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DX71-05B-Mix-P2.doc Rev. 1/24/07

Mixture Design Tutorial(Part 2 Optimization)

IntroductionThis tutorial shows the use of Design-Expert software for optimization of mixtureexperiments. Its based on the data from the preceding tutorial (Part 1 The Basics).You should go back to this section if youve not already completed it. Much of whatsdetailed in this Mixture Design Tutorial (Part 2 Optimization) is a repeat from theMultifactor RSM Tutorial (Part 2 Optimization). If youve already completed thisRSM tutorial, just skip over the bits in this one that you find redundant.

For details on optimization, use the on-line program Help. Also, Stat-Ease provides in-depth training in its Mixture Designs for Optimal Formulations workshop. Call for

information on content and schedules, or better yet, visit our web site atwww.statease.com.

Start the program by finding and double clicking on the Design-Expert software icon.The detergent design, response data and appropriate response models are stored in a filenamed Mix-a.dx7 . To load this file, use the File, Open Design menu item.

File Open dialog box

Once you have found the proper drive, directory and file name, click on Open to loadthe data. To see a description of the file contents, click on the Summary node underthe Design branch at the left of your screen. Drag the left border and open the windowto see the report better. You can also re-size columns with the mouse.

Design-Expert 7.1 Users Guide Mixtur e Design Tutori al Part 2 1


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Design summary

The file you loaded includes analyzed models as well as the raw data for each response.Recall that the formulators chose a three-component simplex design to study a detergentformulation. The components were water, alcohol and urea. The experimenters held allother ingredients constant. They measured two responses: viscosity and turbidity. Youwill optimize the mixture using their analyzed models. From the design status screenyou can see that we modeled viscosity with a quadratic mixture model and turbidity witha special cubic model.

Numerical Optimization

Design-Expert softwares numerical optimization will maximize, minimize or target: A single response A single response, subject to upper and/or lower boundaries on other

responses Combinations of two or more responses.

We will lead you through the latter case: a multiple response optimization. Under theOptimization branch of the program, click on the Numerical node to start the process.

Setting numeric optimization criteria

2 Mixtur e Design Tutori al Part 2 Design-Expert 7.1 Users Guide


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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 responsesdefault to the observed extremes.

Now you get to the crucial phase of numerical optimization: assignment ofOptimization Parameters. The program uses five possibilities for a Goal toconstruct desirability indices (d i):

None (responses only) Maximize, Minimize, Target->, In range, Equal to -> (components only) .

Desirabilities range from zero to one for any given response. The program combines theindividual desirabilities into a single number and then searches for the greatest overalldesirability. A value of one represents the ideal case. A zero indicates that one or moreresponses fall outside desirable limits. Design-Expert uses an optimization methoddeveloped by Derringer and Suich, described by Myers and Montgomery in ResponseSurface Methodology , John Wiley and Sons, New York (available from Stat-Ease).

In this case the components will be allowed to range within their pre-establishedconstraints, but be aware that they can be set to desired goals. For example, since wateris cheap, you could set its goal to maximize.

Options for goals on components

Notice that components can be set equal to a specified level. Leave water at its defaultof in range and click the first response Viscosity . Set its Goal at target-> of 43 .Enter Limits at Lower of 39 and Upper of 48 .



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Setting Target for first response of viscosity

These limits indicate that it is most desirable to achieve the targeted value of 43, butvalues in the range of 39-48 are acceptable. Values outside that range have no (zero)desirability.

Now click on the second response, Turbidity . Select the Goal of minimize , withLimits set at Lower of 800 and Upper of 900 . You must provide both thesethresholds to get the desirability equation to work properly. By default they will be setat the observed response range, in this case 321 to 1122. However, evidently in this casetheres no advantage to getting the detergents turbidity below 800 it already appearsas clear as can be to the consumers eye. On the other hand, when turbidity exceeds 900,

it looks as bad as it gets.

Aiming for minimum on second response of turbidity



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These settings create the following desirability functions:1. Viscosity:

if less than 39, then desirability (d i) equals zero from 39 to 43, d i ramps up from zero to one from 43 to 48, d i ramps back down to zero if greater than 48, then d i equals zero.

2. Turbidity: if less than 800, then d i equals one from 800 to 900, d i ramps down from one to zero if over 900, then d i equals zero.

Do not forget that at your fingertips you will find advice on using sophisticated featuresof Design-Expert software: Press the screen tips button for an overview on NumericalOptimization.

Screen tips at your fingertip

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

Changing Desirabili ty Weights and the (Relative) Importance of Variables

The user can select additional parameters, called weights, for each response. Weightsgive added emphasis to upper or lower bounds or emphasize a target value. With aweight of 1, the d i will vary from 0 to 1 in linear fashion. Weights greater than 1(maximum weight is 10) give more emphasis to the goal. Weights less than 1 (minimumweight is 0.1) give less emphasis to the goal. Weights can be quickly changed bygrabbing (via left mouse-click and drag) the handles (the squares ) on the desirabilityramps. Try pulling the one on the ramp down as shown below.



