Approximate Pareto Optimal Solutions Research Articles

A surrogate-assisted evolution strategy for constrained multi-objective optimization

In many real-world optimization problems, several conflicting objectives must be achieved and optimized simultaneously and the solutions are often required to satisfy certain restrictions or constraints. Moreover, in some applications, the numerical values of the objectives and constraints are obtained from computationally expensive simulations. Many multi-objective optimization algorithms for continuous optimization have been proposed in the literature and some have been incorporated or used in conjunction with expert and intelligent systems. However, relatively few of these multi-objective algorithms handle constraints, and even fewer, use surrogates to approximate the objective or constraint functions when these functions are computationally expensive. This paper proposes a surrogate-assisted evolution strategy (ES) that can be used for constrained multi-objective optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. Such an algorithm can be incorporated into an intelligent system that finds approximate Pareto optimal solutions to simulation-based constrained multi-objective optimization problems in various applications including engineering design optimization, production management and manufacturing. The main idea in the proposed algorithm is to generate a large number of trial offspring in each generation and use the surrogates to predict the objective and constraint function values of these trial offspring. Then the algorithm performs an approximate non-dominated sort of the trial offspring based on the predicted objective and constraint function values, and then it selects the most promising offspring (those with the smallest predicted ranks from the non-dominated sort) to become the actual offspring for the current generation that will be evaluated using the expensive objective and constraint functions. The proposed method is implemented using cubic radial basis function (RBF) surrogate models to assist the ES. The resulting RBF-assisted ES is compared with the original ES and to NSGA-II on 20 test problems involving 2–15 decision variables, 2–5 objectives and up to 13 inequality constraints. These problems include well-known benchmark problems and application problems in manufacturing and robotics. The numerical results showed that the RBF-assisted ES generally outperformed the original ES and NSGA-II on the problems used when the computational budget is relatively limited. These results suggest that the proposed surrogate-assisted ES is promising for computationally expensive constrained multi-objective optimization.

Prediction-Based Multi-Objective Optimization for Oil Purchasing and Distribution with the NSGA-II Algorithm

Due to the uncertainty in oil markets, this paper proposes a novel approach for oil purchasing and distribution optimization by incorporating price and demand prediction, i.e., the prediction-based oil purchasing-and-distribution optimization model. In particular, the proposed method bridges the latest information technology and decision-making technique by introducing the recently proposed information technology (i.e., extreme learning machine (ELM)) into the oil purchasing-and-distribution optimization model. Two main steps are involved: market prediction and planning optimization in the proposed model. In market prediction, the ELM technique is employed to provide fast training time and accurate forecasting results for oil prices and demands. In planning optimization, two objectives of general profit maximization and inventory risk minimization are considered; and the most popular multi-objective evolutionary algorithm (MOEA), nondominated sorting genetic algorithm II (NSGA-II), is implemented to search approximate Pareto optimal solutions. For illustration and verification, the motor gasoline market in the US is focused on as the study sample, and the experimental results demonstrate the superiority of the proposed prediction-based optimization approach over its benchmark models (without market prediction and/or planning optimization), in terms of the highest profit and the lowest risk.

Biogeography-Inspired Multiobjective Optimization and MEMS Design

A new version of the biogeography-based optimization algorithm is proposed in order to consider multiple objectives. In fact, by exploiting non-dominated sorting of habitats, it is possible to approximate Pareto-optimal solutions in the objective space. The optimal shape design of an electrostatic micromotor, which is a benchmark in the Micro Electro-Mechanical Systems design, is considered as the case study.

Multiobjective Optimization Problem of Dieless Incremental Forming

Single Point Incremental Forming (SPIF) technology has been announced in the recentpast to manufacture sheet metal products by using Computer Numerical Control machines (CNC). Ithas been frequently used in different fields like the aeronautics. In incremental forming, materialsare submitted to permanent deformation by cold forming to produce a variety of three complicateddimensional shapes. The final form of the parts in sheet metal forming is highly affected by thespring-back and the pillow effect, occurring when the material is set free of the forming constraints.In this sense, the best solution is to adopt a process of multiobjective optimization in which a set ofnumerical simulations can be achieved on the basis of the box-Behnken experimental design. In thisway, the design variables are wall angle, initial thickness, tool diameter and incremental size. Tostudy the geometric characteristics, a cone-shaped part with circular base is considered. This paperaims to identify an overview of multiobjective design optimization of incremental metal formingparameters in order to minimize objective functions of pillow effect, springback and thinning ratesimultaneously. In an attempt to solve fitness functions, the method of Multiobjective GeneticAlgorithm (MOGA) is developed in this investigation. In this case, we should consider severalpoints of the appropriate process parameters which correspond to the best compromises with respectto several antagonistic objectives. As well as, a generation of the approximate Pareto optimalsolutions is presented in this study.

