In this paper a linear programing approach has been implemented for finding the energy efficient path for prolonging the network lifetime of sensor monitoring systems. For a given set of sensors and spots, a sensor can only monitor a single spot at a time and a spot can be the monitoring range of k-sensors. While using digital signatures on sensor nodes, a linear programming framework is designed to determine the system’s maximum life time. Digital signatures provide node authentication and non-repudiation service in wireless sensor network. Non-repudiation is supported by different types of digital signature schemes. Such schemes have large communication and /or computation cost, which is traditionally infeasible for wireless sensor network. The proposed solution addresses such issue by using a scheduling mechanism. The problem is divided into three steps: (i) using a linear programming approach, determine the system’s maximum life-time and make an assignment matrix. (ii) The assignment matrix is factored into a series of scheduling matrices. (iii) On the basis of schedule matrices that indicate the group of active sensors determine the optimal path of data flow from node-i to node-j.