Distributionally robust optimization for engineering design under uncertainty

This paper addresses the challenge of design optimization under uncertainty when the designer only has limited data to characterize uncertain variables. We demonstrate that the error incurred when estimating a probability distribution from limited data affects the out-of-sample performance (ie, performance under the true distribution) of optimized designs. We demonstrate how this can be mitigated by reformulating the engineering design problem as a distributionally robust optimization (DRO) problem. We present computationally efficient algorithms for solving the resulting DRO problem. The performance of the DRO approach is explored in a practical setting by applying it to an acoustic horn design problem. The DRO approach is compared against traditional approaches to optimization under uncertainty, namely, sample-average approximation and multiobjective optimization incorporating a risk reduction objective. In contrast with the multiobjective approach, the proposed DRO approach does not use an explicit risk reduction objective but rather specifies a so-called ambiguity set of possible distributions and optimizes against the worst-case distribution in this set. Our results show that the DRO designs, in some cases, significantly outperform those designs found using the sample-average or the multiobjective approach.     

Robust closed-loop supply chain network design for perishable goods in agile manufacturing under uncertainty

In this study, a general comprehensive model is proposed for strategic closed-loop supply chain network design under interval data uncertainty. The proposed model considers various assumptions such as multiple periods, multiple products, and multiple supply chain echelons as well as uncertain demand and purchasing cost. In addition, bill of materials for each product is considered via a new approach in management of forward and reverse flows of products for producing new products and reusing or disassembling returned products. Uncertainty of parameters in the proposed model is handled via an interval robust optimisation technique. The model assumptions are well matched with decision making environments of food and high-tech electronics manufacturing industries. The factors that make these two industries similar are time-dependent properties of products such as prices and warehousing lifetime period. The computational results of solving the proposed model via LINGO 8 demonstrate efficiency of the proposed model in dealing with uncertainty in an agile manufacturing context.     

Multifidelity approaches for optimization under uncertainty

It is important to design robust and reliable systems by accounting for uncertainty and variability in the design process. However, performing optimization in this setting can be computationally expensive, requiring many evaluations of the numerical model to compute statistics of the system performance at every optimization iteration. This paper proposes a multifidelity approach to optimization under uncertainty that makes use of inexpensive, low‐fidelity models to provide approximate information about the expensive, high‐fidelity model. The multifidelity estimator is developed based on the control variate method to reduce the computational cost of achieving a specified mean square error in the statistic estimate. The method optimally allocates the computational load between the two models based on their relative evaluation cost and the strength of the correlation between them. This paper also develops an information reuse estimator that exploits the autocorrelation structure of the high‐fidelity model in the design space to reduce the cost of repeatedly estimating statistics during the course of optimization. Finally, a combined estimator incorporates the features of both the multifidelity estimator and the information reuse estimator. The methods demonstrate 90% computational savings in an acoustic horn robust optimization example and practical design turnaround time in a robust wing optimization problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Computing solution spaces for robust design

In some engineering problems, tolerance to variation of design parameters is essential. In an early development phase of a distributed development process for example, the system performance should reach the design goal even under large variations of uncertain component properties. The tolerance to parameter variations may be measured by the size of a solution space on which the system is guaranteed to deliver the required performance. In order to decouple dimensions, the solution space is described as multi‐dimensional box with permissible intervals for each design parameter. An algorithm is presented that computes solution spaces for arbitrary non‐linear high‐dimensional systems. Starting from a design point with required performance, a candidate box is iteratively evaluated and modified. The evaluation is performed by Monte Carlo sampling and Bayesian statistics. The modification algorithm drives the evolution toward increasing box size. Robustness and reliability with respect to the required performance can be assessed without knowledge of the particular kind of uncertainty. Sensitivity to design parameters can be quantified by the widths of solution intervals. Designs failing to meet the performance requirement can be improved by adjusting parameter values to lie within the solution space. The approach is motivated and illustrated by automotive crash design problems. Copyright © 2012 John Wiley & Sons, Ltd.     

