In order for an effective and efficient collaboration among multiple vehicles to be possible, task allocation is essential. In general, task allocation algorithms assume constant costs and attempt to minimize or maximize an expected objective value. However, the optimality and reliability of the results may be greatly affected by uncertainties, such as those of the position of the tasks and the agents. This study addresses such challenges in multi-agent task allocation and proposes a robust task allocation algorithm under the uncertainty in the position of the nodes. To quantify the robustness of the solution, the uncertainty of the cost matrix, and the sensitivity of the solution are considered and evaluated. In order to model the uncertainty, Gaussian approximations are used and the Interval algorithm is employed to investigate the sensitivity of the solution to the uncertainty. For performance evaluation with respect to the travel distance and the value at risk of the proposed methodology, numerical simulations of multi-objective task allocation were carried out.