Using geometric mean to compute robust mixture designs

Mixture experiments involve developing a dedicated formulation for specific applications. We propose the weighted optimality criterion using the geometric mean as the objective function for the genetic algorithms. We generate a robust mixture design using genetic algorithms (GAs) of which the region of interest is an irregularly shaped polyhedral region formed by constraints on proportions of the mixture component. When specific terms in the initial model display unimportant effects, it is assumed that they are removed. The design generation objective requires model robustness across the set of the reduced models of the design. Proposing an alternative way to tackle the problem, we find that the proposed GA designs based on G- or/and IV-efficiency are robust to model misspecification.     

Process capability: a criterion for optimizing multiple response product and process design

In this research, we consider the maximization of process capability as the criterion in product/process design that is used for selecting preferred design factor levels and propose several approaches for single and multiple response performance measure designs. All of these approaches assume that the relationship between a process performance measure and a set of design factors is represented via an estimate of a response surface function. In particular, we develop; (i) criteria for selecting an optimal design, which we call MCpk and MCpm; (h) mathematical programming formulations for maximizing MCpk and MCpm, including formulations for maximizing the desirability index (Harrington, 1965) and for maximizing the standardized performance criteria (Barton and Tsui, 1991) as special cases of the formulation for maximizing MCpk, (iii) formulations for considering cost when maximizing MCpk and MCpm, (iv) a means for assessing propagation of error; (v) a robust design method for assessing design factor effects on residual variance; (vi) a means for assessing the optimality of a proposed solution: and (vii) an original application in the screening of printed circuit board panels.     

Multiobjective Genetic Algorithm Approach to the Economic Statistical Design of Control Charts with an Application to bar and S2 Charts

Control charts are the primary tools of statistical process control. These charts may be designed by using a simple rule suggested by Shewhart, a statistical criterion, an economic criterion, or a joint economic statistical criterion. Each method has its strengths and weaknesses. One weakness of the methods of design listed is their lack of flexibility and adaptability, a primary objective of practical mathematical models. In this article, we explore multiobjective models as an alternative for the methods listed. These provide a set of optimal solutions rather than a single optimal solution and thus allow the user to tailor their solution to the temporal imperative of a specific industrial situation. We present a solution to a well‐known industrial problem and compare optimal multiobjective designs with economic designs, statistical designs, economic statistical designs, and heuristic designs. Copyright © 2012 John Wiley & Sons, Ltd.     

Japanese Manufacturing Techniques: Nine Hidden Lessons in Simplicity

Compound orthogonal arrays (COAs) and single arrays are alternatives to the inner–outer arrays advocated by Taguchi for robust parameter design experiments. A criterion based on the word type pattern and strength of COAs is proposed to select optimal COAs. Single arrays are classified into prodigal single arrays (PSAs) and economical single arrays (ESAs) according to their relative estimation capacities, and various optimality criteria, again based on the word type pattern, are proposed for selecting optimal single arrays. Useful optimal COAs, PSAs, and ESAs are constructed and tabulated as convenient references for experimenters in practice.     

OPTIMUM DESIGN OF LAMINATED COMPOSITE STRUCTURES VIA AMULTILEVEL SUBSTRUCTURING APPROACH

This paper presents a multilevel substructuring and optimization approach to the minimum weight design of laminated composite structures. The optimization process is carried out in a double scheme which consists of optimizations at system and subsystem levels. At the system level of optimization, an optimality criterion method is used to design component thicknesses which minimize structural weight subject to structural behavioral constraints as well as side constraints. At the subsystem level, the structure being divided into several substructures, fiber directions and layer thicknesses of each substructure are determined to minimize its weight subject to component behavioral constraints as well as side constraints. The objective at the subsystem level is accomplished by carrying out the minimization process again in a double scheme where the quasi-Newton method is used at the first sub-level of optimization for the optimal design of fiber directions and an optimality criterion method at the second sub-level for layer thickness design. The optimal solution is obtained by iterating between the different levels of optimization. Appropriate connectivity conditions for linking different levels of optimization are introduced to ensure convergence of solution. The feasibility and application of the present approach is illustrated by an example of the optimal design of a single-cell, three bay, cantilevered boxbeam.     

