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In this paper, we consider the problem of aggregating a collection of transitive fuzzy binary relations in such a way that the aggregation process preserves transitivity. Specifically, we focus our efforts on the characterization of those functions that aggregate a collection of fuzzy binary relations which are transitive with respect to a collection of t-norms preserving the transitivity. We characterize them in terms of triangular triplets. Further, the relationship between triangular triplets, the monotonicity of the aggregation function and an appropriate dominance notion is explicitly stated. Special attention is paid to a few classes of transitive fuzzy binary relations that are relevant in the literature. Concretely, we describe, in terms of triangular triplets, those functions that aggregate a collection of fuzzy pre-orders, fuzzy partial orders, relaxed indistinguishability relations, indistinguishability relations and equalities. A surprising relationship between functions that aggregate transitive fuzzy relations into a T M -transitive fuzzy relation and those that aggregate relaxed indistinguishability relations is shown. A few consequences of the new results are also provided for those cases in which all the t-norms of the given collection are the same. Some celebrated results are retrieved as a particular case from the exposed theory.

The study of consistency in decision-making models is of great importance. Our proposal deals with a special type of extended aggregation functions to guarantee consistency that is based on T-S consistency of linearly ordered fuzzy preference relations (fuzzy ranking for preferences). Necessary conditions to ensure consistency in the aggregation process are provided. Moreover, sufficient conditions to guarantee the existence of extended aggregation functions verifying T-S consistency are presented. Some examples of functions of this type are also given. The equivalence between T-S consistency and the additive consistency for fuzzy preference relations is established.

In many fields of applied sciences, the aggregation of numerical values, in order to get a final one which allows to make a decision, plays a central role. Many times these numerical values represe...

In this contribution we address our attention on the aggregation of partial T-indistinguishability operators (relations) and partial pseudo-metrics. A characterization of those functions that allow to merge a collection of partial T–indistinguishability operators into a new one was provided by Calvo et al. in [10] by means of ( T , T M ) -tuples, but here we present another characterization in terms of ( + , max ⁡ ) -tuples. Also, we analyze the aggregation of a collection ( E i ) i = 1 n of partial T i -indistinguishability operators. Moreover, we provide that a generalized inter–exchange composition functions condition is a sufficient condition to guarantee that a function merges partial T i -indistinguishability operators into a single one. In addition, we give different expressions of those aggregation functions that are object of our study, most of them are defined by means of the additive generators of the corresponding t-norms and another particular function. We see that the functions, that merge partial S-pseudo-metrics into a new one, are related to the functions that aggregate partial pseudo-metrics. Finally, we show the relation between the functions, that merge partial T-indistinguishability operators and the functions that preserve the partial T ⁎ -pseudo-metrics in the aggregation process.

In 1982, E. Trillas introduced the notion of indistinguishability operator with the main aim of fuzzifying the crisp notion of equivalence relation. In the study of such a class of operators, an outstanding property must be stressed. Concretely, there exists a duality relationship between indistinguishability operators and metrics. The aforesaid relationship was deeply studied by several authors that introduced a few techniques to generate metrics from indistinguishability operators and vice-versa. In the last years a new generalization of the metric notion has been introduced in the literature with the purpose of developing mathematical tools for quantitative models in Computer Science and Artificial Intelligence. The aforesaid generalized metrics are known as relaxed metrics. The main purpose of the present paper is to explore the possibility of making explicit a duality relationship between indistinguishability operators and relaxed metrics in such a way that the aforementioned classical techniques to generate both concepts, one from the other, can be extended to the new framework., This is a preprint version of the paper available from https://link.springer.com/article/10.1007/s00500-018-03675-9. This work is also supported by project PGC2018-095709-B-C21 (MCIU/AEI/FEDER, UE), and PROCOE/4/2017 (Govern Balear, 50% P.O. FEDER 2014-2020 Illes Balears).

In this paper we focus our attention on exploring the aggregation of partial T-indistinguishability operators (relations). Concretely we characterize, by means of (T-\(T_{\min }\))-tuples, those functions that allow to merge a collection of partial T-indistinguishability operators into a single one. Moreover, we show that monotony is a necessary condition to ensure that a function aggregates partial T-indistinguishability operators into a new one. We also provide that an inter-exchange composition function condition is a sufficient condition to guarantee that a function aggregates partial T-indistinguishability operators. Finally, examples of this type of functions are also given.

This paper addresses one of the main problems to solve in a multi-robot system, allocating tasks to a set of robots (multi-robot task allocation-MRTA). Among all the approaches proposed in the literature to face up MRTA problem, this paper is focused on swarm-like methods called response threshold algorithms. The task allocation algorithms inspired on response threshold are based on probabilistic Markov chains. In the MRTA problem literature, possibilistic Markov chains have proved to outperform the probabilistic Markov chains when a Max-Min algebra is considered for matrix composition. In this paper we analyze the system behavior when a more general algebra than the Max-Min one is taken for matrix composition. Concretely, we consider the algebra \(([0,1], S_{M},T)\), where \(S_{M}\) denotes the maximum t-conorm and T stands for any t-norm. The performed experiments show how only some well-known t-norms are suitable to allocate tasks and how the possibility transition function parameters are related to the used t-norm.