Design-Expert 8.0使用指南-Mixture Design-Optimization

Rev. 12/8/09

Mixture Design Tutorial

(Part 2 – Optimization)

Introduction

This tutorial shows the use of Design-Expert® software for optimization of mixture

experiments. It’s based on the data from the preceding tutorial (Part 1 – The

Basics). You should go back to that section if you’ve not already completed it.

Much of what’s detailed in this Mixture Design Tutorial (Part 2 – Optimization) is a

repeat of the Multifactor RSM Tutorial (Part 2 – Optimization). If you’ve already

completed that RSM tutorial, simply skip over the areas in this tutorial that you find

redundant.

For details about optimization, use the software’s extensive on-screen program

Help. Also, Stat-Ease provides in-depth training in its workshop titled Mixture

Designs for Optimal Formulations. Call for information on content and schedules,

or better yet, visit our web site at .

Start the program by finding and double clicking the Design-Expert software icon.

The detergent design, response data, and appropriate response models are in a file

named . To load this file, click the Open Design option on the opening

screen.

File Open dialog box

Once you have found the proper drive, directory, and file name, click Open to load

the data. To see a description of the file contents, click the Summary node under

the Design branch at the left of your screen. Drag the left border and open the

window to see the report better. You can also re-size columns with your mouse.

Design summary

The file you just loaded includes analyzed models as well as raw data for each

response. Recall that the formulators chose a three-component simplex lattice

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design to study their detergent formulation. The components are water, alcohol,

and urea. The experimenters held all other ingredients constant. They measured

two responses: viscosity and turbidity. You will now optimize this mixture using

their analyzed models. In the design-status screen above you see we modeled

viscosity using a quadratic mixture model – and turbidity using the special cubic.

Numerical Optimization

Design-Expert software’s numerical optimization maximizes, minimizes, or targets:

 A single response

 A single response, subject to upper and/or lower boundaries on other

responses

 Combinations of two or more responses.

We lead you through the last above case: a multiple-response optimization. Under

the Optimization branch of the program, click the Numerical node to start the

process.

Setting numeric optimization criteria

Setting the Optimization Criteria

Design-Expert allows you to set criteria for all variables, including components and

propagation of error (POE). (We will get to POE later.) The limits for the responses

default to the observed extremes.

Now you reach the crucial phase of numerical optimization: assigning

“Optimization Parameters.” The program uses five possibilities for a “Goal” to

construct desirability indices (di):

 None (responses only)

 Maximize,

 Minimize,

 Target->,

 In range,

 Equal to -> (components only).

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Desirabilities range from zero to one for any given response. The program

combines individual desirabilities into a single number and then searches for the

greatest overall desirability. A value of one represents the ideal case. A zero

indicates that one or more responses fall outside desirable limits. Design-Expert

uses an optimization method developed by Derringer and Suich, described by

Myers, Montgomery and Anderson-Cook in Response Surface Methodology, 3rd

edition, John Wiley and Sons, New York, 2009.

In this case, components are allowed to range within their pre-established

constraints, but be aware they can be set to desired goals. For example, because

water is cheap, you could set its goal to maximize.

Options for goals on components

Notice that components can be set equal to specified levels. Leave water at its “in

range” default and click the first response – Viscosity. Set its Goal to target-> of

43. Enter Limits as Lower of 39 and Upper of 48. Press Tab to set your entries.

Setting Target for first response of viscosity

These limits indicate it is most desirable to achieve the targeted value of 43, but

values in the range of 39-48 are acceptable. Values outside that range have no

(zero) desirability.

Now click the second response – Turbidity. Select its Goal to minimize, with

Limits set at Lower of 800 and Upper of 900. Press Tab to set your entries. You

must provide both these thresholds to get the desirability equation to work

properly. By default they are set at the observed response range, in this case 321 to

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1122. However, evidently in this case there’s no advantage to getting the

detergent’s turbidity below 800 – it already appears as clear as can be to the

consumer’s eye. On the other hand, when turbidity exceeds 900, it looks as bad as

it gets.

Aiming for minimum on second response of turbidity

These settings create the following desirability functions:

1. Viscosity:

 if less than 39, desirability (di) equals zero

 from 39 to 43, di ramps up from zero to one

 from 43 to 48, di ramps back down to zero

 if greater than 48, di equals zero.

