# Fuzzy math program

Some (or all) constraints are fuzzified by replacing the level set, $LaTeX: G_i = \{x \in X : g_i(x) \le 0\}$, with the requirement that $LaTeX: u_G_i(x) \ge a_i$, where $LaTeX: u_G_i$ is a membership function that is given (or derived from primitive membership functions on the level sets of each $LaTeX: g_i$, and $LaTeX: a_i$ is a parameter in $LaTeX: [0,1]$ to be set for each instance of the model. Typically (but not necessarily), $LaTeX: a_i = 1$ means that the constraint must hold, and the lower the value of $LaTeX: a_i$, the more $LaTeX: x$'s are admitted. (This appears similar to the chance-constraint model of stochastic programming, but it is more closely related to goal programming.) While fuzzy sets are used to represent uncertainty, they can also be used to represent preferences, e.g. a requirement, $LaTeX: u_S(x) \ge u_T(x)$ could mean that it is preferred that $LaTeX: x$ be in $LaTeX: S$ at least as much as it is preferred that $LaTeX: x$ be in $LaTeX: T$.