Octagonal Fuzzy Transportation Problem Using Different Ranking Method

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 5, August 2020
@ IJTSRD | Unique Paper ID – IJTSRD31706
Octagonal Fuzzy Transportation
Problem Using
Dr.
1Research Guide, Assistant Professor, Department
Chikkanna Govt
2Research Scholar, Assistant Professor, Department
Sri Shakthi Institute of Engineering
ABSTRACT
Transportation Problem is used on supply and demand of commodities
transported from one source to the different destinations. Finding solution
of Transportation Problems are North
Method and Vogel’s Approximation Method etc. In
Fuzzy Numbers using Transportation problem by Best Candidates Method
and Robust ranking method and Centroid Ranking Technique and Proposed
Ranking Method. A Comparative study is Triangular Fuzzy Numbers and
Trapezoidal Fuzzy Numbers and Octagonal Fuzzy Numbers. The
transportation cost can be minimized by using of Proposed Ranking
Method under Best Candidates Method. The procedure is illustrated with a
numerical example.
KEYWORDS: Transportation problems, Octagonal fuzzy numbers,
Ranking method, BCM method, CRT, PRM, Initial Basic Feasible Solution,
Optimal Solution
1. INTRODUCTION
The central concept in the problem is to find the least total
transportation cost of commodity in different method. In
general, transportation problems are solved with
assumptions that the supply and demand are specified in
precise manner. Intuitionistic fuzzy set is a powerful tool
to deal with such vagueness.
The concept of Fuzzy Sets, proposed by H. A. Taha,
Operations Research- Introduction [8], has been found to
be highly useful to deal with vagueness. Many authors
discussed the solutions of Fuzzy Transportation Problem
(FTP) using various techniques.In 2016, Mrs.
introduced Pentagonal fuzzy. S. Chanas, W. Kolodziejczyk
and A. Machaj, “A fuzzy Approach to the Transportation
Problem [3], A New Algorithm for Finding a Fuzzy
Optimal Solution. K. Prasanna Devi, M. Devi Durga
G.Gokila, Juno Saju [6] introduced Octagonal Fuzzy
Number.
A new method is proposed for finding an optimal solution
for fuzzy transportation problem, in which the cost ,
supplies and Demands are octagonal fuzzy numbers.
Octagonal fuzzy transportation problem BCM method we
get best minimum value of optimal solution.
of Trend in Scientific Research and Development (IJTSRD)
2020 Available Online: www.ijtsrd.com e-
31706 | Volume – 4 | Issue – 5 | July-August
Problem Using Different Ranking Method
Dr. P. Rajarajeswari1, G. Menaka2
Research Guide, Assistant Professor, Department of Mathematics
Chikkanna Govt Arts College, Tirupur, Tamil Nadu, India
Research Scholar, Assistant Professor, Department of Mathematics
f Engineering and Technology, Coimbatore, Tamil
Transportation Problem is used on supply and demand of commodities
transported from one source to the different destinations. Finding solution
of Transportation Problems are North-West Corner Rule, Least Cost
Method and Vogel’s Approximation Method etc. In this paper Octagonal
Fuzzy Numbers using Transportation problem by Best Candidates Method
and Robust ranking method and Centroid Ranking Technique and Proposed
Ranking Method. A Comparative study is Triangular Fuzzy Numbers and
nd Octagonal Fuzzy Numbers. The
transportation cost can be minimized by using of Proposed Ranking
Method under Best Candidates Method. The procedure is illustrated with a
Transportation problems, Octagonal fuzzy numbers, Robust
Ranking method, BCM method, CRT, PRM, Initial Basic Feasible Solution,
How to cite this paper
Rajarajeswari | G. Menaka "Octagonal
Fuzzy Transportation Problem Using
Different Ranking
Method" Published
in International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-5, August 2020, pp.8
www.ijtsrd.com/papers/ijtsrd31706.pdf
Copyright © 20
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org
by/4.0)
The central concept in the problem is to find the least total
transportation cost of commodity in different method. In
general, transportation problems are solved with
assumptions that the supply and demand are specified in
uzzy set is a powerful tool
The concept of Fuzzy Sets, proposed by H. A. Taha,
Introduction [8], has been found to
be highly useful to deal with vagueness. Many authors
nsportation Problem
, Mrs. Kasthuri. B
S. Chanas, W. Kolodziejczyk
and A. Machaj, “A fuzzy Approach to the Transportation
A New Algorithm for Finding a Fuzzy
Prasanna Devi, M. Devi Durga and
introduced Octagonal Fuzzy
A new method is proposed for finding an optimal solution
for fuzzy transportation problem, in which the cost ,
fuzzy numbers. Using
transportation problem BCM method we
get best minimum value of optimal solution.
