape Index is presented and used to support multi-objective trade space exploration. Two approaches for trade space exploration are presented and their advantages are discussed, specifically: (1) using the Pareto shape index for weighting objectives and (2) using the Pareto shape index to rank objective pairs for visualization. By applying the two approaches to two multi-objective problems, the efficiency of using the Pareto shape index for weighting objectives to identify solutions is demonstrated. We also show that using the index to rank objective pairs provides DM with the flexibility to form preferences throughout the process without closely investigating all objective pairs. The limitations and future work are also discussed.

Quantifying the Shape of a Pareto Front in Support of Many-Objective Trade Space Exploration

2016 ◽  
Author(s):  
Mehmet Unal ◽  
Gordon P. Warn ◽  
Timothy W. Simpson
Keyword(s):  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Space Exploration ◽  
Shape Index ◽  
Analytical Geometry ◽  
Conflicting Objectives ◽  
Diminishing Returns ◽  
Final Design ◽  
Trade Space Exploration

Complex design optimization problems typically include many conflicting objectives, and the resulting trade space is comprised of numerous design solutions. To efficiently explore a many-objective trade space, form preferences, and select a final design, one must identify and negotiate tradeoffs between multiple, often conflicting, objectives. Identifying conflicting objective pairs allows decision-makers to concentrate on these objectives when selecting preferred designs from the non-dominated solution set, i.e., the Pareto front. Techniques exist to identify and visualize tradeoffs between these conflicting objectives to support trade space exploration; however, these techniques do not quantify, or differentiate, the shape of the Pareto front, which might be useful information for a decision-maker. More specifically, designers could gain insight from the degree of diminishing returns among solutions on the Pareto front, which can be used to understand the extent of the tradeoffs in the problem. Therefore, the shape of the Pareto front could be used to prioritize exploration of conflicting objective pairs. In this paper, we introduce a novel index that quantifies the shape of the Pareto front to provide information about the degree of diminishing returns. The aim of the index is to help designers gain insight into the underlying tradeoffs in a many-objective optimization problem and support trade space exploration by prioritizing the negotiation of conflicting objectives. The proposed Pareto Shape Index is based on analytical geometry and derived from the coordinates of the Pareto solutions in the n objective trade space. The utility of the Pareto Shape Index in differentiating diminishing returns between conflicting objectives is demonstrated by application to an eight-objective benchmark optimization problem.

Multi-Objective Fish School Search

2015 ◽  
Vol 6 (1) ◽  
pp. 23-40 ◽  
Author(s):  
Carmelo J. A. Bastos-Filho ◽  
Augusto C. S. Guimarães
Keyword(s):  
Multi Objective Optimization ◽  
New Approach ◽  
Fish School ◽  
Multi Objective ◽  
External Archive ◽  
Conflicting Objectives ◽  
Adaptive Schemes ◽  
Convergence Problems ◽  
Pareto Fronts ◽  
Future Work

The authors propose in this paper a very first version of the Fish School Search (FSS) algorithm for Multi-Objective Optimization. The proposal allows the optimization of problems with two or more conflicting objectives. The authors incorporated the dominance concept within the traditional FSS operators, creating a new approach called Multi-objective Fish School Search, MOFSS. They also adapted the barycenter concept deployed in the original FSS, which was replaced by the set of existing solutions in an external archive created to store the non-dominated solutions found during the search process. From their results in the DTLZ set of benchmark functions, the authors observed that the MOFSS obtained a similar performance when compared to well-known and well-established multi-objective swarm-based optimization algorithms. They detected some convergence problems in functions with a high number of local Pareto fronts. However, adaptive schemes can be used in future work to overcome this weakness.

scholarly journals The dilemma between eliminating dominance-resistant solutions and preserving boundary solutions of extremely convex Pareto fronts

2021 ◽  
Author(s):  
Zhenkun Wang ◽  
Qingyan Li ◽  
Qite Yang ◽  
Hisao Ishibuchi
Keyword(s):  
Evolutionary Algorithms ◽  
Coping Strategies ◽  
Test Problem ◽  
Pareto Front ◽  
Optimization Problems ◽  
Feasible Region ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Pareto Fronts ◽  
Boundary Solutions

