Scheduling on parallel identical machines with late work criterion: Offline and online cases
Scheduling on parallel identical machines with late work criterion: Offline and online cases
Xin Chen 0 1
Malgorzata Sterna 0 1
Xin Han 0 1
Jacek Blazewicz 0 1
Xin Chen 0 1
0 School of Software Technology, Dalian University of Technology , Tuqiang Street 321, 116620 Dalian , China
1 Jacek Blazewicz
In the paper, we consider the problem of scheduling jobs on parallel identical machines with the late work criterion and a common due date, both offline and online cases. Since the late work criterion has not been studied in the online mode so far, the analysis of the online problem is preceded by the analysis of the offline problem, whose complexity status has not been formally stated in the literature yet. Namely, for the offline mode, we prove that the twomachine problem is binary NP-hard, and the general case is unary NP-hard. In the online mode we assume that jobs arrive in the system one by one, i.e., we consider the online over list model. We give an algorithm with a competitive ratio being a function of the number of machines, and we prove the optimality of this approach for two identical machines.
Online and offline scheduling; Identical parallel machines; Late work; NP-hardness; Competitive ratio
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2 Institute of Computing Science, Poznan University of
Technology, Piotrowo 2, 60-965 Poznan, Poland
1 Introduction
Time constraints, which can be used to determine feasibility
conditions, as well as to evaluate the quality of feasible
solutions, play an important role in scheduling problems which
can be met in the real world. In the scheduling theory, time
constraints might be modeled with due dates or deadlines and
the quality of schedules is estimated with reference to these
parameters, leading to such criteria as lateness
(e.g.,
McMahon and Florian 1975)
, tardiness
(e.g., Emmons 1969)
, or the
number of tardy jobs
(e.g., Moore 1968)
.
Late work criterion is one of the less explored
objective functions based on due dates. It was first proposed by
Blazewicz (1984)
. Soon a number of research groups focused
on this performance measure, obtaining a set of interesting
results. The single-machine case was considered mainly by
Potts and Van Wassenhove (1991),
Hochbaum and Shamir
(1990)
,
Hariri et al. (1995)
,
Kovalyov et al. (1994)
,
Kethley and Alidaee (2002)
, and, more recently, by
Lin and Hsu
(2005)
,
Zhang and Wang (2005)
, as well as by
Ren et al.
(2009)
. The parallel machines environment was studied by
Blazewicz (1984)
,
Blazewicz and Finke (1987)
, and
Leung
(2004)
, while the dedicated machines environment was
investigated mainly by Blazewicz et al. (2004a, 2005a, 2007) and
by
Leung (2004)
, as well as by
Lin et al. (2006)
. The state
of the art for late work scheduling was presented by
Leung
(2004)
in the context of imprecise computations and then by
Sterna (2011)
. This latter survey shows that the majority of
results obtained for the late work criterion so far concern the
single-machine or shop systems, and not too much attention
has been paid to the parallel machines environment. Besides,
all results presented in the literature concern the offline
version of the mentioned scheduling problems, and no paper is
devoted to the online case.
In this work, we consider the problem of scheduling jobs
on parallel identical machines with the late work criterion and
a common due date, both offline and online versions. For the
offline case, we prove that the problem for two machines
( P2|d j = d|Y ) is binary NP-hard, while for an arbitrary
number of machines ( P|d j = d|Y ), it is unary NP-hard. In
the online model studied in this paper, jobs appear in the
system one by one: when the previous job is scheduled, the
next one may arrive. Since there is an input sequence of
jobs, this model is called in the literature online “over list.”
For the online version of the analyzed scheduling problem
( P|d j = d, onli ne over li st |Y ), we propose an algorithm
with competitive ratio √2m2m−−2m1+1−1 , where m is the number
of machines. Then, we prove the optimality of this method
for two identical machines. More precisely, we show that
when m = 2, the competitive ratio, equal to √5 − 1, is
identical with the lower bound of the problem, so the proposed
algorithm is optimal for P2|d j = d, onli ne over li st |Y .
Moreover, when m → ∞, the competitive ratio converges
to √2, which is constant.
The rest of the paper is organized as follows: Section 2
presents the formal definition of the considered problem and
provides some information on the related work. Section 3
shows that the offline problem is NP-hard. The online case
is investigated in Sect. 4, where an online algorithm with
the constant competitive ratio is proposed together with the
proof of its optimality for two machines. Some conclusions
and future research directions are given in Sect. 5.
2 Problem definition and related work
The problem of scheduling jobs on parallel identical
machines with the late work crite (...truncated)