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A branch-and-bound algorithm for three-machine flowshop scheduling problem to minimize total completion time with separate setup times

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A branch-and-bound algorithm for three-machine flowshop scheduling problem tominimize total...
Article
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European Journal of Operational Research · February 2006
DOI: 10.1016/j.ejor.2004.07.074
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Ali AllahverdiKuwait University
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Fawaz S Al-AnziKuwait University
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A branch-and-bound algorithm for three-machineﬂowshop scheduling problem to minimize totalcompletion time with separate setup times
Ali Allahverdi
a,*
, Fawaz S. Al-Anzi
b,1
a
Department of Industrial and Management Systems Engineering, Kuwait University P.O. Box 5969, Safat, Kuwait
b
Department of Computer Engineering, Kuwait University P.O. Box 5969, Safat, Kuwait
Received 11 November 2003; accepted 28 July 2004Available online 12 March 2005
Abstract
In this paper, we address the three-machine ﬂowshop scheduling problem. Setup times are considered separate fromprocessing times, and the objective is to minimize total completion time. We show that the three-site distributed data-base scheduling problem can be modeled as a three-machine ﬂowshop scheduling problem. A lower bound is developedand a dominance relation is established. Moreover, an upper bound is developed by using a three-phase hybrid heuristicalgorithm. Furthermore, a branch-and-bound algorithm, incorporating the developed lower bound, dominance rela-tion, and the upper bound is presented. Computational analysis on randomly generated problems is conducted to eval-uate the lower and upper bounds, the dominance relation, and the branch-and-bound algorithm. The analysis shows theeﬃciency of the upper bound, and, hence, it can be used for larger size problems as a heuristic algorithm.
2005 Elsevier B.V. All rights reserved.
Keywords:
Scheduling; Distributed database; Flowshop; Total completion time; Dominance relation; Lower and upper bounds;Branch-and-bound
1. Introduction
Many real life problems can be modeled as a three-machine ﬂowshop scheduling problem where setuptimes are treated as separate from processing time. A speciﬁc case is in the area of distributed computing.
0377-2217/$ - see front matter
2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2004.07.074
*
Corresponding author. Tel.: +965 484 1188; fax: +965 481 6137.
E-mail addresses:
allahverdi@kuc01.kuniv.edu.kw (A. Allahverdi), alanzif@eng.kuniv.edu.kw (F.S. Al-Anzi).
1
Fax: +965 481 7451.European Journal of Operational Research 169 (2006) 767–780www.elsevier.com/locate/ejor
Distributed computing is becoming one of the most promising choices for speeding up computations(Elmasri and Navathe, 1999). This is especially true for cases where tasks can easily be broken into sequen-tial related subtasks. In the ﬁeld of databases, it is common to ﬁnd that the business logic of a transaction(job) can be divided into queries that run on separate machines sequentially. The only relation between twoadjacent tasks would be the transfer of results from one subtask to the next one. For this reason, in an
m
site distributed database, queries are typically programmed to execute on diﬀerent sites for eﬃciency rea-sons. Consider a conﬁguration of three sites. An execution of a job is ﬁrst submitted to the machine on site1 for ﬁrst stage of processing. The result of machine processing on site 1 is then transferred through thenetwork to the machine on site 2 for more processing which allows the machine on site 1 to process anew job. When processing of machine 2 is complete, the result is transferred to the machine on site 3 forﬁnal processing which leaves the machine on site 2 ready to process further jobs that have been completedby the machine on site 1. Note that some extra times are needed for submitting a job to the machine on site1 as well as transferring the results to the machines on sites 2 and 3. These times can be observed as timesneeded for setting up jobs for processing on machines on diﬀerent sites.Typically, there is a set of transactions (jobs) grouped in a batch. For the best performance of the dis-tributed system, it is desired to minimize the average completion time of the jobs in that batch since theaverage completion time is a measure of quality of the service of the distributed system.This problem can be modeled as a three-machine ﬂowshop scheduling problem where the objective is tominimize total completion time, and the setup times are considered separate from processing times. An-other application of separate setup can be found in the printing industry, where machine cleaning (setup)time depends on the color of the order. Similar practical situations arise in the chemical, pharmaceutical,food processing, metal processing, and semiconductor industries (Srikar and Ghosh, 1986; Bianco et al.,1988; Bitran and Gilbert, 1990; Uzsoy et al., 1992; Kim and Bobrowski, 1994; and Schaller et al., 2000).
A recent survey on scheduling problems involving setup times is given by Allahverdi et al. (1999).Yoshida and Hitomi (1979) extended the well-known two-machine ﬂowshop scheduling problem to thecase where setup times are separate from processing times and sequence independent with the objective of minimizing makespan. Allahverdi (1995) extended the work of Yoshida and Hitomi to stochastic environ-ments. There are other studies addressing ﬂowshop scheduling problem with the objective function of make-span including Das et al. (1995), Rajendran and Ziegler (1997), and Rios-Mercado and Bard (1998, 1999).
