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Response surface method
- Response surface methodology (RSM) uses statistical techniques to model and analyze problems with response variables influenced by multiple independent variables. The goal is to optimize the response. - RSM has been used since the 1930s and was reviewed in landmark papers in 1966 and 1976. It is commonly used in industries, agriculture, medicine, and other fields to optimize processes and products. - There are two main experimental strategies in RSM - first-order models to initially evaluate relationships between factors and responses, and second-order models to account for curvature and find optimal points if curvature is present.
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- 2. Learning Objectives • History. •Introduction To RSM. • Real Life Example. • Why And When Use RSM. • Experimental Strategy.
- 3. History • In theMead and Pike paper, they move back the origin of RSM to include use of "response curves” dating back into the 1930's. • Then in 1935 Yates work on it. • In the Hill And Hunter Review, they State that • In November 1966, a paper “A Review of Response Surface Methodology” ; A literature was published by “Hill and Hunter. Its purpose was to review the practical applications of RSM in chemical and related fields.
- 4. History • In December1976, another paper “A Review of Response Surface Methodology From A Biometric View Point” was published by Mead and Pike appeared. • With the passage of time many Statisticians work on RSM for Improvement.
- 5. Introduction • Response surfacemethodology (RSM) uses various statistical, graphical, and mathematical techniques to develop, improve, or optimize a process, also use for modeling and analysis of problems if our response variables in influenced by several independent variables. • Main objectives are as follow. – Optimize.(main objective) – Develop. – Improve. (if necessary).
- 6. Real Life Examples •RSM is used in different fields of real life. Like Industries, Agriculture, Electronics, Medical field and many other like this. It is use where we want to get optimum response.
- 7. An Example ofMedical field. • A single tablet is introduced in market after large number of experiments. Suppose a company want to introduce a new pain killer tablet in market. The pharmacist will make the table that will be more effective and has rapid action to kill the pain, will have low price at market for patient ………
- 8. Why And WhenWe Use RSM.
- 9. Experimental Strategy 1. RSMresolve around the assumption that the response is a function of a set of independent(design) variables x1,x2,x3….xk and function can be approximated in some region of polynomial model. 𝑦 = 𝑓 𝑥𝑖 𝑦 = 𝑓 𝑥1, 𝑥2 … … 𝑥 𝑘 Here response variable is “y” that depend on the “k” independent variables.
- 10. Experimental Strategy 2. Ifthe factors are given then directly estimate the effects and interaction of model as describe in figure. 3. And if the factors are unknown then first calculate them by using the Screening method. 4. Estimate The Interaction effect using 1st order model. y = 𝛽0+𝛽1 𝑥1+𝛽2 𝑥2+𝜀
- 11. Experimental Strategy 5. Ifcurvature is found then use the RSM. And 2nd order model will be used to approximate the response variable.
- 12. 6. Make thegraph and find the stationary point. Maximum response, Minimum response or saddle point by using the obtained values of 𝑥1, 𝑥2, 𝑥3 … . 𝑥4.
- 13. Types OF Models Weuse two types of model in RSM. 1. 1st Order Model. 2. 2nd Order Model. When Use Which Model • 1St Order Model. Oftenly in RSM the relationship between response variable and Independent variables is not given. After screening we use 1st order model to find current situation and to find either there is curvature or not. y = 𝛽0+𝛽1 𝑥1+𝛽2 𝑥2+𝜀
- 14. 2nd Order Model Ifwe have find curvature after making fig from the result of 1st order model. Then we use 2nd order model to find our optimum point.
- 15. (I) Sequential NatureOf RSM.
- 16. Sequential Nature OfRSM RSM is sequential procedure. Often, when we are at a point on the response surface that is remote from the optimum, and we want to move rapidly from current point to the optimum point with sequence. If we want the optimum point where the sources are minimum but output is maximum then that is called our optimum point. And we move rapidly toward it.
- 17. (II) Methods ofRSM • There are two methods of RSM to obtain optimum response. And we move toward our optimum point with these two method.. » Method Of Steepest Ascent. » Method Of Steepest Descent.
- 18. Steepest Ascent Method: Thisis a procedure for moving sequentially in the direction of the maximum increase in the response getting optimum response. Steepest Descent Method : If minimization is desired then we call this technique the “method of steepest descent”.
- 19. Steepest Ascent Method •The initial estimate of the optimum operating condition for this will be far from the actual optimum. • In such circumstances, the objective of the experimenter is to move rapidly to the general vicinity(nearest point) of the optimum. We wish to use a simple and economically efficient experimental procedure. When we remote from the optimum, we usually assume that a 1st order model is an adequate approximation to the true surface in a small region of the x’s.
- 20. Steepest Ascent Method Thisis a procedure for moving sequentially in the direction of the maximum increase in the response getting optimum response. 𝑦 = 𝛽𝑜 + 𝑖=1 𝑘 𝛽𝑖 𝑥𝑖
- 21. Steepest Descent Method Ifminimization is desired then we call this technique the “method of steepest descent”. 𝑦 = 𝛽𝑜 + 𝑖=1 𝑘 𝛽𝑖 𝑥𝑖
- 22. Numerical example: Numerical variableCoded variable response Time temp x1 x2 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 5 155 0 0 40.6
- 23. the coded variableare x1= 𝑡𝑖𝑚𝑒−35 5 x2= 𝒕𝒆𝒎𝒑−𝟏𝟓𝟓 5
- 24. A first ordermodel may be fit to these data by least, employing the methods for two level designs, we obtain the following model in the coded variable 𝑦 = 40.44 + 0.775𝑥1 + 0.775𝑥2 Before exploring along the path of steepest ascent, the adequacy of the first order model should be investigated. The 2^2 design with center points allows the experiment to 1. Obtain an estimate of error 2. Check for interactions (cross product terms) in the model
- 25. • the replicatesat the center can be used to calculate an 𝜎2 = (40.3)2+(40.5)2+ 40.7 2+ 40.2 2+ 40.6 2− (202.3)2 5 4 = 0.0430 The first order model assume that the variable 𝑥1 & 𝑥2 have an additive effect on the response. Interaction b/w the variables would be represent by the coefficient 𝛽12 of a cross product term 𝑥1 𝑥2 added to the model. the least square estimate of this coefficient is just one half the interaction effect calculated as in an ordinary 22 factorial design. Or
- 26. • 𝛽12 = 1 4 1∗ 39.3 + 1 ∗ 41.5 + −1 ∗ 40.0 + (−1 ∗ 40.9) = -0.025 The single degree of freedom sum of square for interaction is SS interaction = (−0.1)2 4 =0.0025 Comparing SS interactions to 𝜎2 gives a lack of fit statistics F = 𝑠𝑠 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝜎2 = 0.0025/0.0430 = 0.058 Which is a small ,indicating that interaction is negligible
- 27. Application The most frequentapplications of RSM are in the industrial area. RSM is important in designing formulating and developing and analyzing new specific scientific studying and product. It is also efficient in improvements of existing studies and products Most common application of RSM are in industrial ,biological and clinical sciences, social sciences ,food sciences and physical and engineering sciences