WEBVTT 00:00:00.000 --> 00:00:02.460 SPEAKER: The following content is provided under a Creative 00:00:02.460 --> 00:00:03.760 Commons license. 00:00:03.760 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 00:00:10.090 continue to offer high quality educational resources for free. 00:00:10.090 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.560 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.560 --> 00:00:17.904 at ocw.mit.edu. 00:00:21.383 --> 00:00:22.800 DUANE BONING: So what I want to do 00:00:22.800 --> 00:00:26.550 today is pick up a little bit from last time, 00:00:26.550 --> 00:00:29.970 round our discussion of process optimization 00:00:29.970 --> 00:00:33.690 with a little bit of a different perspective today. 00:00:33.690 --> 00:00:36.540 Last time we talked about response surface methods 00:00:36.540 --> 00:00:41.520 and using those in optimization. 00:00:41.520 --> 00:00:44.970 What I want to do today is especially focus on-- 00:00:47.970 --> 00:00:49.620 geez. 00:00:49.620 --> 00:00:51.870 This arrow is not letting me draw. 00:00:51.870 --> 00:00:56.220 I want to focus on process robustness, 00:00:56.220 --> 00:00:58.430 and that's a little bit of a different perspective. 00:00:58.430 --> 00:01:01.140 So far, most of the optimization we talked about 00:01:01.140 --> 00:01:05.190 was looking at getting a model of the mean response 00:01:05.190 --> 00:01:12.390 and then using that to drive the process to an optimum target, 00:01:12.390 --> 00:01:13.530 if you will. 00:01:13.530 --> 00:01:15.630 So we want to come back a little bit to ideas 00:01:15.630 --> 00:01:19.980 of things like CPK, quality loss function, and so on, 00:01:19.980 --> 00:01:23.760 that start to fold in additional requirements 00:01:23.760 --> 00:01:27.930 or additional goals in the optimization. 00:01:27.930 --> 00:01:30.030 Coming out of this is little bit more 00:01:30.030 --> 00:01:34.260 of an explicit consideration of variation modeling, 00:01:34.260 --> 00:01:36.630 not simply the mean response, but also 00:01:36.630 --> 00:01:38.370 thinking about this assumption that we've 00:01:38.370 --> 00:01:41.280 made that the variance is the same everywhere 00:01:41.280 --> 00:01:44.460 across the process and what to do 00:01:44.460 --> 00:01:48.150 when that is not the case, because in fact, it often 00:01:48.150 --> 00:01:49.910 is not the case. 00:01:49.910 --> 00:01:53.700 So looking at the family of things 00:01:53.700 --> 00:01:57.180 that we might seek to optimize, there 00:01:57.180 --> 00:01:59.160 are really multiple things. 00:01:59.160 --> 00:02:01.290 Right? 00:02:01.290 --> 00:02:08.039 Some of which are easy to make in a quantitative fashion 00:02:08.039 --> 00:02:11.100 assessment of, models of, and then optimize against 00:02:11.100 --> 00:02:14.160 and some that are a little bit more qualitative 00:02:14.160 --> 00:02:17.470 or additional factors. 00:02:17.470 --> 00:02:25.500 So certainly things like quality is the main effect 00:02:25.500 --> 00:02:26.850 that we've been looking at. 00:02:26.850 --> 00:02:29.520 That is to say, we're looking at the response 00:02:29.520 --> 00:02:35.250 of one or more outputs as a function of one or more control 00:02:35.250 --> 00:02:38.700 factors or inputs and then seeking 00:02:38.700 --> 00:02:47.190 to optimize the overall yield, overall output of the process. 00:02:47.190 --> 00:02:49.140 You want to be on target but you also 00:02:49.140 --> 00:02:53.820 want to be as immune as possible to variance. 00:02:53.820 --> 00:02:55.990 And we talked earlier in the term, not so much 00:02:55.990 --> 00:02:58.350 last time on Tuesday, but earlier 00:02:58.350 --> 00:03:03.890 in the term about CPK and quality loss functions. 00:03:03.890 --> 00:03:08.540 But also, of course, another key driver in all manufacturing 00:03:08.540 --> 00:03:14.120 is cost, and there are costs associated with yield, 00:03:14.120 --> 00:03:14.760 of course. 00:03:14.760 --> 00:03:19.380 You're only getting revenue for things that are good. 00:03:19.380 --> 00:03:22.050 But there's also other cost factors, for example, 00:03:22.050 --> 00:03:24.410 picking of different points in the operating space 00:03:24.410 --> 00:03:29.560 may imply different things about cost of running of the process. 00:03:29.560 --> 00:03:32.540 And in fact, one of the important factors of cost 00:03:32.540 --> 00:03:35.300 is associated with rate. 00:03:35.300 --> 00:03:38.870 Very often your process factors or what operating 00:03:38.870 --> 00:03:44.300 point you pick for your process has a direct parts per hour 00:03:44.300 --> 00:03:50.000 or other kind of throughput or rate implication, right? 00:03:50.000 --> 00:03:52.520 In fact, you'll very often have this trade off 00:03:52.520 --> 00:03:56.630 where you might expect if you run things faster, 00:03:56.630 --> 00:03:59.540 your quality might be a little bit reduced. 00:03:59.540 --> 00:04:02.510 Whereas if you run things at a reduced rate, 00:04:02.510 --> 00:04:06.080 you may be able to boost up yield or quality, 00:04:06.080 --> 00:04:07.850 but there's a trade off then in terms 00:04:07.850 --> 00:04:10.190 of how much production you can get out 00:04:10.190 --> 00:04:17.490 or how rapidly you can get that out. 00:04:17.490 --> 00:04:19.709 In many of these cases, if you can actually 00:04:19.709 --> 00:04:25.640 fold all of those down into a direct dollar trade off 00:04:25.640 --> 00:04:28.300 then you can start to formulate an overall cost function 00:04:28.300 --> 00:04:31.480 or objective function, sort of an overall J 00:04:31.480 --> 00:04:35.170 function in our terminology from last time, that might actually 00:04:35.170 --> 00:04:37.330 trade off some of those things. 00:04:37.330 --> 00:04:39.130 But often, it's actually quite difficult 00:04:39.130 --> 00:04:42.730 to get really detailed information 00:04:42.730 --> 00:04:46.340 and build models of all of these factors. 00:04:46.340 --> 00:04:50.880 Now an additional factor is flexibility. 00:04:55.150 --> 00:04:58.420 That's a little bit harder to get quantitative about. 00:04:58.420 --> 00:05:02.740 What do you think we mean by flexibility? 00:05:02.740 --> 00:05:05.950 I mean, we could find the process optimum, 00:05:05.950 --> 00:05:09.048 lock in our equipment, lock down the control knobs, 00:05:09.048 --> 00:05:11.590 and say that's where we're going to operate for the next four 00:05:11.590 --> 00:05:12.700 years. 00:05:12.700 --> 00:05:14.420 That would probably not be very flexible. 00:05:17.550 --> 00:05:22.020 But give me some thoughts on what kind of flexibility 00:05:22.020 --> 00:05:23.950 you might want in a process? 00:05:23.950 --> 00:05:26.058 AUDIENCE: Change over time, like the process is 00:05:26.058 --> 00:05:27.590 easy to modify and change. 00:05:27.590 --> 00:05:28.590 DUANE BONING: Excellent. 00:05:28.590 --> 00:05:32.680 Yeah, so the idea there was change over time. 00:05:32.680 --> 00:05:34.090 So a process. 00:05:34.090 --> 00:05:36.840 So some of those might be goals actually in the equipment 00:05:36.840 --> 00:05:39.970 design or the line design. 00:05:39.970 --> 00:05:42.260 Not necessarily picking an operating point, 00:05:42.260 --> 00:05:45.940 but definitely flexibility is the ability 00:05:45.940 --> 00:05:47.620 to rapidly do changeovers. 00:05:47.620 --> 00:05:49.600 By the way, that often folds into some 00:05:49.600 --> 00:05:51.610 of these other factors, right? 00:05:51.610 --> 00:05:56.710 Can affect cost, quality, rate, change over time, 00:05:56.710 --> 00:06:01.420 how rapidly you do that, can have an impact on throughput. 00:06:01.420 --> 00:06:04.150 Changeover sometimes requires additional steps 00:06:04.150 --> 00:06:10.540 to warm up or prove in the new changeover. 00:06:10.540 --> 00:06:15.560 So furthermore if your equipment requires 00:06:15.560 --> 00:06:17.615 a little bit of adjustment after a changeover, 00:06:17.615 --> 00:06:20.780 it can have a quality impact. 00:06:20.780 --> 00:06:21.590 That's a good one. 00:06:21.590 --> 00:06:22.370 I like that. 00:06:22.370 --> 00:06:23.721 Other kinds of flexibility? 00:06:23.721 --> 00:06:25.346 AUDIENCE: You could [INAUDIBLE] and try 00:06:25.346 --> 00:06:29.200 to make it more insensitive to the exact operating point 00:06:29.200 --> 00:06:32.192 to try to [INAUDIBLE] development, rather, 00:06:32.192 --> 00:06:38.620 something that gives more room for the process 00:06:38.620 --> 00:06:41.050 to [INAUDIBLE] on either side [INAUDIBLE].. 00:06:43.502 --> 00:06:45.460 DUANE BONING: Yeah, so what Neil is just saying 00:06:45.460 --> 00:06:49.960 is you might want to process that is perhaps 00:06:49.960 --> 00:06:53.110 less sensitive at an optimum. 00:06:53.110 --> 00:06:58.720 So this would be less sensitive so that as your inputs vary, 00:06:58.720 --> 00:07:01.470 your outputs would vary less. 00:07:01.470 --> 00:07:06.340 I think that's an important characteristic. 00:07:06.340 --> 00:07:07.960 That's a notion of robustness. 00:07:07.960 --> 00:07:12.460 I'm not sure I would associate that with flexibility, though. 00:07:12.460 --> 00:07:14.150 I think it's a slightly different idea. 00:07:14.150 --> 00:07:15.106 I mean, certainly-- 00:07:19.270 --> 00:07:22.750 I'm trying to think how to think of that as-- 00:07:22.750 --> 00:07:24.400 it may be that what you would like 00:07:24.400 --> 00:07:27.400 is a process where this same characteristic might 00:07:27.400 --> 00:07:31.580 be true of multiple operating points 00:07:31.580 --> 00:07:34.390 so that one could rapidly, rapidly 00:07:34.390 --> 00:07:38.050 tune in the process, rapidly change over the process. 00:07:38.050 --> 00:07:38.740 Yeah. 00:07:38.740 --> 00:07:40.901 AUDIENCE: Different rates. 00:07:40.901 --> 00:07:42.200 Output rates. 00:07:42.200 --> 00:07:44.160 DUANE BONING: Different output rates. 00:07:44.160 --> 00:07:48.580 So what would the benefit of that be? 00:07:48.580 --> 00:07:51.290 AUDIENCE: [INAUDIBLE] demand. 00:07:51.290 --> 00:07:52.950 We don't want to overproduce. 00:08:02.757 --> 00:08:03.840 DUANE BONING: Interesting. 00:08:03.840 --> 00:08:04.380 Sure. 00:08:04.380 --> 00:08:09.230 So rate flexibility might be another characteristic. 00:08:09.230 --> 00:08:14.200 So I think we could identify multiple of these flexibility 00:08:14.200 --> 00:08:15.730 ideas. 00:08:15.730 --> 00:08:19.670 I think these are really important characteristics, 00:08:19.670 --> 00:08:23.510 especially in overall product design as well 00:08:23.510 --> 00:08:26.090 as process design. 00:08:26.090 --> 00:08:28.190 We haven't really talked that much about it. 00:08:28.190 --> 00:08:30.200 We're really focused mostly, in this subject, 00:08:30.200 --> 00:08:34.789 on the manufacturing process control and the quality issues. 00:08:34.789 --> 00:08:41.179 But I did want to have that in there just as a reminder 00:08:41.179 --> 00:08:43.400 that there's a lot of these other kinds of factors 00:08:43.400 --> 00:08:49.780 that are important in a manufacturing setting. 00:08:49.780 --> 00:08:51.460 But let's focus a little bit, look 00:08:51.460 --> 00:08:57.700 at minimizing, well, quality loss 00:08:57.700 --> 00:09:01.330 and thinking a little bit more about other cost factors. 00:09:01.330 --> 00:09:03.940 We've been saying that one of the important goals 00:09:03.940 --> 00:09:07.330 is we want the-- 00:09:07.330 --> 00:09:10.040 I'm not sure I would use x bar here. 00:09:10.040 --> 00:09:11.140 Let's call that y bar. 00:09:11.140 --> 00:09:14.410 If y is my output, I want my y bar output 00:09:14.410 --> 00:09:18.440 to at least hit or come as close as possible to some target. 00:09:18.440 --> 00:09:23.020 So again, if we assume that there is, in fact, some spread 00:09:23.020 --> 00:09:27.070 in the manufacturing process, I can never completely squeeze 00:09:27.070 --> 00:09:28.400 out all the variance. 00:09:28.400 --> 00:09:31.810 We have our typical CPK-like picture 00:09:31.810 --> 00:09:33.700 where we've got some target, we've 00:09:33.700 --> 00:09:36.830 got spec limits, upper and lower, 00:09:36.830 --> 00:09:40.420 and again, we want to maximize yield. 00:09:40.420 --> 00:09:42.610 So one important characteristic of that 00:09:42.610 --> 00:09:46.120 is getting on to target. 00:09:46.120 --> 00:09:52.510 It turns out if you have a fairly rich process-- 00:09:52.510 --> 00:09:57.220 rich in the sense that you've got multiple inputs that you 00:09:57.220 --> 00:09:59.260 can use to affect the process-- 00:09:59.260 --> 00:10:01.300 maybe you also have multiple outputs. 00:10:01.300 --> 00:10:06.160 But for the moment imagine that I've got one output of concern, 00:10:06.160 --> 00:10:13.080 but I've got more than one x control factor or input factor. 