Arquitectura de datos

10 15

100 200 300 400 500 600 700 800

sy

N_TC(cycles) (b)

5.13.3 Signal-to-Noise Ratio Comparison

Before performing this study, the process engineer used the following design parameters: stage temperature= 100^◦C, ultrasonic power= 7 units, bonding force

= 80 gf, bonding time = 50 ms. The combination of levels yields ˆη = 25.10. The optimal levels from this study improve the signal-to-noise ratio by (46.42− 25.10)/25.10= 85%. Therefore, the robustness of reliability against thermal cycling has been increased remarkably.

5.14 ADVANCED TOPICS IN ROBUST DESIGN

In this section we introduce advanced topics, including an alternative to the signal-to-noise ratio, multiple responses, the response surface method, and accel-erated testing, which are related to the subjects discussed earlier in the chapter.

0 20 40 80 70 60 50

30

100 200 300 400 500 600 700 800

my

N_TC(cycles) (c)

100 0 5 10 15

200 300 400

(d) N_TC(cycles)

500 600 700 800 sy

FIGURE 5.25 (a) µy varying withNTC for wire bonds tested at noise factor level 1;

(b) σ_yvarying withN_TCfor wire bonds tested at noise factor level 1; (c) µ_y varying with NTCfor wire bonds tested at noise factor level 2; (d) σyvarying withNTCfor wire bonds tested at noise factor level 2

Recent advancements in these topics are described briefly. The materials in this section are helpful for performing a more efficient robust design.

5.14.1 Alternative to the Signal-to-Noise Ratio

Taguchi (1986, 1987) proposes using the signal-to-noise ratio to measure the robustness of a product performance. This metric has been used extensively in industry because of its simplicity. When response is a smaller-the-better or larger-the-better characteristic, as described in Section 5.11.3, the optimal setting can be found by using a one-step procedure: Select the levels of control factors that maximize the signal-to-noise ratio. This metric is a good choice for engineers.

ADVANCED TOPICS IN ROBUST DESIGN 183

TABLE 5.24 Estimates of Reliability and Signal-to-Noise Ratio for Wire Bonds

Reliability

Run Noise Level 1 Noise Level 2 ˆη

1 0.9911 0.9816 36.66

TABLE 5.25 ANOVA Table for Wire Bond Reliability

A 334.85 2 167.42 81.67

B 772.65 2 386.32 188.45

C 365.02 2 182.51 89.03

D 4.10 2 2.05

(e) (4.10) (2) (2.05)

Total 1476.62 8

As pointed out by Nair (1992), the signal-to-noise ratio has some drawbacks, the major one being that minimizing it does not lead automatically to mini-mization of the quadratic quality loss (5.1) in situations where the variability of a nominal-the-best characteristic is affected by all significant control factors.

The effect of this problem can, however, be mitigated by data transformation through which the variability of the transformed data is made independent of mean adjustment factors (Robinson et al., 2004). Box (1988) proposes the use of lambda plots to identify the transformation that yields the independence.

A measure alternative to the signal-to-noise ratio is the location and dispersion model. For each run in Table 5.9, y_i and ln(s_i²), representing the sample mean and log sample variance over the noise replicates, are used as the measures of location and dispersion. They are

y_i = 1

For a nominal-the-best characteristic, the procedure for obtaining the optimal setting of control factors is the same as that for optimizing the signal-to-noise ratio described in Section 5.11.3. For a smaller-the-better or larger-the-better problem, the procedure consists of two steps:

1. Select the levels of the mean adjustment factors to minimize (or maximize) the location.

2. Choose the levels of the dispersion factors that are not mean adjustment factors to minimize the dispersion.

C. F. Wu and Hamada (2000) and Nair et al. (2002) discuss in greater detail use of the location and dispersion model to achieve robustness.

5.14.2 Multiple Responses

In robust reliability design, the response or quality characteristic may be life, reliability, or performance. If life or reliability is used, the product has a single response. Multiple responses arise when a product has several performance char-acteristics, and some or all of them are equally important. For practical purposes, the robust reliability design described in this chapter uses one characteristic, that most closely reflecting the reliability of the product. Selection of the character-istic is based on engineering judgment, customer expectation, experience, or test data; in Chapter 8 we describe selection of the characteristic.

In some applications, using multiple responses can lead to a better design.

When analyzing multiple response data, for the sake of simplicity, each response is sometimes analyzed separately to determine the optimal setting of design parameters for that response. This naive treatment may work reasonably well if there is little correlation between responses. However, when the multiple responses are highly correlated, a design setting that is optimal for one response may degrade the quality of another. In such cases, simultaneous treatment of the multiple responses is necessary. In the literature, there are two main approaches to handling multiple-response optimization problems: the desirability function method and the loss function approach.

