On intuitionistic fuzzy transportation problem using hexagonal intuitionistic fuzzy numbers

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

DOI : 10.5121/ijfls.2015.5102 15

On Intuitionistic Fuzzy Transportation Problem Using Hexagonal Intuitionistic Fuzzy Numbers

A.Thamaraiselvi1 and R.Santhi

2

1Research scholar, Department of Mathematics, NGM College, Pollachi, India-642001

2Assistant Professor, Department of Mathematics, NGM College, Pollachi, India-642001

ABSTRACT

In this paper we introduce Hexagonal intuitionistic fuzzy number with its membership and non membership

functions. The main objective of this paper is to introduce an Intuitionistic Fuzzy Transportation problem

with hexagonal intuitionistic fuzzy number. The arithmetic operations on hexagonal intuitionistic fuzzy

numbers are performed. Based on this new intuitionistic fuzzy number, we obtain a initial basic feasible

solution and optimal solution of intuitionistic fuzzy transportation problem. The solutions are illustrated

with suitable example.

KEYWORDS

Intuitionistic fuzzy number, Hexagonal Intuitionistic fuzzy number, Hexagonal Intuitionistic Fuzzy

Transportation problem, Initial Basic Feasible Solution, Optimal Solution.

1. INTRODUCTION

The classical transportation problem refers to a special type of linear programming problem in

which a single homogeneous goods kept at various sources to various destinations in such a way

that the total transportation cost is minimum. The basic transportation problem was introduced

and developed by Hitchcock in 1941 in which the transportation costs, the supply and demand

quantities are crisp values. But in the real, the parameters of a transportation problem may be

uncertain due to many uncontrollable factors. To deal such fuzziness in decision making,

Bellmann and Zadeh[3] and Zadeh[12] introduced the concept of fuzziness. Many authors

discussed the solutions of fuzzy transportation problem(FTP) using various techniques. In 1982,

O’heigeartaigh [8] proposed an algorithm to solve FTP with triangular membership function. In

1996, Chanas and Kutcha [4] proposed a method to find the optimal solution to the transportation

problem with fuzzy coefficients. In 2010, Pandian and Natarajan [9] proposed a new algorithm

namely fuzzy zero point method to find optimal solution of a FTP with trapezoidal fuzzy

numbers.

Sometimes the concept of fuzzy set theory is not enough to deal the vagueness in transportation

problems. So intuitionistic fuzzy set (IFS) theory is introduced to deal the transportation

problems. In 1986, the idea of intuitionistic fuzzy sets introduced by Atanassov [1,2] to deal

vagueness or uncertainty. The main advantage of IFSs is that include both the degree of

membership and non membership of each element in the set.


16

In recent years, IFSs play a wide role in decision making in fuzzy environment. In 2012, Gani and

Abbas[6] solved intuitionistic fuzzy transportation problem(IFTP) with triangular membership

function using zero suffix algorithm. Agarval and Gupta proposed a ranking method to solve an

IFTP with generalized trapezoidal IFNs.

In this paper, we introduce IFTP with hexagonal intuitionistic fuzzy demand and supply. And we

obtain an initial basic feasible solution and optimal solution of the same.

The paper is organized as follows: In chapter 2, some basic definitions are given. In chapter 3,

hexagonal intuitionistic fuzzy numbers (HIFNs) are introduced and its basic arithmetic operations

are discussed. In chapter 4, mathematical formulation of Hexagonal IFTP is given. Also the

solution algorithms are given and they are illustrated with numerical examples. Finally, the paper

is concluded in chapter 5.

2. PRELIMINARIES

2.1. Definition (Fuzzy set [FS])

Let � be a nonempty set. A fuzzy set A� of X is defined as A� = �⟨x, μ��x ⟩/x ∈ X� where μ��x is called the membership function which maps each element of � to a value between 0

and 1.

2.2. Definition (Fuzzy Number [FN])

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 atleast one � ∈ ℝ with μ�� = 1. • μ�� is piecewise continuous.

