Decomposition of multidisciplinary optimization problems: formulations and application to a simplified wing design

Several formulations for solving multidisciplinary design optimization (MDO) problems are presented and applied to a test case. Two bi-level hierarchical decomposition approaches are compared with two classical single-level approaches without decomposition of the optimization problem. A methodology to decompose MDO problems and a new formulation based on this decomposition are proposed. The problem considered here for validation of the different formulations involves the shape and structural optimization of a conceptual wing model. The efficiency of the design strategies are compared on the basis of optimization results.     

Decoupled approach to multidisciplinary design optimization under uncertainty

We propose solution methods for multidisciplinary design optimization (MDO) under uncertainty. This is a class of stochastic optimization problems that engineers are often faced with in a realistic design process of complex systems. Our approach integrates solution methods for reliability-based design optimization (RBDO) with solution methods for deterministic MDO problems. The integration is enabled by the use of a deterministic equivalent formulation and the first order Taylor’s approximation in these RBDO methods. We discuss three specific combinations: the RBDO methods with the multidisciplinary feasibility method, the all-at-once method, and the individual disciplinary feasibility method. Numerical examples are provided to demonstrate the procedure. Anukal Chiralaksanakul is currently a full-time lecturer in the Graduate School of Business Administration at National Institute of Development Administration (NIDA), Bangkok, Thailand.     

Uncertainty quantification using evidence theory in multidisciplinary design optimization

Advances in computational performance have led to the development of large-scale simulation tools for design. Systems generated using such simulation tools can fail in service if the uncertainty of the simulation tool's performance predictions is not accounted for. In this research an investigation of how uncertainty can be quantified in multidisciplinary systems analysis subject to epistemic uncertainty associated with the disciplinary design tools and input parameters is undertaken. Evidence theory is used to quantify uncertainty in terms of the uncertain measures of belief and plausibility. To illustrate the methodology, multidisciplinary analysis problems are introduced as an extension to the epistemic uncertainty challenge problems identified by Sandia National Laboratories.After uncertainty has been characterized mathematically the designer seeks the optimum design under uncertainty. The measures of uncertainty provided by evidence theory are discontinuous functions. Such non-smooth functions cannot be used in traditional gradient-based optimizers because the sensitivities of the uncertain measures are not properly defined. In this research surrogate models are used to represent the uncertain measures as continuous functions. A sequential approximate optimization approach is used to drive the optimization process. The methodology is illustrated in application to multidisciplinary example problems.     

Modelling topology optimization problems as linear mixed 0–1 programs

This paper deals with topology optimization of discretized continuum structures. It is shown that a large class of non‐linear 0–1 topology optimization problems, including stress‐ and displacement‐constrained minimum weight problems, can equivalently be modelled as linear mixed 0–1 programs. The modelling approach is applied to some test problems which are solved to global optimality. Copyright © 2003 John Wiley & Sons, Ltd.     

A mixed integer programming for robust truss topology optimization with stress constraints

This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.     

A parallel bi-level multidisciplinary design optimization architecture with convergence proof for general problem

Quite a number of distributed Multidisciplinary Design Optimization (MDO) architectures have been proposed for the optimal design of large-scale multidisciplinary systems. However, just a few of them have available numerical convergence proof. In this article, a parallel bi-level MDO architecture is presented to solve the general MDO problem with shared constraints and a shared objective. The presented architecture decomposes the original MDO problem into one implicit nonlinear equation and multiple concurrent sub-optimization problems, then solves them through a bi-level process. In particular, this architecture allows each sub-optimization problem to be solved in parallel and its solution is proven to converge to the Karush–Kuhn–Tucker (KKT) point of the original MDO problem. Finally, two MDO problems are introduced to perform a comprehensive evaluation and verification of the presented architecture and the results demonstrate that it has a good performance both in convergence and efficiency.     

Industrial waste recycling strategies optimization problem: mixed integer programming model and heuristics

Manufacturers have a legal accountability to deal with industrial waste generated from their production processes in order to avoid pollution. Along with advances in waste recovery techniques, manufacturers may adopt various recycling strategies in dealing with industrial waste. With reuse strategies and technologies, byproducts or wastes will be returned to production processes in the iron and steel industry, and some waste can be recycled back to base material for reuse in other industries. This article focuses on a recovery strategies optimization problem for a typical class of industrial waste recycling process in order to maximize profit. There are multiple strategies for waste recycling available to generate multiple byproducts; these byproducts are then further transformed into several types of chemical products via different production patterns. A mixed integer programming model is developed to determine which recycling strategy and which production pattern should be selected with what quantity of chemical products corresponding to this strategy and pattern in order to yield maximum marginal profits. The sales profits of chemical products and the set-up costs of these strategies, patterns and operation costs of production are considered. A simulated annealing (SA) based heuristic algorithm is developed to solve the problem. Finally, an experiment is designed to verify the effectiveness and feasibility of the proposed method. By comparing a single strategy to multiple strategies in an example, it is shown that the total sales profit of chemical products can be increased by around 25% through the simultaneous use of multiple strategies. This illustrates the superiority of combinatorial multiple strategies. Furthermore, the effects of the model parameters on profit are discussed to help manufacturers organize their waste recycling network.     