The paper is organized as follows, in section 2,
introduction with some basic concepts of Fuzzy
In section 3 introduced Octagonal
proposed algorithm followed by a Numerical example
using BCM method and finally the paper is concluded in
section 4.
2. PRELIMINARIES
2.1. Definition (Fuzzy set
Let X be a nonempty set. A fuzzy set
as‫ܣ‬̅=ሼ൏ ‫,ݔ‬ ߤ஺ ഥ ሺxሻ ൐/‫ݔ‬ ∈ ܺሽ.
membership function, which maps each element of X to a
value between 0 and 1.
2.2. Definition (Fuzzy Number
A fuzzy number is a generalization of a regular real
number and which does not refer to a single value but
rather to a connected a set of possible values, where each
possible value has its weight between 0 and 1. The weight
is called the membership function.
A fuzzy number ‫ܣ‬̅ is a convex normalized fuzzy set on the
real line R such that
There exists at least one x ∈R with
ߤ஺ ഥ ሺxሻߤ஺ ഥ ሺxሻ is piecewise continuous.
of Trend in Scientific Research and Development (IJTSRD)
-ISSN: 2456 – 6470
ugust 2020 Page 8
Different Ranking Method
f Mathematics,
f Mathematics,
Coimbatore, Tamil Nadu, India
How to cite this paper: Dr. P.
Rajarajeswari | G. Menaka "Octagonal
Fuzzy Transportation Problem Using
Different Ranking
Method" Published
in International
Journal of Trend in
Scientific Research
velopment
(ijtsrd), ISSN: 2456-
6470, Volume-4 |
5, August 2020, pp.8-13, URL:
www.ijtsrd.com/papers/ijtsrd31706.pdf
Copyright © 2020 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
The paper is organized as follows, in section 2,
introduction with some basic concepts of Fuzzy definition ,
In section 3 introduced Octagonal Fuzzy Definition and
proposed algorithm followed by a Numerical example
using BCM method and finally the paper is concluded in
ሾࡲࡿሿ)[3]
Let X be a nonempty set. A fuzzy set ‫ܣ‬̅ of Xis defined
ሽ Where ߤ஺ ഥ (x) is called
membership function, which maps each element of X to a
Definition (Fuzzy Numberሾࡲࡺሿ) [3]
A fuzzy number is a generalization of a regular real
number and which does not refer to a single value but
a connected a set of possible values, where each
possible value has its weight between 0 and 1. The weight
is called the membership function.
is a convex normalized fuzzy set on the
R with ߤ஺ ഥ ሺxሻ = 1.
is piecewise continuous.
IJTSRD31706

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
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5
2.3. Definition (Triangular Fuzzy Number [TFN])
A Triangular fuzzy number ‫ܣ‬̅ is denoted by 3 – tuples
(ܽଵ, ܽଶ ,ܽଷ), whereܽଵ, ܽଶ ܽ݊݀ ܽଷare real numbers and
ܽଵ ≤ ܽଶ ≤ ܽଷ with membership function defined as
ߤ஺ ഥ ሺx) =
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬
‫ ݔ‬ − ܽଵ
ܽଶ − ܽଵ
݂‫ܽ ݎ݋‬ଵ ≤ ‫ ݔ‬ ≤ ܽଶ
ܽଷ − ‫ݔ‬
ܽଷ − ܽଶ
݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ
0 ‫ݐ݋‬ℎ݁‫݁ݏ݅ݓݎ‬ ۙ
ۖ
ۘ
ۖ
ۗ
2.4. Definition (Trapezoidal Fuzzy Number [TrFN])
A trapezoidal Fuzzy number is denoted by 4 tuples ‫ܣ‬̅ =
(ܽଵ, ܽଶ,ܽଷ, ܽସ),Where ܽଵ, ܽଶ ,ܽଷ ܽ݊݀ ܽସ are real numbers
and ܽଵ ≤ ܽଶ ≤ ܽଷ ≤ ܽସ with membership function
defined as
ߤ஺ ഥ ሺx) =
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬
‫ ݔ‬ − ܽଵ
ܽଶ − ܽଵ
݂‫ܽ ݎ݋‬ଵ ≤ ‫ ݔ‬ ≤ ܽଶ
1 ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ
ܽସ − ‫ݔ‬
ܽସ − ܽଷ
݂‫ܽ ݎ݋‬ଷ ≤ ‫ ݔ‬ ≤ ܽସ
0 ‫ݐ݋‬ℎ݁‫݁ݏ݅ݓݎ‬ ۙ
ۖ
ۘ
ۖ
ۗ
2.5. Definition (Octagonal Fuzzy Number [OFN])
A Fuzzy Number ‫ܣ‬̅ை஼ is a normal Octagonal Fuzzy Number
denoted by
‫ܣ‬̅ = ሺܽଵ, ܽଶ , ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻, ଼ܽ).
