پارامتراستواری در طراحی محصول و فرایند

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Robust parameter design and process robustness studies + Robust parameter design (RPD): an approach to product realization activities that emphasizes choosing the levels of controllable factors( parameters) in a process or product to achieve two objectives To ensure that the mean of output response is at a desired level or target To ensure that vailability around the Target value is as small as possible * When and PRD study is a conducted on a process it is usually called a process robustness a study * Developed by Genichi Taguchi (1980) DOX 6E Montgomery 1 

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Robust parameter design and process robustness studies A robust design problem usually Focuses on one or more of the following: Designing systems that are insensitive to environmental factors that can affect performance once the system is deployed in the field Designing products so that they are insensitive to the variability transmitted by the components of the system. Designing processes so that they're manufactured product will be as close as possible to the desired target specifications even though some process variables are impossible to control precisely. Determining the operating conditions for a process so that the crucial process characteristics are as close as possible to the desired target values and vailability around this target is minimized. DOX 6E Montgomery 2 

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Taguchi's approach is to construct separate designs in the controllable factors and noise factors, and to cross them. Both designs are based on orthogonal arrays, which include fractional factorial design matrices: an inner array design in the controllable factors an outer array design in the noise factors. The crossed design uses every combination of a treatment in the controllable factors and a treatment in the noise factors. Taguchi's analysis of the resulting data differs from the conventional statistical model. DOX 6E Montgomery 3 

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Robust parameter design and process robustness studies ¢ Before Taguchi (RPD was often done by overdesign-expensive) Controversy about experimental procedure and data analysis methods (Taguchi's my toes are usually inefficient or ineffective) Response surface methodology (RSM) was developed as an approach to the RPD problem Certain types of variables cause variability in the important system response variables( noise variables or uncountable variables) DOX 6E Montgomery 

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CROSSED ARRAY DESIGNS ¢ The original Taguchi methodology for the RPD problem revolved around the use of a statistical design for the controllable variables and another statistical design for the noise variables. Then these two designs were "crossed" This type of experimental design was called a crossed array design. DOX 6E Montgomery 

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CROSSED ARRAY DESIGNS ¢ An important point about the crossed array design is that it provides information about interactions between controllable factors and noise factors. These interactions are crucial to the solution of an RPD problem DOX 6E Montgomery 6 

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Table 12-1. The Leaf Spring Experiment ۸ ‏م ع و‎ E=- E=+ 0 2 = ‏تس‎ 170179781 750,725,712 754 0.090 + = + 8155187383 788,788,744 790 ۰ ۱ 0.001 132 .7.50 ,7.56 ,7.50 7.50 ,1.56 ,7.50 + - + 0.008 1.64 1.56 ,1.15 ,7.63 1.75 ,1.56 ,1.59 ~ - + + 0.074 7.60 7.88 ,8.00 ,7.54 + + - - 8 .۰ 179 8.06 ,8.08 ,769 =$ =$ 0.030 136 7.44 ,7.52 ,7.56 ب + 027 1.66 7.59 ,7.50 ,7,81 7.69 ,7.81 ,7.56 + + 3 + DOX 6E Montgomery 7 

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7 Variability ‏برها‎ * transmited Variability iny is reduced when x =~ Natural variability ing (a} No control x noise interaction {b) Significant control x noise interaction DOX 6E Montgomery 8 

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191 219 204 247 253 241 21.6 242 286 200 242 23 232 215 25 243 22 226 96 198 182 189 214 196 18.6 19.6 221 19.6, 197 26 210 256 147 168 118 BL 99 192 156 18.6 251 198 236 168 113 169 94 19. 189 194 20.0 184 151 93 DOX 6E Montgomery 95 16.2 167 114 18.6 163 191 156 199 156 150 163 183 197 16.2 164 142 161 (a) Inner Array ۸ #8 6 -1 - ‏1د‎ ‎-1 0 0 =] ‏له‎ tl 0 -1 0 0 0 41 0 #۶۲ ot +1 - +1 +1 0 -[ +1 +1 1 Run 

