Digraphs+Theory,+Algorithms+and+Applications_Part34

Digraphs+Theory,+Algorithms+and+Applications_Part34 - 12.2...

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Unformatted text preview: 12.2 Ordering the Vertices of a Digraph of Paired Comparisons 643 pair x,y of vertices (i.e. objects) in D , the arc xy is in D if and only if some experts prefer y to x . The weight of xy is the fraction of the experts that favour x over y . Formally, following Kano and Sakamoto [472, 473], we introduce a digraph of paired comparisons as follows. Let D = ( V,A,² ) be a weighted digraph in which every arc xy has a positive real weight ² ( xy ). A digraph D is called a paired comparison digraph (abbreviated to PCD ) if D satisfies the following conditions: (a) 0 < ² ( xy ) ≤ 1 for every xy ∈ A ; (b) ² ( xy ) + ² ( yx ) = 1 if both xy and yx are arcs; (c) ² ( xy ) = 1 if xy ∈ A but yx / ∈ A . 1 . 7 0 . 3 . 2 . 8 . 1 u v w x . 9 Figure 12.1 A paired comparison digraph H . See Figure 12.1 for an example of a paired comparison digraph. An (un- weighted) digraph D = ( V,A ) can be viewed as a PCD by setting the weight of each arc of D as follows: (i) ² ( xy ) = ² ( yx ) = 0 . 5 if xy,yx ∈ A ; (ii) ² ( xy ) = 1 if xy ∈ A but yx / ∈ A . We call the PCD D = ( V,A,² ) with the weight function ² determined by (i) and (ii) the uniform PCD corresponding to D . The positive (negative ) score of a vertex x ∈ V is σ + ( x ) = X xy ∈ A ² ( xy ) , ( σ- ( x ) = X yx ∈ A ² ( yx ) . ) In Figure 12.1, σ + ( u ) = 0 . 5 and σ- ( u ) = 1 . 5 . A PCD D is not always semicomplete (some pairs of vertices may not be compared). If D is a tournament, then usually the vertices of D are ordered according to their positive score, with the first vertex being of highest positive score. This approach, the score method , is justified by a series of natural ax- ioms (see the paper [647] by Rubinstein). The score method can be naturally used for semicomplete PCDs. When a PCD is not semicomplete, the score 644 12. Additional Topics method may produce results that are not justified from the practical point of view. For example, consider the digraph R = ( V,A ) with V = { 1 , 2 ,...,n } , n ≥ 5, and A = { 12 , 13 }∪{ 41 , 51 ,...,n 1 } . Let R = ( V,A,² ) be the uniform PCD corresponding to R . Even though the positive score of the vertex 1 is maximum, it is against our intuition to order in R the vertex 1 first (i.e. the winner). This raises the question of finding a method of ordering the vertices of an arbitrary PCD, which agrees with the score method for semicomplete PCDs. In Subsection 10.3.3, we studied a method of ordering of the vertices of a weighted digraph D = ( V,A,² ), the feedback set ordering (FSO). Recall that, for an ordering α = ( v 1 ,v 2 ,...,v n ), where n = | V | and { v 1 ,v 2 ,...,v n } = V, an arc ( v i ,v j ) ∈ A is forward (backward) if i < j ( i > j ). In Figure 12.1, for the ordering β = ( u,v,w,x ), uv , ux and wx are (all) forward arcs; vu and xw are backward arcs. An ordering α = ( v 1 ,v 2 ,...,v n ) can be viewed as a bijection from V to { 1 , 2 ,...,n } . Thus, for a vertex x ∈ V , α ( x ) = i if x = v i...
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Digraphs+Theory,+Algorithms+and+Applications_Part34 - 12.2...

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