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Weights change by grabbing handle with mouse

Notice that the Weight now reads 10. Youve now made it much more desirable to getnear the goal of 800 for turbidity. Before moving on from here, re-enter the Upper Weights at its default value of 1 and press the Tab key. This will straighten out to thedesirability ramp.

Importance is a tool for changing the relative priorities for achieving the goals youestablish for some or all of the variables. If you want to emphasize one over the rest, setits importance higher. Design-Expert offers five levels of importance ranging from 1

plus (+) to 5 plus (+++++). For this study, leave the Importance field at +++, amedium setting. By leaving all importance criteria at their default, none of the goals will

be favored over any other.

For an in-depth explanation of the construction of the desirability function, and formulasfor the weights and importance, select Help off the main menu. Then go to Contents and select Optimization . Branch down to the topic of Importance as shown on thescreen-shot below.

Details on Importance found in program Help



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When you are done looking at Help, close out the screen by pressing X at the upper-rightcorner of its screen.

Click the Options button to see how you gain control over how the numericaloptimization will be performed. For example, you can change the number of cycles(searches) per optimization. If you have a very complex combination of responsesurfaces, increasing the number of cycles will give you more opportunities to find theoptimal solution. The Duplicate solution filter establishes the epsilon (minimumdifference) for eliminating essentially identical solutions. The Simplex Fractionspecifies how big the initial steps will be relative to the factor ranges. (The wordsimplex relates to the geometry of the search. For two factors the simplex is anequilateral triangle. By stepping through the three corners the program figures out the

path of steepest ascent. For more detail, go to Help and search on numerical searchalgorithm.)

Another optimization option is to not use random starting points, but rather those in thedesign itself. However, these will be limited to 100 unless you change this default.Leave this and all other options at their default levels shown below. (PS. This screenshot shows underlined letters that show Alt keys for jumping to fields via keystrokesversus mousing.)

Optimization Options Dialog Box

Running the optimization

Start the optimization by clicking on the Solutions icon.



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Numerical Optimization Report on Solutions (Your results may differ )

The program randomly picks a set of conditions from which to start its search fordesirable results. Multiple cycles improve the odds of finding multiple local optimums,some of which will be higher in desirability than others. After grinding through 40cycles 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 doesnt fit in the window, move yourcursor to the left border and drag it open.) In addition to the solutions, the reportincludes a recap of your optimization specifications as well as the random starting pointsfor the search.

The Solutions Tool provides three views of the same optimization. (Drag the tool to a

convenient location on the screen.) Click on the view option Ramps .

Ramps report on numerical optimization (Your results may differ )



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The ramp display combines the individual graphs for easier interpretation. The dot oneach ramp reflects the factor setting or response prediction for that solution. The heightof the dot shows how desirable it is. Press the different solution buttons (1, 2, 3,) andwatch 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)? Click the last solutionon your screen. Does it look something like the one below?

Sub-optimum solution that ranks least desirable

If your search also uncovered this local optimum, you will note that viscosity falls offtarget and turbidity becomes excessive, thus making it less desirable than the option forhigher temperature.

Select the Bar Graph view.

Solution to multiple response optimization - desirability bar graph



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The bar graph shows how well each variable satisfied the criteria and the overallcombined desirability: values near one are good.

Optimization Graphs

Press Graphs to view a contour graph of overall desirability. It now becomes obviousthat at least somewhat desirable formulations fall with three distinct sweet spots.

Desirability contour graph

The screen shot above came from a graph done with graduated colors cool blue forlower desirability and warm yellow for higher. If you just completed part 1 of thistutorial, your graph came up in only one color. This can be easily fixed by right-clickingover the graph and selecting Graph Preferences .

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

Click the Graphs 2 tab and make sure the Contour graph shading option is set atGraduated shading . Then press OK and see what you get.



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Design-Expert software sets a flag at the optimal point. Click back up the Solutionsuntil the flag relocates to the largest sweet spot at the top of the triangular mixture space.To view the responses associated with the desirability, press the down-list arrow forResponse and select Viscosity from the drop-down menu. Then right-click on theflag and Toggle size . Now you see the confidence intervals (CI) on the mean

prediction and the prediction intervals (PI) on the individuals, as well as the compositionat the point (X1, X2, X3). This will be helpful!

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

The colors are neat, but what if you must print the graphs in black and white? This can be easily fixed by right-clicking over the graph and selecting Graph Preferences .Click the Graphs 2 tab and set the Contour graph shading option to Plainbackground .

