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Description
1. Introduction
Taguchi has proposed various performance measures known as Signal-to-Noise (SN) ratios for evaluating the performance of engineering systems (see Taguchi and Wu (1980), and Taguchi (1991a)). Most of them are criticized in the literature (see Nair (1992), Box (1998), Berube and Wu (2000), and Montgomery (2004), among many others). The recommendation was to use data-driven performance measures instead of the SN ratios. Although the SN ratios can be shown to be incorrect under some modeling assumptions, Taguchi's approach has a merit which was not easy to appreciate because of the lack of a rigorous framework. In this article we provide a new perspective to his approach.
A certain type of control factors, namely "adjustment factors", play a crucial role in Taguchi's approach. The SN ratios are supposed to be performance measures independent of these adjustment factors, so that the optimization of the system can be conveniently split into two steps, the first being to maximize the SN ratio and the second being some adjustments using the adjustment factors. Leon et al. (1987) gave a mathematical foundation to this approach and proposed the concept of Performance Measures Independent of Adjustment (PerMIA). Using this concept, SN ratios can be justified under some modeling assumptions, whereas they are inappropriate under other modeling assumptions. However, the notion of adjustment factors is not well defined and thus deriving PerMIAs can be ambiguous (see the discussions by various researchers accompanying the paper of Leon et al. (1987)).
We argue that Taguchi's motivation for using adjustment factors is mainly to simplify the experiment rather than to simplify the optimization. Based on this we propose a criterion for selecting adjustment factors from the set of control factors. The new criterion clarifies several ambiguities about Taguchi's approach and enables one to develop better approaches to robust parameter design.
The article is organized as follows. The new criterion about the adjustment factors is proposed in Section 2. In Section 3, two examples are presented to illustrate the advantages of the new criterion and some concluding remarks are given in Section 4.
2. Adjustment factors
Taguchi's SN ratio for a nominal-the-best characteristic is given by SN = [[mu].sup.2]/[[sigma].sup.2], where [mu] and [[sigma].sup.2] are the mean and variance of the response. In order to minimize the expected value of the quadratic loss function, he proposed to perform the optimization in two steps: first to find the setting of control factors to maximize the SN ratio and then to use an adjustment factor to adjust the mean to target. According to his approach an adjustment factor is a control factor that has a large effect on [mu] but not on the SN ratio. Others, who criticized the use of SN ratios suggested to replace the SN ratio in the first step by [[sigma].sup.2]. Therefore, in their approach an adjustment factor is the one having a large effect on [mu] but not on [[sigma].sup.2]. Clearly the two definitions of an adjustment factor are in contradiction. Then, what is actually an adjustment factor?
The approach of Leon et al. (1987) can be explained as follows (see also the discussion by Easterling (1987)). Let Y be the response and L(Y) be a quality loss function. Divide the control factors into two groups (X, M), where M denotes the set of adjustment factors. Then, the minimization of the E{L(Y)} = R(X, M), where the expectation is taken with respect to the distribution of noise factors, can be done in two steps:
1. Minimize PM(X) = R(X, M*(X)) with respect to X, where M*(X) = arg [min.sub.M] R(X, M). Denote the solution by X*.
2. Adjust M to M*(X*).
PM(X) is called a PerMIA. Leon et al. (1987) further showed that if Y has a multiplicative error model with error depending only on X, then minimizing PM(X) is equivalent to maximizing the SN ratio, whereas if Y has an additive error model with error depending only on X, then minimizing PM(X) is equivalent to minimizing the variance of Y. According to them the adjustment factors are selected from the set of control factors to make the product/process design more flexible. Adjustment factors are easy-to-change factors which... |
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