response surface methodology (rsm)

A second order model is generally used to approximate the response once it is realized that the experiment is close to the optimum response region where a first order model is no longer adequate The second order model is usually sufficient for the optimum region as third order and higher effects

are seldom important The second order regression model takes the following form for factors

(5)

The model contains regression parameters that include coefficients for main

effects ( ) coefficients for quadratic main effects ( ) and coefficients for two

factor interaction effects ( ) A full factorial design with all factors at three levels would provide estimation of all the required regression parameters However full factorial three level designs are expensive to use as the number of runs increases rapidly with the number of factors For

example a three factor full factorial design with each factor at three levels would require

runs while a design with four factors would require runs Additionally these designs will estimate a number of higher order effects which are usually not of much importance to the experimenter Therefore for the purpose of analysis of response surfaces special designs are used that help the experimenter fit the second order model to the response with the use of a minimum number of runs Examples of these designs are the central composite and Box-Behnken designs

Central Composite Designs

Central composite designs are two level full factorial (2 ) or fractional factorial (2 ) designs augmented by a number of center points and other chosen runs These designs are such that they allow the estimation of all the regression parameters required to fit a second order model to a given response The simplest of the central composite designs can be used to fit a second order model to a response

with two factors The design consists of a 2 full factorial design augmented by a few runs at the center point (such a design is shown in Figure 910 (a)) A central composite design is obtained when

runs at four other points - ( ) ( ) ( ) and ( ) are added to this design These points are referred to as axial points or star points and represent runs where all but one of the factors are set at

their mid-levels The number of axial points in a central composite designs having factors is 2

The distance of the axial points from the center point is denoted by and is always specified in terms

of coded values For example the central composite design in Figure 910 (b) has while for

the design of Figure 910 (c) It can be noted that when each factor is run at five

levels ( and ) instead of the three levels of and The reason for running

central composite designs with is to have a rotatable design which is explained next

Earlier we described the response surface method (RSM) objective Under some circumstances a model involving only main effects and interactions may be appropriate to

describe a response surface when

1 Analysis of the results revealed no evidence of pure quadratic curvature in the response of interest (ie the response at the center approximately equals the average of the responses at the factorial runs)

2 The design matrix originally used included the limits of the factor settings available to run the process

Equations for quadratic and cubic models

In other circumstances a complete description of the process behavior might require a quadratic or cubic model

Quadratic

Cubic

These are the full models with all possible terms rarely would all of the terms be needed in an application

Quadratic models almost always sufficient for industrial applications

If the experimenter has defined factor limits appropriately andor taken advantage of all the tools available in multiple regression analysis (transformations of responses and factors for example) then finding an industrial process that requires a third-order model is highly unusual Therefore we will only focus on designs that are useful for fitting quadratic models As we will see these designs often provide lack of fit detection that will help determine when a higher-order model is needed

General quadratic surface types

Figures 39 to 312 identify the general quadratic surface types that an investigator might encounter

FIGURE 39 A Response Surface

Peak


Hillside

FIGURE 311 A Response Surface Rising Ridge


SaddleFactor Levels for Higher-Order Designs

Possible behaviors of responses as functions of factor settings

Figures 313 through 315 illustrate possible behaviors of responses as functions of factor settings In each case assume the value of the response increases from the bottom of the figure to the top and that the factor settings increase from left to right

FIGURE 313 Linear Function

FIGURE 314 Quadratic Function

FIGURE 315 Cubic Function

A two-level experiment with center points can detect but not fit quadratic effects

If a response behaves as in Figure 313 the design matrix to quantify that behavior need only contain factors with two levels -- low and high This model is a basic assumption of simple two-level factorial and fractional factorial designs If a response behaves as in Figure 314 the minimum number of levels required for a factor to quantify that behavior is three One might logically assume that adding center points to a two-level design would satisfy that requirement but the arrangement of the treatments in such a matrix confounds all quadratic effects with each other While a two-level design with center points cannot estimate individual pure quadratic effects it can detect them effectively

Three-level factorial design

A solution to creating a design matrix that permits the estimation of simple curvature as shown in Figure 314 would be to use a three-level factorial design Table 321 explores that possibility

Four-level factorial design

Finally in more complex cases such as illustrated in Figure 315 the design matrix must contain at least four levels of each factor to characterize the behavior of the response adequately

3-level factorial designs can fit quadratic models but they require many runs when there are more than 4 factors

TABLE 321 Three-level Factorial Designs

Numberof Factors

Treatment Combinations

3k Factorial

Number of CoefficientsQuadratic Empirical

Model

2 9 63 27 104 81 155 243 216 729 28

Fractional factorial designs created to avoid such a large number of runs

Two-level factorial designs quickly become too large for practical application as the number of factors investigated increases This problem was the motivation for creating `fractional factorial designs Table 321 shows that the number of runs required for a 3k factorial becomes unacceptable even more quickly than for 2k designs The last column in Table 321 shows the number of terms present in a quadratic model for each case

Number of runs large even for modest number of factors

With only a modest number of factors the number of runs is very large even an order of magnitude greater than the number of parameters to be estimated when k isnt small For example the absolute minimum number of runs required to estimate all the terms present in a four-factor quadratic model is 15 the intercept term 4 main effects 6 two-factor interactions and 4 quadratic terms

