RESPONSE SURFACE METHODOLOGY (R S M) Par Mariam MAHFOUZ

Remember that: General Planning Part I A - Introduction to the RSM method B - Techniques of the RSM method C - Terminology D - A review of the method of least squares Part II A - Procedure to determine optimum conditions – Steps of the RSM method B – Illustration of the method with an example

Part II

A - Procedure to determine optimum conditions steps of the method This method permits to find the settings of the input variables which produce the most desirable response values. The set of values of the input variables which result in the most desirable response values is called the set of optimum conditions.

Steps of the method The strategy in developing an empirical model through a sequential program of experimentation is as follows: The simplest polynomial model is fitted to a set of data collected at the points of a first-order design. If the fitted first-order model is adequate, the information provided by the fitted model is used to locate areas in the experimental region, or outside the experimental region, but within the boundaries of the operability region, where more desirable values of the response are suspected to be.

In the new region, the cycle is repeated in that the first-order model is fitted and testing for adequacy of fit. If nonlinearity in the surface shape is detected through the test for lack of fit of the first-order model, the model is upgraded by adding cross-product terms and / or pure quadratic terms to it. The first-order design is likewise augmented with points to support the fitting of the upgraded model.

If curvature of the surface is detected and a fitted second-order model is found to be appropriate, the second-order model is used to map or describe the shape of the surface, through a contour plot, in the experimental region. If the optimal or most desirable response values are found to be within the boundaries of the experimental region, then locating the best values as well as the settings of the input variables that produce the best response values.

7. Finally, in the region where the most desirable response values are suspected to be found, additional experiments are performed to verify that this is so.

B- Illustration of the method with an example For simplicity of presentation we shall assume that there is only one response variable to be studied although in practice there can be several response variables that are under investigation simultaneously.

Two controlled Factors Experience Two controlled Factors Chemical reaction One response Temperature (X1) percent yield Time (X2) An experimenter, interested in determining if an increase in the percent yield is possible by varying the levels of the two factors.

And two repetitions at each point Two levels of temperature: 70° and 90°. Two levels of time: 30 sec and 90 sec. Four different design points Four temperature-time settings (factorial combinations) And two repetitions at each point The total number of observations is N = 8

Detail The response of interest is the percent yield, which is a measure of the purity of the end product. The process currently operates in a range of percent purity between 55 % and 75 %, but it is felt that a higher percent yield is possible.

x1 and x2 are the coded variables which are defined as: Design 1 Original variables Coded variables Percent yield Temperature X1 (C°) Time X2 (sec.) x1 x2 Y 70 30 -1 49.8 48.1 90 1 57.3 52.3 65.7 69.4 73.1 77.8 x1 and x2 are the coded variables which are defined as:

Representation of the first design

The remaining term, , represents random error in the yield values. First-order model Expressed in terms of the coded variables, the observed percent yield values are modeled as: The remaining term, , represents random error in the yield values. The eight observed percent yield values, when expressed as function of the levels of the coded variables, in matrix notation, are: Y = X  + 

Vector of response values Matrix form Vector of error terms Vector of response values Matrix of the design = + Vector of unknown parameters

Estimations The estimates of the coefficients in the first-order model are found by solving the normal equations: The estimates are: The fitted first-order model in the coded variables is:

ANOVA table – design 1 Source Degrees of freedom d.f. Sum of squares SS Mean square F Model 2 864.8125 432.4063 63.71 Residual 5 33.9363 6.7873 Lack of fit 1 2.1013 0.264 Pure error 4 31.8350 7.9588 Test of adequacy

Individual tests of parameters To do that the Student-test is used. For the test of: we have And for we have Each of the null hypotheses is rejected at the  = 0.05 level of significance owing to the calculated values, 3.73 and 10.65, being greater in absolute value than the tabled value, T5;0.025 =2.571.

Conclusion of the first analysis The first – order model is adequate. That both temperature and time have an effect on percent yield. Since both b1 and b2 are positive, the effects are positive. Thus, by raising either the temperature or time of reaction, this produced a significant increase in percent yield.

Second stage of the sequential program At this point, the experimenter quite naturally might ask: If additional experiments can be performed At what settings of temperature and time should the additional experiments be run?” To answer this question, we enter the second stage of our sequential program of experimentation.

Contour plots The fitted model: can now be used to map values of the estimated response surface over the experimental region. This response surface is a hyper-plane; their contour plots are lines in the experimental region. The contour lines are drawn by connecting two points (coordinate settings of x1 and x2) in the experimental region that produce the same value of

In the figure above are shown the contour lines of the estimated planar surface for percent yield corresponding to values of = 55, 60, 65 and 70 %.

