Selecting the best task to execute (task allocation problem) is one of the main problems in multi-robot systems. Typical ways to address this problem are based on swarm intelligence and very especially using the so-called response threshold method. In the aforementioned method a robot has a certain probability of executing a task which depends on a task threshold response and a task stimulus. Nevertheless, response threshold method cannot be extended in a natural way to allocate more than two tasks when the theoretical basis is provided by probability theory. Motivated by this fact, this paper leaves the probabilistic approach to the problem and provides a first theoretical framework towards a possibilistic approach. Thus, task allocation problem is addressed using fuzzy Markov chains instead of probabilistic processes. This paper demonstrates that fuzzy Markov chains associated to a task allocation problem can converge to a stationary stage in a finite number of steps. In contrast, the probabilistic processes only can converge asymptotically, i.e. the number of steps is not bounded in general. Moreover, fuzzy Markov chains predicts in a better way the future behavior of the system in the presence of vagueness when measuring distances. The simulations performed confirm the theoretical results and show how the number of steps needed to get a stable state with fuzzy Markov chains is reduced more than 10 times and the system’s behavior prediction can be improved more than a 60% compared to probabilistic approaches.

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