Robust optimization of foam-filled thin-walled structure based on sequential Kriging metamodel

Deterministic optimization has been successfully applied to a range of design problems involving foam-filled thin-walled structures, and to some extent gained significant confidence for the applications of such structures in automotive, aerospace, transportation and defense industries. However, the conventional deterministic design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, a robust design methodology is presented in this paper to address the effects of parametric uncertainties of foam-filled thin-walled structure on design optimization, in which different sigma criteria are adopted to measure the variations. The Kriging modeling technique is used to construct the corresponding surrogate models of mean and standard deviation for different crashworthiness criteria. A sequential sampling approach is introduced to improve the fitness accuracy of these surrogate models. Finally, a gradient-based sequential quadratic program (SQP) method is employed from 20 different initial points to obtain a quasi-global robust optimum solution. The optimal solutions were verified by using the Monte Carlo simulation. The results show that the presented robust optimization method is fairly effective and efficient, the crashworthiness and robustness of the foam-filled thin-walled structure can be improved significantly.     

一类新型动态多目标鲁棒进化优化方法

传统动态多目标优化问题（Dynamic multi-objective optimization problems,DMOPs）的求解方法,通常需要在新环境下,通过重新激发寻优过程,获得适应该环境的Pareto最优解.这可能导致较高的计算代价和资源成本,甚至无法在有限时间内执行该优化解.由此,提出一类寻找动态鲁棒Pareto最优解集的进化优化方法.动态鲁棒Pareto解集是指某一时刻下的Pareto较优解可以以一定稳定性阈值,逼近未来多个连续动态环境下的真实前沿,从而直接作为这些环境下的Pareto解集,以减小计算代价.为合理度量Pareto解的环境适应性,给出了时间鲁棒性和性能鲁棒性定义,并将其转化为两类鲁棒优化模型.引入基于分解的多目标进化优化方法和无惩罚约束处理方法,构建了动态多目标分解鲁棒进化优化方法.特别是基于移动平均预测模型实现了未来动态环境下适应值的多维时间序列预测.基于提出的两类新型性能评价测度,针对8个典型动态测试函数的仿真实验,结果表明该方法得到满足决策者精度要求,且具有较长平均生存时间的动态鲁棒Pareto最优解.     

Robust and stable flow shop scheduling with unexpected arrivals of new jobs and uncertain processing times

In real scheduling problems, some disruptions and unexpected events may occur. These disruptions cause the initial schedule to quickly become infeasible and non-optimal. In this situation, an appropriate rescheduling method should be used. In this paper, a new approach has been proposed to achieve stable and robust schedule despite uncertain processing times and unexpected arrivals of new jobs. This approach is a proactive–reactive method which uses a two-step procedure. In the first step an initial robust solution is produced proactively against uncertain processing times using robust optimization approach. This initial robust solution is more insensitive against the fluctuations of processing times in future. In the next step, when an unexpected disruption occurs, an appropriate reactive method is adopted to deal with this unexpected event. In fact, in the second step, the reactive approach determines the best modified sequence after any unexpected disruption based on the classical objective and performance measures. The robustness measure is implemented in the reactive approach to increase the performance of the real schedule after disruption. Computational results indicate that this method produces better solutions in comparison with four classical heuristic approaches according to effectiveness and performance of solutions.     

Evolutionary robust optimization in production planning – interactions between number of objectives,sample size and choice of robustness measure

We aim to find robust solutions in optimization settings where there is uncertainty associated with the operating/environmental conditions, and the fitness of a solution is hence best described by a distribution of outcomes. In such settings, the nature of the fitness distribution (reflecting the performance of a particular solution across a set of operating scenarios) is of potential interest in deciding solution quality, and previous work has suggested the inclusion of robustness as an additional optimization objective. However, there has been limited investigation of different robustness criteria, and the impact this choice may have on the sample size needed to obtain reliable fitness estimates. Here, we investigate different single and multi-objective formulations for robust optimization, in the context of a real-world problem addressed via simulation-based optimization. For the (limited evaluation) setting considered, our results highlight the value of an explicit robustness criterion in steering an optimizer towards solutions that are not only robust (as may be expected), but also associated with a profit that is, on average, higher than that identified by standard single-objective approaches. We also observe significant interactions between the choice of robustness measure and the sample size employed during fitness evaluation, an effect that is more pronounced for our multi-objective models.     

