J Optim Theory Appl (2017) 174:927–944
Polynomial Time Approximation Scheme for Two
Parallel Machines Scheduling with a Common Due Date
to Maximize Early Work
· Kateryna Czerniachowska
Received: 2 January 2017 / Accepted: 24 July 2017 / Published online: 2 August 2017
© The Author(s) 2017. This article is an open access publication
Abstract We study the scheduling problem with a common due date on two parallel
identical machines and the total early work criterion. The problem is known to be NP-
hard. We prove a few dominance properties of optimal solutions of this problem. Their
proposal was inspired by the results of some auxiliary computational experiments.
Test were performed with the dynamic programming algorithm and list algorithms.
Then, we propose the polynomial time approximation scheme, based on structuring
problem input. Moreover, we discuss the relationship between the early work criterion
and the related late work criterion. We compare the computational complexity and
approximability of scheduling problems with both mentioned objective functions.
Keywords Scheduling · Parallel machines · Early work · Late work · Polynomial
time approximation scheme
Mathematics Subject Classiﬁcation 90B35 · 68Q25
The scheduling theory provides models and methods helpful in solving practical
problems (cf., e.g., [1,2]). To cover different application ﬁelds, several performance
measures have been proposed in the literature, which allow for modeling various
Institute of Computing Science, Poznan University of Technology, Piotrowo 2,
60-965 Poznan, Poland