Performing experiments along the path of steepest ascent To describe the method of steepest ascent mathematically, we begin by assuming the true response surface can be approximated locally with an equation of a hyper-plane Data are collected from the points of a first-order design and the data are used to calculate the coefficient estimates to obtain the fitted first-order model Estimated response function

subject to the constraint The next step is to move away from the center of the design, a distance of r units, say, in the direction of the maximum increase in the response. By choosing the center of the design in the coded variable to be denoted by O(0, 0, …, 0), then movement away from the center r units is equivalent to find the values of which maximize subject to the constraint Maximization of the response function is performed by using Lagrange multipliers. Let where  is the Lagrange multiplier.

To maximize subject to the above-mentioned constraint, first we set equal to zero the partial derivatives i=1,…,k and Setting the partial derivatives equal to zero produces: i =1,…,k, and The solutions are the values of xi satisfying or i = 1,…,k, where the value of  is yet to be determined. Thus the proposed next value of xi is directly proportional to the value of bi.

Let us the change in Xi be noted by i , and the change in xi be noted by i. The coded variables is obtained by these formulas where (respectively si) is the mean (respectively the standard deviation) of the two levels of Xi . Thus , then or

Let us illustrate the procedure with the fitted first-order model: that was fitted early to the percent yield values in our example. To the change in X2, 2=45 sec. corresponds the change in x2, 2=45/30=1.5 units. In the relation , we can substitute i to xi: , thus and 1 = 0.526, so 1=0.526*10=5.3°C .

Observed percent yield Points along the path of steepest ascent and observed percent yield values at the points Temperature X1 (°C) Time X2 (sec.) Observed percent yield Base 80.0 60 i 5.3 45 Base + i 85.3 105 74.3 Base + 1.5 i 87.95 127.5 78.6 Base + 2 I 90.6 150 83.2 Base + 3 I 95.9 195 84.7 Base + 4 i 101.2 240 80.1

Design two: For this design the coded variables are defined as: Sequence of experimental trials performed in moving to a region of high percent yield values Design two: For this design the coded variables are defined as: x1 x2 X1 X2 % yield -1 85.9 165 82.9; 81.4 +1 105.9 87.4; 89.5 225 74.6; 77.0 84.5; 83.1 95.9 195 84.7; 81.9

The model is jugged adequate The fitted model corresponding to the group of experiments of design two is: The corresponding analysis of variance is: Source d.f. SS MS F Model 2 162.745 81.372 42.34 Residual 7 13.455 1.922 Lack of fit 2.345 1.173 0.53 Pure error 5 11.110 2.222 Total (variations) 9 176.2 The model is jugged adequate

sequence of experimental trials that were performed in the direction two: Steps x1 x2 X1 X2 % yield 1 Center + i +1 - 0.77 105.9 171.9 89.0 2 Center + 2 i +2 - 1.54 115.9 148.8 90.2 3 Center + 3 i +3 - 2.31 125.9 125.7 87.4 4 Center + 4 i +4 - 3.08 135.9 102.6 82.6

Retreat to center + 2 i and proceed in direction three Steps x1 x2 X1 X2 % yield 5 Replicated 2 +2 - 1.54 115.9 148.8 91.0 6 +3 - 0.77 125.9 171.9 93.6 7 +4 135.9 195 96.2 8 +5 0.77 145.9 218.1 92.9

Set up design three using points of steps 6, 7, and 8 along with the following two points: x1 x2 X1 X2 % yield 9 +3 0.77 125.9 218.1 91.7 10 +5 - 0.77 145.9 171.9 92.5 11 Replicated 7 +4 135.9 195 97.0

Design three was set up using the point at step 7 as its center Design three was set up using the point at step 7 as its center. It includes steps 6 – 11. If we redefine the coded variables: and then the fitted first-order model is: The corresponding analysis of variance table is:

ANOVA table source d.f. SS MS F Model 2 0.5650 0.2825 0.04 Residual 3 22.1833 7.3944 total 5 22.7483 It is obvious from the ANOVA table that the least model does not explain a significant amount of the overall variation in the percent yield values, and it is necessary to fit a curved surface.

Fitting a second-order model A second-order model in k variables is of the form: The number of terms in the model above is p=(k+1)(k+2)/2; for example, when k=2 then p=6. Let us return to the chemical reaction example of the previous section. To fit a second-order model (k=2), we must perform some additional experiments.

Central composite rotatable design Suppose that four additional experiments are performed, one at each of the axial settings (x1,x2): These four design settings along with the four factorial settings: (-1,-1); (-1,1); (1,-1); (1,1) and center point comprise a central composite rotatable design. The percent yield values and the corresponding nine design settings are listed in the table below:

Central composite rotatable design

Percent yield values at the nine points of a central composite rotatable design

The fitted second-order model, in the coded variables, is: The analysis is detailed in this table, using the RSREG procedure in the SAS software:

SAS output 1

SAS output 2

Response surface and the contour plot

More explanations The contours of the response surface, showing above, represent predicted yield values of 95.0 to 96.5 percent in steps of 0.5 percent. The contours are elliptical and centered at the point (x1; x2)=(- 0.0048; - 0.0857) or (X1; X2)=(135.85°C; 193.02 sec). The coordinates of the centroid point are called the coordinates of the stationary point. From the contour plot we see that as one moves away from the stationary point, by increasing or decreasing the values of either temperature or time, the predicted percent yield (response) value decreases.

