The ordinal consistency of an incomplete reciprocal preference relation (original) (raw)
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Consistency of Reciprocal Preference Relations
2007 Ieee International Fuzzy Systems Conference, 2007
The consistency of reciprocal preference relations is studied. Consistency is related with rationality, which is associated with the transitivity property. For fuzzy preference relations many properties have been suggested to model transitivity and, consequently, consistency may be measured according to which of these different properties is required to be satisfied. However, we will show that many of them are not appropriate for reciprocal preference relations. We put forward a functional equation to model consistency of reciprocal preference relations, and show that self-dual uninorms operators are the solutions to it. In particular, Tanino's multiplicative transitivity property being an example of such type of uninorms seems to be an appropriate consistency property for fuzzy reciprocal preferences.
Combining preference relations: Completeness and consistency
2007
We introduce two criteria for judging "goodness" of the result when combining preference relations in information systems: completeness and consistency. Completeness requires that the result must be the union of all preference relations, while consistency requires that the result must be an acyclic relation. In other words, completeness requires that the result contain all pairs appearing in the preference relations, and only those pairs; while consistency requires that for every pair (x, y) in the result, it must be able to decide which of x and y is preferred to the other. Obviously, when combining preference relations, there is little hope for the result to satisfy both requirements. In this paper, we classify the various methods for combining preference relations, based on the degree to which the result satisfies completeness and consistency. Our results hold independently of the nature of preference relations (quantitative or qualitative); and also independently of the preference elicitation method (i.e. whether the preference relations are obtained by the system using query-log analysis or whether the user states preferences explicitly). Moreover, we assume no constraints whatsoever on the preference relations themselves (such as transitivity, strict ordering and the like).
2021
Consistency is an important issue that causes wide public concern of decision-makers in the decision-making process. The lack of consistency in preference relations results in a vague solution. The main goal of this paper is to achieve the consistent intuitionistic multiplicative preference relation using a graphical approach. We have proposed two different characterizations of the consistency for intuitionistic multiplicative preference relation(IMPR). In the first approaches, we propose an algorithm to achieve the consistency of IMPR by using the cycles of various length in a directed graph. The second approach proves isomorphism between the set of IMPRs and the set of asymmetric multiplicative preference relations. That result is explored to use the methodologies developed for asymmetric multiplicative preference relations to the case of IMPRs and achieve the consistency of asymmetric multiplicative preference relation using a directed graph. Sometimes the decision maker may not ...
Incomplete preference relations: An upper bound condition
In decision making, consistency in fuzzy preference relations is associated with the study of transitivity property. While using additive consistency property to complete incomplete preference relations, the preference values found may lie outside the interval [0, 1] or the resultant relation may itself be inconsistent. This paper proposes a method that avoids inconsistency and completes an incomplete preference relation using an upper bound condition. Additionally, the paper extends the upper bound condition for multiplicative reciprocal preference relations. The proposed methods ensure that if (n − 1) preference values are provided by an expert, such that they satisfy the upper bound condition, then the preference relation is completed such that the estimated values lie inside the unit interval [0, 1] in the case of preference relations and [1/9, 9] in the case of multiplicative preference relation. Moreover, the resultant preference relation obtained using the proposed method is transitive.
2014
The experts may have difficulty in expressing all their preferences over alternatives or criteria, and produce the incomplete linguistic preference relation. Consistency plays an important role in estimating unknown values from an incomplete linguistic preference relation. Many methods have been developed to obtain a complete linguistic preference relation based on additive consistency, but some unreasonable values may be produced in the estimation process. To overcome this issue, we propose a new characterisation about multiplicative consistency of the linguistic preference relation, present an algorithm to estimate missing values from an incomplete linguistic preference relation, and establish a decision support system for aiding the experts to complete their linguistic preference relations in a more consistent way. Some examples are also given to illustrate the proposed methods.
International Journal of Computational Intelligence Systems
This paper comprises a new iterative method for multi-person decision making based on multiplicative consistency with incomplete reciprocal preference relations (IRPRs). Additionally, multiplicative transitivity property of reciprocal preference relation (RPR) is used at the first level to estimate the unknown preference values and get the complete preference relation, then it is confirmed to be multiplicative consistent by using transitive closure formula. Following this, expert's weights are evaluated by merging consistency and trust weights. The consistency weights against the experts are evaluated through multiplicative consistency investigation of the preferences given by each expert, while trust weights play the role to measure the level of trust for an expert. The consensus process determines whether the selection procedure should start or not. If it results in negative, the feedback mechanism is used to enhance the consensus degree. At the end, a numerical example is given to demonstrate the efficiency and practicality of the proposed method.
Iterative algorithms for improving consistency of intuitionistic preference relations
Journal of the Operational Research Society, 2014
Consistency of preference relations is an important research topic in decision making with preference information. The existing research about consistency mainly focuses on multiplicative preference relations, fuzzy preference relations and linguistic preference relations. Intuitionistic preference relations, each of their elements is composed of a membership degree, a non-membership degree and a hesitation degree, can better reflect the very imprecision of preferences of decision makers. There has been little research on consistency of intuitionistic preference relations up to now, and thus, it is necessary to pay attention to this issue. In this paper, we first propose an approach to constructing the consistent (or approximate consistent) intuitionistic preference relation from any intuitionistic preference relation. Then we develop a convergent iterative algorithm to improve the consistency of an intuitionistic preference relation. Moreover, we investigate the consistency of intuitionistic preference relations in group decision making situations, and show that if all individual intuitionistic preference relations are consistent, then the collective intuitionistic preference relation is also consistent. Moreover, we develop a convergent iterative algorithm to improve the consistency of all individual intuitionistic preference relations. The practicability and effectiveness of the developed algorithms is verified through two examples.
Interval weight generation approaches for reciprocal relations
Applied Mathematical Modelling, 2014
In this paper, we propose methods to derive interval weight vectors from reciprocal relations for reflecting the inconsistency when decision makers provide preferences over alternatives (or criteria). Several goal programming models are established to minimize the inconsistency based on multiplicative and additive consistency, respectively. Especially, if we obtain a crisp weight vector from a reciprocal relation, then it is consistent. Then, we extend the proposed methods to incomplete reciprocal relations and interval reciprocal relations and develop the corresponding models to derive interval weight vectors. Several examples are also given to compare the developed methods with the existing ones.