International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 03, March 2019, pp. 284-289. Article ID: IJMET_10_03_029
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=3
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
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A NEW HEURISTIC ALGORITHM TO
DETERMINE MORE THAN ONE SEQUENCE IN
PERMUTATION FLOW SHOP SCHEDULING BY
USING HARMONIC TRIANGLE
B. Dhanasakkaravarthi
Research Scholar, Sathyabama Institute of Science and Technology, India, Chennai-119
Dr. A.Krishnamoorthy
Professor, School of Mechanical Engineering, Sathyabama Institute of Science and
Technology, India, Chennai-119
ABSTRACT
In this paper we present a new heuristic algorithm to minimize the total
completion time (Makespan) in permutation flow shop scheduling of ‘n’ jobs and ‘m’
machines by using harmonic triangle. In any shop floor, the major responsibility of
process planning engineer is to process the ‘n’ number of jobs in ‘m’ machines within
the due date. It can be achieved by optimal sequence of processing the jobs. Many
classical heuristics procedures were proposed starting from Johnson’s algorithm to
find optimal or near optimal sequence for job completion.In this research, an attempt
is made to propose a new heuristic by using Harmonic triangle. Also, the new
heuristic is compared with fewother popular heuristics like CDS, Palmer, RA and
Gupta Heuristics and the efficacy of new heuristic is analysed.
Keywords Permutation Flow shop Scheduling, Heuristics, Harmonic
Triangle,Sequencing.
Cite this Article B. Dhanasakkaravarthi and Dr. A.Krishnamoorthy, A New Heuristic
Algorithm to Determine More Than One Sequence in Permutation Flow Shop
Scheduling By Using Harmonic Triangle, International Journal of Mechanical
Engineering and Technology, 10(3), 2019, pp. 284-289.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=3
1. INTRODUCTION
In flow shop scheduling the performance of the processing of „n‟ jobs with „m‟ machines is
evaluated by multiple criteria, like make span, lateness, earliness (EDD), average time of jobs
in machines. The main objective of our research is focused to find the sequence of jobs to
reduce the total completion time (Makespan) by using harmonic triangle. Johnson‟s [1]
proposed an algorithm to find the optimal solution for 2 machines with „n‟ jobs and further
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284
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A New Heuristic Algorithm to Determine More Than One Sequence in Permutation Flow Shop
Scheduling By Using Harmonic Triangle
this algorithm is extended for 3 machines with „n‟ jobs. The condition for extended Johnson‟s
algorithm for 3 machines is:
If mini ti1 ≥ max I ti2
(Or)
If mini ti3 ≥ max I ti2
If any one condition is satisfied, the 3 machines problems are converted to 2 machines
with „n‟ jobs.The problem of minimizing the makespan is NP hard, therefore certain
assumptions are made given by Baker[2]:
 All the jobs are independent and available processing time is zero initially
 All the machines are readily available
 All the jobs are processed at each machine at one time
 Pre-emption is not allowed.
Many classical heuristics have been developed to optimize the make span in permutation
flow shop scheduling problem. [3] Palmer proposed the slope index method to find the near
optimal solution. Campbell [4] proposed the CDS and RA algorithmwas proposed by
Dannenbring [5] to minimize the makesspan and they resemble greatly the Johnson‟s
algorithm. In real situations, NEH [6] algorithm proposed by Nawaz et al. gives the better
performance than any other heuristic. Baskar and Anthony Xavior [7, 9, 11] proposed a few
new heuristic algorithmsone based on the Pascal‟s triangle, another based on dummy
machines and few more variants of NEH. They analysed the heuristics to determine the
sequence for minimizing the make span in FSSP using Taillard [8] and Vallada‟s [10]
problems. In this paper we propose a new heuristic based on Harmonic triangle and analysed
the algorithm using two problems, one with 4 jobs and 4 machines and the other with 10 jobs
and 10 machines.
