## 6 Conclusions and Recommendation  
We have jointly optimized three variables namely; transmission rate, transmission power and transmission delay needed by the SUs to transmit in a CRSN amidst a number of constraints including interference, BER, boundary constraints on the variables, channel power and rate constraints. The resulting channel selection problem is a multi-objective mixed integer non-linear programming  
Figure 5: Effect of network size and packet size on the total interference.  
Figure 6: The effect of increasing network size and packet size, on the total Bit Error Rate (BER) of the network.  
problem. The resulting problem has been solved using a branch-and-cut polyhedral approach using MATLAB together with the state-of-the-art BARON software. We have analyzed the optimal values of the problem i.e., the total transmission rate, total transmission power and total transmission delay, with changing values of packet size, network size, interference and BER. Our set up also puts into consideration the effect of shadowing during transmission.  
We have observed that in general, an increase in network size and packet size leads to an increase in total rate, total power, total transmission delay, total interference and total BER. It is worth noting that we do not compare with any other work because other scholars have only optimized one or two variables but not all the three as we have done in this study.  
For all the other cases, the rate, delay and power generally increase with increasing size of network. It has also been observed that an increase in packet size generally leads to an increase in total rate, power and delay. The choice of cases 1, 2, 3 and 4 generally leads to different optimal values, albeit in the same direction. It is therefore the choice of the system manager/user to decide the choice of case, depending on where more weight is needed.  
For future work, the authors are already trying to devise a heuristic technique to solve the problem for a larger set of parameter values and network sizes. In addition, further modifications to the studied problem are being considered to incorporate ideas like SU prioritization, more constraints, etc.  
Conceptualization: B.S.M., F.M. and T.P.; methodology: B.S.M. and F.M.; software, B.S.M., F.M. and T.P.; validation, B.S.M., F.M. and J.S.O.; formal analysis, B.S.M., F.M. and T.P.; investigation, B.S.M. and F.M.; resources, T.P.; data curation, B.S.M. and F.M.; writing--original draft preparation, B.S.M., J.S.O. and F.M.; writing--review and editing, B.S.M., F.M., J.S.O. and T.P.; supervision, T.P. and J.S.O.; project administration, T.P.; funding acquisition, T.P. All authors have read and agreed to the published version of the manuscript.  
This research was supported by Hunan province science and technology project funds (2018 TP1036), Science and Technology Planning Project of Guangdong Province of China (2017B010111005) and partly by the College of Computing and Information Sciences, Makerere University. We would like to also acknowledge the financial support of NORAD (Agreement Number UGA-13/0018). The authors would like to acknowledge the three anonymous reviewers for their extremely valuable inputs. The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} CRSN & Cognitive Radio Sensor Network \\ PU & Primary user \\ SU & Secondary user \\ WSN & Wireless Sensor Network \\ SNR & Signal-to-Noise-Ratio \\ SINR & Signal-to-Interference-Noise-Ratio \\ BER & Bit Error Rate \\ MINLP & Mixed Integer Non-Linear Programming \\ \end{tabular}