When ranking objects (like chemicals, geographical sites, river sections, etc.) by multicriteria analysis, it is in most cases controversial and difficult to find a common scale among the criteria of concern. Therefore, ideally, one should not resort to such artificial additional constraints. The theory of partially ordered sets (or posets for short) provides a solid formal framework for the ranking of objects without assigning a common scale and/or weights to the criteria, and therefore constitutes a valuable alternative to traditional approaches. In this paper, we aim to give a comprehensive literature review on the topic. First we formalize the problem of ranking objects according to some predefined criteria. In this theoretical framework, we focus on several algorithms and illustrate them on a toy example. To conclude, a more realistic real-world application shows the power of some of the algorithms considered in this paper.
It is well known that the linear extension majority (LEM) relation of a poset of size n ≥ 9 can contain cycles. In this paper we are interested in obtaining minimum cutting levels αm such that the crisp relation obtained from the mutual rank probability relation by setting to 0 its elements smaller than or equal to αm, and to 1 its other elements, is free from cycles of length m. In a first part, theoretical upper bounds for αm are derived using known transitivity properties of the mutual rank probability relation. Next, we experimentally obtain minimum cutting levels for posets of size n ≤ 13. We study the posets requiring these cutting levels in order to have a cycle-free strict cut of their mutual rank probability relation. Finally, a lower bound for the minimum cutting level α4 is computed. To accomplish this, a family of posets is used that is inspired by the experimentally obtained 12-element poset requiring the highest cutting level to avoid cycles of length 4.
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