Unmanned Aerial Vehicles (UAVs) are being extensively deployed in numerous Livestock Management applications such as cattle disease diagnosis, counting and behavioral monitoring from videos and images captured by drones. The paper focuses on one increasingly important family of applications, UAV-assisted cattle monitoring applications where the objective is to remotely acquire some health state information from IoT nodes attached to the herding cattle. Such livestock data acquisition applications have many challenges. One of these challenges, which is the focus of this paper, is the problem that the target cattle position may not be known precisely, and might be defined with a large area. To address this issue, we design a formulation of this UAV-cattle search path problem as a mathematical optimization problem and show how it can be derived from other well-known formulation and related literature. A Mixed-Integer linear Programming (MILP) formulation is introduced to minimize the expected search time while covering all the search area to efficiently locate the animal. This formulation exploits a cattle position probability distribution map. The results show that the suggested approach yields excellent results using the existing MILP solvers.