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A probabilistic non-dominated sorting genetic algorithm (P-NSGA) for multi-objective optimization under uncertainty is presented. The purpose of this algorithm is to create a tight coupling between the optimization and uncertainty procedures, use all of the possible probabilistic information to drive the optimizer, and leverage high-performance parallel computing.

This algorithm is a generalization of a classical genetic algorithm for multi-objective optimization (NSGA-II) by Deb et al. The proposed algorithm relies on the use of all possible information in the probabilistic domain summarized by the cumulative distribution functions (CDFs) of the objective functions. Several analytic test functions are used to benchmark this algorithm, but only the results of the Fonseca-Fleming test function are shown. An industrial application is presented to show that P-NSGA can be used for multi-objective shape optimization of a Formula 1 tire brake duct, taking into account the geometrical uncertainties associated with the rotating rubber tire and uncertain inflow conditions.

This algorithm is shown to have deterministic consistency (i.e. it turns back to the original NSGA-II) when the objective functions are deterministic. When the quality of the CDF is increased (either using more points or higher fidelity resolution), the convergence behavior improves. Since all the information regarding uncertainty quantification is preserved, all the different types of Pareto fronts that exist in the probabilistic framework (e.g. mean value Pareto, mean value penalty Pareto, etc.) are shown to be generated a posteriori. An adaptive sampling approach and parallel computing (in both the uncertainty and optimization algorithms) are shown to have several fold speed-up in selecting optimal solutions under uncertainty.

There are no existing algorithms that use the full probabilistic distribution to guide the optimizer. The method presented herein bases its sorting on real function evaluations, not merely measures (i.e. mean of the probabilistic distribution) that potentially do not exist.

This paper presents a methodology based on Multiobjective Genetic Algorithms (MOGAs) for the design of electrical engineering systems. MOGAs allow one to optimize multiple heterogeneous criteria in complex systems, but also simplify couplings and sensitivity analysis by determining the evolution of design variables along the Pareto‐optimal front.

To illustrate the use of MOGAs in electrical engineering, the optimal design of an electromechanical system has been investigated. A rather simplified case study dealing with the optimal dimensioning of an inverter – permanent magnet motor – reducer – load association is carried out to demonstrate the interest of the approach. The purpose is to simultaneously minimize two objectives: the global losses and the mass of the system. The system model is described by analytical model and we use the MOGA called NSGA‐II.

From the extraction of Pareto‐optimal solutions, MOGAs facilitate the investigation of parametric sensitivity and the analysis of couplings in the system. Through a simple but typical academic problem dealing with the optimal dimensioning of a inverter – permanent magnet motor – reducer – load association, it has been shown that this multiobjective a posteriori approach could offer interesting outlooks in the global optimization and design of complex heterogeneous systems. The final choice between all Pareto‐optimal configurations can be a posteriori done in relation to other issues which have not been considered in the optimization process. In this paper, we illustrate this point by considering the cogging torque for the final decision.

We have proposed an original quantitative methodology based on correlation coefficients to characterize the system interactions.

In an attempt to solve multiobjective optimization problems, many traditional methods scalarize an objective vector into a single objective by a weight vector. In these cases, the obtained solution is highly sensitive to the weight vector used in the scalarization process and demands a user to have knowledge about the underlying problem. Moreover, in solving multiobjective problems, designers may be interested in a set of Pareto‐optimal points, instead of a single point. In this paper, Pareto‐based Continuous Evolutionary Algorithms for Multiobjective Optimization problems having continuous search space are introduced. These algorithms are based on Continuous Evolutionary Algorithms, which were developed by the authors to solve single‐objective optimization problems with a continuous function and continuous search space efficiently. For multiobjective optimization, a progressive reproduction operator and a niche‐formation method for fitness sharing and a storing process for elitism are implemented in the algorithm. The operator and the niche formulation allow the solution set to be distributed widely over the Pareto‐optimal tradeoff surface. Finally, the validity of this method has been demonstrated through some numerical examples.

The purpose of this work is to apply and compare surrogate-assisted and multi-fidelity, multi-objective optimization (MOO) algorithms to simulation-based aerodynamic design exploration.

The three algorithms for multi-objective aerodynamic optimization compared in this work are the combination of evolutionary algorithms, design space reduction and surrogate models, the multi-fidelity point-by-point Pareto set identification and the multi-fidelity sequential domain patching (SDP) Pareto set identification. The algorithms are applied to three cases, namely, an analytical test case, the design of transonic airfoil shapes and the design of subsonic wing shapes, and are evaluated based on the resulting best possible trade-offs and the computational overhead.

