This article investigates the multi-robot task assignment problem with complex precedence constraints, where multiple robots need to visit a set of target locations respecting the prescribed order/sequence precedence constraints. These precedence constraints include strong precedence constraints and weak precedence constraints: 1) a strong precedence constraint between two target locations implies that the same robot should uninterruptedly visit these two locations, and 2) a weak precedence constraint between two target locations implies that the time for visiting one target location should be earlier/later than the other one. The objective is to minimize the time for the last target location to be visited while satisfying all precedence constraints. First, it is analyzed that the studied multi-robot task assignment problem is NP-hard. A lower bound on the optimal solution is constructed based on graph theory to measure the proximity of a suboptimal solution to the optimal one. Then, an improved multi-population genetic algorithm is proposed, in which two crossover operators tailored for precedence constraints and an adaptive termination strategy are designed. Simulation results show that the designed algorithm exhibits advantages in solving the multi-robot task assignment problem with complex precedence constraints compared with the existing iterative auction algorithm, adaptive large neighborhood search algorithm, and the co-evolutionary multi-population genetic algorithm.