BI-CRITERIA IN TWO-MACHINE SCHEDULING UNDER
RENTAL SITUATION
Laxmi Narain
Associate Professor
Department of Mathematics, Acharya Narendra Dev College,
University of Delhi, Delhi, India
laxmi_narain_2004@yahoo.com
ABSTRACT
This paper studies bi-criteria in 2-machine flow-shop
problems under rental policy III. In this paper, Policy III is
modified. Here second machine will not be taken on rent at
times when the first job is completed on first machine but 2 nd
machine will be taken on rent subject to minimum total
elapsed time. The objective is: obtain a sequence which
minimizes total rental cost subject to minimum total elapsed
time. A simple algorithm is developed using Johnson
technique. The algorithm is illustrated through a numerical
example.
Keywords
Flow-shop Scheduling, Rental Cost, Rental Time, Elapsed
Time, Idle Time
1. INTRODUCTION
A study on scheduling literature has revealed the desirability
of an optimal schedule being evaluated by more than one
performance measures or criteria. Various authors [3-7, 9-19]
have studied the flow-shop problems having more than one
optimization measures. Gupta and Dudek [7] strongly
recommended the use of combination of criteria total flowtime and total elapsed time. Narain and Bagga [12] studied njob, m-machine special flow-shop problems which give
minimum possible mean flowtime while minimizing total
elapsed time. Narain and Bagga [9] determine the sequence
which minimizes the total elapsed time subject to zero total
idle time of machines i.e., machines should not remain idle
once they start the first job. Narain and Bagga [10] studies njob, m-machines flowshop problems when processing times of
jobs on various machines follow certain conditions and the
objective is to obtain a sequence which minimizes total
elapsed time under no-idle constant. Narain and Bagga [11]
studied n-job, 2-machine flowshop problem and provided an
algorithm for obtaining a sequence which gives minimum
possible mean flowtime under no-idle constraint.
In flow-shop problem, situation can occur in practice when
one has got the assignment but does not have one's own
machines or does not have enough money for the purchase of
machines, under these circumstances, may take machines on
rent to complete the assignment. Minimization of total rental
cost of machines will be the criterion in these types of
situations.
The following renting policies generally exist:
Policy I: All the machines are taken on rent at one time and
are returned also at one time.
Policy II: All the machines are taken on rent at one time and
are returned as and when they are no longer
required.
Policy III: All the machines are taken on rent as and when
they are required and are returned as and when
they are no longer required for processing.
Bagga [1] studied three-machine problem under Policy I and
provide a sequence which minimize the total rental cost of
machines. Under Policy II, for three-machine flow-shop
problem, Bagga and Ambika [2] provided a Branch-andBound algorithm.
This paper studies bi-criteria in 2-machine flow-shop
problems under rental Policy III. In this paper, Policy III is
modified. Here second machine will not be taken on rent at
times when the first job is completed on first machine but 2 nd
machine will be taken on rent subject to minimum total
elapsed time. Here, the objective is to obtain the sequence
which minimize the total elapsed time and provides the total
rental cost as minimum as possible. For any sequence S,
Total rental cost of machines
2
=
n
[ p
i, j
( S )  Ii , j (S )]  cj
j 1 i 1
Where pi,j(S) is the processing time of ith job of sequence S on
machine Mj, Ii,j(S) is the idle time of machine Mj for ith job of
sequence S and Cj is rental cost per unit time of machine M j.
Here, the processing times p i,j(S) and rental cost Cj(S) are
constant. Therefore, we can only reduce idle times I i,j(S). Idle
time on 1st machine is zero. To reduce idle time on 2nd
machine, we delay the time of renting of 2 nd machine to
process jobs. We have obtained a simple and efficient
algorithm, without using Branch-and-Bound technique, which
provides optimal sequence having minimum total rental cost
without altering the total elapsed time. The algorithm is
illustrated through a numerical example.
2. MATHEMATICAL FORMULATION
Notations:
S
:
Sequence of jobs 1, 2,…, n.
= max (Zm+1,1, max (Zm+1,1, Zm,2 )) + pm+1,2
Mj
:
Machine j; j=1, 2, 3.
= max (Zm+1,1, Zm,2 ) + pm+1,2
ith
pi,j(S)
:
Processing time of
machine Mj.
job of sequence S on
= Zm+1,2
Ii,j(S)
:
Idle time of machine Mj for ith job of
sequence S.
