A methodology for planning experiments in robust product and process design

The recently completed volume of Quality and Reliability Engineering International is its 25th anniversary. The mission of the journal is to provide information on new developments in many aspects of the discipline to professionals with research articles, review papers, and applications and case study papers. It is appropriate to look back over this anniversary volume and identify some important themes, highlight some of the significant papers that contribute to those themes, and discuss what we hope to see in the future. Statistical Process Control (SPC) and related topics continue to be an important area of research and applications. The journal published many noteworthy papers in this area, including the papers on adaptive procedures by Li et al.1 and Kooli and Liman2, run-to-run control3, integration of SPC with other techniques4, 5, design of control charts6, 7, and methods for identifying the change point in high-yield processes8. Reliability engineering and the associated statistical methods continue to be a key focus area for the journal. Koka et al.9 present an interdisciplinary approach to reliability in design and Samet10 discusses fault-tolerant computing. Accelerated life testing techniques are discussed by Liu and Tang11 and Pan12. Remenyte-Prescott and Andrews13 present a new real-time method for the efficient analysis of non-coherent fault trees. Anderson-Cook et al.14 investigate resource allocation for a complex system with aging components. Designed experiments and robust design are an integral part of both product design and process improvement. Methodology for robust product or process design is considered by Shaibu et al.15, Steenackers et al.16 and Rodriguez et al.17. Hazelrigg18 discusses many interesting aspects of engineering design. Other extremely useful papers include Gaudard et al.19 on the interaction between data mining and designed experiments, Guo et al.20 on the general balanced metric for mixed-level fractional factorial designs, Wesley et al.21 on prediction variance in split-plot designs, Guo et al.22 on optimal fold-over designs for mixed-level fractional factorial designs, and Jones and Johnson23 on designs for computer experiments. The journal also presented several interesting case studies, including Monroe and Pan24, Carmignani25, Magniez et al.26, Johnson et al.27 and Aksoy and Orbak28. Case studies are an important part of this journal. We have a high standard for these papers. They must represent the best of contemporary practice either through the application of new methodology or the application of existing methods in unique environments. Quality and reliability engineers are knowledge workers. They accumulate, generate and deploy knowledge with the express objective of improving products, processes, services and society. Case studies facilitate the generation and widespread adoption of new methods or provide insight about how existing methods can be used in novel situations. These types of papers can perform a valuable service to our profession. We encourage potential authors of case studies to contribute to our journal.

This paper presents a method for assembly evaluation. The method uses two evaluation criteria, robustness and variation analysis, and is supported by a software tool developed by the authors. The robustness evaluation aims at detecting design and assembly solutions that are sensitive to variation and may cause problems during production. Using this method in early product and process design phases helps to find more robust concepts, resulting in shorter production start-up time and better precision. The method’s use is exemplified in a concept study of the assembly process of the door to the body of a (fictitious) jeep. The study shows that the proposed method can be used to obtain an objective comparison between different concepts. This comparison includes both general robustness and the expected variation in the critical dimensions. The results can be used, together with economical and practical aspects, to determine which concept is best suited for the assembly process. The software used is implemented in the MS Windows environment and has an JGES interface that enables the user to import CAD geometry from an arbitrary CAD system. It can perform different types of robustness evaluations as well as traditional variation analyses.

With the increasing desire for products suitable for widely varying markets worldwide, this study offers insight into capabilities associated with successful robust design in global product markets. These robust design capabilities (i.e., the possibility for success under varying circumstances or scenarios) are a potential organizational response to rapid change and uncertainty, which also improve the likelihood of product acceptance on a global basis. From literature, executive interviews, and anecdotal evidence, four capabilities associated with robust product design are derived: (1) functional; (2) aesthetic; (3) technological; and (4) quality based. A model is proposed and an empirical test conducted that considers the moderating influence of environmental uncertainty on the relationship between these robust capabilities and firm performance. The findings suggest that the use of robust design capabilities are affected by uncertainty and have an important influence on firm performance and speed to market. Specifically, the product development process tends to be characterized by aesthetic and technological robust design capabilities in more certain environments and functional robust design capabilities in more uncertain environments when seeking to improve firm performance. Alternatively, technological design capabilities in more certain environments and functional design capabilities in more uncertain environments are associated with improved speed to market.