An interval approach to robust design with parameter uncertainty

In robust design, it is common to estimate empirical models that relate an output response variable to controllable input variables and uncontrollable noise variables from experimental data. However, when determining the optimal input settings that minimise output variability, parameter uncertainties in noise factors and response models are typically neglected. This article presents an interval robust design approach that takes parameter uncertainties into account through the confidence regions for these unknown parameters. To avoid obtaining an overly conservative design, the worst and best cases of mean squared error are both adopted to build an optimisation approach. The midpoint and radius of the interval are used to measure the location and dispersion performances, respectively. Meanwhile, a data-driven method is applied to obtain the relative weights of the location and dispersion performances in the optimisation approach. A simulation example and a case study using automobile manufacturing data from the dimensional tolerance design process are used to demonstrate the effectiveness of the proposed approach. The proposed approach of considering both uncertainties is shown to perform better than other approaches.     

A robust optimization methodology for preliminary aircraft design

This article focuses on a robust optimization of an aircraft preliminary design under operational constraints. According to engineers' know-how, the aircraft preliminary design problem can be modelled as an uncertain optimization problem whose objective (the cost or the fuel consumption) is almost affine, and whose constraints are convex. It is shown that this uncertain optimization problem can be approximated in a conservative manner by an uncertain linear optimization program, which enables the use of the techniques of robust linear programming of Ben-Tal, El Ghaoui, and Nemirovski [Robust Optimization, Princeton University Press, 2009]. This methodology is then applied to two real cases of aircraft design and numerical results are presented.     

A bi-objective robust model for emergency resource allocation under uncertainty

Emergency resource allocation constitutes one of the most critical elements of response operations in the field of emergency management. This paper addresses an emergency resource allocation problem which involves multiple competing affected areas and one relief resource centre under supply shortage and uncertainty in the post-disaster phase. In humanitarian situations, both the efficiency and fairness of an allocation policy have a considerable influence on the effectiveness of emergency response operations. Thus, we formulate a bi-objective robust emergency resource allocation (BRERA) model which tries to maximise efficiency as well as fairness under different sources of uncertainties. To obtain decision-makers’ most preferred allocation policy, we propose a novel emergency resource allocation decision method which consists of three steps: (1) develop a bi-objective heuristic particle swarm optimisation algorithm to search the Pareto frontier of the BRERA model; (2) select a coefficient to measure fairness; and (3) establish a decision method based on decision-makers’ preference restricted by the fairness coefficient. Finally, a real case study taken from the 5 December 2008 Wenchuan Earthquake demonstrates the effectiveness of the proposed method through numerical results. The solution and model robustness are also analysed.     

Decision making with epistemic uncertainty under safety constraints: An application to seismic design

The problem of accounting for epistemic uncertainty in risk management decisions is conceptually straightforward, but is riddled with practical difficulties. Simple approximations are often used whereby future variations in epistemic uncertainty are ignored or worst-case scenarios are postulated. These strategies tend to produce sub-optimal decisions. We develop a general framework based on Bayesian decision theory and exemplify it for the case of seismic design of buildings. When temporal fluctuations of the epistemic uncertainties and regulatory safety constraints are included, the optimal level of seismic protection exceeds the normative level at the time of construction. Optimal Bayesian decisions do not depend on the aleatory or epistemic nature of the uncertainties, but only on the total (epistemic plus aleatory) uncertainty and how that total uncertainty varies randomly during the lifetime of the project.     

Interval-parameter robust quadratic programming for water quality management under uncertainty

Effective planning of water quality management is important for facilitating sustainable socio-economic development in watershed systems. An interval-parameter robust quadratic programming (IRQP) method is developed by incorporating techniques of robust programming and interval quadratic programming within a general optimization framework. The IRQP improves upon existing quadratic programming methods, and can tackle uncertainties presented as interval numbers and fuzzy sets as well as their combinations. Moreover, it can deal with nonlinearities in the objective function such that economies-of-scale effects can be reflected. The developed method is applied to a case study of a water quality management under uncertainty. A number of decision alternatives are generated based on the interval solutions as well as the projected applicable conditions. They represent multiple decision options with various environmental and economic considerations. Willingness to accept a low economic revenue will guarantee satisfying the water quality requirements. A strong desire to acquire a high benefit will run the risk of violating environmental criteria.     

An enhanced fuzzy robust optimization model for regional solid waste management under uncertainty

In this study, an enhanced fuzzy robust optimization (EFRO) model is proposed for supporting regional solid waste management under uncertainty. This model is an extended version of robust optimization from a stochastic to a fuzzy environment, and novel in the following two aspects: (1) it uses multiple algorithms to tackle fuzzy constraints according to their characteristics; and (2) it incorporates fuzzy violation variables into the model, which could effectively reflect the trade-off between system economy and reliability. The regional waste management of the City of Dalian, China, was used as a case study for demonstration. A variety of solutions was obtained under various weight coefficients and confidence levels. From the case study, it was found that EFRO could help decision makers to design desired waste management alternatives under complex uncertainties. The successful application of EFRO in the studied real case is expected to be a good example for solid waste management in many other cities.     