The results show that all three algorithms yield comparable best possible trade-offs for all the test cases. For the aerodynamic test cases, the multi-fidelity Pareto set identification algorithms outperform the surrogate-assisted evolutionary algorithm by up to 50 per cent in terms of cost. Furthermore, the point-by-point algorithm is around 27 per cent more efficient than the SDP algorithm.

The novelty of this work includes the first applications of the SDP algorithm to multi-fidelity aerodynamic design exploration, the first comparison of these multi-fidelity MOO algorithms and new results of a complex simulation-based multi-objective aerodynamic design of subsonic wing shapes involving two conflicting criteria, several nonlinear constraints and over ten design variables.

The purpose of this paper is to develop an optimal approach for optimizing the dynamic behavior of incremental linear actuators.

First, a parameterized design model is built. Second, a dynamic model is implemented. This model takes into account the thrust force computed from a finite element model. Finally, the multiobjective optimization approach is applied to the dynamic model to optimize control as well as design parameters.

The Pareto front resulting from the optimization approach (or the parallel optimization approach,) is better than the Pareto, which is obtained from the only application of MultiObjective Genetic Algorithm (MOGA) method (or parallel MOGA with the same number of optimization approach objective function evaluations). The only use of MOGA can reach the region near an optimal Pareto front, but it consumes more computing time than the multiobjective optimization approach. At each flowchart stage, parallelization leads to a significant reduction of computing time which is halved when using two-core machine.

In order to solve the multiobjective problem, a hybrid algorithm based on MOGA is developed.

In recent decades, development of effective methods for optimizing a set of conflicted objective functions has been absorbing an increasing interest from researchers. This refers to the essence of real-life engineering systems and complex natural mechanisms which are generally multi-modal, non-convex and multi-criterion. Until now, several deterministic and stochastic methods have been proposed to cope with such complex systems. Advanced soft computational methods such as evolutionary games (cooperative and non-cooperative), Pareto-based techniques, fuzzy evolutionary methods, cooperative bio-inspired algorithms and neuro-evolutionary systems have effectively come to the aid of researchers to build up efficient paradigms with application to vector optimization. The paper aims to discuss this issue.

A novel hybrid algorithm called synchronous self-learning Pareto strategy (SSLPS) is presented for the sake of vector optimization. The method is the ensemble of evolutionary algorithms (EA), swarm intelligence (SI), adaptive version of self-organizing map (CSOM) and a data shuffling mechanism. EA are powerful numerical optimization algorithms capable of finding a global extreme point over a wide exploration domain. SI techniques (the swarm of bees in our case) can improve both intensification and robustness of exploration. CSOM network is an unsupervised learning methodology which learns the characteristics of non-dominated solutions and, thus, enhances the quality of the Pareto front.

To prove the effectiveness of the proposed method, the authors engage a set of well-known benchmark functions and some well-known rival optimization methods. Additionally, SSLPS is employed for optimal design of shape memory alloy actuator as a nonlinear multi-modal real-world engineering problem. The experiments show the acceptable potential of SSLPS for handling both numerical and engineering multi-objective problems.

To the author’s best knowledge, the proposed algorithm is among the rare multi-objective methods which fosters the use of automated unsupervised learning for increasing the intensity of Pareto front (while preserving the diversity). Also, the research evaluates the power of hybridization of SI and EA for efficient search.

The purpose of the paper is to show that the a posteriori analysis of the Pareto front associated with a given design problem facilitates the task of the decision maker and possibly helps to identify innovative solutions. The idea is to investigate the similarities existing among non‐dominated solutions.

A permanent‐magnet alternator for automotive applications is considered as case study. The design problem exhibits six design variables and two energy‐related objective functions. A suitable sampling of the objective space is made and non‐dominated solutions, located along an L‐shaped front, are approximated. Results are assessed by means of a successive optimization using NSGA‐II algorithm.

From the approximated Pareto front, three optimal devices have been selected and remapped in the design space in order to compare their performance. This is done in terms of iron and copper losses, material costs, rated voltage, and air‐gap induction. Moreover, making the NSGA‐II start from the knee‐point of the front, it is shown that a direct approximation of the two sub‐fronts is possible.

In this paper, a method to sort out the optimal solutions located along the Pareto front is proposed as a possible criterion of decision making; so doing, previously unpredicted solutions might be identified.

