Development of a multidisciplinary–multiresponse robust design optimization model

Robust design is an efficient process improvement methodology that combines experimentation with optimization to create systems that are tolerant to uncontrollable variation. Most traditional robust design models, however, consider only a single quality characteristic, yet customers judge products simultaneously on a variety of scales. Additionally, it is often the case that these quality characteristics are not of the same type. To addresses these issues, a new robust design optimization model is proposed to solve design problems involving multiple responses of several different types. In this new approach, noise factors are incorporated into the robust design model using a combined array design, and the results of the experiment are optimized using a new approach that is formulated as a nonlinear goal programming problem. The results obtained from the proposed methodology are compared with those of other robust design methods in order to examine the trade-offs between meeting the objectives associated with different optimization approaches.     

Batch sequential design to achieve predictive maturity with calibrated computer models

Sequential experiment design strategies have been proposed for efficiently augmenting initial designs to solve many problems of interest to computer experimenters, including optimization, contour and threshold estimation, and global prediction. We focus on batch sequential design strategies for achieving maturity in global prediction of discrepancy inferred from computer model calibration. Predictive maturity focuses on adding field experiments to efficiently improve discrepancy inference. Several design criteria are extended to allow batch augmentation, including integrated and maximum mean square error, maximum entropy, and two expected improvement criteria. In addition, batch versions of maximin distance and weighted distance criteria are developed. Two batch optimization algorithms are considered: modified Fedorov exchange and a binning methodology motivated by optimizing augmented fractional factorial skeleton designs.     

Response surface-based robust parameter design optimization with both qualitative and quantitative variables

The response surface-based robust parameter design, with its extensive use of optimization techniques and statistical tools, is known as an effective engineering design methodology for improving production processes, when input variables are quantitative on a continuous scale. In many engineering settings, however, there are situations where both qualitative and quantitative variables are considered. In such situations, traditional response surface designs may not be effective. To rectify this problem, this article lays out a foundation by embedding those input variables into a factorial design with pseudo-centre points. A 0–1 mixed-integer nonlinear programming model is then developed and the solutions found using three optimization tools, namely the outer approximation method, the branch-and-bound technique and the hybrid branch-and-cut algorithm, are compared with traditional counterparts. The numerical example shows that the proposed models result in better robust parameter design solutions than the traditional models.     

Stochastic analysis and robust optimization for a deck lid inner panel stamping

FE-simulation and optimization are widely used in the stamping process to improve design quality and shorten development cycle. However, the current simulation and optimization may lead to non-robust results due to not considering the variation of material and process parameters. In this study, a novel stochastic analysis and robust optimization approach is proposed to improve the stamping robustness, where the uncertainties are involved to reflect manufacturing reality. A meta-model based stochastic analysis method is developed, where FE-simulation, uniform design and response surface methodology (RSM) are used to construct meta-model, based on which Monte-Carlo simulation is performed to predict the influence of input parameters variation on the final product quality. By applying the stochastic analysis, uniform design and RSM, the mean and the standard deviation (SD) of product quality are calculated as functions of the controllable process parameters. The robust optimization model composed of mean and SD is constructed and solved, the result of which is compared with the deterministic one to show its advantages. It is demonstrated that the product quality variations are reduced significantly, and quality targets (reject rate) are achieved under the robust optimal solution. The developed approach offers rapid and reliable results for engineers to deal with potential stamping problems during the early phase of product and tooling design, saving more time and resources.     

Characterization of sequential optimization strategies in concurrent product and process design: A dual-team approach

In team-based design optimization, one type of workflow is relevant to sequential design decision-making; that is, one team's design decision goes after another team's in an alternate fashion. However, the strategies in use for sequential optimization significantly affect the final design solutions. Conventionally, the over-the-wall strategy and the Stackelberg solution strategy from game theory are extensively used for sequential optimization. In this paper, these strategies are extended to cover more design case scenarios in sequential optimization. A dual-team approach is presented to model, in particular, concurrent product and process design (CPPD) using a bi-objective optimization formalism in which a team acts as a decision maker towards a design objective. By differentiating the role of a team in sequential optimization, a set of sequential optimization strategies is provided for CPPD applications. Four CPPD models are accordingly derived to account for four case scenarios in CPPD. The implementation-related algorithms are also presented, along with an example illustration, to support computational design executions.     

