A positive n×n matrix, R=(rij), is said to be reciprocal if its entries verify rji=1/rij>0 for all 1•i, j•n. In the context of the analytic hierarchy process, where such matrices arise from the pairwise comparison of n≥2 decision alternatives on an arbitrary ratio scale, Saaty (in Journal of Mathematical Psychology, 1977) proposed to use μ=(λmax-n)/(n-1)≥0, a linear transform of the Perron eigenvalue λmax of R, as a measure of the cardinal consistency in an agent’s responses and posed the problem of determining how it might vary as a function of the rij’s. The authors also suggested that an upper bound could be found for that consistency index when the entries of R are restricted to take their values in a bounded set. Both of these questions are answered here using classical results from linear algebra.
