# Optimizing Rule Weights to Improve FRBS Clustering in Wireless Sensor Networks
## 1 Introduction
Thus, if the algorithm that determines the CH selection runs at a unique point like the BS, this is a centralized method. Conversely, if each node promotes itself as a CH depending on an internal processing that considers the data gathered by that node, the clustering is defined as a distributed method. Thus, in cases where the awareness of the whole network is high, centralized clustering algorithms usually perform significantly better than distributed ones. However, the acquisition of data often requires more complex processes of information interchange and the use of additional hardware like GPS devices in the nodes to obtain their precise location. The way in which centralized and distributed clustering methods select CHs can be achieved by stochastic or heuristic methods or by some kind of artificial intelligence technique like fuzzy control or genetic algorithms, among others. For example, ref. [16] presented a centralized method that uses two heuristic equations to classify nodes: the first is based on coverage and the second on node connectivity. In [17], the authors used a particle swarm optimization (PSO) strategy to select the CHs. In addition, machine learning techniques are widely applied to WSN management, as pointed out in [18]. Fuzzy rule-based systems (FRBS) are the most common approach in these kinds of applications. In [19], the CH candidates were sorted based on a Type-II fuzzy system. In [20], an expert system was used with three variables to choose the best nodes to become CHs. Finally, as can be observed in Figure 1, the BS in a WSN application can be in a fixed location or it can move along a sensing path to cover all the nodes in the network. It must be noted that BS mobility allows more efficient data gathering, because it can be closer to the contributing nodes or CHs. With this approach, they will spend less energy and it also can cover a larger area [21]. However, the visits of the BS to a group of nodes must be carefully synchronized with the instant that they are ready to send data and considering the distance that the BS has to move. As was previously mentioned, fuzzy control systems are usually applied in clustering methods [22]. However, the design of the knowledge base is a demanding task because it depends on the expert knowledge of the research team. In addition, when the number of variables and fuzzy sets are high, the definition of precise rules becomes a challenging problem. This often results in rules that have minimal effect in the developed method, while they waste computational time in the fuzzy engine. Moreover, in the case of applying clustering for WSN lifetime maximization, there are three instants that are usually studied [23]: the instant when the first node depletes its battery (or dies), called first node dies (FND); the \(round\) when half of the nodes are dead, or half of the nodes have died (HND); and the moment when the last alive node dies, or last node dies (LND). Thus, when a clustering method tries to maximize those parameters, one may be improved but the other two parameters could be negatively affected. This is a common problem of multi-objective optimization in WSNs [24] (the Pareto front), which degrades WSN performance. Consequently, a research team should make certain trade-offs according to the application they are studying. In particular, they should choose which of the parameters (FND, HND, or LND) is more important for the application. Therefore, in this paper, we present the optimization of the knowledge base of a Type-1 fuzzy control system adapted to the particular performance metric (FND, HND, or LND) that is expected to be improved. The proposed clustering method was applied in a WSN that was deployed in a rectangular area whose \(skip\) value was equal to 1. The clustering method employed was centralized with a BS in a fixed location for the entire duration of the application.