Response Surface.pptx

Response Surface Methodology

1. Response Surface Methodology ( RSM )

2. What is RSM? RSM can be defines as a statistical method that uses quantitative data from appropriate experiments to determine & simultaneously solve multivarient equations

3. Why RSM? Critical factors are known Region of interest , where factor levels influencing product is known Factors vary continuously through- out the experimental range tested A mathematical function relates the factors to the measured response The response defined by the function is a smooth curve

4. illustration the growth of a plant is affected by a certain amount of water x1 and sunshine x2. The plant can grow under any combination of treatment x1and x2. Therefore, water and sunshine can vary continuously. When treatments are from a continuous range of values, then a Response Surface Methodology is useful for developing, improving, and optimizing the response variable. In this case, the plant growth y is the response variable, and it is a function of water and sunshine. It can be expressed as y = f (x1, x2) + e

5. First Order Model If the response can be defined by a linear function of independent variables, then the approximating function is a first-order model. A first-order model with 2 independent variables can be expressed as

6. Second Order Model If there is a curvature in the response surface, then a higher degree polynomial should be used. The approximating function with 2 variables is called a second-order model:

7. second degree equation

8. RSM In general all RSM problems use either one or the mixture of the both of these models. In order to get the most efficient result in the approximation of polynomials the proper experimental design must be used to collect data. Once the data are collected, the Method of Least Square is used to estimate the parameters in the polynomials. The response surface designs are types of designs for fitting response surface.

9. Objectives Therefore, the objective of studying RSM can be accomplish by understanding the topography of the response surface (local maximum, local minimum, ridge lines), and finding the region where the optimal response occurs. The goal is to move rapidly and efficiently along a path to get to a maximum or a minimum response so that the response is optimized.

10. Experimental Design First Degree Models A. Equiradial designs for 2 factors B. Equiradial designs for More tha n 2 factors – the Simplex Design

11. Experimental Design Second Degree Models Spherical Domain A. Composite Experimental Designs B. Uniform Shell ( Doehlert ) Designs C. Hybrid & Related Desings D. Equiradial Design for 2 factors – Regular Pentagon E. Box Behnken Designs

12. Experimental Design Cubic Shaped Domain A. Standard Design with 3 levels - Factorial Designs 3 K - Central Composite Design ( α = 1) - Box Behnken Desing B. Non Standard Design

13. Equiradial Design for 2 factors

14. The Simplex Design

15. Central Composite Desing Three distinct sets of experimental runs: A factorial design in the factors studied, each having two levels; A set of center points , experimental runs whose values of each factor are the medians of the values used in the factorial portion. A set of axial points (star point), Each factor is sequentially placed at ±α and all other factors are at zero.

16. Central Composite Desing

17. CC   Design Type Terminology Comments Circumscribed CCC CCC designs are the original formed CCD. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. Enlarging an existing factorial or fractional factorial design with star points can produce this design.  Inscribed CCI CCI design uses the factor settings as the star points and creates a factorial or fractional factorial design within those limits (in other words, a CCI design is a scaled down CCC design with each factor level of the CCC design divided by to generate the CCI design). This design also requires 5 levels of each factor.  Face Centered CCF In this design the star points are at the center of each face of the factorial space, so = ± 1. This variety requires 3 levels of each factor.

18. Comparison of 3 CC design CCC explores the largest process space and the CCI explores the smallest process space. Both the CCC and CCI are rotatable designs, but the CCF is not. Both the CCC and CCI are require 5 level for each factor while CCF is require 3 level for each factor.

19. Number of Factors Factorial Portion Scaled Value for Relative to ±1 2 2 2 2 2/4 = 1.414 3 2 3 2 3/4 = 1.682 4 2 4 2 4/4 = 2.000 5 2 5-1 2 4/4 = 2.000 5 2 5 2 5/4 = 2.378 6 2 6-1 2 5/4 = 2.378 6 2 6 2 6/4 = 2.828

20. Design matrix for 2 factor BLOCK X1 X2 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 1 2 -1.414 2 1.414 2 0  -1.414 2 1.414 2 2 Total Runs = 12

21. Design matrix for 3 factor CCC (CCI) Rep X 1 X 2 X 3 1 -1 -1 -1 1 +1 -1 -1 1 -1 +1 -1 1 +1 +1 -1 1 -1 -1 +1 1 +1 -1 +1 1 -1 +1 +1 1 +1 +1 +1 1 -1.682 1 1.682 1 -1.682 1 1.682 1 -1.682 1 1.682 6 Total Runs = 20

22. Box- Behnken Design The Box- Behnken design is an independent quadratic design in that it does not contain an surrounded factorial or fractional factorial design. In this design the treatment combinations are at the midpoints of edges of the process space and at the center . These designs are rotatable (or near rotatable) and require 3 levels of each factor.

