# Goal program

### From Glossary

A *goal* is neither a constraint nor an objective because it is neither a firm
requirement nor a function that must be at an extreme value. In English, a
goal represents a relation that is desired, but it may be violated
if there is adequate compensation. For example, we may have a
budget of $1000, and we prefer to operate within it, but if
spending just one more dollar yields a major improvement in
operations, we would consider spending $1001.

A goal program is put into a standard form mathematical program in two steps:

- Add the variable, , and the constraints,
and , to measure the
*level of violation*. (If the goal were , instead of , the added constraint would be , where and are (non-negative) levels of violation below and above the goal, respectively.) - Add the
*penalty term*to the objective: , where , and is strictly increasing -- i.e., and for some imply .

The resulting mathematical program represents the goal program. If the goal is satisfied, ; otherwise, the penalty term, , reflects "adequate compensation" for the violation.