Comparison of designs in presence of a possible correlation in observations

We often assume the standard linear model with uncorrelated observations for comparison of designs without realizing a possible presence of correlation in observations. In this paper we present several change of variance functions including the one given in Zhou (2001) for comparing designs in presence of possible correlation in observations. We find a design by minimizing one of our proposed change of variance functions in a simple response surface setup. We then compare its performance with all variance design, all bias design, and the design making the average variance equal to the average squared bias. We also compare a second order rotatable design with a non-rotatable design. The rotatable design is better than the non-rotatable design with respect to A-, D-, and E- optimality criterion functions under the standard linear model with uncorrelated observations. We observe that the rotatable design may not perform better than the non-rotatable design with respect to the change of variance functions. We present some important properties of the change of variance functions. We find that the A-optimum designs may perform poorly with respect to a change of variance function.     

Robust Optimization of Experimentally Derived Objective Functions

In the design or improvement of systems and processes, the objective function is often a performance response surface estimated from experiments. A standard approach is to identify the levels of the design variables that optimize the estimated model. However, if the estimated model varies from the true model due to random error in the experiment, the resulting solution may be quite far from optimal. We consider all points in the confidence intervals associated with the estimated model and construct a minimax deviation model to find a robust solution that is resistant to the error in the estimated empirical model. We prove a reduction theorem to reduce the optimization model to a tractable, finite, mathematical program. The proposed approach is applied to solve for a robust order policy in an inventory problem and is compared with the canonical approach using a Monte Carlo simulation.     

Optimal design of multi-response experiments using semi-definite programming

Optimal design of multi-response experiments for estimating the parameters of multi-response linear models is a challenging problem. The main drawback of the existing algorithms is that they require the solution of many optimization problems in the process of generating an optimal design that involve cumbersome manual operations. Furthermore, all the existing methods generate approximate design and no method for multi-response n-exact design has been cited in the literature. This paper presents a unified formulation for multi-response optimal design problem using Semi-Definite Programming (SDP) that can generate D-, A- and E-optimal designs. The proposed method alleviates the difficulties associated with the existing methods. It solves a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space. We generate both approximate and n-exact designs for multi-response models by solving SDP models with integer variables. Another advantage of the proposed method lies in the amount of computation time taken to generate an optimal design for multi-response models. Several test problems have been solved using an existing interior-point based SDP solver. Numerical results show the potentials and efficiency of the proposed formulation as compared with those of other existing methods. The robustness of the generated designs with respect to the variance-covariance matrix is also investigated.     

Decomposition method for solving optimal material orientation problems

The strain energy density is considered as a measure of the stiffness/flexibility of the composite structure. A methodology for determining the stationary points of the strain energy density in anisotropic solids is developed. The methodology proposed is based on new problem formulation, derivation and analysis of optimality conditions, and decomposition method. The optimal material orientation problem is formulated in terms of strains. The optimality conditions derived cover different material symmetries, linear and also some non-linear material models. The complexity analysis of the optimality conditions has been performed. The proposed approach allows to divide the solution of the optimal material orientation problem into less complicated subtasks.     

Robust inversion method for jointly estimating parameters and variance components from heterogeneous monitoring data

In this paper, a novel inversion method is proposed for jointly robust estimation of parameters and variance components from disjunctive groups of observations affected by outliers. This method, named robust non-negative variance component estimation (RVCE), is a coupling of variance component estimation (VCE) technique with a robust estimation method, developed to cope with outliers and to avoid negative variance, leading thus, to an estimation reliable enough. The principle of RVCE method is based on the refinement of the stochastic model via an equivalent weight matrix established from the original measurement weight matrix and an adapted full weight matrix with hard rejection to outliers. This last one is derived from the robust standardized residuals, using a highly robust estimator, as an initial solution of the inverse problem, and a cut-off value adapted to sort out the good observations from the bad ones. Furthermore, because the original weight matrix is partly known, the integration of the VCE technique plays a key role to reach an optimal solution and to provide valuable information on the precision of the estimates. The performance of the proposed method is demonstrated by considering a rockfill dam as an example, where the material parameters and variance components are jointly estimated from geotechnical and geodetic measurements. The results of comparison study between RVCE method with other methods such as the classical NN-VCE, RIMCO, least squares and the combined Huber’s M-estimator with VCE (HVCE) for various configuration options have highlighted the pertinence of the proposed method.     

Allocation of variance reduction targets under the influence of supplier interaction

A common quality improvement strategy used by manufacturers is to periodically allocate quality improvement targets among their suppliers. We propose a formal modelling and optimization approach for assessing quality improvement targets for suppliers. In this approach it is understood that a manufacturer's quality improvement results from reductions in supplier process variances, which occurs only through investments in learning. A constrained nonlinear optimization model is developed for determining an optimal allocation of variance reduction target that minimizes expected total cost, where the relationship between performance measures and the set of design parameters is generally represented by second-order polynomial functions. An example in the fabrication of a tyre tread compound is used both to demonstrate the implementation of our proposed models as well as to provide an empirical comparison of optimal learning rates for different functional relationships between the performance measures and the set of design parameters.     

