Solving Transportation Problems with Hexagonal Fuzzy Numbers Using Best Candidates Method and Different Ranking Techniques

Dr. M. S. Annie Christi Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 2, (Part - 4) February 2016, pp.76-81
www.ijera.com 76|P a g e
Solving Transportation Problems with Hexagonal Fuzzy
Numbers Using Best Candidates Method and Different Ranking
Techniques
Dr. M. S. Annie Christi, Mrs. Malini. D,
Department of Mathematics, Providence College for Women, Coonoor.
ABSTRACT
In this paper, we introduce a Fuzzy Transportation Problem (FTP) in which the values of transportation costs are
represented as hexagonal fuzzy numbers. We use the Best candidate method to solve the FTP. The Centroid
ranking technique is used to obtain the optimal solution.
Keywords:Fuzzy transporation problem, Hexagonal fuzzy number, Best candidate method and centroid
ranking.
I. PRELIMINARIES:
1.1. Definition (Fuzzy set [FS]):[8] [3]
Let X be a nonempty set . A fuzzy set Ã of X is defined as Ã = {(x,𝜇 𝐴(𝑥))/𝑥 ∈ 𝑋} where 𝜇 𝐴(𝑥) is called
the membership function which maps each element of X to a value between 0 and 1.
1.2. Definition (Fuzzy Number [FN]):[8]
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. This
weight is called the membership function.
A fuzzy number is a convex normalized fuzzy set on the real line R such that:
• There exist at least one 𝑥 ∈ 𝑅with 𝜇 𝐴(𝑥) = 1
• 𝜇 𝐴(𝑥) is piecewise continuous
1.3. Definition (Triangular fuzzy numbers [TFN]): [8]
For a triangular fuzzy number A(x), it can be represented by A (a, b, c; 1) with the membership function
μ(x) is given by
(𝑥−𝑎)
(𝑏−𝑎)
, a ≤ x ≤ b
𝜇 𝑥 = 1 , x= b
(𝑐−𝑥)
(𝑐−𝑏)
, c ≤ x ≤ d
0 , otherwise
1.4. Definition (Trapezoidal fuzzy numbers [TrFN]): [8]
A Fuzzy number A͂ defined on the universal set of real numbers R denoted as A͂ = (a, b, c, d; 1) is said to be
a trapezoidal fuzzy number if its membership function 𝜇 𝐴= (x) is given by
(𝑥−𝑎)
(𝑏−𝑎)
, a ≤ x ≤ b
𝜇 𝐴(𝑥) = 1 , b ≤ x ≤ c
(𝑑−𝑥)
(𝑑−𝑐)
, c ≤ x ≤ d
0 , otherwise
RESEARCH ARTICLE OPEN ACCESS

1.5. Definition (Hexagonal fuzzy numbers [HFN]): [5] [8]
A Fuzzy number AH is denoted by AH = (a1, a2, a3, a4, a5, a6) where a1, a2, a3, a4, a5and a6the real numbers
and its membership function are given below:
2. RANKING OF HEXAGONAL FUZZY NUMBER: [7] [9]
The parametric methods of comparing fuzzy numbers, especially in fuzzy decision making theory are more
efficient than non-parametric methods. Cheng’s centroid point method [6], Chu and Tsao’smethod [7],
Abbasbandy and Assady’s[1] sign-distance method was all non-parametric and was applicable only for normal
fuzzy numbers. The non-parametric methods for comparing fuzzy numbers have some drawbacks in practice.
A͂ cis a scaling of fuzzy number defined by
A͂ c =
3√3
4
(𝑎1+ 𝑎3+𝑎6) k + (𝑎2+ 𝑎4+𝑎5) 1-k
3. THE BEST CANDIDATES METHOD (BCM) HAS THE FOLLOWING SOLUTION
STEPS: [1][2]
Step1:
Prepare the BCM matrix. If the matrix unbalanced, we balance it and don't use the added row or column
candidates in our solution procedure.
Step2:
Select the best candidates that are 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 3
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 optimal solution.
4. PROPOSED METHOD: [1][2]
In this study, we proposed a new solving method for transportation problems by using BCM. The proposed
method must operate the as following:
Step1:
We must check the matrix balance, if the total supply is equal to the total demand, then the matrix is
balanced and also apply Step 2. If the total supply is not equal to the total demand, then we add a dummy row or

