Cost effective simulation-based multiobjective optimization in the performance of an internal combustion engine

Solving real-life engineering problems can be difficult because they often have multiple conflicting objectives, the objective functions involved are highly nonlinear and they contain multiple local minima. Furthermore, function values are often produced via a time-consuming simulation process. These facts suggest the need for an automated optimization tool that is efficient (in terms of number of objective function evaluations) and capable of solving global and multiobjective optimization problems. In this article, the requirements on a general simulation-based optimization system are discussed and such a system is applied to optimize the performance of a two-stroke combustion engine. In the example of a simulation-based optimization problem, the dimensions and shape of the exhaust pipe of a two-stroke engine are altered, and values of three conflicting objective functions are optimized. These values are derived from power output characteristics of the engine. The optimization approach involves interactive multiobjective optimization and provides a convenient tool to balance between conflicting objectives and to find good solutions.     

Preference-based Pareto optimization in certain and noisy environments

In this article a method for including a priori preferences of decision makers into multicriteria optimization problems is presented. A set of Pareto-optimal solutions is determined via desirability functions of the objectives which reveal experts’ preferences regarding different objective regions. An application to noisy objective functions is not straightforward but very relevant for practical applications. Two approaches are introduced in order to handle the respective uncertainties by means of the proposed preference-based Pareto optimization. By applying the methods to the original and uncertain Binh problem and a noisy single cut turning cost optimization problem, these approaches prove to be very effective in focusing on different parts of the Pareto front of the ori-ginal problem in both certain and noisy environments.     

Archery Algorithm: A Novel Stochastic Optimization Algorithm for Solving Optimization Problems

Finding a suitable solution to an optimization problem designed in science is a major challenge. Therefore, these must be addressed utilizing proper approaches. Based on a random search space, optimization algorithms can find acceptable solutions to problems. Archery Algorithm (AA) is a new stochastic approach for addressing optimization problems that is discussed in this study. The fundamental idea of developing the suggested AA is to imitate the archer's shooting behavior toward the target panel. The proposed algorithm updates the location of each member of the population in each dimension of the search space by a member randomly marked by the archer. The AA is mathematically described, and its capacity to solve optimization problems is evaluated on twenty-three distinct types of objective functions. Furthermore, the proposed algorithm's performance is compared vs. eight approaches, including teaching-learning based optimization, marine predators algorithm, genetic algorithm, grey wolf optimization, particle swarm optimization, whale optimization algorithm, gravitational search algorithm, and tunicate swarm algorithm. According to the simulation findings, the AA has a good capacity to tackle optimization issues in both unimodal and multimodal scenarios, and it can give adequate quasi-optimal solutions to these problems. The analysis and comparison of competing algorithms’ performance with the proposed algorithm demonstrates the superiority and competitiveness of the AA.     

Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.     

A local convergence analysis of bilevel decomposition algorithms

Multidisciplinary design optimization (MDO) problems are engineering design problems that require the consideration of the interaction between several design disciplines. Due to the organizational aspects of MDO problems, decomposition algorithms are often the only feasible solution approach. Decomposition algorithms reformulate the MDO problem as a set of independent subproblems, one per discipline, and a coordinating master problem. A popular approach to MDO problems is bilevel decomposition algorithms. These algorithms use nonlinear optimization techniques to solve both the master problem and the subproblems. In this paper, we propose two new bilevel decomposition algorithms and analyze their properties. In particular, we show that the proposed problem formulations are mathematically equivalent to the original problem and that the proposed algorithms converge locally at a superlinear rate. Our computational experiments illustrate the numerical performance of the algorithms.     

A multiple surrogate assisted evolutionary algorithm for optimization involving iterative solvers

In many real-world optimization problems, the underlying objective and constraint function(s) are evaluated using computationally expensive iterative simulations such as the solvers for computational electro-magnetics, computational fluid dynamics, the finite element method, etc. The default practice is to run such simulations until convergence using termination criteria, such as maximum number of iterations, residual error thresholds or limits on computational time, to estimate the performance of a given design. This information is used to build computationally cheap approximations/surrogates which are subsequently used during the course of optimization in lieu of the actual simulations. However, it is possible to exploit information on pre-converged solutions if one has the control to abort simulations at various stages of convergence. This would mean access to various performance estimates in lower fidelities. Surrogate assisted optimization methods have rarely been used to deal with such classes of problem, where estimates at various levels of fidelity are available. In this article, a multiple surrogate assisted optimization approach is presented, where solutions are evaluated at various levels of fidelity during the course of the search. For any solution under consideration, the choice to evaluate it at an appropriate fidelity level is derived from neighbourhood information, i.e. rank correlations between performance at different fidelity levels and the highest fidelity level of the neighbouring solutions. Moreover, multiple types of surrogates are used to gain a competitive edge. The performance of the approach is illustrated using a simple 1D unconstrained analytical test function. Thereafter, the performance is further assessed using three 10D and three 20D test problems, and finally a practical design problem involving drag minimization of an unmanned underwater vehicle. The numerical experiments clearly demonstrate the benefits of the proposed approach for such classes of problem.     

