Abstract
Scheduling jobs with release and due dates on a single machine is a classical strongly NP-hard combination optimization problem. It has not only immediate real-life applications but also it is effectively used for the solution of more complex multiprocessor and shop scheduling problems. Here, we propose a general method that can be applied to the scheduling problems with job release times and due-dates. Based on this method, we carry out a detailed study of the single-machine scheduling problem, disclosing its useful structural properties. These properties give us more insight into the complex nature of the problem and its bottleneck feature that makes it intractable. This method also helps us to expose explicit conditions when the problem can be solved in polynomial time. In particular, we establish the complexity status of the special case of the problem in which job processing times are mutually divisible by constructing a polynomial-time algorithm that solves this setting. Apparently, this setting is a maximal polynomially solvable special case of the single-machine scheduling problem with non-arbitrary job processing times.
Highlights
Scheduling jobs with release and due-dates on single machine is a classical strongly NP-hard combination optimization problem according to Garey and Johnson [1]
In Part 1, an algorithmic framework for a single machine environment and a common due-date oriented objective function, the maximum job lateness, is presented, whereas, in Part 2, the framework is finished to a polynomial-time algorithm for the special case of the problem with mutually divisible job processing times
Among all jobs released by a given scheduling time, one with the minimum due-date is assigned to the machine
Summary
Scheduling jobs with release and due-dates on single machine is a classical strongly NP-hard combination optimization problem according to Garey and Johnson [1]. We establish the complexity status of the special case of the problem in which job processing times are mutually divisible by constructing a polynomial-time algorithm that solves this setting. In Part 1, an algorithmic framework for a single machine environment and a common due-date oriented objective function, the maximum job lateness, is presented, whereas, in Part 2, the framework is finished to a polynomial-time algorithm for the special case of the problem with mutually divisible job processing times. Part 2 consists of Sections 8–11, and is devoted to the version of the general single-machine scheduling problem with mutually divisible job processing times (under the assumption that the optimality condition of Section 7 is not satisfied).
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