Control of robust design in multiobjective optimization under uncertainties

In design and optimization problems, a solution is called robust if it is stable enough with respect to perturbation of model input parameters. In engineering design optimization, the designer may prefer a use of robust solution to a more optimal one to set a stable system design. Although in literature there is a handful of methods for obtaining such solutions, they do not provide a designer with a direct and systematic control over a required robustness. In this paper, a new approach to robust design in multiobjective optimization is introduced, which is able to generate robust design with model uncertainties. In addition, it introduces an opportunity to control the extent of robustness by designer preferences. The presented method is different from its other counterparts. For keeping robust design feasible, it does not change any constraint. Conversely, only a special tunable objective function is constructed to incorporate the preferences of the designer related to the robustness. The effectiveness of the method is tested on well known engineering design problems.     

Robust optimization with convex model considering bounded constraints on performance variation

In robust optimization, double-looped structures are often adopted where the outer loop is used to seek for the optimal design and the optimization performed in the inner loop is for the robustness assessment of the candidate solutions. However, the double-looped techniques usually will lead to a significant increase in computational efforts. Therefore, in this paper, a new robustness index is developed to handle bounded constraints on performance variation where no optimization run is required for the robustness evaluation work in the inner loop. The computation of this new index is based on the sensitivity Jacobian matrix of the system performances with respect to the uncertainties and it can quantitatively measure the maximal allowable magnitude of system variations. By introducing this index, the robust design problem can be reformulated as a deterministic optimization with robustness indices requirements. Two numerical examples are tested to show the effectiveness and efficiency of the proposed approach, whose solutions and computational efforts are compared to those from a double-looped approach proposed in previous literature.     

Evacuation Transportation Planning Under Uncertainty: A Robust Optimization Approach

This paper considers evacuation via surface transportation networks in an uncertain environment. We focus on demand uncertainty which can lead to significant infeasibility cost during evacuation, where loss of life or property may appear. We develop a robust linear programming model based on a robust optimization approach where hard constraints are guaranteed within an appropriate uncertainty set. The robust counterpart solutions have been shown tractable. We show that the robustness in evacuation is important and a robust solution outperforms a nominal deterministic solution in both quality and feasibility.     

Robust Optimization Model for a Dynamic Network Design Problem Under Demand Uncertainty

This paper describes a robust optimization approach for a network design problem explicitly incorporating traffic dynamics and demand uncertainty. In particular, we consider a cell transmission model based network design problem of the linear programming type and use box uncertainty sets to characterize the demand uncertainty. The major contribution of this paper is to formulate such a robust network design problem as a tractable linear programming model and demonstrate the model robustness by comparing its solution performance with the nominal solution from the corresponding deterministic model. The results of the numerical experiments justify the modeling advantage of the robust optimization approach and provide useful managerial insights for enacting capacity expansion policies under demand uncertainty.     

A systematic method for network topology reconfiguration with limited link additions

As a promising approach to improve network survivability, reliability and flexibility, topology reconfiguration is extremely important for modern networked infrastructures. In particular, for an existing network and the limited link addition resources, it is valuable to determine how to optimally allocate the new link resources, such that the resulting network is the most robust and efficient. In this paper, we investigate the problem of network topology reconfiguration (NTR) optimization with limited link additions. A dynamic robustness metric is developed to quantitatively characterize the robust connectivity and the efficiency under either random or targeted attack. We show that the NTR optimization with limited link additions is NP-hard. Therefore, to approximately solve the problem, we develop a preferential configuration node-protecting cycle (PCNC) method for sequential link additions. Analysis showed that PCNC method provides an approximate optimal solution under the dynamic robustness metric when compared with the optimal solution found by exhaustive search. Simulation results also showed that PCNC method effectively improves the network robustness and communication efficiency at the cost of least added link resources.     