2. HARMONIC TRIANGLE
Harmonic triangle is similar to Pascal‟s triangle and the rule for generating the harmonic
triangle is by adding two consecutive entries to give the entry between them in the row above.
To get the next term, subtract the next term from the corresponding term on the row above.
Moreover, it is possible to work downwards row by row because the entry at the left hand end
of the nth row is 1/n. The entries in the harmonic are similar to Pascal‟s triangle,involving the
binomial coefficients. The harmonic triangle for rth entry in the nth row is given by:
0Hr
1/1
1Hr
1/2 1/2
2Hr
1/3 1/6 1/3
3Hr
1/4 1/12 1/12 1/4
4Hr
1/5 1/20 1/30 1/20 1/5
5Hr
1/6 1/30 1/60 1/60 1/30 1/6
Mathematically,
H(n,r) =
(
(
)(
)
)
(
(
=
)
)(
)
The formula for harmonic triangle is given by: H (n, r) + H (n,r+1) = H(n-1,r)
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B. Dhanasakkaravarthi and Dr. A.Krishnamoorthy
3. PROPOSED HEURISTICS BASED ON HARMONIC TRIANGLE
The new heuristic for minimization of make span can be revealed by considering the example
given [12] by Panneerselvam as shown in Table 1. There are 4 jobs to be processed with 4
machines with minimum job completion time. Before adopting this procedure, Gupta‟s
heuristic gives the solution by converting the 4 machines into a 2 machine problem. He added
first 3 machines and makes it as a single machine and the last machine is left out to get the
first machine. For obtaining second machine, the first machine is left out and added the
remaining machines. Finally,Johnson‟s multistage rule is applied to find the sequence.
Table 1 .Four Machine problem
Job
1
2
3
4
I
4
3
1
3
Processing time in machine
II
III
3
7
7
2
2
4
4
3
IV
8
5
7
2
Let us now explain the procedure using the numeric example for the new heuristic as
follows:
Step 1
4 jobs are to be processed in 4 machines by assuming that all the jobs are available with
zero processing time. ie, here n=4, m=4.
Step 2
By using Harmonic triangle,select (m-2)Hr [3] , the elements are; 1/3,1/6,1/3.
Step 3
Now, the machining times are multiplied with harmonic triangle elements as selected in
the earlier step and added together. For example:
t11 = 1/3(4) + 1/6(3) +1/3(7), last machine time is left out (i.e., 8)
t12 = 1/3(3) + 1/6(7) +1/3(8), first machine time is left out (i.e., 4)
Similarly, for all the remaining jobs with processing time t21,t22,t31,t32,t41 & t42can be
computed.
Step 4
Now, the problem is reduced to 4 jobs with 2 machine problem and Johnson‟s algorithm is
used to find the sequence and corresponding makespan for the given problem.
Calculate all the job timings t11 to t42:
t11 = 1/3(4) + 1/6(3) +1/3(7) = 25
t12 = 1/3(3) + 1/6(7) +1/3(8) = 29
t21 = 1/3(3) + 1/6(7) +1/3(2) = 17
t22 = 1/3(7) + 1/6(2) +1/3(5) = 26
t31 = 1/3(1) + 1/6(2) +1/3(4) = 12
t32 = 1/3(2) + 1/6(4) +1/3(7) = 22
t41 = 1/3(3) + 1/6(4) +1/3(3) = 16
t42 = 1/3(4) + 1/6(3) +1/3(2) = 15
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A New Heuristic Algorithm to Determine More Than One Sequence in Permutation Flow Shop
Scheduling By Using Harmonic Triangle
Step 5
Now, the processing sequence can be obtained by using Johnson‟s Algorithm, because the
4 machine problem is reduced to 2 machine problem and given in Table 2.
