The ever-increasing integration of non-dispatchable distributed generation, i.e., renewable energy sources (RES), arises new challenges in the field of power system's reliability. Distribution network reconfiguration (DNR) is a cost-effective approach for the distribution system operator (DSO) that wishes to enhance system's reliability without infrastructure upgrades. This paper introduces a novel path-based mixed-integer second-order cone programming model to optimally solve the reliability-oriented DNR problem. The DSO's objectives that are optimized are: a) improvement of distribution system's reliability indices and b) minimization of power losses. The proposed model is enriched with a scenario-based stochastic programming formulation that considers multiple levels of load and RES production. The standard 33-nodes distribution system and a real-world 83-nodes distribution system are employed to prove the efficiency and applicability of the model. Firstly, the multi-objective nature of the reliability-oriented DNR problem is investigated by conducting a sensitivity analysis, which reveals a trade-off region between reliability indices and power losses. Moreover, the obtained results show different global optimal solutions when the variability of load and RES production is considered. This highlights the importance of considering a scenario-based approach for load and RES production when solving the reliability-oriented DNR problem.