How are linear constraints different than component bounds in a mixtures design?

Linear constraints are the upper and lower bounds on a function of components in a mixture design. Setting these limits helps to define your design space and lets your experiment make the best use of testing resources.

In contrast, a component bound puts upper and lower limits on individual components.

To illustrate the difference, a chemical company manufactures an epoxy that includes two hardening agents, Hardener A and Hardener B. Chemists at the company know that their product must contain at least 5% hardening agents to be viable. Additionally, the quality of the product degrades when the mixture includes more than 15% hardening agents. These requirements imply the following bounds on the individual components:
• 0 Hardener A 0.15
• 0 Hardener B 0.15
However, these bounds are interdependent. For example, if Hardener A makes up 2% of the mixture, then Hardener B must make up at least 3% and no more than 13%. The constraint is for the combination of the two components, not only the individual components. Thus, the linear constraint is:
• 0.05 (Hardener A + Hardener B) 0.15
To specify a linear constraint in Minitab, you must state the following values:
• The lower bound
• The upper bound
• The coefficient for each of the components in the mixture
In this case, the epoxy mixture contains two other ingredients—Adhesive A and Adhesive B—so the following equation would state the linear constraint.
• 0.05 (1*Hardener A) + (1*Hardener B) + (0*Adhesive A) + (0*Adhesive B) 0.15

Because the amount of adhesive is not considered in the constraint it receives a coefficient of 0.

Now suppose the chemist wanted to put one more constraint on the mixture. The constraint is that whatever the amount of total hardener, there must be at least twice as much Hardener A as Hardener B. The chemist would specify a separate linear constraint with this equation:
• 0.0 (1*Hardener A) + (-2*Hardener B) + (0*Adhesive A) + (0*Adhesive B)
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