Table 2 .Two machine problem
Processing time in machine
Job
I
II
1
25
29
2
17
26
3
12
22
4
16
15
Based on Johnson‟s Algorithm, the sequence obtained is 3-2-1-4 with a make span of 31
units.Now, consider a random problem with 10 jobs to be processed in 10different machines
(Table 3). In this case n=10 and m=10.
In this problem, we select (m-2)Hr = 8H8 and the corresponding harmonic triangle
elements are 1/9, 1/72,1/252,1/504,1/630,1/504,1/252,1/72,1/9. Now, the problem is reduced
to 10 jobs with 2 machines by following the same procedure as in earlier example as shown in
Table 4.
Table-3 Example of 10 jobs and 10machines
Job
1
2
3
4
5
6
7
8
9
10
M/c1
5
2
1
7
6
3
7
5
7
4
M/c2
2
6
2
5
6
7
2
1
8
3
M/c3
3
4
2
6
1
5
4
7
6
5
M/c4
5
2
1
3
8
2
6
1
9
8
M/c5
7
6
3
2
6
2
5
7
1
3
M/c6
9
2
7
3
4
1
5
3
8
1
M/c7
7
5
2
2
3
5
1
6
2
3
M/c8
8
2
5
4
9
3
2
6
1
8
M/c9
2
6
4
2
6
2
5
2
6
3
M/c10
7
1
4
2
4
6
2
2
6
7
Table 4. 10 Machines reduced to Two Machines
Job
M/c1
M/c2
1
2
3
4
5
6
7
2256
2654
1737
2953
4009
1873
3625
2931
2413
2003
2402
3276
3974
1565
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B. Dhanasakkaravarthi and Dr. A.Krishnamoorthy
8
9
10
2383
4125
2482
1302
4487
3274
The computations are carried out taking the same denominator while adding the weighted
machining times. Now, the problem is reduced to „n‟ jobs „2‟ machine similar to Johnson‟s
algorithm with the sequence obtained is 3-6-1-10-9-5-4-2-7-8 with a make span of 96 units.
4. COMPARISON OF OTHER HEURISTICS
The performance is compared with other classical heuristics and presented in Table 5.
Table 5. Comparison of Performance
Algorithm
Palmers slope index
Gupta‟s Heuristics
RA Heuristics
CDS Heuristics
Baskar Heuristic (by using Pascal‟s Triangle)
Proposedalgorithm (by Harmonic triangle)
Sequence
3-1-10-5-6-2-8-7-9-4
3-6-10-1-5-9-8-2-7-4
3-10-1-5-9-8-6-2-7-4
3-6-10-1-9-5-8-2-7-4
3-8-1-5-9-7-2-10-4-6
3-6-1-10-9-5-4-2-7-8
Make span
99
103
97
102
103
96
It is observed that the make span determined by the new heuristic is found to be better
when compared with any other heuristic. RA heuristic is a close runner with a make span of
96 units.
The make span deviation for the better solution is
(103-96)/ 96 = 7.2%, which is significant.
5. CONCLUSION
In this paper, a new heuristic algorithm is proposed based on harmonic triangle to process the
„n‟ jobs in „m‟ machine and it is analysed by using case studies. The results were compared
with other heuristics and it shows good results. The author is presently working to find the
effectiveness of the proposed heuristic by using well known Tailard bench mark problems
which are 120 in numbers.
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[1]
[2]
[3]
[4]
[5]
[6]
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H. G. Campbell, R. A. Dudek, and M. L. Smith, A Heuristic Algorithm for the n Job m
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D.G.Dannenbring, “An Evaluation of Flow-Shop Sequencing Heuristics”, Management
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M. Nawaz, E. Enscore Jr. and I. Ham, “A Heuristic Algorithm for the m-Machine, nJob Flow-shop Sequencing Problem”, OMEGA, The International Journal of
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A New Heuristic Algorithm to Determine More Than One Sequence in Permutation Flow Shop
Scheduling By Using Harmonic Triangle
[7]
[8]
[9]
[10]
[11]
[12]
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