GMAA: A DSS Based on the Decision Analysis Methodology - Application Survey and Further Developments - Efficient Decision Support Systems: Practice and Challenges from Current to Future - page 280

4. Dominance measuring methods
A recent approach for dealing with incomplete information about weights within MAUT is
to use information about each alternative's intensity of dominance, known as dominance
measuring methods .
Let us consider a group decision-making problem with n attributes ( X i , i =1,…, n ) and m
alternatives ( A j , j =1,…, m ), where incomplete information about input parameters has been
incorporated into the decision-making process as follows:
 Alternative performances under uncertainty ( x i j  [ x i jL , x i jU ], i =1,…, n; j= 1,…, m .
 Imprecision concerning utility function assessment ( u i (•) [ u i L (•), u i U (•)], i =1,…, n ),
where u i L (•) and u i U (•) are the lower and the upper utility functions of the attribute X i ,
and
 Imprecision concerning weights, which is represented by weights intervals ( w i 
[ w i L , w i U ], i =1,…, n ).
Given two alternatives A k and A l , alternative A k dominates A l if D' kl ≥ 0, D' kl being the
optimum value of the optimization problem (Puerto et al., 2000),



n n



 k l

 

D  min u A ( )  u A ( )  w u ( x )  w u ( x )

 

kl k l i i i i i i

  i  1 i  1

L U

wwwi  

 
 

, ,..., n

i i i

kL k kU

(2)

xxxi , ,..., n

.

i i i

lL l lU

st ..

xxxi , ,..., n

i i i

L L k UkU

ux ( )  ux ( )  u x ( ), i  ,..., n

i i i i i i

LlL l UlU

ux () ()  ux  u x ( , i  ,..., n

i i i i i i

This concept of dominance is called pairwise dominance and leads to the so-called dominance
matrix :

  D ...

...
...
...
...

D D 

11 1 m 




1 1 m

 

DD D D

 

21 22 2 m 1 2 m

 

D   DD D D

31 32 3 m 1 3 m



 

 

DD D 

 

mm 1 2 m  1

where D kl = D' kl /Σ w i and D' kl and w i are the optimum values for the objective function and
weights in problem (2), respectively. The dominance matrix is the starting point of
dominance measuring methods.
The first two dominance measuring methods were proposed in (Ahn & Park, 2008). In the
first, denoted AP1 (Ahn and Park 1), alternatives are ranked according to a dominating
measure

m

   D .

k kj

j  1, j k

The higher this dominating measure is the more preferred the alternative will be, because
the sum of the intensity of one alternative dominating the others will also be greater. In the