In the last years, the interest in the notion of fuzzy metric has been growing in such a way that many works have focused their efforts on the study of their topological properties and their applications to Engineering problems. However, the applicability of fuzzy metrics is limited due to lack of examples in the literature. Motivated, on the one hand, by these facts and, on the other hand, by the fact that most of the instances of fuzzy metrics in the literature are constructed from classical metrics, in this paper we introduce two new techniques which allow us to construct systematically fuzzy metrics from metrics in such a way that the celebrated classical method for constructing indistinguishability operators from metrics is retrieved as a particular case. Hence, we construct strong fuzzy metrics from a given classical one considering continuous Archimedean t-norms and the pseudo-inverse of their additive generators acting on the metric modified by a positive real function. Moreover, we extend this technique tackling the particular case of the minimum t-norm, which is continuous but non-Archimedean. In such a construction, two non-negative real functions are now involved in order to modify the classical metric and one of them must be superadditive. In this case, the fuzzy metric obtained is not strong in general. Furthermore, the new methods are illustrated by means of different examples which, in addition, show that some celebrated examples of fuzzy metrics can be retrieved as a particular case through them. Finally, in the light of the developed theory, an open problem about strong fuzzy metrics is solved completing the partial solutions that can be found in the literature.

It is a well-known fact that the topology induced by a fuzzy metric is metrizable. Nevertheless, the problem of how to obtain a classical metric from a fuzzy one in such a way that both induce the same topology is not solved completely. A new method to construct a classical metric from a fuzzy metric, whenever it is defined by means of an Archimedean t-norm, has recently been introduced in the literature. Motivated by this fact, we focus our efforts on such a method in this paper. We prove that the topology induced by a given fuzzy metric M and the topology induced by the metric constructed from M by means of such a method coincide. Besides, we prove that the completeness of the fuzzy metric space is equivalent to the completeness of the associated classical metric obtained by the aforementioned method. Moreover, such results are applied to obtain fuzzy versions of two well-known classical fixed point theorems in metric spaces, one due to Matkowski and the other one proved by Meir and Keeler. Although such theorems have already been adapted to the fuzzy context in the literature, we show an inconvenience on their applicability which motivates the introduction of these two new fuzzy versions.

Response-threshold methods stand out among the different developed swarm-like methodologies that address the task allocation problem, which must be faced in multi-robot systems in order to assign to each robot the best task to perform at each instant of time. In many real missions the tasks have associated deadlines. However, the literature only contains a few swarm methodologies, and thus response-threshold methods, tackling tasks with deadlines. Motivated by this fact, in this paper, we propose a new task allocation strategy inspired by response-threshold methods which deals with tasks with time deadlines, models stimuli using fuzzy sets and, in addition, in which each robot makes the decision about the best task to perform through the celebrated Bellman-Zadeh fuzzy optimization technique. An extensive number of simulations have been carried out in order to evaluate the quantitative performance of the swarm system based on the new approach. The results confirm that the proposed mathematical approach is able to model the evolution of the system when tasks with deadlines are under consideration. We have also observed competitive performance on a fleet of real robots, which corroborates the results derived from the simulations.