International Journal of Engineering and Advanced Technology (IJEAT)
ISSN: 2249 – 8958, Volume-9 Issue-3, February 2020
2360
Retrieval Number: C5741029320/2020©BEIESP
DOI: 10.35940/ijeat.C5741.029320
Published By:
Blue Eyes Intelligence Engineering
& Sciences Publication
Abstract: In this paper, a new method is proposed for solving
octagonal fuzzy sequencing problem using new ranking method
(taking k = 0.5) for converting crisp problem for finding Job
sequence of two machine n jobs. The optimality for the completion
of the task and idle time for each machine is obtained by solving
corresponding sequencing problem using Johnsons’s rule. It is
illustrated with a numerical example.
Index Terms: Octagonal Fuzzy Number, Fuzzy sequencing
problem, Fuzzy Optimal sequencing, Octagonal fuzzy
membership function, New Ranking method & Johnson’s rule.
I. INTRODUCTION
Since the development of fuzzy set by Zadeh [3] in 1965,
there have been many research studies applying it in various
domains. It was applied in decision making process by
Bellman and Zadeh [1]. Fuzzy environment has been applied
in the domain of engineering applications, for solving linear
programming problems techniques as stated in Fang. S.C[2].
Octagonal fuzzy Job sequencing problem is applicable in
many real world situations. In many industrial engineering
units, it is essential to economize the cost of production by
saving idle time of machinery by utilizing them optimally. In
order to find total elapsed time in job sequencing, Johnson`s
method plays a vital role in solving job sequencing problems.
Job sequencing is defined as finding the most appropriate
order of performing jobs out of a number of operations to be
performed that can be assigned to a number of facilities for
optimum utilization.
II. PRELIMINARIES
In this part, usage of definition of a fuzzy set, crisp set,
octagonal fuzzy number and its pictorial representations of
octagonal fuzzy number, basic operations, new ranking
method of fuzzy number definition and its properties are
presented
A. Definition:
“A Fuzzy set A is defined as the set of ordered pairs (x,
μ
A
(x)), where x is an element of the universe of discourse U
and μ
A
(x) is the membership function that attributes to each
X∈U a real number∈[0,1] describing the degree to which X
belongs to the set.”
Revised Manuscript Received on February 14, 2020.
Althada Ramesh Babu*, Research Scholar, Department of
Mathematics, JNTUA, Ananthapuramu, Andhra Pradesh, INDIA &
Associate Professor, AS&H Dept. Sasi Institute of Technology &
Engineering, Tadepalligudem, W G Dt. A.P. Pin: 534101- INDIA Email:
rameshbabu.althada@sasi.ac.in
B. Rama Bhupala Reddy, Professor of Mathematics, KSRM College
of Engineering, Kadapa, A.P. India, E-mail reddybrb@gmail.com
B. Definition:
“A crisp set is a special case of Fuzzy set, in which the
membership function takes only two values 0 and 1.”
C. Definition:
“An ordered number L
~
= (l
1
, l
2
, l
3
, l
4
, l
5
, l
6
, l
7
, l
8
) is to be an
octagonal fuzzy number, where l
1
, l
2
, l
3
, l
4
, l
5
, l
6
, l
7
, l
8
are real
numbers from real line with its fuzzy membership function
) ( ~ x
L
is defined” [4]
8
8 7
7 8
8
7 6
6 5
5 6
5
5 4
4 3
3 4
3
3 2
2 1
1 2
1
1
~
, 0
,
,
, ) 1 (
, 1
, ) 1 (
,
,
, 0
) (
l x
l x l
l l
x l
k
l x l k
l x l
l l
x l
k k
l x l
l x l
l l
l x
k k
l x l k
l x l
l l
l x
k
l x
x
L
D. Graphical representation of a octagonal fuzzy
number[4]
where 0 < k < 1
E. Basic operations:
“Let L
~
= (l
1
, l
2
, l
3
, l
4
, l
5
, l
6
, l
7
, l
8
) and M
~
= (m
1
, m
2
, m
3
, m
4
,
m
5
, m
6
, m
7
, m
8
) be two octagonal fuzzy numbers, then its
mathematical operations are defined as [4]
i: M L
~ ~
= (l
1
, l
2
, l
3
, l
4
, l
5
, l
6
, l
7
, l
8
) (m
1
, m
2
, m
3
, m
4
, m
5
,
m
6
, m
7
, m
8
) = (l
1
+m
1
, l
2
+m
2
,
l
3
+m
3
, l
4
+m
4
, l
5
+m
5
, l
6
+m
6
,
l
7
+m
7
, l
8
+m
8
)
Solving Octagonal Fuzzy Sequencing Problem
using New Ranking Method
Althada Ramesh Babu, B Rama Bhupal Reddy