00:10:13.080 --> 00:10:16.630 Conceptually do you think there's 00:10:16.630 --> 00:10:20.440 going to be more than one solution to or just 00:10:20.440 --> 00:10:24.070 one solution to a combination of those input factors 00:10:24.070 --> 00:10:27.730 to hit the particular output? 00:10:27.730 --> 00:10:31.580 If I only had one input and one output, 00:10:31.580 --> 00:10:33.160 seems like most likely I'm only going 00:10:33.160 --> 00:10:35.300 to have one setting for that input that 00:10:35.300 --> 00:10:38.450 is going to correspond to a maximum on that output. 00:10:38.450 --> 00:10:43.830 But if I've got multiple ones and they have some influence, 00:10:43.830 --> 00:10:46.130 you got an interesting situation here, right? 00:10:46.130 --> 00:10:51.380 You end up with multiple solutions, if you will, 00:10:51.380 --> 00:10:54.310 to the maximum output problem. 00:10:54.310 --> 00:10:57.610 That is to say you may have contours 00:10:57.610 --> 00:11:02.830 in the multidimensional space, an x1, x2, y space 00:11:02.830 --> 00:11:06.230 that all achieve the same output. 00:11:06.230 --> 00:11:08.270 Turns out that's wonderful. 00:11:08.270 --> 00:11:11.430 It means, well, yes, you can match to target. 00:11:11.430 --> 00:11:14.990 But now that second input or the additional inputs 00:11:14.990 --> 00:11:17.780 give you a little bit of freedom to try 00:11:17.780 --> 00:11:23.010 to achieve other goals as well. 00:11:23.010 --> 00:11:25.790 Now that's assuming that sort of the match to target 00:11:25.790 --> 00:11:29.180 is the paramount goal, and getting 00:11:29.180 --> 00:11:32.060 close to that is pretty close to a paramount goal because 00:11:32.060 --> 00:11:35.510 of the direct effect on yield and quality loss, 00:11:35.510 --> 00:11:38.120 but it wouldn't necessarily have to be so. 00:11:38.120 --> 00:11:39.800 But here I'm sort of assuming you've 00:11:39.800 --> 00:11:41.870 got to get to target first, and then 00:11:41.870 --> 00:11:46.130 after that it's an opportunity to choose your operating 00:11:46.130 --> 00:11:49.370 point to do other things. 00:11:49.370 --> 00:11:51.950 To do things like minimize the cycle time 00:11:51.950 --> 00:11:55.190 to get as fast a rate as possible 00:11:55.190 --> 00:11:59.120 or to start to fold in other qualitative 00:11:59.120 --> 00:12:02.680 or quantitative cost drivers. 00:12:02.680 --> 00:12:04.630 A couple of simple examples might 00:12:04.630 --> 00:12:07.720 be at different operating points you might actually 00:12:07.720 --> 00:12:14.770 use more input materials or, say, manufacturing 00:12:14.770 --> 00:12:17.020 gases in a plasma reactor. 00:12:17.020 --> 00:12:18.640 At different operating points, you 00:12:18.640 --> 00:12:22.630 might be more or less efficient in the use of energy 00:12:22.630 --> 00:12:25.160 or other material resources. 00:12:25.160 --> 00:12:28.670 Another factor might be things like at different operating 00:12:28.670 --> 00:12:32.510 points, it has different side effects on the equipment, 00:12:32.510 --> 00:12:35.764 like wear of a machining tool. 00:12:35.764 --> 00:12:36.620 OK? 00:12:36.620 --> 00:12:42.380 So these are places where you can start to add those in. 00:12:42.380 --> 00:12:45.970 And that actually means that you really 00:12:45.970 --> 00:12:49.990 would like to have multivariable models in the sense 00:12:49.990 --> 00:12:54.910 of some output y that in fact, does let you trade off 00:12:54.910 --> 00:12:57.413 two or more inputs. 00:12:57.413 --> 00:12:59.080 And this is just a visualization of what 00:12:59.080 --> 00:13:01.250 I was just describing earlier. 00:13:01.250 --> 00:13:02.860 Here's a very simple linear model, 00:13:02.860 --> 00:13:05.970 two inputs with an interaction. 00:13:05.970 --> 00:13:11.960 And now let's say that the target value for y is 5. 00:13:11.960 --> 00:13:15.590 Well, that plane intersects with my response surface 00:13:15.590 --> 00:13:18.740 along some line, and you can sort of 00:13:18.740 --> 00:13:21.050 see that dashed line there. 00:13:21.050 --> 00:13:26.810 That indicates any combination of the x1 and x2 factors 00:13:26.810 --> 00:13:29.750 that can achieve that output. 00:13:29.750 --> 00:13:36.670 And so now, in fact, to find and decide what x1 and x2 00:13:36.670 --> 00:13:38.710 I'm going to use, I actually need 00:13:38.710 --> 00:13:41.140 to think about some other criteria. 00:13:41.140 --> 00:13:42.610 Right? 00:13:42.610 --> 00:13:45.940 I guess if it doesn't matter, you can pick any of them 00:13:45.940 --> 00:13:51.960 and you might randomly pick. 00:13:51.960 --> 00:13:54.630 I guess randomly picking is about the only thing you 00:13:54.630 --> 00:13:57.900 can do where you're not, I think, intuitively 00:13:57.900 --> 00:14:02.010 applying some additional filter or some additional criteria 00:14:02.010 --> 00:14:03.810 to the x1 and x2. 00:14:03.810 --> 00:14:06.750 So again, some of these might be picking x1 and x2 based 00:14:06.750 --> 00:14:11.630 on things like cost or rate. 00:14:11.630 --> 00:14:14.050 What are some other things that you 00:14:14.050 --> 00:14:21.580 might use to decide on which point along this space 00:14:21.580 --> 00:14:25.155 you would actually like to pick for an operating point? 00:14:25.155 --> 00:14:25.655 Yeah? 00:14:25.655 --> 00:14:27.140 AUDIENCE: Minimum variability. 00:14:27.140 --> 00:14:28.590 DUANE BONING: Minimum variability. 00:14:28.590 --> 00:14:30.920 Good. 00:14:30.920 --> 00:14:34.010 And we will talk about that for sure. 00:14:34.010 --> 00:14:35.740 Excellent. 00:14:35.740 --> 00:14:42.472 AUDIENCE: If you were having [INAUDIBLE] x1 but not x2, 00:14:42.472 --> 00:14:48.790 then you will use x1 to control most of the [INAUDIBLE].. 00:14:48.790 --> 00:14:50.230 DUANE BONING: Interesting. 00:14:50.230 --> 00:14:53.060 Yeah, so that's getting at the notion of flexibility, 00:14:53.060 --> 00:14:56.320 so the idea was if I had some other product that 00:14:56.320 --> 00:14:58.600 also was using x1 and x2, there might 00:14:58.600 --> 00:15:03.190 be value in keeping one of those variables constant 00:15:03.190 --> 00:15:05.920 or using it to optimize all of them 00:15:05.920 --> 00:15:10.660 and then focusing in and letting one variable be more associated 00:15:10.660 --> 00:15:12.100 with the flexibility, if you will, 00:15:12.100 --> 00:15:16.090 the thing that you use to tune in across different products 00:15:16.090 --> 00:15:18.730 or different processes. 00:15:18.730 --> 00:15:20.290 I like that. 00:15:20.290 --> 00:15:20.958 Yep? 00:15:20.958 --> 00:15:22.250 AUDIENCE: Physical constraints. 00:15:22.250 --> 00:15:26.282 Your machine might not be able to go that fast. 00:15:26.282 --> 00:15:28.240 DUANE BONING: Physical constraints, absolutely. 00:15:28.240 --> 00:15:32.830 So depends on whether these are rate or other things in terms 00:15:32.830 --> 00:15:35.620 of whether the machine can go that fast, 00:15:35.620 --> 00:15:39.010 but certainly you may have explored 00:15:39.010 --> 00:15:42.220 a space in these coded variables in x1 and x2 00:15:42.220 --> 00:15:47.140 that cover this broad range, but perhaps your equipment really 00:15:47.140 --> 00:15:50.810 doesn't want to run out here at one of the extreme low points. 00:15:50.810 --> 00:15:56.220 So that might actually drive you away from these extreme points 00:15:56.220 --> 00:16:01.870 or maybe it drives you towards one of those extreme points. 00:16:01.870 --> 00:16:04.860 In general, I think-- 00:16:04.860 --> 00:16:07.410 let's see, I believe one of the case 00:16:07.410 --> 00:16:09.690 studies-- it's either case study two or three. 00:16:09.690 --> 00:16:12.990 Dave Hardt will come in and talk about run by run control, 00:16:12.990 --> 00:16:15.180 a little bit of use of some of these kinds of models 00:16:15.180 --> 00:16:16.860 for feedback control. 00:16:16.860 --> 00:16:22.110 And in general, an observation that comes out of that is you 00:16:22.110 --> 00:16:24.390 might want to avoid operating at one 00:16:24.390 --> 00:16:28.110 of the boundaries of your process, right? 00:16:28.110 --> 00:16:31.470 If you were, in fact, picking one of these points that's 00:16:31.470 --> 00:16:35.795 at the edge of the space that you used to build your model, 00:16:35.795 --> 00:16:37.170 well, you're not really sure what 00:16:37.170 --> 00:16:40.950 happens if you adjust x to a little bit beyond that space. 00:16:40.950 --> 00:16:44.430 You might hope that the process continues to be well behaved 00:16:44.430 --> 00:16:47.370 and extrapolate out into that region. 00:16:47.370 --> 00:16:50.040 That's a little bit scarier to do. 00:16:50.040 --> 00:16:53.670 But in general, if you pick a point that's 00:16:53.670 --> 00:16:58.530 closer to somewhere in the interior, 00:16:58.530 --> 00:17:01.500 almost maybe even the center of your space, 00:17:01.500 --> 00:17:04.050 that gives you the maximum freedom 00:17:04.050 --> 00:17:07.950 in an active control setting to make small modifications 00:17:07.950 --> 00:17:15.270 to your x1, x2 factors to drive the process back onto target. 00:17:15.270 --> 00:17:18.990 So I would be inclined to also try 00:17:18.990 --> 00:17:22.230 to sort of keep my factors in a place 00:17:22.230 --> 00:17:26.700 where I know I can keep moving them to control the process. 00:17:26.700 --> 00:17:28.493 AUDIENCE: [INAUDIBLE],, doesn't it 00:17:28.493 --> 00:17:30.160 make sense that if possible you actually 00:17:30.160 --> 00:17:32.616 need to test further out because you could 00:17:32.616 --> 00:17:35.936 have more optimal conditions if you 00:17:35.936 --> 00:17:39.038 could, say, push the boundary? 00:17:39.038 --> 00:17:39.830 DUANE BONING: Yeah. 00:17:39.830 --> 00:17:41.550 So the question-- yeah, yeah. 00:17:41.550 --> 00:17:44.640 The question is kind of a general one 00:17:44.640 --> 00:17:47.910 about response surface models and optimal points. 00:17:47.910 --> 00:17:51.530 Is if you are on some boundary-- 00:17:51.530 --> 00:17:56.970 well, first off if I have linear models, 00:17:56.970 --> 00:18:00.150 very often you get to the constraints 00:18:00.150 --> 00:18:02.880 being a boundary especially if I fold in something, 00:18:02.880 --> 00:18:05.370 some additional factor. 00:18:05.370 --> 00:18:08.370 And so you could always ask the question, 00:18:08.370 --> 00:18:10.680 should I extend my experimental space 00:18:10.680 --> 00:18:12.750 and explore a little bit further? 00:18:12.750 --> 00:18:17.760 And the answer is if your variables naturally 00:18:17.760 --> 00:18:22.200 can go beyond your initial guess of the range of exploration, 00:18:22.200 --> 00:18:23.580 absolutely. 00:18:23.580 --> 00:18:25.560 And as we saw last time, for example, 00:18:25.560 --> 00:18:28.290 in incremental or iterative optimization, 00:18:28.290 --> 00:18:30.270 you may have actually already-- 00:18:30.270 --> 00:18:32.520 I guess I'm sort of assuming you've already 00:18:32.520 --> 00:18:39.450 explored the space to cover the optimum region as much as you 00:18:39.450 --> 00:18:41.400 can. 00:18:41.400 --> 00:18:45.500 That's a good observation as well. 00:18:45.500 --> 00:18:46.000 Good. 00:18:46.000 --> 00:18:46.850 Any others? 00:18:46.850 --> 00:18:47.570 Any other ideas? 00:18:51.000 --> 00:18:51.540 OK. 00:18:51.540 --> 00:18:52.040 Good. 00:18:52.040 --> 00:18:52.540 Good. 00:18:52.540 --> 00:18:55.290 I did want to hit on some of these. 00:18:55.290 --> 00:18:57.600 Now we'll come back to this minimum variance 00:18:57.600 --> 00:19:02.160 because that is a great idea, right? 00:19:02.160 --> 00:19:05.490 So far here this picture is still looking at, even 00:19:05.490 --> 00:19:08.820 in this picture, it's still looking at just output 00:19:08.820 --> 00:19:10.980 and matching that output to the target, 00:19:10.980 --> 00:19:14.970 that shaded plane is hitting the target. 00:19:14.970 --> 00:19:19.560 And a bigger goal [INAUDIBLE],, even just focused on quality 00:19:19.560 --> 00:19:25.560 is overall yield and minimizing either the effective variation 00:19:25.560 --> 00:19:27.640 or minimizing the variation. 00:19:27.640 --> 00:19:32.970 So if we go back to our quality and variation equation, 00:19:32.970 --> 00:19:37.380 my output can vary due to a couple of different factors. 00:19:37.380 --> 00:19:44.940 We said alpha are sort of noise factors, 00:19:44.940 --> 00:19:47.660 and these are the actual process settings 00:19:47.660 --> 00:19:49.655 if you will or control factors. 00:19:53.320 --> 00:19:55.470 And so we can see that they'll have 00:19:55.470 --> 00:19:59.730 some sensitivity of the process to each 00:19:59.730 --> 00:20:01.140 of these kinds of factors. 