column as needed to make supply is equal to the demand. So the transportation costs in this row or column will
be assigned to zero.
Step2:
Appling BCM to determine the best combination that is to produce the lowest total weight of the costs,
where is one candidate for each row and column.
Step3:
Identify the row with the smallest cost candidate from the chosen combination. Then allocate the demand
and the supply as much as possible to the variable with the least unit cost in the selected row or column. Also,
we should adjust the supply and demand by crossing out the row/column to be then assigned to zero. If the row
or column is not assigned to zero, then we check the selected row if it has an element with lowest cost
comparing to the determined element in the chosen combination, then we elect it.
5. Numerical example
Consider the following problem with hexagonal fuzzy numbers:
D1 D2 D3 Supply
S1 (3,7,11,15,19, 24) (3,5,7,9,10,12) (11,14,17,21,25,30) (7,9,11,13,16,20)
S2 (3,5,7,9,10,12) (5,7,10,13,17,21) (7,9,11,14,18,22) (6,8,11,14,19,25)
S3 (7,9,11,14,18,22) (2,3,4,6,7,9) (5,7,8,11,14,17) (9,11,13,15,18,20)
Demand (6,9,12,15,20,25) (6,7,9,11,13,16) (10,12,14,16,20,24)
Using the ranking given above the given transportation problem can be reduced as follows:
D1 D2 D3 Supply
S1 54.16 78.72 50.40
S2 49.23 53.91 55.20
S3 53.91 57.42
Demand 58.33 41.30 63.39
From the above selected candidates we elect the best among them.
The optimal solution using best candidate method is given as follows:
(31.44)*(41.30) + (78.72)*(9.1) + (31.44)*(55.20) + (53.91)*(3.13) + (41.31)*(54.29) =6161.77
D1 D2 D3 Supply
S1 54.16
41.30
9.1
78.72
50.40
41.30
0
S2
55.20
49.23 53.91
55.20
0
S3
3.13
53.91
20.91
54.29 57.42
54.29
0
Demand
58.33
3.13
0
41.30
0
63.39
9.1
0
31.44
31.44
44
20.91 41.31
11
31.44
31.44
44
41.31
11

6. RANKING OF HEXAGONAL FUZZY NUMBERS: [6] [5] [8]
6.1. Proposed centroid ranking Method:
Fig.1 Generalized hexagonal fuzzy number
The centroid of a hexagonal fuzzy number is considered to be the balancing point of the hexagon
(Fig.1).Divide the hexagonal into three plane figures .These three plane figures are a Triangle ABQ, Hexagon
CDERQB and again a triangle REF respectively. The circumcenter of the centroids of these three plane figures
is taken as the point of reference to define the ranking of generalized Hexagonal fuzzy numbers. Let the centroid
of the three plane figures be G1, G2, G3, respectively.
The centroid of the three plane figures isG1=
𝑎1+𝑎2+𝑎3
3
,
𝑤
6
,
G2 =
𝑎2+2𝑎3+2𝑎4+𝑎5
6
,
𝑤
2
and G3 =
𝑎4+𝑎5+𝑎6
3
,
𝑤
6
respectively.
Equation of the line G1, G3 is y =
𝑤
6
and G2 does not lie on the line G1, G3. Thus G1, G2 and G3 are non collinear
and they form a triangle. We define the centroid 𝐺 𝐴̃ 𝐻
(𝑥0, 𝑦0) of the triangle with vertices G1, G2 and G3 of the
generalized trapezoidal fuzzy number
𝐴̃ 𝐻= (𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6; 𝑤) as
𝐺 𝐴̃ 𝐻
(𝑥0, 𝑦0) =
2𝑎1+3𝑎2+4𝑎3+4𝑎4+3𝑎5+2𝑎6
18
,
5𝑤
18
--------- (1)
The ranking function of the generalized hexagonal fuzzy number 𝐴̃ 𝐻= (𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6; 𝑤) which maps the
set of all fuzzy numbers to a set of real numbers is definedas:
R (𝐴̃ 𝐻) = (𝑥0)(𝑦0) =
2𝑎1+3𝑎2+4𝑎3+4𝑎4+3𝑎5+2𝑎6
18
5𝑤
18
------- (2)
This is the area between the centroid of the centroids 𝐺 𝐴̃ 𝐻
(𝑥0, 𝑦0) as defined in (1) and (2) the original point.
The mode of the generalized hexagonal fuzzy number 𝐴̃ 𝐻= (𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6; 𝑤) is defined as
Mode =
1
2
(𝑎3 + 𝑎4)𝑑𝑥
𝑤
0
--------------- (3)
The divergence of the generalized hexagonal fuzzy number 𝐴̃ 𝐻= (𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6; 𝑤) is defined as
Divergence = (𝑎6 − 𝑎1)𝑑𝑥
𝑤
0
-------------- (4)
The left spread of the generalized hexagonal fuzzy number 𝐴̃ 𝐻= (𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6; 𝑤) is defined as
Left spread𝑙𝑠 = (𝑎2 − 𝑎1)𝑑𝑥
𝑤
2
0
+ (𝑎3 − 𝑎2)𝑑𝑥
𝑤
𝑤
2