TREATING CONSTRAINTS AS OBJECTIVES FOR SINGLE-OBJECTIVE EVOLUTIONARY OPTIMIZATION

This paper presents a new approach to handle constraints using evolutionary algorithms. The new technique treats constraints as objectives, and uses a multiobjective optimization approach to solve the re-stated single-objective optimization problem. The new approach is compared against other numerical and evolutionary optimization techniques in several engineering optimization problems with different kinds of constraints. The results obtained show that the new approach can consistently outperform the other techniques using relatively small sub-populations, and without a significant sacrifice in terms of performance.     

Packaging optimization using the dynamic vector fields method

In this paper, a novel packaging optimization method for convex objects is presented. This method solves the packaging optimization problem through dynamic simulation of object positions and rotations over time. Object positions and orientations are determined by dynamic vector fields, which accelerate objects according to optimization objectives or physical effects between objects or their environment. Using these vector fields, any number of objectives or effects can be accounted for, and this scalability allows the method to potentially be employed to solve a wide variety of engineering packaging optimization problems. The current implementation, as presented in this paper, represents the foundation of the method that future improvements will build upon and is currently limited to the analysis of convex objects. Three basic vector fields are presented to solve packing density maximization problems: the first maximizes packing density, the second prevents collisions between objects, and the third optimally orients objects relative to each other. Collisions between objects are relaxed in this method, allowing objects to pass through each other, which provides the potential for reduced initial condition dependence and has shown promising results thus far. Several test problems are presented and solved, demonstrating the method and its ability to generate optimal solutions.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by 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.     

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.     

Simulation optimization for revenue management of airlines with cancellations and overbooking

This paper develops a model-free simulation-based optimization model to solve a seat-allocation problem arising in airlines. The model is designed to accommodate a number of realistic assumptions for real-world airline systems—in particular, allowing cancellations of tickets by passengers and overbooking of planes by carriers. The simulation–optimization model developed here can be used to solve both single-leg problems and multi-leg or network problems. A model-free simulation–optimization approach only requires a discrete-event simulator of the system along with a numerical optimization method such as a gradient-ascent technique or a meta-heuristic. In this sense, it is relatively “easy” because alternative models such as dynamic programming or model-based gradient-ascent usually require more mathematically involved frameworks. Also, existing simulation-based approaches in the literature, unlike the one presented here, fail to capture the dynamics of cancellations and overbooking in their models. Empirical tests conducted with our approach demonstrate that it can produce robust solutions which provide revenue improvements over heuristics used in the industry, namely, EMSR (Expected Marginal Seat Revenue) for single-leg problems and DAVN (Displacement Adjusted Virtual Nesting) for networks.     

一种基于近似模型管理的多目标优化方法

针对实际工程优化问题,为了提高效率和减少近似模型带来的误差,提出一种基于模型管理的多目标优化方法。利用加强径向基插值函数在整个寻优区域内构造目标和约束的近似模型,结合微型多目标遗传算法寻找当前非支配解。通过模型管理方法更新近似模型,并控制由于近似模型带来的误差和更新次数,最后将误差控制在一定范围内的多个非支配解当作实际问题的解。在测试函数中验证了此方法的效率及非支配解的精度和分布的均匀性。最后成功应用于车身薄壁构件的耐撞性优化中,表明了可用于求解复杂的工程优化问题。     

An improvement of Kriging based sequential approximate optimization method via extended use of design of experiments

When Kriging is used as a meta-model for an inequality constrained function, approximate optimal solutions are sometimes infeasible in the case where they are active at the constraint boundary. This article explores the development of a Kriging-based meta-model that enhances the constraint feasibility of an approximate optimal solution. The trust region management scheme is used to ensure the convergence of the approximate optimal solution. The present study proposes a method of enhancing the constraint feasibility in which the currently infeasible design is replaced by the most feasible-usable design during the sequential approximate optimization process. An additional convergence condition is also included to reinforce the design accuracy and feasibility. Latin hypercube design and (2n+1) design are used as tools for design of experiments. The proposed approach is verified through a constrained mathematical function problem and a number of engineering optimization problems to support the proposed strategies.     

