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.

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.

Introduction of a Tradeoff Index for Efficient Trade Space Exploration

2015 ◽  
Author(s):  
Mehmet Unal ◽  
Gordon Warn ◽  
Timothy W. Simpson
Keyword(s):  
Approximate Solution ◽  
Design Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Space Exploration ◽  
High Dimensional ◽  
Two Dimensional ◽  
Complex Design ◽  
Trade Space Exploration ◽  
Tradeoff Index

The development of many-objective evolutionary algorithms has facilitated solving complex design optimization problems, that is, optimization problems with four or more competing objectives. The outcome of many-objective optimization is often a rich set of solutions, including the non-dominated solutions, with varying degrees of tradeoff amongst the objectives, herein referred to as the trade space. As the number of objectives increases, exploring the trade space and identifying acceptable solutions becomes less straightforward. Visual analytic techniques that transform a high-dimensional trade space into two-dimensional (2D) presentations have been developed to overcome the cognitive challenges associated with exploring high-dimensional trade spaces. Existing visual analytic techniques either identify acceptable solutions using algorithms that do not allow preferences to be formed and applied iteratively, or they rely on exhaustive sets of 2D representations to identify tradeoffs from which acceptable solutions are selected. In this paper, an index is introduced to quantify tradeoffs between any two objectives and integrated into a visual analytic technique. The tradeoff index enables efficient trade space exploration by quickly pinpointing those objectives that have tradeoffs for further exploration, thus reducing the number of 2D representations that must be generated and interpreted while allowing preferences to be formed and applied when selecting a solution. Furthermore, the proposed index is scalable to any number of objectives. Finally, to illustrate the utility of the proposed tradeoff index, a visual analytic technique that is based on this index is applied to a Pareto approximate solution set from a design optimization problem with ten objectives.

scholarly journals Many-Objective Genetic Programming for Job-Shop Scheduling

2021 ◽  
Author(s):  
◽  
Atiya Masood
Keyword(s):  
Local Search ◽  
Pareto Front ◽  
Job Shop ◽  
Optimization Problems ◽  
Reference Points ◽  
Experimental Results ◽  
Dispatching Rules ◽  
Model Driven ◽  
Conflicting Objectives ◽  
Mean Function

<p>The Job Shop Scheduling (JSS) problem is considered to be a challenging one due to practical requirements such as multiple objectives and the complexity of production flows. JSS has received great attention because of its broad applicability in real-world situations. One of the prominent solutions approaches to handling JSS problems is to design effective dispatching rules. Dispatching rules are investigated broadly in both academic and industrial environments because they are easy to implement (by computers and shop floor operators) with a low computational cost. However, the manual development of dispatching rules is time-consuming and requires expert knowledge of the scheduling environment. The hyper-heuristic approach that uses genetic programming (GP) to solve JSS problems is known as GP-based hyper-heuristic (GP-HH). GP-HH is a very useful approach for discovering dispatching rules automatically.  Although it is technically simple to consider only a single objective optimization for JSS, it is now widely evidenced in the literature that JSS by nature presents several potentially conflicting objectives, including the maximal flowtime, mean flowtime, and mean tardiness. A few studies in the literature attempt to solve many-objective JSS with more than three objectives, but existing studies have some major limitations. First, many-objective JSS problems have been solved by multi-objective evolutionary algorithms (MOEAs). However, recent studies have suggested that the performance of conventional MOEAs is prone to the scalability challenge and degrades dramatically with many-objective optimization problems (MaOPs). Many-objective JSS using MOEAs inherit the same challenge as MaOPs. Thus, using MOEAs for many-objective JSS problems often fails to select quality dispatching rules. Second, although the reference points method is one of the most prominent and efficient methods for diversity maintenance in many-objective problems, it uses a uniform distribution of reference points which is only appropriate for a regular Pareto-front. However, JSS problems often have irregular Pareto-front and uniformly distributed reference points do not match well with the irregular Pareto-front. It results in many useless points during evolution. These useless points can significantly affect the performance of the reference points-based algorithms. They cannot help to enhance the solution diversity of evolved Pareto-front in many-objective JSS problems. Third, Pareto Local Search (PLS) is a prominent and effective local search method for handling multi-objective JSS optimization problems but the literature does not discover any existing studies which use PLS in GP-HH.  To address these limitations, this thesis's overall goal is to develop GP-HH approaches to evolving effective rules to handle many conflicting objectives simultaneously in JSS problems.  To achieve the first goal, this thesis proposes the first many-objective GP-HH method for JSS problems to find the Pareto-fronts of nondominated dispatching rules. Decision-makers can utilize this GP-HH method for selecting appropriate rules based on their preference over multiple conflicting objectives. This study combines GP with the fitness evaluation scheme of a many-objective reference points-based approach. The experimental results show that the proposed algorithm significantly outperforms MOEAs such as NSGA-II and SPEA2.  To achieve the second goal, this thesis proposes two adaptive reference point approaches (model-free and model-driven). In both approaches, the reference points are generated according to the distribution of the evolved dispatching rules. The model-free reference point adaptation approach is inspired by Particle Swarm Optimization (PSO). The model-driven approach constructs the density model and estimates the density of solutions from each defined sub-location in a whole objective space. Furthermore, the model-driven approach provides smoothness to the model by applying a Gaussian Process model and calculating the area under the mean function. The mean function area helps to find the required number of the reference points in each mean function. The experimental results demonstrate that both adaptive approaches are significantly better than several state-of-the-art MOEAs.  To achieve the third goal, the thesis proposes the first algorithm that combines GP as a global search with PLS as a local search in many-objective JSS. The proposed algorithm introduces an effective fitness-based selection strategy for selecting initial individuals for neighborhood exploration. It defines the GP's proper neighborhood structure and a new selection mechanism for selecting the effective dispatching rules during the local search. The experimental results on the JSS benchmark problem show that the newly proposed algorithm can significantly outperform its baseline algorithm (GP-NSGA-III).</p>