Graph preferences set to plain background

While youre at preferences, go to Graphs 1 tab and click Show 2D grid lines .

Show grid lines option



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Finally, to make the graph really plain, go to the Fonts & Colors tab, choose ContourBackground , click Edit Color and select white off the grid and press OK.

Graph changed to black and white with grid lines

There are many other options on this and other tabs for Graph preferences. Look themover if you like and then press OK to see what the options specified by this tutorial dofor your contour plot. If you like, look at the optimal turbidity response as well.

To look at the desirability surface in three dimensions, click again on Response andchoose Desirability . Then select View , 3D Surface from the main menu.

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



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Preferences . On the Graphs 1 tab change the Graph resolution to High . ThenOn the Graphs 2 tab change the 3D graph drawing method to Std errorshading .

Changing graph preferences for the 3D plot

Press OK for the new graph preferences. Then to get the view shown below, go to theRotation tool and change the horizontal control h to 350 and v to 85 . Click on thegraph for a spectacular view!

Shaded 3D desirability plot at high resolution

Now you can see two high ridges where desirability can be maintained at a maximumlevel over a range of compositions. The other peak is less desirable.

When you have more than three components to plot, Design-Expert software uses thecomposition 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 thisthree-dimensional space you may find several optimums, which require multipletriangular slices, one for each optimum.

The best way for pointing out the most desirable formulation for your mixture is todemonstrate it on your computer screen or with the output projected for a largeraudience. In this case, youd best shift back to the default colors and other displayschemes. Do this by right-clicking and selecting Graph Preferences and pressing theDefault buttons for the Graphs 1 , Graphs 2 and Fonts and Colors (Colors sideonly).



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Adding Propagation of Error (POE) to the Optimization

If you have prior knowledge of the variation in your component amounts, thisinformation can be fed into Design-Expert software. Then you can generate propagationof error (POE) plots that show how that error is transmitted to the response. Look forcompositions that minimize the transmitted variation, thus creating a formula thatsrobust to slight variations in the measured amounts.

Start by clicking on the Design node on the left side of the screen to get back to thedesign layout. Then select View , Column Info Sheet . Enter the followinginformation into the Std. Dev. column: Water: 0.08 , Alcohol: 0.06 , Urea: 0.06 , asshown on the screen below.

Column Info Sheet with standard deviations filled in

Now you can calculate the propagation of error by generating their graphs for eachresponse. First, click on the Viscosity analysis node and press the Model Graphs

button. Next, select View , Propagation of Error , which previously was grayed out.Also choose the 3D Surface view. Now your screen should match whats shown

below.

3D view of the POE graph



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The surface reaches a minimum where the least amount of error is transmitted, or propagated, to the viscosity response. These minima occur at the flat regions on themodel graphs where formulations will be most robust to varying amounts ofcomponents.

Click on 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 thesurface best.

POE surface for turbidity

For additional details on POE, attend Stat-Eases DOE for DFSS, Variation by Designworkshop.

Now that youve found optimum conditions for the two responses, lets go back and addcriteria for the propagation of error. Click on the Numerical optimization node. Selectthe 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)



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Select POE (Turbidity) and set its Goal also to minimize with Limits of Lower at90 and Upper of 120 .

Criteria for POE (Turbidity)

Now click on the Solutions button to generate new solutions with the additionalcriteria. The number 1 solution represents the formulation that best achieves the targetvalue of 43 for viscosity and minimizes turbidity, while at the same time finds the spotwith 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 thecriteria you entered. In this case, the number 2 solution, which you may or may not getdue to the random nature of the optimization, increases the water level (presumablycheaper) and reduces turbidity, so it may actually be preferred by the formulators.



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Viewing Trace Plots from Optimal Point

Press ahead to the numerical optimization Graphs to look at the contour plot fordesirability. It will not look much different that before, because adding criteria on POEhad only a small impact on the result. However, this will be a good time to get a feel forthe sensitivity of responses around the optimum point. See this by changing theResponse to Viscosity .

Changing the response to be viewed from the optimum point

Then select View , Trace and click solutions 2 and beyond to see how it shifts withvarying 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 thenumerical search algorithm employed by Design-Expert software).

Trace plot viewed from optimal point

Take a look at the trace for the other response turbidity. It looks even moreinteresting!



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Graphical Optimization

By shading out regions that fall outside of specified contours, you can identify adesirable sweet spots for each response windows where all specifications can be met.In this case, the response specifications are:

39 < Viscosity < 48 POE (Viscosity) < 8 Turbidity < 900 POE (Turbidity) < 120

To overlay the plots for all these responses, click on the Graphical optimization node.For the Viscosity response enter a Lower limit of 39 and an Upper limit of 48 .

Setting criteria for Graphical optimization: Viscosity response

Skip the POE for the moment. Click the Turbidity response and enter an Upper limitof 900 . Do not enter a lower limit it will not be needed for the graphical optimization.