AbstractIt has been acknowledged that dominance-resistant solutions (DRSs) extensively exist in the feasible region of multi-objective optimization problems. Recent studies show that DRSs can cause serious performance degradation of many multi-objective evolutionary algorithms (MOEAs). Thereafter, various strategies (e.g., the $$\epsilon $$ ϵ -dominance and the modified objective calculation) to eliminate DRSs have been proposed. However, these strategies may in turn cause algorithm inefficiency in other aspects. We argue that these coping strategies prevent the algorithm from obtaining some boundary solutions of an extremely convex Pareto front (ECPF). That is, there is a dilemma between eliminating DRSs and preserving boundary solutions of the ECPF. To illustrate such a dilemma, we propose a new multi-objective optimization test problem with the ECPF as well as DRSs. Using this test problem, we investigate the performance of six representative MOEAs in terms of boundary solutions preservation and DRS elimination. The results reveal that it is quite challenging to distinguish between DRSs and boundary solutions of the ECPF.

The Importance of Training for Interactive Trade Space Exploration: A Study of Novice and Expert Users

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
David Wolf ◽  
Jennifer Hyland ◽  
Timothy W. Simpson ◽  
Xiaolong (Luke) Zhang
Keyword(s):  
Space Exploration ◽  
Design Decision ◽  
Support Design ◽  
Computing Power ◽  
Training Video ◽  
Engine Combustion ◽  
Interactive Data ◽  
Effectiveness And Efficiency ◽  
Trade Space Exploration ◽  
Future Work

Thanks to recent advances in computing power and speed, engineers can now generate a wealth of data on demand to support design decision-making. These advances have enabled new approaches to search multidimensional trade spaces through interactive data visualization and exploration. In this paper, we investigate the effectiveness and efficiency of interactive trade space exploration strategies by conducting human subject experiments with novice and expert users. A single objective, constrained design optimization problem involving the sizing of an engine combustion chamber is used for this study. Effectiveness is measured by comparing the best feasible design obtained by each user, and efficiency is assessed based on the percentage of feasible designs generated by each user. Results indicate that novices who watch a 5-min training video before the experiment obtain results that are not significantly different from those obtained by expert users, and both groups are statistically better than the novices without the training video in terms of effectiveness and efficiency. Frequency and ordering of the visualization and exploration tools are also compared to understand the differences in each group’s search strategy. The implications of the results are discussed along with future work.

On the Use of Problem-Specific Candidate Generators for the Hybrid Optimization of Multi-Objective Production Engineering Problems

2009 ◽  
Vol 17 (4) ◽  
pp. 527-544 ◽  
Author(s):  
K. Weinert ◽  
A. Zabel ◽  
P. Kersting ◽  
T. Michelitsch ◽  
T. Wagner
Keyword(s):  
Pareto Front ◽  
Mold Temperature ◽  
Milling Process ◽  
Production Engineering ◽  
General Applicability ◽  
Multi Objective ◽  
Conflicting Objectives ◽  
Objective Space ◽  
Engineering Problems ◽  
Five Axis

In the field of production engineering, various complex multi-objective problems are known. In this paper we focus on the design of mold temperature control systems, the reconstruction of digitized surfaces, and the optimization of NC paths for the five-axis milling process. For all these applications, efficient problem-specific algorithms exist that only consider a subset of the desirable objectives. In contrast, modern multi-objective evolutionary algorithms are able to cope with many conflicting objectives, but they require a long runtime due to their general applicability. Therefore, we propose hybrid algorithms for the three applications mentioned. In each case, the problem-specific algorithms are used to determine promising initial solutions for the multi-objective evolutionary approach, whose variation concepts are used to generate diversity in the objective space. We show that the combination of these techniques provides great benefits. Since the final solution is chosen by a decision maker based on this Pareto front approximation, appropriate visualizations of the high-dimensional solutions are presented.

scholarly journals A Survey of Multi-Objective Sequential Decision-Making

2013 ◽  
Vol 48 ◽  
pp. 67-113 ◽  
Author(s):  
D. M. Roijers ◽  
P. Vamplew ◽  
S. Whiteson ◽  
R. Dazeley
Keyword(s):  
Decision Making ◽  
Pareto Front ◽  
Optimal Solution ◽  
Multiple Objectives ◽  
Sequential Decision Making ◽  
Sequential Decision ◽  
Multi Objective ◽  
Scalarization Function ◽  
Future Work ◽  
Single Objective