Whereܽଵ, ܽଶ , ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻ ܽ݊݀ ଼ܽ are real numbers
and its membership functionߤ஺ ഥ ሺx) is given below,
ߤ஺ ഥ ሺx) =
‫ە‬
ۖ
ۖ
ۖ
ۖ
ۖ
‫۔‬
ۖ
ۖ
ۖ
ۖ
ۖ
‫ۓ‬
0 ݂‫ݔ ݎ݋‬ < ܽଵ
݇ ൬
‫ ݔ‬ − ܽଵ
ܽଶ − ܽଵ
൰ ݂‫ܽ ݎ݋‬ଵ ≤ ‫ ݔ‬ ≤ ܽଶ
݇ ݂‫ܽ ݎ݋‬ଶ ≤ ‫ ݔ‬ ≤ ܽଷ
݇ + ሺ1 − ݇) ሺ
‫ ݔ‬ − ܽଷ
ܽସ − ܽଷ
) ݂‫ܽ ݎ݋‬ଷ ≤ ‫ ݔ‬ ≤ ܽସ
1 ݂‫ܽ ݎ݋‬ସ ≤ ‫ ݔ‬ ≤ ܽହ
݇ + ሺ1 − ݇) ൬
ܽ଺ − ‫ݔ‬
ܽ଺ − ܽହ
൰ ݂‫ܽ ݎ݋‬ହ ≤ ‫ ݔ‬ ≤ ܽ଺
݇ ݂‫ܽ ݎ݋‬଺ ≤ ‫ ݔ‬ ≤ ܽ଻
݇ሺ
଼ܽ − ‫ݔ‬
଼ܽ − ܽ଻
) ݂‫ܽ ݎ݋‬଻ ≤ ‫ ݔ‬ ≤ ଼ܽ
0 ݂‫ݔ ݎ݋‬ > ଼ܽ ۙ
ۖ
ۖ
ۖ
ۖ
ۖ
ۘ
ۖ
ۖ
ۖ
ۖ
ۖ
ۗ
3.1. BEST CANDIDATES METHOD (BCM) :[2]
Step1:
Prepare the BCM matrix, If the matrix unbalanced, then the
matrix will be balanced without using the added row or
column candidates in solution procedure.
Step2:
Select the best candidate that is for minimizing problems
to the minimum cost, and maximizing profit to the
maximum cost. Therefore, this step can be done by
electing the best two candidates in each row. If the
candidate repeated more than two times, then the
candidate should be elected again. As well as, the columns
must be checked such that if it is not have candidates so
that the candidates will be elected for them. However, if the
candidate is repeated more than one time, the elect it
again.
Step3:
Find the combinations by determining one candidate for
each row and column, this should be done by starting from
the row that have the least candidates, and then delete that
row and column. If there are situations that have no
candidate for some rows or columns, then directly elect the
best available candidate. Repeat Step 2 by determining the
next candidate in the row that started from. Compute and
compare the summation of candidates for each
combination. This is to determine the best combination
that gives the optimalsolution.
3.2. ROBUST RANKING TECHNIQUES:[11]
Robust ranking technique which satisfy compensation,
linearity, and additively properties and provides results
which are consist human intuition. If ã is a fuzzy number
then the Robust Ranking is defined by,
R(ã) = ‫׬‬ ሺ0.5) ሺ ܽఈ
௅ଵ
଴
, ܽఈ
௎
) ݀ߙ.
Where ሺ ܽఈ
௅
, ܽఈ
௎
) is the ߙ level cut of the fuzzy number ã.
[ܽఈ
௅
, ܽఈ
௎
] = {[(b-a) ߙ + ܽ], [݀ − ሺ݀ − ܿ)ߙ], [ሺ݂ − ݁)ߙ +
݁], [ℎ − ሺℎ − ݃)ߙ]}
In this paper we use this method for ranking the objective
values. The Robust ranking index R(ã) gives the
representative value of fuzzy number ã.