00:20:01.140 --> 00:20:05.410 If I tweak my x factor or u factor a little bit, 00:20:05.410 --> 00:20:09.960 my y is going to change some amount and normally I can 00:20:09.960 --> 00:20:14.260 control or I have power over those process settings, 00:20:14.260 --> 00:20:17.460 those options, but sometimes I don't. 00:20:17.460 --> 00:20:20.310 Sometimes I can dial it in, but I don't always 00:20:20.310 --> 00:20:22.650 get exactly what I dialed in. 00:20:22.650 --> 00:20:27.780 So there might be differences in sensitivity of the output 00:20:27.780 --> 00:20:30.600 where little deviations in x, especially 00:20:30.600 --> 00:20:33.720 with a non-linear or polynomial kind of model, 00:20:33.720 --> 00:20:38.400 might actually imply different amounts of delta y 00:20:38.400 --> 00:20:41.200 at different operating points. 00:20:41.200 --> 00:20:45.835 So that's an example where our assumption about-- 00:20:45.835 --> 00:20:47.460 let me just get some of you down here-- 00:20:50.190 --> 00:20:52.980 equal variance everywhere in the process space 00:20:52.980 --> 00:20:56.100 might actually not apply. 00:20:56.100 --> 00:20:59.350 But also, we have these noise factors. 00:20:59.350 --> 00:21:05.240 We have these variation sources, uncontrollable factors, 00:21:05.240 --> 00:21:07.210 things that we can't completely eliminate, 00:21:07.210 --> 00:21:10.480 and we might want to find an operating 00:21:10.480 --> 00:21:16.580 point where we minimize the sensitivity to those noise 00:21:16.580 --> 00:21:18.030 factors. 00:21:18.030 --> 00:21:18.930 OK? 00:21:18.930 --> 00:21:20.730 So one way of thinking about this as we 00:21:20.730 --> 00:21:24.180 look back at just a sensitivity kind of analysis, 00:21:24.180 --> 00:21:30.210 where can I look at either using my response surface 00:21:30.210 --> 00:21:35.400 model, which usually has this in it, 00:21:35.400 --> 00:21:40.200 and minimize some sensitivity metric on that 00:21:40.200 --> 00:21:44.370 or start to try to deal a little bit with these unmodeled noise 00:21:44.370 --> 00:21:46.020 factors. 00:21:46.020 --> 00:21:47.970 One approach here we're going to talk about 00:21:47.970 --> 00:21:50.900 is, what if I want to include those in the model? 00:21:50.900 --> 00:21:52.770 I actually want to understand how 00:21:52.770 --> 00:21:58.790 variance may change as a function of operating point. 00:21:58.790 --> 00:22:01.370 This is going to be a little bit overcoming the assumptions 00:22:01.370 --> 00:22:03.370 that we made. 00:22:03.370 --> 00:22:05.000 Now how can I get minimum variation? 00:22:05.000 --> 00:22:07.400 I mean, so one approach here is think about it 00:22:07.400 --> 00:22:09.860 from the sensitivity point of view. 00:22:09.860 --> 00:22:13.100 The other is think about it from a CPK perspective. 00:22:15.840 --> 00:22:16.470 You know? 00:22:16.470 --> 00:22:19.860 And we talked about needing to center the process as one 00:22:19.860 --> 00:22:24.480 part of that, but the other part would be the variance part. 00:22:24.480 --> 00:22:28.590 And again, here if I'm doing anything other 00:22:28.590 --> 00:22:31.680 than mean centering, I have to know how the variance actually 00:22:31.680 --> 00:22:34.020 depends on anything. 00:22:34.020 --> 00:22:37.500 Our assumptions all along in most of what we've done so far, 00:22:37.500 --> 00:22:41.940 ANOVA and other kinds of tools that we've looked at, 00:22:41.940 --> 00:22:45.060 have sort of assumed that the variance was constant. 00:22:45.060 --> 00:22:49.207 We want to back off and relax that as well. 00:22:49.207 --> 00:22:51.540 And another perspective that we'll look for a little bit 00:22:51.540 --> 00:22:56.070 here is to look back at the quality loss function, which 00:22:56.070 --> 00:22:57.510 can fold in together. 00:22:57.510 --> 00:22:59.310 Remember the quality loss function 00:22:59.310 --> 00:23:06.750 that says, as I deviate my output from my target I 00:23:06.750 --> 00:23:09.840 incur more and more loss? 00:23:09.840 --> 00:23:16.230 It actually has a penalty, not just it's good as long 00:23:16.230 --> 00:23:19.890 as it's above the lower spec limit 00:23:19.890 --> 00:23:23.310 and below the upper spec limit, but as I deviate 00:23:23.310 --> 00:23:28.440 from the overall target I have continuing incremental worse 00:23:28.440 --> 00:23:31.920 behavior in the product. 00:23:31.920 --> 00:23:34.320 So that's sort of a generalization a little bit 00:23:34.320 --> 00:23:41.940 of the focus away from just the target to also the variance. 00:23:41.940 --> 00:23:45.460 So what this is saying is for many of these approaches, 00:23:45.460 --> 00:23:51.640 whether you look at it from a CPK or a quality loss function, 00:23:51.640 --> 00:23:53.970 you need to know more about the variance 00:23:53.970 --> 00:23:57.270 or you need to know more about the influence of the noise 00:23:57.270 --> 00:24:01.650 factors in order to really get to process robustness. 00:24:01.650 --> 00:24:03.300 With what we've done so far already, 00:24:03.300 --> 00:24:05.130 you could do a little bit of robustness 00:24:05.130 --> 00:24:06.930 just with the response surface models 00:24:06.930 --> 00:24:11.520 just in terms of reaction to modelled input, 00:24:11.520 --> 00:24:15.240 but that's a pretty weak notion of robustness. 00:24:15.240 --> 00:24:17.340 That's only looking at some of the variances. 00:24:19.890 --> 00:24:22.440 Well, let's explore the CPK one. 00:24:22.440 --> 00:24:25.270 This is actually all just reminder. 00:24:25.270 --> 00:24:32.230 We've said CPK is a nice measure that looks at two effects, 00:24:32.230 --> 00:24:34.260 right? 00:24:34.260 --> 00:24:44.270 So the CP has an effect from the deviation 00:24:44.270 --> 00:24:49.130 from the mean of your process where the true mean is 00:24:49.130 --> 00:24:52.250 and how far away it is from your spec limits, 00:24:52.250 --> 00:24:55.310 so that's sort of a centering or a target point, 00:24:55.310 --> 00:24:58.340 and then it also has a notion of folding 00:24:58.340 --> 00:25:01.340 in the spread in your process, right? 00:25:01.340 --> 00:25:07.850 So one could, in fact, formulate something like CP or CPK 00:25:07.850 --> 00:25:14.030 as your cost function, your penalty function, your J, 00:25:14.030 --> 00:25:16.710 and try to build a model-- 00:25:16.710 --> 00:25:20.300 either use your model or build a separate model-- 00:25:20.300 --> 00:25:24.710 and minimize some cost. 00:25:24.710 --> 00:25:28.970 Well, in this case, maximize the CPK because a bigger CPK, 00:25:28.970 --> 00:25:32.780 we said, corresponds to yield. 00:25:32.780 --> 00:25:34.760 In fact, we'd use different formulas. 00:25:34.760 --> 00:25:40.610 You could actually relate a CPK to a fraction of nonconforming 00:25:40.610 --> 00:25:46.890 parts given different values. 00:25:46.890 --> 00:25:48.330 Another approach would actually be 00:25:48.330 --> 00:25:52.560 to extend that almost directly and say, 00:25:52.560 --> 00:25:55.440 what I'd like to do is actually use J directly 00:25:55.440 --> 00:25:57.930 or a model of CPK. 00:25:57.930 --> 00:26:02.700 Build a response surface model, if you will, of CPK, 00:26:02.700 --> 00:26:09.000 and that would be a way to fold in both mean information 00:26:09.000 --> 00:26:12.280 and process variance information. 00:26:14.860 --> 00:26:19.210 What are some gotchas in trying to do this? 00:26:19.210 --> 00:26:20.440 What are some difficulties? 00:26:24.280 --> 00:26:27.920 We said one of our purposes here, it might be in fact, 00:26:27.920 --> 00:26:34.020 pick your process operating point to maximize CPK. 00:26:34.020 --> 00:26:36.910 Why not just directly use that? 00:26:36.910 --> 00:26:39.120 Use this function right here. 00:26:39.120 --> 00:26:41.430 There's my function, I want to maximize 00:26:41.430 --> 00:26:49.680 K. Let me broaden the question. 00:26:49.680 --> 00:26:52.520 What are some good things or some bad things about this? 00:26:52.520 --> 00:26:53.020 Yeah? 00:26:53.020 --> 00:26:55.890 AUDIENCE: [INAUDIBLE]. 00:26:55.890 --> 00:26:56.940 DUANE BONING: OK. 00:26:56.940 --> 00:26:57.810 I'm sorry? 00:26:57.810 --> 00:27:00.870 AUDIENCE: [INAUDIBLE] is going to depend on y. 00:27:00.870 --> 00:27:03.180 DUANE BONING: OK, so the comment there-- 00:27:03.180 --> 00:27:05.370 by the way, you need to try to speak loudly. 00:27:05.370 --> 00:27:07.800 I'm trying to repeat the questions or the answers 00:27:07.800 --> 00:27:11.470 as much as I can, but I think it's a little hard for people 00:27:11.470 --> 00:27:11.970 to hear. 00:27:11.970 --> 00:27:15.215 So I'm sorry, now I lost the question-- or the answer. 00:27:15.215 --> 00:27:16.590 AUDIENCE: One of the things might 00:27:16.590 --> 00:27:19.000 be that the sigma might depend on mu 00:27:19.000 --> 00:27:22.470 or [INAUDIBLE] may depend on [INAUDIBLE].. 00:27:27.293 --> 00:27:28.960 DUANE BONING: So one of the complexities 00:27:28.960 --> 00:27:35.470 you said is this might actually be a function of the process 00:27:35.470 --> 00:27:36.980 conditions. 00:27:36.980 --> 00:27:37.480 Yes. 00:27:37.480 --> 00:27:40.270 In fact I would even almost say, this approach 00:27:40.270 --> 00:27:45.970 here where I built a quality overall CPK 00:27:45.970 --> 00:27:48.820 and if I were able to actually evaluate 00:27:48.820 --> 00:27:52.240 CPK at each of my process operating points, 00:27:52.240 --> 00:27:57.460 that might actually be a way to incorporate or accommodate 00:27:57.460 --> 00:28:02.020 a change in variance or a change in standard deviation. 00:28:02.020 --> 00:28:06.430 Whereas in some sense, if I build a response surface 00:28:06.430 --> 00:28:10.210 model up front with the assumption of a fixed variance, 00:28:10.210 --> 00:28:13.990 it might then be hard to actually use that fixed model 00:28:13.990 --> 00:28:17.600 with just J as a cost function. 00:28:17.600 --> 00:28:21.040 So these are two slightly different ideas, 00:28:21.040 --> 00:28:23.830 and I'm glad you mentioned that because-- 00:28:23.830 --> 00:28:25.720 I mean, this idea, the first idea 00:28:25.720 --> 00:28:30.100 up here is basically saying, I've already built my response 00:28:30.100 --> 00:28:31.760 model, maybe for y. 00:28:31.760 --> 00:28:35.560 I might, might, talk about a response model for s, 00:28:35.560 --> 00:28:43.500 and then I just use the model or use CPK as my cost 00:28:43.500 --> 00:28:45.090 that I'm minimizing. 00:28:45.090 --> 00:28:47.610 The second idea, in contrast down here, 00:28:47.610 --> 00:28:51.690 is build a new response variable and actually 00:28:51.690 --> 00:28:57.570 build a model directly of CPK as a function of my input 00:28:57.570 --> 00:28:58.870 conditions. 00:28:58.870 --> 00:29:01.080 And then I optimize on that. 00:29:01.080 --> 00:29:05.400 They're slightly different ideas because in that case, 00:29:05.400 --> 00:29:08.460 in some sense this case is, perhaps, assuming 00:29:08.460 --> 00:29:10.620 we're mostly having a fixed s. 00:29:10.620 --> 00:29:14.740 This one, as I said, can perhaps more easily accommodate 00:29:14.740 --> 00:29:20.348 a process varying standard deviation. 00:29:23.280 --> 00:29:27.060 There's one other potential challenge 00:29:27.060 --> 00:29:29.680 with the model down below. 00:29:29.680 --> 00:29:34.505 Well, either approach, really, with these two functions. 00:29:34.505 --> 00:29:37.896 What's something that's a little nasty about these functions? 00:29:45.680 --> 00:29:50.240 They have things like a min function in them. 00:29:50.240 --> 00:29:52.550 A minimum. 00:29:52.550 --> 00:29:59.885 And minimum between two variables 00:29:59.885 --> 00:30:03.950 can be very non-linear, so it may not 00:30:03.950 --> 00:30:05.720 be that smooth of a function. 00:30:05.720 --> 00:30:08.720 There may be a point at which you actually 00:30:08.720 --> 00:30:13.910 start to break more to the left side or to the right side 00:30:13.910 --> 00:30:18.300 and have not a continuously varying parameter. 00:30:18.300 --> 00:30:22.490 So it can be a little bit of a challenge 00:30:22.490 --> 00:30:25.880 from a modeling point of view as well as in optimization. 00:30:25.880 --> 00:30:28.840 It's kind of a nasty function. 00:30:28.840 --> 00:30:29.530 OK? 00:30:29.530 --> 00:30:33.820 And in fact, that nonlinear nasty behavior 00:30:33.820 --> 00:30:37.420 with its possible discontinuity is-- 00:30:37.420 --> 00:30:38.940 let's see, I want to-- 00:30:38.940 --> 00:30:42.040 ope, I guess I don't have it in here. 00:30:42.040 --> 00:30:45.910 It's kind of one of the main drivers for the whole notion 00:30:45.910 --> 00:30:50.620 of the quality loss function, that E of L, 00:30:50.620 --> 00:30:57.100 that quality loss function with a quadratic dependence 00:30:57.100 --> 00:31:03.520 on deviation from the mean is a nice way of folding in together 00:31:03.520 --> 00:31:08.410 in a more continuously varying function, 00:31:08.410 --> 00:31:12.430 both deviation from target and effects of variance. 