Toward the Use of Pareto Performance Solutions and Pareto Robustness Solutions for Multi-Objective Robust Optimization Problems

2012 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Oscar Brito Augusto
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Robust Solutions ◽  
Multi Objective ◽  
Robust Optimization Problem ◽  
Design Solutions ◽  
Further Development

For Multi-Objective Robust Optimization Problem (MOROP), it is important to obtain design solutions that are both optimal and robust. To find these solutions, usually, the designer need to set a threshold of the variation of Performance Functions (PFs) before optimization, or add the effects of uncertainties on the original PFs to generate a new Pareto robust front. In this paper, we divide a MOROP into two Multi-Objective Optimization Problems (MOOPs). One is the original MOOP, another one is that we take the Robustness Functions (RFs), robust counterparts of the original PFs, as optimization objectives. After solving these two MOOPs separately, two sets of solutions come out, namely the Pareto Performance Solutions (PP) and the Pareto Robustness Solutions (PR). Make a further development on these two sets, we can get two types of solutions, namely the Pareto Robustness Solutions among the Pareto Performance Solutions (PR(PP)), and the Pareto Performance Solutions among the Pareto Robustness Solutions (PP(PR)). Further more, the intersection of PR(PP) and PP(PR) can represent the intersection of PR and PP well. Then the designer can choose good solutions by comparing the results of PR(PP) and PP(PR). Thanks to this method, we can find out the optimal and robust solutions without setting the threshold of the variation of PFs nor losing the initial Pareto front. Finally, an illustrative example highlights the contributions of the paper.

A Two-Phase Complete Algorithm for Multi-Objective Distributed Constraint Optimization

2014 ◽  
Vol 18 (4) ◽  
pp. 573-580
Author(s):  
Alexandre Medi ◽  
◽  
Tenda Okimoto ◽  
Katsumi Inoue ◽  
◽  
...  
Keyword(s):  
Branch And Bound ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Branch And Bound Algorithm ◽  
Constraint Optimization ◽  
Distributed Constraint Optimization ◽  
Multi Objective ◽  
Multi Agent ◽  
Distributed Constraint Optimization Problem

A Distributed Constraint Optimization Problem (DCOP) is a fundamental problem that can formalize various applications related to multi-agent cooperation. Many application problems in multi-agent systems can be formalized as DCOPs. However, many real world optimization problems involve multiple criteria that should be considered separately and optimized simultaneously. A Multi-Objective Distributed Constraint Optimization Problem (MO-DCOP) is an extension of a mono-objective DCOP. Compared to DCOPs, there exists few works on MO-DCOPs. In this paper, we develop a novel complete algorithm for solving an MO-DCOP. This algorithm utilizes a widely used method called Pareto Local Search (PLS) to generate an approximation of the Pareto front. Then, the obtained information is used to guide the search thresholds in a Branch and Bound algorithm. In the evaluations, we evaluate the runtime of our algorithm and show empirically that using a Pareto front approximation obtained by a PLS algorithm allows to significantly speed-up the search in a Branch and Bound algorithm.

An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

2021 ◽  
Vol 21 (4) ◽  
Author(s):  
Tingting Xia ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Computational Time ◽  
Pareto Solutions ◽  
Worst Case ◽  
Multi Objective ◽  
Pareto Points ◽  
Single Objective

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design. To find robust Pareto fronts, multi-objective robust optimization (MORO) methods with inner–outer optimization structures usually have high computational complexity, which is a critical issue. Generally, in design problems, robust Pareto solutions lie somewhere closer to nominal Pareto points compared with randomly initialized points. The searching process for robust solutions could be more efficient if starting from nominal Pareto points. We propose a new method sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points where MOOPs with uncertainties are solved in two stages. The deterministic optimization problem and robustness metric optimization are solved in the first stage, where nominal Pareto solutions and the robust-most solutions are identified, respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and robust-most points. The proposed SARPF method can reduce a significant amount of computational time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values, which can reduce computational efforts further. The global solvers, NSGA-II for multi-objective cases and genetic algorithm (GA) for single-objective cases, are used in corresponding optimization processes. Three examples with the comparison with results from the previous method are presented to demonstrate the applicability and efficiency of the proposed method.

