1、L . M. Lye,DOE Course,1,Design and Analysis of Multi-Factored Experiments,Response Surface Methodology (RSM),L . M. Lye,DOE Course,2,Introduction to Response Surface Methodology (RSM),Best and most comprehensive reference:R. H. Myers and D. C. Montgomery (2002): Response Surface Methodology: Process
2、 and Product Optimization Using Designed Experiments, John Wiley and Sons.Best software:Design-Expert Version 6 or 7- Statease Inc.Available at Minitab also has DOE and RSM capabilities,L . M. Lye,DOE Course,3,RSM: Introduction,Primary focus of previous discussions is factor screeningTwo-level facto
3、rials, fractional factorials are widely usedRSM dates from the 1950s (Box and Wilson, 1951) Early applications in the chemical industryCurrently RSM is widely used in quality improvement, product design, uncertainty analysis, etc.,L . M. Lye,DOE Course,4,Objective of RSM,RSM is a collection of mathe
4、matical and statistical techniques that are useful for modeling and analysis in applications where a response of interest is influenced by several variables and the objective is to optimize the response.Optimize maximize, minimize, or getting to a target.Or, where a nonlinear model is warranted when
5、 there is significant curvature in the response surface.,L . M. Lye,DOE Course,5,Uses of RSM,To determine the factor levels that will simultaneously satisfy a set of desired specification (e.g. model calibration)To determine the optimum combination of factors that yield a desired response and descri
6、bes the response near the optimum To determine how a specific response is affected by changes in the level of the factors over the specified levels of interest,L . M. Lye,DOE Course,6,Uses of RSM (cont),To achieve a quantitative understanding of the system behavior over the region tested To find con
7、ditions for process stability = insensitive spot (robust condition)To replace a more complex model with a much simpler second-order regression model for use within a limited range replacement models, meta models, or surrogate models. E.g. Replacing a FEM with a simple regression model.,L . M. Lye,DO
8、E Course,7,Example,Suppose that an engineer wishes to find the levels of temperature (x1) and feed concentration (x2) that maximize the yield (y) of a process. The yield is a function of the levels of x1 and x2, by an equation: Y = f (x1, x2) + e If we denote the expected response by E(Y) = f (x1, x
9、2) = ,L . M. Lye,DOE Course,8,then the surface represented by: = f (x1, x2)is called a response surface.,The response surface maybe represented graphically using a contour plot and/or a 3-D plot. In the contour plot, lines of constant response (y) are drawn in the x1, x2, plane.,L . M. Lye,DOE Cours
10、e,9,L . M. Lye,DOE Course,10,These plots are of course possible only when we have two factors. With more than two factors, the optimal yield has to be obtained using numerical optimization methods.In most RSM problems, the form of the relationship between the response and the independent variables i
11、s unknown. Thus, the first step in RSM is to find a suitable approximation for the true relationship between Y and the Xs.,L . M. Lye,DOE Course,11,If the response is well modeled by a linear function of the independent variables, then the approximating function is the first-order model (linear):Y =
12、 b0 + b1 x1 + b2 x2 + + bk xk + eThis model can be obtained from a 2k or 2k-p design.If there is curvature in the system, then a polynomial of higher degree must be used, such as the second-order model:Y = b0 + Sbi xi + Sbii x2i + SSbij xi xj + eThis model has linear + interaction + quadratic terms.
13、,L . M. Lye,DOE Course,12,Many RSM problems utilize one or both of these approximating polynomials. The response surface analysis is then done in terms of the fitted surface. The 2nd order model is nearly always adequate if the surface is “smooth”.If the fitted surface is an adequate approximation (
14、high R2) of the true response function, then analysis of the fitted surface will be approximately equivalent to analysis of the actual system (within bounds).,L . M. Lye,DOE Course,13,Types of functions,Figures 1a through 1c on the following pages illustrate possible behaviors of responses as functi
15、ons of factor settings. In each case, assume the value of the response increases from the bottom of the figure to the top and that the factor settings increase from left to right.,L . M. Lye,DOE Course,14,Types of functions,Figure 1aLinear function,Figure 1bQuadratic function,Figure 1cCubic function
16、,L . M. Lye,DOE Course,15,If a response behaves as in Figure 1a, the design matrix to quantify that behavior need only contain factors with two levels - low and high. This model is a basic assumption of simple two-level factorial and fractional factorial designs. If a response behaves as in Figure 1
17、b, the minimum number of levels required for a factor to quantify that behavior is three.,L . M. Lye,DOE Course,16,One might logically assume that adding center points to a two-level design would satisfy that requirement, but the arrangement of the treatments in such a matrix confounds all quadratic
18、 effects with each other. While a two-level design with center points cannot estimate individual pure quadratic effects, it can detect them effectively. A solution to creating a design matrix that permits the estimation of simple curvature as shown in Figure 1b would be to use a three-level factoria
19、l design. Table 1 explores that possibility.Finally, in more complex cases such as illustrated in Figure 1c, the design matrix must contain at least four levels of each factor to characterize the behavior of the response adequately.,L . M. Lye,DOE Course,17,Table 1: 3 level factorial designs,No. of
20、factors # of combinations(3k) Number of coefficients 2 9 6 3 27 10 4 81 15 5 243 21 6 729 28The number of runs required for a 3k factorial becomes unacceptable even more quickly than for 2k designs. The last column in Table 1 shows the number of terms present in a quadratic model for each case.,L .