00:31:12.430 --> 00:31:15.860 So we'll come back to that, but in fact, 00:31:15.860 --> 00:31:19.150 that's one of the main reasons why the quality loss 00:31:19.150 --> 00:31:24.370 function is used and has an advantage is 00:31:24.370 --> 00:31:28.420 it has a smoother behavior, a behavior that drives 00:31:28.420 --> 00:31:31.990 towards an optimum that pulls in or folds in both 00:31:31.990 --> 00:31:33.580 of those effects and doesn't have 00:31:33.580 --> 00:31:39.580 these discontinuities or not quite as much of the nastiness 00:31:39.580 --> 00:31:42.100 of these min functions. 00:31:42.100 --> 00:31:43.750 OK? 00:31:43.750 --> 00:31:49.480 OK, so I guess this discussion and everything 00:31:49.480 --> 00:31:52.340 we've been leading up to is really starting to say, 00:31:52.340 --> 00:31:55.240 OK, look. 00:31:55.240 --> 00:32:00.580 In order to be able to fold in both deviations from target 00:32:00.580 --> 00:32:03.700 but also variation, we need better ways 00:32:03.700 --> 00:32:07.480 to model the variance. 00:32:07.480 --> 00:32:10.090 In particular, the variance that might not 00:32:10.090 --> 00:32:15.010 be constant at your entire process base 00:32:15.010 --> 00:32:19.560 but may vary as a function or depend on a operating point. 00:32:19.560 --> 00:32:20.830 OK? 00:32:20.830 --> 00:32:26.010 Now remember, in some cases we've 00:32:26.010 --> 00:32:30.180 been careful to remind ourselves that there 00:32:30.180 --> 00:32:34.470 were assumptions on constant variance in our approaches. 00:32:34.470 --> 00:32:39.090 But sometimes we haven't even been explicitly mentioning 00:32:39.090 --> 00:32:42.030 that, but I want to highlight that some of the methods we've 00:32:42.030 --> 00:32:44.400 been using and developed actually 00:32:44.400 --> 00:32:47.520 implicitly continued that assumption 00:32:47.520 --> 00:32:50.550 that sigma squared or the variance 00:32:50.550 --> 00:32:52.780 was constant throughout the space. 00:32:52.780 --> 00:32:57.030 So in ANOVA, the simple ANOVA that we did, in order 00:32:57.030 --> 00:33:00.300 to decide if an effect is real we 00:33:00.300 --> 00:33:02.940 were always comparing a fixed offset 00:33:02.940 --> 00:33:06.240 to natural variance in the process 00:33:06.240 --> 00:33:10.410 and we were always pulling the estimate of variance 00:33:10.410 --> 00:33:14.680 across all operating points equally. 00:33:14.680 --> 00:33:15.400 OK? 00:33:15.400 --> 00:33:18.010 So we were enforcing an assumption 00:33:18.010 --> 00:33:23.290 that there was only one natural underlying process 00:33:23.290 --> 00:33:24.490 variance at work. 00:33:24.490 --> 00:33:28.750 And then we were trying to say, are there variances associated 00:33:28.750 --> 00:33:32.410 with fixed offsets that are larger 00:33:32.410 --> 00:33:35.310 than the replicate noise? 00:33:35.310 --> 00:33:39.610 So that was in there and we were careful in talking about that. 00:33:39.610 --> 00:33:45.940 But another place was actually in regression model fitting. 00:33:45.940 --> 00:33:51.460 When we formed a least squares regression problem 00:33:51.460 --> 00:33:54.520 and then solved it, that actually equally 00:33:54.520 --> 00:33:57.580 weights all of our data and all of our deviations 00:33:57.580 --> 00:34:01.870 from the model prediction equally 00:34:01.870 --> 00:34:07.740 and essentially, is also a equal variance assumption. 00:34:07.740 --> 00:34:09.469 Think of it this way. 00:34:09.469 --> 00:34:12.219 If I had, in fact-- 00:34:12.219 --> 00:34:12.719 wait. 00:34:12.719 --> 00:34:13.850 Let me draw it. 00:34:13.850 --> 00:34:20.300 If I in fact had different operating points or locations 00:34:20.300 --> 00:34:23.150 where the variance was very different 00:34:23.150 --> 00:34:28.520 and I'm trying to fit a line through this, but over here 00:34:28.520 --> 00:34:32.400 I have a huge spread. 00:34:32.400 --> 00:34:35.610 It may in fact be the case that I 00:34:35.610 --> 00:34:39.389 have less confidence or a wider spread 00:34:39.389 --> 00:34:42.719 here in picking a point that corresponds 00:34:42.719 --> 00:34:46.469 to where that line is going to go 00:34:46.469 --> 00:34:50.400 than in other regions of this space. 00:34:50.400 --> 00:34:56.380 Now they can still appeal to best estimates, 00:34:56.380 --> 00:35:02.250 but in terms of confidence in the use of each of your data 00:35:02.250 --> 00:35:06.090 points, there are alternative regression approaches, 00:35:06.090 --> 00:35:10.500 weighted regression approaches that might say in regions where 00:35:10.500 --> 00:35:12.600 I have less variance-- 00:35:12.600 --> 00:35:15.060 where I have more confidence, I know those data points 00:35:15.060 --> 00:35:18.000 are more solid-- 00:35:18.000 --> 00:35:21.300 I might actually give them more influence 00:35:21.300 --> 00:35:26.380 in the weighting in the regression than other points 00:35:26.380 --> 00:35:28.780 that I'm not so sure about. 00:35:28.780 --> 00:35:32.320 And so weighted regression approaches are available, 00:35:32.320 --> 00:35:34.960 and probably the most common weighted regression 00:35:34.960 --> 00:35:38.830 is to apply a weight that's inversely proportional 00:35:38.830 --> 00:35:43.180 to the variance associated with each of the operating 00:35:43.180 --> 00:35:47.140 regions in your space. 00:35:47.140 --> 00:35:47.650 OK? 00:35:47.650 --> 00:35:52.540 So just wanted to highlight that there are 00:35:52.540 --> 00:35:56.710 approaches that go beyond this assumption. 00:35:56.710 --> 00:35:59.200 We've been assuming that in both our regression 00:35:59.200 --> 00:36:04.570 analysis and our ANOVA analysis that we have one variant. 00:36:04.570 --> 00:36:07.780 I think I alluded to a book earlier this semester 00:36:07.780 --> 00:36:12.290 called Analysis of Messy Data. 00:36:12.290 --> 00:36:15.650 It's a wonderful book. 00:36:15.650 --> 00:36:18.420 Or actually, there's a series of books. 00:36:18.420 --> 00:36:20.360 There's either two or three of them. 00:36:20.360 --> 00:36:23.150 But one of the examples of messy data 00:36:23.150 --> 00:36:25.430 is when the variance is not constant 00:36:25.430 --> 00:36:30.500 and there's one of the volumes in that series is focused, 00:36:30.500 --> 00:36:32.780 I believe, on analysis of variance 00:36:32.780 --> 00:36:38.040 when you don't have constant variance throughout your space. 00:36:38.040 --> 00:36:42.410 So there are places you can look when things get messy. 00:36:42.410 --> 00:36:44.550 OK. 00:36:44.550 --> 00:36:46.980 OK, so what do we do? 00:36:46.980 --> 00:36:50.130 We've been assuming that but in reality, 00:36:50.130 --> 00:36:51.570 I think we've made the point. 00:36:51.570 --> 00:36:55.560 The process variation may in fact vary or depend 00:36:55.560 --> 00:36:57.480 on our operating point. 00:36:57.480 --> 00:37:00.090 I've already talked about the impact 00:37:00.090 --> 00:37:01.800 in our variance equation. 00:37:01.800 --> 00:37:06.420 If I had imperfect control on a control factor, 00:37:06.420 --> 00:37:08.820 I may have different sensitivity-- 00:37:08.820 --> 00:37:13.380 that dy, du-- especially with a non-linear model can vary. 00:37:13.380 --> 00:37:15.870 But it's also just simply the case 00:37:15.870 --> 00:37:18.090 that the effect of these noise factors 00:37:18.090 --> 00:37:21.000 may be such that I have different variants 00:37:21.000 --> 00:37:22.780 at different operating points. 00:37:22.780 --> 00:37:26.100 So what are some approaches? 00:37:26.100 --> 00:37:31.920 All I've said so far is variance may depend on your process 00:37:31.920 --> 00:37:33.210 settings. 00:37:33.210 --> 00:37:37.600 How are you going to study that and understand it? 00:37:37.600 --> 00:37:40.870 Well, let's do design of experiments 00:37:40.870 --> 00:37:44.230 but also include an explicit model 00:37:44.230 --> 00:37:47.840 for variance or standard deviation. 00:37:47.840 --> 00:37:51.640 So all we're doing here is simply saying treat variants 00:37:51.640 --> 00:37:55.030 as another process output. 00:37:55.030 --> 00:37:58.990 And so it is possible to include another response 00:37:58.990 --> 00:38:07.810 variable, either s squared or s or some variance model 00:38:07.810 --> 00:38:13.660 and do our typical DOE here on our input factors, 00:38:13.660 --> 00:38:15.790 but now get enough data that I can actually 00:38:15.790 --> 00:38:19.000 build a model of variance at each of my operating points. 00:38:21.800 --> 00:38:26.710 Can I possibly do that without replication? 00:38:26.710 --> 00:38:27.430 No. 00:38:27.430 --> 00:38:33.610 I have to have replicates at each of my design points, 00:38:33.610 --> 00:38:34.990 each of my DOE points. 00:38:34.990 --> 00:38:37.240 I have to gather multiple replicates in order 00:38:37.240 --> 00:38:42.130 to have an estimate of the noise at that replicate point. 00:38:42.130 --> 00:38:47.580 So fundamentally, you have to do more 00:38:47.580 --> 00:38:51.900 than a non-replicated, full factorial 00:38:51.900 --> 00:38:53.730 DOE like we were talking about that just 00:38:53.730 --> 00:38:56.040 explores the operating space. 00:38:56.040 --> 00:38:59.520 You've got to get more data if you want a model variance. 00:38:59.520 --> 00:39:02.350 You always have to have much more data. 00:39:02.350 --> 00:39:06.750 So you have to pick some number of replicates 00:39:06.750 --> 00:39:09.600 at each of your design points. 00:39:09.600 --> 00:39:12.480 And now this goes back to the tools 00:39:12.480 --> 00:39:14.400 we talked about earlier in the term 00:39:14.400 --> 00:39:17.790 in terms of deciding what sample size you need in order 00:39:17.790 --> 00:39:20.220 to be able to have different confidence 00:39:20.220 --> 00:39:24.360 intervals on your estimate of variance or standard deviation. 00:39:24.360 --> 00:39:24.870 Right? 00:39:24.870 --> 00:39:27.660 So depending on how close, how accurate 00:39:27.660 --> 00:39:32.940 you want to be on your estimate of a variance, that 00:39:32.940 --> 00:39:35.350 will influence how many replicates you need. 00:39:35.350 --> 00:39:35.850 Yeah? 00:39:35.850 --> 00:39:38.326 AUDIENCE: To get this at each level, you do [INAUDIBLE].. 00:39:43.877 --> 00:39:44.710 DUANE BONING: Right. 00:39:44.710 --> 00:39:45.220 Right. 00:39:45.220 --> 00:39:46.070 Absolutely. 00:39:46.070 --> 00:39:49.240 So the comment or question was, to do 00:39:49.240 --> 00:39:50.890 this you actually need replicates 00:39:50.890 --> 00:39:53.510 at all of the points. 00:39:53.510 --> 00:39:54.010 Yeah. 00:39:54.010 --> 00:39:59.890 So in some sense, the DOE that we were talking about, 00:39:59.890 --> 00:40:01.540 this would be, say, full factorial, 00:40:01.540 --> 00:40:03.640 and then if I picked my center point 00:40:03.640 --> 00:40:07.960 we talked about doing something where I had replicates only 00:40:07.960 --> 00:40:09.460 at the center points. 00:40:09.460 --> 00:40:12.910 And that's wonderful if you're continuing 00:40:12.910 --> 00:40:14.710 to assume that there's only one variance 00:40:14.710 --> 00:40:16.007 and it applies everywhere. 00:40:16.007 --> 00:40:17.590 In fact, that's kind of the assumption 00:40:17.590 --> 00:40:21.130 we made, even without necessarily 00:40:21.130 --> 00:40:23.110 having good evidence for it. 00:40:23.110 --> 00:40:29.200 But clearly here if I want a model of both 00:40:29.200 --> 00:40:31.440 the average output but the variance at each point, 00:40:31.440 --> 00:40:33.147 I mean replicates at each point. 00:40:33.147 --> 00:40:34.980 And you would normally want to do that where 00:40:34.980 --> 00:40:38.520 you have the same number of replicates at each point. 00:40:38.520 --> 00:40:44.850 It's possible you could do it without a balanced experiment 00:40:44.850 --> 00:40:46.920 where you had different numbers of replicates. 00:40:46.920 --> 00:40:54.120 Then it gets really messy because then your confidence 00:40:54.120 --> 00:40:57.600 interval for your estimate of variance at each of your points 00:40:57.600 --> 00:40:58.650 is different. 00:40:58.650 --> 00:41:03.180 You'd like to have at least the same confidence 00:41:03.180 --> 00:41:07.710 interval or the same number of replicates. 00:41:07.710 --> 00:41:13.890 Now there is some danger in some of these kinds of models. 00:41:13.890 --> 00:41:17.780 I wouldn't quite say danger, I would say 00:41:17.780 --> 00:41:21.340 some approximation going on. 00:41:21.340 --> 00:41:24.560 So for example, one thing that makes me a little bit 00:41:24.560 --> 00:41:27.380 uncomfortable is actually building a model directly 00:41:27.380 --> 00:41:32.300 of standard deviation, because estimating directly 00:41:32.300 --> 00:41:38.150 standard deviation is difficult. It's got bias in it. 00:41:38.150 --> 00:41:40.950 Whereas building a model, perhaps, for example, 00:41:40.950 --> 00:41:44.510 of variance, you can use the data 00:41:44.510 --> 00:41:46.820 and get an unbiased estimate of variance. 