2. Turbidity:

 if less than 800, di equals one

 from 800 to 900, di ramps down from one to zero

 if over 900, di equals zero.

Do not forget that at your fingertips you will find advice about using sophisticated

features of Design-Expert software: Press the screen tips icon

for an overview

about Numerical Optimization.

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Screen tips at your fingertips

Close out Tips by pressing X at the upper-right corner of its screen.

Changing Desirability Weights and the (Relative) Importance of Variables

The user can select additional parameters, called “weights,” for each response.

Weights give added emphasis to upper or lower bounds, or emphasize a target

value. With a weight of 1, di varies from 0 to 1 in linear fashion. Weights greater

than 1 (maximum weight is 10) give more emphasis to goals. Weights less than 1

(minimum weight is 0.1) give less emphasis to goals. Weights can be quickly

changed by ‘grabbing’ (via left mouse-click and drag) the handles (the squares ▫)

on the desirability ramps. Try pulling the handle on the ramp down as shown

below.

Weights change by grabbing handle with mouse

Notice that Weight now reads 10. You’ve made it much more desirable to get near

the turbidity goal of 800. Before moving on, re-enter Upper Weights to its default

value of 1 and press the Tab key. This straightens the desirability ramp.

“Importance” is a tool for changing relative priorities for achieving goals you

establish for some or all of the variables. If you want to emphasize one variable

over the rest, set its importance higher. Design-Expert offers five levels of

importance ranging from 1 plus (+) to 5 pluses (+++++). For this study, leave

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Importance at +++, a medium setting. By leaving all importance criteria at their

defaults, none of the goals is favored over any other.

For an in-depth explanation of constructing desirability functions, and formulas for

weights and importance, select Help from the main menu. Then go to Contents

and select Optimization, then Numerical Optimization “Statistical Details.”

Branch down to the topic of Importance as shown on the screen shot below.

Details about Importance found in program Help

When you finish viewing Help, close the screen by pressing X at the upper-right

corner of its screen.

Click the Options button to see how to gain control over how numerical

optimization is performed. For example, using optimization you can change the

number of cycles (searches). If you have a very complex combination of response

surfaces, increasing the number of cycles gives you more opportunities to find the

optimal solution. The Duplicate solution filter establishes epsilon (minimum

difference) for eliminating essentially identical solutions. Simplex Fraction

specifies how big the initial steps are relative to factor ranges. (The word “simplex”

relates to search geometry. For two factors, the simplex is an equilateral triangle.

By stepping through the three corners, the program figures out the path of steepest

ascent. For more detail, go to Help and search “numerical search algorithm.”)

Another optimization option is to not use random starting points, but rather those

points in the design itself. However, these are limited to 100 unless you change

default. Leave this and all other options at their default levels shown below. Click

OK or Cancel (PS. This screen shot shows underlined letters that indicate Alt keys

for jumping to fields via keystrokes versus mousing. The underlining occurs when

you press Alt.)

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Optimization Options Dialog Box

Running the optimization

Start the optimization by clicking the Solutions icon.

Numerical Optimization Report on Solutions (Your results may differ)

The program randomly picks a set of conditions from which to start its search for

desirable results. Multiple cycles improve the odds of finding multiple local

optimums, some of which will be higher in desirability than others. After grinding

through 40 cycles of optimization (starting from the 10 design points plus 30 more

at random), Design-Expert sorts the results for you. It shows the best solution first

in report format. You get a summary of all the cycles. (If the report doesn’t fit in

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the window, move your cursor to the left border and drag it open.) In addition to

solutions, the report includes a recap of your optimization specifications as well as

the random starting points for the search.

The floating Solutions Tool palette provides three views of the same optimization.

(Drag the tool to a convenient location on your screen.) Click view option Ramps.

Ramps report on numerical optimization (Your results may differ)

The ramp display combines individual graphs for easier interpretation. The dot on

each ramp reflects the factor setting or response prediction for that solution. The

height of the dot shows how desirable it is. Press the different solution buttons (1,

2, 3,…) and watch the dots. The red ones representing the component levels move

around quite a bit, but do the responses remain within their goals (desirability of

1)? Near the graph’s top, click the last solution (solution 13 in this case) on your

screen. Does your solution look something like the one below?

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Sub-optimum solution that ranks least desirable

If your search also uncovered the above local optimum, note that viscosity falls off

target and turbidity becomes excessive, thus making it less desirable than the

option for higher temperature.

