DESIGN OF EXPERIMENT ERT 427 Response Surface Methodology (RSM) Miss Hanna Ilyani Zulhaimi

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1 + DESIGN OF EXPERIMENT ERT 427 Response Surface Methodology (RSM) Miss Hanna Ilyani Zulhaimi

2 + Outline n Definition of Response Surface Methodology n Method of Steepest Ascent n Second-Order Response Surface n Response Surface Characterization n Location of Stationary Point n Experimental Design for Fitting First Order and Second Order Model (CCD and Box-Behnken Design) n Blocking in Response Surface Design

3 + Response Surface Methods and Designs n Primary focus of previous chapters is factor screening Ø Two-level factorials, fractional factorials are widely used n Response Surface Methodology (RSM) is useful for the modeling and analysis of programs in which a response of interest is influenced by several variables. n Objective of RSM is optimization n For example: Find the levels of temperature (x 1 ) and pressure (x 2 ) to maximize the yield (y) of a process. y = f (x 1, x 2 )+ε

4 + Example of Response Surface Graph

5 + (continued) n RSM a sequential procedure n The objective is to lead the experimenter rapidly and efficiently along a path of improvement toward the general vicinity of the optimum. n First-order model goes to Second-order model n Climb a hill

6 + Response Surface Model

7 + Method of Steepest Ascent n Assume that the first-order model is an adequate approximation to the true surface in a small ragion of the x s. n A p r o c e d u r e f o r m o v i n g sequentially from an initial guess towards to region of the optimum. n Based on the fitted first-order model n Steepest ascent is a gradient procedure n The path of steepest ascent is the regression coefficients Figure 11.4: First-order response surface and path of steepest ascent

8 + Example 11.1: n A chemical engineer is interested in determining the operating conditions that maximize the yield of a process. Two controllable variables: reaction time (ξ1) and reaction temperature (ξ2). The engineer is currently operating the process with reaction time of 35 minutes and temperature of 155 F, which result in yield around 40 percent. Because it is unlikely that this region contains the optimum, she fits a firstorder model and applies the method of steepest ascent.

9 + (continue ) The design employed was a 22 factorial design with 6ive centerpoints

10 + (Continued) n A first-order model may be fit to these data by least squares.

11 + n To move away from the design center, the point (x 1 =0,x 2 =0) along the path of steepest ascent, we would move units in the x 1 direction for every units in the x 2 direction. If we decide to use 5 minutes of reaction time as the basic step size, then. Δx 1 = Δx 2 = [(0.325/0.775)Δx 1 ] = n The engineer computes points along this path and observes yields at these points until a decrease in response is noted.the results are shown in Table1-3.The Steps are shown in both coded and natural variables.

12 The step size is 5 minutes of reaction time and 2 degrees F We can see from the table1-3,increases in response are observed through the tenth step. However the eleventh step produces a decrease in yield. Therefore, another first-order model must be fit in the general vicinity of the point( ξ 1 = 85, ξ 2 =175) 2 12

13 + Data for second First-Order Model The first- order model fit to the coded data in Table 1-4 is y= x X 2

14 + Analysis of data for second first-model

15 + Summary of Steepest Ascent q Points on the path of steepest ascent are proportional to the magnitudes of the model regression coefficients. q The direction depends on the sign of the regression coefficient. q Step-by-step procedure:

16 + The Second-Order Model n When the experimenter is relative closed to the optimum, the second- order model is used to approximate the response. n It is used to 6ind the stationary point. n Determine whether the stationary point is a point of maximum or minimum response or a saddle point. n There is a lot of empirical evidence that they work very well. These models are used widely in practice. n Optimization is easy.

17 + Characterization of the Response Surface n Find the stationary point n Find what type of surface we have Ø Graphical Analysis Ø Canonical Analysis n Determine the sensitivity of the response variable to the optimum value Ø Canonical Analysis

18 + Second Order Model: Stationary Point n Writing the second-order model in matrix notation, we have: b x y b B x B b x Bx x b x s 1 s ' ˆ ˆ 2 1 ˆ 2 / ˆ ˆ 2 / ˆ 2 / ˆ ˆ and ˆ ˆ ˆ,, ' ' ˆ ˆ s kk k k k x k x x y + = = = = = + + = β β β β β β β β β β β ç Stationary point

19 + Second Order Model: Stationary Point n In derivatives form: n Stationary point represents: Ø Maximum Point Ø Minimum Point Ø Saddle Point (minimax)

20 + Response Surface with Maximum Point

21 + Response Surface with minimum point

22 + Response Surface with Saddle point

23 + Canonical Analysis The canonical form: the {λ i } are the e i g e n v a l u e s o r characteristic roots of the matrix B

24 + Canonical Analysis n The nature of the response surface can be determined from the stationary point & the signs and magnitudes of the {λi }. Ø all positive: a minimum is found Ø all negative: a maximum is found Ø mixed: a saddle point is found n The response surface is steepest in the direction (canonical) corresponding to the largest absolute eigenvalue

25 + In Example 11.2 n We continue in Example However, we try to transfer first-order model with second-order. n Central composite design (CCD) was chosen (Table 11.6 & Figure 11.10). n The coefficients are as in Table 11.7 Ø Find the stationary point, x s Ø Change coded variable to natural variables (ξ 1 = 87 minutes; ξ 2 = F) Ø Characterize response surface by using canonical analysis Ø Rewrite the fitted model in canonical form:

26 + Fitting Design for First-Order Model n Suppose we have first-order model; n The class of orthogonal includes 2k full and fractional design in which main effects are not aliased with each other. Addition of center points is usually a good idea

27 + Fitting Design for Second-Order Model n Suppose we have second-order model; n Design for second-order model includes: Ø Central Composite Design Ø Box-Behnken design Ø Face-centered design Ø Equiradial design Ø small composite design & hybrid design

28 + Central Composite Design n The CCD consists of a 2 k factorial with n F factorial runs, 2k axial or star runs, and n c center runs.

29 + Rotatable CCD,α= nf 1/4 u A second-order response surface design is rotatable if the variance of the predicted response V[yˆ(x)] is the same at all the points of x that are at the same distance form the design center: α = (n F ) 1/4 u Gives the same prediction in all directions.

30 + Spherical CCD, α= (k) 1/2 n All factorial runs and axial runs have the same distance to the center: α = (k) 1/2 n Rotatability is a spherical property. However, it is not important to have exact rotatability to have spherical. n Spherical CCD: puts all the factorial and axial points on the surface of a sphere of radius k.

31 + Box Behnken Design n Three-level designs for small number of runs. n All points are on the distance 2 1/2 = from the center. n No points at the vertices. n Either rotatable or nearly rotatable.

32 + Blocking in Response Surface Methodology n It is often necessary to consider blocking to eliminate nuisance. n For second-order design to block orthogonally, two conditions must be satisfied: 1. Each block must be a first-order orthogonal design; that is: 2. The fraction of the total sum of square for each variable contributed by every block must be equal to the fraction of of the total observation that occur in the block; that is:

33 + Blocking in Response Surface Methodology n In general, when two blocks are required blocking, there should be an axial block and a factorial block. n For three blocks, the factorial block is divided into two blocks and the axial block is not split.

34 + CCD of Two Factors, 2 Blocks

35 + CCD of Three Factors, 3 Blocks

36 + Thank you J