00:41:46.820 --> 00:41:54.250 Whereas you're susceptible to slight biases if you go in 00:41:54.250 --> 00:41:56.930 and directly model standard deviation. 00:41:56.930 --> 00:42:00.420 But for the most part, people pretty much ignore those. 00:42:00.420 --> 00:42:03.950 And are basically-- 00:42:03.950 --> 00:42:07.400 I think assuming that those kinds of biases 00:42:07.400 --> 00:42:09.530 get washed out in the noise because I 00:42:09.530 --> 00:42:11.150 don't have so many replicates. 00:42:14.230 --> 00:42:17.240 They tend not to be all that careful with it, 00:42:17.240 --> 00:42:21.002 but I would just put a little side note here. 00:42:21.002 --> 00:42:23.210 We should be a little bit careful about the influence 00:42:23.210 --> 00:42:26.210 of bias in these factors. 00:42:26.210 --> 00:42:28.740 And in fact, I think in one of the case studies 00:42:28.740 --> 00:42:33.680 we'll talk about approaches for modeling variance 00:42:33.680 --> 00:42:35.630 and some of the biases that can creep in. 00:42:38.240 --> 00:42:41.770 But the main latch here, the main lever 00:42:41.770 --> 00:42:44.260 for being able to get some power over the problem 00:42:44.260 --> 00:42:48.160 is simply doing a DOE that includes explicit modeling 00:42:48.160 --> 00:42:52.360 of the variance so that one can build up 00:42:52.360 --> 00:42:54.970 some kind of a variance or a standard deviation 00:42:54.970 --> 00:42:56.680 response surface. 00:42:56.680 --> 00:43:01.420 And then you can use that in combination 00:43:01.420 --> 00:43:04.210 with your output to form a cost function that 00:43:04.210 --> 00:43:10.100 seeks to minimize in some way both of those factors. 00:43:10.100 --> 00:43:12.350 So for example, now we can go back. 00:43:12.350 --> 00:43:18.340 We can go back to that surface that we had. 00:43:18.340 --> 00:43:23.110 That x1, x2 surface where I found the line or some 00:43:23.110 --> 00:43:26.980 function that minimized the error in my output 00:43:26.980 --> 00:43:28.550 from the target. 00:43:28.550 --> 00:43:33.610 But then I can explore over that within my x1 and x2 constrained 00:43:33.610 --> 00:43:35.770 space and look for the place where 00:43:35.770 --> 00:43:38.590 I've got the minimum variance. 00:43:38.590 --> 00:43:39.160 Right? 00:43:39.160 --> 00:43:41.320 So if I took back-- 00:43:44.840 --> 00:43:47.900 if I go back here to slide five-- 00:43:47.900 --> 00:43:49.790 I just heard a bell. 00:43:49.790 --> 00:43:52.830 Is everything still working there? 00:43:52.830 --> 00:43:55.010 You still have the slides in Singapore? 00:43:55.010 --> 00:43:58.900 AUDIENCE: No, we don't We don't have the slides, 00:43:58.900 --> 00:44:00.073 it was just disconnected. 00:44:00.073 --> 00:44:01.240 DUANE BONING: Yeah, I just-- 00:44:01.240 --> 00:44:05.722 AUDIENCE: You might want to share the desktop again. 00:44:05.722 --> 00:44:06.430 DUANE BONING: OK. 00:44:15.683 --> 00:44:17.740 Yeah, it looked like it dropped the call. 00:44:17.740 --> 00:44:22.600 So while that's calling, so the basic point was 00:44:22.600 --> 00:44:26.440 if I have an additional response surface 00:44:26.440 --> 00:44:29.530 for variance or standard deviation, 00:44:29.530 --> 00:44:31.960 the simplest use of that is simply 00:44:31.960 --> 00:44:34.150 to still do my optimization on the target 00:44:34.150 --> 00:44:39.130 and then just look up or use this to disambiguate my xy 00:44:39.130 --> 00:44:40.854 parameters. 00:44:40.854 --> 00:44:46.950 AUDIENCE: When you have [INAUDIBLE] experiment 00:44:46.950 --> 00:44:49.327 that we're doing, [INAUDIBLE]? 00:44:55.868 --> 00:44:56.660 DUANE BONING: Yeah. 00:44:56.660 --> 00:44:58.970 So the question was, let's say you only did 00:44:58.970 --> 00:45:02.180 four replicates at each point. 00:45:02.180 --> 00:45:04.040 Well, you ask the question, how good 00:45:04.040 --> 00:45:06.430 is your estimate of variance at that point? 00:45:06.430 --> 00:45:08.810 And you know how to calculate that, right? 00:45:08.810 --> 00:45:12.140 And your point is with only four replicates, 00:45:12.140 --> 00:45:15.132 it's kind of iffy about how good you're-- 00:45:15.132 --> 00:45:19.160 you're only going to have resolving power 00:45:19.160 --> 00:45:23.390 to be able to see really big differences in variance. 00:45:23.390 --> 00:45:25.610 If it's small differences in variance 00:45:25.610 --> 00:45:27.410 with that number of replicates, you're 00:45:27.410 --> 00:45:30.020 not going to really be able to disambiguate. 00:45:30.020 --> 00:45:34.160 So in essence with that small number-- oh, thank you-- 00:45:34.160 --> 00:45:37.100 that small number of replicates, you're really just looking 00:45:37.100 --> 00:45:39.410 for gross variance differences. 00:45:39.410 --> 00:45:43.740 Which if they exist, you definitely want to use. 00:45:43.740 --> 00:45:47.570 Now, if you're looking for really trying 00:45:47.570 --> 00:45:53.660 to get a good process that has the ability to take advantage 00:45:53.660 --> 00:45:56.540 of small variance differences-- because 00:45:56.540 --> 00:45:59.330 over many, many thousands or millions of parts, 00:45:59.330 --> 00:46:06.530 even small variances can really make a difference. 00:46:06.530 --> 00:46:08.840 Then you'd want to run more replicates 00:46:08.840 --> 00:46:13.920 in your design of experiments. 00:46:13.920 --> 00:46:16.770 OK, are we back at Singapore now? 00:46:16.770 --> 00:46:19.320 You have the slides there? 00:46:19.320 --> 00:46:20.280 AUDIENCE: No we don't. 00:46:20.280 --> 00:46:21.520 AUDIENCE: They had it and then they dropped it. 00:46:21.520 --> 00:46:23.187 AUDIENCE: Yeah, we can't see the slides. 00:46:30.572 --> 00:46:31.280 DUANE BONING: OK. 00:46:31.280 --> 00:46:34.175 So this one, it thinks we're in a call. 00:46:39.360 --> 00:46:40.410 Should I unshare? 00:46:40.410 --> 00:46:41.800 What should I-- 00:46:41.800 --> 00:46:43.425 AUDIENCE: No, I think it's coming back. 00:46:43.425 --> 00:46:44.883 DUANE BONING: Oh, it's coming back? 00:46:44.883 --> 00:46:46.040 AUDIENCE: [INAUDIBLE]. 00:46:46.040 --> 00:46:46.540 Yeah, yeah. 00:46:46.540 --> 00:46:47.010 It's back. 00:46:47.010 --> 00:46:47.850 DUANE BONING: OK, great. 00:46:47.850 --> 00:46:48.240 Great. 00:46:48.240 --> 00:46:48.740 Great. 00:46:51.000 --> 00:46:51.500 Good. 00:46:57.940 --> 00:47:02.530 Great, so here again, the first and easiest idea 00:47:02.530 --> 00:47:13.450 for combining models of both mean with a y and variance 00:47:13.450 --> 00:47:18.310 is use the variance to simply pick 00:47:18.310 --> 00:47:22.990 which of the x1, x2 combinations such that you're on target, 00:47:22.990 --> 00:47:25.540 give you minimum variance. 00:47:25.540 --> 00:47:29.050 Now you can also do that directly 00:47:29.050 --> 00:47:31.390 and form an overall optimization, 00:47:31.390 --> 00:47:33.160 and this is just working through that. 00:47:33.160 --> 00:47:35.140 This is the direct approach where 00:47:35.140 --> 00:47:38.740 if I have a response surface for my mean 00:47:38.740 --> 00:47:42.940 and a response surface for my variance, 00:47:42.940 --> 00:47:44.710 I have a constraint here first off 00:47:44.710 --> 00:47:51.370 that I need an optimum point that satisfies 00:47:51.370 --> 00:47:53.680 my output hitting a target. 00:47:53.680 --> 00:47:56.290 And then I can simply solve together, 00:47:56.290 --> 00:48:00.670 pick some x that satisfies that, and plug that 00:48:00.670 --> 00:48:03.040 into the s surface. 00:48:03.040 --> 00:48:08.030 Continuing on, substitute that in into the s surface 00:48:08.030 --> 00:48:13.450 so that you basically get some function 00:48:13.450 --> 00:48:19.420 where I need to find the x sub 2 that minimizes my variance. 00:48:19.420 --> 00:48:23.530 And then I can do a direct analytic variance minimization 00:48:23.530 --> 00:48:28.710 to solve for x2 and once I have x2, I can solve back for x1. 00:48:28.710 --> 00:48:33.550 So all I'm saying there is you can just algebraically use 00:48:33.550 --> 00:48:35.110 your two response surfaces. 00:48:35.110 --> 00:48:38.620 If they are simple and linear then 00:48:38.620 --> 00:48:44.650 it's nice and easy to drive that towards where 00:48:44.650 --> 00:48:50.050 you have the best y hat that's on target 00:48:50.050 --> 00:48:51.640 that also is minimum variance. 00:48:56.410 --> 00:49:01.690 Another approach, once I have models for both-- 00:49:01.690 --> 00:49:04.090 again, if I have a model for the mean response 00:49:04.090 --> 00:49:06.550 and a model for variance-- 00:49:09.300 --> 00:49:12.240 is think about other cost functions that 00:49:12.240 --> 00:49:17.340 combine both of these effects. 00:49:17.340 --> 00:49:23.690 So this is a bit more general because it 00:49:23.690 --> 00:49:27.800 allows us to look at, say, the quality loss function 00:49:27.800 --> 00:49:33.235 and have terms that penalize us for deviation from-- 00:49:39.200 --> 00:49:46.220 I don't-- well, penalizes for deviations from the target. 00:49:46.220 --> 00:49:49.700 I'm just looking at this and saying, I don't know why we-- 00:49:49.700 --> 00:49:53.160 let's just stick with y here everywhere. 00:49:53.160 --> 00:49:54.770 I don't know why we switched to x. 00:49:58.620 --> 00:50:03.120 So when I have deviations from my optimum point from the mean, 00:50:03.120 --> 00:50:04.500 that's going to cost me something 00:50:04.500 --> 00:50:08.850 and then I also have a term associated with the variance. 00:50:08.850 --> 00:50:15.370 And if both of those are functions of my input 00:50:15.370 --> 00:50:20.350 parameters, now I'm finding again the best input parameter 00:50:20.350 --> 00:50:23.600 that minimizes the overall cost function. 00:50:23.600 --> 00:50:25.890 Now there's a subtle difference here between-- 00:50:25.890 --> 00:50:30.310 or more power between one of these approaches. 00:50:30.310 --> 00:50:33.280 This approach versus the direct solution 00:50:33.280 --> 00:50:36.400 approach in the previous slide. 00:50:36.400 --> 00:50:38.290 The direct solution on the previous slide 00:50:38.290 --> 00:50:44.400 basically said, I have two outputs and two inputs 00:50:44.400 --> 00:50:47.940 and I'm trying to pull those two things 00:50:47.940 --> 00:50:52.220 into one overall optimum. 00:50:52.220 --> 00:50:52.720 Yes? 00:50:52.720 --> 00:50:53.250 AUDIENCE: Professor? 00:50:53.250 --> 00:50:54.417 DUANE BONING: Yes, question? 00:50:54.417 --> 00:50:56.937 AUDIENCE: I think we cannot see-- 00:50:56.937 --> 00:50:59.520 I think we can see a slide but then we cannot see any drawing, 00:50:59.520 --> 00:51:03.330 and actually we are now stuck at slide five. 00:51:03.330 --> 00:51:04.320 DUANE BONING: Oh, OK. 00:51:04.320 --> 00:51:06.870 So that doesn't sound like it's changing there. 00:51:09.397 --> 00:51:11.230 AUDIENCE: We don't really know which slide-- 00:51:11.230 --> 00:51:12.730 DUANE BONING: Yeah, I'm on slide 14. 00:51:15.300 --> 00:51:17.115 Yeah, I'm sorry? 00:51:17.115 --> 00:51:18.990 AUDIENCE: What we have right now is slide 11. 00:51:29.312 --> 00:51:30.770 DUANE BONING: Yeah, so I'm not sure 00:51:30.770 --> 00:51:32.960 if the sharing is working right. 00:51:32.960 --> 00:51:35.420 Are there just long delays? 00:51:35.420 --> 00:51:37.393 Should I unshare and reshare? 00:51:37.393 --> 00:51:40.231 AUDIENCE: Yeah, I would do that first and then [INAUDIBLE].. 00:51:44.961 --> 00:51:46.020 DUANE BONING: Oh, OK. 00:51:46.020 --> 00:51:47.790 Well, let me reshare here. 00:52:19.495 --> 00:52:20.940 Oh, we killed the coffee. 00:52:26.890 --> 00:52:29.740 OK, do you guys see slide 14 there in Singapore yet? 00:52:37.240 --> 00:52:38.992 AUDIENCE: Yeah, we can see it right now. 00:52:38.992 --> 00:52:39.700 DUANE BONING: OK. 00:52:39.700 --> 00:52:41.710 Great. 00:52:41.710 --> 00:52:44.140 So let's see if this works. 00:52:44.140 --> 00:52:47.500 We'll try continuing here. 00:52:47.500 --> 00:52:51.040 The point I was going to make is if I go back to slide 13-- 00:52:51.040 --> 00:52:52.630 see if you got 13 now. 00:52:52.630 --> 00:52:53.680 Did 13 come up? 00:52:57.160 --> 00:52:58.720 AUDIENCE: We can't see on the screen, 00:52:58.720 --> 00:53:00.475 but we can see behind your projectors. 00:53:00.475 --> 00:53:01.350 DUANE BONING: Oh, OK. 00:53:01.350 --> 00:53:03.040 So that's what we're using in the-- 00:53:03.040 --> 00:53:03.898 all right. 00:53:03.898 --> 00:53:05.190 AUDIENCE: Yeah, it's all right. 00:53:05.190 --> 00:53:08.970 DUANE BONING: OK, we'll just use the projector then. 00:53:08.970 --> 00:53:11.700 The point I was making here on slide 12 and 13 00:53:11.700 --> 00:53:15.760 is in some sense I've got two outputs and two inputs. 