$\mathbb{T}-$Relative Fuzzy Linear Programming for $\mathbb{T}-$Relative Fuzzy Target Coverage Problems

Abstract

Optimal set covering problems are commonplace in communication, remote sensing, logistics, image processing, and network fields [3]. Thus, studies on determining optimal covering sets (sensors) of points (targets) in a region have emerged recently. One characteristic of these studies is the consideration of cases where a target is considered fully covered when it falls within a coverage area ("Boolean" coverage). Consequently, optimality solutions/methods/algorithms founded on this coverage scheme are usually too restrictive and (or) precise and so are not suitable for many complex and real life situations, which are most times plagued with ambiguity, vagueness, imprecision and approximate membership of points and (or) covering sets. Fuzzy structures have proven to be suitable for the representation and analysis of such complex systems with many successful applications. Although fuzzy sets generalizes a set, a more recent generalization for both and its related concepts is the Relative fuzzy set [1] which gives a dynamic fuzzy representation to sets. $\mathbb{T}-$Relative Fuzzy fixed points results of $\mathbb{T}-$Relative fuzzy maps were studied in [5] and recently, the concept of $\mathbb{T}-$Relative fuzzy linear programming [6] was introduced as a generalization of fuzzy linear programming. The results were applied to generalize the Boolean set based covering problems in literature to a $\mathbb{T}-$Relative fuzzy Boolean coverage one. Although, Shan et al. in [15] and others [16] - [21] have given a probabilistic coverage consideration but this lacks subjectivity in representing vagueness and imprecision inherent in most systems. In this present article the Linear Programming (LP) formulation of “A Computational Physics-based Algorithm for Target Coverage Problems" by Jordan Barry and Christopher Thron is generalized by considering a fuzzy and relative fuzzy target coverage instead of the crisp set Boolean coverage. Also we introduce the Fuzzy Linear Programming (FLP) and the $\mathbb{T}-$Relative Fuzzy Linear Programming (RFLP) for the set coverage problem which allows for ascertaining dynamic optimality with aspiration levels.