2.3. Definition (Triangular Fuzzy Numbers [TFN])

A triangular fuzzy number �� is denoted by 3-tuples ��, ��, �� where ��, �� are real

numbers and �� ≤ �� ≤ �� with membership function defined as

μ�� ="#$#%� − �� − �� , '()�� ≤ � ≤ �� − �� − �� , '()�� ≤ � ≤ ��

0,(+ℎ-)./0-1


17

2.4. Definition (Trapezoidal Fuzzy Numbers [TrFN])

A trapezoidal fuzzy number[1] is denoted by 4-tuples �� = ��, ��, ��, �2 where ��, ��, �� 2 are real numbers and �� ≤ �� ≤ �� ≤ �2 with membership function defined as

μ�� ="#$#%� − �� − �� , '()�� ≤ � ≤ ��1,'()�� ≤ � ≤ ��2 − ��2 − �� , '()�� ≤ � ≤ �2

0,(+ℎ-)./0-1

2.5. Definition (Hexagonal fuzzy number [HFN])

A hexagon fuzzy number [10] � HA is specified by 6-tuples ��3 = ��, ��, ��, �2, �4, �5 where ��, ��, ��, �2, �4�� 5 are real numbers and �� ≤ �� ≤ �� ≤ �2 ≤ �4 ≤ �5 its membership

function is given below,

μ�� =

"####$####%12 7 � − �� − ��8 ,'()�� ≤ � ≤ ��12 + 12 7 � − �� − ��8 ,'()�� ≤ � ≤ ��1,'()�� ≤ � ≤ �21 − 12 7 � − �2�4 − �28 ,'()�2 ≤ � ≤ �412 7 �5 − ��5 − �48 ,'()�4 ≤ � ≤ �5

0,(+ℎ-)./0-

1

2.6. Definition (Intuitionistic Fuzzy set [IFS])

Let � be a nonempty set. An Intuitionistic fuzzy set �� : of X is defined as �� : = �⟨x, μ�� ;�x , ν�� ;�x ⟩/x ∈ X� where μ�� ;�x andν�� ;�x ] are membership and non

membership functions such that μ�� ;�x , ν�� ;�x : X → [0,1] and 0 ≤ C�� ;�� + ν�� ;�� ≤ 1 for all � ∈ X.

2.7. Definition (Intuitionistic Fuzzy Number [IFN])

An Intuitionistic fuzzy subset �� : = D⟨x, μ�� ;�x , ϑ�� ;�x ⟩/x ∈ � F of the real line � is called an

IFN if the following conditions hold:

(i) There exists m ∈ ℝ such that μ�� ;�m = 1 and ν�� ;�m = 0

(ii) μ� is a continuous function from ℝ → [0,1] such that 0 ≤ μ�� ;�x + ν�� ;�x ≤ 1 for all x ∈ X.

(iii) The membership and non membership functions of �� : are in the following form:


18

µ�� ; ="#$#%0, − ∞ < � ≤ ��'�� ,�� ≤ � ≤ ��1,� = ��J�� ,�� ≤ � ≤ ��0,�� ≤ � < ∞

1 ν�� ; ="#$#%1, − ∞ < � ≤ ��′' ′�� ,��′ ≤ � ≤ ��0,� = ��J′�� ,�� ≤ � ≤ ��′1,��′ ≤ � < ∞

1

Where ', ' ′, J, J′ are functions fromℝ → [0,1], ' and J′are strictly increasing

functions and J and ' ′ are strictly decreasing functions with the conditions 0 ≤ '�� + ' ′�� ≤ 1 and 0 ≤ J�� + J′�� ≤ 1 .

2.8. Definition (Triangular Intuitionistic Fuzzy Numbers [TIFN])

A triangular Intuitionistic fuzzy number ��: is denoted by ��: = ��, ��, �� , ��K, ��, ��K where ��K ≤ �� ≤ �� ≤ �� ≤ ��K with the following membership μ�� ;�x and non membership function

ν�� ;�x :

µ�� ="#$#%� − �� − �� , '()�� ≤ � ≤ �� − �� − �� , '()�� ≤ � ≤ ��

0,(+ℎ-)./0-1 ν��

="#$#%�� − �� − ��′ , '()�� ≤ � ≤ ��x − ��′ − �� , '()�� ≤ � ≤ ��

1,(+ℎ-)./0-1

2.9. Definition (Trapezoidal Intuitionistic Fuzzy Numbers [TrIFN])

A trapezoidal Intuitionistic fuzzy number is denoted by ��: = ��, ��, ��, �2 , ��K, ��, ��, �2K where ��K ≤ �� ≤ �� ≤ �� ≤ �2 ≤ �2K with membership and non membership functions are

defined as follows

μ�� ="#$#%� − �� − �� , '()�� ≤ � ≤ ��1,'()�� ≤ � ≤ ��2 − ��2 − �� , '()�� ≤ � ≤ �2

