Analysis of a linear programming heuristic for scheduling unrelated parallel machines

Each of n jobs is to be processed without interruption on one of m unrelated parallel machines. The objective is to minimize the maximum completion time. A heuristic method is presented, the first stage of which uses linear programming to form a partial schedule leaving at most m 1 jobs unscheduled: the second stage schedules these m1 jobs using an enumerative method. For m 2 3, it is shown that the heuristic has a (best possible) worst-case performance ratio of 2 and has a computational requirement which is polynomial in n although it is exponential in tn. For m= 2, it is shown that the heuristic has a (best possible) worst-case performance ratio of (1 +11x)/2 and requires linear time. A modified version of the heuristic is presented for m=2 which is shown to hake a (best possible) worst-case performance ratio of 3/2 while still requiring linear time.