𝑙𝑠 = (𝑎3 − 𝑎1)𝑑𝑥
𝑤
0
----------------- (5)
The right spread of the generalized hexagonal fuzzy number 𝐴̃ 𝐻= (𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6; 𝑤) is defined as
Right spread 𝑟𝑠= (𝑎6 − 𝑎5)𝑑𝑥
𝑤
2
0
+ (𝑎5 − 𝑎4)𝑑𝑥
𝑤
𝑤
2
𝑟𝑠 = (𝑎6 − 𝑎4)𝑑𝑥
𝑤
0
----------------- (6)
6.2. Numerical example:
Consider the following problem with hexagonal fuzzy numbers:
D1 D2 D3 Supply
S1 (3,7,11,15,19, 24) (3,5,7,9,10,12) (11,14,17,21,25,30) (7,9,11,13,16,20)
S2 (3,5,7,9,10,12) (5,7,10,13,17,21) (7,9,11,14,18,22) (6,8,11,14,19,25)
S3 (7,9,11,14,18,22) (2,3,4,6,7,9) (5,7,8,11,14,17) (9,11,13,15,18,20)
Demand (6,9,12,15,20,25) (6,7,9,11,13,16) (10,12,14,16,20,24)
Using the ranking given above the given transportation problem can be reduced as follows:
D1 D2 D3 Supply
S1 3.64 5.42 3.47
S2 3.36 2.95 3.75
S3 2.95 3.96
Demand 3.96 2.84 4.38
From the above selected candidates we elect the best among them.
D1 D2 D3 Supply
S1
0.21
3.64
2.84
0.42
5.42
3.47
3.26
0.42
0
S2
3.75
3.36 2.95
3.75
0
S3 2.95
3.96
3.96
0
Demand
3.96
0.21
0
2.84
0
4.38
3.98
0
The optimal solution using best candidate method is given as follows:
= (3.64)(0.26)+(2.15)(2.84)+(5.42)(0.42)+(2.15)(3.75)+(2.82)(3.96) = 28.3765
7. Conclusion:
In this paper, the transportation costs, sources and demands are considered as fuzzy quantifiers which are
represented by Hexagonal fuzzy numbers. By using Best Candidate method, the numerical shows that we can
get the optimal total cost. The fuzzy transportation problem canbe transformed into crisp transportation problem
by using the centroid ranking of fuzzy numbers. This gives a way for solving various fuzzy transportation
problems in a simpler form.
2.15
2.15
1.42 2.82
2.15
2.15
1.42
2.82

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