Preemptive Scheduling on Uniform Parallel Machines with Controllable Job Processing Times

Abstract

In this paper, we provide a unified approach to solving preemptive scheduling problems with uniform parallel machines and controllable processing times. We demonstrate that a single criterion problem of minimizing total compression cost subject to the constraint that all due dates should be met can be formulated in terms of maximizing a linear function over a generalized polymatroid. This justifies applicability of the greedy approach and allows us to develop fast algorithms for solving the problem with arbitrary release and due dates as well as its special case with zero release dates and a common due date. For the bicriteria counterpart of the latter problem we develop an efficient algorithm that constructs the trade-off curve for minimizing the compression cost and the makespan.

Handling Scheduling Problems with Controllable Parameters by Methods of Submodular Optimization

“…For problem Q|p(j) = p(j) − x(j), pmtn, C(j) ≤ d|W , it is shown in [22] how to find the actual processing times in O(nm + n log n) time. For the latter problem, Shakhlevich and Strusevich [29] use the SO reasoning to design an algorithm of the same running time and extend it to solving a bicriteria problem Q|p(j) = p(j) − x(j), pmtn| (C max , W ). The best known algorithms for solving problems Q|p(j) = p(j) − x(j), pmtn| (C max , W ) and Q|p(j) = p(j) − x(j), pmtn, C(j) ≤ d|W are discussed in Section 4.2 and in Section 4.3, respectively; their time complexity is O(nm log m) and O(n log n).…”

“…Such a statement (in different terms) was first formulated in [8] and [10]. For the problems on parallel machines, the corresponding representation of the total processing capacity in the form of a set function is defined in [29]. For all versions of the problem, with a single or parallel machines, the set function ϕ is submodular.…”

“…A systematic development of a general framework for solving the SCPT problems via submodular methods has been initiated by Shakhlevich and Strusevich [28,29] and further advanced in [30]. As a result, a powerful toolkit of the SO techniques [5,26] can be used for design and justification of algorithms for solving a wide range of the SCPT problems.…”

Literature Review of Single Machine Scheduling Problem with Uniform Parallel Machines

“…Shakhlevich and Strusevich [89] have provided a unified approach to solve preemptive scheduling problems with uniform parallel machines and controllable processing times. They demonstrated that a single criterion problem of minimizing total compression cost subject to the constraint that all due dates should be met can be formulated in terms of maximizing a linear function over a generalized polymatriod.…”

A Submodular Optimization Approach to Bicriteria Scheduling Problems with Controllable Processing Times on Parallel Machines

“…A systematic development of a general framework for solving scheduling problems with controllable processing times via submodular methods has been initiated by Shakhlevich and Strusevich [14,15] and further advanced by Shakhlevich et al [16] and Shioura et al [17]. This paper makes an additional contribution to the development of this approach.…”