A Novel Approach to Simultaneous Robust Design of Product Parameters and Tolerances Using Quality Loss and Multivariate ANOVA Concepts

Design and development of high quality products are of utmost importance to any production plant. Product design consists of parameter design and tolerance design, which affect the product performances and the manufacturing costs, respectively. Most products involve more than one quality feature. Design and development of such products raise multi‐response surface problems in which it is necessary to determine the optimal values of parameters and the tolerances for all responses simultaneously. In this research, an approach for simultaneous robust parameter and tolerance design is proposed to deal with multi‐response problems. The proposed method employs quality loss concept and one‐way multivariate analysis of variance. Two simulation studies are performed to validate the applicability of the proposed method. Research findings show that the proposed method performs better in quality improvement as well as in cost reduction than the existing methods. The variances of the responses are also lower than those of the other methods, that is, the proposed method results in a more robust approach to product design. Copyright © 2016 John Wiley & Sons, Ltd.     

An integrated approach to the facilities and material handling system design

The facility layout problem involves the optimal location of manufacturing facilities into a workshop. The classical approach to the layout design is carried out in two separate steps: the first step is the construction of the block layout, i.e. the location of the departments into the workshop, and the second step is the design of the material handling system. The separate optimization of these two aspects of the problem leads to solutions that can be far from the total optimum. In this paper, an integrated approach to the facilities and material handling system design is proposed. Referring to a physical model, named the bay structure , and to a unidirectional AGV system, a genetic approach is proposed to individuate the locations of the departments, the positions of the pickup/delivery stations and the direction of the flow-path. The minimization of material handling cost is adopted as optimality criterion.     

ROBUST DESIGN FOR IMPROVED VEHICLE HANDLING UNDER A RANGE OF MANEUVER CONDITIONS

In this article, a robust design procedure is applied to achieve improved vehicle handling performance as an integral part of simulation-based vehicle design. Recent developments in the field of robust design optimization and the techniques for creating global approximations of design behaviors are applied to improve the computational efficiency of robust vehicle design built upon sophisticated vehicle dynamic simulations. The approach is applied to the design of a M916A1 6-wheel tractor/M870A2 3-axle semi-trailer. The results illustrate that the proposed procedure is effective for preventing the rollover of ground vehicles as well as for identifying a design that is not only optimal against the worst maneuver condition but is also robust with respect to a range of maneuver inputs. Furthermore, a comparison is made between a statistical approach and a bi-level optimization approach in terms of their effectiveness in solving robust design problems     

A Prediction Region‐based Approach to Model Uncertainty for Multi‐response Optimization

Multi‐response optimization methods rely on empirical process models based on the estimates of model parameters that relate response variables to a set of design variables. However, in determining the optimal conditions for the design variables, model uncertainty is typically neglected, resulting in an unstable optimal solution. This paper proposes a new optimization strategy that takes model uncertainty into account via the prediction region for multiple responses. To avoid obtaining an overly conservative design, the location and dispersion performances are constructed based on the best‐case strategy and the worst‐case strategy of expected loss. We reveal that the traditional loss function and the minimax/maximin strategy are both special cases of the proposed approach. An example is illustrated to present the procedure and the effectiveness of the proposed loss function. The results show that the proposed approach can give reasonable results when both the location and dispersion performances are important issues. Copyright © 2015 John Wiley & Sons, Ltd.     

A kriging metamodel-assisted robust optimization method based on a reverse model

The goal of robust optimization methods is to obtain a solution that is both optimum and relatively insensitive to uncertainty factors. Most existing robust optimization approaches use outer–inner nested optimization structures where a large amount of computational effort is required because the robustness of each candidate solution delivered from the outer level should be evaluated in the inner level. In this article, a kriging metamodel-assisted robust optimization method based on a reverse model (K-RMRO) is first proposed, in which the nested optimization structure is reduced into a single-loop optimization structure to ease the computational burden. Ignoring the interpolation uncertainties from kriging, K-RMRO may yield non-robust optima. Hence, an improved kriging-assisted robust optimization method based on a reverse model (IK-RMRO) is presented to take the interpolation uncertainty of kriging metamodel into consideration. In IK-RMRO, an objective switching criterion is introduced to determine whether the inner level robust optimization or the kriging metamodel replacement should be used to evaluate the robustness of design alternatives. The proposed criterion is developed according to whether or not the robust status of the individual can be changed because of the interpolation uncertainties from the kriging metamodel. Numerical and engineering cases are used to demonstrate the applicability and efficiency of the proposed approach.     

