For many companies it is important to protect their physical and intellectual property in an efficient and economically viable manner. Thus, specialized security companies are delegated to guard private and public property. These companies have to control a typically large number of buildings, which is usually done by teams of security guards patrolling different sets of buildings. Each building has to be visited several times within given time windows and tours to patrol these buildings are

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... nned over a certain number of periods (days). This problem is regarded as the Districting and Routing Problem for Security Control. Investigations have shown that small time window violations do not really matter much in practice but can drastically improve solution quality. When softening time windows of the original problem, a new subproblem arises where the minimum time window penalty for a given set of districts has to be found for each considered candidate route: What are optimal times for the individual visits of objects that minimize the overall penalty for time window violations? We call this Optimal Arrival Time Problem. In this paper, we investigate this subproblem in particular and first give an exact solution approach based on linear programming. As this method is quite time-consuming we further propose a heuristic approach based on greedy methods in combination with dynamic programming. The whole mechanism is embedded in a large neighborhood search (LNS) to seek for solutions having minimum time window violations. Results show that using the proposed heuristic method for determining almost optimal starting times is much faster, allowing substantially more LNS iterations yielding in the end better overall solutions.