The upper and lower bounds on a function of components in a mixture design. Setting these limits helps to define your design space and focuses your experiment to make the best use of testing resources.
Whereas component bounds place upper and lower limits on individual components, linear constraints put limits on functions of the components.
For example, a chemical company makes an epoxy that includes two hardening agents, Hardener A and Hardener B. They know that their product must contain at least 5% hardening agents to be viable, while using more than 15% begins to degrade the quality of the product. This implies the following bounds on the individual components:
0.0 <= Hardener A <= 0.15
0.0 <= Hardener B <= 0.15
However, these bounds are interdependent: if, for example, Hardener A makes up 2% of the mixture, then Hardener B must make up at least 3% and no more than 13%. In other words, it is the combination of the two ingredients that is being limited, not just 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 upper and lower bounds as well as coefficient values for each of the components in the mixture. In this case, the epoxy mixture contains two other ingredient- Adhesive A and Adhesive B- so we would use the following equation to 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 impose another constraint on the mixture; that regardless of the amount of total hardener, there must always be twice as much Hardener A as Hardener B. He would need specify a separate linear constraint with this equation:
0.0 <= (1*Hardener A) + (-2*Hardener B) + (0*Adhesive A) + (0*Adhesive B) <= 0.0