Connectivity after disasters has become a critical problem in the management of modern cities. This comes from the need of the decision-makers to ensure urgent medical attention by providing access to hospitals and also to other relevant services needed by the population. Managing congestion could help maintain some routes operative even in complex scenarios such as natural disasters, terrorist attacks, protests or riots. Recent advances in humanitarian logistics have faced this problem using different modeling approaches, but by principally focusing on the response phase. In this paper, firstly, we propose a penalized variant of an existing mathematical model for the robust s-t path problem with investments. With the aim to solve the robust several-to-one path problem with investments, and due to the high complexity of this new problem, a heuristic to solve it is proposed. Moreover, this approach allows us to improve travel times in both specific paths and a set of routes in a systemic framework. The new problem and the proposed heuristic are illustrated by using an example, which corresponds to a typical city network, that provides a concrete vision about the potential application of the framework. Lastly, some managerial insights are given when the analysis of results is exhibited in the example network