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

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

A robust optimization approach to overall profit efficiency with data uncertainty: application on bank industry

In this paper, we consider a framework of data envelopment analysis (DEA) to measure the overall profit efficiency of decision-making units (DMUs) subject to inputs and outputs uncertainty. Under uncertain conditions, classic methods can lead to unrealistic solutions in practice. In this work, robust optimization is proposed to incorporate uncertainty into measuring the overall profit efficiency. In a robust optimization model, it is supposed that uncertain parameters belong to a specified set with a solution that is efficient for all possible uncertainty outcomes while it is not optimal for a given value of the parameters. We show that the overall profit efficiency score may not always occur in an optimistic case and the decision maker can obtain the overall profit efficiency score corresponding to a value in the uncertainty set. The results of the experiment on bank data show that a robust overall profit efficiency score provides a significant improvement in the performance, as the uncertainty increases.

Abbreviations: DEA: data envelopment analysis; DMUs: decision-making units; CRS: constant returns to scale; VRS: variable returns to scale; ROP: robust optimization problem; RC: robust counterpart; ROPE: robust overall profit efficiency; OOPE: optimistic overall profit efficiency; GAMS: generalized algebraic modeling system     

A new method for estimating missing values for a genetic algorithm used in robust design

This article presents an improved genetic algorithm (GA), which finds solutions to problems of robust design in multivariate systems with many control and noise factors. Since some values of responses of the system might not have been obtained from the robust design experiment, but may be needed in the search process, the GA uses response surface methodology (RSM) to estimate those values. In all test cases, the GA delivered solutions that adequately adjusted the mean of the responses to their corresponding target values and with low variability. The GA found more solutions than the previous versions of the GA, which makes it easier to find a solution that may meet the trade-off among variance reduction, mean adjustment and economic considerations. Moreover, RSM is a good method for estimating the mean and variance of the outputs of highly non-linear systems, which makes the new GA appropriate for optimizing such systems.     

Sequential algorithms for structural design optimization under tolerance conditions

A deterministic optimization usually ignores the effects of uncertainties in design variables or design parameters on the constraints. In practical applications, it is required that the optimum solution can endure some tolerance so that the constraints are still satisfied when the solution undergoes variations within the tolerance range. An optimization problem under tolerance conditions is formulated in this article. It is a kind of robust design and a special case of a generalized semi-infinite programming (GSIP) problem. To overcome the deficiency of directly solving the double loop optimization, two sequential algorithms are then proposed for obtaining the solution, i.e. the double loop optimization is solved by a sequence of cycles. In each cycle a deterministic optimization and a worst case analysis are performed in succession. In sequential algorithm 1 (SA1), a shifting factor is introduced to adjust the feasible region in the next cycle, while in sequential algorithm 2 (SA2), the shifting factor is replaced by a shifting vector. Several examples are presented to demonstrate the efficiency of the proposed methods. An optimal design result based on the presented method can endure certain variation of design variables without violating the constraints. For GSIP, it is shown that SA1 can obtain a solution with equivalent accuracy and efficiency to a local reduction method (LRM). Nevertheless, the LRM is not applicable to the tolerance design problem studied in this article.     

