An improved weighting method with multibounds formulation and convex programming for multicriteria structural optimization

This paper presents an improved weighting method for multicriteria structural optimization. By introducing artificial design variables, here called as multibounds formulation (MBF), we demonstrate mathematically that the weighting combination of criteria can be transformed into a simplified problem with a linear objective function. This is a unified formulation for one criterion and multicriteria problems. Due to the uncoupling of involved criteria after the transformation, the extension and the adaptation of monotonic approximation‐based convex programming methods such as the convex linearization (CONLIN) or the method of moving asymptotes (MMA) are made possible to solve multicriteria problems as efficiently as for one criterion problems. In this work, a multicriteria optimization tool is developed by integrating the multibounds formulation with the CONLIN optimizer and the ABAQUS finite element analysis system. Some numerical examples are taken into account to show the efficiency of this approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Rule-based optimization and multicriteria decision support for packaging a truck chassis

Trucks are highly individualized products where exchangeable parts are flexibly combined to suit different customer requirements, this leading to a great complexity in product development. Therefore, an optimization approach based on constraint programming is proposed for automatically packaging parts of a truck chassis by following packaging rules expressed as constraints. A multicriteria decision support system is developed where a database of truck layouts is computed, among which interactive navigation then can be performed. The work has been performed in cooperation with Volvo Group Trucks Technology (GTT), from which specific rules have been used. Several scenarios are described where the methods developed can be successfully applied and lead to less time-consuming manual work, fewer mistakes, and greater flexibility in configuring trucks. A numerical evaluation is also presented showing the efficiency and practical relevance of the methods, which are implemented in a software tool.     

Explicit model of dual programming and solving method for a class of separable convex programming problems

An objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.     

A solution selection approach to multiresponse surface optimization based on a clustering method

In Multi-Response Surface Optimization (MRSO), responses are often in conflict. To obtain a satisfactory compromise, the preference information of a Decision Maker (DM) on the tradeoffs among the responses should be considered. One of the promising alternatives is a posterior preference articulation approach. It first generates nondominated solutions and then makes the DM select the best one from the nondominated solutions. In this article, a solution selection approach is presented. It takes the posterior approach and employs a clustering method to aid the selection process of the DM. The DM can obtain the satisfactory compromise solution easily by the proposed method.     

An alternating optimization approach for mixed discrete non-linear programming

This article contributes to the development of the field of alternating optimization (AO) and general mixed discrete non-linear programming (MDNLP) by introducing a new decomposition algorithm (AO-MDNLP) based on the augmented Lagrangian multipliers method. In the proposed algorithm, an iterative solution strategy is proposed by transforming the constrained MDNLP problem into two unconstrained components or units; one solving for the discrete variables, and another for the continuous ones. Each unit focuses on minimizing a different set of variables while the other type is frozen. During optimizing each unit, the penalty parameters and multipliers are consecutively updated until the solution moves towards the feasible region. The two units take turns in evolving independently for a small number of cycles. The validity, robustness and effectiveness of the proposed algorithm are exemplified through some well known benchmark mixed discrete optimization problems.     

A fast method for binary programming using first‐order derivatives,with application to topology optimization with buckling constraints

We present a method for finding solutions of large‐scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimization subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiffness matrix with respect to the density of an element in the finite‐element setting. Results are presented for a number of two‐dimensional test problems.Copyright © 2012 John Wiley & Sons, Ltd.     

Duality of regularizations and hullfunctions for mathematical programming problems

For the study of mathematical programming problems and solution methods the duality theory forms a powerful tool. There are also some concepts ofregularization andstabilization of a given problem for a better behavior in practical solution procedures. The aim of this paper is the investigation of duality aspects of such regularizations and the forming ofhullfunctions on the other hand. Applications for handling of so-calledill-posed problems (Eremin) using some parametrizations of the original problem will emphasize the importance for practical numerical methods, especially. This results will inspire some applications to solution methods for parametric and multicriteria optimization.     