Select the Bar Graph view from the solutions tool.

Solution to multiple-response optimization – desirability bar graph

The above bar graph shows how well each variable satisfies the criteria and the

overall combined desirability: values near one are good.

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Optimization Graphs

Press Graphs to view a contour graph of overall desirability. It now becomes

obvious that at least somewhat desirable formulations fall with three distinct

‘sweet spots’ as indicated by the three graduated color areas within the blue

background.

Desirability contour graph

The screen shot above came from a graph done showing graduated colors – cool

blue for lower desirability and warm yellow for higher. If you just completed part 1

of this tutorial, your graph came up in only one color. This can be easily fixed by

right-clicking over the graph and selecting Graph Preferences, then Surface

Graphs.

Graph preferences via right-click menu – Selecting graduated (color) shading

Make sure Contour graph shading is set to Graduated. Press OK and see what

you get.

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Design-Expert software sets a flag at the optimal point for solution 13. Click

through the numbered Solutions choices atop your screen until the flag relocates to

the largest sweet spot (the one with the largest area) at the top of the triangular

mixture space. To view the responses associated with this desirability (sweet

spot), press the droplist arrow for Response and select Viscosity. Right-click the

flag and press Toggle size. Now you see confidence intervals (CI) on the mean

prediction and prediction intervals (PI) on the individuals, as well as the

composition at point (X1, X2, X3). This is helpful!

Viscosity contour plot (with optimum flagged on largest sweet spot)

The colors look nice, but what if you must print the graphs in black and white? This

is easily fixed by right-clicking over the graph and selecting Graph Preferences.

Click the Surface Graphs tab and set Contour graph shading to Plain

background.

Graph preferences set to plain background

While you’re at preferences, checkmark-on Show contour grid lines.

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Show grid lines option

Finally, to make your graph truly plain, go to the Fonts & Colors tab, choose

Contour Background under Colors, click Edit Color and select white from the

Color grid. Press OK to close the color palette.

Graph changed to black and white with grid lines

There are many other options on this and other Graph preferences tabs. Look them

over if you like and then press OK to see how options specified by this tutorial

affect your contour plot. If you like, look at the optimal turbidity response as well.

To view the desirability surface in three dimensions, again click Response and

choose Desirability. Then from the floating Graphs Tool select 3D Surface.

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3D view of desirability at default resolution in color

In this case the desirability surface is a bit too jaggedy for the colors to provide

much advantage for interpretation, so right-click over the graph and select Graph

Preferences. On the Graphs 1 tab change the Graph resolution to Very High.

Changing to very high resolution for the 3D plot

Press OK for the new graph preferences. Then to get the view shown below, select

View, Show Rotation tool and change the horizontal control h to 30 and v to 50.

Click on the graph for a spectacular view!

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3D desirability plot at high resolution

Now you can see one high ridge where desirability can be maintained at a

maximum level over a range of compositions. Another high point can be achieved,

but it requires sharp control of the composition. The other peak is less desirable

(lower).

When you have more than three components to plot, Design-Expert software uses

the composition at the optimum as the default for the remaining constant axes. For

example, if you design for four components, the experimental space is a

tetrahedron. Within this three-dimensional space you may find several optimums,

which require multiple triangular “slices,” one for each optimum.

The best way for pointing out the most desirable formulation for your mixture is to

demonstrate it on your computer screen or with the output projected for a larger

audience. In this case, you’d best shift back to the default colors and other display

schemes. Do this by right-clicking and selecting Graph Preferences and pressing

the Default buttons for the Graphs 1, Graphs 2 and Fonts and Colors (Colors

side only).

Adding Propagation of Error (POE) to the Optimization

If you have prior knowledge of the variation in your component amounts, this

information can be fed into Design-Expert software. Then you can generate

propagation of error (POE) plots showing how that error transmits to the response.

Look for compositions that minimize transmitted variation, thus creating a formula

that’s robust to slight variations in the measured amounts.

Start by clicking the Design node on the left side of the screen to get back to the

design layout. Select Column Info Sheet from the floating Design Tool palette.

Enter the following information into the Std. Dev. column: Water: 0.08, Alcohol:

0.06, Urea: 0.06, as shown on the screen below.