Biobjective Optimization of Well Placement: Algorithm, Validation, and Field Testing

SPE Journal ◽  
2021 ◽  
pp. 1-28
Author(s):  
Faruk Alpak ◽  
Vivek Jain ◽  
Yixuan Wang ◽  
Guohua Gao
Keyword(s):  
Multiobjective Optimization ◽  
Objective Function ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Field Testing ◽  
Efficient Implementation ◽  
Field Development ◽  
Biobjective Optimization ◽  
Single Objective

Summary We describe the development and validation of a novel algorithm for field-development optimization problems and document field-testing results. Our algorithm is founded on recent developments in bound-constrained multiobjective optimization of nonsmooth functions for problems in which the structure of the objective functions either cannot be exploited or are nonexistent. Such situations typically arise when the functions are computed as the result of numerical modeling, such as reservoir-flow simulation within the context of field-development planning and reservoir management. We propose an efficient implementation of a novel parallel algorithm, namely BiMADS++, for the biobjective optimization problem. Biobjective optimization is a special case of multiobjective optimization with the property that Pareto points may be ordered, which is extensively exploited by the BiMADS++ algorithm. The optimization algorithm generates an approximation of the Pareto front by solving a series of single-objective formulations of the biobjective optimization problem. These single-objective problems are solved using a new and more efficient implementation of the mesh adaptive direct search (MADS) algorithm, developed for nonsmooth optimization problems that arise within reservoir-simulation-based optimization workflows. The MADS algorithm is extensively benchmarked against alternative single-objective optimization techniques before the BiMADS++ implementation. Both the MADS optimization engine and the master BiMADS++ algorithm are implemented from the ground up by resorting to a distributed parallel computing paradigm using message passing interface (MPI) for efficiency in industrial-scaleproblems. BiMADS++ is validated and field tested on well-location optimization (WLO) problems. We first validate and benchmark the accuracy and computational performance of the MADS implementation against a number of alternative parallel optimizers [e.g., particle-swarm optimization (PSO), genetic algorithm (GA), and simultaneous perturbation and multivariate interpolation (SPMI)] within the context of single-objective optimization. We also validate the BiMADS++ implementation using a challenging analytical problem that gives rise to a discontinuous Pareto front. We then present BiMADS++ WLO applications on two simple, intuitive, and yet realistic problems, and a model for a real problem with known Pareto front. Finally, we discuss the results of the field-testing work on three real-field deepwater models. The BiMADS++ implementation enables the user to identify various compromise solutions of the WLO problem with a single optimization run without resorting to ad hoc adjustments of penalty weights in the objective function. Elimination of this “trial-and-error” procedure and distributed parallel implementation renders BiMADS++ easy to use and significantly more efficient in terms of computational speed needed to determine alternative compromise solutions of a given WLO problem at hand. In a field-testing example, BiMADS++ delivered a workflow speedup of greater than fourfold with a single biobjective optimization run over the weighted-sumsobjective-function approach, which requires multiple single-objective-function optimization runs.

scholarly journals Improving the Performance of Visual Steering Commands for Multi-Dimensional Trade Space Exploration

2008 ◽  
Author(s):  
Christopher D. Congdon ◽  
Daniel E. Carlsen ◽  
Timothy W. Simpson ◽  
Jay D. Martin
Keyword(s):  
Optimization Problems ◽  
Space Exploration ◽  
Engineering Application ◽  
Region Of Interest ◽  
Monte Carlo Sampling ◽  
Test Problems ◽  
Design Decision ◽  
Support Design ◽  
Exploration Process ◽  
Trade Space Exploration

Designers perform many tasks when developing new products and systems, and making decisions may be among the most important of these tasks. The trade space exploration process advocated in this work provides a visual and intuitive approach for formulating and solving single- and multi-objective optimization problems to support design decision-making. In this paper, we introduce an advanced sampling method to improve the performance of the visual steering commands that have been developed to explore and navigate the trade space. This method combines speciation and crowding operations used within the Differential Evolution (DE) algorithm to generate new samples near the region of interest. The accuracy and diversity of the resulting samples are compared against simple Monte Carlo sampling as well as the current implementation of the visual steering commands using a suite of test problems and an engineering application. The proposed method substantially increases the efficiency and effectiveness of the sampling process while maintaining diversity within the trade space.

scholarly journals MultiObjective Genetic Modified Algorithm (MOGMA)

2012 ◽  
Vol 12 (2) ◽  
pp. 23-33
Author(s):  
Elica Vandeva
Keyword(s):  
Genetic Algorithms ◽  
Multiobjective Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multiobjective Optimization Problem ◽  
Multiobjective Optimization Problems ◽  
Modified Algorithm

Abstract Multiobjective optimization based on genetic algorithms and Pareto based approaches in solving multiobjective optimization problems is discussed in the paper. A Pareto based fitness assignment is used − non-dominated ranking and movement of a population towards the Pareto front in a multiobjective optimization problem. A MultiObjective Genetic Modified Algorithm (MOGMA) is proposed, which is an improvement of the existing algorithm.

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