Compound Noise - MIT Department of Mechanical Engineering

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388 J. SINGH ET AL. As discussed in a recent review paper by Robinson et al. 2 , research efforts in robust design have led to novel performance measures, new experimental designs, and alternatives to Taguchi’s methods building upon response surface methodology. One of the principal goals of recent research has been to reduce run sizes while still attaining good outcomes. The focus of this paper will be compound noise, which is also intended to reduce run sizes. Compound noise is a technique in which multiple noise factors are varied simultaneously as if they were a single noise factor. In most documented applications, an outer array of noise factors is replaced by the compound noise factor being varied between two levels. Taguchi 1 and Phadke 3 suggested forming compound noise factors based on the directionality of the noise factor effects thereby creating two conditions that represent, in some sense, opposite extremes. Du et al. 4 suggested forming compound noises based on the conditions of the two ‘most probable points of inverse reliability’. This approach to compounding is better able to account for skew in the distribution of system performance. This new concept of compound noise, like Taguchi’s, is based on two ‘extreme settings’ of noise factors. Hou 5 studied the conditions that will make compound noise yield robust settings for systems. Hou said ‘extreme settings should exist for compound noise to work’. We find below that compound noise can be effective even when extreme settings do not exist. The conditions mentioned in ‘compound noise factor theory’ turn out to be the sufficient conditions. In later sections we extend the analysis to determine conditions under which compound noise will predict a robust setting. Hou’s formulation was limited to systems that had active effects up to two-factor interactions. We extend the formulation to systems that can have active effects up to three-factor interactions. Compound noise can be considered as an extension of supersaturated designs (SSD). This concept initially originated with a paper by Satterthwaite 6 . SSDs were assumed to offer a potentially useful way to investigate many factors with few experiments. Holcomb and Carlyle 7 discussed the construction and evaluation of SSDs. Holcomb et al. 8 outlined the analysis of SSDs. Compound noise is an unbalanced SSD. Allen and Bernshteyn 9 discussed the advantages of unbalanced SSDs in terms of performance and affordability. Heyden et al. 10 argued that SSDs can be used to estimate variance of response, which can be used as a measure of robustness rather than using it to find main effects. SSDs ‘do not allow estimation of the effects of the individual factors because of confounding between the main effects’. However, ‘estimation of the separate factor effects is not necessarily required’ in improving robustness. Using compound noise as a robust design method we try to estimate the robustness of the system at a given control factor setting. The setting that improves this estimate is taken as the predicted robust setting. The aim of this study is to explore the effectiveness of compound noise as a robust design method. Engineering systems show certain regularities, one of which is the hierarchical ordering principle (see Wu and Hamada 11 and Chipman et al. 12 ). We classified systems based on the hierarchical ordering principle. The classes are as follows. • Strong hierarchy systems are those that have only main effects and two-factor interactions active. Some small three-factor interactions might be present in such systems, but they are not active. • Weak hierarchy systems are those that also have active three-factor interactions. We apply compound noise to three strong hierarchy systems and three weak hierarchy systems and analyze robustness gain. There are three main questions we want to address in this paper. • Why is compound noise effective in achieving a robust setting in certain cases, but ineffective in other cases • How can we measure the effectiveness of a compound noise strategy • Do we need to know the directionality of noise factors to use compound noise We analyze a compound noise factor strategy for six case studies and explore reasons for the effectiveness or ineffectiveness of the strategy. We then present the conditions for compound noise to work. These conditions are an extension of Hou 5 . First, we present conditions for strong and weak hierarchy systems. Second, we present the effectiveness of the compound noise strategy in real scenarios using fractional factorial arrays for control Copyright c○ 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398 DOI: 10.1002/qre
COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 389 Table I. Results from full factorial control factor array and two-level compound noise Matching of all control Matching of Measure Existence of factors with control factors of sparsity extreme Number of their robust with their Improvement System Hierarchy of effects settings (Hou 5 ) replications setting (%) robust setting (%) ratio (%) Op Amp Strong 0.105 Existing 100 96 99.2 99.2 PNM Strong 0.132 Non-existent 100 98 99 99.5 Journal bearing Strong 0.28 Existing 100 97.8 98.8 98.84 CSTR Weak 0.647 Non-existent 100 10 55.17 58.4 Temperature Weak 0.725 Non-existent 100 5 29.25 30.65 Control circuit Slider crank Weak 0.119 Existing 100 95 98 98.5 and noise factors. Finally, we present a procedure to determine an outer array for robust design experiments, which can be used by practitioners. CASE STUDIES We analyzed six case studies. Three of them exhibited strong hierarchy: passive neuron model (PNM; Tawfik and Durand 13 ), operational amplifier (Op Amp; Phadke 3 ) and journal bearing: half Sommerfeld solution (Hamrock et al. 14 ). Three case studies exhibited weak hierarchy: the continuous-stirred tank reactor (CSTR; Kalagnanam and Diwekar 15 ), the temperature control circuit (Phadke 3 ) and the slider crank (Gao et al. 16 ). For each case study we ran full factorial control and noise factor crossed array experiments. These runs were used to determine the effect coefficients for the main effects, two-factor interactions and three-factor interactions. Lenth’s method (Lenth 17 ) was employed to determine the active effects. The full factorial results were also used to determine the most robust setting for all six systems. In order to formulate compound noise for robust design experiments, we found the directionality of noise factors on the system response for all six systems. For each system we ran a full factorial design in control factors crossed with two-level compound noise. Pure experimental error was introduced into the system’s response. The error was of the order of the active main effects. From Lenth’s method, we can estimate the average magnitude of active main effects. Pure experimental error introduced into the system’s response was normally distributed with zero mean and standard deviation as the average magnitude of active main effects. The compound noise strategy performed extremely well for the PNM, Op Amp, journal bearing and slider crank. Table I the shows results from the robust design experiments. The third column of Table I shows the measure of sparsity of effects, which is the ratio of active effects in a system to the total number of effects that can be present in the system. The sparsity of effects means that among several experimental effects examined in any experiment, a small fraction will usually prove to be significant (Box and Meyer 18 and Wu and Hamada 11,19 ). The sixth column of Table I shows how well the predicted robust setting from compound noise experiments matches with the actual robust setting as found by carrying out full factorial control and noise array experiments. The seventh column of Table I shows the percentage of control factors whose robust setting matched for full factorial and compound noise experiments. At the predicted robust setting of control factors from compound noise experiments, we found the corresponding variance of the system’s response from full factorial experiments. Full factorial experiments yield the optimal variance of the system’s response and the average variance of the response at all control factor settings. With this knowledge, variance can be improved from the average value to the optimal value, which is the maximum possible improvement. For compound noise experiments the variance of the response can be improved from the average value to the variance at the predicted robust setting. The ratio of achieved Copyright c○ 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398 DOI: 10.1002/qre
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Compound Noise - MIT Department of Mechanical Engineering

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