Ensemble of Surrogates for Dual Response Surface Modeling in Robust Parameter Design

The robust parameter design of industrial processes and products on the basis of the concept of building quality into a design has attracted much attention from researchers and practitioners for many years, and several methods have been studied in the research community. Dual response surface methodology is one of the most commonly used approaches for simultaneously optimizing the mean and the variance of response in quality engineering. Nevertheless, when the relationship between influential input factors and output quality characteristics of a process is very complex (e.g. highly nonlinear and noisy), traditional approaches have their limitations. In this article, we introduced support vector regression, kriging model, and radial basis function, which are commonly used in computer experiments, into robust parameter design, and especially introduced a new strategy that builds the dual response surface using the ensemble of surrogates, which can provide a more robust approximation model. We demonstrated the advantages of kriging, support vector regression, radial basis function, and the ensemble of surrogates by reinvestigating the dual response approach on the basis of parametric, nonparametric, and semiparametric approaches, and a simulation experiment is studied. The results show that our presented models can achieve more desirable results than parametric, nonparametric, and semiparametric approaches in terms of fitting and predictive accuracy, and the optimal operating conditions recommended by our presented models are similar to those recommended in literature, which indicates the validation of our presented models. Copyright © 2012 John Wiley & Sons, Ltd.     

Multidisciplinary design and optimization of an air launched satellite launch vehicle using a hybrid heuristic search algorithm

A multidisciplinary design and optimization strategy for a multistage air launched satellite launch vehicle comprising of a solid propulsion system to low earth orbit with the implementation of a hybrid heuristic search algorithm is proposed in this article. The proposed approach integrated the search properties of a genetic algorithm and simulated annealing, thus achieving an optimal solution while satisfying the design objectives and performance constraints. The genetic algorithm identified the feasible region of solutions and simulated annealing exploited the identified feasible region in search of optimality. The proposed methodology coupled with design space reduction allows the designer to explore promising regions of optimality. Modules for mass properties, propulsion characteristics, aerodynamics, and flight dynamics are integrated to produce a high-fidelity model of the vehicle. The objective of this article is to develop a design strategy that more efficiently and effectively facilitates multidisciplinary design analysis and optimization for an air launched satellite launch vehicle.     

An optimality criterion method for dynamic optimization of structures

This paper presents an optimality criterion method for the determination of the least weight design of a structural system which satisfies a specific frequency requirement plus upper and lower bounds on the design variables. The design algorithm is an iterative solution of the Kuhn–Tucker optimality criterion based on choosing a single value of the Lagrange multiplier which minimizes the sum of the squares of residuals. The method has been applied to a variety of structures. Results assert that the method is capable of locating the optimal design in a small number of redesign cycles. The method avoids the scaling of design variables. It can treat non-structural masses and is applicable to. structural elements with a wide variety of size-stiffness. The procedure has been completely automated in a computer program on an IBM-PC microcomputer.     

Pitfalls of using a single criterion for selecting experimental designs

For surrogate construction, a good experimental design (ED) is essential to simultaneously reduce the effect of noise and bias errors. However, most EDs cater to a single criterion and may lead to small gains in that criterion at the expense of large deteriorations in other criteria. We use multiple criteria to assess the performance of different popular EDs. We demonstrate that these EDs offer different trade‐offs, and that use of a single criterion is indeed risky. In addition, we show that popular EDs, such as Latin hypercube sampling (LHS) and D‐optimal designs, often leave large regions of the design space unsampled even for moderate dimensions. We discuss a few possible strategies to combine multiple criteria and illustrate them with examples. We show that complementary criteria (e.g. bias handling criterion for variance‐based designs and vice versa) can be combined to improve the performance of EDs. We demonstrate improvements in the trade‐offs between noise and bias error by combining a model‐based criterion, like the D‐optimality criterion, and a geometry‐based criterion, like LHS. Next, we demonstrate that selecting an ED from three candidate EDs using a suitable error‐based criterion helped eliminate potentially poor designs. Finally, we show benefits from combining the multiple criteria‐based strategies, that is, generation of multiple EDs using the D‐optimality and LHS criteria, and selecting one design using a pointwise bias error criterion. The encouraging results from the examples indicate that it may be worthwhile studying these strategies more rigorously and in more detail. Copyright © 2007 John Wiley & Sons, Ltd.