3.3. RANKING OF GENERALIZED OCTAGONAL FUZZY
NUMBERS:[12]
Ranking of fuzzy numbers are very important task to
reduce the more numbers. Nowadays number of proposed
ranking techniques is available. Ranking methods map
fuzzy number directly into the real line i.e. which associate
every fuzzy number with a real number.
The centroid point of an octagon is considered to be the
balancing point of the trapezoid. Divide the octagon into
three plane figures. These three plane figures are a
trapezium ABCR, a hexagon RCDEFS, and again a trapezium
SFGH respectively. Let the centroid of three plane figures
be ‫ܩ‬1, ‫ܩ‬2, ܽ݊݀ ‫ܩ‬3 respectively.
The centroid of these centroids ‫ܩ‬1,‫ܩ‬2,ܽ݊݀ ‫ܩ‬3 is taken as a point
of reference to define the ranking of generalized octagon
fuzzy numbers. The reason for selecting this point as a
point of reference is that each centroid point ‫ܩ‬1 of a
trapezoid ABCR, ‫ܩ‬2 of a hexagon RCDEFS and ‫ܩ‬3 of a
trapezoid SFGH are balancing point of each individual
plane figure and the centroid o these centroids points is
much more balancing point for a general octagonal fuzzy
number.
Consider a generalized octagonal fuzzy number ‫ܣ‬௢ =
(ܽଵ, ܽଶ , ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻, ଼ܽ ; ‫.)ݓ‬ The centroid of these plane
figures is
‫ܩ‬1 = (
௔భାଶ௔మ
ଷ
+
௔మା௔య
ଶ
+
ଶ௔యା௔ర
ଷ
+
௪
଺
+
௪
ସ
+
௪
଺
)
‫ܩ‬1 = (
ଶ௔భା଻௔మ ା଻௔యାଶ௔ర
ଵ଼
,
଻௪
ଷ଺
)
Similarly,
‫ܩ‬3 = (
ଶ௔ఱା଻௔లା଻௔ళାଶ௔ఴ
ଵ଼
,
଻௪
ଷ଺
)
‫ܩ‬2 = (
௔భାଶ௔ర ାଶ௔ఱା௔ల
଺
,
௪
ଶ
)
The equation of the line‫ܩ‬1, ‫ܩ‬3 is ‫ݕ‬ =
௪
ଶ
, and does not lie on
the line ‫ܩ‬1 and ‫ܩ‬3. Thus ‫ܩ‬1 , ‫ܩ‬2 ܽ݊݀ ‫ܩ‬3 are non-collinear and
theyform a triangle.

The ranking function of the generalized hexagonal fuzzy
number ‫ܣ‬௢ = (ܽଵ, ܽଶ , ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻, ଼ܽ :; ‫)ݓ‬ which maps
the set of all fuzzy numbers to a set of real numbers is
defined as
R(‫ܣ‬௢) = ‫ܩ‬஺ሺ‫ݔ‬଴, ‫ݕ‬଴) =
(
ଶ௔భା଻௔మ ାଵ଴௔యା଼௔ర ା଼௔ఱାଵ଴௔ల ା଻௔ళାଶ௔ఴ
ହସ
,
଼௪
ଶ଻
)
3.4. PROPOSED RANKING METHOD:
The ranking function of Octagonal Fuzzy Number (OIFN)
‫ܣ‬௢ =ሺܽଵ, ܽଶ ,ܽଷ , ܽସ , ܽହ, ܽ଺, ܽ଻,଼ܽ) maps the set of all Fuzzy numbers to a set of real numbers
defined as
ܴሺ‫ܣ‬௢) = ሺ
2ܽଵ + 3ܽଶ + 4ܽଷ + 5 ܽସ + 5ܽହ + 4ܽ଺ + 3ܽ଻ + 2଼ܽ
28
,
7‫ݓ‬
28
)
3.5. COMPARISON ROBUST RANKING, CRT AND PRM METHODS:[1]
Solutions of Fuzzy Transportation Problem Using Best Candidates Method and Different Ranking Techniques [1], The
transportation costs per pump on different routes, rounded to the closest dollar using Triangular Fuzzy Numbers and
Trapezoidal Fuzzy Numbers are given below. Now this results are compared with the results for Octagonal Fuzzy
Numbers.
