Most sensor devices are equipped with local batteries, placing a limitation on available energy, assuming that replacement of batteries is not feasible. This constraint limits the operation time of sensor networks. In many scenarios, sensors are operating in remote or dangerous areas, and it is impractical to replace or recharge the battery of the sensors. In some scenarios, such as machine monitoring or medical monitoring, it is more convenient to install new sensors than to replace the battery of the sensor nodes. The focus of this article is to design routing methods to achieve the maximum operation time of sensor networks (the number of rounds to transmit data) under the constraint of battery sources. Therefore, the main contribution of this research is not to change any characteristic of the network infrastructure, but to define the interactions between nodes to achieve better energy efficiency. K E Y W O R D S battery sources, computing, multihop routing, operation time, power transmission, sensor networks 1 INTRODUCTION Some applications, for example in military, require sensors to be small in size and have short transmission ranges to reduce the chances of being detected. These requirements cause further constraints in CPU speed, memory, and battery lifetime. 1 As the lifetime of a sensor node is closely related to its battery lifetime, reducing the energy usage of a sensor by two often results in a reduction of sensor installation cost by 50%. This means that all aspects related to the energy usage must be designed very efficiently. In this article, the energy efficiency of wireless ad hoc sensor networks (WASNs), in which traffic can be transmitted from any source to any destination and there is no base station, is examined. Section 2 first studies the energy dissipation model inside a sensor. Section 3 proves the complexity theory of the selection of energy efficient paths and then proposes heuristic algorithms for the problem. Also, a broadcasting scheme to eliminate the overhearing energy of the neighboring nodes is analyzed. Finally, simulation results are presented. In the results, three heuristic routing methods are studied: shortest path (SP), SP including neighboring nodes (SP_N), and SP of remaining energy (SP_RE). As the energy dissipation by neighboring nodes is quite significant, a prebroadcast method is used to eliminate the energy dissipation. The performance of SP and SP_RE with the prebroadcast scheme is then examined and compared to that of SP and SP_RE without the broadcast scheme. This is shown in Figure 1. 2 ENERGY DISSIPATION SOURCES INSIDE A SENSOR Sensor nodes dissipate energy in different ways when they are in different modes: active, idle, or sleep mode. 2,3 When a node is in sleep mode, it turns off its radio, and the node cannot hear any action of other nodes in the network. The energy dissipation in this mode is very low which is often known as background energy (P sleep). This mode also includes some small energy dissipation for sensing data or an event. In idle mode, the no...