A multiobjective mesh optimization framework for mesh quality improvement and mesh untangling

We propose a multiobjective mesh optimization framework for mesh quality improvement and mesh untangling. Our framework combines two or more competing objective functions into a single objective function to be solved using one of various multiobjective optimization methods. Methods within our framework are able to optimize various aspects of the mesh such as the element shape, element size, associated PDE interpolation error, and number of inverted elements, but the improvement is not limited to these categories. The strength of our multiobjective mesh optimization framework lies in its ability to be extended to simultaneously optimize any aspects of the mesh and to optimize meshes with different element types. We propose the exponential sum, objective product, and equal sum multiobjective mesh optimization methods within our framework; these methods do not require articulation of preferences. However, the solutions obtained satisfy a sufficient condition of weak Pareto optimality. Experimental results show that our multiobjective mesh optimization methods are able to simultaneously optimize two or more aspects of the mesh and also are able to improve mesh qualities while eliminating inverted elements. We successfully apply our methods to real‐world applications such as hydrocephalus treatment and shape optimization. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiobjective optimization with an adaptive weight determination scheme using the concept of hyperplane

Finding an optimum design that satisfies all performances in a design problem is very challenging. To overcome this problem, multiobjective optimization methods have been researched to obtain Pareto optimum solutions. Among the different methods, the weighted sum method is widely used for its convenience. However, since the different weights do not always guarantee evenly distributed solutions on the Pareto front, the weights need to be determined systematically. Therefore, this paper presents a multiobjective optimization using a new adaptive weight determination scheme. Solutions on the Pareto front are gradually found with different weights, and the values of these weights are adaptively determined by using information from the previously obtained solutions' positions. For an n-objective problem, a hyperplane is constructed in n -dimensional space, and new weights are calculated to find the next solutions. To confirm the effectiveness of the proposed method, benchmarking problems that have different types of Pareto front are tested, and a topology optimization problem is performed as an engineering problem. A hypervolume indicator is used to quantitatively evaluate the proposed method, and it is confirmed that optimized solutions that are evenly distributed on the Pareto front can be obtained by using the proposed method.     

Computational fluid dynamics and interactive multiobjective optimization in the development of low-emission industrial boilers

A CFD-based model is applied to study emission formation in a bubbling fluidized bed boiler burning biomass. After the model is validated to a certain extent, it is used for optimization. There are nine design variables (nine distinct NH₃ injections in the selective non-catalytic reduction process) and two objective functions (which minimize NO and NH₃ emissions in flue gas). The multiobjective optimization problem is solved using the reference-point method involving an achievement scalarizing function. The interactive reference-point method is applied to generate Pareto optimal solutions. Two inherently different optimization algorithms, viz. a genetic algorithm and Powell's conjugate-direction method, are applied in the solution of the resulting optimization problem. It is shown that optimization connected with CFD is a promising design tool for combustion optimization. The strengths and weaknesses of the proposed approach and of the methods applied are discussed from the point of view of a complex real-world optimization problem.     

A method for simulation based optimization using radial basis functions

We propose an algorithm for the global optimization of expensive and noisy black box functions using a surrogate model based on radial basis functions (RBFs). A method for RBF-based approximation is introduced in order to handle noise. New points are selected to minimize the total model uncertainty weighted against the surrogate function value. The algorithm is extended to multiple objective functions by instead weighting against the distance to the surrogate Pareto front; it therefore constitutes the first algorithm for expensive, noisy and multiobjective problems in the literature. Numerical results on analytical test functions show promise in comparison to other (commercial) algorithms, as well as results from a simulation based optimization problem.     

Handling constraints using multiobjective optimization concepts

In this paper, we propose a new constraint‐handling technique for evolutionary algorithms which we call inverted‐shrinkable PAES (IS‐PAES). This approach combines the use of multiobjective optimization concepts with a mechanism that focuses the search effort onto specific areas of the feasible region by shrinking the constrained search space. IS‐PAES also uses an adaptive grid to store the solutions found, but has a more efficient memory‐management scheme than its ancestor (the Pareto archived evolution strategy for multiobjective optimization). The proposed approach is validated using several examples taken from the standard evolutionary and engineering optimization literature. Comparisons are provided with respect to the stochastic ranking method (one of the most competitive constraint‐handling approaches used with evolutionary algorithms currently available) and with respect to other four multiobjective‐based constraint‐handling techniques. Copyright© 2004 John Wiley & Sons, Ltd.     

