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Procedure for Finding the Direction of Maximum Improvementthe estimated direction of steepest ascent, given by the gradient of , if the objective is to maximize Y; 1.. the estimated dire

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There are two main decisions an engineer must make in Phase I:

determine the search direction;

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Procedure for Finding the Direction of Maximum Improvement

the estimated direction of steepest ascent, given by the gradient of , if the objective is

to maximize Y;

1

the estimated direction of steepest descent, given by the negative of the gradient of , if

the objective is to minimize Y.

The direction of the gradient, g, is given by the values of the parameter estimates, that is, g' =

(b1, b2, , b k ) Since the parameter estimates b1, b2, , b k depend on the scaling convention forthe factors, the steepest ascent (descent) direction is also scale dependent That is, two

experimenters using different scaling conventions will follow different paths for processimprovement This does not diminish the general validity of the method since the region of thesearch, as given by the signs of the parameter estimates, does not change with scale An equal

variance scaling convention, however, is recommended The coded factors x i, in terms of the

factors in the original units of measurement, X i, are obtained from the relation

This coding convention is recommended since it provides parameter estimates that are scaleindependent, generally leading to a more reliable search direction The coordinates of the factorsettings in the direction of steepest ascent, positioned a distance from the origin, are givenby:

Solution is a

simple

equation

This problem can be solved with the aid of an optimization solver (e.g., like the solver option

of a spreadsheet) However, in this case this is not really needed, as the solution is a simpleequation that yields the coordinates

To see the details that explain this equation, see Technical Appendix 5A

Example: Optimization of a Chemical Process

5.5.3.1.1 Single response: Path of steepest ascent

http://www.itl.nist.gov/div898/handbook/pri/section5/pri5311.htm (3 of 6) [5/1/2006 10:31:05 AM]

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by search

example

It has been concluded (perhaps after a factor screening experiment) that the yield (Y, in %) of a

chemical process is mainly affected by the temperature (X 1, in C) and by the reaction time

(X 2, in minutes) Due to safety reasons, the region of operation is limited to

Factor levels The process is currently run at a temperature of 200 C and a reaction time of 200 minutes A

process engineer decides to run a 22 full factorial experiment with factor levels at

ANOVA table The corresponding ANOVA table for a first-order polynomial model, obtained using the

DESIGN EASE statistical software, is

SUM OF MEAN FSOURCE SQUARES DF SQUARE VALUE PROB>F

MODEL 503.3035 2 251.6517 4.810 0.0684CURVATURE 8.1536 1 8.1536 0.1558 0.7093RESIDUAL 261.5935 5 52.3187

LACK OF FIT 37.6382 1 37.6382 0.6722 0.4583 PURE ERROR 223.9553 4 55.9888

COR TOTAL 773.0506 85.5.3.1.1 Single response: Path of steepest ascent

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model

It can be seen from the ANOVA table that there is no significant lack of linear fit due to aninteraction term and there is no evidence of curvature Furthermore, there is evidence that thefirst-order model is significant Using the DESIGN EXPERT statistical software, we obtain theresulting model (in the coded variables) as

Then from the equation above for the predicted Y response, the coordinates of the factor levels

for the next run are given by:

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Compute the partials and equate them to zero

Solve two

equations in

two unknowns

These two equations have two unknowns (the vector x and the scalar ) and thus can be solved

yielding the desired solution:

or, in non-vector notation:

Multiples of

the direction

of the

gradient

From this equation we can see that any multiple of the direction of the gradient (given by

) will lead to points on the steepest ascent direction For steepest descent, use instead

-b i in the numerator of the equation above

5.5.3.1.1 Single response: Path of steepest ascent

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5 Process Improvement

5.5 Advanced topics

5.5.3 How do you optimize a process?

5.5.3.1 Single response case

5.5.3.1.2 Single response: Confidence region for search

The direction given by the gradient g' = (b0, b2, , b k) constitutes only a single (point) estimate

based on a sample of N runs If a different set of N runs were conducted, these would provide

different parameter estimates, which in turn would give a different gradient To account for thissampling variability, Box and Draper gave a formula for constructing a "cone" around thedirection of steepest ascent that with certain probability contains the true (unknown) systemgradient given by The width of the confidence cone is useful to assess howreliable an estimated search direction is