21、M. Lye,DOE Course,18,Problems with 3 level factorial designs,With only a modest number of factors, the number of runs is very large, even an order of magnitude greater than the number of parameters to be estimated when k isnt small. For example, the absolute minimum number of runs required to estima
22、te all the terms present in a four-factor quadratic model is 15: the intercept term, 4 main effects, 6 two-factor interactions, and 4 quadratic terms. The corresponding 3k design for k = 4 requires 81 runs.,L . M. Lye,DOE Course,19,Considering a fractional factorial at three levels is a logical step
23、, given the success of fractional designs when applied to two-level designs. Unfortunately, the alias structure for the three-level fractional factorial designs is considerably more complex and harder to define than in the two-level case. Additionally, the three-level factorial designs suffer a majo
24、r flaw in their lack of rotatabilityMore on rotatability later.,L . M. Lye,DOE Course,20,Sequential Nature of RSM,Before going on to economical designs to fit second-order models, lets look at how RSM is carried out in general.RSM is usually a sequential procedure. That is, it done in small steps to
25、 locate the optimum point, if thats the objective. This is not always the only objective.The analogy of climbing a hill is appropriate here (especially if it is a very foggy day)!,L . M. Lye,DOE Course,21,Sequential Nature of RSM (continue),When we are far from the optimum (far from the peak) there
26、is little curvature in the system (slight slope only), then first-order model will be appropriate. The objective is to lead the experimenter rapidly and efficiently to the general vicinity of the optimum. Once the region of the optimum has been found, a more elaborate model such a second-order model
27、 may be employed, and an analysis performed to locate the optimum.,L . M. Lye,DOE Course,22,L . M. Lye,DOE Course,23,The eventual objective of RSM is to determine the optimum operating conditions for the system or to determine a region of the factor space in which operating specifications are satisf
28、ied. The word “Optimum” in RSM is used in a special sense. The “hill climbing” procedures of RSM guarantee convergence to a local optimum only. In terms of experimental designs, when we are far from optimum, a simple 2k factorial experiment would allow us to fit a first-order model. As we get nearer
29、 to the peak, we can check for curvature by adding center-points to the 2k factorial.,L . M. Lye,DOE Course,24,If curvature is significant, we may now be in the vicinity of the peak and we use a more elaborate design (e.g. a CCD) to fit a second-order model to “capture” the optimum.,L . M. Lye,DOE C
30、ourse,25,Method of Steepest Ascent,The method of steepest ascent is a procedure for moving sequentially along the path of steepest ascent (PSA), that is, in the direction of the maximum increase in the response. If minimization is desired, then we are talking about the method of steepest descent. Fo
31、r a first-order model, the contours of the response surface is a series of parallel lines. The direction of steepest ascent is the direction in which the response y increases most rapidly. This direction is normal (perpendicular) to the fitted response surface contours.,L . M. Lye,DOE Course,26,Firs
32、t-order response and PSA,L . M. Lye,DOE Course,27,Path of Steepest Ascent (PSA),The PSA is usually the line through the center of the region of interest and normal to the fitted surface contours. The steps along the path are proportional to the regression coefficients bi. The actual step size would
33、depend on the experimenters knowledge of the process or other practical considerations.,L . M. Lye,DOE Course,28,For example, consider the first-order model:y = 40.00 + 0.775 x1 + 0.325 x2For steepest ascent, we move 0.775 unit in the x1 direction for every 0.325 unit in the x2 direction. Thus the P
34、SA passes through the center (0, 0) and has a slope of 0.375/0.775.,L . M. Lye,DOE Course,29,If say 1 unit of x1 is actually equal to 5 minutes in actual units, and 1 unit of x2 is actually equal to 5 F, the PSA are Dx1 = 1.00 and Dx2 = (0.375/0.775) Dx2 = 0.42 = 2.1 F. Therefore, you will move alon
35、g the PSA by increasing time by 5 minutes and temperature by 2 F. An actual observation on yield will be determined at each point.,L . M. Lye,DOE Course,30,Experiments are then conducted along the PSA until no further increase in the response is observed. Then a new first-order model may be fit, a n
