Ordinal algorithms for parallel machine scheduling

The minimization of maximum completion time for scheduling n jobs on m identical parallel machines is an NP-hard problem for which many excellent heuristic algorithms have been developed. In this paper, the problem is investigated under the assumption that only limited information about the jobs is available. Specifically, processing times are not known for the jobs; rather, the ordering of the jobs by processing time is known. For the cases of two and three parallel machines, algorithms which cannot be improved upon with respect to worst case performance ratio are developed. For the case of four parallel machines, an algorithm which is near optimal with respect to worst case performance ratio is developed. For arbitrary m, an algorithm which produces solutions whose value is at most five-thirds times the optimal value is presented. Finally, it is shown that as the number of machines gets arbitrarily large, the best possible ordinal algorithm has worst case performance ratio of at least 3/2. The problem of minimizing the maximum completion time for scheduling n jobs on m identical parallel machines (which is denoted by P II Cmax as in [5]), and its well-known relatives the partition problem and the subset-sum problem, have fascinated many researchers for several decades. Many excellent algorithms have been developed, and almost any (pragmatically) reasonable level of performance may be obtained in modest computer time using various approximation schemes. An excellent review may be found in [5]. The problem P 11 Cmax is defined as follows: Given a set J = { 1 ..... n} of n jobs, where job i has non-negative processing time ai, partition the job set into m subsets J1 .... , J,, so as to minimize the maximum sum of processing times of the jobs in any of the subsets.