Determining the coordinates of the stationary point A near stationary region is defined as a region where the surface slopes (or gradients along the variables axes) are small compared to the estimate of experimental error. The stationary point of a near stationary region is the point at which the slope of the response surface is zero when taken in all direction. The coordinates of the stationary point are calculated by differentiating the estimated response equation with respect to each xi, equating these derivatives to zero, and solving the resulting k equations simultaneously.

Remember that the fitted second-order model in k variables is: To obtain the coordinates of the stationary point, let us write the above model using matrix notation, as:

where and

Some details The partial derivatives of with respect to x1, x2, …, xk are :

More details Setting each of the k derivatives equal to zero and solving for the values of the xi, we find that the coordinate of the stationary point are the values of the elements of the kx1 vector x0 given by: At the stationary point, the predicted response value, denoted by , is obtained by substituting x0 for x:

Return to our example The fitted second-order model was: so the stationary point is: In the original variables, temperature and time of the chemical reaction example, the setting at the stationary point are: temperature=135.85°C and time=193.02 sec. And the predicted percent yield at the stationary point is:

Moore details Note that the elements of the vector x0 do not tell us anything about the nature of the surface at the stationary point. This nature can be a minimum, a maximum or a mini_max point. For each of these cases, we are assuming that the stationary point is located inside the experimental region. When, on the other hand, the coordinates of the stationary point are outside the experimental region, then we might have encountered a rising ridge system or a falling ridge system, or possibly a stationary ridge.

Nest Step The next step is to turn our attention to expressing the response system in canonical form so as to be able to describe in greater detail the nature of the response system in the neighborhood of the stationary point.

The canonical Equation of a Second-Order Response System The first step in developing the canonical equation for a k-variable system is to translate the origine of the system from the center of the design to the stationary point, that is, to move from (x1,x2,…,xk)=(0,0,…,0) to x0. This is done by defining the intermediate variables (z1,z2,…,zk)=(x1-x10,x2-x20,…,xk-xk0) or z=x-x0. Then the second-order response equation is expressed in terms of the values of zi as:

This transformation is a rotation of the zi axes to form the wi axes. Now, to obtain the canonical form of the predicted response equation, let us define a set of variables w1,w2,…,wk such that W’=(w1,w2,…,wk) is given by where M is a kxk orthogonal matrix whose columns are eigenvectors of the matrix B. The matrix M has the effect of diagonalyzing B, that is, where 1,2,…,k are the corresponding eigenvalues of B. The axes associated with the variables w1,w2,…,wk are called the principal axes of the response system. This transformation is a rotation of the zi axes to form the wi axes.

It is easy to see that if 1,2,…,k are: So we obtain the canonical equation: The eigenvalues i are real-valued (since the matrix B is a real-valued, symmetric matrix) and represent the coefficients of the terms in the canonical equation. It is easy to see that if 1,2,…,k are: 1) All negative, then at x0 the surface is a maximum. 2) All positive, then at x0 the surface is a minimum. 3) Of mixed signs, that is, some are positive and the others are negative, then x0 is a saddle point of the fitted surface. The canonical equation for the percent yield surface is:

Moore details The magnitude of the individual values of the i tell how quickly the surface height changes along the Wi axes as one moves away from x0. Today there are computer software packages available that perform the steps of locating the coordinates of the stationary point, predict the response at the stationary point, and compute the eigenvalues and the eigenvectors.

For example, the solution for optimum response generated from PROC RSREG of the Statistical Analysis System (SAS) for the chemical reaction data, is in following table:

Contours and optimal direction New series of experiments Recapitulate Process to optimize Contours and optimal direction Input and output variables Experiments in the Optimal direction Experimental and Operational regions Locate a new Experimental region Series of experiments New series of experiments Yes Fitting First-order model Fitting First-order model Model Adequate ? Model Adequate ? Yes Fitting a Second-order model No No

Doing this: Coordinates of the stationary point. Description of the shape of the response surface near the stationary point by contour plots. Canonical analysis. If needed, Ridge analysis (not detailed here).

Field of use of the method In agriculture In food industry In pharmaceutical industry In all kinds of the light and heavy industries In medical domain Etc….

Bibliography André KHURI and John CORNELL: “Response Surfaces – Designs and Analyses” , Dekker, Inc., ASQC Quality Press, New York. Irwin GUTTMAN: “Linear Models: An Introduction”, John Wiley & Sons, New York. George BOX, William HUNTER & J. Stuart HUNTER: “Statistics for experimenters: An Introduction to Design, Data Analysis, and Model Building” , John Wiley & Sons, New York. George BOX & Norman DRAPPER: “Empirical Model-Building and Response Surfaces” , John Wiley & Sons, New York.

Thank you Questions?