Desirability Function Method This method, proposed by Derringer and Suich (1980) and modified by Del Casttillo et al. (1996), turns a multiple-response problem into a single-response case using a desirability function. The function is given by

D = [d1(y1)× d2(y2)× · · · × dm(ym)]^1/m, (5.50) where di (i= 1, 2, . . . , m) is the desirability of response yi, m the number of responses, and D the total desirability. Now the response of the product is D, which is a larger-the-better characteristic. In the context of robust design described in this chapter, the signal-to-noise ratio should be computed from the values ofD. Then the robust design is to choose the optimal setting of control factors that maximizes the signal-to-noise ratio.

ADVANCED TOPICS IN ROBUST DESIGN 185

The desirability for each response depends on the type of response. For a nominal-the-best response, the individual desirability is

di =

wheremi, Li, andHi are the target and minimum and maximum allowable values of y, respectively, and wL and wH are positive constants. These two constants are equal if the value of a response smaller than the target is as undesirable as a value greater than the target.

For a smaller-the-better response, the individual desirability is

di=

where w is a positive constant and Li is a small enough number.

For a larger-the-better response, the individual desirability is

di=

where w is a positive constant and Hi is a large enough number.

The desirability of each response depends on the value of the exponent. The choice of the value is arbitrary and thus subjective. In many situations it is difficult to specify meaningful minimum and maximum allowable values for a smaller-the-better or larger-smaller-the-better response. Nevertheless, the method has found many applications in industry (see, e.g., Dabbas et al., 2003; Corzo and Gomez, 2004).

Loss Function Approach Loss function for multiple responses, described by Pignatiello (1993), Ames et al. (1997), and Vining (1998), is a natural extension of the quality loss function for a single response. The loss function for a nominal-the-best response is

L= (Y − my)^TK(Y− my), (5.54) where Y= (y1, y2, . . . , ym) is the response vector, my = (my1, my2, . . . , mym) is the target vector, and K is am× m matrix of which the elements are constants.

The values of the constants are related to the repair and scrap cost of the product and may be determined based on the functional requirements of the product. In general, the diagonal elements of K measure the weights of them responses, and the off-diagonal elements represent the correlations between these responses.

Like the single-response case, the loss function (5.54) can be extended to mea-sure the loss for a smaller-the-better or larger-the-better response (Tsui, 1999).

For a smaller-the-better response, we replace the fixed targetmyi with zero. For a larger-the-better response, the reciprocal of the response is substituted into (5.54) and treated as the smaller-the-better response.

If Y has a multivariate normal distribution with mean vector µ and vari-ance–covariance matrix, the expected loss can be written as

E(L)= (µ − my)^TK(µ− my)+ trace(K), (5.55) where µ and  are the functions of control factors and noise factors and can be estimated from experimental data by multivariate analysis methods. The methods are described in, for example, Johnson (1998).

The simplest approach to obtaining the optimal setting of control factors is to directly minimize the expected loss (5.55). The direct optimization approach is used by, for example, Romano et al. (2004). Because the approach may require excessive time to find the optimal setting when the number of control factors is large, some indirect but more efficient optimization procedures have been pro-posed. The most common one is the two-step approach, which finds its root in Taguchi’s two-step optimization approach for a single response. The approach first minimizes an appropriate variability measure and then brings the mean response on its target. Pignatiello (1993) and Tsui (1999) describe this two-step approach in detail.

5.14.3 Response Surface Methodology

The experimental analysis described in this chapter may lead to local optima because of the lack of informative relationship between response and control fac-tors. A more effective design of experiment is the response surface methodology (RSM), which is known as a sequential experimental technique. The objective of RSM is to ascertain the global optimal setting of design parameters by establish-ing and analyzestablish-ing the relationship between response and experimental variables.

The variables may include both control and noise factors. The RSM experiment usually begins with a first-order experiment aimed at establishing the first-order relationship given by

y= β0+

n i=1

βixi + e, (5.56)

where y is the response, xi the experimental variable,e the residual error, βi the coefficient representing the linear effect ofxi to be estimated from experimental data, andn the number of experimental variables. Once the relationship is built, a search must be conducted over the experimental region to determine if a curvature of the response is present. If this is the case, a second-order experiment should be conducted to build and estimate the second-order relationship given by

y= β0+

ADVANCED TOPICS IN ROBUST DESIGN 187

whereβi is the coefficient representing the linear effect ofxi,βij the coefficient representing the linear-by-linear interaction betweenxiandxj, andβii the coeffi-cient representing the quadratic effect ofxi. The optimum region for experimental variables is solved by differentiating y in (5.57) with respect to xi and setting it zero. C. F. Wu and Hamada (2000) and Myers and Montgomery (2002), for example, describe in detail the design and analysis of RSM experiments.

