# Mobile Data-Mule Optimal Path Planning for Wireless Sensor Networks
## 1 Introduction  
Wireless Sensor Networks (WSNs) [1] have been used in sensing automation tasks combining sensing, data processing, and communication capabilities in a unified platform. The data collected by each node are further relayed via the network towards the base station, which demands the utilization of several relay nodes and the integration of multi-hopping routing protocols [2, 3]. In [4], an alternative to static-relay nodes is suggested, using Mobile Elements (MoEls) as data MULEs (Mobile Ubiquitous Local area network Extensions), which travel within the sparse deployment of the sensor nodes and collect their stored data. This data-collection method by MoEls within a WSN is known as \"data harvesting\" and is subject to the hardware limitations of sensor nodes, such as the nodes' limited storage capacity and the requirement for increased node lifetime. Since the frequency of data generation in sensor nodes depends on the event occurrence frequency, it can result in buffer overflow, whereas on the other hand, the adoption of a multi-hopping routing protocol may result in high energy consumption rates. The aforementioned work was extended in [5], where the robotic data mules' routes were assigned for gathering data in an efficient manner. Therefore, an MoEl motion planning is extended to the Mobile Element Scheduling (MES) problem [6]. This is defined as the problem of scheduling the visits of an MoEl to sensor nodes with the objective being the elimination of data loss due to the sensors' hardware limitations. In [7, 8], the examined problem is defined as the Message Ferrying scheme, where each node communicates only with the message ferries, thus avoiding any long-distance communication and increasing the lifetime for the WSN. Furthermore, this lifetime is further extended as the MoEls can easily be recharged periodically at a base station, as opposed to sensor nodes, which do not possess energy replenishment capabilities. In addition, in the case of disconnected networks, the use of MoEls may be the only viablesolution to ensure the network's connectivity. In this case, data mules are incorporated in a WSN in order to extend the functionality within the network, since the mobility of these systems accounts for the filling of the connectivity gaps inside the monitored area. Research in data harvesting within a WSN using an MoEl, includes routing optimality and fair resource management. These requirements impose the adoption of adaptive sink mobility schemes [9; 10], and path planning optimization problems are formulated to ensure reduced data delivery latency and adjustment of the sensor nodes' transmission ranges, so as the total sensor node energy consumption is minimized [11; 12]. In order to extract the locations that need to be visited by MoEls, the Traveling Salesman Problem (TSP) [5; 13; 14] is employed. Similar optimization approaches used in path planning are the Orienteering Problem (OP) [15; 16] and the Vehicle Routing Problem (VRP) [17; 18]. Other relevant works that focus on the MoEls' scheduling for data harvesting in a WSN formulate the problem as a Mixed Integer Linear Programming (MILP) optimization problem [6; 19; 20], or follow greedy-like approaches [21]. If a MoEl is used for data harvesting purposes, it is expected that the nodes, which store data with a higher rate or collect critical measurements, have to be visited more frequently by the MoEl. The procedure used in this work eliminates the loss of the sensors' stored data or measurements due to the sensors' limited storage capacity. In [22], the time-window constraint is introduced in the formulation of OP, which is also the case as the MoEl visits a number of locations, albeit within specific time spaces. Furthermore, in this formulation, there is no constraint that restricts the total duration of the route as a time budget \(B\). The maximized profit function indirectly measures only the visits to the static locations, relying on a heuristic algorithm that takes into account neither the route length nor its duration.