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

Abstract

In this paper, we propose a family of indistinguishability operators, that we have called Yager Possibilitic Response Functions (YPRFs for short), as an appropriate tool for allocating tasks to a collective of agents. In order to select the best agent to carry out each task, we have used the so-called response threshold method, where each agent decides the next task to perform following a probabilistic Markov process and, in addition, involves a response function which models how appropriate the task is for the agent. In previous works, we developed a new response threshold method which incorporates the use of indistinguishability operators as response functions and possibility theory instead of probability, for task allocation from a very general perspective without taking into account the specific characteristics of the agents except their limitations to carry out a task. Such an allocation is modelled by means of possibilistic, instead of probabilisitic, Markov chains. We show that possibilistic Markov chains outperform its probabilistic counterparts for the aforementioned propose. All the indistinguishability operators considered in previous papers were not able to take into account the agents’ restrictions for moving from a task to another one, or equivalently to carry out a task instead of another one. In order to avoid this handicap, we introduce a new kind of response functions, YPRFs, which are modelled by means of indistinguishability operators obtained via Yager t-norms. This new type of response functions drops to zero when an agent, due to its limitations, is not able to execute a task and, therefore, is able to model a generic multi-agent system with restrictions. The performed simulation, under Matlab, allows us to compare the results obtained using the new YPRFs with those obtained applying celebrated response functions also generated via indistinguishability operators (that we call Original Possibilitic Response Functions, OPRFs for short). Moreover, the results confirm that the YPRFs are able to take into account agent’s restrictions while the OPRFs are not able. Finally, in the light of the experimental results, we can confirm that those systems modelled.

Modular Quasi-Pseudo Metrics and the Aggregation Problem

“…In light of the preceding notions, we can recall the aforementioned characterization, the proof of which can be found in [18]. F(0 n ) = 0 and F is monotone and transforms n-triangular triplets into a 1-triangular triplet.…”

“…Motivated by the growing interest in the distance aggregation problem, a characterization of those functions that merge a collection of modular (pseudo-)metrics has been provided in [18]. With the aim of recalling such a result, we introduce the notion of a modular (pseudo-)metric aggregation function.…”

“…On account of [18], given n ∈ N, a function F : [0, +∞] n → [0, +∞] is a modular (pseudo-)metric aggregation function provided that, for each collection of modular (pseudo-)metrics {w i } n i=1 defined on X, the function F • w is a modular (pseudo-)metric on X, where w :]0, +∞[×X × X → [0, +∞] n is given by w(λ, x, y) = (w 1 (λ, x, y), . .…”

“…According to [18], in the case of modular metrics, the preceding characterization can be stated as follows: Theorem 6. Let n ∈ N, and let F : [0, +∞] n → [0, +∞] be a function.…”

“…Of course, every modular metric aggregation function is always a modular pseudometric aggregation function (this fact was stated as Corollary 1 in [18]). The following instance, which can be found in Example 1 in [18], shows that the reciprocal implication does not hold true. Consider the function F : [0, +∞] n → [0, +∞] defined by F(a) = 0 for all a ∈ [0, +∞] n .…”