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Column Info Sheet with standard deviations filled in

Now you can calculate propagation of error by generating graphs for each

response. First, click the Viscosity analysis node and press the Model Graphs

button. Next, select View, Propagation of Error, which previously was grayed

out. Also choose 3D Surface view. Now your screen should match what’s shown

below.

3D view of the POE graph

The surface reaches a minimum where the least amount of error is transmitted

(propagated) to the viscosity response. These minima occur at flat regions on

model graphs where formulations are most robust to varying amounts of

components.

Click the Turbidity node, press the Model Graphs button and select View,

Propagation of Error and look at its 3D Surface. Rotate it so you can see the

surface best.

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POE surface for turbidity

For additional details about POE, attend Stat-Ease’s Robust Design and Tolerance

Analysis workshop.

Now that you’ve found optimum conditions for the two responses, let’s go back and

add criteria for the propagation of error. Click the Numerical optimization node.

Select POE (Viscosity) and establish a Goal to minimize with Limits of Lower at

5 and Upper of 8.

Set goal and limits for POE (Viscosity)

Select POE (Turbidity) and set its Goal also to minimize with Limits of Lower at

90 and Upper of 120.

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Criteria for POE (Turbidity)

Now click the Solutions button to generate new solutions with the additional

criteria. The number 1 solution represents the formulation that best achieves the

target value of 43 for viscosity and minimizes turbidity, while at the same time

finds the spot with the minimum POE (most robust to slight variations in the

component amounts).

Solutions Generated with Added POE Criteria (Your results may differ)

Be sure to review the alternative solutions, which may be nearly as good based on

the criteria you entered. In this case, the number 2 solution, which you may or may

not get due to the random nature of the optimization, increases the water level

(presumably cheaper) and reduces turbidity, so it may actually be preferred by the

formulators.

Viewing Trace Plots from Optimal Point

Continue on to the numerical optimization Graphs to look at the desirability

contour plot. It does not look much different from before because adding POE

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criteria had only a small impact on the result. However, this is a good time to get a

feel for the sensitivity of responses around the optimum point. Observe this by

changing Response to Viscosity.

Changing the response to be viewed from the optimum point

Select Trace from the Graphs Tool palette. Select Cox from the Trace Graph

palette. Click Solutions 2 and beyond to see how graphs shift with varying

optimal reference points. Return to Solution 1, which may produce the trace

pictured below (remember that your results may vary due to random elements in

the numerical search algorithm used by Design-Expert software).

Trace plot viewed from optimal point

Take a look at the trace for the other response – turbidity. It looks even more

interesting!

Graphical Optimization

By shading out regions that fall outside of specified contours, you can identify

desirable sweet spots for each response – windows of opportunity where all

specifications can be met. In this case, response specifications are:



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39 < Viscosity < 48

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





POE (Viscosity) < 8

Turbidity < 900

POE (Turbidity) < 120

To overlay plots of all these responses, click the Graphical optimization node. For

the Viscosity response, if the following values are not already pre-set, enter a

Lower limit of 39 and an Upper limit of 48.

Setting criteria for Graphical optimization: Viscosity response

Click the POE(Viscosity) response. If the following value is not already pre-set,

enter an Upper limit of 8. Do not enter a lower limit – it will not be needed for the

graphical optimization when simply minimizing.

Graphical criterion for POE of viscosity

Press forward to the Turbidity response and, if the following value is not already

pre-set, enter an Upper limit of 900. This again is a minimization, so don’t enter a

lower limit.

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Setting criteria for turbidity

Click the POE(Turbidity) response and, if the following value is not already pre-set, enter an Upper limit of 120.

The criterion for POE of turbidity

Press the Graphs button to produce the “overlay” plot. If the feasible regions are

not already colored by default, visually improve the graph by adding yellow (or the

color of your choice) to the contour background via Graphs Preferences.

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Adding visual appeal to contour background

Your graph should now appear similar to that below.

Graphical optimization

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Notice that regions not meeting your specifications are grayed out, leaving

(hopefully!) an operating window or “sweet spot.”

Notice the flag remains planted at the optimum. That’s handy! This Design-Expert

display may not look as fancy as 3D desirability, but it is very useful to show

windows of operability where requirements simultaneously meet critical

properties. Grayed areas on the graphical optimization plot do not meet selection

criteria. The clear “window” shows where you can set factors to satisfy

requirements for both responses.