A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach

This article proposes a multi-objective mixed-integer model to optimize the location of hubs within a hub network design problem under uncertainty. The considered objectives include minimizing the maximum accumulated travel time, minimizing the total costs including transportation, fuel consumption and greenhouse emissions costs, and finally maximizing the minimum service reliability. In the proposed model, it is assumed that for connecting two nodes, there are several types of arc in which their capacity, transportation mode, travel time, and transportation and construction costs are different. Moreover, in this model, determining the capacity of the hubs is part of the decision-making procedure and balancing requirements are imposed on the network. To solve the model, a hybrid solution approach is utilized based on inexact programming, interval-valued fuzzy programming and rough interval programming. Furthermore, a hybrid multi-objective metaheuristic algorithm, namely multi-objective invasive weed optimization (MOIWO), is developed for the given problem. Finally, various computational experiments are carried out to assess the proposed model and solution approaches.     

Robust H 8 output feedback control for general nonlinear systems with structured uncertainty

Abstract

In this paper, the robust H ⁸ output feedback control problem for general nonlinear systems with L ₂‐norm‐bounded structured uncertainties is considered. Sufficient conditions for the solvability of robust performance synthesis problems are represented in terms of two Hamilton‐Jacobi inequalities with n independent variables. Based on these conditions, a state space characterization of a robust H ⁸ output feedback controller solving the considered problem is proposed. An example is provided for illustration.     

关于一类不确定非线性系统的鲁棒控制

对于一类同时具有匹配不确定性和非匹配不确定性的非线性系统,结合反馈线性化理论、矩阵不等式、李亚普诺夫稳定性理论和变结构控制,提出了一种鲁棒控制器的设计方法,使得闭环系统是渐近稳定的.     

I-VFRP: An interval-valued fuzzy robust programming approach for municipal waste-management planning under uncertainty

In this study, an interval-valued fuzzy robust programming (I-VFRP) model has been developed and applied to municipal solid-waste management under uncertainty. The I-VFRP model can explicitly address system uncertainties with multiple presentations, and can directly communicate the waste manager's confidence gradients into the optimization process, facilitating the reflection of weak or strong confidence when subjectively estimating parameter values. Parameters in the I-VFRP model can be represented as either intervals or interval-valued fuzzy sets. Thus, variations of the waste manager's confidence gradients over defining parameters can be effectively handled through interval-valued membership functions, leading to enhanced robustness of the optimization efforts. The results of a theoretical case study indicate that useful solutions for planning municipal solid-waste-management practices can be generated. The waste manager's confidence gradients over various subjective judgments can be directly incorporated into the modeling formulation and solution process. The results also suggest that the proposed methodology can be applied to practical problems that are associated with complex and uncertain information.     

A robust optimisation model for hybrid remanufacturing and manufacturing systems under uncertain return quality and market demand

In remanufacturing research, most researchers predominantly emphasised on the recovery of whole product (core) rather than at the component level due to its complexity. In contrast, this paper addresses the challenges to focus on remanufacturing through component recovery, so as to solve production planning problems of hybrid remanufacturing and manufacturing systems. To deal with the uncertainties of quality and quantity of product returns, the processing time of remanufacturing, remanufacturing costs, as well as market demands, a robust optimisation model was developed in this research and a case study was used to evaluate its effectiveness and efficiency. To strengthen this research, a sensitivity analysis of the uncertain parameters and the original equipment manufacturer’s (OEM’s) pricing strategy was also conducted. The research finding shows that the market demand volatility leads to a significant increase in the under fulfilment and a reduction in OEM’s profit. On the other hand, recovery cost reduction, as endogenous cost saving, encourages the OEM to produce more remanufactured products with the increase in market demand. Furthermore, the OEM may risk profit loss if they raise the price of new products, and inversely, they could gain more if the price of remanufactured products is raised.     

