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In wireless sensor networks, one of the key challenge is to achieve minimum energy consumption in order to maximize network lifetime. In fact, lifetime depends on many parameters: the topology of the sensor network, the data aggregation regime in the network, the channel access schemes, the routing protocols, and the energy model for transmission. In this paper, we tackle the routing challenge for maximum lifetime of the sensor network. We introduce a novel linear programming approach to the maximum lifetime routing problem. To the best of our knowledge, this is the first mathematical programming of the maximum lifetime routing problem that considers the operation modes of the node. Solution of the linear programming gives the upper analytical bound for the network lifetime. In order to illustrate the application of the optimization model, we solved the problem for different parameter settings for an example network.