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

In this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs. The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered. The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs

We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented. The research is funded by the Danish Natural Science Research Council and the Danish Research Council for Technology and Production Sciences.     

Integrating reliability optimization into chemical process synthesis

The growing need to achieve high availability for large integrated chemical process systems demands higher levels of system reliability at the operational stage. In these circumstances, it has become critical to consider the reliability aspects of a system and its components at the design stage. Traditional reliability/availability analysis methods and maintenance optimization frameworks, commonly applied at the design stage, are limited in their application, as in most of these methods the designer is required to specify the process system components, their connectivity and their reliabilities a priori. As a result, these traditional methods do not provide the flexibility to reconfigure a process or select initial reliabilities of equipment in a way that maximizes the inherent plant availability at the design stage. In this paper, we developed an optimization framework by combining the reliability optimization and process synthesis challenges and the combined optimization problem is posed as a mixed integer non-linear programming optimization problem. The proposed optimization framework features an expected profit objective function, which takes into account the trade-off between initial capital investment and the annual operational costs by supporting appropriate estimation of revenues, investment cost, raw material and utilities cost, and maintenance cost as a function of the system and its component availability. The effectiveness and usefulness of the proposed optimization framework is demonstrated for the synthesis of the hydrodealkylation process (HDA) process.     

A mixed integer goal programming model for discrete values of design requirements in QFD

A main feature of quality function deployment (QFD) planning process is to determine target values for the design requirements (DRs) of a product, with a view to achieving a higher level of overall customer satisfaction. However, in real world applications, values of DRs are often discrete instead of continuous. Therefore, a mixed integer linear programming (MILP) model considering discrete data is suggested. As opposed to the existing literature, the fulfilment levels of DRs are assumed to have a piece-wise linear relationship with cost; because, constraints of technology and resource rarely provides a linear relationship in manufacturing systems. In the proposed MILP model, we considered customer satisfaction as the only goal. But, QFD process may be necessary to optimise cost and technical difficulty goals as well as customer satisfaction. Therefore, by developing the MILP model with multi-objective decision making (MODM) approach, a novel mixed integer goal programming (MIGP) model is proposed to optimise these goals simultaneously. Finally, MILP model solution turns out to be a more realistic approach to real applications because piece-wise linear relationship is taken into account. The solution of MIGP model provided different alternative results to decision makers according to usage of the lexicographic goal programming (LGP) approach. The applicability of the proposed models in practice is demonstrated with a washing machine development problem.     

A discrete hybrid differential evolution algorithm for solving integer programming problems

Differential evolution (DE) is one of the most prominent new evolutionary algorithms for solving real-valued optimization problems. In this article, a discrete hybrid differential evolution algorithm is developed for solving global numerical optimization problems with discrete variables. Orthogonal crossover is combined with DE crossover to achieve crossover operation, and the simplified quadratic interpolation (SQI) method is employed to improve the algorithm's local search ability. A mixed truncation procedure is incorporated in the operations of DE mutation and SQI to ensure that the integer restriction is satisfied. Numerical experiments on 40 test problems including seventeen large-scale problems with up to 200 variables have demonstrated the applicability and efficiency of the proposed method.     

反渗透系统中膜元件位置优化的0-1整数规划算法

通过膜元件参数对系统产水含盐量的相关分析得出了相应的系统灵敏度参数,进而形成了反渗透系统膜元件优化排列的0-1整数规划模型.通过对整数规划的求解与膜元件全排列的系统模拟计算,验证了膜元件优化排列的数值优势与统计优势.     

复杂系统的多学科设计优化综述

从设计和分析的本质出发,结合复杂系统的特点,通过分析传统设计优化流程在面对复杂系统时存在的困难和缺陷,指出多学科设计优化(multidisciplinary design optimization,MDO)方法是解决复杂系统设计优化问题的一种有效措施.在此基础上,介绍了多学科优化方法的基本思想,总结了子系统耦合方式及MDO在处理耦合时的基本方法,归纳了MDO的知识框架和主要研究内容.最后在现有研究成果的基础上,对MDO今后的研究提出了几点参考意见.     