Dileepan and Sen (1991) addressed the same problem, but with the objective of minimizing maximumlateness. They presented dominance relations along with a lower bound to be used in a branch-and-boundalgorithm. They also developed two heuristic algorithms. Allahverdi and Al-Anzi (2002) indicated that themultimedia data objects scheduling problem for WWW Applications can be modeled as a two-machineﬂowshop problem of minimizing maximum lateness with separate setup times. They established three dom-inance relations and proposed four heuristics. They showed that their proposed heuristics outperform theones developed by Dileepan and Sen.The two-machine ﬂowshop problem was shown to be NP-hard (Gonzalez and Sahni, 1978) when theobjective is to minimize mean or total completion time instead of makespan, even for the case where setuptimes are neglected. This means that it is highly unlikely to ﬁnd a polynomial algorithm to solve the prob-lem. Therefore, researchers have focused on the development of branch-and-bound algorithms for the meanor total ﬂowtime criterion problem. As a result, several lower bounds, dominance relations, and versions of branch-and-bound algorithm have been proposed for the case where setup times are ignored (Ahmadi andBaghci, 1990; Cadambi and Sathe, 1983; and Della Croce et al., 2002).
Bagga and Khurana (1986) addressed the two-machine ﬂowshop problem to minimize total completiontime when considering setup times as separate from processing times and sequence-independent. Theydeveloped one dominance relation and a lower bound for the problem. Allahverdi (2000) also consideredthe same problem of Bagga and Khurana. He established two more dominance relations for the problem.He also presented a branch-and-bound algorithm incorporating the dominance relations that he and Bagga
768
A. Allahverdi, F.S. Al-Anzi / European Journal of Operational Research 169 (2006) 767–780
and Khurana established. Moreover, he presented three heuristics for the problem. Al-Anzi and Allahverdi(2001) showed that the three-tired client–server database internet connectivity problem is equivalent to thetwo-machine ﬂowshop problem with separate setup times, and hence the results of Bagga and Khurana(1986) and Allahverdi (2000) can be used for the three-tired client–server database internet connectivity
problem when the objective is to minimize total completion time. Al-Anzi and Allahverdi (2001) also
proposed heuristics for the problem and discussed the computational complexity of the proposed andthe previous heuristics. Al-Anzi and Allahverdi (2001) showed that their heuristics outperform the previous
ones.The literature reveals that the ﬂowshop problem with separate setup times and total completion timecriterion has been limited to the two-machine ﬂowshop. In this paper, we extend the problem to thethree-machine ﬂowshop problem. We develop a lower bound, an upper bound, and a dominance relation.Furthermore, we propose a branch-and-bound algorithm that incorporates these bounds and the relation.We evaluate the lower and upper bounds, the dominance relation, and the branch-and-bound algorithmthrough computational experiments. Moreover, we establish the conditions under which an optimal solu-tion can be obtained.Problem formulation is presented in the next section. The dominance relation, the lower bound, the opti-mum sequence, the upper bound, the branch-and-bound algorithm, the computational experiments, andthe conclusions are presented in Sections 3–9, respectively.
2. Formulation
Let
t
j,k
: the processing time of job
j
(
j
= 1,2,
. . .
,
n
) on machine
k
(
k
= 1,2,3),
s
j,k
: the setup time of job
j
on machine
k
,
C
j,k
: the completion time of job
j
on machine
k
,TCT: total completion time.Also let [
j
] denote the job in position
j
. Therefore,
t
[
j
],
k
denotes the processing time of the job in position
j
.
s
[
j
],
k
and
C
[
j
],
k
are deﬁned similarly.Let ST
[
j
],
k
denote the sum of the setup and processing times of jobs in positions 1,2,
. . .
,
j
on machine
k
,i.e.,ST
½
j
;
k
¼
X
jr
¼
1
ð
s
½
r
;
k
þ
t
½
r
;
k
Þ
;
j
¼
1
;
2
;
. . .
;
n
and
k
¼
1
;
2
;
3
:
Let
d
½
j
¼
ST
½
j
;
1
ð
ST
½
j
1
;
2
þ
s
½
j
;
2
Þ
;
j
¼
1
;
2
;
. . .
;
n
;
ð
1
Þ
where ST
[0],2
= 0. Let IT
[
j
],2
denote total idle time on the second machine until the job in position
j
on themachine is completed. It is known that (see Allahverdi, 2000):IT
½
j
;
2
¼
max
f
0
;
d
½
1
;
d
½
2
;
. . .
;
d
½
j
g
:
ð
2
Þ
Therefore,
C
½
j
;
2
¼
ST
½
j
;
2
þ
IT
½
j
;
2
:
ð
3
Þ
Now let
/
½
j
¼
IT
½
j
;
2
þ
ST
½
j
;
2
ð
ST
½
j
1
;
3
þ
s
½
j
;
3
Þ
;
j
¼
1
;
2
;
. . .
