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Article Excerpt
1. Introduction
Continuous quality improvement has become widely recognized by many industries as a critical concept in maintaining a competitive advantage in the marketplace. It is also recognized that quality improvement activities are most efficient and cost-effective when implemented during the design stage. Based on this awareness, Taguchi (1986) introduced a systematic method for applying experimental design, which has become known as robust design. The primary goal of this method is to determine the best design factor settings by minimizing performance variability and product bias, i.e., the deviation from the target value of a product. Because of their practicability in reducing the inherent uncertainty associated with design factors and system performance, the widespread application of robust design techniques has resulted in significant improvements in product quality, manufacturability and reliability at low cost.
There is little disagreement among researchers and practitioners about Taguchi's basic philosophy. Steinberg and Bursztyn (1998) provided a comprehensive discussion on Taguchi's offline quality control and showed that the use of noise factors can significantly increase the power for detecting factors with dispersion effects, when noise factors are explicitly modeled in the analysis. However, the ad hoc robust design methods suggested by Taguchi remain controversial due to various mathematical flaws. The controversy surrounding Taguchi's assumptions, experimental design and statistical analysis has been well addressed by Leon et al. (1987), Box (1988), Box et al (1988), Nair (1992) and Tsui (1992). Consequently, researchers have closely examined alternative approaches using well-established statistical and optimization tools. Vining and Myers (1990) introduced the dual-response approach based on Response Surface Methodology (RSM) as an alternative for modeling process relationships by separately estimating the response functions of process mean and variance, thereby achieving the primary goal of robust design by minimizing the process variance while adjusting the process mean at the target. Del Castillo and Montgomery (1993) and Copeland and Nelson (1996) showed that the optimization technique used by Vining and Myers (1990) does not always guarantee optimal robust design solutions, and proposed standard non-linear programming techniques, such as the generalized reduced gradient method and the Nelder--Mead simplex method, which can provide better robust design solutions. The modified dual-response approaches using fuzzy theory were further developed by Khattree (1996) and Kim and Lin (1998). However, Lin and Tu (1995), pointing out that the robust design solutions obtained from the dual-response model may not necessarily be optimal since this model forces the process mean to be located at the target value, proposed the mean-squared error model, relaxing the zero-bias assumption. While allowing some process bias, the resulting process variance is less than or at most equal to the variance obtained from the Vining-Myers model; hence, the mean-squared error model may provide better (or at least equal) robust design solutions unless the zero-bias assumption must be met. Further modifications to the mean-squared error model have been discussed by Jayaram and Ibrahim (1999), Cho et al. (2000), Kim and Cho (2000, 2002), Yue (2002), Park and Cho (2003), Miro-Quesada and Del Castillo (2004), Cho and Park (2005), Govindaluri and Cho (2005), Shin and Cho (2005, 2006) and Lee et al. (2007). Along this line, Myers et al. (2005), Park and Cho (2005) and Robinson et al. (2006) developed modified dual-response models using a generalized linear model, a robust design model using the weighted least-square method for unbalanced data and a robust design model using a generalized linear mixed model for non-normal quality characteristics, respectively. As for an experimental strategy, Kovach and Cho (2005, 2006, 2007, 2008) and Kovach et al (2008) studied D-optimal robust design problems by minimizing the variance of the regression coefficients. Ginsburg and Ben-Gal (2006) then developed a new optimality criterion, called the [V.sub.s] optimality, which minimizes the variance of the optimal solution by prioritizing the estimation of various model coefficients; thereby, estimating coefficients more accurately at each experimental stage. It is well known that estimated empirical models are often subject to random error. In order to obtain a more precise robust design solution in the presence of the error, Xu and Albin (2003) developed a model which can be resistant to the error by considering all points in the confidence intervals associated with the estimated model. When multiple quality characteristics are considered, those characteristics are often correlated. Govindaluri and Cho (2007) investigated the effect of correlations of quality characteristics on robust design solutions. Finally, Egorov et al. (2007) and Kovach et al. (2008) studied optimal robust design solutions by using the indirect optimization algorithm and physical programming, respectively.
Most of the robust design models discussed above find robust design solutions either by requiring a zero process bias (i.e., prioritizing process bias over process variability) or by assigning an equal weight to the process bias and the process variability which is a main concept in the mean-squared error approach. In real-world industrial settings, however, a decision maker often needs a balance between the two process parameters such as process mean and process variability. Reasonable control of process variability can apparently be achieved only at the expense of sacrificing the process bias, and vice versa. In order to find trade-off solutions between the two process parameters, this situation can be considered a biobjective robust design problem for which Pareto solutions (i.e., solutions corresponding to a particular trade-off of different objectives) of the two process parameters need to be provided. Tang and Xu (2002), Koksoy and Doganaksoy (2003) and Ding et al. (2004) provided the Pareto solutions for these two parameters using an additive mean-squared error model based on the concept of a weighted sum which minimizes [omega][([^.[mu]](x)-[[tau]).sup.2] + (1-omega])[[^.[sigma]].sup.2](x) where [omega], [^.[mu]](x), [[^.[sigma]].sup.2](x) and [tau] are a weight, an estimated function of process mean, an estimated function of process variance and a desired target value, respectively. In fact, this simple weighted sum approach is often used by researchers, and is considered one of the standard optimization techniques (see Steuer (1986)). However, care must be exercised when this approach is applied to robust design problems. In most such problems, second-order models are often adequate for representing [^.[mu]](x) and [[^.[sigma]].sup.2](x), as evidenced by Vining and Myers (1990), Del Castillo and Montgomery (1993), Lin and Tu (1995), Tang and Xu (2002), Koksoy and Doganaksoy (2003) and Shin and Cho (2005). A closer look at the mean-squared error model they used, however, reveals that ([^.[mu]](x)-[tau]).sup.2] then becomes a fourth-order function, which is often neither convex nor concave. When an optimizing function is of a higher order than second, it is known that obtaining non-dominated solutions is usually unlikely (see, Tind and Wiecek (1999), Messac et al. (2000) and Mattson and Messac (2003)). As a result, the robust design solutions using the weighted sum model...
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