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10 Half-normal % probability 1 1 ۱ 1 1 822 818 8 0.06 3.00 ممع Figure 12-2 Half-normal plot of effect, mean free height response. DOX 6E Montgomery 

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ANALYSIS OF THE CROSSED ARRAY DESIGN * we summarize the data from a crossed array experiment with two statistics: the average of each observation in the inner array across all runs in the outer array and a summary statistic that attempted to combine information about the mean and variance, called the signal-to-noise ratio result in (1) the mean as close as possible to the desired target and (2) a maximum value of the signal-to-noise ratio. DOX 6E Montgomery 11 

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* Amore appropriate analysis for a crossed array design is to model the mean and variance of the response directly, where the sample mean and sample variance for each observation in the inner array is computed across all runs in the outer array. Consequently, choosing the levels of the controllable variables to optimize the mean and simultaneously minimize the variability is a valid approach. DOX 6E Montgomery 12 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH * If we wish to consider a first-order model involving the controllable variables, a logical model is Y= Bo ۵ + Bory + Brot titi + yang + Oye + © model, incorporating both controllable and noise variables, is often called a response model. Unless at least one of the regression coefficients 6,, and 6,, is nonzero, there will be no robust design problem DOX 6E Montgomery 13 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH An important advantage of the response model approach is that both the controllable factors and the noise: factors can be placed in a single experimental design; that is, the inner and outer array structure of the Taguchi approach can be avoided. We usually call the design containing both controllable and noise factors a combined array design DOX 6E Montgomery 14 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH we assume that noise variables are random variables, although they are controllable for purposes of an experiment. Specifically, we assume that the noise variables are expressed in coded units, that they have expected value zero, variance 62,, and if there are several noise variables, they have zero covariances. Under these assumptions, it is easy to finda model *.~ 5 5 taking DOX 6E Montgomery 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH ¢ Now the variance of y can be obtained by applying the variance operator across this last expression (without R). The resulting variance MVC y) = o2(y1 + ‏تروق + ریق‎ + 0? DOX 6E Montgomery 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH * 1. The mean and variance models involve only the controllable variables. * 2. it also involves the interaction regression coeflcients between the controllable and noise variables. This is how the noise variable influences the response. ¢ 3. The variance model is a quadratic function of the controllable variables * 4. The variance model (apart from 67)is just the square of the slope of the fitted response model in the direction of the noise variable DOX 6E Montgomery 17 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH ¢ To use these models operationally, we would ¢ 1. Perform an experiment and fit an appropriate response model, such as Equation * 2. Replace the unknown regression coefficients in the mean and variance models with their least squares estimates from the response model and replace 6? in the variance model by the residual mean square found when fitting the response model. ¢ 3. Optimize the mean and variance model using the standard multiple response optimization methods DOX 6E Montgomery 18 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH ٠ It is very easy to generalize these results. Suppose that there are k controllable variables and r noise variables. We will write the general response model involving these variables ‏مور‎ z) = f(x) + h(x, 2) + ‏ع‎ DOX 6E Montgomery 19 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH ¢ where f (x) is the portion of the model that involves only the controllable variables and h(x, z) are the terms involving the main effects of the noise factors and the interactions between the controllable and noise factors. ‏ماس مسي لصاف الوم زور[‎ lig A(x, 2) = y ye; + > > 8 yx) ¢ then the mean r7*-" *~*“- ~esponse is just ELy(%, 2)] = fo) DOX 6E Montgomery 20 

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COMBINED ARRAY DESIGNS AND THE RESPONSE MODEL APPROACH * then the mean model for the response is just EL yx, 2] = fx) respoi VX) = 2 il * and tl ۱ ‏تج‎ ap de ae DOX 6E Montgomery 21 

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CHOICE OF DESIGNS * The selection of the experimental design is a very important aspect of an RPD problem. Generally, the combined array approach will result in smaller designs that will be obtained with a crossed array. Also, the response modeling approach allows direct incorporation of the controllable factor-noise factor interactions, which is usually superior to direct mean and variance modeling. If all of the design factors are at two levels, a resolution V design is a good choice for an RPD study DOX 6E Montgomery 22