An H-minimax approach to the design of robust control systems, part II: All solutions, all-pass form solutions and the ‘best’ solution

We characterize all solutions to a robustness optimization problem as the solutions of a two-parameter interpolation problem. From this characterization it is easy to show that an all-pass form solution always exists as long as a solution exists. We also study the possibility of using non-all-pass form solutions and by introducing other optimization objectives (motivated by improvements in disturbance rejection and robust stability) we search for the 'best' solution.     

Reformulation based MaxSAT robustness

The presence of uncertainty in the real world makes robustness a desirable property of solutions to constraint satisfaction problems (CSP). A solution is said to be robust if it can be easily repaired when unexpected events happen. This issue has already been addressed in the frameworks of Boolean satisfiability (SAT) and Constraint Programming (CP). Most existing works on robustness implement search algorithms to look for robust solutions instead of taking the declarative approach of reformulation, since reformulation tends to generate prohibitively large formulas, especially in the CP setting. In this paper we consider the unaddressed problem of robustness in weighted MaxSAT, by showing how robust solutions to weighted MaxSAT instances can be effectively obtained via reformulation into pseudo-Boolean formulae. Our encoding provides a reasonable balance between increase in size and performance, as shown by our experiments in the robust resource allocation framework. We also address the problem of flexible robustness, where some of the breakages may be left unrepaired if a totally robust solution does not exist. In a sense, since the use of SAT and MaxSAT encodings for solving CSP has been gaining wide acceptance in recent years, we provide an easy-to-implement new method for achieving robustness in combinatorial optimization problems.     

Crashworthiness design of vehicle by using multiobjective robust optimization

Although deterministic optimization has to a considerable extent been successfully applied in various crashworthiness designs to improve passenger safety and reduce vehicle cost, the design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on multiple crashworthiness criteria, where several different sigma criteria are adopted to measure the variations. As an example, a full front impact of vehicle is considered with increase in energy absorption and reduction of structural weight as the design objectives, and peak deceleration as the constraint. A multiobjective particle swarm optimization is applied to generate robust Pareto solution, which no longer requires formulating a single cost function by using weighting factors or other means. From the example, a clear compromise between the Pareto deterministic and robust designs can be observed. The results demonstrate the advantages of using multiobjective robust optimization, with not only the increase in the energy absorption and decrease in structural weight from a baseline design, but also a significant improvement in the robustness of optimum.     

Robust optimization to secure urban bulk water supply against extreme drought and uncertain climate change

Urban bulk water systems supply water with high reliability and, in the event of extreme drought, must avoid catastrophic economic and social collapse. In view of the deep uncertainty about future climate change, it is vital that robust solutions be found that secure urban bulk water systems against extreme drought. To tackle this challenge an approach was developed integrating: 1) a stochastic model of multi-site streamflow conditioned on future climate change scenarios; 2) Monte Carlo simulation of the urban bulk water system incorporated into a robust optimization framework and solved using a multi-objective evolutionary algorithm; and 3) a comprehensive decision space including operating rules, investment in new sources and source substitution and a drought contingency plan with multiple actions with increasingly severe economic and social impact. A case study demonstrated the feasibility of this approach for a complex urban bulk water supply system. The primary objective was to minimize the expected present worth cost arising from infrastructure investment, system operation and the social cost of “normal” and emergency restrictions. By introducing a second objective which minimizes either the difference in present worth cost between the driest and wettest future climate change scenarios or the present worth cost for driest climate scenario, the trade-off between efficiency and robustness was identified. The results show that a significant change in investment and operating strategy can occur when the decision maker expresses a stronger preference for robustness and that this depends on the adopted robustness measure. Moreover, solutions are not only impacted by the degree of uncertainty about future climate change but also by the stress imposed on the system and the range of available options.     