00:53:15.760 --> 00:53:21.450 I can directly solve the x1 and x2 to achieve both a y 00:53:21.450 --> 00:53:24.510 on target and then a minimum s. 00:53:24.510 --> 00:53:33.690 If I had three x's or four input factors, this approach-- 00:53:33.690 --> 00:53:37.620 well, I might have additional power 00:53:37.620 --> 00:53:40.530 in terms of picking points. 00:53:40.530 --> 00:53:44.940 But this approach actually can-- 00:53:44.940 --> 00:53:48.210 I guess it can work as well if my variance actually 00:53:48.210 --> 00:53:50.610 is different. 00:53:50.610 --> 00:53:53.140 It has a unique minimum. 00:53:53.140 --> 00:53:54.660 But the point I wanted to make here 00:53:54.660 --> 00:53:59.230 is that with minimizing of the quality loss function, 00:53:59.230 --> 00:54:01.950 it's a little bit more natural in the output 00:54:01.950 --> 00:54:04.170 and can combine or led to explore 00:54:04.170 --> 00:54:07.890 over a different mixture of input factors a little bit more 00:54:07.890 --> 00:54:09.160 naturally. 00:54:09.160 --> 00:54:10.930 OK? 00:54:10.930 --> 00:54:12.960 So it's a little bit more general than assuming 00:54:12.960 --> 00:54:15.390 immediately I've just got an x1 and an x2 00:54:15.390 --> 00:54:21.180 and I'm directly solving for the outputs individually. 00:54:21.180 --> 00:54:23.970 What it's letting me do is essentially combine them 00:54:23.970 --> 00:54:27.060 into a single cost function, and it also 00:54:27.060 --> 00:54:31.500 gives another little bit of flexibility 00:54:31.500 --> 00:54:34.470 that is an important difference between this 00:54:34.470 --> 00:54:37.140 and the previous approach. 00:54:37.140 --> 00:54:38.850 In the previous approach, we said 00:54:38.850 --> 00:54:45.280 the constraint is your output is exactly on target. 00:54:45.280 --> 00:54:46.180 Is that true here? 00:54:55.280 --> 00:54:56.330 No. 00:54:56.330 --> 00:55:01.430 When we're minimizing this combined penalty function 00:55:01.430 --> 00:55:05.120 with two terms in it it's actually a little bit more 00:55:05.120 --> 00:55:07.190 flexible in another important way, which 00:55:07.190 --> 00:55:12.320 is to say it's allowing you to maybe pick a point that's 00:55:12.320 --> 00:55:17.690 off target slightly because it's a lower variance 00:55:17.690 --> 00:55:24.920 point and overall you'll win in terms of yield or quality loss. 00:55:24.920 --> 00:55:27.100 So it's a little bit more flexible in letting 00:55:27.100 --> 00:55:30.220 you make that trade off between variance 00:55:30.220 --> 00:55:32.800 and being exactly on target. 00:55:32.800 --> 00:55:36.820 It doesn't presuppose that your mean on target 00:55:36.820 --> 00:55:40.800 is absolutely your best point. 00:55:40.800 --> 00:55:44.220 So I like the quality loss formalism a little bit 00:55:44.220 --> 00:55:50.020 if, in fact, I have a variance that also depends 00:55:50.020 --> 00:55:55.670 on what operating point I'm at, what x factors 00:55:55.670 --> 00:55:58.720 and therefore also what y point I'm at. 00:55:58.720 --> 00:56:02.110 It lets me combine those and trade those off 00:56:02.110 --> 00:56:04.720 in a natural way. 00:56:04.720 --> 00:56:08.030 Now by the way, the classic quality loss function, 00:56:08.030 --> 00:56:13.960 they also use a constant factor between the variance 00:56:13.960 --> 00:56:15.490 and being off target. 00:56:15.490 --> 00:56:17.710 A generalization of that is you can actually 00:56:17.710 --> 00:56:22.290 use different weighting factors in those two cases. 00:56:22.290 --> 00:56:27.660 You may actually have, I don't know, a customer that says, 00:56:27.660 --> 00:56:31.220 I really want my mean to be-- 00:56:31.220 --> 00:56:39.680 maybe your customer actually has some reason 00:56:39.680 --> 00:56:44.940 to wait or be more concerned about variance. 00:56:44.940 --> 00:56:48.950 He likes a tighter variance, but actually has-- 00:56:48.950 --> 00:56:53.430 as long as a mean is somewhere in my spec region, I'm fine. 00:56:53.430 --> 00:56:57.230 It's kind of an odd combination, but there might be reasons 00:56:57.230 --> 00:57:00.860 where you might weight variance or being 00:57:00.860 --> 00:57:03.000 near target a little bit differently, 00:57:03.000 --> 00:57:05.720 and so that's one nice thing about this minimizing 00:57:05.720 --> 00:57:07.640 of the expected loss. 00:57:07.640 --> 00:57:08.960 You can actually weight those. 00:57:11.740 --> 00:57:14.960 OK, so what are some challenges here when the variance is 00:57:14.960 --> 00:57:15.460 varying? 00:57:21.690 --> 00:57:24.690 One challenge with minimizing or variance [INAUDIBLE] 00:57:24.690 --> 00:57:27.850 that we've already talked about is you have to model it. 00:57:27.850 --> 00:57:29.580 And what I want to explore a little bit 00:57:29.580 --> 00:57:31.650 are some of the response surface model 00:57:31.650 --> 00:57:33.930 or approaches for actually building up 00:57:33.930 --> 00:57:37.690 models of these dependencies. 00:57:37.690 --> 00:57:38.550 Are you guys back? 00:57:38.550 --> 00:57:41.550 I just heard another bell. 00:57:41.550 --> 00:57:43.700 I'm just going to keep going. 00:57:43.700 --> 00:57:45.200 AUDIENCE: No, we're not but it's OK. 00:57:45.200 --> 00:57:46.242 We can see the projector. 00:57:46.242 --> 00:57:49.290 DUANE BONING: OK, we'll continue with that. 00:57:49.290 --> 00:57:52.140 I'll ignore the little bells and the birds chirping and whatnot. 00:57:55.150 --> 00:57:57.450 Now as we start to look at ways to build 00:57:57.450 --> 00:57:59.873 the model of the process we can return 00:57:59.873 --> 00:58:01.290 to the question of what is it that 00:58:01.290 --> 00:58:04.290 causes non-constant variance. 00:58:04.290 --> 00:58:09.090 And part of the effect is that our noise factors 00:58:09.090 --> 00:58:12.150 may have different influence at different parts 00:58:12.150 --> 00:58:14.140 of the operating space. 00:58:14.140 --> 00:58:17.250 So one approach for building the model 00:58:17.250 --> 00:58:20.520 and trying to get places that are robust, 00:58:20.520 --> 00:58:24.030 that is to say insensitive to variation, 00:58:24.030 --> 00:58:29.070 is actually explicitly play with some of these noise 00:58:29.070 --> 00:58:31.680 factors, additional factors in our process 00:58:31.680 --> 00:58:35.400 that we're not using directly to control the process 00:58:35.400 --> 00:58:42.000 and try to explore and model what effect those noise factors 00:58:42.000 --> 00:58:44.550 have directly on the process. 00:58:44.550 --> 00:58:45.060 OK? 00:58:45.060 --> 00:58:49.950 And so that gets to, essentially, 00:58:49.950 --> 00:58:53.520 an approach where we can do an extension 00:58:53.520 --> 00:58:57.390 to our design of experiments where we explicitly 00:58:57.390 --> 00:59:02.110 look not just at, say, our main control factors. 00:59:02.110 --> 00:59:08.070 So these might be our A and B control factors. 00:59:08.070 --> 00:59:11.820 But at each of our design points, what we might do 00:59:11.820 --> 00:59:16.920 is not just do pure replicates but actually take 00:59:16.920 --> 00:59:21.780 some of these other factors, things we can kind of change 00:59:21.780 --> 00:59:24.480 in the process but in normal operation 00:59:24.480 --> 00:59:27.250 we would not want to be. 00:59:27.250 --> 00:59:29.740 Maybe it's a machine setting that 00:59:29.740 --> 00:59:33.220 has a big cost associated with changing it 00:59:33.220 --> 00:59:34.830 or a big time lag in it. 00:59:34.830 --> 00:59:40.730 It's a noise factor or some other factor. 00:59:40.730 --> 00:59:42.730 What we might want to do instead of just running 00:59:42.730 --> 00:59:46.900 pure replicates is actually look at these additional noise 00:59:46.900 --> 00:59:50.470 factors and explicitly, in the design of experiments, 00:59:50.470 --> 00:59:53.230 tweak them, play with them, bury them 00:59:53.230 --> 00:59:58.180 in a small region around that particular operating point. 00:59:58.180 --> 01:00:02.200 And what that does is back to our variation equation. 01:00:02.200 --> 01:00:05.860 That's explicitly saying, I know what my alpha factors are, 01:00:05.860 --> 01:00:09.280 these noise factors, I'm going to actually modify 01:00:09.280 --> 01:00:12.340 them a little bit and build a model that 01:00:12.340 --> 01:00:19.130 gives me information about sensitivity to those factors. 01:00:19.130 --> 01:00:23.980 So this is referred to as inner and outer factors. 01:00:23.980 --> 01:00:28.060 What we've got if I do the same thing at each of these corner 01:00:28.060 --> 01:00:30.310 points, I've got the inner design 01:00:30.310 --> 01:00:33.760 and then I've also got an outer design 01:00:33.760 --> 01:00:35.787 where I play with these additional factors 01:00:35.787 --> 01:00:36.745 at each of the corners. 01:00:39.930 --> 01:00:44.220 This, by the way, gets very close to approaches often 01:00:44.220 --> 01:00:46.200 referred to as Taguchi approaches-- 01:00:46.200 --> 01:00:55.290 after one of the big spokesmen for explicitly thinking about 01:00:55.290 --> 01:00:58.770 dealing with noise and modeling of noise and optimizing 01:00:58.770 --> 01:01:01.380 processes to be robust to noise-- 01:01:01.380 --> 01:01:08.530 where you've got this inner and outer factor mixed design. 01:01:08.530 --> 01:01:10.930 So let's see how this works. 01:01:10.930 --> 01:01:12.840 Here's a very simple example. 01:01:12.840 --> 01:01:18.550 We've got two main factors in our inner array. 01:01:18.550 --> 01:01:21.360 So if I do just a full factorial in that, 01:01:21.360 --> 01:01:23.940 we've got my four combinations. 01:01:23.940 --> 01:01:24.720 Right? 01:01:24.720 --> 01:01:29.100 And then here, instead of running pure replicates, 01:01:29.100 --> 01:01:32.130 what we're doing is an outer array 01:01:32.130 --> 01:01:36.570 where we vary the levels in another little full factorial 01:01:36.570 --> 01:01:41.010 around each of those corner points. 01:01:41.010 --> 01:01:43.680 And so now you can imagine that that's giving us 01:01:43.680 --> 01:01:48.150 an ability to model the mean response, again, 01:01:48.150 --> 01:01:55.500 at that corner point, but also a notion of variance. 01:01:55.500 --> 01:01:59.760 If I just treat those as pure replicates 01:01:59.760 --> 01:02:03.030 and build a model or an estimate of variance, 01:02:03.030 --> 01:02:08.490 that gives me kind of a funny, slightly different way 01:02:08.490 --> 01:02:10.200 of thinking about the variant. 01:02:10.200 --> 01:02:13.200 It's actually a perturbed variant, 01:02:13.200 --> 01:02:18.180 but I'm thinking of that as in my normal operation 01:02:18.180 --> 01:02:20.610 those might be noise factors that I'm not explicitly 01:02:20.610 --> 01:02:23.130 controlling and might randomly be picked 01:02:23.130 --> 01:02:25.410 at one of those settings and so I'm 01:02:25.410 --> 01:02:28.050 going to treat them, for the purpose of my process 01:02:28.050 --> 01:02:35.100 optimization, for the purpose of robustness, as overall noise. 01:02:35.100 --> 01:02:35.940 Right? 01:02:35.940 --> 01:02:39.150 So in essence, it's lumping together these other factors 01:02:39.150 --> 01:02:45.420 in order to explicitly and proactively explore 01:02:45.420 --> 01:02:51.060 the space in those factors and get an estimate of variance 01:02:51.060 --> 01:02:55.900 at each of our operating points, at each of our corner points. 01:02:55.900 --> 01:02:56.530 OK? 01:02:56.530 --> 01:02:59.080 So then you can build a model. 01:02:59.080 --> 01:03:02.170 You could build a model both of the average output 01:03:02.170 --> 01:03:03.850 and of the variance. 01:03:03.850 --> 01:03:06.970 Another part of the methodology popularized 01:03:06.970 --> 01:03:10.720 by Taguchi is to say, I don't want two outputs 01:03:10.720 --> 01:03:13.270 that I have to somehow then optimize separately. 01:03:13.270 --> 01:03:16.660 I want to fold these things in together into one combined cost 01:03:16.660 --> 01:03:19.840 function that I then optimize. 01:03:19.840 --> 01:03:23.470 And the particular cost function that he refers to or uses 01:03:23.470 --> 01:03:28.180 is a signal to noise ratio, where what you're trying to do 01:03:28.180 --> 01:03:32.410 is get on target or maximize some output, 01:03:32.410 --> 01:03:35.140 but also then minimize variance. 01:03:35.140 --> 01:03:41.750 And he actually has multiple different factors, 01:03:41.750 --> 01:03:45.310 but one can imagine building a signal to noise ratio 01:03:45.310 --> 01:03:51.880 that looks at the ratio of output mean to variant. 01:03:51.880 --> 01:03:56.560 You take the log so that it becomes more of a linear model 01:03:56.560 --> 01:03:58.210 and form a signal to noise. 