We study how to design a worst-case robust training sequence for multiple-input multiple-output (MIMO) channel estimation. We consider mean-squared error of channel estimates as the figure of merit which is a function of second-order statistics of the MIMO channel, i.e., channel covariance matrix, in order to optimize training sequences under a total power constraint. In practical applications, the channel covariance matrix is not known perfectly. Thus the main aspect of our design is to improve robustness of the training sequences against possible uncertainties in the available channel covariance matrix. Using a deterministic uncertainty model, we formulate a robust training sequence design as a minimax optimization problem where we take such imperfections into account. We investigate the robust design problem assuming the general case of an arbitrarily correlated MIMO channel and a non-empty compact convex uncertainty set. We prove that such a problem admits a globally optimal solution by exploiting the convex-concave structure of the objective function, and propose numerical algorithms to address the robust training design problem. We proceed the analysis by considering multiple-input single-output (MISO) channels and Kronecker structured MIMO channels along with unitarily-invariant uncertainty sets. For these scenarios, we show that the problem is diagonalized by the eigenvectors of the nominal covariance matrices so that the robust design is significantly simplified from a complex matrix-variable problem to a real vector-variable power allocation problem. For the MISO channel, we provide closed-form solutions for the robust training sequences with the uncertainty sets defined by the spectral norm and nuclear norm.

The Taguchi robust design method traditionally deals with single‐characteristic problems. Various methods have been developed for extending the Taguchi single‐characteristic robust design method to the case of multi‐characteristic robust design problems. However, most of those methods have shortcomings in that they do not properly consider the variance–covariance structures among performance characteristics and/or do not preserve the original properties of the Taguchi signal‐to‐noise ratio for single‐characteristic robust design problems. To overcome these shortcomings, this paper develops a multivariate loss function approach to multi‐characteristic robust design problems with an appropriately defined signal‐to‐noise ratio. Its performance is evaluated using simulated examples, and the results indicate that it generally outperforms existing representative methods for correlated as well as uncorrelated experimental data. Copyright © 2016 John Wiley & Sons, Ltd.

The purpose of robust design optimization is to minimize variations in design performances and therefore to make the design insensitive to uncertainties. Current robust design methods fall into two types — probabilistic robust design and worst-case (interval) robust design. The former method is used when random variables are involved. In this method, robustness is measure by standard deviations of design performances. The later method is used when uncertainties are represented by intervals. The widths of design performances are then used to measure robustness. In many engineering application, both random variables and interval variables may exist simultaneously. In this paper, a general approach to robust design optimization is proposed. The generality comes from the ability to handle both random and interval variables. To alleviate the computational burden, we employ a previously developed general robustness assessment method — semi-second-order Taylor expansion method, to evaluate the maximum and minimum standard deviations of a design performance. An efficient integration strategy of the general robustness assessment and optimization is proposed. With the integration strategy, the number of function calls can be reduced while good accuracy is maintained. A robust shaft design problem is given for demonstration.

A hierarchical optimization approach to robust design of energy supply systems based on a mixed-integer linear model

During the hot forging process, the shaping property and forging quality will fluctuate because of die wear, manufacturing tolerance, dimensional variation caused by temperature and the different friction conditions, etc. In order to control this variation in performance and to optimize the process parameters, a robust design method is proposed in this paper, based on the finite element method for the hot forging process. During the robust design process, the Taguchi method is the basic robust theory. The finite element analysis is incorporated in order to simulate the hot forging process. In addition, in order to calculate the objective function value, an orthogonal design method is selected to arrange experiments and collect sample points. The ANOVA method is employed to analyze the relationships of the design parameters and design objectives and to find the best parameters. Finally, a case study for the gear hot forging process is conducted. With the objective to reduce the forging force and its variation, the robust design mathematical model is established. The optimal design parameters obtained from this study indicate that the forging force has been reduced and its variation has been controlled.

Discrete-event simulation is a common tool for the analysis of semiconductor manufacturing systems. With the aid of a simulation model, and in conjunction with sensitivity analysis and metamodeling techniques, robust design can be performed to optimize a system. Robust design problems often include integer decision variables. This paper shows a graphical approach to robust design that is effective in the presence of discrete or qualitative variables. The graphical robust design methodology was applied to a backend semiconductor manufacturing process. Changes in specific resource capacities and product mix were examined to determine their effect on the level and variance of cycle time and work in process.