Aggregate Objective Functions and Pareto Frontiers: Required Relationships and Practical Implications

This paper addresses the problem of capturing Pareto optimal points on non-convex Pareto frontiers, which are encountered in nonlinear multiobjective optimization problems in computational engineering design optimization. The emphasis is on the choice of the aggregate objective function (AOF) of the objectives that is employed to capture Pareto optimal points. A fundamental property of the aggregate objective function, the admissibility property, is developed and its equivalence to the coordinatewise increasing property is established. Necessary and sufficient conditions for such an admissible aggregate objective function to capture Pareto optimal points are derived. Numerical examples illustrate these conditions in the biobjective case. This paper demonstrates in general terms the limitation of the popular weighted-sum AOF approach, which captures only convex Pareto frontiers, and helps us understand why some commonly used AOFs cannot capture desirable Pareto optimal points, and how to avoid this situation in practice. Since nearly all applications of optimization in engineering design involve the formation of AOFs, this paper is of direct theoretical and practical usefulness.     

Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

Function-transformation methods for multi-objective optimization

It is useful with multi-objective optimization (MOO) to transform the objective functions such that they all have similar units and orders of magnitude. This article evaluates various transformation methods using simple example problems. Viewing these methods as different means to restrict function values sheds light on how the methods perform. The weighted sum approach for MOO is used to study how well different methods aid in depicting the Pareto optimal set. Whereas using unrestricted weights is well suited for providing a single solution that reflects preferences, it is found that using a convex combination of functions is desirable when generating the Pareto set. In addition, it is shown that some transformation methods are detrimental to the process of generating a diverse spread of points, and criteria are proposed for determining when the methods fail to generate an accurate representation of the Pareto set. Advantages of using a simple normalization–modification are demonstrated.     

Shape design of internal cooling passages within a turbine blade

The article concerns the optimization of the shape and location of non-circular passages cooling the blade of a gas turbine. To model the shape, four Bezier curves which form a closed profile of the passage were used. In order to match the shape of the passage to the blade profile, a technique was put forward to copy and scale the profile fragments into the component, and build the outline of the passage on the basis of them. For so-defined cooling passages, optimization calculations were carried out with a view to finding their optimal shape and location in terms of the assumed objectives. The task was solved as a multi-objective problem with the use of the Pareto method, for a cooling system composed of four and five passages. The tool employed for the optimization was the evolutionary algorithm. The article presents the impact of the population on the task convergence, and discusses the impact of different optimization objectives on the Pareto optimal solutions obtained. Due to the problem of different impacts of individual objectives on the position of the solution front which was noticed during the calculations, a two-step optimization procedure was introduced. Also, comparative optimization calculations for the scalar objective function were carried out and set up against the non-dominated solutions obtained in the Pareto approach. The optimization process resulted in a configuration of the cooling system that allows a significant reduction in the temperature of the blade and its thermal stress.     

Norm methods and partial weighting in multicriterion optimization of structures

Methods for generating Pareto optimal solutions to a multicriterion optimization problem are considered. The norm methods based on the scalarization of the original multicriterion problem by using the l-norm are discussed in a unified form and a parametrization suitable for different interactive design systems is suggested. In addition, an alternative approach which, instead of scalarization, reduces the dimension of the multicriterion problem is proposed. This is called the partial weighting method and it can beinterpreted as a generalization of the traditional scalarization technique where the weighted sum of the criteria is used as the objective function. The first of these two approaches (norm method) is very flexible from a designer's point of view and it can be applied also in non-convex cases to the determination of the Pareto optimal set whereas the latter (partial weighting method) is especially suitable for problems where the number of criteria is large. Throughout the article several illustrative truss examples are presented to augment the scanty collection of multicriterion problems treated in the literature of optimum structural design.     

Optimizing the Hot Isostatic Pressing Process

A genetic algorithm (GA) is used to optimize the hot isostatic pressing (HIPing) process for beryllium powder. The GA evaluates a HIPing model with different processing schedules in an effort to minimize temperature, pressure, processing time, ramp rates, grain growth, and distance to target relative density. It is shown that this is a constrained, multiobjective, noisy, optimization problem to which the GA is able to evolve a large number of viable solutions. However, for the GA to work in such a large multidimensional search space, it is suggested that the constraints be treated as objectives and then penalize the Pareto ranking for each constraint violated. This approach is necessary because a large-dimensional objective space naturally results in most members being Pareto rank 1.     

