This paper explores the idea of exploiting the mobility of data collection points (sinks) for the purpose of increasing the lifetime of a wireless sensor network with energy-constrained nodes. We give a novel linear programming formulation for the joint problems of determining the movement of the sink and the sojourn time at different points in the network that induce the maximum network lifetime. Differently from previous solutions, our objective function maximizes the overall network lifetime (here defined as the time till the first node "dies" because of energy depletion) rather than minimizing the energy consumption at the nodes. For wireless sensor networks with up to 256 nodes our model produces sink movement patterns and sojourn times leading to a network lifetime up to almost five times that obtained with a static sink. Simulation results are performed to determine the distribution of the residual energy at the nodes over time. These results confirm that energy consumption varies with the current sink location, being the nodes more drained those in the proximity of the sink. Furthermore, the proposed solution for computing the sink movement results in a fair balancing of the energy depletion among the network nodes.