Response surface methodology.pptx

Response surface methodology with method models and numerical.

1. presentation By: ROLL NUMBER: Rakhshanda kousar STAT51F20S013

2. Response Surface Methodology (RSM): History: Response surface methodology was developed by BOX and WILSON (1951) to improve the production processes in the chemical industries. What is the RSM? In the statistics the response surface methodology explore the relationship between the several explanatory variables and one or more response variables. Response surface methodology (RSM) uses various statistical, graphical and mathematical techniques to develop, improve or optimize a process, also use for modeling and analysis of problem if our response variable in effect by several independents variable.

3. Objectives : Optimize. (main objective) Develop Improve (if necessary) Real life example: .RSM is used in different fields of real life. Like industries ,agriculture, electronics, medical field and many other like this.

4. Methods of Response Surface Methodology: onse RSM is often a sequential procedure when we are at a point on the response surface that is remote from the optimum. Thus in this method, move rapidly from the current point to the optimum point (sources are minimum, but the output is maximum) with a sequence. There are basically two methods of RSM to obtain the optimum point. 1. Steepest ascent method 2. Steepest descent Method

5. Method of Steepest Ascent: It is a process of moving sequentially toward the maximum increase in the response to the optimum response is called steepest ascent method

6. Method of steepest descent: Use the method of steepest descent to minimize the response. To move from the current position ̅y=50 to ̅y =10 in a faster way, use the method of steepest descent.

7. Response Surface Modeling The independent variables x1, x2,x3… xk in RSM, and the assumption is that response is a function of a set of independent variables. While the function can be represented in some regions of the polynomial model. y= f(xi) + Ɛ (simple linear model) y= f(x1, x2,x3… xk ) + Ɛ ( multiple linear model) y is the response or dependent variable and k independent variable. If the factors are known, directly estimate the effects and also the interactions. Otherwise, use a screening method to calculate the unknown factors. Estimate the interaction effect using the 1st order model.

8. y= β0+β1x 1 + β2x 2 + Ɛ Ɛ-Error β0, β1, β2 are the constants.  If the curvature exists, then use RSM. Use 2nd order model to approximate the response variable. y= β0+β1x1+ β2x 2 + β12 x 1 x 2 + β11x 1 2 + β22 x 2 2 + Ɛ

9. Response Surface Modeling Example of Response Surface Modeling Example: A researcher wants to install air-conditioning in a room, and they are estimating the efficiency of the room. The efficiency of the room depends upon the temperature as well as the volume of the room. Models: efficiency = β0+β1(temperature )+ Ɛ -----> simple efficiency = β0+β1(temperature )+ β2(volume) + Ɛ --->multi

10. Method for estimation of second order Model : There are two commonly used methods : Central composite design (CCD) Box-Behnken Design (BBD) Both these methods are used for the development of a second order response surface

11. Second order response surface Design:

12.   Central composite Design (CCD): Central composite design is also known as Box-Wilson Design. The design include both the factorial and center points used for the estimation of curvature . In case of CCD the experimental runs can be divide into three distinct categories Factorial Runs Central Runs and Axial Runs

13. Box-Behnken Design: The box-Behnken design is most commonly used method for second order response methodology. Box-Behnken design is developed by George E.P Box and Donald Behnken in 1960. It is also known as three level design. It has three levels and each level has different factor. Box-Behnken designs are used to generate higher-order response surfaces using fewer required runs than a normal factorial technique Used in different fields such as engineering, chemistry , and biology for desired outcomes.

14. Difference between CCD and BBD: Central Composite Design: CCD has factorial, axial and central points. CCD used for the estimation of extreme points. Used for large sample size. More efficient than three level design Box-Behnken Design: BBD has factorial and central points. BBD used for estimation of central points. Used for small sample size. More efficient than full factorial design

15. Numerical Question of 1 st order model: A chemical engineer is interested in determining the operating conditions that maximize the yield of a process. Two controllable variables influence proves yield: reaction time and reaction temperature. The engineer is currently operating the process with the rection time of 35 minutes and a temperature of F, which result in yields of around 40 percent. Because it is unlikely that this region contains the optimum, she fits a first-order model.  

16. Natural Variables Coded Variables Response y 30 150 -1 -1 39.3 30 160 -1 1 40.0 40 150 1 -1 40.9 40 160 1 1 41.5 35 155 0 0 40.3 35 155 0 0 40.5 35 155 0 0 40.7 35 155 0 0 40.2 35 155 0 0 40.  

17. Fit the regression model of coadded values µ= = = 40.44 = = =0.775 = = = 0.325 Now, putting the values in model.  

18. Now calculate model sum of square Effect of A = -1+a-b+ab = -39.3+40.0-40.9+41.5 = 1.3 Effect of B = -1-a+b+ab = -39.3-40.0+40.9+41.5 =3.1 SSA = = = 0.4225 SSB = = =2.4025 Model ( , ) = SSA + SSB Model ( , ) =0.4225+2.4025 = 2.825  

19. Now calculate total sum of squar e C.F = TSS = + - =14724.78 – 14721.78 TSS = 3.002  

20. Calculate Interaction = = = - 0.25   Sum of Square of (Interaction) = 0.0025  

21. Calculate Quadratic Effect = =40.425 40.425 – 40.46 = = 40.46 = - 0.035 Sum of Square of = =0.0027  

22. Estimation of Error = = = 0.0430 Now calculate F values F= = = 0.058 F= = = 0.063  

23. Calculate Error Sum of Square ESS= TSS – Model ( , ) - - ESS= 3.002 – 2.8250 – 0.0025 – 0.0027 ESS= 0.172  

24. ANOVA Table : SOV SS DF MS F Model 2.8250 1+1= 2 1.4125 47.83 Interaction 0.0025 1 0.0025 0.058 Quadratic 0.0027 1 0.0027 0.063 Error 0.1720 4 0.0430 Total 3.0022 8 SOV SS DF MS F 2.8250 1+1= 2 1.4125 47.83 Interaction 0.0025 1 0.0025 0.058 Quadratic 0.0027 1 0.0027 0.063 Error 0.1720 4 0.0430 Total 3.0022 8