Using Lagrangean Relaxation to Minimize
the (Weighted) Numb er of Late Jobs
on a Single Machine
Stephane Dauzere-Peres
a
Marc Sevaux
b
a
Department of Finance and Management Science
Norwegian School of Economics and Business Administration
Helleveien 30
N-5035 Bergen-Sandviken, Norway
b
Department of Automatic Control and Pro duction Engineering
Ecole des Mines de Nantes
La Chantrerie, BP 20722
F-44307 Nantes Cedex 03, France
E-mail:
f
Stephane.Dauzere-Peres, Marc.Sevaux
g
@emn.fr
June 28, 1999
Abstract
This pap er tackles the general single machine scheduling problem, where jobs
have dierent release and due dates and the ob jective is to minimize the weighted
number of late jobs. The notion of
master sequence
is rst introduced,
i.e.
, a se-
quence that contains at least an optimal sequence of jobs on time. This
master
sequence
is used to derive an original mixed-integer linear programming formula-
tion. By relaxing some constraints, it is possible to design a Lagrangean relaxation
algorithm which gives both lower and upp er bounds. The sp ecial case where jobs
have equal weights is analyzed. Computational results are presented and, although
the duality gap becomes larger with the number of jobs, it is p ossible to solve
problems of more than 100 jobs.
1 Intro duction
A set of
n
jobs
f
J
1
; ::; J
n
g
, sub ject to
release dates
r
i
and
due dates
d
i
,haveto be scheduled
on a single machine. The
processing time
of jobs on the machine is denoted by
p
i
, and
on leave from IRCyN/Ecole des Mines de Nantes
1
aweight
w
i
is associated to each job. The machine can only pro cess one job at a time.
Ascheduled job completed b efore its due date is said to be
early
or
on time
, and
late
otherwise. The ob jective is to minimize the weighted number of late jobs, or equivalently
to maximize the weighted number of early jobs. A well-known and important remark is
that there is always an optimal schedule in which late jobs are sequenced after all the
early jobs.
This single-machine scheduling problem, noted 1
j
r
j
j
P
w
j
U
j
in the standard classica-
tion, is strongly
NP
-Hard [8]. When all weights are equal (1
j
r
j
j
P
U
j
), the problem re-
mains
NP
-Hard, but b ecomes polynomially solvable if all release dates are equal (1
jj
P
U
j
)
[9](
O
(
n
log
n
)), or if release and due dates are similarly ordered (
r
i
<r
j
)
d
i
d
j
8
(
J
i
;J
j
)) [6] (
O
(
n
2
)), [7] (
O
(
n
log
n
)). However, some exact approaches have recently
been proposed for this problem [1] [5]. Lawler [7] showed that the Moore's algorithm ([9])
could b e applied when pro cessing times and weights are aggeeable,
i.e.
,
p
i
<p
j
)
w
i
w
j
8
(
J
i
;J
j
). Finally, branch-and-bound pro cedures have been developed to solve the case
where all release dates are equal (1
jj
P
w
j
U
j
) in [12] and [11]. To our knowledge, no
algorithm has been prop osed to solve the general problem 1
j
r
j
j
P
w
j
U
j
.
In this paper, based on the notion of
master sequence i.e.
, a sequence from which
an optimal sequence can be extracted, a new mixed-integer linear programming formula-
tion is introduced. Using this formulation, a Lagrangean relaxation algorithm is derived.
Lagrangean relaxation is a p owerful optimization tool from which heuristic iterative al-
gorithms can be designed, where both upper and lower bounds are determined at every
iteration. It is thus p ossible to always know the maximum gap between the b est so-
lution found and the optimal solution, and stop the algorithm when this gap is small
enough. One condition that is often associated to the eciency of Lagrangean relaxation
approaches is to relax as few constraints as possible, in order to obtain go od b ounds when
solving the relaxed problem. This is why our formulation compares very favorably to
other known ones (see [4] for a study of classical formulations for this problem). Only
one constrainttype, coupling variables of dierent jobs, needs to b e relaxed to obtain an
easily solvable problem, that can be solved indep endently for each job.
