# When are upper and lower bounds necessary in a mixtures design?

By default, Minitab generates settings for an unconstrained design, that is, the lower bound is zero and the upper bound is one for all of the components. However, in some mixture experimentation, lower bounds and/or upper bounds are necessary on some or all the components.
• Lower bounds are necessary when any of the components must be in the mixture. For example, lemonade must contain lemon juice.
• Upper bounds are necessary when the mixture cannot contain more than a specified proportion of an ingredient. For example, a cake mixture cannot contain more than 5% baking powder.

Constrained designs (those in which you specify lower or upper bounds) produce coefficients that are highly correlated. Usually, you can reduce the correlations between the coefficients by transforming the components to pseudo-components.

## How Minitab calculates achievable upper bounds when you change the default lower bound

When you change the default lower bounds of one component, the achievable upper values on the other components are also adjusted downward. Specifically, the achievable upper bound (Ui) for component i is:
• Ui = Total for mixture - [L1 + L2 + ... + L(i-1) + L(i+1) + ... + Lq] where L is the lower bound and q is the number of components

Minitab prints out both the specified lower bounds and achievable upper bounds. As an example, suppose you have 3 components and a total mixture amount of 10. If you entered the following lower bounds, L1 = 0, L2 = 2 and L3 = 1, then the achievable upper bounds are U1 = 7, U2 = 9 and U3 = 8.

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