Optimization of Pressure Vessel Under Thermo-Elastic Condition

In this work, a pressure vessel under thermo-elastic condition is modeled which is useful for industries like refineries, power plant etc. The combined effect of internal pressure and steady-state temperature gradient is considered. A pressure vessel problem is formulated to minimize its total cost subjected to the constraints that can ensure safe design. The problem is solved using elitist Genetic Algorithm (GA). Further, simulations on stress against thickness are carried out, which reveal that the maximum shear stress gets reduced up to a certain thickness. After that, it starts increasing due to an increase in the compressive hoop/circumferential stress under thermo-elastic condition. The bi-objective optimization problem is then formulated by minimizing the total cost and the maximum shear stress simultaneously. The bi-objective problem is solved using elitist non-dominated sorting genetic algorithm. The cost effective to the safe trade-off solutions are generated. These approximate Pareto-optimal solutions are evolved in two clusters in which the solution from one of the clusters is more preferable for practical and feasible pressure vessel design. The post-optimization analysis of results suggests that the bi-objective optimization study offers more valuable insight of the problem than the single-objective optimization.

Multiobjective Optimization of Single Point Incremental Forming: Comparison between Isotropic and Combined Hardening Behavior

Single point incremental forming (SPIF) of sheet metal is a promising process to produce small batch production and prototyping. This process consists of a controlled process of displacement performed on a three-axis CNC milling machine. In former work, the most critical factors which affected single point incremental forming process were found to be formed shape, tool size, material type, material thickness and incremental step size. The present work is focused on an optimization strategy of (SPIF) process determined by a numerical study based on finite element analyses (FEA) according to a Box-Behnken Design of Experiments. Two types of hardening behaviour laws of material are used: isotropic and combined isotropic-kinematic hardening behaviour. To do so, a set of numerical simulations are carried out for an aluminum truncated cone as geometry of a benchmark model. The simulation results include some decisions about the mechanical resistance and geometrical quality of the parts such as the thickness distribution and the magnitude of springback. In this paper, the main objective is to present an overview of multiobjective design optimization of process parameters in single point incremental forming operation in order to minimize the sheet thinning rate and the springback simultaneously. In this investigation, the steps of optimization procedure include the using of Box-Behnken experimental design for sample producing, response surface model for coarse fitting and a developed Multiobjective Genetic Algorithm (MOGA) for exact solving of fitness functions. The results show that these methods are able to determine all the best possible compromise with respect several antagonistic objectives as well as generate the approximate Pareto optimal solutions. So these will make it possible to choose the appropriate process parameters according to the objectives functions to be minimized and consequently the improvement of the products formed by the process of incremental forming.

기계학습을 이용한 파레토 프런티어의 생성

Evolutionary algorithms have been applied to multi-objective optimization problems by approximation methods using computational intelligence. Those methods have been improved gradually in order to generate more exactly many approximate Pareto optimal solutions. The paper introduces a new method using support vector machine to find an approximate Pareto frontier in multi-objective optimization problems. Moreover, this paper applies an evolutionary algorithm to the proposed method in order to generate more exactly approximate Pareto frontiers. Then a decision making with two or three objective functions can be easily performed on the basis of visualized Pareto frontiers by the proposed method. Finally, a few examples will be demonstrated for the effectiveness of the proposed method.

Utilizing expected improvement and generalized data envelopment analysis in multi-objective genetic algorithms

Meta-heuristic methods such as genetic algorithms (GA) and particle swarm optimization (PSO) have been extended to multi-objective optimization problems, and have been observed to be useful for finding good approximate Pareto optimal solutions. In order to improve the convergence and the diversity in the search of solutions using meta-heuristic methods, this paper suggests a new method to make offspring by utilizing the expected improvement (EI) and generalized data envelopment analysis (GDEA). In addition, the effectiveness of the proposed method will be investigated through several numerical examples in comparison with the conventional multi-objective GA and PSO methods.

An Improved Particle Swarm Optimization for Solving Bilevel Multiobjective Programming Problem

An improved particle swarm optimization (PSO) algorithm is proposed for solving bilevel multiobjective programming problem (BLMPP). For such problems, the proposed algorithm directly simulates the decision process of bilevel programming, which is different from most traditional algorithms designed for specific versions or based on specific assumptions. The BLMPP is transformed to solve multiobjective optimization problems in the upper level and the lower level interactively by an improved PSO. And a set of approximate Pareto optimal solutions for BLMPP is obtained using the elite strategy. This interactive procedure is repeated until the accurate Pareto optimal solutions of the original problem are found. Finally, some numerical examples are given to illustrate the feasibility of the proposed algorithm.

Trade-off analysis approach for interactive nonlinear multiobjective optimization

When solving multiobjective optimization problems, there is typically a decision maker (DM) who is responsible for determining the most preferred Pareto optimal solution based on his preferences. To gain confidence that the decisions to be made are the right ones for the DM, it is important to understand the trade-offs related to different Pareto optimal solutions. We first propose a trade-off analysis approach that can be connected to various multiobjective optimization methods utilizing a certain type of scalarization to produce Pareto optimal solutions. With this approach, the DM can conveniently learn about local trade-offs between the conflicting objectives and judge whether they are acceptable. The approach is based on an idea where the DM is able to make small changes in the components of a selected Pareto optimal objective vector. The resulting vector is treated as a reference point which is then projected to the tangent hyperplane of the Pareto optimal set located at the Pareto optimal solution selected. The obtained approximate Pareto optimal solutions can be used to study trade-off information. The approach is especially useful when trade-off analysis must be carried out without increasing computation workload. We demonstrate the usage of the approach through an academic example problem.