36、ew direction of steepest ascent determined, and further experiments conducted in that direction until the experimenter feels that the process is near the optimum (peak of hill is within grasp!).,L . M. Lye,DOE Course,31,Yield vs steps along the PSA,L . M. Lye,DOE Course,32,The steepest ascent would
37、terminate after about 10 steps with an observed response of about 80%. Now we move on to the next step. Fit another first-order model with a new center (where step 10 is) and check whether there is a new PSA. Repeat until peak is near. See flowchart on the next slide.,L . M. Lye,DOE Course,33,Flowch
38、art for RSM,L . M. Lye,DOE Course,34,Steps in RSM,Fit linear model/planar models using two-level factorialsFrom results, determine PSA (Descent)Move along path until no improvement occursRepeat steps 1 and 2 until near optimal (change of direction is possible)Fit quadratic model near optimal in orde
39、r to determine curvature and find peak. This phase is often called “method of local exploration”Run confirmatory tests,L . M. Lye,DOE Course,35,Steps in RSM,L . M. Lye,DOE Course,36,With well-behaved functions with a single peak or valley, the above procedure works very well. It becomes more difficu
40、lt to use RSM or any other optimization routine when the surface has many peaks, ridges, and valleys.,Response surface with many peaks and valleys,L . M. Lye,DOE Course,37,Multiple Objectives,With more than 2 factors, it is more difficult to determine where the optimal is. There may be several possi
41、ble “optimal” points and not all are desirable. Whatever the final choice of optimal factor levels, common sense and process knowledge must be your guide. It is also possible to have more than one response variable with different objectives (sometimes conflicting). For these cases, a weighting syste
42、m may be used to for the various objectives.,L . M. Lye,DOE Course,38,Methods of Local Exploration,The method of steepest ascent, in addition to fitting first-order model, must provide additional information that will eventually identify when the first-order model is no longer valid.This information
43、 can come only from additional degrees of freedom which are used to measure “lack of fit” in some way. This means additional levels and extra data points.It is rare to go more than 5 levels for even the most complex response surfaces.,L . M. Lye,DOE Course,39,Consider the 2nd order model:Y = b0 + b1
44、 x1 + b2 x2 + + bk xk + b11 x12 + b22 x22 + + bkk xk2+ b12 x1 x2 + + b1k x1 xk + + b23 x2 x3 + + bk-1,k xk-1 xk + e- EQN (1)To be able to fit a 2nd order model like EQN (1), there must be least three levels and enough data points.,L . M. Lye,DOE Course,40,Designs for fitting 2nd order models,Two ver
45、y useful and popular experimental designs that allow a 2nd order model to be fit are the:Central Composite Design (CCD)Box-Behnken Design (BBD)Both designs are built up from simple factorial or fractional factorial designs.,L . M. Lye,DOE Course,41,3-D views of CCD and BBD,L . M. Lye,DOE Course,42,C
46、entral Composite Design (CCD),Each factor varies over five levels Typically smaller than Box-Behnken designs Built upon two-level factorials or fractional factorials of Resolution V or greaterCan be done in stages factorial + centerpoints + axial pointsRotatable,L . M. Lye,DOE Course,43,General Stru
47、cture of CCD,2k Factorial + 2k Star or axial points + nc Centerpoints The factorial part can be a fractional factorial as long as it is of Resolution V or greater so that the 2 factor interaction terms are not aliased with other 2 factor interaction terms. The “star” or “axial” points in conjunction
48、 with the factorial and centerpoints allows the quadratic terms (bii) to be estimated.,L . M. Lye,DOE Course,44,Generation of a CCD,Factorial points + centerpoints,Axial points,L . M. Lye,DOE Course,45,Axial points are points on the coordinate axes at distances “a” from the design center; that is, w
49、ith coordinates: For 3 factors, we have 2k = 6 axial points like so:(+a, 0, 0), (-a, 0, 0), (0, +a, 0), (0, -a, 0), (0, 0, +a), (0, 0, -a)The “a” value is usually chosen so that the CCD is rotatable. At least one point must be at the design center (0, 0, 0). Usually more than one to get an estimate of “pure error”. See earlier 3-D figure. If the “a” value is 1.0, then we have a face-centered CCD Not rotatable but easier to work with.,
