## 1 Introduction
If the deployment problem considers a single-objective focus, the most relevant performance metric is selected as the optimization objective, while the remaining metrics are treated as constraints. This single-objective approach may be biased in real-world applications because it is supposed the importance of one metric to the detriment of others. A Multi-Objective (MO) approach means a more realistic focus, where several conflicting objectives are simultaneously optimized [12]. On this basis, this paper studies how to efficiently deploy energy-harvesting RNs in previously-established static T-WSNs by following an ST network model. This problem is known as the Relay Node Placement Problem (RNPP). To this end, the authors apply a set of MO metaheuristics to solve two formulations of the same RNPP with a different number of objectives: two and three. The goal of this analysis is to identify potential robust MO techniques, which could be recommended as a general solving-method for problems as the one considered in this work. To the best of our knowledge, this is the first work which performs this type of study for a WSN deployment problem. The following tasks are performed through this study:* A formal statement is provided for the two formulations of the RNPP addressed. The first formulation optimizes Average Energy Cost (AEC) and Average Sensitivity Area (ASA). The second formulation optimizes the two previous objectives and Network Reliability (NR). * The authors apply eight MO metaheuristics specially adapted for solving the problem addressed. The algorithms are from the three main groups in the field [13]: Evolutionary Algorithms (EAs), Swarm Intelligence Algorithms (SIAs), and Trajectory Algorithms (TAs). Concretely, the authors consider three EAs: Non-dominated Sorting Genetic Algorithm II (NSGA-II) [14], Strength Pareto Evolutionary Algorithm 2 (SPEA2) [15], and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) [16]. Three SIAs: Multi-Objective Artificial Bee Colony (MO-ABC) [17], Multi-Objective Firefly Algorithm (MO-FA) [18], and Multi-Objective Gravitational Search Algorithm (MO-GSA) [19]. A TA: Multi-Objective Variable Neighbourhood Search Algorithm (MO-VNS) [20]. A modified approach of MO-VNS and called MO-VNS* is also considered, as will be discussed in Section 5. * The previously introduced MO metaheuristics are applied for solving the two formulations of the RNPP while optimizing a freely available dataset proposed by ourselves in [21]. This is because standard data sets do not exist for the problem definition. Instead, authors in the recent scientific literature consider non-public or randomly generated data sets. Thus, the results obtained in this paper could be replicated or improved by other authors in future works. * The results obtained are analysed through an accepted statistical methodology to determine if there is any metaheuristic which provides a significantly better performance for each formulation. As a result, we could conclude if there is any metaheuristic which provides a robust performance independently of the problem complexity and the number of objectives. The rest of this paper is structured as follows. Section 2 discusses the related work. Section 3 describes the WSN model considered. Section 4 defines the two optimization problems. Section 5 discusses the MO metaheuristics considered. Section 6 includes the solving strategy. Section 7 exposes the experimental results. Finally, conclusions and future lines of research are left for Section 8.