Editing Dempster–Shafer theory (section)

==Relational measures==
In considering preferences one might use the [[partial order]] of a [[lattice (order)|lattice]] instead of the [[total order]] of the real line as found in Dempster–Schafer theory. Indeed, [[Gunther Schmidt]] has proposed this modification and outlined the method.<ref>[[Gunther Schmidt]] (2006) [https://link.springer.com/content/pdf/10.1007%2F11828563_23.pdf Relational measures and integration], [[Lecture Notes in Computer Science]] # 4136, pages 343−57, [[Springer books]]</ref>

Given a set of criteria ''C'' and a [[bounded lattice]] ''L'' with ordering ≤, Schmidt defines a '''relational measure''' to be a function ''&mu;'' from the [[power set]] of ''C'' into ''L'' that respects the order ⊆ on <math>\mathbb{P}</math>(''C''): 
:<math>A \subseteq B \implies \mu(A) \leq \mu(B)</math>
and such that ''&mu;'' takes the empty subset of <math>\mathbb{P}</math>(''C'') to the least element of ''L'', and takes ''C'' to the greatest element of ''L''.

Schmidt compares ''&mu;'' with the belief function of Schafer, and he also considers a method of combining measures generalizing the approach of Dempster (when new evidence is combined with previously held evidence). He also introduces a ''relational integral'' and compares it to the [[Choquet integral]] and [[Sugeno integral]]. Any relation ''m'' between ''C'' and ''L'' may be introduced as a "direct valuation", then processed with the [[calculus of relations]] to obtain a ''possibility measure'' ''&mu;''.