The demand for autonomous logistics service robots requires an efficient task scheduling system in order to optimise cost and time for the robot to complete its tasks. This paper presents a Genetic algorithm (GA) based task scheduling system for a ground mobile robot that is able to find a global near-optimal travelling path to complete a logistics task of pick-and-deliver items at various locations. In this study, the chromosome representation and the fitness function of GA is carefully designed to cater for a single load logistics robotic task. Two variants of GA crossover are adopted to enhance the performance of the proposed algorithm. The performance of the scheduling is compared and analysed between the proposed GA algorithms and a conventional greedy algorithm in a virtual map and a real map environments that turns out the proposed GA algorithms outperform the greedy algorithm by 40% to 80% improvement.