# Binary Spring Search Algorithm for Solving Various Optimization Problems
## 8 Conclusions  
There are many optimization problems that must be solved by using a suitable method. Different heuristic optimization algorithms have been proposed to overcome the shortcomings of traditional methods, such as Linear Programming (LP), non-linear LP, and differential programming. Most of these algorithms are population-based using the randomness of natural phenomena. A heuristic optimization algorithm called Binary Spring Search Algorithm (\(BSSA\)) is proposed, which uses laws of the spring force law. The proposal was mathematically modeled, and its efficiency was evaluated using 23 standard test functions. These test functions were selected from three different types: unimodal, multimodal, and multimodal with fixed dimension test functions to evaluate different aspects of the proposed algorithm. Seven optimization algorithms (binary genetic algorithm, binary particle swarm optimization, binary gravitational search algorithm, binary dragonfly algorithm, binary bat algorithm, and binary grasshopper optimization algorithm) were compared to evaluate the performance of the proposed algorithm. Compared to the other algorithms, in all cases, the \(BSSA\) produces nearly optimal solutions. [PERSON]'s rank test was used to further analyze the performance of the \(BSSA\). The results obtained from this test show the clear superiority of the proposed algorithm in the three different types of test functions. The overall results of all test functions (F1-F23) show that the \(BSSA\) is significantly superior to the other algorithms and ranks first among them. Based on the optimization results and the [PERSON] rank test results, it is clear that the proposed \(BSSA\) performs well in solving optimization problems and is more competitive than similar algorithms.  
Conceptualization, M.D., Z.M., and A.D.; methodology, M.D., R.A.R.-M.; software, M.D., Z.M., A.D., G.D., and N.N.; validation, O.P.M., N.N., and A.E.; formal analysis, O.P.M., R.M.-M.; investigation, R.A.R.-M.; resources, J.M.G., A.E.; data curation, R.M.-M.; writing--original draft preparation, M.D., Z.M., and A.D.; writing--review and editing, O.P.M., R.A.R.-M., R.M.-M., G.D., and J.M.G.; visualization; supervision, M.D. and Z.M.; project administration, A.D.; funding acquisition, R.M.-M., R.A.R.-M. All authors have read and agreed to the published version of the manuscript.  
The current project was funded by Tecnologico de Monterrey and FEMSA Foundation (grant CAMPUSCITY project).  
The authors declare no conflict of interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & **Test Function** & & **BGA** & **BPSO** & **BGSA** & **BDA** & **BBA** & **BMOA** & **BGOA** & **BSSA** \\ \hline \multirow{2}{*}{1} & Unimodal & & & & & & & & & \\  & (F1–F7) & [PERSON] value & 39 & 39 & 42 & 46 & 38 & 26 & 14 & 7 \\ \hline \multirow{2}{*}{2} & [PERSON] rank & 5 & 5 & 6 & 7 & 4 & 3 & 2 & **1** \\ \hline \multirow{2}{*}{2} & Multimodal & & & & & & & & & \\  & (F8–F13) & [PERSON] value & 25 & 21 & 36 & 40 & 28 & 31 & 22 & 6 \\ \hline \multirow{2}{*}{3} & [PERSON] rank & 4 & 2 & 7 & 8 & 5 & 6 & 3 & **1** \\ \hline \multirow{3}{*}{3} & Multimodal with fixed & & & & & & & & & \\  & dimensions & [PERSON] value & 52 & 31 & 42 & 44 & 44 & 34 & 29 & **10** \\ \hline \multirow{3}{*}{3} & [PERSON] rank & 7 & 3 & 5 & 6 & 6 & 4 & 2 & **1** \\ \cline{1-1}  & & [PERSON] value & 116 & 91 & 120 & 130 & 110 & 91 & 65 & **23** \\ \hline \multirow{3}{*}{4} & [PERSON] rank & 5 & 3 & 6 & 7 & 4 & 3 & 2 & **1** \\ \hline \end{tabular}
\end{table}
Table 8: Results of the [PERSON] rank test.