The principle of RSM can be applied to improve the optimality of the design setting obtained from ANOVA or graphical response analysis (K. Yang and Yang, 1998). In the context of robust design presented in this chapter, the response y in (5.57) is the signal-to-noise ratio. If there exists an interaction or quadratic effect, the relationship between signal-to-noise ratio and control factors may be modeled by (5.57), where y is replaced with η. The model contains 1+ 2n + n(n− 1)/2 parameters. To estimate the parameters, the experimental run must have the size of at least 1+ 2n + n(n − 1)/2, and each factor must involve at least three levels. The use of some orthogonal arrays, such as L9(3⁴) and L27(3¹³), satisfies the requirements; thus, it is possible to continue response surface analysis for the experimental design. The optimal setting may be obtained through the use of standard methods for response surface analysis as described in C. F. Wu and Hamada (2000) and Myers and Montgomery (2002).

RSM assumes that all variables are continuous and derivatives exist. In prac-tice, however, some design parameters may be discrete variables such as the type of materials. In these situations, (5.57) is still valid. But it cannot be used to determine the optima because the derivative with respect to a discrete variable is not defined. To continue the response surface analysis, we suppose that there are n1 discrete variables and n2 continuous variables, where n1+ n2 = n. Because the optimal levels of the n1 discrete factors have been determined in previous analysis by using the graphical response method or ANOVA, the response sur-face analysis is performed for then2continuous variables. Since the levels ofn1

variables have been selected, only the parameter settings that contain combina-tions of the selected levels of then1 variables can be used for response surface analysis. In general, the number of such settings is

ws = N

whereN is the run size of an inner array and qiis the number of levels of discrete variablexi. Refer to Table 5.3, for example. L9(3⁴) is used to accommodate four design parameters, one of which is assumed to be discrete and assigned to column 1. Suppose that ANOVA has concluded that level 1 is the optimal level for this variable. Thenws = 9/3 = 3, because only the responses from runs 4, 5, and 6 can apply to our response surface analysis.

Excluding discrete variables from modeling, (5.57) includes only n2 con-tinuous variables and has 1+ 2n2+ n2(n2− 1)/2 parameters to be estimated.

Therefore, we require thatws ≥ 1 + 2n2+ n2(n2− 1)/2. This requirement is fre-quently unachievable whenn1≥ 2. Thus, in most situations the response surface analysis is applicable when only one discrete design parameter is involved.

5.14.4 Accelerated Testing in Robust Design

Life or performance degradation is the primary response of an experiment designed for improving robustness and reliability. The experiment may yield few failures or little degradation at censoring time, when the levels of noise factors are within the normal use spectrum. In these situations it is difficult or impossible to perform data analysis and find the truly optimal setting of design parameters. Clearly, obtaining more life data or degradation information is necessary. This may be accomplished by carefully applying the concept of accelerated testing. Although accelerated testing has been studied and applied extensively, it is seldom discussed in the context of robust design aimed at improving reliability.

To produce a shorter life and more degradation in testing, it is natural to think of elevating noise factor levels as is usually done in a typical accelerated test. Then the accelerated test data are analyzed to draw conclusions about the optimal setting of control factors. In the context of robust design, however, the conclusions may be faulty if the accelerating noise factors interact with control factors. Without loss of generality, we assume that a robust design concerns one control factor and one noise factor. If the life has a location-scale distribution, the location parameterµ can be written as

µ= β0+ β1x+ β2z+ β12xz, (5.59) where x is the control factor, z the noise factor, and β0,β1,β2, andβ12 are the coefficients to be estimated from experimental data.

The acceleration factor between the life at noise level z1 and that at noise levelz2 is

Af = exp(µ1)

exp(µ2) = exp(µ1− µ2), (5.60) where Af is the acceleration factor and µi is the location parameter at noise level i. Chapter 7 presents in detail definition, explanation, and computation of the acceleration factor. For a given level of control factor, the acceleration factor between noise levelsz1andz2is obtained by substituting (5.59) into (5.60). Then we have

Af = exp[(z1− z2)(β2+ β12x)], (5.61) which indicates that the acceleration factor is a function of the control factor level. This is generally true when there are interactions between control factors and accelerating noise factors.

Accelerated test data may lead to a falsely optimal setting of design parameters if an acceleration factor depends on the level of control factor. This is illustrated by the following arguments. For the sake of convenience, we still assume that robust design involves one design parameter and one noise factor. The control factor has two levels: x0 and x1. The noise factor also has two levels: z1 and z2, wherez1 is within the normal use range andz2 is an elevated level. Letyij

(i= 0, 1; j = 1, 2) denote the life at xi and zj. Also, let Af 0 and Af 1 denote