Global optimization for signomial discrete programming problems in engineering design

Signomial discrete programming (SDP) problems arise frequently in a variety of real applications. Although many optimization techniques have been developed to solve an SDP problem, they use too many binary variables to reformulate the problem for finding a globally optimal solution or can only derive a local or an approximate solution. This article proposes a global optimization method to solve an SDP problem by integrating an efficient linear expression of single variable discrete functions and convexification techniques. An SDP problem can be converted into a convex mixed-integer programming problem solvable to obtain a global optimum. Several illustrative examples are also presented to demonstrate the usefulness and effectiveness of the proposed method.     

Mixed integer simulation optimization for optimal hydraulic fracturing and production of shale gas fields

Optimal development of shale gas fields involves designing a most productive fracturing network for hydraulic stimulation processes and operating wells appropriately throughout the production time. A hydraulic fracturing network design—determining well placement, number of fracturing stages, and fracture lengths—is defined by specifying a set of integer ordered blocks to drill wells and create fractures in a discrete shale gas reservoir model. The well control variables such as bottom hole pressures or production rates for well operations are real valued. Shale gas development problems, therefore, can be mathematically formulated with mixed-integer optimization models. A shale gas reservoir simulator is used to evaluate the production performance for a hydraulic fracturing and well control plan. To find the optimal fracturing design and well operation is challenging because the problem is a mixed integer optimization problem and entails computationally expensive reservoir simulation. A dynamic simplex interpolation-based alternate subspace (DSIAS) search method is applied for mixed integer optimization problems associated with shale gas development projects. The optimization performance is demonstrated with the example case of the development of the Barnett Shale field. The optimization results of DSIAS are compared with those of a pattern search algorithm.     

A new optimization algorithm for solving complex constrained design optimization problems

This article presents the performance of a very recently proposed Jaya algorithm on a class of constrained design optimization problems. The distinct feature of this algorithm is that it does not have any algorithm-specific control parameters and hence the burden of tuning the control parameters is minimized. The performance of the proposed Jaya algorithm is tested on 21 benchmark problems related to constrained design optimization. In addition to the 21 benchmark problems, the performance of the algorithm is investigated on four constrained mechanical design problems, i.e. robot gripper, multiple disc clutch brake, hydrostatic thrust bearing and rolling element bearing. The computational results reveal that the Jaya algorithm is superior to or competitive with other optimization algorithms for the problems considered.     

A mixed integer linear programming solution for single hoist multi-degree cyclic scheduling with reentrance

This article considers single hoist multi-degree cyclic scheduling problems with reentrance. Time window constraints are also considered. Firstly, a mixed integer programming model is formulated for multi-degree cyclic hoist scheduling without reentrance, referred to as basic lines in this article. Two valid inequalities corresponding to this problem are also presented. Based on the model for basic lines, an extended mixed integer programming model is proposed for more complicated scheduling problems with reentrance. Phillips and Unger's benchmark instance and randomly generated instances are applied to test the model without reentrance, solved using the commercial software CPLEX. The efficiency of the model is analysed based on computational time. Moreover, an example is given to demonstrate the effectiveness of the model with reentrance.     

Improved genetic algorithm for mixed-discrete-continuous design optimization problems

An improved genetic algorithm (IGA) is presented to solve the mixed-discrete-continuous design optimization problems. The IGA approach combines the traditional genetic algorithm with the experimental design method. The experimental design method is incorporated in the crossover operations to systematically select better genes to tailor the crossover operations in order to find the representative chromosomes to be the new potential offspring, so that the IGA approach possesses the merit of global exploration and obtains better solutions. The presented IGA approach is effectively applied to solve one structural and five mechanical engineering problems. The computational results show that the presented IGA approach can obtain better solutions than both the GA-based and the particle-swarm-optimizer-based methods reported recently.     

Solving large-scale fixed cost integer linear programming models for grid-based location problems with heuristic techniques

Grid-based location problems (GBLPs) can be used to solve location problems in business, engineering, resource exploitation, and even in the field of medical sciences. To solve these decision problems, an integer linear programming (ILP) model is designed and developed to provide the optimal solution for GBLPs considering fixed cost criteria. Preliminary results show that the ILP model is efficient in solving small to moderate-sized problems. However, this ILP model becomes intractable in solving large-scale instances. Therefore, a decomposition heuristic is proposed to solve these large-scale GBLPs, which demonstrates significant reduction of solution runtimes. To benchmark the proposed heuristic, results are compared with the exact solution via ILP. The experimental results show that the proposed method significantly outperforms the exact method in runtime with minimal (and in most cases, no) loss of optimality.