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In this paper we explore the mobility of a mobile sink in a wireless sensor network (WSN) to prolong the network lifetime. Since the mechanical movement of mobile sink is driven by petrol and/or electricity, the total travel distance of the mobile sink should be bounded. To minimize the data loss during the transition of the mobile sink from its current location to its next location, its moving distance must be restricted. Also, considering the overhead on a routing tree construction at each sojourn location of the mobile sink, it is required that the mobile sink sojourns for at least a certain amount of time at each of its sojourn locations. The distance constrained mobile sink problem in a WSN is to find an optimal sojourn tour for the mobile sink such that the sum of sojourn times in the tour is maximized, subject to the above mentioned constraints. In this paper we first formulate the problem as a mixed integer linear programming (MILP). Due to its NP-hardness, we then devise a novel heuristic for it. We finally conduct extensive experiments by simulations to evaluate the performance of the proposed algorithm in terms of network lifetime. The experimental results demonstrate that the solution delivered by the proposed heuristic is nearly optimal which is comparable with the one by solving the MILP formulation but with much shorter running time.