Nonpreemptive Scheduling with Stochastic Precedence Constraints

Abstract

At first we consider two well-known deterministic single-machine scheduling problems (for the basic concepts and results from deterministic scheduling we refer to Lawler et al. 1982). In the min-max problem 1 ∣prec∣f max, “1” stands for a single machine, “prec” means that the precedence constraints for the jobs are given by an acyclic directed graph, and the objective function f max to be minimized is the maximum of nondecreasing functions f v where f v (t) is the cost arising when job v is completed at time t. The min-sum problem 1∣prec∣Σ w v C v says that the weighted sum of the completion times C v of the jobs v (the so-called weighted flow-time) is to be minimized and the precedence constraints are given by an outtree. Both problems can be solved in polynomial time. For more general precedence constraints or more general objective functions, single-machine min-sum scheduling problems become NP-hard. For example, the problems 1∣prec∣Σ C v (where all job weights are equal to one) and 1∣∣Σw v T v (where there are no precedence constraints for the jobs and T v is the tardiness of job v) are NP-hard. Thus in deterministic scheduling, min-sum problems seem to be less tractable than min-max problems.