DX8-05B-Mix-P2.docxx 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
odds
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 www.statease.
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 Mix-a.dxp.  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
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 maximize s, 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 (d i):
∙None (responses only)
∙Maximize,
∙Minimize,
∙Target->,
∙In range,
∙Equal to -> (components only).
DX8-05B-Mix-P2.docxx Rev. 12/8/09 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
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 (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, d i equals zero.
2. Turbidity:
∙if less than 800, d i equals one
∙from 800 to 900, d i ramps down from one to zero
∙if over 900, d i equals zero.
about Numerical Optimization.
DX8-05B-Mix-P2.docxx Rev. 12/8/09
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, d i 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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