Optimal ballast water exchange sequence design using symmetrical multitank strategy

The quality of the ballast water exchange scheme affects many aspects of a ship, including intact stability, structural strength, building and operational expenses, etc. This paper focuses on developing an optimization methodology for large-scale sequential ballast water exchange. The method of transversely symmetrical ballast water tanks being exchanged simultaneously is adopted, and the potentials of multiple tanks being exchanged at a time are explored. The mathematical model for the symmetrical ballast water exchange problem is built by minimizing the occurred maximum trims, maximum hull girder bending moments, and shearing forces as objectives, and the related safety assessment criteria as constraints. A multiobjective genetic algorithm with some operators specially designed for the problem is presented to obtain a set of approximate Pareto-optimal solutions for engineers to choose from. Finally, a real case study on the symmetrical ballast water exchange of a 176,000 DWT bulk carrier is conducted to illustrate the effectiveness of the proposed approach.

A simulated annealing algorithm to find approximate Pareto optimal solutions for the multi-objective facility layout problem

In this article, we consider the facility layout problem which combines the objective of minimization of the total material handling cost and the maximization of total closeness rating scores. Multi-objective optimization is the way to consider the two objectives at the same time. A simulated annealing (SA) algorithm is proposed to find the non-dominated solution (Pareto optimal) set approximately for the multi-objective facility layout problem we tackle. The Pareto optimal sets generated by the proposed algorithm was compared with the solutions of the previous algorithms for multi-objective facility layout problem. The results showed that the approximate Pareto optimal sets we have found include almost all the previously obtained results and many more approximate Pareto optimal solutions.

Genetic local search for multi-objective flowshop scheduling problems

This paper addresses flowshop scheduling problems with multiple performance criteria in such a way as to provide the decision maker with approximate Pareto optimal solutions. Genetic algorithms have attracted the attention of researchers in the nineties as a promising technique for solving multi-objective combinatorial optimization problems. We propose a genetic local search algorithm with features such as preservation of dispersion in the population, elitism, and use of a parallel multi-objective local search so as intensify the search in distinct regions. The concept of Pareto dominance is used to assign fitness to the solutions and in the local search procedure. The algorithm is applied to the flowshop scheduling problem for the following two pairs of objectives: (i) makespan and maximum tardiness; (ii) makespan and total tardiness. For instances involving two machines, the algorithm is compared with Branch-and-Bound algorithms proposed in the literature. For such instances and larger ones, involving up to 80 jobs and 20 machines, the performance of the algorithm is compared with two multi-objective genetic local search algorithms proposed in the literature. Computational results show that the proposed algorithm yields a reasonable approximation of the Pareto optimal set.

A partial enumeration heuristic for multi-objective flowshop scheduling problems

This paper addresses the flowshop scheduling problem with multiple performance objectives in such a way as to provide the decision maker with approximate Pareto optimal solutions. It is well known that the partial enumeration constructive heuristic NEH and its adaptations perform well for single objectives such as makespan, total tardiness and flowtime. In this paper, we develop a similar heuristic using the concept of Pareto dominance when comparing partial and complete schedules. The heuristic is tested on problems involving combinations of the above criteria. For the two-machine case, and the pairs of objectives: (i) makespan and maximum tardiness, (ii) makespan and total tardiness, the heuristic is compared with branch-and-bound algorithms proposed in the literature. For two and more than two machines, and the criteria combinations considered in this article, the heuristic performance is tested against constructive heuristics reported in the literature. By means of an illustrative example, it is shown that a genetic algorithm from the literature performs better when starting from heuristic solutions rather than random solutions.

Evolutionary model selection in unsupervised learning

Feature subset selection is important not only for the insight gained from determining relevant modeling variables but also for the improved understandability, scalability, and possibly, accuracy of the resulting models. Feature selection has traditionally been studied in supervised learning situations, with some estimate of accuracy used to evaluate candidate subsets. However, we often cannot apply supervised learning for lack of a training signal. For these cases, we propose a new feature selection approach based on clustering. A number of heuristic criteria can be used to estimate the quality of clusters built from a given feature subset. Rather than combining such criteria, we use ELSA, an evolutionary local selection algorithm that maintains a diverse population of solutions that approximate the Pareto front in a multi-dimensional objective space. Each evolved solution represents a feature subset and a number of clusters; two representative clustering algorithms, K-means and EM, are applied to form the given number of clusters based on the selected features. Experimental results on both real and synthetic data show that the method can consistently find approximate Pareto-optimal solutions through which we can identify the significant features and an appropriate number of clusters. This results in models with better and clearer semantic relevance.