We propose a novel method to generate sufficiently unpredictable routes by varying the arrival time at each customer, while minimizing transportation costs. By removing the previous arrival time slots at each customer from the solution space, the problem becomes a Vehicle Routing Problem with Multiple Time Windows (VRPMTW) in which every customer has a set of time windows in which it is still available for service. Because of the reformulation into a VRPMTW with a rolling horizon, our approach is easier, more efficient and more powerful than existing methods. Since waiting times are not allowed a new method is proposed to check if a route is time window feasible. To allow time window violations during the local search, four different penalty methods are proposed and compared in terms of solution quality and computational time. The routing problem is solved using an iterated granular tabu search which finds new best-known solutions for all benchmark instances from the literature. The proposed method reduces average distance with 28% and computational time with 91%. A case study is performed on data from a Cash in Transit company that transfers valuable goods to banks and ATMs. For security and legal regulations they have to use varying routes and computational experiments show the savings potential of the proposed solution approach and quantify the trade-off between arrival time diversification and transportation costs.