An approach to multi‐start clustering for global optimization with non‐linear constraints

This paper describes a multi‐start with clustering strategy for use on constrained optimization problems. It is based on the characteristics of non‐linear constrained global optimization problems and extends a strategy previously tested on unconstrained problems. Earlier studies of multi‐start with clustering found in the literature have focused on unconstrained problems with little attention to non‐linear constrained problems. In this study, variations of multi‐start with clustering are considered including a simulated annealing or random search procedure for sampling the design domain and a quadratic programming (QP) sub‐problem used in cluster formation. The strategies are evaluated by solving 18 non‐linear mathematical problems and six engineering design problems. Numerical results show that the solution of a one‐step QP sub‐problem helps predict possible regions of attraction of local minima and can enhance robustness and effectiveness in identifying local minima without sacrificing efficiency. In comparison to other multi‐start techniques found in the literature, the strategies of this study can be attractive in terms of the number of local searches performed, the number of minima found, whether the global minimum is located, and the number of the function evaluations required. Copyright © 2002 John Wiley & Sons, Ltd.     

A Multi‐Objective Design Approach for the c Chart Considering Taguchi Loss Function

The present paper proposes a multi‐objective design approach for the c chart, considering in the optimization process of the chart parameters both the statistical and the economic objectives. In particular, the minimization of the hourly total quality related costs is the considered objective to carry out the economic goal, whereas the statistical objective is reached by the minimization the out‐of‐control average run length of the chart. A mixed integer non‐linear constrained mathematical model is formulated to solve the treated multi‐objective optimization problem, whereas the Pareto optimal frontier is described by the ε‐constraint method. In order to show the employment of the proposed approach, an illustrative example is developed and the related considerations are given. Finally, some sensitivity analysis is also performed to investigate the effects of operative and costs parameters on the chart performance. Copyright © 2013 John Wiley & Sons, Ltd.     

Novel orthogonal simulated annealing with fractional factorial analysis to solve global optimization problems

A classical simulated annealing (SA) method is a generic probabilistic and heuristic approach to solving global optimization problems. It uses a stochastic process based on probability, rather than a deterministic procedure, to seek the minima or maxima in the solution space. Although the classical SA method can find the optimal solution to most linear and nonlinear optimization problems, the algorithm always requires numerous numerical iterations to yield a good solution. The method also usually fails to achieve optimal solutions to large parameter optimization problems. This study incorporates well-known fractional factorial analysis, which involves several factorial experiments based on orthogonal tables to extract intelligently the best combination of factors, with the classical SA to enhance the numerical convergence and optimal solution. The novel combination of the classical SA and fractional factorial analysis is termed the orthogonal SA herein. This study also introduces a dynamic penalty function to handle constrained optimization problems. The performance of the proposed orthogonal SA method is evaluated by computing several representative global optimization problems such as multi-modal functions, noise-corrupted data fitting, nonlinear dynamic control, and large parameter optimization problems. The numerical results show that the proposed orthogonal SA method markedly outperforms the classical SA in solving global optimization problems with linear or nonlinear objective functions. Additionally, this study addressed two widely used nonlinear functions, proposed by Keane and Himmelblau to examine the effectiveness of the orthogonal SA method and the presented penalty function when applied to the constrained problems. Moreover, the orthogonal SA method is applied to two engineering optimization design problems, including the designs of a welded beam and a coil compression spring, to evaluate the capacity of the method for practical engineering design. The computational results show that the proposed orthogonal SA method is effective in determining the optimal design variables and the value of objective function.     

Process optimal design in non‐isothermal,steady‐state metal forming by the finite element method

A new approach to process optimal design in non‐isothermal, steady‐state metal forming is presented. In this approach, the optimal design problem is formulated on the basis of the integrated thermo‐mechanical finite element process model so as to cover a wide class of the objective functions and to accept diverse process parameters as design variables, and a derivative‐based approach is adopted as a solution technique. The process model, the formulation for process optimal design, and the schemes for the evaluation of the design sensitivity, and an iterative procedure for design optimization are described in detail. The validity of the schemes for the evaluation of the design sensitivity is examined by performing a series of numerical tests. The capability of the proposed approach is demonstrated through applications to some selected process design problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Convergence speed in multi‐objective metaheuristics: Efficiency criteria and empirical study