COMPARISON OF ROBUST RANKING AND CRT:[1]
S.NO Fuzzy Numbers Methods Robust Ranking CRT Technique
1
Triangular
Fuzzy
Numbers
BCM 59,905 10,936.33
VAM 61,010 10,962.23
NWCR 65,275 12,010.61
LCM 66,515 12,211.77
2 Trapezoidal Fuzzy Numbers
BCM 59,850 9,049.66
VAM 59,950 10,156.5
NWCR 65,315 10,072.43
LCM 65,210 9,690.87
3.6. Numerical Example:
BANGALORE PUNE NEWDELHI KOLKATA SUPPLY
KOREA
(71,72,73,74,
75, 76,77,78)
(66,67,68,69,
70,71,72,74)
(80,81,82,83,
84,85,86,88)
(78,79,80,81,
82,83,84,85)
(86,87,88,89,
90,91,92,94)
JAPAN
(82,83,84,85,
86,87,88,90)
(76,78,80,82,
83,84,85,86)
(92,93,94,95,
96,97,98,100)
(84,85,86,87,
88,89,90,92
(130,135,140,145,
150,155,160,170)
UK
(98,99,100,101,
102,103,104,106)
(86,87,88,89,
90,91,92,94)
(132,133,134,135,
136,137,138,140)
(114,115,116,118,
120,122,124,126)
(210,215,220,225,
230,235,240,250)
LUPTON
(96,97,98,99,
100,101,102,103)
(92,93,94,95,
96,97,98,100)
(110,112,113,114,
115,116,118,120)
(100,102,104,106,
108,112,116,118
(140,145,150,160,
170,180,190,200)
DEMAND
(90,92,94,96,
98,100,105,110)
(180,190,195,200,
205,210,215,220)
(160,165,170,175,
180,185,190,195)
(120,130,140,150,
160,170,180,190)
ΣDemand = Σ Supply

Using the Robust Ranking Technique the above problem can be reduced as follows:
BANGALORE PUNE NEW DELHI KOLKATA SUPPLY
KOREA 74.5 69.5 83.5 81.5 89.5
JAPAN 85.5 81.5 95.5 87.5 148
UK 101.5 89.5 135.5 119 228
LUPTON 99.5 95.5 114.5 108 166.5
DEMAND 98 201.5 177.5 155
Applying VAM method, Table corresponding to initial basic feasible solution is = 59042.75
Applying North West Corner method, Table corresponding to initial basic feasible solution is = 64862.5
Applying LCM method, Table corresponding to initial basic feasible solution is = 65942
Applying BCM method, Table corresponding to Optimal solution is = 63223
Using the Centroid Ranking Method the above problem can be reduced as follows:
KOREA 21.6 20.1 23.8 23.6 25.9
JAPAN 24.8 23.7 27.7 25.3 42.8
UK 29.4 25.9 33.2 46.3 66.0
LUPTON 43.5 27.7 33.2 31.3 55.2
DEMAND 28.3 58.7 51.4 44.9
Σ Demand ≠ Σ Supply
BANGALORE PUNE NEW DELHI KOLKATA DUMMY SUPPLY
KOREA 21.6 20.1 23.8 23.6 0 25.9
JAPAN 24.8 23.7 27.7 25.3 0 42.8
UK 29.4 25.9 33.2 46.3 0 66.0
LUPTON 43.5 27.7 33.2 31.3 0 55.2
DEMAND 28.3 58.7 51.4 44.9 6.61
Σ Demand = Σ Supply
Applying VAM method, Table corresponding to initial basic feasible solution is = 4027.18
Applying North West Corner method, Table corresponding to initial basic feasible solution is = 5162.26
Applying LCM method, Table corresponding to initial basic feasible solution is = 5249.66
Applying BCM method, Table corresponding to Optimal solution is = 5249.99
Using Proposed Ranking Method the above problem can be reduced as follows:
KOREA 18.6 17.4 20.9 20.4 22.3
JAPAN 21.4 20.5 23.9 23.5 36.9
UK 25.4 22.4 33.9 29.8 56.9
LUPTON 24.9 23.9 28.6 27 41.5
DEMAND 24.4 50.5 44.3 38.6
Σ Demand ≠ Σ Supply
KOREA 18.6 17.4 20.9 20.4 22.3
