# A Variable Neighbourhood Search-Based Algorithm for the Transit Route Network Design Problem
## 4 Conclusions  
The TRNDP has long attracted research attention, with multiple algorithms proposed over the last five decades for its efficient solution. The principal contribution of this work is the development of a VNS-based algorithm for the TRNDP problem. To the best of our knowledge, this is the first study to apply such an algorithm to the TRNDP. Due to the high complexity of the TRNDP, the options for modifying solutions without disproportionally increasing the computational cost are limited, as discussed in detail in [PERSON] [56]. Meanwhile, we found that simple neighbourhood moves and efficient stagnation prevention can yield very high-quality results. Interestingly, our algorithm either matched or outperformed the highly efficient hyper-heuristics proposed by [PERSON] et al. [46] in the instance by [PERSON] while overall producing superior results out of all approaches. Particularly for the 7- and 8-route cases, improvements were attained in terms of direct demand coverage, which is crucial for the profitability and public perception of public transport networks. These results empirically corroborate the theoretical potential of VNS-based algorithms for complex routing problems, which extends to the TRNDP.  
Moving forward, the proposed algorithm, owing to its simplicity and efficiency, can be used within more complex methodological frameworks to enhance computational performance when handling other variants of the TRNDP. For instance, bi-level formulations for the Electric TRNDP can be efficiently solved by combining VNS moves with mathematical programming. Moreover, the algorithm can be suitably modified for dealing with other routing problems arising in public transport, such as demand-responsive systems and modular transit systems.  
Conceptualisation, C.I., I.T. and G.B.; methodology, C.I. and I.T.; software, C.I. and I.T.; validation, C.I. and I.T.; formal analysis, C.I. and I.T; investigation, I.T.; resources, C.I.; data curation, C.I.; writing--original draft preparation, C.I.; writing--review and editing, I.T. and G.B.; visualisation, C.I.; supervision, G.B. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
\begin{table}
\begin{tabular}{c c} \hline
**Rout \#** & **Node Sequence** \\ \hline
1 & 21, 6, 13, 0, 19, 17, 28, 7, 4, 3, 1, 23, 20, 14, 9 \\ \hline
2 & 26, 0, 28, 16, 27, 29, 2, 10, 7, 14, 11, 17, 19, 8, 12 \\ \hline
3 & 23, 20, 7, 27, 10, 21, 6, 13, 18, 22, 17, 11, 3, 1, 9 \\ \hline
4 & 17, 28, 25, 11, 3, 9, 1, 4, 7, 16, 6, 13, 18, 12, 8 \\ \hline
5 & 15, 27, 7, 20, 14, 11, 17, 12, 19, 22, 25, 28, 16, 10, 21 \\ \hline
6 & 17, 28, 16, 15, 5, 21, 2, 29, 27, 7, 14, 1, 9, 23, 24 \\ \hline
7 & 16, 2, 29, 27, 7, 4, 24, 14, 11, 17, 22, 18, 13, 6, 10 \\ \hline
8 & 9, 1, 24, 7, 10, 15, 5, 6, 13, 0, 25, 11, 14, 23, 4 \\ \hline
9 & 1, 3, 11, 17, 19, 12, 8, 26, 0, 18, 22, 25, 7, 27, 29 \\ \hline
10 & 21, 5, 6, 16, 7, 14, 23, 9, 3, 11, 17, 22, 0, 26, 8 \\ \hline
11 & 2, 15, 6, 13, 18, 22, 19, 8, 12, 0, 25, 7, 20, 24, 4 \\ \hline
12 & 8, 12, 19, 17, 11, 14, 23, 9, 1, 3, 24, 7, 2, 29, 15 \\ \hline \end{tabular}
\end{table}
Table 10: Best route set for _Mumford0_.