## 1 Introduction  
Sensor nodes are affordable, extremely small sensing devices made possible by advances in wireless communication technology [1]. These sensor nodes can self-organize and establish a wireless sensor network (WSN) once they are deployed. WSNs can be used to detect inaccessible and hazardous locations where traditional sensors are nearly impossible to deploy. Detecting the location is a difficult task, but it is a crucial characteristic in many WSN applications. Locating all of the sensor nodes efficiently to obtain the precise location of an occurrence is a critical challenge. Surveillance, environment monitoring [2], tracking of moving objects, and forest fire detection [3] are just a few of the applications that demand precise location determination. These applications necessitate the stationing of an abundance of sensor nodes in the field, imposing size and cost limits on sensor nodes. Because of these limitations, all current sensor nodes have limited processing and battery power. Furthermore, due to their deployment in inaccessible places, power sources cannot be replaced or regenerated. This necessitates making the most use of sensor node processing and battery power to extend the WSN's overall lifespan. A lot of WSN applications necessitate knowing the exact location where data is sensed. Sensor nodes can be manually placed at known locations or all sensor nodes can be equipped with GPS [4; 5] or other position-finding hardware to find the location of the sensed data. Due to its deployment in difficult areas, manual placement of sensor nodes is not practical, whereas placing a GPS or location-finding system on each node will increase the cost of WSN deployment, making it economically unviable [6]. A variety of alternative strategies have been proposed in the literature [7; 8; 9; 10], in which a restricted amount of location-aware nodes, termed as anchor nodes or beacon nodes, are randomly distributed throughout the network while location-unaware nodes, called target nodes, use anchor nodes to detect their location. Range-based and range-free localization techniques are two types of anchor-node-based localization systems. Range-based approaches are accurate [11], but they necessitate the installation of additional hardware on sensor nodes, making them an expensive solution. Noise and multipath fading have an impact on these as well. Range-free techniques, on the other hand, are less expensive because only connectivity information is needed for location prediction. These approaches are widely used due to their inexpensive cost, although they provide poor localization accuracy when compared to range-based techniques. Localization can be precise in the case of stationary nodes, but it is more complicated in the case of movable nodes. In this research, a modified learning enthusiasm-based teaching-learning-based optimization (mLebTLBO) algorithm [12] is used to deploy a single anchor along with hexagonal projection to project the anchor nodes virtually in six diverse orientations to target the undiscovered nodes. Two of these virtual anchors will be chosen at the time when the targeted node comes inside the span of the anchor node, as a minimum of three anchors is necessary to identify the target node's 2D positions. These three anchor nodes have preliminary awareness of their co-ordinates, which helps in identifying the unknown target node position. The mLebTLBO algorithm is used to assess the effectiveness of the 2D WSN localization problem utilizing a variety of meta-heuristics. The primary issue with teaching-learning-based optimization (TLBO) is that during the teacher or learner phases, all learners have an equal opportunity to gain knowledge. In fact, learners are distinct individuals who are passionate about learning. Higher levels of learning excitement make the learners more concentrated while they are studying, which gives them a fantastic chance to learn from others. Conversely, those who are less enthusiastic in learning have a relatively less likelihood of learning from others.