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This paper considers the deterministic vehicle routing problem with service time requirements for delivery. Service requests are also available at different times known at the initial time of route planning. This paper presents an approach based on the generation of service sequences (routes) using randomness. Both the single-vehicle and the multiple-vehicle cases are studied. Our approach is validated using random-generated data and compared against the optimal solution obtained by mathematical programming for small-sized instances, as well as against known lower bounds for medium to large-sized instances. Results show that our approach is competitive with reference to the value of the objective function, and requires less computational time in comparison with the exact resolution procedure.