Model-based design principles have received considerable attention in biotechnology and the chemical industry over the last two decades. However, parameter uncertainties of first-principle models are critical in model-based design and have led to the development of robustification concepts. Various strategies have been introduced to solve the robust optimization problem. Most approaches suffer from either unreasonable computational expense or low approximation accuracy. Moreover, they are not rigorous and do not consider robust optimization problems where parameter correlation and equality constraints exist. In this work, we propose a highly efficient framework for solving robust optimization problems with the so-called point estimation method (PEM). The PEM has a fair trade-off between computational expense and approximation accuracy and can be easily extended to problems of parameter correlations. From a statistical point of view, moment-based methods are used to approximate robust inequality and equality constraints for a robust process design. We also apply a global sensitivity analysis to further simplify robust optimization problems with a large number of uncertain parameters. We demonstrate the performance of the proposed framework with two case studies: (1) designing a heating/cooling profile for the essential part of a continuous production process; and (2) optimizing the feeding profile for a fed-batch reactor of the penicillin fermentation process. According to the derived results, the proposed framework of robust process design addresses uncertainties adequately and scales well with the number of uncertain parameters. Thus, the described robustification concept should be an ideal candidate for more complex (bio)chemical problems in model-based design.

Finite element simulation has become an important tool of process and production design in various fields, especially in the automotive industry. The calculation of forming processes in the early concept phase of new cars allows virtual adaptions, which can reduce costs of later phases in the product development significantly. Therefore, the precise characterization and modelling of the material behavior is necessary to ensure a robust and reliable numerical process design. The mechanical properties of numerous materials are highly influenced by the rolling or extrusion direction in the production process. This necessitates the characterization of materials in different loading directions. However, depending on the dimensional aspects of the semi-finished product, the manufacturing of specimens can be challenging or even impossible. Thus, in this investigation, an innovative, indirect approach for the identification of the Lankford coefficient in transversal direction is presented. Based on numerical and experimental data of layer compression tests the Lankford coefficient is determined by inverse modelling of the resulting specimen contour. Due to the characteristics of the layer compression test, it can even be used for semi-finished products with small transversal dimensions like extruded profiles. The presented methodology is on the one hand verified by conventional uniaxial tensile tests for aluminum as well as steel blank material. On the other hand it is used to determine Lankford coefficients for an aluminum extrusion hollow profile and the inversely identified material model is validated by comparison of strain distributions of experimental and numerical square tube bending tests.

An LMI approach to design robust fault detection filter for uncertain LTI systems

This paper considers a robust FIR precoder design problem for frequency-selective multiple-input multiple-output (MIMO) wireless systems under transmit power constraints with imperfect channel knowledge at the transmitter. The robust design problem requires finding the optimal power constrained FIR precoder for the worst-case channel model within a neighborhood of the estimated channel. The neighborhood is formed by placing a bound on Kullback-Leibler (KL) divergence between the actual and estimated channels. We show that the optimal power constrained FIR precoder designed for the estimated channel under the assumption of perfect channel knowledge is robust when channel uncertainties are measured by the KL divergence metric.

Optimal design towards enhancement of board-level thermomechanical reliability of wafer-level chip-scale packages

In designing consumer durables such as appliances and power tools, it is important to account for variations in product performance across different usage situations and conditions. Since the specific usage of the product and the usage conditions can vary, the resultant variations in product performance also can impact consumer preferences for the product. Therefore, any new product that is designed should be robust to these variations—both in product performances and consumer preferences. This article refers to a robust product design as a design that has (1) the best possible (engineering and market) performance under the worst‐case variations and (2) the least possible sensitivity in its performance under the variations. Achieving these robustness criteria, however, implies consideration of a large number of design factors across multiple functions. This article's objectives are (1) to provide a tutorial on how variations in product performance and consumer preferences can be incorporated in the generation and comparison of design alternatives and (2) to apply a multi‐objective genetic algorithm (MOGA) that incorporates multifunction criteria in order to identify better designs while incorporating the robustness criteria in the selection process. Since the robustness criteria is based on variations in engineering performance as well as consumer preferences, the identified designs are robust and optimal from different functional perspectives, a significant advantage over extant approaches that do not consider robustness issues from multifunction perspectives. This study's approach is particularly useful for product managers and product development teams, who are charged with developing prototypes. They may find the approach helpful for obtaining customers' buy‐in as well as internal buy‐in early on in the product development cycle and thereby for reducing the cost and time involved in developing prototypes. This study's approach and its usefulness are illustrated using a case‐study application of prototype development for a handheld power tool.