Abstract
In this paper we characterize optimal schedules for scheduling problems with parallel machines and unit processing times by providing necessary and sufficient conditions of optimality. We show that the optimality conditions for parallel machine scheduling are equivalent to detecting negative cycles in a specially defined graph. For a range of the objective functions, we give an insight into the underlying structure of the graph and specify the simplest types of cycles involved in the optimality conditions. Using our results we demonstrate that the optimality check can be performed by faster algorithms in comparison with existing approaches based on sufficient conditions.
Highlights
Finding optimal schedules is the primary goal of scheduling and the main stream of research often deals with sufficient conditions of optimality that play the key role in the design of solution algorithms and in proving their correctness
Using standard three field notation, the scheduling problems we consider are denoted as P|r j, p j = 1|F (C, w), where the first field represents identical parallel machines, the second field specifies job requirements and the third field is the objective function of type (a), (b) or (c)
In this paper we have studied the necessary and sufficient optimality conditions for various problems with parallel identical machines and unit job processing times
Summary
Finding optimal schedules is the primary goal of scheduling and the main stream of research often deals with sufficient conditions of optimality that play the key role in the design of solution algorithms and in proving their correctness. The objective is to find a feasible schedule C which minimizes a given non-decreasing function F (C, w) depending on job completion times and job weights w = (w j )nj=1, which are assumed to be positive. Using standard three field notation, the scheduling problems we consider are denoted as P|r j , p j = 1|F (C, w), where the first field represents identical parallel machines, the second field specifies job requirements and the third field is the objective function of type (a), (b) or (c). The new algorithms based on the necessary and sufficient conditions outperform those that find optimal schedules
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