Figure 5.4 shows such a cone for the steepest ascent direction in an experiment with two factors

If the cone is so wide that almost every possible direction is inside the cone, an experimentershould be very careful in moving too far from the current operating conditions along the path ofsteepest ascent or descent Usually this will happen when the linear fit is quite poor (i.e., when the

R2 value is low) Thus, plotting the confidence cone is not so important as computing its width

If you are interested in the details on how to compute such a cone (and its width), see TechnicalAppendix 5B

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FIGURE 5.4: A Confidence Cone for the Steepest Ascent Direction in an Experiment with 2

C jj is the j-th diagonal element of the matrix (X'X)-1 (for j = 1, , k these values are all equal if

the experimental design is a 2k-p factorial of at least Resolution III), and X is the model matrix of

the experiment (including columns for the intercept and second-order terms, if any) Any

operating condition with coordinates x' = (x1, x2, , x k) that satisfies this inequality generates a

direction that lies within the 100(1- )% confidence cone of steepest ascent if

5.5.3.1.2 Single response: Confidence region for search path

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or inside the 100(1- )% confidence cone of steepest descent if&

A measure of "goodness" of a search direction is given by the fraction of directions excluded by

the 100(1- )% confidence cone around the steepest ascent/descent direction (see Box andDraper, 1987) which is given by:

with T k-1 () denoting the complement of the Student's-t distribution function with k-1 degrees of

freedom (that is, T k-1 (x) = P(t k-1 x)) and F , k-1, n-p denotes an percentage point of the F

distribution with k-1 and n-p degrees of freedom, with n-p denoting the error degrees of freedom.

The value of represents the fraction of directions included by the confidence cone Thesmaller is, the wider the cone is, with Note that the inequality equation and the

"goodness measure" equation are valid when operating conditions are given in coded units

since F0.05,1,6 = 5.99 Thus 71.05% of the possible directions from the current operating point are

excluded with 95% confidence This is useful information that can be used to select a step length.The smaller is, the shorter the step should be, as the steepest ascent direction is less reliable Inthis example, with high confidence, the true steepest ascent direction is within this cone of 29%

of possible directions For k=2, 29% of 360o = 104.4o, so we are 95% confident that our estimatedsteepest ascent path is within plus or minus 52.2o of the true steepest path In this case, we shouldnot use a large step along the estimated steepest ascent path

5.5.3.1.2 Single response: Confidence region for search path

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5.5 Advanced topics

5.5.3.1.3 Single response: Choosing the step

improvement The procedure is given below A similar approach is obtained by choosing increasing values of in

The following is the procedure for selecting the step length.

Choose a step length Xj (in natural units of measurement) for some factor

j Usually, factor j is chosen to be the one engineers feel more comfortable

varying, or the one with the largest |bj| The value of Xj can be based on the width of the confidence cone around the steepest ascent/descent

direction Very wide cones indicate that the estimated steepest ascent/descent direction is not reliable, and thus Xj should be small This usually occurs when the R2 value is low In such a case, additional experiments can be conducted in the current experimental region to obtain a better model fit and

a better search direction.

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with sj denoting the scale factor used for factor j (e.g., sj = rangej/2).

The following is an example of the step length selection procedure.

For the chemical process experiment described previously , the process engineer selected X2 = 50 minutes This was based on process engineering

considerations It was also felt that X2 = 50 does not move the process too

far away from the current region of experimentation This was desired since the R2 value of 0.6580 for the fitted model is quite low, providing a not very reliable steepest ascent direction (and a wide confidence cone, see Technical

Procedure: Conducting Experiments Along the Direction of Maximum Improvement

Given current operating conditions = (X1, X2, , Xk) and a step size X'

= ( X1, X2, , Xk), perform experiments at factor levels X0 + X, X0

+ 2 X, X0 + 3 X, as long as improvement in the response Y (decrease or

increase, as desired) is observed.

1

Once a point has been reached where there is no further improvement, a new

first-order experiment (e.g., a 2k-p fractional factorial) should be performed with repeated center runs to assess lack of fit If there is no significant evidence of lack of fit, the new first-order model will provide a new search direction, and another iteration is performed as indicated in Figure 5.3.

Otherwise (there is evidence of lack of fit), the experimental design is augmented and a second-order model should be fitted That is, the experimenter should proceed to "Phase II".

