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On weights estimation in Multiple Criteria Decision Analysis 55efficient solutions for this problem. Using these results, we prove that the geometric-meanmethod for deriving weights, i.e.provides efficient solutions for (X).⎛ ⎞n∏x i = ⎝ a ij⎠j=11n, i = 1, . . . , n (5)A geometric characterization of efficiency is also been obtained, based on the graph introducednext.Definition 1. Given y ∈ R N , let G(y) be the digraph G(y) := ({1, 2, . . . , N}, E(y)),(i, j) ∈ E(y) iff i ≠ j and y i − y j ≥ log(a ij )The above-mentioned geometric characterization is provided in the following result (werecall that a directed graph is said to be strongly connected if for all pairs of nodes (i, j), i ≠ j,there exist directed paths from i to j and from j to i).Theorem 2. Vector x ∈ R n ++ is efficient for (X) if and only if G(log(x)) is strongly connected.A similar characterization can be obtained for weakly efficient solutions. We recall thatx ∗ ∈ R n ++ is said to be weakly efficient for (X) if and only if no x ∈ Rn ++ exists with∣ ∣ x i ∣∣∣ x ∗ ∣∣∣∣i∣ − a ij