01:03:58.210 --> 01:04:00.550 And so here, for example in the picture, 01:04:00.550 --> 01:04:05.620 is an example of an overall integrated cost function, 01:04:05.620 --> 01:04:11.500 a signal to noise ratio function where you might want 01:04:11.500 --> 01:04:15.220 the maximum signal to noise-- 01:04:15.220 --> 01:04:17.860 larger output, smaller variance-- and this 01:04:17.860 --> 01:04:22.330 would drive you towards an optimum point in your space 01:04:22.330 --> 01:04:30.300 based on where you're going to achieve that notion of best. 01:04:30.300 --> 01:04:32.560 OK? 01:04:32.560 --> 01:04:34.640 So there's a couple of ideas on this slide. 01:04:34.640 --> 01:04:36.970 One here is simply this notion of how 01:04:36.970 --> 01:04:39.970 you do the design with the inner and outer array 01:04:39.970 --> 01:04:46.120 and then the other is this idea of a natural way of combining 01:04:46.120 --> 01:04:49.870 both variants and target different than the expected 01:04:49.870 --> 01:04:53.230 loss function, but kind of related to it 01:04:53.230 --> 01:04:55.460 are the signal to noise ratios. 01:04:55.460 --> 01:04:57.040 And if you look in the literature, 01:04:57.040 --> 01:05:02.050 there's different ones depending on what kind of objectives 01:05:02.050 --> 01:05:04.270 you have qualitatively. 01:05:04.270 --> 01:05:08.320 For example, if you really want to be close to the nominal 01:05:08.320 --> 01:05:11.140 and be insensitive to the noise, then 01:05:11.140 --> 01:05:14.770 that cost function that we just described on the previous slide 01:05:14.770 --> 01:05:16.300 is a pretty good one. 01:05:16.300 --> 01:05:19.900 There's other places where maybe your main driver is really 01:05:19.900 --> 01:05:21.010 trying to get as-- 01:05:24.200 --> 01:05:25.280 do I have this right? 01:05:25.280 --> 01:05:26.270 I may have this-- 01:05:26.270 --> 01:05:31.460 no, I think I have this right because of the negative there. 01:05:31.460 --> 01:05:34.040 You might have a slightly different signal to noise ratio 01:05:34.040 --> 01:05:36.950 if you're more driven towards you 01:05:36.950 --> 01:05:41.000 want a large output, a large y, or another one 01:05:41.000 --> 01:05:43.880 if you want a small y, you want to minimize y. 01:05:43.880 --> 01:05:47.210 So you can still formulate these things in slightly different 01:05:47.210 --> 01:05:48.320 signal to noise metrics. 01:05:50.860 --> 01:05:55.000 So let's explore this inner and outer array design 01:05:55.000 --> 01:05:56.740 approach a little bit more. 01:05:56.740 --> 01:06:01.240 This is often referred to as a crossed ray method. 01:06:01.240 --> 01:06:04.850 One question that comes up is, gee, 01:06:04.850 --> 01:06:07.900 looks like an awful lot of experiments. 01:06:07.900 --> 01:06:10.910 So we do want to understand how many experiments there are. 01:06:10.910 --> 01:06:16.200 Are there ways to minimize that or reduce that? 01:06:16.200 --> 01:06:19.440 And overall, it's clear that I've 01:06:19.440 --> 01:06:23.190 got something that requires both control factor 01:06:23.190 --> 01:06:28.590 tests in the inner array, but also noise factor 01:06:28.590 --> 01:06:32.220 tests associated with additional variations 01:06:32.220 --> 01:06:35.850 right around each of the points. 01:06:35.850 --> 01:06:37.520 OK? 01:06:37.520 --> 01:06:41.210 And it can actually be even worse 01:06:41.210 --> 01:06:43.790 than just sort of what I've drawn here, 01:06:43.790 --> 01:06:49.220 because if our goal is really to get to an optimum point that 01:06:49.220 --> 01:06:55.470 hits the target and really is an optimum point in a target, 01:06:55.470 --> 01:06:58.950 you may even need something more than just a full factorial 01:06:58.950 --> 01:07:02.970 on your x1 and x2 parameters. 01:07:02.970 --> 01:07:05.340 Think back to what we talked about on Tuesday. 01:07:05.340 --> 01:07:07.050 We said if you're trying to drive 01:07:07.050 --> 01:07:11.480 the process to an optimum y, you might 01:07:11.480 --> 01:07:13.950 have locally linear models. 01:07:13.950 --> 01:07:17.900 But if you really have an optimum y-- 01:07:17.900 --> 01:07:21.440 not just at some target, but some overall optimum-- 01:07:21.440 --> 01:07:25.910 what you need to get to is some natural maximum or minimum 01:07:25.910 --> 01:07:28.280 point, which is a place where you've got something 01:07:28.280 --> 01:07:31.380 like, say, quadratic curvature. 01:07:31.380 --> 01:07:35.420 So ultimately your model for y, if you really 01:07:35.420 --> 01:07:38.930 are trying to get to a true optimum it 01:07:38.930 --> 01:07:46.660 may require some notion of a quadratic model, at least 01:07:46.660 --> 01:07:50.110 in terms of your output as a function of your input. 01:07:50.110 --> 01:07:52.990 So one approach was it might actually 01:07:52.990 --> 01:07:59.300 require, for example, a three level model. 01:07:59.300 --> 01:08:02.560 And in that case, things get bad really fast, right? 01:08:02.560 --> 01:08:06.490 If I really needed a true three level model 01:08:06.490 --> 01:08:13.720 on each of my x1, x2 inputs, now I go as 3 to the K sub C 01:08:13.720 --> 01:08:16.720 where K sub C is the number of control factors, 01:08:16.720 --> 01:08:19.100 K sub N is the number of noise factors. 01:08:19.100 --> 01:08:21.160 So that could grow really, really fast. 01:08:21.160 --> 01:08:23.380 Fortunately we know I don't necessarily 01:08:23.380 --> 01:08:28.050 have to do a full three level factorial. 01:08:28.050 --> 01:08:30.899 There are other approaches like central composite 01:08:30.899 --> 01:08:35.580 that can build me a quadratic model in the output 01:08:35.580 --> 01:08:41.040 without needing so much replication. 01:08:41.040 --> 01:08:44.370 I don't have to have necessarily full three 01:08:44.370 --> 01:08:47.819 levels on all of my factors because I don't really have 01:08:47.819 --> 01:08:51.590 that much interaction going on. 01:08:51.590 --> 01:08:53.260 But the point here is if I'm really 01:08:53.260 --> 01:08:57.670 trying to explore the space for building up 01:08:57.670 --> 01:09:00.640 my estimate of my output, I might well 01:09:00.640 --> 01:09:04.100 need a quadratic model in that. 01:09:04.100 --> 01:09:06.590 Do you think it gets that bad for the noise factors? 01:09:06.590 --> 01:09:10.264 Do I need a quadratic kind of model for my noise factors? 01:09:14.290 --> 01:09:16.170 In other words, why not? 01:09:16.170 --> 01:09:18.800 If I look at this little corner point, 01:09:18.800 --> 01:09:21.102 why don't I need a quadratic? 01:09:23.640 --> 01:09:25.380 Should I include a center point there? 01:09:25.380 --> 01:09:29.460 Should I include multiple exploration around that 01:09:29.460 --> 01:09:30.029 as well? 01:09:30.029 --> 01:09:33.750 Do I need a quadratic model of variance at each corner? 01:09:40.350 --> 01:09:45.060 What are we using this exploration 01:09:45.060 --> 01:09:48.920 in this corner array to do? 01:09:48.920 --> 01:09:51.710 We're just using it to exercise these noise factors 01:09:51.710 --> 01:09:55.040 and build up an aggregate into one estimate of variance 01:09:55.040 --> 01:09:56.430 at that point. 01:09:56.430 --> 01:10:03.920 We are rarely actually trying to build a local quadratic model. 01:10:03.920 --> 01:10:05.840 I'm just using it-- 01:10:05.840 --> 01:10:10.760 think of it almost as structured replicates at the corner. 01:10:10.760 --> 01:10:16.520 I don't really need that kind of resolution. 01:10:16.520 --> 01:10:20.120 So you typically will not see things 01:10:20.120 --> 01:10:23.600 like doing a central composite design at each corner, 01:10:23.600 --> 01:10:25.010 something like that. 01:10:25.010 --> 01:10:30.020 Something very simple like a factorial, 01:10:30.020 --> 01:10:33.240 either full or even fractional factorial. 01:10:33.240 --> 01:10:37.190 This is a beautiful place for fractional factorial 01:10:37.190 --> 01:10:38.540 in those corner points. 01:10:38.540 --> 01:10:41.810 If I had eight noise factors, I do not really 01:10:41.810 --> 01:10:47.230 need to do the eighth exploration 01:10:47.230 --> 01:10:50.110 at each of my corner points because I don't really 01:10:50.110 --> 01:10:52.840 care if it's a third or fourth or sixth order 01:10:52.840 --> 01:10:55.810 interaction between noise factors, what 01:10:55.810 --> 01:10:56.860 I would like to do-- 01:10:56.860 --> 01:10:58.780 I don't care about confounding. 01:10:58.780 --> 01:11:01.060 Confounding is great, no problem. 01:11:01.060 --> 01:11:03.130 I want to lump them all together and just get 01:11:03.130 --> 01:11:07.000 one estimate, in fact, of overall noise variance there. 01:11:07.000 --> 01:11:08.830 So very reduced models-- 01:11:08.830 --> 01:11:10.742 whoops. 01:11:10.742 --> 01:11:13.490 I didn't want to do that. 01:11:13.490 --> 01:11:15.920 Very reduced models. 01:11:15.920 --> 01:11:18.920 Heavily fractional factorial in those corner points 01:11:18.920 --> 01:11:22.390 are very effective for this purpose. 01:11:26.300 --> 01:11:28.720 Another approach that we can use here 01:11:28.720 --> 01:11:34.680 is we could actually go beyond that notion of just lumping 01:11:34.680 --> 01:11:37.500 it all together just trying to get an estimate for variance 01:11:37.500 --> 01:11:42.360 at that point and actually treat my control factors 01:11:42.360 --> 01:11:46.470 and my noise factors almost on equal basis. 01:11:46.470 --> 01:11:50.340 So the purposes of our design, why distinguish them? 01:11:50.340 --> 01:11:54.570 I might actually want a functional dependence 01:11:54.570 --> 01:11:59.070 on the noise factor, maybe because that 01:11:59.070 --> 01:12:01.650 allows me to sort of more naturally 01:12:01.650 --> 01:12:05.470 fold it all in together into one big experiment. 01:12:05.470 --> 01:12:08.580 So in that case, you can easily imagine 01:12:08.580 --> 01:12:12.790 building a single integrated model that 01:12:12.790 --> 01:12:15.670 has linear interaction, maybe even 01:12:15.670 --> 01:12:20.320 quadratic dependencies on my control factors. 01:12:20.320 --> 01:12:23.710 But also that has terms in it that 01:12:23.710 --> 01:12:30.990 look like essentially different parts of the problem 01:12:30.990 --> 01:12:33.390 that we might be interested in. 01:12:33.390 --> 01:12:36.360 One of those would be terms like this. 01:12:36.360 --> 01:12:39.032 I've got a gamma times z1. 01:12:39.032 --> 01:12:40.480 Well, what's that doing? 01:12:40.480 --> 01:12:43.260 That's telling me essentially the sensitivity. 01:12:43.260 --> 01:12:47.460 That's Dy to Dz1. 01:12:47.460 --> 01:12:51.240 That's one of these alpha noise factors. 01:12:51.240 --> 01:12:54.090 I directly have a measure there of sensitivity 01:12:54.090 --> 01:12:58.030 to noise, that noise factor at that point. 01:12:58.030 --> 01:12:59.580 So that might be a very useful thing 01:12:59.580 --> 01:13:03.450 to know and actually model and distinguish. 01:13:03.450 --> 01:13:07.970 A second one are terms that look like this. 01:13:07.970 --> 01:13:10.510 Here's an x1 and z1. 01:13:10.510 --> 01:13:13.330 What's this telling me? 01:13:13.330 --> 01:13:16.120 Well, this is telling me there's an interaction 01:13:16.120 --> 01:13:20.480 between my x variable and that noise factor. 01:13:20.480 --> 01:13:23.950 So I know, depending on where I pick my x variable, 01:13:23.950 --> 01:13:28.730 I've got a different sensitivity to noise at that point. 01:13:28.730 --> 01:13:32.200 So that's kind of breaking that part of the overall response 01:13:32.200 --> 01:13:35.680 out and letting you explicitly say, yes, now I 01:13:35.680 --> 01:13:37.450 understand that interaction. 01:13:37.450 --> 01:13:40.780 I understand how my noise interacts 01:13:40.780 --> 01:13:42.910 with that particular factor and I 01:13:42.910 --> 01:13:46.330 can use that to help guide my optimization 01:13:46.330 --> 01:13:48.650 and selection of input points. 01:13:48.650 --> 01:13:53.440 So there is some value in going ahead and doing 01:13:53.440 --> 01:13:59.870 an integrated effect and noise response surface 01:13:59.870 --> 01:14:04.400 approach where you can build up an integrated model together 01:14:04.400 --> 01:14:05.270 in here. 01:14:05.270 --> 01:14:07.190 And then you can use the same techniques 01:14:07.190 --> 01:14:10.610 that we talked about last time, about aliasing and worrying 01:14:10.610 --> 01:14:13.940 about which interaction factors and how high your order 01:14:13.940 --> 01:14:16.280 interaction really needs so you can 01:14:16.280 --> 01:14:18.950 go ahead and get reduced models and fold some of those 01:14:18.950 --> 01:14:20.100 together. 01:14:20.100 --> 01:14:22.940 But it essentially allows us to explicitly think 01:14:22.940 --> 01:14:28.940 about both control factor responses, 01:14:28.940 --> 01:14:32.005 but interaction with the noise factor. 01:14:32.005 --> 01:14:32.505 Yeah? 01:14:32.505 --> 01:14:34.710 AUDIENCE: Do we have a control on triggering 01:14:34.710 --> 01:14:36.078 the level of noise? 