Negative priorities in the analytic hierarchy process

In decision-making, there are often criteria that are opposite in direction to other criteria as in benefits (B) versus costs (C), and in opportunities (O) versus risks (R), and sometimes need to be distinguished by using negative numbers. In making paired comparisons of alternatives with respect to a benefits criterion, one always uses the fundamental scale of positive absolute values of the analytic hierarchy process to estimate how much more benefits an alternative yields than the another alternative with which it is compared, puts the final values in the idealized mode of the AHP and synthesizes the results for the criteria under benefits. One does the same for a costs criterion to determine how much more one alternative costs than another, forms the ideal and synthesizes for the costs criteria. Similarly for opportunities and risks. One then needs to combine the four sets of priorities to get the overall ranking of the alternatives. Several different ways are described in the paper for doing this. A fundamental problem in the process of combining the B, O, C, and R had to be solved first and was done by the first author using ratings rather than paired comparisons in an earlier work done in 1999 described in this paper and used to deal with combining priorities that are opposite in direction. It is pointed out in the paper that each of the positive or negative priorities need not have a symmetric opposite value, because the opposite criterion may not exist in practice.