Solving optimization problems using a reduced number of objective function evaluations is an open issue in the design of multi‐objective optimization metaheuristics. The usual approach to analyze the behavior of such techniques is to choose a benchmark of known problems, to perform a predetermined number of function evaluations, and then, apply a set of performance indicators in order to assess the quality of the solutions obtained. However, this sort of methodology does not provide any insights of the efficiency of each algorithm. Here, efficiency is defined as the effort required by a multi‐objective metaheuristic to obtain a set of non‐dominated solutions that is satisfactory to the user, according to some pre‐defined criterion. Indeed, the type of solutions of interest to the user may vary depending on the specific characteristics of the problem being solved. In this paper, the convergence speed of seven state‐of‐the‐art multi‐objective metaheuristics is analyzed, according to three pre‐defined efficiency criteria. Our empirical study shows that SMPSO (based on a particle swarm optimizer) is found to be the best overall algorithm on the test problems adopted when considering the three efficiency criteria. Copyright © 2010 John Wiley & Sons, Ltd.     

A multi‐objective optimization using distribution characteristics of reference data for reverse engineering

As different industries produce similar products, engineers tend to analyze the products of competitors and adopt the excellent merits in their current products. This process is called reverse engineering. There can be multiple target characteristics in reverse engineering. In many cases, the improved design from reverse engineering usually keeps the data distribution characteristics of the competitors unless the developed product is a fully new creative design. The distribution should be considered in reverse engineering. Therefore, the reverse engineering process can be modeled as multi‐objective optimization considering data distribution. Recently, Taguchi developed the Mahalanobis Taguchi system (MTS) technique to minimize the Mahalanobis distance (MD), which is defined by a multi‐objective function with data distribution. However, the MTS technique has the limit that the new design is not better than the mean values of the competitors. In this research, a function named as the skewed Mahalanobis distance (SMD) is proposed to overcome the drawbacks of the MTS technique. SMD is a new distance scale defined by multiplying the skewed value of a design point to MD. SMD is used instead of MD and the method is named the SMD method. The SMD method can always give a unique Pareto optimum solution. To verify the efficiency of the SMD method, a non‐convex mathematical example, a cantilever beam, and a practical automobile suspension system are optimized. Copyright © 2010 John Wiley & Sons, Ltd.     

Topology optimization of fluid problems using genetic algorithm assisted by the Kriging model

A non‐gradient‐based approach for topology optimization using a genetic algorithm is proposed in this paper. The genetic algorithm used in this paper is assisted by the Kriging surrogate model to reduce computational cost required for function evaluation. To validate the non‐gradient‐based topology optimization method in flow problems, this research focuses on two single‐objective optimization problems, where the objective functions are to minimize pressure loss and to maximize heat transfer of flow channels, and one multi‐objective optimization problem, which combines earlier two single‐objective optimization problems. The shape of flow channels is represented by the level set function. The pressure loss and the heat transfer performance of the channels are evaluated by the Building‐Cube Method code, which is a Cartesian‐mesh CFD solver. The proposed method resulted in an agreement with previous study in the single‐objective problems in its topology and achieved global exploration of non‐dominated solutions in the multi‐objective problems. © 2016 The Authors International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd     

A MODIFIED DEMPSTER-SHAFER THEORY FOR MULTICRITERIA OPTIMIZATION

Abstract

A new methodology, based on a modified Dempster-Shafer (DS) theory, is proposed for solving multicriteria design optimization problems. It is well known that considerable amount of computational information is acquired during the iterative process of optimization. Based on the computational information generated in each iteration, an evidence-based approach is presented for solving a multiobjective optimization problem. The method handles the multiple design criteria, which are often conflicting and non-commensurable, by constructing belief structures that can quantitatively evaluate the effectiveness of each design in the range 0 to 1. An overall satisfaction function is then defined for converting the original multicriteria design problem into a single-criterion problem so that standard single-objective programming techniques can be employed for the solution. The design of a mechanism in the presence of seven design criteria and eighteen design variables is considered to illustrate the computational details of the approach. This work represents the first attempt made in the literature at applying DS theory for numerical engineering optimization.     

