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Dear Jacob,
 
thanks for your clarification.
 
But what if in your example you come up with three different incidence  
matrices
connecting all pairs of nodes (i,j) , the first representing connection  
values
with respect to Q, the second S, the third E, respectively?
To select some overall 'optimal' way for a choosing a path, one would then  
also need
a loss (or gain, if you prefer) function that weighs the optimality  values
with respect to the individual criteria.
 
I feel that it is necessary to avoid evaluating n^d ( n-number of paths, d-  
dimension
of criteria) items in searching for an optimal path ...
Using several incidence matrices, you would need n*d  evaluations.
 
Best regards,
 
Axel

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