;
n
;
ð
4
Þ
A. Allahverdi, F.S. Al-Anzi / European Journal of Operational Research 169 (2006) 767–780
769
where ST
[0],3
= 0. Similarly, if IT
[
j
],3
denotes the total idle time on the third machine until the job in position
j
on the machine is completed. It can be shown thatIT
½
j
;
3
¼
max
f
0
;
/
½
1
;
/
½
2
;
. . .
;
/
½
j
g
:
ð
5
Þ
Hence,
C
½
j
;
3
¼
ST
½
j
;
3
þ
IT
½
j
;
3
:
ð
6
Þ
Once the completion times of jobs on the last (third) machine are known, then,TCT
¼
X
n j
¼
1
C
½
j
;
3
:
ð
7
Þ
It should be noted that throughout this paper we only consider permutation ﬂowshops.
3. A dominance relation
Dominance relations are common in the scheduling literature (Chu, 1992; Bagga and Khurana, 1986;Allahverdi, 2000; and Allahverdi and Al-Anzi, 2002). They are mainly used in implicit enumeration tech-
niques such as branch-and-bound algorithms. In this section a dominance relation is developed for ourproblem.Considerexchangingthepositionsoftwoadjacentjobsonathree-machineﬂowshopinasequence
p
1
thathas job
i
in an arbitrary position
s
and job
j
in position
s
+ 1. Consider another sequence that is obtainedfrom the sequence
p
1
by only interchanging jobs
i
and
j
. Call the sequence obtained from
p
1
as
p
2
, i.e.,
p
1
=
. . .
,
i
,
j
,
. . .
and
p
2
=
. . .
,
j
,
i
,
. . .
If it is shown that TCT(
p
2
)
6
TCT(
p
1
), then, sequence
p
2
would be no worsethan sequence
p
1
, and, therefore, job
j
precedes job
i
in a sequence that minimizes total completion time.The following four lemmas will be used for the proof of Theorem 1, which speciﬁes a dominance relationfor our problem. In Lemma 1, it is shown that
d
[
r
]
values for both sequences of
p
1
and
p
2
are the same for allpositions except
s
and
s
+ 1.
Lemma 1.
For both sequences
p
1
and
p
2
,
d
[r]
(
p
1
) =
d
[r]
(
p
2
) for r = 1,2,
. . .
,
s
1,
s
+ 2,
s
+ 3,
. . .
,n.
Proof.
It is obvious that
d
[
r
]
(
p
1
) =
d
[
r
]
(
p
2
) for
r
= 1,2,
. . .
,
s
1 since both sequences have the same jobs inthese positions. Furthermore,
d
[
r
]
(
p
1
) =
d
[
r
]
(
p
2
) for
r
=
s
+ 2,
s
+ 3,
. . .
,
n
since again both sequences havethe same jobs in these positions, and both include the jobs in positions
s
and
s
+ 1. Observe that thesum of processing and setup times is taken into account and hence the order is not important.
h
In the following Lemma, it is shown that
/
[
r
]
value for sequence
p
2
is less than or equal to that of
p
1
forall positions except
s
and
s
+ 1 if a certain condition is satisﬁed.
Lemma 2.
If
max{
d
[
s
]
(
p
2
),
d
[
s
+1]
(
p
2
)
}
6
max{
d
[
s
]
(
p
1
),
d
[
s
+1]
(
p
1
)
}
, then
/
[r]
(
p
1
) =
/
[r]
(
p
2
) for r = 1,2,
. . .
,
s
1, and furthermore
/
[r]
(
p
2
)
6
/
[r]
(
p
1
) for r =
s
+ 2,
s
+ 3,
. . .
,n.
Proof.
It is clear that
/
[
r
]
(
p
1
) =
/
[
r
]
(
p
2
) for
r
= 1,2,
. . .
,
s
1 since both sequences have the same jobs inthese positions. Now, it follows by deﬁnition of
/
[
r
]
that for
r
=
s
+ 2,
s
+ 3,
. . .
,
n
/
½
r
ð
p
2
Þ
/
½
r
ð
p
1
Þ ¼
max
f
0
;
d
½
1
ð
p
2
Þ
;
. . .
;
d
½
s
ð
p
2
Þ
;
d
½
s
þ
1
ð
p
2
Þ
;
. . .
;
d
½
r
ð
p
2
Þg
max
f
0
;
d
½
1
ð
p
1
Þ
;
. . .
;
d
½
s
ð
p
1
Þ
;
d
½
s
þ
1
ð
p
1
Þ
. . .
;
d
½
r
ð
p
1
Þg
:
If max{
d
[
s
]
(
p
2
),
d
[
s
+1]
(
p
2
)}
6
max{
d
[
s
]
(
p
1
),
d
[
s
+1]
(
p
1
)}, then by Lemma 1,
/
[
r
]
(
p
2
)
6
/
[
r
]
(
p
1
).
h
770
A. Allahverdi, F.S. Al-Anzi / European Journal of Operational Research 169 (2006) 767–780

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