Robust structural damage identification based on multi‐objective optimization

Inverse analysis for structural damage identification often involves an optimization process that minimizes the discrepancies between the computed responses and the measured responses. Conventional single‐objective optimization approach defines the objective function by combining multiple error terms into a single one, which leads to a weaker constraint in solving the identification problem. A multi‐objective approach is proposed, which minimizes multiple error terms simultaneously. Its non‐domination‐based convergence provides a stronger constraint that enables robust identification of damages with lower false‐negative detection rate. Another merit of the proposed approach is quantified confidence in damage detection through processing Pareto‐optimal solutions. Numerical examples that simulate static testing are provided to compare the proposed approach with conventional formulation based on single‐objective optimization. Copyright © 2009 John Wiley & Sons, Ltd.     

A Hopfield Neural Network Approach to the Dual Response Problem

The application of neural networks to optimization problems has been an active research area since the early 1980s. Unconstrained optimization, constrained optimization and combinatorial optimization problems have been solved using neural networks. This study presents a new approach using Hopfield neural networks (HNNs) for solving the dual response system (DRS) problems. The major aim of the proposed method is to produce a string of solutions, rather than a ‘one‐shot’ optimum solution, to make the trade‐offs available between the mean and standard deviation responses. This gives more flexibility to the decision‐maker in exploring alternative solutions. The proposed method has been tested on two examples. The HNN results are very close to those obtained by using the NIMBUS (Nondifferentiable Interactive Multiobjective Bundle‐based Optimization System) algorithm. Choosing an appropriate solution method for a certain multi‐objective optimization problem is not easy, as has been made abundantly clear. Unlike the NIMBUS method, the HNN approach does not set any specific assumptions on the behaviour or the preference structure of the decision maker. As a result, the proposed method will still work and generate alternative solutions whether or not the decision maker has enough time and capabilities for co‐operation. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Accelerating adaptive trade‐off model using shrinking space technique for constrained evolutionary optimization

Adaptive trade‐off model (ATM) is a constraint‐handling mechanism proposed recently. The main advantages of this model are its simplicity and adaptation. Moreover, it can be easily embedded into evolutionary algorithms for solving constrained optimization problems. This paper proposes a novel method for constrained optimization, which aims at accelerating the ATM using shrinking space technique. Eighteen benchmark test functions and five engineering design problems are used to test the performance of the method proposed. Experimental results suggest that combining the ATM with the shrinking space technique is very beneficial. The method proposed can promptly converge to competitive results without loss of the quality and the precision of the final results. Performance comparisons with some other state‐of‐the‐art approaches from the literature are also presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A Risk‐based Approach to Managing Performance‐based Maintenance Contracts

The global trend towards performance‐based maintenance contracting has presented new challenges to maintenance service providers as they are compensated or penalized based on performance outcomes instead of time and materials consumed during maintenance service. The problem becomes more complex when uncertainties exist in reliability performance and maintenance activities of technical systems. In this paper, a general framework for managing performance‐based maintenance contract under risks is proposed. We illustrate our approach with an application in a multi‐echelon multi‐system spare parts control problem. Several different performance measures are considered and a probabilistic constrained optimization problem is formulated from the perspective of the service provider. Hybrid simulation/analytic heuristics are proposed to solve the problem based on the monotonic properties of performance measures. This approach is flexible and can be applied to a wide range of problems with similar properties. Numerical example shows that the probability of violating performance requirements is high if the risk is overlooked. We also provide guidelines on how to apply this approach in practice. Copyright © 2016 John Wiley & Sons, Ltd.     

A modified version of a T‐Cell Algorithm for constrained optimization problems

In this paper, we present a heuristic inspired on the T‐Cell model of the immune system (i.e. an artificial immune system). The proposed approach (called T‐Cell) is used for solving constrained (numerical) optimization problems, and is validated using several test functions taken from the specialized literature on evolutionary optimization. Additionally, several engineering optimization problems are also used for assessing the performance of the proposed approach. The results are compared with respect to approaches representative of the state‐of‐the‐art in constrained evolutionary optimization. Copyright © 2010 John Wiley & Sons, Ltd.     

