## 3 Problem Solution  
A question is worth answering, before presenting the problem solution: what is expected from an effective network lifetime optimization process? According to the guidelines reported in [21], one must strike a balance between the minimization of the wasted energy, which is the unused residual energy at the network death moment, and the minimization of the energy spent to report the sensory information to the sink node. The authors of [21] complement their guidelines by indicating that this can be accomplished by exploring channel state information (CSI) and residual energy information (REI). However, one must recall that estimating the CSI would render more complex devices and additional energy consumption. In the approach described in what follows, non-real-time REI is explored, and the CSI is not, conferring to it a simpler implementation. In the optimization strategy proposed in this paper, the energy balance is achieved by dynamically controlling the communication activity levels of the sensor nodes in a constant attempt to trade two goals: (i) make all sensor nodes consume the same amount of energy during communication, thus making them die together and yield the least possible wasted energy; (ii) prioritize longer lifetimes by relaxing on the equalization of energy consumptions to allow for bursty transmissions followed by silent periods. When the equalization of energy consumptions is privileged, the lifetime improvement is penalized. However, the communication with the sink node is more frequent, thus avoiding the waste of bandwidth caused by the bursty transmission mode when there is not enough data to send to the sink node at the very moment that it is allowed to transmit, yet avoiding the storage of a high amount of data until the next burst transmission is permitted in the bursty operation mode. Figure 4: Pictorial representation of the maximum energy consumptions of five sensor nodes during \(100\) frames, for \(\rho=0.999\) (**a**) and \(\rho=0.98\) (**b**), for \(B_{\min}=0.1\) and \(B_{\max}=5\). Then, in each optimization event, the objectives are to minimize the residual energies so that the unused energy is minimized and to minimize what we call the discounted residual energies. These discounted energies are formed by subtracting the potential maximum energy consumptions of the sensor nodes from their residual energies. As a consequence, higher residual energies combined with lower potential consumptions (better channels) will force the optimization to assign high activity levels, i.e., the corresponding sensor nodes will be granted burst-like transmissions. On the other hand, low residual energies combined with high potential consumptions (bad channels) will be associated with very low activity levels, i.e., the corresponding sensor nodes will be practically inactive while this situation remains. Mathematically, in the \(k\)-th optimization event, a bi-criterion optimization problem is established in which the objective function is formed by the weighted sum of two objective functions, that is,  
\[f_{0}\left[\mathbf{s}_{f}(k)\right]=w_{1}\max\{\mathbf{s}_{f}(k)\}+w_{2}\max \{\mathbf{s}_{f}(k)-\mathbf{b}_{f}(k)\},f=1,2,\ldots,F. \tag{3}\]  
The variables of the optimization problem are the columns \(\mathbf{s}_{f}(k)\) of the residual energy matrix \(\mathbf{S}(k)\) and the implicit activity levels, which are the columns \(\mathbf{x}_{f}(k)\) of the matrix \(\mathbf{X}(k)\). The roles of the weights \(w_{1}\geq 0\) and \(w_{2}\geq 0\) are as follows: when \(w_{1}/w_{2}>1\), the optimization acts in favor of equalized residual energies at the cost of reduced lifetime improvements; on the other hand, when \(w_{2}/w_{1}>1\), the optimization tends to assign high activity levels to nodes with better channels and high residual energies, sometimes even disabling those under bad channels and having low residual energies, thus bringing the possibility of larger lifetime improvements.