The master sequence is introduced in Section 2, and the resulting mixed-integer linear
programming formulation is given and discussed in Section 3. Section 4 shows how the
size of the master sequence, and thus the size of the model, can b e reduced. Section 5
presents the Lagrangean relaxation algorithm, and Section 6 improves the algorithm. The
non-weighted case is studied in more details in Section 7. Numerical results on a large
set of test instances are given and discussed in Section 8. Finally, some conclusions and
persp ectives are drawn in Section 9.
2
2 The master sequence
In the remainder of this paper, because we are only interested in sequencing jobs on time
(late jobs can be set after the jobs on time), the sequence of jobs will mean the sequence
of
early
jobs. Many results in this paper are based on the following theorem.
Theorem 1
There is always an optimal sequence of jobs on time that solves the problem
1
j
r
j
j
P
w
j
U
j
, in which every job
J
j
is sequenced just after a job
J
i
such that either condition
(1)
d
i
<d
j
, or (2)
d
i
d
j
and
r
k
r
j
8
J
k
sequencedbefore
J
j
, holds, or equivalently
condition (3)
d
i
d
j
and
9
J
k
sequencedbefore
J
j
such that
r
k
>r
j
is not satised.
Proof:
The proof go es by showing that, by construction, it is p ossible to change any
optimal sequence into an optimal sequence that satises the conditions (1) or (2).
Suppose that wehave a sequence in which some (or all) ready jobs do not satisfy one of
the conditions. Starting from the b eginning of the sequence, nd the rst pair of jobs
(
J
i
;J
j
) in the sequence that does not satisfy the two conditions,
i.e.
, for which condition
(3) holds. If
t
i
and
t
j
denote the start times of the two jobs, the latter condition ensures
that, after interchanging the two jobs,
J
j
can start at
t
i
(since
9
J
k
sequenced before
J
j
such that
r
j
<r
k
t
i
). Hence,
J
i
will end at the same time than
J
j
before the interchange
(
t
i
+
p
i
+
p
j
), and thus will still b e on time (since
t
i
+
p
i
+
p
j
d
j
d
i
).
The interchange should b e repeated if
J
j
and the new job just b efore it do not satisfy
conditions (1) or (2), until one of these conditions is satised for
J
j
and the job just b efore
it, or
J
j
is sequenced rst.
The procedure is rep eated for all jobs until the conditions are satised for all jobs. Because
once a job has been moved, it will never go back again, one knows that the procedure will
not b e repeated more than
n
times,
i.e.
, takes a nite amount of time.
2
We will denote by
S
the subset of sequences in which jobs satisfy the conditions in
Theorem 1. In the sequel, we will only be interested in sequences in
S
, since we know
that it always contains an optimal sequence.
Prop osition 1
If, in a sequenceof
S
, job
J
j
is after jobs
J
i
such that
r
j
<r
i
, then there
is at least a job
J
i
such that
d
i
<d
j
.
Proof:
By contradiction, if all jobs
J
i
before
J
j
such that
r
j
<r
i
verify
d
i
d
j
, then
none of the conditions (1) and (2) is satised. Thus, the sequence is not in
S
.
2
Corollary 1
If, for every job
J
i
such that
r
j
<r
i
,condition
d
j
d
i
holds, then, in every
sequenceof
S
(i.e., in an optimal sequence), job
J
j
is sequencedbefore al l jobs
J
i
.
3
Corollary 2
If, for every job
J
j
such that
d
j
<d
i
,condition
r
j
r
i
holds, then, in every
sequenceof
S
(i.e., in an optimal sequence), job
J
i
is sequenced after all jobs
J
j
.
Wewanttoshow that it is p ossible to derive what will be called a
master sequence
and denoted by
, and which \contains" every sequence in
S
. Corollary 1 implies that
there is only one position for
J
j
in the master sequence, and Corollary 2 that there is only
one p osition for
J
i
.
Example 1
Let us consider a 5-job problem with the data of Table 1.