An improvement of Kriging based sequential approximate optimization method via extended use of design of experiments

When Kriging is used as a meta-model for an inequality constrained function, approximate optimal solutions are sometimes infeasible in the case where they are active at the constraint boundary. This article explores the development of a Kriging-based meta-model that enhances the constraint feasibility of an approximate optimal solution. The trust region management scheme is used to ensure the convergence of the approximate optimal solution. The present study proposes a method of enhancing the constraint feasibility in which the currently infeasible design is replaced by the most feasible-usable design during the sequential approximate optimization process. An additional convergence condition is also included to reinforce the design accuracy and feasibility. Latin hypercube design and (2n+1) design are used as tools for design of experiments. The proposed approach is verified through a constrained mathematical function problem and a number of engineering optimization problems to support the proposed strategies.     

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.     

A comparison between different robust optimum design approaches: Application to tuned mass dampers

This paper is focused on the comparison between different approaches in structural optimization. More precisely, the conventional deterministic optimum design, based on the assumption that the only source of uncertainty concerns the forcing input, is compared to robust single-objective and multi-objective optimum design methods.The analysis is developed by considering as case of study a single-degree-of-freedom system with uncertain parameters, subject to random vibrations and equipped with a tuned mass damper device (TMD). The optimization problem concerns the selection of TMD mechanical characteristics able to enlarge the efficiency of the strategy of vibration reduction.Results demonstrate the importance of performing a robust optimum design and show that the multi-objective robust design methodology provides a significant improvement in performance stability, giving a better control of the design solution choice.     

Multiobjective robust optimization method for drawbead design in sheet metal forming

It is recognized that fracture and wrinkling in sheet metal forming can be eliminated via an appropriate drawbead design. Although deterministic multiobjective optimization algorithms and finite element analysis (FEA) have been applied in this respect to improve formability and shorten design cycle, the design could become less meaningful or even unacceptable when considering practical variation in design variables and noises of system parameters. To tackle this problem, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on drawbead design, where the six sigma principle is adopted to measure the variations, a dual response surface method is used to construct surrogate model and a multiobjective particle swarm optimization is developed to generate robust Pareto solutions. In this paper, the procedure of drawbead design is divided into two stages: firstly, equivalent drawbead restraining forces (DBRF) are obtained by developing a multiobjective robust particle swarm optimization, and secondly the DBRF model is integrated into a single-objective particle swarm optimization (PSO) to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust. Thus, the presented method provides an effective solution to geometric design of drawbead for improving product quality.     

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.     

Strategies for planning experiments using orthogonal arrays and confounding tables

Genichi Taguchi has popularized a robust design method which employs experimental design techniques to help identify the levels of design factors to improve the quality of products and manufacturing processes. Experimental design techniques are extremely effective for identifying improved factor levels in problems that involve a large number of factors. Taguchi's success in getting engineers to use experimental design techniques is due, at least in large part, to his use of tools and techniques that simplify the experiment planning process. Recognizing the advantages of this approach, this paper proposes a new set of tools, confounding tables, which offer more guidance to experimenters. Confounding tables provide a clear and systematic representation of confounding relationships. They are simple and useful tools for constructing experiment plans, and they enable users easily to evaluate the confounding patterns of a completed plan. We show how confounding tables provide more information than Taguchi's linear graphs, and are useful for a large class of experiment plans.     

Multi-objective robust design of energy-absorbing components using coupled process–performance simulations

The stochastic uncertainties associated with the material, process and product are represented and propagated to process and performance responses. A finite element-based sequential coupled process–performance framework is used to simulate the forming and energy absorption responses of a thin-walled tube in a manner that both material properties and component geometry can evolve from one stage to the next for better prediction of the structural performance measures. Metamodelling techniques are used to develop surrogate models for manufacturing and performance responses. One set of metamodels relates the responses to the random variables whereas the other relates the mean and standard deviation of the responses to the selected design variables. A multi-objective robust design optimization problem is formulated and solved to illustrate the methodology and the influence of uncertainties on manufacturability and energy absorption of a metallic double-hat tube. The results are compared with those of deterministic and augmented robust optimization problems.     

