Indistinguishability Operators Applied to Task Allocation Problems in Multi-Agent Systems

Abstract

Featured Application: Indistinguishability can provide a new family of response functions that could be applied to implement a new generation of swarm-like methods to carry out missions, like for example, allocate tasks to a set of robots in different environments. Actually, this paper provides a first implementation of this kind of systems.Abstract: In this paper we show an application of indistinguishability operators to model response functions. Such functions are used in the mathematical modeling of the task allocation problem in multi-agent systems when the stimulus, perceived by the agent, to perform a task is assessed by means of the response threshold model. In particular, we propose this kind of operators to represent a response function when the stimulus only depends on the distance between the agent and a determined task, since we prove that two celebrated response functions used in the literature can be reproduced by appropriate indistinguishability operators when the stimulus only depends on the distance to each task that must be carried out. Despite the fact there is currently no systematic method to generate response functions, this paper provides, for the first time, a theoretical foundation to generate them and study their properties. To validate the theoretical results, the aforementioned indistinguishability operators have been used to simulate, with MATLAB, the allocation of a set of tasks in a multi-robot system with fuzzy Markov chains.

Indistinguishability Operators via Yager t-norms and Their Applications to Swarm Multi-Agent Task Allocation

“…In contrast, the YPRFs, proposed in this paper, are an excellent candidate to model the transitions when agent's restrictions are under consideration in the aforementioned scenarios. Moreover, in all cases, we will show that fuzzy Markov chains outperform their probabilistic counterparts in the spirit of [11]. Thus, to our best knowledge, this paper states for the first time that indistinguishability operators are an appropriate mathematical tool to deal with the agents restrictions in task allocation problems, and shows how the modelling based on their use and fuzzy Markov chains outperform the classical swarm-like approaches.…”

“…Motivated by these handicaps, a new possibilistic theoretical formalism for implementing the RTM algorithms based on possibilistic response functions and the use of fuzzy Markov chains was proposed in [8]. Later on, the relationship between the possibilistic response functions and indistinguishability operators was explored in [11].…”

“…In the light of the advantages provided by the use of possibility theory in RTMs applied to task allocation problems, the relationship between transition possibilities given by (2) and indistinguishability operators was studied in [11]. Concretely, it was showed that indistinguishability operators can be an appropriate mathematical tool to model transition possibilities (possibilistic response functions) and, in addition, a formal method to generate possibilistic response functions was stated in the same reference.…”

“…We refer the reader to [10] for the basics about fuzzy Markov chains. Furthermore, in [11], it was proved the suitability of indistinguishability operators as mathematical tool to model possibilistic response functions (see [12] for a detailed treatment of indistinguishability operators). Thanks to indistinguishability operators, a formal method to generate possibilistic response functions was introduced in the same reference.…”

“…All indistinguishability operators considered in [11] present a common characteristic, which is that they always take values strictly greater than zero. Since an indistinguishability operator is identified with a response function which, at the same time, provides the possibility transition, the fact that it takes values strictly greater than zero can be interpreted as follows: an agent is always able to transit from its current state (or task) to any other one.…”