The deviation variables that we have utilized in order to formulate alternative regression models as LPs have other uses as well. Let us consider a generic optimization problem over a feasible set S. A standard complication of real-life decision problems is that there is not just one criterion to evaluate the quality of a solution, but many. Say that such criteria are represented by functions fi(x), i = 1, …, M. We would like to maximize some of these functions, whereas others should be minimized. Needless to say, alternative criteria are usually in conflict with each other, and there is no easy way to assess a satisfactory tradeoff. In the next section we consider one possible approach to multiobjective optimization, and in this section we consider a possible alternative, based on setting desirable targets, or goals, for each objective.
Even though it is impossible to find a solution optimizing all the criteria at the same time, we might be able to find values such that we would be satisfied with them. Let us denote these target values, or goals, by 
. This means that if we could find a solution x ∈ S, such that
we would be satisfied. Most likely, even this is impossible to accomplish, but we might settle for a solution with minimal deviations from prescribed goals. So, let us introduce deviations 
and 
from goals, associated with penalties 
and 
, respectively. A goal programming model may be stated as follows:
If the functions fi are linear, then this will be an LP problem. The difficulty, of course, is setting goals and penalties. For a given function fi, we need not set two positive penalties. If the function is linked to a profit, we need not penalize a profit larger than a threshold goal, but we should penalize only under achievements.
Fig. 12.10 Illustrating the concept of dominated solution.
The flexibility in setting goals and penalties might look quite confusing, but goal programming might be a useful framework for building an interactive decision support tool for an experienced user.
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