The lines that mark the high or low boundaries on the responses can be identified

with a mouse-click. Notice that the contour and its label change color for easy

identification. Click outside the graph to reset the contour and its label to the

original color.

Let’s say someone wonders whether the 900 maximum for turbidity can be

decreased. What will this do to the operating window? Find out by clicking the 900

turbidity contour line – you know you’ve got it when it turns red. Then drag the

contour until it reaches a value of approximately 750. Finally right-click over this

contour, select Set contour value and enter 750.

Setting the turbidity contour value

Press OK to get the 750 contour level. Right-click over the flag and select Delete

Flag to make it easier to see how this change in turbidity affects the sweet spots.

Notice the smaller sweet spot has disappeared.

Changing the specification on turbidity to 750 maximum

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Graphical optimization works great for three components, but as the number

increases, it becomes more and more tedious. Once you find solutions much more

quickly by using the numerical optimization feature, return to the graphical

optimization and produce outputs for presentation purposes.

Response Prediction at the Optimum

Click the Point Prediction node (middle left on your screen). Notice it defaults to

your first solution.

Point prediction set to Solutions 1 (yours may be different)

Save the Data to a File

Now that you’ve invested all this time into setting up the optimization for this

design, it is wise to save your work. Click the File menu item and select Save As.

Save As selection

You can now specify the File name (we suggest tut-MIX-opt) to Save as type

*.dxp” in the Data folder for Design-Expert (or wherever you want to Save in).

Final Comments

We feel that numerical optimization provides powerful insights when combined

with graphical analysis. Numerical optimization becomes essential when

investigating many components with many responses. However, computerized

optimization does work very well in the absence of subject-matter knowledge. For

example, a naive user may define impossible optimization criteria. The result will

be zero desirability everywhere! To avoid this, try setting broad, acceptable ranges.

Narrow down the ranges as you gain knowledge about how changing factor levels

affect responses. Often, you will need to make more than one pass to find the “best”

factor levels that satisfy constraints on several responses simultaneously.

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Using Design-Expert software allows you to explore the impact of changing

multiple components on multiple responses – and to find maximally desirable

solutions quickly via numerical optimization. For your final report, finish up with a

graphical overlay plot at the optimum “slice.” (Don’t forget you can set goals on the

components themselves. For example, in this case it might be wise to try

maximizing the amount of cheap water.)

Learn more about mixture design methods at our workshop titled Mixture Designs

for Optimal Formulations. To get the latest class schedule, give Stat-Ease a call.

Also, we appreciate your questions and comments on Design-Expert software. Call

us, send an annotated fax of output, write us a letter, or send us an e-mail. You will

find our contact information and email links at .

Postscript: Adding a Cost Equation

In the comments above, we suggested you consider maximizing the cheapest

ingredient – water in this case. Conversely, you may have an incredibly expensive

material in your formulation that obviously needs to be minimized. With only a

small amount of effort, you can set up cost as a response to be included in Design-Expert’s numerical optimization.

Re-open the file. In the Design branch, right-click the last response

column. From the menu, select Insert Response, After This Column.

Inserting a new response

Next, right-click the new untitled response header and select Equation Only.

Equation only option

In the dialog box enter Response Name as Cost and Response units in $/kg.

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Specifying name and units for new response

Press Model equation and enter with no spaces .5b+.2c (alcohol at $0.50 per kilo

and urea at $0.20 cents – assume water costs practically nothing).

Entering the cost equation

Press OK to accept the equation and OK again to calculate costs for all

formulations in this mixture design. To make these more presentable, right-click

the Cost column header, select Edit Info, and change Format to 0.00. Press OK.

Costs calculated and formatted

Now, under the Analysis branch, click the Cost node to bring up the model graph

directly – no modeling is necessary because you already entered the deterministic

equation.

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Contour plot of cost

The water shows blue due to it being so cheap.

This sets the stage to include cost in your multiple response optimization. As

pictured below, go to Optimization node Numerical, select Cost and set its Goal

to minimize.

Minimizing cost

Pressing Solutions at this stage only tells you what you already know: The lowest

cost formula is at the greatest amount of water within the specified constraints. Re-enter the goals for viscosity and turbidity if you like, but it really isn’t necessary

now. Wait until you do your own mixture design and then make use of this

postscript tip to take costs into account.

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