Mathematical Modeling With Rsm

Mathematical

Modeling

with

RSM

Response Surface Methodology [RSM] is a simple and powerful method for modeling with more than two variables. It results in a surface in a 3D With RSM, You can also do optimisation with very little additional Then why RSM has not been popular? Well, RSM has been mountains of statistical methods very few people read or learn. There are software modules in math packages to do RSM fast.! Linear Models In models , we have used two variables x and y, andand connect use the by function: a function: Y = f (X) Now let us extend this to two factors or independent variables: Z = f (X,Y) Take a simple example: The yield of a chemical process (z) may depend on both temperature (x) and time (t): We write: Yield Z = f (x,y)

The simplest model we can have is th elinear the linear model: model Z=a+bX+CY [equation 1] The example could be the process of baking cakes: the baking time would depend on the temperature for a given le In this model equation, note that the effect of x and of y are "additive." The two factors x and y do not interact , but act separately. Take the simple anology: You add milk and sugar sugar toatocup a cup of black of black coffee. coffee. the coffee, while sugar adds sweetness. We assume that milk does not add sweetness, and sugar does not any creaminess. [In practice, milk can add vey small amount of So we have two variables x and y affecting z, the output or response Response Surfaces A response surface shows the response or output ,here z. We can plot z in th e vertical axis and x and y in then tow the horizonatl two horizontal axis.The axes.3D will show the model, in this case a flat plane. For a given value of yield or z, we get a line or point in the plane. Experimental Design We have to You may have to do experiments; For this linear model, FORM A SIMPLE TABLE:

find the values of the three constants or or collect data from the field. we can collect data for two temperatures and two different times an To illustrate: T1= 40 DEG T2 = 50 DEG t1= 10 minsy1 y3 t2= 40 min y2 y4

Now we can set up algebraic equations with three values of y1,y2 and y3 Solve them for a, b and c. y1= a + b. 40 + 10 c y2= a + b40 + 40 c y3 = a + b 50 + 10 c Solve for a , b and c. We can use y4 as additional check.

Now we

can plot the falt latplane. plane.

Quadratic Model This model is mor epowerful and ,in fact, most RSM models are of this type. To th elinear model [equation 1] , we can add "quadratic terms' ---that is, x squared, y squared and x.y term. The model equation beocmes: Z = a + b x + ccyy + d x.x + e.y.y + f x.y [Equation 2] We have added three terms: x.x, y.y and x.y The third term is interesting: this term models the interaction between the response or yield. We can plot z in the vertical axis and X andy and Yininthe theehorizontal horizontalaxis. axis. which may exhibit maximum or minumum points for yield.

Experimental Design In this model equation, equation 2, we have six constants or coefficients We determine these by setting up experiments of collecting data from the Specifically we can determine the yield z at three different temperatures two different times at each temperature. [Alternatively you can also do at two temperatures and three different times at each temperature.You do t that is faster or that is cheaper or less energy consuming.--the choice is yours!] Let us put the data ina tabular form: To illustrate: t1=10 min t2= 40 mins T1=40 deg y1 y4 T2= 50 degy2 y5 T3 = 60 degy3 y6 We set up the six equations for y1 ,y2….y6. Then solve for the six constants: a,b,c,d,e,f When we do this, we may find that some of the constants are that we can ignore the terms. If d is close to zero, it means that x squared term does not operate. Most important, look for the constant 'f'.If it is close to zero, it means that between temp and time. The model equation reduces to: f' =0 --> z= a + bx + cy + d x.x + e y.y z = a + (bx + dx.x) + ( c y + e y.y) We can separate out the model into two partsand treat as single vriable e variable model model for a speicifed value of x Response surface With a ,b,c,d,e,f , you can plot the Response Surface. helps you to do that in a few seconds. Study the curved surface. a maximum point or min point. These points give th eoptimal optimal mis mix of x and y to achieve ,in this c high yield. Advanced Models 1 Beyond quadtratic models, we can have models with three variables: N = a + bx + cy + d z To depict this in 3 D models, we pick a particular value of N and plot x,y,and z as a surface. These are like level surfaces or contour lines in 2D maps. 2 The simple

linear model

can be used for z = f( x,y) = a + bx + cy Here, y need not be just the varaible ,but other functions.

more complex functions:

Thus, then

y = log (t) z = f( x,y) = a + bx + cy z = a + bx + c log t z= a+ b (exp x) + c (log y)

3 Another example: Applications 1 A brewer finds the following data for the alcohol yield in his brewing Time 10 hours Time 100 hours Temp 50 deg C 10 20 temp 60 deg C 40 52 Using the linear model

Subtracting:

Subtracting; Now we find 'a':

find the constants z = a + bx + cy 10 = a + 50b + 10 C 40 = a + 60 b + 10 c b=3 10 = a + 50b + 10 C 20 = a + 50 b +100 c (-10 = -90 c) c= 1/9 10 = a + 50b + 10 C 10 = a + 150 + 10/9 a= -140-10/9 -141.11

a,b and c

in the model:

Z = -141.111+3 x + y/9 Can we predict yield for temp = 60 deg z= -141.11 + 3 x 60 + (1/9) 100 50 50 This value is close to what the brewer So, our linear model is quite adequate.

labor 2 labor 3 Use a

and time= 100 hours

found.

z=52%

2 A manager finds the following data between cost of a steel product grades of steel {one ordinary low carbon steel, another free-machining steel] the raw material cost and the labour cost. material = 5 material 6 20$ 35 25$ 50

model which is linear with labor cost ,but is a square of material cost: z = a + b x.x + c y 20 = a +25b + 2c 25= a + 25b +3c Subtract (-5 = -c) c= 5 c= 5 20 = a+ 25b +10 35= a+ 36b + 10 Subtract 15 = 11b b= 15/11 1.36 Solve for a: 20 = a + 15/11 *25 +10 a= 20 -(15 * 25)/11 + 10 -24.09

Thus:

z = -24.1 +

1.36 x.x + 5 y

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ethod for modeling with diagram, very easy to interpret. effort. buried under It is getting popular,because:

cess (z) may depend on both

nd on the temperature for a given level of baking.

f black coffee. Milk adds creaminess to add sweetness, and sugar does not add sweetness, but let us ignore that.!] separately.

It a given value of yield or z,

coefficients a,b and c.

peratures and two different times and find 4 values of yiels.ields.

erms' ---that is,

x and y, towards We will get a curved surface.

a to f , . from the field. temperatures [called 'levels' ] and times at each temperature.You do the scheme

zero or very small ared term does not operate. there is no interaction

ariable model Math software Do you find for

h three variables:

s a surface.

e complex functions:

N = f (x,y,z)

kettle.

t of a steel product

with two different

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