An efficient metamodeling approach for uncertainty quantification of complex systems with arbitrary parameter probability distributions

This paper proposes an efficient metamodeling approach for uncertainty quantification of complex system based on Gaussian process model (GPM). The proposed GPM‐based method is able to efficiently and accurately calculate the mean and variance of model outputs with uncertain parameters specified by arbitrary probability distributions. Because of the use of GPM, the closed form expressions of mean and variance can be derived by decomposing high‐dimensional integrals into one‐dimensional integrals. This paper details on how to efficiently compute the one‐dimensional integrals. When the parameters are either uniformly or normally distributed, the one‐dimensional integrals can be analytically evaluated, while when parameters do not follow normal or uniform distributions, this paper adopts the effective Gaussian quadrature technique for the fast computation of the one‐dimensional integrals. As a result, the developed GPM method is able to calculate mean and variance of model outputs in an efficient manner independent of parameter distributions. The proposed GPM method is applied to a collection of examples. And its accuracy and efficiency is compared with Monte Carlo simulation, which is used as benchmark solution. Results show that the proposed GPM method is feasible and reliable for efficient uncertainty quantification of complex systems in terms of the computational accuracy and efficiency. Copyright © 2016 John Wiley & Sons, Ltd.     

IFQP: A hybrid optimization method for filter management in fluid power systems under uncertainty

An interval-fuzzy quadratic programming (IFQP) method is developed for the assessment of filter allocation and replacement strategies in fluid power systems (FPS) under uncertainty. It can directly handle uncertainties expressed as interval values and/or fuzzy sets that exist in the left-hand and right-hand sides of constraints, as well as in the objective function. Multiple control variables are used to tackle independent uncertainties in the model's right-hand sides and thus optimize the overall satisfaction of the system performance. The IFQP method is applied to a case of planning filter allocation and replacement strategies under uncertainty for an FPS with a single circuit. A piecewise linearization approach is firstly employed to convert the nonlinear FPS problem into a linear one. The generated decision alternatives can help decision makers to identify desired policies for contamination control under various total costs, satisfaction degrees, and system-failure risks under different contaminant-ingression/generation rates.     

Robust optimization based on the Propagation of Variance method for analytic design models

The robust optimisation is performed in the preliminary design phase dealing with analytic models. The analytic models come either from the finite element models or from the physical laws approximation. The variability on the design parameters is defined using random variables identified by their first two Moments, the Mean and the Standard deviation. A robust design approach is proposed that determines whether a robust design solution exists or not to the given design problem. This approach combines a reformulation of the analytic model with the new design specifications. It integrates the parameter uncertainties (Mean and Standard deviation) and a deterministic optimisation algorithm (SQP algorithm). The Means and the Standard deviation are computed using the Propagation of Variance method. The engineering application of an electrical actuator design is introduced and used to show the implementation and the effectiveness of the proposed robust approach.     

Enhancing resilience assessment and bus bridging service design for large-scale rail transit systems under disruptions

This paper proposes a more computationally efficient approach for resilience assessment of rail transit system under disruptions. An improved linear programming model is developed to depict commuter flows and estimate system statuses. To address the computational challenge caused by the complexity of system, a four-step approach is proposed based on the proposed commuter flow model. In the first step, Origin-Destination (OD) pairs are divided into smaller groups and their flows under normal conditions are estimated by the proposed model separately, with the assumption that the railway capacity is sufficient relative to demand. Next, overall system statuses under normal conditions, including commuters on each train and spare capacities of each train are calculated by integrating results obtained in the first step. In the third step, system statuses under disruptions are estimated. In this step, we assume that unaffected commuters will not change their routes and flows of all affected commuters are estimated by a modified commuter model with given spare space of trains. Based on these outputs, several critical measures are introduced and calculated to quantify the resilience, resistance, and recovery ability of rail network systematically. We also demonstrate how our approach could be used to facilitate design and evaluation of bus bridging service. The proposed approach is demonstrated on the core part of Hangzhou rail transit network.     

Sequential optimization with particle splitting-based reliability assessment for engineering design under uncertainties

The evaluation of probabilistic constraints plays an important role in reliability-based design optimization. Traditional simulation methods such as Monte Carlo simulation can provide highly accurate results, but they are often computationally intensive to implement. To improve the computational efficiency of the Monte Carlo method, this article proposes a particle splitting approach, a rare-event simulation technique that evaluates probabilistic constraints. The particle splitting-based reliability assessment is integrated into the iterative steps of design optimization. The proposed method provides an enhancement of subset simulation by increasing sample diversity and producing a stable solution. This method is further extended to address the problem with multiple probabilistic constraints. The performance of the particle splitting approach is compared with the most probable point based method and other approximation methods through examples.