A Variant of Multi-task n-vehicle Exploration Problem: Maximizing Every Processor’s Average Profit

We discuss a variant of the multi-task n-vehicle exploration problem. Instead of requiring an optimal permutation of vehicles in every group, the new problem requires all vehicles in a group to arrive at the same destination. Given n tasks with assigned consume-time and profit, it may also be viewed as a maximization of every processor’s average profit. Further, we propose a new kind of partition problem in fractional form and analyze its computational complexity. By regarding fractional partition as a special case, we prove that the average profit maximization problem is NP-hard when the number of processors is fixed and it is strongly NP- hard in general. At last, a pseudo-polynomial time algorithm for the average profit maximization problem and the fractional partition problem is presented, using the idea of the pseudo-polynomial time algorithm for the classical partition problem.