Reliability-based optimal structural design by the decoupling approach

A decoupling approach for solving optimal structural design problems involving reliability terms in the objective function, the constraint set or both is discussed and extended. The approach employs a reformulation of each problem, in which reliability terms are replaced by deterministic functions. The reformulated problems can be solved by existing semi-infinite optimization algorithms and computational reliability methods. It is shown that the reformulated problems produce solutions that are identical to those of the original problems when the limit-state functions defining the reliability problem are affine. For nonaffine limit-state functions, approximate solutions are obtained by solving series of reformulated problems. An important advantage of the approach is that the required reliability and optimization calculations are completely decoupled, thus allowing flexibility in the choice of the optimization algorithm and the reliability computation method.     

A human‐like numerical technique for design of engineering systems

Because of the necessity for considering various creative and engineering design criteria, optimal design of an engineering system results in a highly‐constrained multi‐objective optimization problem. Major numerical approaches to such optimal design are to force the problem into a single objective function by introducing unjustifiable additional parameters and solve it using a single‐objective optimization method. Due to its difference from human design in process, the resulting design often becomes completely different from that by a human designer. This paper presents a novel numerical design approach, which resembles the human design process. Similar to the human design process, the approach consists of two steps: (1) search for the solution space of the highly‐constrained multi‐objective optimization problem and (2) derivation of a final design solution from the solution space. Multi‐objective gradient‐based method with Lagrangian multipliers (MOGM‐LM) and centre‐of‐gravity method (CoGM) are further proposed as numerical methods for each step. The proposed approach was first applied to problems with test functions where the exact solutions are known, and results demonstrate that the proposed approach can find robust solutions, which cannot be found by conventional numerical design approaches. The approach was then applied to two practical design problems. Successful design in both the examples concludes that the proposed approach can be used for various design problems that involve both the creative and engineering design criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

MONSS: A multi-objective nonlinear simplex search approach

This article presents a novel methodology for dealing with continuous box-constrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple elements of the Pareto optimal set. The search is directed by a well-distributed set of weight vectors, each of which defines a scalarization problem that is solved by deforming a simplex according to the movements described by Nelder and Mead's method. Considering an MOP with n decision variables, the simplex is constructed using n+1 solutions which minimize different scalarization problems defined by n+1 neighbor weight vectors. All solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a proper representation of the Pareto optimal front. In this article, it is shown that a well-designed strategy using just mathematical programming techniques can be competitive with respect to the state-of-the-art multi-objective evolutionary algorithms against which it was compared.     

Modification of DIRECT for high-dimensional design problems

DIviding RECTangles (DIRECT), as a well-known derivative-free global optimization method, has been found to be effective and efficient for low-dimensional problems. When facing high-dimensional black-box problems, however, DIRECT's performance deteriorates. This work proposes a series of modifications to DIRECT for high-dimensional problems (dimensionality d>10). The principal idea is to increase the convergence speed by breaking its single initialization-to-convergence approach into several more intricate steps. Specifically, starting with the entire feasible area, the search domain will shrink gradually and adaptively to the region enclosing the potential optimum. Several stopping criteria have been introduced to avoid premature convergence. A diversification subroutine has also been developed to prevent the algorithm from being trapped in local minima. The proposed approach is benchmarked using nine standard high-dimensional test functions and one black-box engineering problem. All these tests show a significant efficiency improvement over the original DIRECT for high-dimensional design problems.     

A probabilistic computational framework for bridge network optimal maintenance scheduling

This paper presents a probabilistic computational framework for the Pareto optimization of the preventive maintenance applications to bridges of a highway transportation network. The bridge characteristics are represented by their uncertain reliability index profiles. The in/out of service states of the bridges are simulated taking into account their correlation structure. Multi-objective Genetic Algorithms have been chosen as numerical tool for the solution of the optimization problem. The design variables of the optimization are the preventive maintenance schedules of all the bridges of the network. The two conflicting objectives are the minimization of the total present maintenance cost and the maximization of the network performance indicator. The final result is the Pareto front of optimal solutions among which the managers should chose, depending on engineering and economical factors. A numerical example illustrates the application of the proposed approach.     

Sequential interactive evolution for finding high-quality topologies

Finding a diverse set of high-quality (HQ) topologies for a single-objective optimization problem using an evolutionary computation algorithm can be difficult without a reliable measure that adequately describes the dissimilarity between competing topologies. In this article, a new approach for enhancing diversity among HQ topologies for engineering design applications is proposed. The technique initially selects one HQ solution and then searches for alternative HQ solutions by performing an optimization of the original objective and its dissimilarity with respect to the previously found solution. The proposed multi-objective optimization approach interactively amalgamates user articulated preferences with an evolutionary search so as sequentially to produce a set of diverse HQ solutions to a single-objective problem. For enhancing diversity, a new measure is suggested and an approach to reducing its computational time is studied and implemented. To illustrate the technique, a series of studies involving different topologies represented as bitmaps is presented.