QFD optimization using linear physical programming

A new approach to quality function deployment (QFD) optimization is presented. The approach uses the linear physical programming (LPP) technique to maximize overall customer satisfaction in product design. QFD is a customer-focused product design method which translates customer requirements into product engineering characteristics. Because market competition is multidimensional, companies must maximize overall customer satisfaction by optimizing the design of their products. At the same time, all constraints (e.g. product development time, development cost, manufacturing cost, human resource in design and production, etc.) must be taken into consideration. LPP avoids the need to specify an importance weight for each objective in advance. This is an effective way of obtaining optimal results. Following a brief introduction to LPP in QFD, the proposed approach is described. A numerical example is given to illustrate its application and a sensitivity analysis is carried out. Using LPP in QFD optimization provides a new direction for optimizing the product design process.     

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 new approach to solving stochastic programming problems with recourse

A new numerical approach to the solution of two-stage stochastic linear programming problems is described and evaluated. The approach avoids the solution of the first-stage problem and uses the underlying deterministic problem to generate a sequence of values of the first-stage variables which lead to successive improvements of the objective function towards the optimal policy. The model is evaluated using an example in which randomness is described by two correlated factors. The dynamics of these factors are described by stochastic processes simulated using lattice techniques. In this way, discrete distributions of the random parameters are assembled. The solutions obtained with the new iterative procedure are compared with solutions obtained with a deterministic equivalent linear programming problem. It is concluded that they are almost identical. However, the computational effort required for the new approach is negligible compared with that needed for the deterministic equivalent problem.     

A moving superimposed finite element method for structural topology optimization

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary‐based moving superimposed finite element method (s‐version FEM or S‐FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S‐FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co‐ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S‐FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S‐FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well‐recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S‐FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.     

A deterministic global approach for mixed-discrete structural optimization

This study proposes a novel approach for finding the exact global optimum of a mixed-discrete structural optimization problem. Although many approaches have been developed to solve the mixed-discrete structural optimization problem, they cannot guarantee finding a global solution or they adopt too many extra binary variables and constraints in reformulating the problem. The proposed deterministic method uses convexification strategies and linearization techniques to convert a structural optimization problem into a convex mixed-integer nonlinear programming problem solvable to obtain a global optimum. To enhance the computational efficiency in treating complicated problems, the range reduction technique is also applied to tighten variable bounds. Several numerical experiments drawn from practical structural design problems are presented to demonstrate the effectiveness of the proposed method.     

Limit analysis and convex programming: A decomposition approach of the kinematic mixed method

This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776—value to be compared with the best published lower bound 3.772—by succeeding in solving non‐linear optimization problems with millions of variables and constraints. Copyright © 2008 John Wiley & Sons, Ltd.     

A novel minimum weight formulation of topology optimization implemented with reanalysis approach

In this paper, we develop an efficient diagonal quadratic optimization formulation for minimum weight design problem subject to multiple constraints. A high-efficiency computational approach of topology optimization is implemented within the framework of approximate reanalysis. The key point of the formulation is the introduction of the reciprocal-type variables. The topology optimization seeking for minimum weight can be transformed as a sequence of quadratic program with separable and strictly positive definite Hessian matrix, thus can be solved by a sequential quadratic programming approach. A modified sensitivity filtering scheme is suggested to remove undesirable checkerboard patterns and mesh dependence. Several typical examples are provided to validate the presented approach. It is observed that the optimized structure can achieve lighter weight than those from the established method by the demonstrative numerical test. Considerable computational savings can be achieved without loss of accuracy of the final design for 3D structure. Moreover, the effects of multiple constraints and upper bound of the allowable compliance upon the optimized designs are investigated by numerical examples.     

Multiple sequence alignment using modified dynamic programming and particle swarm optimization

Abstract

Dynamic Programming (DP) is widely used in Multiple Sequence Alignment (MSA) problems. However, when the number of the considered sequences is more than two, multiple dimensional DP may suffer from large storage and computational complexities. Often, progressive pairwise DP is employed for MSA. However, such an approach also suffers from local optimum problems. In this paper, we present a hybrid algorithm for MSA. The algorithm combines the pairwise DP and the particle swarm optimization (PSO) techniques to overcome the above drawbacks. In the algorithm, pairwise DP is used to align sequences progressively and PSO is employed to avoid the result of alignment being trapped into local optima. Several existing MSA tools are employed for comparison. The experimental results show excellent performance of the proposed algorithm.     