Sequential decision-making problems with multiple objectives arise naturally in practice and pose unique challenges for research in decision-theoretic planning and learning, which has largely focused on single-objective settings. This article surveys algorithms designed for sequential decision-making problems with multiple objectives. Though there is a growing body of literature on this subject, little of it makes explicit under what circumstances special methods are needed to solve multi-objective problems. Therefore, we identify three distinct scenarios in which converting such a problem to a single-objective one is impossible, infeasible, or undesirable. Furthermore, we propose a taxonomy that classifies multi-objective methods according to the applicable scenario, the nature of the scalarization function (which projects multi-objective values to scalar ones), and the type of policies considered. We show how these factors determine the nature of an optimal solution, which can be a single policy, a convex hull, or a Pareto front. Using this taxonomy, we survey the literature on multi-objective methods for planning and learning. Finally, we discuss key applications of such methods and outline opportunities for future work.

scholarly journals Stochastic Multi-objective Optimisation of Exoskeleton Structures

2020 ◽  
Vol 187 (3) ◽  
pp. 822-841
Author(s):  
Anna Reggio ◽  
Rita Greco ◽  
Giuseppe Carlo Marano ◽  
Giuseppe Andrea Ferro
Keyword(s):  
Genetic Algorithm ◽  
Structural Damage ◽  
Pareto Front ◽  
Gaussian White Noise ◽  
Economic Cost ◽  
Numerical Application ◽  
Multi Objective ◽  
Input Design ◽  
Design Variables ◽  
Pareto Fronts

AbstractIn this study, a structural optimisation problem, addressed through a stochastic multi-objective approach, is formulated and solved. The problem deals with the optimal design of exoskeleton structures, conceived as vibration control systems under seismic loading. The exoskeleton structure is assumed to be coupled to an existing primary inner structure for seismic retrofit: the aim is to limit the dynamic response of the primary structure to prevent structural damage. A non-stationary filtered Gaussian white noise stochastic process is taken as the seismic input. Design variables pertain to the mechanical properties (stiffness, damping) of the exoskeleton structure. Two concurrent and competing objective functions are introduced, in order to take into account not only safety performance but also economic cost considerations. The resulting trade-off is solved searching the Pareto front by way of a controlled elitist genetic algorithm, derived from the Non-dominated Sorting Genetic Algorithm-II. Sensitivities of Pareto fronts and Pareto optimal sets to different system parameters are finally investigated by way of a numerical application.

Multi-objective Optimization of a Regenerative Rotorcraft Powerplant: Trade-off Between Overall Engine Weight and Fuel Economy

2015 ◽  
Vol 137 (12) ◽  
Author(s):  
Fakhre Ali ◽  
Konstantinos Tzanidakis ◽  
Ioannis Goulos ◽  
Vassilios Pachidis ◽  
Roberto d'Ippolito
Keyword(s):  
Fuel Economy ◽  
Pareto Front ◽  
Design Space Exploration ◽  
Design Space ◽  
Space Exploration ◽  
Design Parameters ◽  
Optimization Strategy ◽  
Multi Objective Optimization ◽  
Trade Off ◽  
Multi Objective

A computationally efficient and cost effective simulation framework has been implemented to perform design space exploration and multi-objective optimization for a conceptual regenerative rotorcraft powerplant configuration at mission level. The proposed framework is developed by coupling a comprehensive rotorcraft mission analysis code with a design space exploration and optimization package. The overall approach is deployed to design and optimize the powerplant of a reference twin-engine light rotorcraft, modeled after the Bo105 helicopter, manufactured by Airbus Helicopters. Initially, a sensitivity analysis of the regenerative engine is carried out to quantify the relationship between the engine thermodynamic cycle design parameters, engine weight, and overall mission fuel economy. Second, through the execution of a multi-objective optimization strategy, a Pareto front surface is constructed, quantifying the optimum trade-off between the fuel economy offered by a regenerative engine and its associated weight penalty. The optimum sets of cycle design parameters obtained from the structured Pareto front suggest that the employed heat effectiveness is the key design parameter affecting the engine weight and fuel efficiency. Furthermore, through quantification of the benefits suggested by the acquired Pareto front, it is shown that the fuel economy offered by the simple cycle rotorcraft engine can be substantially improved with the implementation of regeneration technology, without degrading the payload-range capability and airworthiness (one-engine-inoperative) of the rotorcraft.

