Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results their pairwise comparisons according two criteria, subject constraints on ratings. formulate as bi-objective optimization constrained matrix approximation in Chebyshev sense logarithmic scale. The is approximate comparison matrices for each criterion simultaneously by common consistent unit rank, which determines vector represent and solve framework tropical (idempotent) algebra, deals with theory applications idempotent semirings semifields. solution involves introduction parameters that minimum values error thereby describe Pareto frontier problem. then reduces parametrized inequality. necessary sufficient conditions solutions inequality serve derive All inequality, correspond frontier, are taken complete Pareto-optimal apply these decision interest present illustrative examples.