Pareto Frontier Based Concept Selection Under Uncertainty,with Visualization

In a recent publication, we presented a new multiobjective decision-making tool for use in conceptual engineering design. In the present paper, we provide important developments that support the next phase in the evolution of the tool. These developments, together with those of our previous work, provide a concept selection approach that capitalizes on the benefits of computational optimization. Specifically, the new approach uses the efficiency and effectiveness of optimization to rapidly compare numerous designs, and characterize the tradeoff properties within the multiobjective design space. As such, the new approach differs significantly from traditional (non-optimization based) concept selection approaches where, comparatively speaking, significant time is often spent evaluating only a few points in the design space. Under the new approach, design concepts are evaluated using a so-calleds-Pareto frontier; this frontier originates from the Pareto frontiers of various concepts, and is the Pareto frontier for thesetof design concepts. An important characteristic of the s-Pareto frontier is that it provides a foundation for analyzing tradeoffs between design objectives and the tradeoffs between design concepts. The new developments presented in this paper include; (i) the notion ofminimally representingthe s-Pareto frontier, (ii) the quantification of concept goodness using s-Pareto frontiers, (iii) the development of an interactive design space exploration approach that can be used to visualizen-dimensional s-Pareto frontiers, and (iv) s-Pareto frontier-based approaches for considering uncertainty in concept selection. Simple structural examples are presented that illustrate representative applications of the proposed method.     

Hierarchical multiobjective routing model in Multiprotocol Label Switching networks with two service classes – a Pareto archive strategy

The article begins by reviewing a two-level hierarchical multicriteria routing model for Multiprotocol Label Switching networks with two service classes (QoS, i.e. with Quality of Service requirements, and Best Effort services) and alternative routing, as well as the foundations of a heuristic resolution approach, previously proposed by the authors. Afterwards a new variant of this heuristic approach, which includes a Pareto archive strategy, is described. In this archive, non-dominated solutions obtained throughout the heuristic are kept. At the end of the main procedure of the heuristic, these solutions are evaluated and a final solution for the routing problem is chosen using a reference point-based approach. The application of this procedure to two test networks will show, with analytic and discrete-event simulation models, that, in certain initial conditions, this approach provides improvements in the final results concerning the top-level objective functions, especially in more ‘difficult’ situations detected through sensitivity analysis.     

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.     

Pareto navigator for interactive nonlinear multiobjective optimization

We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one’s preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.     

Evolutionary optimization for passive gamma tomography reconstruction of circular inclusions in nuclear waste vaults

In this study we investigate numerically the possibility of reconstructing the interior details of a nuclear waste vault using photon flux measurements. The passive gamma ray tomography method is employed in order to simultaneously retrieve the intensity and the location of strongly radioactive components. After reformulating the problem as an optimization problem the objective functional obtained is found to have a multi‐modal non‐smooth structure. Therefore, evolutionary algorithms are used in order to minimize the objective functional in order to determine the radioactive components. Circular inclusions are considered and detailed numerical results are presented for both exact and noisy data. Copyright © 2007 John Wiley & Sons, Ltd.     

A network-wide exact optimization approach for multiobjective routing with path protection in multiservice multiprotocol label switching networks

A multiobjective routing model for multiprotocol label switching networks with multiple service types and path protection is presented in this article. The routing problem is formulated as a biobjective integer program, where the considered objectives are formulated according to a network-wide optimization approach, i.e. the objective functions of the route optimization problem depend explicitly on all traffic flows in the network. A disjoint path pair is considered for each traffic trunk, which guarantees protection to the associated connection. A link-path formulation is proposed for the problem, in which a set of possible pairs of paths is devised in advance for each traffic trunk. An exact method (based on the classical constraint method for solving multiobjective problems) is developed for solving the formulated problem. An extensive experimental study, with results on network performance measures in various randomly generated networks, is also presented and discussed.     

A new multi-objective particle swarm optimizer using empirical movement and diversified search strategies

Most real-world optimization problems involve the optimization task of more than a single objective function and, therefore, require a great amount of computational effort as the solution procedure is designed to anchor multiple compromised optimal solutions. Abundant multi-objective evolutionary algorithms (MOEAs) for multi-objective optimization have appeared in the literature over the past two decades. In this article, a new proposal by means of particle swarm optimization is addressed for solving multi-objective optimization problems. The proposed algorithm is constructed based on the concept of Pareto dominance, taking both the diversified search and empirical movement strategies into account. The proposed particle swarm MOEA with these two strategies is thus dubbed the empirical-movement diversified-search multi-objective particle swarm optimizer (EMDS-MOPSO). Its performance is assessed in terms of a suite of standard benchmark functions taken from the literature and compared to other four state-of-the-art MOEAs. The computational results demonstrate that the proposed algorithm shows great promise in solving multi-objective optimization problems.