Multi-objective optimization of space station logistics strategies using physical programming

This study extends a previously proposed single-objective optimization formulation of space station logistics strategies to multi-objective optimization. The four-objective model seeks to maximize the mean utilization capacity index, total utilization capacity index, logistics robustness index and flight independency index, aiming to improve both the utilization benefit and the operational robustness of a space station operational scenario. Physical programming is employed to convert the four-objective optimization problem into a single-objective problem. A genetic algorithm is proposed to solve the resulting physical programming-based optimization problem. Moreover, the non-dominated sorting genetic algorithm-II is tested to obtain the Pareto-optimal solution set and verify the Pareto optimality of the physical programming-based solution. The proposed approach is demonstrated with a notional one-year scenario of China's future space station. It is shown that the designer-preferred compromise solution improving both the utilization benefit and the operational robustness is successfully obtained.     

A dual algorithm for the topology optimization of non‐linear elastic structures

Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non‐linear structures. The perimeter constraint is used to make the topology problem well‐posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two‐dimensional although the loading and the geometry are three‐dimensional. Copyright © 2008 John Wiley & Sons, Ltd.     

An uncertain optimization method for overall ballistics based on stochastic programming and a neural network surrogate model

A nonlinear stochastic programming method is proposed in this article to deal with the uncertain optimization problems of overall ballistics. First, a general overall ballistic dynamics model is achieved based on classical interior ballistics, projectile initial disturbance calculation model, exterior ballistics and firing dispersion calculation model. Secondly, the random characteristics of uncertainties are simulated using a hybrid probabilistic and interval model. Then, a nonlinear stochastic programming method is put forward by integrating a back-propagation neural network with the Monte Carlo method. Thus, the uncertain optimization problem is transformed into a deterministic multi-objective optimization problem by employing the mean value, the standard deviation, the probability and the expected loss function, and then the sorting and optimizing of design vectors are realized by the non-dominated sorting genetic algorithm-II. Finally, two numerical examples in practical engineering are presented to demonstrate the effectiveness and robustness of the proposed method.     

Multi-objective design optimization of reconfigurable machine tools: a modified fuzzy-Chebyshev programming approach

A reconfigurable manufacturing system (RMS) is designed for rapid adjustment of functionalities in response to market changes. A RMS consists of a number of reconfigurable machine tools (RMTs) for processing different jobs using different processing modules. The potential benefits of a RMS may not be materialized if not properly designed. This paper focuses on RMT design optimization considering three important yet conflicting factors: configurability, cost and process accuracy. The problem is formulated as a multi-objective model. A mechanism is developed to generate and evaluate alternative designs. A modified fuzzy-Chebyshev programming (MFCP) method is proposed to achieve a preferred compromise of the design objectives. Unlike the original fuzzy-Chebyshev programming (FCP) method which imposes an identical satisfaction level for all objectives regardless of their relative importance, the MFCP respects their priority order. This method also features an adaptive satisfaction-level-dependent process to dynamically adjust objective weights in the search process. A particle swarm optimization algorithm (PSOA) is developed to provide quick solutions. The application of the proposed approach is demonstrated using a reconfigurable boring machine. Our computational results have shown that the combined MFCP and PSOA algorithm is efficient and robust. The advantages of the MFCP over the original FCP are also illustrated based on the results.     

QPSchur: A dual, active-set, Schur-complement method for large-scale and structured convex quadratic programming

We describe an active-set, dual-feasible Schur-complement method for quadratic programming (QP) with positive definite Hessians. The formulation of the QP being solved is general and flexible, and is appropriate for many different application areas. Moreover, the specialized structure of the QP is abstracted away behind a fixed KKT matrix called K_o and other problem matrices, which naturally leads to an object-oriented software implementation. Updates to the working set of active inequality constraints are facilitated using a dense Schur complement, which we expect to remain small. Here, the dual Schur complement method requires the projected Hessian to be positive definite for every working set considered by the algorithm. Therefore, this method is not appropriate for all QPs. While the Schur complement approach to linear algebra is very flexible with respect to allowing exploitation of problem structure, it is not as numerically stable as approaches using a QR factorization. However, we show that the use of fixed-precision iterative refinement helps to dramatically improve the numerical stability of this Schur complement algorithm. The use of the object-oriented QP solver implementation is demonstrated on two different application areas with specializations in each area; large-scale model predictive control (MPC) and reduced-space successive quadratic programming (with several different representations for the reduced Hessian). These results demonstrate that the QP solver can exploit application-specific structure in a computationally efficient and fairly robust manner as compared to other QP solver implementations.