Generating robust and flexible job shop schedules using genetic algorithms

The problem of finding robust or flexible solutions for scheduling problems is of utmost importance for real-world applications as they operate in dynamic environments. In such environments, it is often necessary to reschedule an existing plan due to failures (e.g., machine breakdowns, sickness of employees, deliveries getting delayed, etc.). Thus, a robust or flexible solution may be more valuable than an optimal solution that does not allow easy modifications. This paper considers the issue of robust and flexible solutions for job shop scheduling problems. A robustness measure is defined and its properties are investigated. Through experiments, it is shown that using a genetic algorithm it is possible to find robust and flexible schedules with a low makespan. These schedules are demonstrated to perform significantly better in rescheduling after a breakdown than ordinary schedules. The rescheduling performance of the schedules generated by minimizing the robustness measure is compared with the performance of another robust scheduling method taken from literature, and found to outperform this method in many cases.     

Evaluating solution sets of <Emphasis Type="Italic">a posteriori</Emphasis> solution techniques for bi-criteria combinatorial optimization problems

The quality of an approximate solution for combinatorial optimization problems with a single objective can be evaluated relatively easily. However, this becomes more difficult when there are multiple objectives. One potential approach to solving multiple criteria combinatorial optimization problems when at least one of the single objective problems is NP-complete, is to use an a posteriori method that approximates the efficient frontier. A common difficulty in this type of approach, however, is evaluating the quality of approximate solutions, since sets of multiple solutions should be evaluated and compared. This necessitates the use of a comparison measure that is robust and accurate. Furthermore, a robust measure plays an important role in metaheuristic optimization for tuning various parameters for evolutionary algorithms, simulated annealing, etc., which are frequently employed for multiple criteria combinatorial optimization problems. In this paper, the performance of a new measure, which we call Integrated Convex Preference (ICP) is compared to that of other measures appearing in the literature through numerical experiments—specifically, we use two a posteriori solution techniques based on genetic algorithms for a bi-criteria parallel machine scheduling problem and evaluate their performance (in terms of solution quality) using different measures. Experimental results show that the ICP measure evaluates the solution quality of approximations robustly (i.e., similar to visual comparison results) while other alternative measures can misjudge the solution quality. We note that the ICP measure can be applied to other non-scheduling multiple objective combinatorial optimization problems, as well.     

A note on robustness tolerances for combinatorial optimization problems

We consider so-called generic combinatorial optimization problem, where the set of feasible solutions is some family of nonempty subsets of a finite ground set with specified positive initial weights of elements, and the objective function represents the total weight of elements of the feasible solution. We assume that the set of feasible solutions is fixed, but the weights of elements may be perturbed or are given with errors. All possible realizations of weights form the set of scenarios.A feasible solution, which for a given set of scenarios guarantees the minimum value of the worst-case relative regret among all the feasible solutions, is called a robust solution. The maximum percentage perturbation of a single weight, which does not destroy the robustness of a given solution, is called the robustness tolerance of this weight with respect to the solution considered.In this paper we give formulae for computing the robustness tolerances with respect to an optimal solution obtained for some initial weights and we show that this can be done in polynomial time whenever the optimization problem is polynomially solvable itself.     