The goal of this study is to carry out multi‐objective optimization by considering minimization of surface roughness (Ra) and build time (T) in selective laser sintering (SLS) process, which are functions of “build orientation”. Evolutionary algorithms are applied for this purpose. The performance comparison of the optimizers is done based on statistical measures. In order to find truly optimal solutions, local search is proposed. An important task of decision making, i.e. the selection of one solution in the presence of multiple trade‐off solutions, is also addressed. Analysis of optimal solutions is done to gain insight into the problem behavior.

The minimization of Ra and T is done using two popular optimizers – multi‐objective genetic algorithm (non‐dominated sorting genetic algorithm (NSGA‐II)) and multi‐objective particle swarm optimizers (MOPSO). Standard measures from evolutionary computation – “hypervolume measure” and “attainment surface approximator” have been borrowed to compare the optimizers. Decision‐making schemes are proposed in this paper based on decision theory.

The objects are categorized into groups, which bear similarity in optimal solutions. NSGA‐II outperforms MOPSO. The similarity of spread and convergence patterns of NSGA‐II and MOPSO ensures that obtained solutions are (or are close to) Pareto‐optimal set. This is validated by local search. Based on the analysis of obtained solutions, general trends for optimal orientations (depending on the geometrical features) are found.

A novel and systematic way to address multi‐objective optimization decision‐making post‐optimal analysis is shown. Simulations utilize experimentally derived models for roughness and build time. A further step could be the experimental verification of findings provided in this study.

This study provides a thorough methodology to find optimal build orientations in SLS process. A route to decipher valuable problem information through post‐optimal analysis is shown. The principles adopted in this study are general and can be extended to other rapid prototyping (RP) processes and expected to find wide applicability.

This paper is a distinct departure from past studies in RP and demonstrates the concepts of multi‐objective optimization, decision‐making and related issues.

The purpose of this paper is to study the use of outsourcing as a mechanism to cope with supply chain uncertainty, more specifically, how to deal with sudden arrival of higher priority jobs that require immediate processing, in an in-house manufacturer's facility from the perspective of outsourcing. An operational level schedule of production and distribution of outsourced jobs to the manufacturer's facility should be determined for the subcontractor in order to achieve overall optimality.

The problem is of bi-criteria in that both the transportation cost measured by number of delivery vehicles and schedule performance measured by jobs’ delivery times. In order to obtain the problem's Pareto front, we propose dynamic programming (DP) heuristic solution procedure based on integrated decision making, and population-heuristic solution procedures using different encoding schemes based on sequential decision making. Computational studies are designed and carried out by randomly generating comparative variations of numerical problem instances.

By comparing several existing performance metrics for the obtained Pareto fronts, it is found that DP heuristic outperforms population-heuristic in both solutions diversity and proximity to optimal Pareto front. Also in population-heuristic, sub-range keys representation appears to be a better encoding scheme for the problem than random keys representation.

This study contributes to the limited yet important knowledge body on using outsourcing approach to coping with possible supply chain disruptions in production scheduling due to sudden customer orders. More specifically, we used modeling methodology to confirm the importance of collaboration with subcontractors to effective supply chain risk management.

The purpose of this paper is to introduce an effective methodology of obtaining Perot-optimal solutions when combining system dynamics (SD) and multi-objective optimization (MOO) for supply chain problems.

This paper proposes a new approach that combines SD and MOO within a simulation-based optimization framework for generating the efficient frontier for supporting decision making in supply chain management (SCM). It also addresses the issue of the curse of dimensionality, commonly found in practical optimization problems, through design space reduction.

The integrated MOO and SD approach has been shown to be very useful for revealing how the decision variables in the Beer Game (BG) affect the optimality of the three common SCM objectives, namely, the minimization of inventory, backlog, and the bullwhip effect (BWE). The results from the in-depth BG study clearly show that these three optimization objectives are in conflict with each other, in the sense that a supply chain manager cannot minimize the BWE without increasing the total inventory and total backlog levels.

Having a methodology that enables effective generation of optimal trade-off solutions, in terms of computational cost, time as well as solution diversity and intensification, assist decision makers in not only making decision in time but also present a diverse and intense solution set to choose from.

This paper presents a novel supply chain MOO methodology to assist in finding Pareto-optimal solutions in a more effective manner. In order to do so the methodology tackles the so-called curse of dimensionality by reducing the design space and focussing the search of the optimization to regions of inters. Together with design space reduction, it is believed that the integrated SD and MOO approach can provide an innovative and efficient approach for the design and analysis of manufacturing supply chain systems in general.