This paper introduces a new type of Security Games (SG) played on a plane with targets moving along predefined straight line trajectories and its respective Mixed Integer Linear Programming (MILP) formulation. Three approaches for solving the game are proposed and experimentally evaluated: application of an MILP solver to finding exact solutions for small-size games, MILP-based extension of recently published zero-sum SG approach to the case of generalsum games for finding approximate solutions of medium-size games, and the use of Memetic Algorithm (MA) for mediumsize and large-size game instances, which are beyond MILP’s scalability. Utilization of MA is, to the best of our knowledge, a new idea in the field of SG. The novelty of proposed solution lies specifically in efficient chromosome-based game encoding and dedicated local improvement heuristics. In vast majority of test cases with known equilibrium profiles, the method leads to optimal solutions with high stability and approximately linear time scalability. Another advantage is an iteration-based construction of the system, which makes the approach essentially an anytime method. This property is of paramount importance in case of restrictive time limits, which could hinder the possibility of calculating an exact solution. On a general note, we believe that MA-based methods may offer a viable alternative to MILP solvers for complex games that require application of approximate solving methods.