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In this paper, we consider a two-hop decode-and-forward relay network where relay nodes have heterogeneous channel conditions. We propose a new fair and minimal power allocation scheme where a relay node is selected to forward packets based on thresholds for channel conditions of relay nodes and the selected relay node uses an adaptive power control to meet a given data rate requirement. The individual threshold values of relay nodes are determined based on the following three criteria: i) the average amounts of power consumption at relay nodes are all equal, ii) given outage probability and data rate requirements are guaranteed to support quality of service (QoS), and iii) the average total power consumption at all relay nodes is minimized. To determine threshold values for relay nodes we first mathematically model the proposed scheme and derive the average power consumption of each relay node. We then formulate an optimization problem to determine the threshold values that satisfy the above-mentioned three criteria. To solve the optimization problem, we rigorously analyze the performance of the proposed scheme, and show that the threshold values can be uniquely determined. Numerical and simulation results are provided to demonstrate various properties and the benefits of the proposed scheme.