Cj
:
Rental cost per unit time of machine Mj.
Hj(S)
:
The time when Mj is taken on rent for
sequence S.
Zi,j(S)
:
Completion time of ith job of sequence S
on machine Mj.
Z′i,j(S)
:
Completion time of ith job of sequence S
on machine Mj when Mj starts processing
jobs at time Hj(S).
T2(S)
:
Total time for which M2 is required when
M2 starts processing jobs at time H2(S).
M2 at time H2 >
i = 1, 2,…, n and j= 1, 2, 3.
increase.
Therefore, the result holds for k = m+1 also.
Hence, by mathematical induction this theorem holds for all k,
where k = 1, 2, ..., n.
Under Policy III, machine is taken on rent when it is required
and is returned as soon as it completes the last job. By
Theorem 2.1; the starting of processing jobs at time H2 on M2
will reduce the idle time of M2 to zero and M2 will be required
only for time equivalent to the sum of the processing times of
all the jobs on it. Therefore, total rental cost of M 2 will be
minimum (least). Total rental cost of M1 will always be
minimum (least), since idle time of M1 is always zero. It is
obvious from Theorem 2.1, that if we start processing jobs on
n
Let n jobs require processing over three machines M1, M2 and
M3 in the order M1→ M2→ M3.
I
i,2
, then total elapsed time will
i 1
The following algorithm provides the procedure to obtain the
sequence which gives minimum possible rental cost while
minimizing total elapsed time in 2-machine flow-shop
problem under Policy III.
Theorem 2.1: If we start processing jobs on M2 at time
k
H2 =
I
i,2
, then Zk,2 will remain unaltered.
i 1
3. ALGORITHM
Algorithm 3.1:
Proof: Let Z′i,2 be the completion time of ith job on machine
M2 when M2 starts processing jobs at time H2. The proof of
the theorem is based on the method of mathematical
induction.
For k = 1;
Step 1:
Obtain the sequence having minimum total elapsed
time by Johnson algorithm. Let this sequence be S.
Step 2:
Compute total elapsed time Zn,2(S).
Step 3:
Compute rental time H2 of M2 for sequence S
Z′1,2 = H2 + p1,2
n
H2 = Zn,2(S) -
1
=
I
i,2
p
i,2
i 1
+ p1,2
i 1
Step 4:
Compute total rental cost for sequence S
= p1,1 + p1,2
n
R(S) =
= Z1,2
 pi ,1 × C1 +
i 1
n
p
i,2
× C2
i 1
Therefore, the result holds for k = 1.
Let the result holds for k = m
4. EXAMPLE
For k = m+1;
Example 4.1:
Z′m+1,2 = max (Zm+1,1, Z′m,2 ) + pm+1,2
m
p
= max (Zm+1,1, H2 +
i,2
) + pm+1,2
i 1
m 1
= max (Zm+1,1,
I
i,2
i 1
I
i,2
i 1
p
i,2
) + pm+1,2
Jobs
i 1
m
= max (Zm+1,1,
Table 1: Processing Times of Jobs on Machines
m
+
Consider 5-Job, 2-Machine flow-shop
problem with processing times (in hours) as given in Table 1
and the rental costs per unit time for machines M1 and M2 are
Rs 100 and Rs 200 respectively.
m
+
p
i,2
+ Im+1,2) + pm+1,2
i 1
= max (Zm+1,1, Zm,2 + max (Zm+1,1- Zm,2 , 0)) + pm+1,2
Machines
M1
M2
1
3
5
2
9
2
3
7
4
4
15
11
5
5
6
Operational Research Society, U. K. 43(9) (1992), 871884.
Applying Algorithm 3.1;
Step 1: Provides sequence S =1-5-4-3-2
Step 2: To obtain minimum total elapsed time, the completion
time In-Out of the sequence S is given in table 2.
Table 2: Completion Times In-Out.
Jobs
Machines
M1
M2
In-Out
In-Out
2
0-3
3
3-8
8-14
4
8-23
23-34
5
23-30
34-38
1
30-39
39-41
[9] Narain, L. and Bagga, P.C. 2003. Minimizing total
elapsed time subject to zero idle time of machines in n3
flow-shop problem. Indian Journal of Pure & Applied
Mathematics. 34 (2003), 219-228.
