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J.
Opl Res. Soc. Vol. 45,
No.5,
pp. 589-594
Printed
in
Great Britain. All rights reserved
0160-5682/94 $9.00+0.00
Copyright
©
1994
Operational Research Society Ltd
Single Machine Earliness-Tardiness Scheduling
Problems Using the
Equal-Slack
Rule
CEYDA
OGUZ
and
CEMAL
DINCER
Bilkent University, Turkey
The purpose of this paper
is
to analyse a special case of the non-pre-emptive single machine
scheduling problem where the distinct due dates for each job are related to processing times
according to the Equal-Slack rule. The scheduling objective
is
to minimize the sum
of
earliness and
tardiness penalties. After determining some properties
of
the problem, the unrestricted case
is
shown
to be equivalent to a polynomial time solvable problem, whereas the restricted case
is
shown to be
NP-hard, and suggestions are made for further research.
Key words: machine scheduling
INTRODUCTION
Recently, the single-machine scheduling problem with earliness and tardiness penalties has
attracted enormous attention from researchers. Surveys can be found in Baker and Scudder
1
,
Cheng and
Gupta
2
and Sen and
Gupta
3
•
The objective function of a scheduling problem with
earliness and tardiness penalties
is
consistent with the just-in-time concept in which both early
and late completion of jobs from their due dates are prohibited. Very briefly, while early jobs
result in inventory holding costs, late jobs result in penalties, such as loss
of
customer
goodwill and loss of the orders. Therefore, minimizing the earliness and tardiness penalties
has important practical applications. So far, most
of
the literature concentrates on problems
with a common due date for all jobs. But, from a practical point of view, it
is
meaningful to
have distinct due dates for every job.
Unfortunately, the general problem with distinct due dates
is
one
of
the NP-complete
scheduling problems. This paper will present a special case
of
the general problem. The next
section introduces the notation used in the paper and some properties of the special case are
determined in the section after, where the problem
is
split into two
cases-unrestricted
and
restricted. After showing the equivalence of the unrestricted case to a polynomially solvable
problem, the NP-hardness for the restricted case
is
presented.
NOTATION
The machine scheduling problem studied in this
paper
requires n independent jobs
~
(j
= 1,
...
,
n)
to be processed on a single machine with the following assumptions: (1) all
jobs are available for processing at time zero; (2) the single machine can process
at
most one
job
~
at a time; and (3) no pre-emption
is
allowed.
Throughout the paper, it
is
assumed
that
a processing time
Pj
for each
~
(1) =
{p
1
,
p
2
,
••.
,
Pn}
), a target starting time aj by which
~
should ideally be started, a due date dj
by which
~
should ideally be completed
(dj
= aj +
pj),
and a cost function jj:
rJZ
~
(Q,
indicating the costs incurred
as
a function of the completion time of
~'
can be specified for
each
~·
It
is
assumed that all data, except
J;,
are non-negative integers. Given a processing
order, the starting time
Sj
(J
=
{S1o
S
2
,
•••
, Sn}), the completion time Cj =
Sj
+
pj,
the
tardiness
~
= max {0, Cj -
dj},
and the earliness
Ej
= max {0,
dj
-
Cj}
can be computed for
each
~·
Correspondence:
C.
Oguz, Department
of
Management, Hong Kong Polytechnic, Hung Hom, Kowloon, Hong Kong
589
Journal
of
the Operational Research Society Vol.
45,
No. 5
In the analysis of the given scheduling problem, the following additional notation
is
used:
1T
and a denote sequences
of
jobs;
7T(i)
and
a(i)
denote the ith job in the sequence;
z(a) denotes the value
of
optimality criterion for the schedule of a and z(a) =
L}=
1
(Ej
+
Ij);
t represents the set of jobs that complete before the target starting time;
g represents the set of jobs that start exactly
on
or
after the target starting time.
In
this paper, we follow the terminology used in Lawler et
al.
4
in order to identify the
scheduling problems defined by the above formulation.
Garey
et
al.
5
have shown that
lldjiL}=
1
(Ej
+ Ij)
is
NP-hard by a reduction from the
Even-Odd
Partition problem. They presented an efficient algorithm to find the optimal
schedule for fixed job sequencing. This algorithm inserts idle times between jobs in the given
sequence. They also developed a polynomial algorithm for the problem when all jobs have
equal processing times.
Since
lldj1Lj':.
1
(Ej
+ Ij)
is
NP-hard in its general form, a special case, which
is
one of the
models given by Baker and
Scudder\
is
analysed. Namely, problems in which distinct due
dates are related to processing times according to the Equal-Slack rule are considered. This
means that distinct due dates are given
as
dj =
pj
+
q,
q > 0 V
j.
Hence, the problem can be
stated
as
lldj
=
pj
+
qiL}=t
(Ej
+ Ij). Since this problem permits distinct due dates that relate
to processing times, it allows a nice structure for the optimal solutions in a special case.
For
further analysis, the notation
LJ!=
1
1Cj-
djl will be used instead of
LJ!=
1
(Ej
+ Ij) since
minimizing the sum of unweighted earliness and tardiness penalties
is
equivalent to minimiz-
ing the sum
of
absolute deviation
of
completion times from respective distinct due dates.
Furthermore, it is easy to observe the following.
Observation 1
Minimizing ICj- djl
is
equivalent to minimizing
ISj-
aJ
Proof
By definition, dj = aj + Pj. Substituting this into I
cj
- djIyields the following:
I
cj
- dj I = I
cj
- (aj +
pj)
I = I
cj
-
pj
-
aJ
The definition sj =
cj-
pj,
gives ICj- djl =
ISj-
ajl· Hence, minimizing ICj- djl
is
equiva-
lent to minimizing
ISj
- ajl·
Observation 2
If
dj =
pj
+ q then aj = q V
j.
Proof
This result
is
obtained easily after substituting dj = Pj + q into aj = dj - Pj
as
follows:
aj = dj - Pj = Pj + q - Pj = q ·
Observation 2 means that each job with dj =
pj
+ q has a common target starting time,
namely
q.
Observation 3
lldj
=
pj
+
qiL}=t
(Ej
+ Ij)
is
equivalent to
llaj
=
qiLJ!=
1
1Sj-
aJ
590