A mixed-integer simulation-based optimization approach with surrogate functions in water resources management

Efficient and powerful methods are needed to overcome the inherent difficulties in the numerical solution of many simulation-based engineering design problems. Typically, expensive simulation codes are included as black-box function generators; therefore, gradient information that is required by mathematical optimization methods is entirely unavailable. Furthermore, the simulation code may contain iterative or heuristic methods, low-order approximations of tabular data, or other numerical methods which contribute noise to the objective function. This further rules out the application of Newton-type or other gradient-based methods that use traditional finite difference approximations. In addition, if the optimization formulation includes integer variables the complexity grows even further. In this paper we consider three different modeling approaches for a mixed-integer nonlinear optimization problem taken from a set of water resources benchmarking problems. Within this context, we compare the performance of a genetic algorithm, the implicit filtering algorithm, and a branch-and-bound approach that uses sequential surrogate functions. We show that the surrogate approach can greatly improve computational efficiency while locating a comparable, sometimes better, design point than the other approaches.     

Reliability‐based robust design optimization: A multi‐objective framework using hybrid quality loss function

In this globally competitive business environment, design engineers are constantly striving to establish new and effective tools and techniques to ensure a robust and reliable product design. Robust design (RD) and reliability‐based design approaches have shown the potential to deal with variability in the life cycle of a product. This paper explores the possibilities of combining both approaches into a single model and proposes a hybrid quality loss function‐based multi‐objective optimization model. The model is unique because it uses a hybrid form of quality loss‐based objective function that is defined in terms of desirable as well as undesirable deviations to obtain efficient design points with minimum quality loss. The proposed approach attempts to optimize the product design by addressing quality loss, variability, and life‐cycle issues simultaneously by combining both reliability‐based and RD approaches into a single model with various customer aspirations. The application of the approach is demonstrated using a leaf spring design example. Copyright © 2009 John Wiley & Sons, Ltd.     

Horsetail matching: a flexible approach to optimization under uncertainty*

It is important to design engineering systems to be robust with respect to uncertainties in the design process. Often, this is done by considering statistical moments, but over-reliance on statistical moments when formulating a robust optimization can produce designs that are stochastically dominated by other feasible designs. This article instead proposes a formulation for optimization under uncertainty that minimizes the difference between a design's cumulative distribution function and a target. A standard target is proposed that produces stochastically non-dominated designs, but the formulation also offers enough flexibility to recover existing approaches for robust optimization. A numerical implementation is developed that employs kernels to give a differentiable objective function. The method is applied to algebraic test problems and a robust transonic airfoil design problem where it is compared to multi-objective, weighted-sum and density matching approaches to robust optimization; several advantages over these existing methods are demonstrated.     

A new harmony search algorithm for solving mixed–discrete engineering optimization problems

In general design optimization problems, it is usually assumed that the design variables are continuous. However, many practical problems in engineering design require considering the design variables as integer or discrete values. The presence of discrete and integer variables along with continuous variables adds to the complexity of the optimization problem. Very few of the existing methods can yield a globally optimal solution when the objective functions are non-convex and non-differentiable. This article presents a mixed–discrete harmony search approach for solving these nonlinear optimization problems which contain integer, discrete and continuous variables. Some engineering design examples are also presented to demonstrate the effectiveness of the proposed method.     

Concurrent optimization of the design and manufacturing stages of product development

The problem of concurrent optimization of the design and the process planning stages when a new product is developed is addressed. The paper advocates for a simultaneous approach rather than the traditional sequential one. A mathematical representation of this approach is given for these two stages. A mathematical programming technique is used to find the optimal values of the design and the process characteristics. The objective function is a quality loss function. The constraints are the customer requirements, the product's specification limits, the parts’ dimensional limits and the process capability. The traditional sequential approach of concurrent engineering is compared with the proposed simultaneous approach. A parametric analysis of the objective function is performed by applying an interactive multi-objective goal programming technique. A numerical example of a low-pass electrical circuit is given. It is shown that the proposed approach leads to better efficient solutions than the sequential approach. The decision-maker interacts with the optimization process and can choose the efficient solution that best satisfies the company's needs.