Preemptive models of scheduling with controllable processing times and of scheduling with imprecise computation: A review of solution approaches

Abstract

a b s t r a c tThis paper provides a review of recent results on scheduling with controllable processing times. The stress is on the methodological aspects that include parametric flow techniques and methods for solving mathematical programming problems with submodular constraints. We show that the use of these methodologies yields fast algorithms for solving problems on single machine or parallel machines, with either one or several objective functions. For a wide range of problems with controllable processing times we report algorithms with the running times which match those known for the corresponding problems with fixed processing times. As a by-product, we present the best possible algorithms for a number of problems on parallel machines that are traditionally studied within the body of research on scheduling with imprecise computation.

Approximation algorithms for process systems engineering

“…Process scheduling involves tasks with variable processing times to determine the batch sizes. Scheduling with controllable processing times is an active operations research area dealing with this setting (Shabtay and Steiner, 2007;Shioura et al, 2018). In TCS, analogous investigations have taken place in the context of speed scaling where a processing unit may modify its speed to save energy and task processing times are decision variables (Albers, 2010;Albers et al, 2017;Angel et al, 2019;Bampis et al, 2015Bampis et al, , 2016Bampis et al, , 2018Bansal et al, 2007;Yao et al, 1995).…”

Scheduling problems with controllable processing times and a common deadline to minimize maximum compression cost

“…The area of Scheduling with Controllable Processing Times (SCPT ) has been active since the 1980s; see surveys [21,26], as well as the review [31] that covers the most recent developments in the area. In particular, it is demonstrated in [31] that the SCPT models include those models that have been studied within another popular area of research known as Scheduling with Imprecise Computation (SIC), see [10,18] for the corresponding reviews.…”

“…The problems with the objective are mainly addressed in the mainstream study of SCPT, while those with the max objective are studied within the SIC literature. The most recent survey [31] indicates that in the most general setting, when the jobs have arbitrary release dates and arbitrary deadlines, algorithms developed for the problems with the objective for all machine environments and for the max objective on parallel machines are not-improvable, since they are as fast as the algorithms for the counterparts with fixed processing times. If the jobs have a common deadline, for the problems of minimizing algorithms are known that perform faster than in the general case.…”

“…O (n log n) [9,13] O (n log n) [30] O n 2 [11] P O n 3 [13] O n 3 [19] O n 3 [31] Q O mn 3 [4] O mn 3 [19] O mn 3 [31] be a single machine (α = 1), m parallel identical machines (α = P), or m parallel uniform machines (α = Q). In the middle field, the presence of r ( j) means that the jobs may have individual release date (otherwise, they have a common zero release date).…”

“…McCormick [19] reduces problems with the objective to finding the maximum flow in networks in which capacities of some arcs linearly depend on a parameter, individual for each of these arcs. To deal with the max objective, Shioura et al [31] demonstrate that the corresponding SCPT problems can be reduced to solving lexicographic flow sharing problems in special networks. In these networks, capacities of some arcs depend on a common parameter and the required flow sharing problems can be solved by methods developed in [7], which results into the algorithms Table 2 Running times for solving feasibility problems α|r ( j), C( j) ≤ d, pmtn|− and problems α|r ( j),…”