JAPAN 21.4 20.5 23.9 23.5 36.9
UK 25.4 22.4 33.9 29.8 56.9
LUPTON 24.9 23.9 28.6 27 41.5
DUMMY 0 0 0 0 0.3
DEMAND 24.4 50.5 44.3 38.6
Σ Demand = Σ Supply
Applying VAM method, Table corresponding to initial basic feasible solution is
KOREA 18.6 17.4 [22.3] 20.9 20.4 22.3
JAPAN 21.4 20.5 23.9 [36.9] 23.5 36.9
UK 25.4 [24.4] 22.4 [28.2] 33.9 29.8 [4.3] 56.9
LUPTON 24.9 23.9 28.6 [7.1] 27 [34.4] 41.5
DUMMY 0 0 0 [0.3] 0 0.3
DEMAND 24.4 50.5 44.3 38.7
Minimum Cost = 22.3× 17.4 + 36.9 × 25.3 + 24.4 × 25.4 + 28.2 × 22.4 + 29.8 × 4.3 + 7.1 × 28.6 + 34.4 ×27+0 × 0.3
= 3781.37

Applying North West Corner method, Table corresponding to initial basic feasible solution is
KOREA 18.6 [22.3] 17.4 20.9 20.4 22.3
JAPAN 21.4 [2.1] 20.5 [34.8] 23.9 23.5 36.9
UK 25.4 22.4 [15.7] 33.9 [41.2] 29.8 56.9
LUPTON 24.9 23.9 28.6 [3.1] 27 [38.4] 41.5
DUMMY 0 0 0 0 [0.3] 0.3
DEMAND 24.4 50.5 44.3 38.6
Minimum Cost = 22.3× 18.6 + 2.1 × 21.4 + 34.8 × 20.5 + 15.7 × 22.4 + 41.2 × 33.9 + 3.1 × 28.6 + 38.4 ×27+0 × 0.3
= 4046.94
Applying LCM method, Table corresponding to initial basic feasible solution is
KOREA 18.6 17.4 [22.3] 20.9 20.4 22.3
JAPAN 21.4 [8.7] 20.5 [28.2] 23.9 23.5 36.9
UK 25.4 22.4 33.9 [44.3] 29.8 [12.6] 56.9
LUPTON 24.9 [15.4] 23.9 28.6 27 [26.1] 41.5
DUMMY 0 [0.3] 0 0 0 0.3
DEMAND 24.4 50.5 44.3 38.6
Minimum Cost = 22.3× 17.4 + 8.7 × 21.4 + 28.2 × 20.5 + 44.3 × 33.9 + 12.6 × 29.8 + 15.4 × 24.9 + 26.1 ×27+0 × 0.3
= 4117.71
Applying BCM method, Table corresponding to Optimal solution is
KOREA 18.6 17.4 [22.3] 20.9 20.4 22.3
JAPAN 21.4 [24.4] 20.5 [12.5] 23.9 23.5 36.9
UK 25.4 22.4 [15.7] 33.9 [2.5] 29.8 [38.7] 56.9
LUPTON 24.9 [15.4] 23.9 28.6 [41.5] 27 41.5
DUMMY 0 [0.3] 0 0 [0.3] 0 0.3
DEMAND 24.4 50.5 44.3 38.6
S.NO Ranking Methods Methods
Triangular
Fuzzy Numbers
Trapezoidal Fuzzy
Numbers
Octagonal Fuzzy
Numbers
1
Robust
Ranking
BCM 59,905 59,850 59,790
VAM 61,010 59,950 59,042.75
NWCR 65,275 65,315 64,862.5
LCM 66,515 65,210 64,942
2 CRT Technique
BCM 10,936.33 9,049.66 5,249.99
VAM 10,962.23 10,156.5 4,027.18
NWCR 12,010.61 10,072.43 5,162.26
LCM 12,211.77 9,690.87 5,249.66
3 Proposed Ranking Method
BCM 3,938.74
VAM 3,781.37
NWCR 4,046.94
LCM 4,117.71
Minimum Cost = 22.3× 17.4 + 24.4 × 21.4 + 12.5 × 20.5 + 15.7 × 22.4 + 33.9 × 2.5 + 29.8 × 38.7 + 41.5 ×28.6 +0 × 0.3
= 3938.74
4. CONCLUSION:
In this paper, We took the example of Fuzzy
Transportation Problem Using Best Candidates Method ,
where the result arrived at using Octagonal Fuzzy
Numbers are more cost effective then Best Candidate
Method .we discussed finding Initial Basic solution and
Optimal Solution for Octagonal Fuzzy Transportation. The
transportation cost can be minimized by using of
Proposed Ranking Method under Best Candidates Method.
It is concluded that Octagonal Fuzzy Transportation
method proves to be minimum cost of Transportation.
REFERENCES:
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