01:14:36.078 --> 01:14:36.870 DUANE BONING: Yeah. 01:14:36.870 --> 01:14:39.520 So the big assumption in here is, in fact, 01:14:39.520 --> 01:14:42.480 many of these noise factors in the design of experiments, 01:14:42.480 --> 01:14:46.170 while you're doing your upfront process optimization, 01:14:46.170 --> 01:14:49.348 that you actually can vary them. 01:14:49.348 --> 01:14:50.390 AUDIENCE: That's a noise. 01:14:50.390 --> 01:14:51.543 [INAUDIBLE] 01:14:51.543 --> 01:14:53.460 DUANE BONING: Yeah, it's kind of a funny term. 01:14:53.460 --> 01:14:56.580 Yeah, I'm glad you raised that question. 01:14:56.580 --> 01:15:00.810 The question is we've got these noise factors and the term 01:15:00.810 --> 01:15:02.410 calling the noise factors-- 01:15:02.410 --> 01:15:05.070 one strong interpretation of noise 01:15:05.070 --> 01:15:07.410 is it's something I have no control over. 01:15:07.410 --> 01:15:08.970 This is a weaker notion. 01:15:08.970 --> 01:15:11.130 This is basically saying there are 01:15:11.130 --> 01:15:16.440 things like in the normal operation of the equipment 01:15:16.440 --> 01:15:22.140 I can't vary them or in the normal operation 01:15:22.140 --> 01:15:24.840 they will be naturally varied over 01:15:24.840 --> 01:15:31.950 and I can't set and specify and keep it at that noise factor 01:15:31.950 --> 01:15:32.740 at one of these. 01:15:32.740 --> 01:15:36.610 So here's what I'm thinking as a good example. 01:15:36.610 --> 01:15:39.900 Temperature, room temperature, say. 01:15:39.900 --> 01:15:43.530 So you know in normal operation throughout the day, 01:15:43.530 --> 01:15:46.110 the temperature from the morning to the afternoon 01:15:46.110 --> 01:15:52.320 might be varying in your factory and you've only got a limit. 01:15:52.320 --> 01:15:54.630 You can't really completely control that. 01:15:54.630 --> 01:15:58.480 Maybe you're operating in a place where it has a big range 01:15:58.480 --> 01:16:01.870 and you don't have any control over it. 01:16:01.870 --> 01:16:04.770 So what you would want to do is understand that while you're 01:16:04.770 --> 01:16:06.870 doing your process design. 01:16:06.870 --> 01:16:11.040 Treat it as a noise factor, know it's impact, 01:16:11.040 --> 01:16:13.920 but then in actual operation, I can't pick my temperature. 01:16:13.920 --> 01:16:16.140 It's going to just vary, but you would 01:16:16.140 --> 01:16:19.290 like your process to be as robust as possible 01:16:19.290 --> 01:16:21.330 to that factor. 01:16:21.330 --> 01:16:23.040 So that's the kind of noise factor 01:16:23.040 --> 01:16:27.060 here is things that you can either actively 01:16:27.060 --> 01:16:29.070 control during the design of experiments 01:16:29.070 --> 01:16:33.690 or naturally sample from during the design of experiments. 01:16:33.690 --> 01:16:35.640 Other things would be operator. 01:16:39.270 --> 01:16:41.890 So that's a very good question. 01:16:41.890 --> 01:16:45.000 I'm glad you asked that. 01:16:45.000 --> 01:16:45.570 OK. 01:16:45.570 --> 01:16:49.230 Once you've got a noise response kind of surface, 01:16:49.230 --> 01:16:53.430 now what you can also do is build outputs. 01:16:53.430 --> 01:16:55.170 Not only as a function of-- 01:16:58.120 --> 01:17:00.940 mean responses as a function of your input 01:17:00.940 --> 01:17:05.950 where you would essentially ignore the noise factors, 01:17:05.950 --> 01:17:08.290 but also if you just simply take the variance 01:17:08.290 --> 01:17:10.930 of this relationship that gives you 01:17:10.930 --> 01:17:16.450 a model for the variance of the response when 01:17:16.450 --> 01:17:21.160 under the assumption that x are constant. 01:17:23.970 --> 01:17:27.770 So in other words, what we do is we can build two of these, 01:17:27.770 --> 01:17:33.920 one under the assumption such that my noise factors 01:17:33.920 --> 01:17:35.310 are at the mean. 01:17:35.310 --> 01:17:38.120 So I'm basically just lumping all of those 01:17:38.120 --> 01:17:41.750 into some overall noise from my noise factors. 01:17:41.750 --> 01:17:44.030 And then in my variance case, I just 01:17:44.030 --> 01:17:50.480 look at the terms that relate to the interactions assuming 01:17:50.480 --> 01:17:54.710 some x is relatively constant. 01:17:54.710 --> 01:17:58.160 So we ignore these terms or lump all these terms 01:17:58.160 --> 01:18:02.810 into just the overall gamma and then include 01:18:02.810 --> 01:18:08.900 some of the interaction terms with my z. 01:18:08.900 --> 01:18:11.120 So essentially what this lets us do 01:18:11.120 --> 01:18:15.800 if we built an overall response surface is 01:18:15.800 --> 01:18:19.970 separate out and explore both the mean response and also 01:18:19.970 --> 01:18:25.790 variance as we try to pick an x1 and our x2 to minimize 01:18:25.790 --> 01:18:29.120 the sensitivity to those noise factors. 01:18:29.120 --> 01:18:30.500 Yes, there was a question? 01:18:30.500 --> 01:18:32.047 Singapore? 01:18:32.047 --> 01:18:33.505 AUDIENCE: Prof, do we need to check 01:18:33.505 --> 01:18:36.560 the r squared before calculating the mean and then the variance. 01:18:36.560 --> 01:18:39.030 Whatever the r squared is, it's not very good. 01:18:39.030 --> 01:18:40.040 It's [? small. ?] 01:18:40.040 --> 01:18:40.450 DUANE BONING: Yeah. 01:18:40.450 --> 01:18:40.950 Yeah. 01:18:40.950 --> 01:18:44.240 So this sort of goes without saying. 01:18:44.240 --> 01:18:48.110 I was just assuming that you use all of the normal response 01:18:48.110 --> 01:18:51.270 surface modeling methodology. 01:18:51.270 --> 01:18:55.430 You have to check significance of each of these terms. 01:18:55.430 --> 01:18:58.110 You would only include terms if they are significant. 01:18:58.110 --> 01:19:00.020 So it's more than just checking r squared, 01:19:00.020 --> 01:19:02.150 you really are running ANOVA. 01:19:02.150 --> 01:19:05.220 Does that noise factor have an effect or not? 01:19:05.220 --> 01:19:07.670 If not, you just lump it into an overall aspect. 01:19:07.670 --> 01:19:10.685 So all of the typical model construction 01:19:10.685 --> 01:19:11.810 you would still want to do. 01:19:14.710 --> 01:19:19.030 So anyway, the point here is now if you explicitly 01:19:19.030 --> 01:19:21.190 do a response surface model, now you can 01:19:21.190 --> 01:19:23.680 build models of the variance. 01:19:23.680 --> 01:19:26.980 This is just an explosion or expansion 01:19:26.980 --> 01:19:31.660 of the variance surface in general 01:19:31.660 --> 01:19:34.570 you will observe because of this squaring 01:19:34.570 --> 01:19:36.400 in the variance expression. 01:19:36.400 --> 01:19:40.990 It typically is quadratic in your x1 and x2, which is nice 01:19:40.990 --> 01:19:44.950 because then if you're trying to minimize that variance if it's 01:19:44.950 --> 01:19:47.830 quadratic in x1 and x2, it does tend to drive 01:19:47.830 --> 01:19:50.320 towards a nice minimum point. 01:19:53.300 --> 01:19:55.730 I have one quick example and then we'll be done. 01:19:55.730 --> 01:19:58.808 This is alluding back to our robust bending 01:19:58.808 --> 01:20:00.350 where I've got a punch and I'm trying 01:20:00.350 --> 01:20:02.990 to get a bend in a piece of sheet metal, 01:20:02.990 --> 01:20:05.720 and we might have a couple of control factors 01:20:05.720 --> 01:20:11.670 that again, we can use to optimize the process. 01:20:11.670 --> 01:20:14.100 Like how deep I push the punch and what 01:20:14.100 --> 01:20:17.010 the width of the width of the initial punch dye 01:20:17.010 --> 01:20:20.160 is, but here's an example of a couple additional noise 01:20:20.160 --> 01:20:21.060 factors. 01:20:21.060 --> 01:20:25.590 It may be we know that some of the incoming material 01:20:25.590 --> 01:20:30.210 will have a range of yield points and thicknesses. 01:20:30.210 --> 01:20:33.510 Again, I'm not going to be able to necessarily select those. 01:20:33.510 --> 01:20:36.540 I want to process it, it's going to be robust to those. 01:20:36.540 --> 01:20:38.250 But in a design of experiments, I 01:20:38.250 --> 01:20:41.250 can certainly sample from different incoming material 01:20:41.250 --> 01:20:44.410 types that explore that. 01:20:44.410 --> 01:20:47.250 So I could then do a design of experiments 01:20:47.250 --> 01:20:57.720 where I vary both yield point and thickness of the input. 01:20:57.720 --> 01:21:00.570 I can build up a model of z, I can build up 01:21:00.570 --> 01:21:04.650 a model of the angle, I can go in and do estimation 01:21:04.650 --> 01:21:11.150 of both my main effect on my control factors, 01:21:11.150 --> 01:21:13.040 on interaction terms. 01:21:13.040 --> 01:21:16.250 I could then estimate the size of those coefficients, 01:21:16.250 --> 01:21:18.890 go in and make sure that they are significant, 01:21:18.890 --> 01:21:21.380 and overall get down to say something 01:21:21.380 --> 01:21:25.940 that looks like a reduced model where I might 01:21:25.940 --> 01:21:27.680 be looking and saying, aw, OK. 01:21:27.680 --> 01:21:31.610 That little beta 1, 2, 3 is insignificant. 01:21:31.610 --> 01:21:36.320 I'm not going to include an x1, x2, x3 factor, for example, 01:21:36.320 --> 01:21:39.110 in the model. 01:21:39.110 --> 01:21:41.430 Here's the example of the reduced order model. 01:21:41.430 --> 01:21:46.130 In this case, the mean surface is nice and linear 01:21:46.130 --> 01:21:49.810 with an the interaction term. 01:21:49.810 --> 01:21:53.640 So we could go and do optimization purely 01:21:53.640 --> 01:21:58.680 on the mean surface trying to hit some target angle, 01:21:58.680 --> 01:22:01.540 but I can also look at the variant surface. 01:22:01.540 --> 01:22:05.950 And in this case, the variant surface has-- 01:22:05.950 --> 01:22:09.580 it is a quadratic, it's just a weakly quadratic. 01:22:09.580 --> 01:22:13.330 But you can see a very, very strong dependence 01:22:13.330 --> 01:22:15.730 in the variance on this x2 factor. 01:22:15.730 --> 01:22:19.500 I think that was the thickness factor. 01:22:19.500 --> 01:22:21.690 Very big differences in variance. 01:22:21.690 --> 01:22:24.930 And so then I could combine those two in different ways 01:22:24.930 --> 01:22:27.750 to form an optimization and try to get 01:22:27.750 --> 01:22:30.240 to a good robust operating point and that 01:22:30.240 --> 01:22:32.640 might be a Taguchi signal to noise 01:22:32.640 --> 01:22:35.970 or it might be an expected loss function. 01:22:35.970 --> 01:22:38.520 In this case here, I've plotted out 01:22:38.520 --> 01:22:43.080 what the expected loss surface is as a function of x1 and x2, 01:22:43.080 --> 01:22:45.900 and if you're seeking to minimize that you can start 01:22:45.900 --> 01:22:50.820 to see a region there where you would want to be in terms 01:22:50.820 --> 01:22:55.320 of the x1, x2 to have a good on target or close 01:22:55.320 --> 01:22:59.260 to target and a low variance process. 01:22:59.260 --> 01:23:00.123 OK. 01:23:00.123 --> 01:23:01.540 I'm going to skip over this slide. 01:23:01.540 --> 01:23:05.560 It's basically just talking about a summary 01:23:05.560 --> 01:23:08.800 of doing the combined response surface 01:23:08.800 --> 01:23:11.530 model with a little bit of an assessment of maybe 01:23:11.530 --> 01:23:13.180 the number of different design points 01:23:13.180 --> 01:23:15.480 that might be required in those cases. 01:23:15.480 --> 01:23:18.760 But what I wanted to do here is just conclude and give you 01:23:18.760 --> 01:23:23.860 the high level perspective here is we did lots of machinery 01:23:23.860 --> 01:23:26.200 on response surface modeling talking mostly 01:23:26.200 --> 01:23:30.460 about mean output, but that same machinery can also 01:23:30.460 --> 01:23:34.900 be used with some extra ideas of these inner and outer arrays 01:23:34.900 --> 01:23:36.940 for also modeling variance when it 01:23:36.940 --> 01:23:39.830 varies as a function of operating point. 01:23:39.830 --> 01:23:42.070 And then we can combine those together 01:23:42.070 --> 01:23:47.800 to try to get processes that are robust to not only 01:23:47.800 --> 01:23:50.350 selection of operating point, but also to these kinds 01:23:50.350 --> 01:23:52.220 of noise factors. 01:23:52.220 --> 01:23:57.310 So with that, I'll conclude and we'll see you all on Tuesday. 01:23:57.310 --> 01:24:00.718 I think on Tuesday we'll talk about nested variances, 01:24:00.718 --> 01:24:02.510 so it's going to be a little bit different. 01:24:02.510 --> 01:24:04.802 We're going to pull back from the design of experiments 01:24:04.802 --> 01:24:10.780 and talk a little bit more about interesting variance structures 01:24:10.780 --> 01:24:13.510 often arising in places like semiconductor manufacturing 01:24:13.510 --> 01:24:14.210 and elsewhere. 01:24:14.210 --> 01:24:16.320 So we'll see you on Tuesday.