Sequential robust optimization of a V-bending process using numerical simulations

The coupling of finite element simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a generally applicable strategy for modeling and efficiently solving robust optimization problems based on time consuming simulations. Noise variables and their effect on the responses are taken into account explicitly. The robust optimization strategy consists of four main stages: modeling, sensitivity analysis, robust optimization and sequential robust optimization. Use is made of a metamodel-based optimization approach to couple the computationally expensive finite element simulations with the robust optimization procedure. The initial metamodel approximation will only serve to find a first estimate of the robust optimum. Sequential optimization steps are subsequently applied to efficiently increase the accuracy of the response prediction at regions of interest containing the optimal robust design. The applicability of the proposed robust optimization strategy is demonstrated by the sequential robust optimization of an analytical test function and an industrial V-bending process. For the industrial application, several production trial runs have been performed to investigate and validate the robustness of the production process. For both applications, it is shown that the robust optimization strategy accounts for the effect of different sources of uncertainty onto the process responses in a very efficient manner. Moreover, application of the methodology to the industrial V-bending process results in valuable process insights and an improved robust process design.     

基于响应面和支持向量机的产品健壮设计方法

提出一种基于集成替代模型的产品健壮设计方法．该方法采用响应面和支持向量机来逼近产品质量特性与其影响因素的关系模型,保证了在有限样本条件下的建模效率和精度．为了提高产品的抗干扰能力和可靠性,在同时考虑设计目标的健壮性和设计约束的可行健壮性的基础上,建立了基于分位数的产品健壮设计优化模型．最后通过太阳能灌溉系统的健壮设计实例,验证了该方法的有效性和实用性．     

Bad-scenario-set Robust Optimization Framework With Two Objectives for Uncertain Scheduling Systems

This paper proposes a robust optimization framework generally for scheduling systems subject to uncertain input data, which is described by discrete scenarios. The goal of robust optimization is to hedge against the risk of system performance degradation on a set of bad scenarios while maintaining an excellent expected system performance. The robustness is evaluated by a penalty function on the bad-scenario set. The bad-scenario set is identified for current solution by a threshold, which is restricted on a reasonable-value interval. The robust optimization framework is formulated by an optimization problem with two conflicting objectives. One objective is to minimize the reasonable value of threshold, and another is to minimize the measured penalty on the bad-scenario set. An approximate solution framework with two dependent stages is developed to surrogate the biobjective robust optimization problem. The approximation degree of the surrogate framework is analyzed. Finally, the proposed bad-scenario-set robust optimization framework is applied to a scenario job-shop scheduling system. An extensive computational experiment was conducted to demonstrate the effectiveness and the approximation degree of the framework. The computational results testified that the robust optimization framework can provide multiple selections of robust solutions for the decision maker. The robust scheduling framework studied in this paper can provide a unique paradigm for formulating and solving robust discrete optimization problems.      

An efficient procedure for structural reliability-based robust design optimization

This paper focuses on the development of an optimization tool with the aim to obtain robust and reliable designs in short computational time. The robustness measures considered here are the expected value and standard deviation of the performance function involved in the optimization problem. When using these robustness measures combined, the search of optimal design appears as a robust multiobjective optimization (RMO) problem. Reliable design addresses uncertainties to restrict the structural probability of failure. The mathematical formulation for the reliability based robust design optimization (RBRDO) problem is obtained by adding a reliability based constraint into the RMO problem. As both, statistics calculations and the reliability analysis could be very costly, approximation technique based on reduced-order modeling (ROM) is also incorporated in our procedure. The selected ROM is the proper orthogonal decomposition (POD) method, with the aim to produce fast outputs considering structural non-linear behavior. Moreover, to obtain RBRDO designs with reduced CPU time we propose others developments to be added in the integrated tool. They are: Probabilistic Collocation Method (PCM) to evaluate the statistics of the structural responses and, also, an approximated reliability constraints procedure based on the Performance Measure Approach (PMA) for reliability constraint assessment. Finally, Normal-Boundary Intersection (NBI) or Normalized Normal-Constraint (NNC) multiobjective optimization techniques are employed to obtain fast and even spread Pareto robust designs. To illustrate the application of the proposed tool, optimization studies are conducted for a linear (benchmark) and nonlinear trusses problems. The nonlinear example consider different loads level, exploring the material plasticity. The integrated tool prove to be very effective reducing the computational time by up to five orders of magnitude, when compared to the solutions obtained via classical standard approaches.