scholarly journals Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 453
Author(s):  
Nicolas Dupin ◽  
Frank Nielsen ◽  
El-Ghazali Talbi
Keyword(s):  
Dynamic Programming ◽  
Pareto Front ◽  
Dynamic Programming Algorithm ◽  
Efficient Solutions ◽  
Programming Algorithm ◽  
Planar Case ◽  
Multi Objective ◽  
Isolated Points ◽  
Time Specific ◽  
Pareto Fronts

With many efficient solutions for a multi-objective optimization problem, this paper aims to cluster the Pareto Front in a given number of clusters K and to detect isolated points. K-center problems and variants are investigated with a unified formulation considering the discrete and continuous versions, partial K-center problems, and their min-sum-K-radii variants. In dimension three (or upper), this induces NP-hard complexities. In the planar case, common optimality property is proven: non-nested optimal solutions exist. This induces a common dynamic programming algorithm running in polynomial time. Specific improvements hold for some variants, such as K-center problems and min-sum K-radii on a line. When applied to N points and allowing to uncover M<N points, K-center and min-sum-K-radii variants are, respectively, solvable in O(K(M+1)NlogN) and O(K(M+1)N2) time. Such complexity of results allows an efficient straightforward implementation. Parallel implementations can also be designed for a practical speed-up. Their application inside multi-objective heuristics is discussed to archive partial Pareto fronts, with a special interest in partial clustering variants.

A Many-Objective Evolutionary Algorithm Based on Two-Phase Selection

2021 ◽  
Vol 01 ◽  
Author(s):  
Erchao Li ◽  
Li-sen Wei
Keyword(s):  
Optimization Algorithm ◽  
Pareto Front ◽  
Multi Objective Optimization ◽  
Two Phase ◽  
Phase Selection ◽  
Testing Problem ◽  
Evolutionary Selection ◽  
Multi Objective ◽  
Pareto Fronts ◽  
Better Than

Aims: The main purpose of this paper is to achieve good convergence and distribution in different Pareto fronts. Background: Research in recent decades has appeared that evolutionary multi-objective optimization can effectively solve multi-objective optimization problems with no more than 3 targets. However, when solving MaOPs, the traditional evolutionary multi-objective optimization algorithm is difficult to balance convergence and diversity effectively. In order to solve these problems, many algorithms have emerged, which can be roughly divided into three types: decomposition-based, index-based, and dominance relationship-based. In addition, many algorithms introduce the idea of clustering into the environment. However, there are some disadvantages to solving different types of MaOPs. In order to take advantage of the above algorithms, this paper proposes a many-objective optimization algorithm based on two-phase evolutionary selection. Objective: To verify the comprehensive performance of the algorithm on the testing problem of different Pareto front, 18 examples of regular PF problems and irregular PF problems are used to test the performance of the algorithm proposed in this paper. Method: This paper proposes a two-phase evolutionary selection strategy. The evolution process is divided into two phases to select individuals with good quality. In the first phase, the convergence area is constructed by indicators to accelerate the convergence of the algorithm. In the second phase, the parallel distance is used to map the individuals to the hyperplane, and the individuals are clustered according to the distance on the hyperplane, and then the smallest fitness in each category is selected. Result: For regular Pareto front testing problems, MaOEA/TPS performed better than RVEA 、PREA 、CAMOEA and One by one EA in 19,21,30,26 cases, respectively, while it was only outperformed by RVEA 、PREA 、CAMOEA and One by one EA in 8,5,1,6 cases. For irregular front testing problem, MaOEA/TPS performed better than RVEA 、PREA 、CAMOEA and One by one EA in 20,17,25,21 cases, respectively, while it was only outperformed by RVEA 、PREA 、CAMOEA and One by one EA in 6,8,1,6 cases. Conclusion: The paper proposes a many-objective evolutionary algorithm based two phase selection, termed MaOEA/TPS, for solving MaOPs with different shapes of Pareto fronts. The results show that MaOEA/TPS has quite a competitive performance compared with the several algorithms on most test problems. Other: Although the algorithm in this paper has achieved good results, the optimization problem in the real environment is more difficult, so applying the algorithm proposed in this paper to real problems will be the next research direction.

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