In determining irrigation releases to hit a target closely, we suggested that the "distance" measure be the difference between the release and the target. This would be an especially meaningful objective if the targets were not far in value from one another. It may be, however, that the targets may be very different from, say. the very early part of the growing season to a late month of fast growth prior to harvest.
Suppose that the early month target is 5000 and the late month target is 10,000. If the release Y, is 4000 in the early month and 9000 in the late month, we count these as equivalent failures in several of the models that we built. Yet the first is a 20% shortage, and the second is a 10% shortage. Perhaps they are not equivalent. We need to put the two measures of difference on a common scale so that we are calculating impacts correctly.
Our suggestion for the two models described below is that the distance be calculated as the difference between release and the target, divided by the target. In this way, deviations are normalized; that is, a difference of 1000 when the target is 5000 is equivalent to a difference of 2000 when the target is 10,000.
a. Set up in standard form the constraints and objective for minimizing the sum of the normalized absolute deviations from targets. That is, minimize the sum of the ab-solute values of the fractional deviations from targets. Assume the constraints of the basic model are already in place. Be sure to note that your variables are non-negative.
b. Set up the objective for minimizing the sum of the squared fractional deviations from targets. Again, the constraints of the basic model are already in place. Indicate how this problem should be solved; name the technique.