RSM - Response Surface Methodology Author: John M. Cimbala,...

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Response Surface Methodology Author: John M. Cimbala, Penn State University Latest revision: 11 February 2008 Introduction Taguchi design arrays were introduced previously as a way to hunt intelligently for an optimum point. Taguchi’s design arrays are not infallible, however, and may not always be practical to set up, since they work best with fairly large changes of the parameters. There are other methods with which one can search or hunt for an optimum. One common method is called response surface methodology ( RSM ). RSM is particularly useful for optimizing a running system . For example, suppose a power plant is running continuously. RSM can be used to optimize the efficiency or power output without large changes in the operating parameters, while continuing the operation without interruption . The main advantage of RSM is that it can be done “ on the fly ” with small changes of the parameters. A good reference for most of this material is Design and Analysis of Experiments , 4th ed., D. C. Montgomery, John Wiley and Sons, QA279.M66 1996, Ch. 14. Overview of the RSM Method Suppose y = y ( a , b , c , . ..), where y is the outcome or result or response that is to be optimized, and there are n parameters, a , b , c , . .., which can be varied. In these notes, it is assumed that the optimum y is the maximum y . A similar analysis can be performed for minimizing y . The goal of RSM is to efficiently hunt for the optimum values of a , b , c , . .. such that y is maximized . RSM works by the method of steepest ascent , in which the parameters are varied in the direction of maximum increase of the response until the response no longer increases. RSM is best illustrated by examples. First, some simple qualitative examples are given with 1 and 2 parameters ( n = 1 and n = 2), followed by a quantitative example with three parameters ( n = 3). Example for n = 1 : [ Note : This is only a qualitative example.] Given: Response y = y ( a ). Here, n = 1 since there is only one parameter in the problem. The experiments begin at some arbitrary operating point, i.e., at some value of a , a = a 0 . To do: Apply RSM to find the optimum value of y , i.e., find the value of parameter a where y is a maximum. Solution: a y a 0 o Imagine a plot of y as a function of a , with initial operating condition a 0 indicated by the red dot on the plot to the right. o Note that in an actual experiment, such a plot would not be available, but is used here for illustrative purposes only. o With only one parameter ( n = 1) as in this simple example, all one needs to know is whether to search to the left or to the right. One of these is the direction of steepest ascent . o To find this direction of steepest ascent, we measure y at some data points in the vicinity of a 0 , as illustrated by the blue dots in the plot to the right. Two other data points, in addition to that at a = a 0 itself, are sufficient for this simple case.
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RSM - Response Surface Methodology Author: John M. Cimbala,...

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