23. Box- Behnken Design

24. Box- Behnken Design Box- Behnken designs are response surface designs, specially made to require only 3 levels, coded as -1, 0, and +1. Box- Behnken designs are available for 3 to 10 factors. It is formed by combining two-level factorial designs with incomplete block designs. This procedure creates designs with desirable statistical properties but, most importantly, with only a fraction of the experimental trials required for a three-level factorial. Because there are only three levels, the quadratic model was found to be appropriate. In this design three factors were evaluated, each at three levels, and experiment design were carried out at all seventeen possible combinations.

25. CCC (CCI) CCF Box- Behnken Rep X 1 X 2 X 3 Rep X 1 X 2 X 3 Rep X 1 X 2 X 3 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 1 +1 -1 -1 1 +1 -1 -1 1 +1 -1 1 -1 +1 -1 1 -1 +1 -1 1 -1 +1 1 +1 +1 -1 1 +1 +1 -1 1 +1 +1 1 -1 -1 +1 1 -1 -1 +1 1 -1 -1 1 +1 -1 +1 1 +1 -1 +1 1 +1 -1 1 -1 +1 +1 1 -1 +1 +1 1 -1 +1 1 +1 +1 +1 1 +1 +1 +1 1 +1 +1 1 -1.682 1 -1 1 -1 -1 1 1.682 1 +1 1 +1 -1 1 -1.682 1 -1 1 -1 +1 1 1.682 1 +1 1 +1 +1 1 -1.682 1 -1 3 1 1.682 1 +1         6 6         Total Runs = 20 Total Runs = 20 Total Runs = 15

26. Case study of Box Behnken Experimental Design

27. Batch Code Coded value Actual value X 1 X 2 X 3 X 1 X 2 X 3 B 1 -1 1 10 15 25 B 2 -1 -1 20 5 15 B 3 1 -1 30 5 25 B 4 -1 -1 10 10 15 B 5 -1 1 10 10 35 B 6 -1 1 20 5 35 B 7 1 1 30 10 35 B 8 -1 -1 10 5 25 B 9 20 10 25 B 10 1 1 30 15 25 B 11 1 -1 20 15 15 B 12 1 1 20 15 25 B 13 1 -1 30 10 15 B 14 20 10 25 B 15 20 10 25 B 16 20 10 25 B 17 20 10 25 Coded and actual values of Box-Behnken design The amount of HPMC K4M (X 1 ), amount of Carbopol 934P (X 2 ) and amount of Sodium alginate (X 3 ) were selected as independent variables.

28. Batch X 1 (%) X 2 (%) X 3 (%) FLT  SD (sec) TFT  SD (hr) t 50  SD (hr) n  SD B1 10 15 25 2  1 2.5  0.35 13.1  0.03 0.48  0.02 B2 20 5 15 9  2 10.0  0.41 12.5  0.06 0.57  0.01 B3 30 5 25 4  2 24.0  0.29 13.3  0.04 0.52  0.03 B4 10 10 15 11  2 4.2  0.32 12.0  0.07 0.60  0.02 B5 10 10 35 5  2 5.3  0.28 11.9  0.04 0.65  0.07 B6 20 5 35 3  2 24.0  0.34 14.8  0.08 0.52  0.01 B7 30 10 35 26  4 5.6  0.35 14.7  0.05 0.51  0.01 B8 10 5 25 4  2 8.0  0.44 12.0  0.01 0.39  0.02 B9 20 10 25 3  2 2.5  0.22 15.8  0.02 0.44  0.01 B10 30 15 25 15  3 4.4  0.14 12.0  0.04 0.52  0.03 B11 20 15 15 33  4 3.6  0.26 12.8  0.03 0.62  0.02 B12 20 15 25 15  4 4.9  0.16 11.1  0.02 0.47  0.04 B13 30 10 15 3  1 24.0  0.36 11.3  0.05 0.36  0.06 B14 20 10 25 24  3 4.8  0.18 10.5  0.04 0.50  0.01 B15 20 10 25 10  2 6.8  0.45 15.0  0.07 0.48  0.04 B16 20 10 25 6  2 7.0  0.0.36 13.2  0.06 0.70  0.03 B17 20 10 25 12  2 4.2  0.26 13.2  0.03 0.45  0.02

29. 29 Multiple Regression It is an extension of linear regression in which we wish to relate a response, Y dependent variables to more than one independent variable Linear Regression Y = A+ BY Multiple Regression Y = b o + b 1 X 1 + b 2 X 2 +…. X1, X2, …. Represent factors which influence the response

30. 30 Y = bo + b1X1 + b2X2 + b3X3… Y is response i.e. dissolution time Xi is independent variable bo is the intercept bi is regression coefficient for the i th independent variable X1, X2, X3.. Are the levels of variables

31. The Polynomial equation generated by this experimental design is described as: Yi = b0 + b1x1 +b2x2 + b3x3 + b12x1x2 + b13 x1x3 + b23x2x3 + b11x1 2 +b22x2 2 + b33x3 2 Where Y i is the dependent variable b is the intercept; b i , b ij and b ijk represents the regression coefficients X i represents the level of independent variables which were selected from the preliminary experiments.