3-8
[10] Narain, L. and Bagga, P. C. 2004. Flow-shop/no-idle
scheduling to minimize total elapsed time. Journal of
Global Optimization. 33(2004), 349-367.
=39 – 28 = 13
[13] Sen, T. and Gupta, S.K. 1983. A branch and bound
procedure to solve a bi- criterion scheduling problem.
AIIE Transactions. 15(1983), 84-88.
= 9400 (Rupees).
Using Algorithm3.1;
[14] Sen, T. and Dileepan, P. 1999. A bicriterion scheduling
problem involving total flowtime and total tardiness.
Journal of Information and Optimization Sciences. 20(2)
(1999), 155-170
= 13 x 200 = 2600 (Rupees).
5. CONCLUSION
We have proved a theorem to find out the times at which
machines should be taken on rent so that total elapsed does
not change when we delay the processing of jobs on
machines. By using this theorem a simple and efficient
Algorithm is developed. From given example it is clear that
this Algorithm is very important if the rent of the machines is
high.
REFERENCES
[1] Bagga, P.C. 1969. Sequencing in a rental situation. Jr. of
Canadian Operations Research Society. 7(1969), 152153.
[2] Bagga, P.C. and Ambika Bhambani. 1996. Minimizing
rental costs in three-machine sequencing problem. Jr. of
Indian Association for Productivity, Quality and
Reliability. 21(1996), 73-77.
[3] Bagga, P.C. and Ambika Bhambani. 1996. Bicriteria in
flowshop
scheduling
problems.
Journal
of
Combinatorics, Information and System Sciences. 21(34)(1996).
[4] Bagga, P.C. and Ambika Bhambani. 2002. Bicriteria in
special flowshop problems. Indian Journal Pure and
Applied Mathematics. 33(4) (2002), 435-441
[5] Chandersekharan
Rajendran.
1992.
production scheduling problem with
[11] Narain, L. and Bagga, P. C. 2005. Flow-shop/no-idle
scheduling to minimize mean flow-time. ANZIAM.
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[12] Narain, L. and Bagga, P. C. 2006. Bi-criteria in n × m
flow-shop problems. IAPQR Transactions. 31(2006), 5771.
= 38 x 100 + 28 x 200
= 3800 +5600
Money saving
[7] Gupta, J.N.D. and Dudek, R.A. 1971. Optimality criteria
for flowshop schedule. AIIE Transactions. 3(3) (1971),
199-205.
[8] Johnson, S.M. 1954. Optimal two and three stage
production schedule with set up times included. Naval
Research Logistics Quarterly. 1(1954), 61-68.
Step 3: Rental time of machine M2 = 41 – (5+2+4+11+6)
Step 4: Total Rental cost
[6] Dileepan, P. and Sen, T. 1988. Bicriterion state
scheduling research for a single machine. Omega. 16
(1988), 53-59.
Two-stage
bicriteria.
[15] Sen, T. and Gupta, S.K. 1983. A branch and bound
procedure to solve a bicriterion scheduling problem.
AIIE Transactions. 15(1983), 84-88.
[16] Sen, T. and Raiszadeh, F. 1997. An algorithm
minimize total flowtime and maximum job lateness
the two-machine flowshop system. Journal
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16.
to
in
of
9-
[17] Sen, T., Raiszadeh, F. and Dileepan, P. 1988. A branchand-bound approach to the
bicriterion
scheduling
problem involving total flowtime and range of lateness.
Management Science. 34(2) (1988), 254-260.
[18] Van Wassenhove, L.N. and Gelders, L.F. 1980. Solving a
bicriterion scheduling problem. European Journal of
Operational Research. 4(1980), 42-48.
[19] Van Wassenhove, L.N. and Baker, K.R. 1982. A
bicriterion approach to time/cost trade-off in sequence.
European Journal of Operational Research. 11 (1982)
48-54.
AUTHOR’S BIOGRAPHY
Dr laxmi Narain is currently an Associate Professor in
Department of Mathematics, Acharya Narendra Dev College,
University of Delhi, Delhi, India. He has specialization in
Optimization Technique. He has done Ph.D. in Flow-shop
Sequencing Problem from Department of Mathematics,
University of Delhi under the Guidance of Dr. P. C. Bagga.
He has published research papers in National and
International Journals.