Jobs
J
1
J
2
J
3
J
4
J
5
r
i
0 5 8 12 14
p
i
8 6 5 6 10
d
i
16 26 24 22 32
Table 1: Data for a 5-job problem
Considering sequences in
S
, and because of Corollary 1, one knows that
J
1
is set before
al l jobs (conditions
r
1
<r
i
and
d
1
<d
i
are satised for every job
J
i
6
=
J
1
), and al l jobs
are set before
J
5
(conditions
r
i
<r
5
and
d
i
<d
5
are satised for every job
J
i
6
=
J
5
).
Hence, in the master sequence
, job
J
1
wil l be set rst and job
J
5
last.
The master sequence has the following form:
=(
J
1
;J
2
;J
3
;J
2
;J
4
;J
3
;J
2
;J
5
)
Every sequence of jobs in
S
can beconstructedfrom
. In this example, they are
numerous sequences or early jobs (more than 40). For instance, the subset of sequences
containing 5 jobs is:
f
(
J
1
;J
2
;J
3
;J
4
;J
5
)
;
(
J
1
;J
2
;J
4
;J
3
;J
5
)
;
(
J
1
;J
3
;J
2
;J
4
;J
5
)
;
(
J
1
;J
3
;J
4
;J
2
;J
5
)
;
(
J
1
;J
4
;J
3
;J
2
;J
5
)
g
One can check that each of these sequences is includedin
S
.
Prop osition 2
In the master sequence, if
r
i
<r
j
and
d
i
>d
j
, then thereisaposition
for
J
i
before
J
j
and a position for
J
i
after
J
j
.
Proof:
Because
r
i
<r
j
, Condition (2) in Theorem 1 is satised for the pair of jobs
(
J
i
;J
j
), and because
d
i
>d
j
, Condition (1) is satised for the pair (
J
j
;J
i
). Hence, there
is a position in the master sequence for
J
i
before and after
J
j
.
2
Hence, there must be a position in the master sequence for
J
i
after every job
J
j
such
that
r
i
<r
j
and
d
i
>d
j
. This shows that there will be
at most
n
(
n
+1)
2
positions in the
master sequence.
4
Corollary 3
If, for every job
J
j
such that
r
i
<r
j
, the condition
d
i
d
j
holds, then there
is only one position for job
J
i
in the master sequence.
Corollary 3 shows that, when release and due dates are similarly ordered (as in Kise
et al.
[6]), the master sequence will be the sequence of jobs in increasing order of their
release dates (or due dates if some jobs have equal release dates). In the non-weighted case
(
w
i
=1,
8
J
i
), the problem is then p olynomially solvable using the algorithm prop osed in
[6] (in
O
(
n
2
)) or in [7] (in
O
(
n
log
n
)).
An interesting and important prop erty of the master sequence is a kind of transitivity
property. If job
J
i
is set b efore and after
J
j
in the master sequence b ecause either
Condition (1) or (2) of Theorem 1 holds, and if
J
j
is set before and after
J
k
in the
master sequence b ecause either Condition (1) or (2) holds, then either Condition (1) or
(2) of Theorem 1 holds and
J
i
is set before and after
J
k
in the master sequence.
The algorithm to create the master sequence
is sketched b elow. We suppose that
the jobs are pre-ordered in non-decreasing order of their release dates, and
J
denotes the
set of jobs already sequenced. Moreover, to speed up the algorithm, jobs added in
J
are
ordered on non-decreasing order of their due dates.
FOR every job
J
i
2
J
DO
[
J
i
J
J
[
J
i
FOR every job
J
j
2
J
such that
d
j
d
i
DO
[
J
j
The algorithm has a time complexityof
O
(
n
2
). The job set at position
k
in
is
denoted
(
k
). The number of positions in the master sequence is denoted by
P
. Recall
that
P
n
(
n
+1)
2
. Actually,
P
will only be equal to its upper bound if the job with the
smallest release date has also the largest due date, the job with the second smallest release
date has the second largest due date, and so on (see Proposition 2). This is clearly a very
special case and, in practical exp eriments,
P
will be much smaller than
n
(
n
+1)
2
.
3 A new mixed-integer linear programming formula-
tion
Based on the master sequence, one can derive the following mo del:
5