Setting criteria for turbidity



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Then click on the Graphs button to produce the overlay plot.

Graphical optimization

Notice that regions not meeting your specifications are shaded out, leaving (hopefully!)an operating window or sweet spot. (If you still see the grid lines you added earlier,right-click over the graph, select Graph Preferences, go to Graphs 1 tab and click thecheck mark off Show grid lines. These just clutter up the graph.)

Notice that the flag remains planted at the optimum. Thats handy! This display byDesign-Expert may not look as fancy as the 3D of desirability but it can be very usefulto show windows of operability where requirements simultaneously meet the critical

properties. The shaded areas on the graphical optimization plot do not meet theselection criteria. The clear window shows where you can set the factors to satisfy therequirements on both responses.

Click back to Criteria and for the POE(Viscosity) enter an Upper limit of 8 and forPOE(Turbidity) an Upper limit of 120 . (Neither of these minimized responses need alower limit.) Then click back to Graphs . The lines that mark the high or low

boundaries on the responses can be identified via a mouse-click. Notice that the contourand its label change color for easy identification.

Lets say that someone wonders whether the 900 maximum for turbidity can bedecreased. What will this do to the operating window? Find out by dragging the 900turbidity contour until it reaches a value of approximately 750 . Then right-click overthis contour and select Set contour value and enter 750 .

Caution: Having changed the criteria, the flagged optimum is no longer valid. To get itright you would have to go back to the Numerical optimization option and enter any newresponse criteria there.



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Setting the turbidity contour value

Press OK to get the cleaner contour level. Right-click over the flag and select Deleteflag to make it easier to see how this change in turbidity affects the sweet spots. Noticethat the smaller one has now disappeared.

Changing the specification on turbidity to 750 maximum

Graphical optimization works great for three components, but as the number increases, it becomes more and more tedious. You will find solutions much more quickly by usingthe numerical optimization feature. Then return to the graphical optimization and

produce outputs for presentation purposes.

Response Prediction at the Optimum

Click on the Point Prediction node (lower left on your screen). Notice how it nowdefaults to the first solution.



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Point prediction set to solution #1

Save the Data to a File

Now that youve invested all this time into setting up the optimization for this design, itwould be prudent to save your work. Click on the File menu item and select Save As .

Save As selection

You can then specify the File name (we suggest tut-MIX-opt) to Save as type *.dx7in 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 withgraphical analysis. Numerical optimization becomes essential when you investigatemany components with many responses. However, computerized optimization will notwork very well in the absence of subject matter knowledge. For example, a naive usermay define impossible optimization criteria. The result will be zero desirabilityeverywhere! To avoid this, try setting broad acceptable ranges. Narrow them down asyou gain knowledge about how changing factor levels affect the responses. Often, youwill need to make more than one pass to find the best factor levels to satisfyconstraints on several responses simultaneously.

With Design-Expert software, you can explore the impact of changing multiplecomponents on multiple responses, and find maximally desirable solutions quickly vianumerical optimization. (Dont forget that you can set goals on the componentsthemselves for example, in this case it might be wise to try maximizing the amount ofcheap water.) For your final report, finish up with a graphical overlay plot at theoptimum slice.



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Learn more about methods of mixture design at our Mixture Designs for OptimalFormulations workshop. To get the latest class schedules, give Stat-Ease a call. Also,we appreciate your questions and comments on Design-Expert software. Call us, sendan annotated fax of output, write us a letter or send us an e-mail. You will find ourcontact information and email links at www.statease.com .

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 yourformulation that obviously needs to be minimized. However, with only a small amountof effort, you can set up cost as a response to be included in Design-Experts numericaloptimization.

Re-open the Mix-a.dx7 and in the Design layout right-click the last response columnSelect from the menu Insert Response , After this co lumn .

Inserting a new response

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

Equation only option

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

http://www.statease.com/http://www.statease.com/


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Specifying name and units for new response

Press the Model equation button and enter .5b+.2c (alcohol at $0.50 per kilo and ureaat $0.20 cents assume water costs practically nothing).

Entering in the cost equation

Press OK to accept the equation and OK again to calculate costs for all formulations inthis mixture design. To make these more presentable, right-click the Cost column, selectEdit Info and change the Format to 0.00 . Then press OK.

Costs calculated and formatted

Now under the Analysis node click the Cost to bring up the model graph directly nomodeling 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 tominimize .

Minimizing cost

Pressing Solutions at this stage only tells you what you already know: The lowest costformula will be at the greatest amount of water within the specified constraints. Re-enterthe goals for viscosity and turbidity if you like, but it really isnt necessary now. Waituntil you do your own mixture design and then make use of this postscript tip to takecosts into account.

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