The corresponding 3k design for k = 4 requires 81 runsComplex alias Considering a fractional factorial at three levels is a logical

structure and lack of rotatability for 3-level fractional factorial designs

step given the success of fractional designs when applied to two-level designs Unfortunately the alias structure for the three-level fractional factorial designs is considerably more complex and harder to define than in the two-level case

Additionally the three-level factorial designs suffer a major flaw in their lack of `rotatabilityRotatability of Designs

Rotatability is a desirable property not present in 3-level factorial designs

In a rotatable design the variance of the predicted values of y is a function of the distance of a point from the center of the design and is not a function of the direction the point lies from the center Before a study begins little or no knowledge may exist about the region that contains the optimum response Therefore the experimental design matrix should not bias an investigation in any direction

Contours of variance of predicted values are concentric circles

In a rotatable design the contours associated with the variance of the predicted values are concentric circles Figures 316 and 317 (adapted from Box and Draper `Empirical Model Building and Response Surfaces page 485) illustrate a three-dimensional plot and contour plot respectively of the `information function associated with a 32 design

Information functionThe information function is

with V denoting the variance (of the predicted value )

Each figure clearly shows that the information content of the design is not only a function of the distance from the center of the design space but also a function of direction

Graphs of the information function for a rotatable quadratic design

Figures 318 and 319 are the corresponding graphs of the information function for a rotatable quadratic design In each of these figures the value of the information function depends only on the distance of a point from the center of the space

FIGURE 316 Three-Dimensional Illustration

for the Information Function of a 32 Design

FIGURE 317 Contour Map of the Information

Function for a 32Design


of the Information Function for a Rotatable

Quadratic Design for Two Factors

FIGURE 319 Contour Map of the Information Function for a Rotatable Quadratic Design for

Two Factors

Classical Quadratic DesignsCentral composite and Box-Behnken designs

Introduced during the 1950s classical quadratic designs fall into two broad categories Box-Wilson central composite designs and Box-Behnken designs The next sections describe these design classes and their properties

Optimization of Biodiesel Production by Response Surface Methodology and Genetic Algorithm

Singhal Richa

CSIR-Indian Institute of Petroleum Dehradun

Seth Prateek


Bangwal Dinesh


Kaul Savita


(Received 15 September 2011 accepted 18 April 2012)

Abstract

The biodiesel production from alkali-catalyzed transesterification of karanja oil was investigated In this study the

effect of three parameters ie reaction temperature catalyst concentration and molar ratio of methanol to oil on

biodiesel yield was studied Central composite design (CCD) along with response surface methodology (RSM)

was used for designing experiments and estimating the quadratic response surface Catalyst concentration was

found to have a negative effect on biodiesel yield whereas molar ratio showed positive effect Temperature and

molar ratio showed significant interaction effect The reaction conditions were optimized for maximum response

ie biodiesel yield from RSM The program for the RSM model coupled with genetic algorithm (GA) was

developed for predicting the optimized process parameters for maximum biodiesel yield to obtain a global optimal

solution The results were found to be similar from both of the methods

In statistics a central composite design is an experimental design useful in response surface

methodology for building a second order (quadratic) model for the response variable without needing

to use a complete three-level factorial experiment

After the designed experiment is performed linear regression is used sometimes iteratively to obtain

results Coded variables are often used when constructing this design

In this work a standard RSM design called CCD wasapplied to study the variables for preparing the activatedcarbon from rice husk (RHAC) The CCD consists of threekinds of runs which are the 2n factorial runs 2(n) axial runsand six center runs where n is the number of variables Thepreparation variables used were activation temperature (x1)activation time (x2) and IR (x3) indicating that altogether 20experiments for this procedure as calculated from (4)1050416 1050416 21050416 1050416 2n 1050416 n1050416 1050416 21050416 1050416 2 3 1050416 6 1050416 20 105041641050416where N is the total number of experiments requiredThese three variables together with their respectiveranges were chosen based on literature and prelimin


describe a response surface when

1 Analysis of the results revealed no evidence of pure quadratic curvature in the response of interest (ie the response at the center approximately equals the average of the responses at the factorial runs)

2 The design matrix originally used included the limits of the factor settings available to run the process

Equations for quadratic and cubic models

In other circumstances a complete description of the process behavior might require a quadratic or cubic model

Quadratic

Cubic







Peak


Hillside

















Numberof Factors


3k Factorial


Model

2 9 63 27 104 81 155 243 216 729 28


























Two Factors




Singhal Richa


Seth Prateek


Bangwal Dinesh


Kaul Savita



Abstract























Peak


Hillside

















Numberof Factors


3k Factorial


Model

2 9 63 27 104 81 155 243 216 729 28


























Two Factors




Singhal Richa


Seth Prateek


Bangwal Dinesh


Kaul Savita



Abstract




























Numberof Factors


3k Factorial


Model

2 9 63 27 104 81 155 243 216 729 28


























Two Factors




Singhal Richa


Seth Prateek


Bangwal Dinesh


Kaul Savita



Abstract





































Two Factors




Singhal Richa


Seth Prateek


Bangwal Dinesh


Kaul Savita



Abstract

















response surface methodology (rsm)

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