An evolutionary algorithm to generate alternatives (EAGA) for engineering optimization problems

Typically for a real optimization problem, the optimal solution to a mathematical model of that real problem may not always be the ‘best’ solution when considering unmodeled or unquantified objectives during decision-making. Formal approaches to explore efficiently for good but maximally different alternative solutions have been established in the operations research literature, and have been shown to be valuable in identifying solutions that perform expectedly well with respect to modeled and unmodeled objectives. While the use of evolutionary algorithms (EAs) to solve real engineering optimization problems is becoming increasingly common, systematic alternatives-generation capabilities are not fully extended for EAs. This paper presents a new EA-based approach to generate alternatives (EAGA), and illustrates its applicability via two test problems. A realistic airline route network design problem was also solved and analyzed successfully using EAGA. The EAGA promises to be a flexible procedure for exploring alternative solutions that could assist when making decisions for real engineering optimization problems riddled with unmodeled or unquantified issues.     

考虑双目标的JIT生产系统看板数量设定

针对现有JIT系统看板数量决策问题研究多以单目标为主的不足,提出了一种基于实验设计的双目标JIT生产系统看板数量设定方法。该方法同时考虑了高订单满足率和低系统平均在制品水平的双目标优化,以B公司CR油嘴JIT生产系统为例,建立了该JIT生产线的Witness仿真模型以实现数据的收集,以各看板循环回路的看板数量和看板容量进行水平设定,并进行正交实验设计及数据的直观分析处理,然后采用全因子实验的方法,基于帕累托最优的思想获得生产系统看板数量帕累托最优解,形成近似最优看板数量组合的帕累托最优前沿。生产管理人员可根据不同的生产计划和绩效目标从组合中选择合适的看板数量。最后的研究结果验证了该方法的可行性和有效性。     

An Efficient Adaptive Sequential Methodology for Expensive Response Surface Optimization

The preset response surface methodology (RSM) designs are commonly used in a wide range of process and design optimization applications. Although they offer ease of implementation and good performance, they are not sufficiently adaptive to reduce the required number of experiments and thus are not cost effective for applications with high cost of experimentation. We propose an efficient adaptive sequential methodology based on optimal design and experiments ranking for response surface optimization (O‐ASRSM) for industrial experiments with high experimentation cost, limited experimental resources, and requiring high design optimization performance. The proposed approach combines the concepts from optimal design of experiments, nonlinear optimization, and RSM. By using the information gained from the previous experiments, O‐ASRSM designs the subsequent experiment by simultaneously reducing the region of interest and by identifying factor combinations for new experiments. Given a given response target, O‐ASRSM identifies the input factor combination in less number of experiments than the classical single‐shot RSM designs. We conducted extensive simulated experiments involving quadratic and nonlinear response functions. The results show that the O‐ASRSM method outperforms the popular central composite design, the Box–Behnken design, and the optimal designs and is competitive with other sequential response surface methods in the literature. Furthermore, results indicate that O‐ASRSM's performance is robust with respect to the increasing number of factors. Copyright © 2012 John Wiley & Sons, Ltd.     

Thermoeconomic optimization of a refrigeration plant

In this paper, thermoeconomic theory is applied to the economic optimization of a conventional refrigeration plant, aimed at minimizing its overall operation and amortization cost.Thermal systems cannot always be optimized by means of mathematical or numerical techniques, because a complete model of the plant is not always available, and, in any case, mathematical difficulties are often great, even for not particularly complex systems, and the help of computerized algorithms is needed.In this paper, a simplified cost minimization methodology is applied, based on the so-called Theory of Exergetic Cost, here utilized to evaluate the economic costs of all the internal flows and products of the installation. As shown in the paper, once these costs have been calculated, a design configuration not far from the real global optimum can be obtained by means of a sequential, local optimization of the system, carried out unit by unit, that is, breaking down the global problem into a sequence of simpler problems.In the paper, the case of a very simple plant is considered to develop a numerical example, and, in spite of the approximations introduced to simplify the optimization procedure, the results obtained show acceptable accuracy when compared with those provided by a conventional and more complex optimization methodology.