Goal program

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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:

  1. Add the variable, LaTeX: v, and the constraints, LaTeX: G(x) - v \le 0 and LaTeX: v \ge 0, to measure the level of violation. (If the goal were LaTeX: H(x) = 0, instead of LaTeX: G(x) \le 0, the added constraint would be LaTeX: H(x) + u - v = 0, where LaTeX: u and LaTeX: v are (non-negative) levels of violation below and above the goal, respectively.)
  2. Add the penalty term to the objective: LaTeX: P(v), where LaTeX: P(0)=0, and LaTeX: P is strictly increasing -- i.e., LaTeX: v' \ge v and LaTeX: v'_i > v_i for some LaTeX: i imply LaTeX: P(v') > P(v).

The resulting mathematical program represents the goal program. If the goal is satisfied, LaTeX: v=0; otherwise, the penalty term, LaTeX: P(v), reflects "adequate compensation" for the violation.

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