2

5.5.3.1.3 Single response: Choosing the step length

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Example: Experimenting Along the Direction of Maximum Improvement

Step 2: new

factorial

experiment

Step 2:

A new 22 factorial experiment is performed with X' = (189.5, 350) as the origin.

Using the same scaling factors as before, the new scaled controllable factors are:

Five center runs (at X1 = 189.5, X2 = 350) were repeated to assess lack of fit The

experimental results were:

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DESIGN-EASE statistical software, is

SUM OF MEAN F SOURCE SQUARES DF SQUARE VALUE PROB > F

MODEL 505.300 2 252.650 4.731 0.0703 CURVATURE 336.309 1 336.309 6.297 0.0539 RESIDUAL 267.036 5 53.407

LACK OF FIT 93.857 1 93.857 2.168 0.2149 PURE ERROR 173.179 4 43.295

COR TOTAL 1108.646 8 From the table, the linear effects (model) is significant and there is no evidence of lack of fit However, there is a significant curvature effect (at the 5.4% significance level), which implies that the optimization should proceed with Phase II; that is, the fit and optimization of a second-order model.

5.5.3.1.3 Single response: Choosing the step length

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5.5 Advanced topics

5.5.3.1.4 Single response: Optimization when there is

adequate quadratic fit

in the response A second-order polynomial can be used as a local approximation of the response

in a small region where, hopefully, optimal operating conditions exist However, while aquadratic fit is appropriate in most of the cases in industry, there will be a few times when aquadratic fit will not be sufficiently flexible to explain a given response In such cases, the analystgenerally does one of the following:

Uses a transformation of Y or the Xi's to improve the fit

Once a linear model exhibits lack of fit or when significant curvature is detected, the experimental

design used in Phase I (recall that a 2k-p factorial experiment might be used) should be augmented

with axial runs on each factor to form what is called a central composite design This

experimental design allows estimation of a second-order polynomial of the form

Using some graphics software, obtain a contour plot of the fitted response If the number of

factors (k) is greater than 2, then plot contours in all planes corresponding to all the possible pairs of factors For k greater than, say, 5, this could be too cumbersome (unless

the graphic software plots all pairs automatically) In such a case, a "canonical analysis" ofthe surface is recommended (see Technical Appendix 5D)

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Example: Second Phase Optimization of Chemical Process

Experimental

results for

axial runs

Recall that in the chemical experiment, the ANOVA table, obtained from using an experiment run

around the coordinates X1 = 189.5, X2 = 350, indicated significant curvature effects Augmenting

the 2 2 factorial experiment with axial runs at to achieve a rotatable centralcomposite experimental design, the following experimental results were obtained:

ANOVA table The corresponding ANOVA table for the different effects, based on the sequential sum of squares

procedure of the DESIGN-EXPERT software, is

SUM OF MEAN FSOURCE SQUARES DF SQUARE VALUE PROB > FMEAN 51418.2 1 51418.2

Linear 1113.7 2 556.8 5.56 0.024Quadratic 768.1 3 256.0 7.69 0.013Cubic 9.9 2 5.0 0.11 0.897RESIDUAL 223.1 5 44.6

Linear 827.9 6 138.0 3.19 0.141Quadratic 59.9 3 20.0 0.46 0.725Cubic 49.9 1 49.9 1.15 0.343PURE ERROR 173.2 4 43.3

ROOT ADJ PREDSOURCE MSE R-SQR R-SQR R-SQR PRESSLinear 10.01 0.5266 0.4319 0.2425 1602.025.5.3.1.4 Single response: Optimization when there is adequate quadratic fit

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Quadratic 5.77 0.8898 0.8111 0.6708 696.25Cubic 6.68 0.8945 0.7468 -0.6393 3466.71

The quadratic model has a larger p-value for the lack of fit test, higher adjusted R2, and a lowerPRESS statistic; thus it should provide a reliable model The fitted quadratic equation, in codedunits, is

A contour plot of this function (Figure 5.5) shows that it appears to have a single optimum point

in the region of the experiment (this optimum is calculated below to be (-.9285,.3472), in coded

x 1 , x 2 units, with a predicted response value of 77.59)

FIGURE 5.5: Contour Plot of the Fitted Response in the Example

5.5.3.1.4 Single response: Optimization when there is adequate quadratic fit

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