0,(+ℎ-)./0-1ν�� =

"#$#%� − �� − �� , '()�� ≤ � ≤ ��1,'()�� ≤ � ≤ ��2 − ��2 − �� , '()�� ≤ � ≤ �2

0,(+ℎ-)./0-1


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3. HEXAGONAL INTUITIONISTIC FUZZY NUMBER

3.1. Definition (Hexagonal Intuitionistic fuzzy number [HIFN])

A hexagonal intuitionistic fuzzy number is specified by ��3: = ��, ��, ��, �2, �4, �5 , ��K, ��K, ��, �2, �4K, �5K where ��, ��, ��, �2, �4, �5, ��K, ��K, �4K�� 5K are real numbers such that ��K ≤ �� ≤ ��K ≤ �� ≤�� ≤ �2 ≤ �4 ≤ �4K ≤ �5 ≤ �5K and its membership and non membership functions are given

below,

μ�� =

"###$###% �� L MNOPOQNOPR ,'()�� ≤ � ≤ ��

��+ �� L MNOQOSNOQR ,'()�� ≤ � ≤ ��1,'()�� ≤ � ≤ �21 − �� L MNOTOUNOTR ,'()�2 ≤ � ≤ �4�� L OVNMOVNOUR ,'()�4 ≤ � ≤ �50,(+ℎ-)./0-

1and

ν�� =

"####$####% 1 −12W � − ��′��′ − ��′Y ,'()��′ ≤ � ≤ ��′

12 7 �� − �� − ��′8 ,'()��′ ≤ � ≤ ��0,'()�� ≤ � ≤ �212 7 � − �2�4′ − �28 ,'()�2 ≤ � ≤ �4′

12 + 12W � − �4′�5′ − �4′Y ,'()�4′ ≤ � ≤ �5′1,(+ℎ-)./0-

1

3.2. Graphical representation of Hexagonal Intuitionistic fuzzy numbers

Figure 1.Hexagonal

IFN ��3: = ��, ��, ��, �2, �4, �5 , ��K, ��K, ��, �2, �4K, �5K


20

3.3. Arithmetic operations on Hexagonal Intuitionistic fuzzy numbers

Let ��3: = ��, ��, ��, �2, �4, �5 , ��K, ��K, ��, �2, �4K, �5K and

Z�3: = �[�, [�, [�, [2, [4, [5 , �[�K, [�K, [�, [2, [4K, [5K be two HIFNs. Then

• ��3: + Z�3: = �� + [�, �� + [�, �� + [�, �2 + [2, �4 + [4, �5 + [5 , ��K + [�K, ��K + [�K, �� + [�, �2 + [2, �4K + [4K, �5K + [5K .

• ��3: − Z�3: = �� − [5, �� − [4, �� − [2, �2 − [�, �4 − [�, �5 − [� , ��K − [5K, ��K − [4K, �� − [2, �2 − [�, �4K − [�K, �5K − [�K .

• ��3: ∗ Z� 3: = �]�, ]�, ]�, ]2, ]4, ]5 where ]� = ^/�/^_^��[�, ��[5, �5[�, �5[5 ]� = ^/�/^_^��[�, ��[4, �4[�, �4[4 ]� = ^/�/^_^��[�, ��[2, �2[�, �2[2 ]2 = ^��/^_��[�, ��[2, �2[�, �2[2 ]4 = ^��/^_^��[�, ��[4, �4[�, �4[4 ]5 = ^��^_^��[�, ��[5, �5[�, �5[5

• `��3: = a�`��, `��, `��, `�2, `�4, `�5 , �`��K, `��K, `��, `�2, `�4K, `�5K /'` > 0�`�5, `�4, `�2, `��, `��, `�� , �`�5K, `�4K, `�2, `��, `��K, `��K /'` < 01

3.4. Ranking of Hexagonal Intuitionistic fuzzy numbers

The ranking function[11] of a HIFN ��3: = ��, ��, ��, �2, �4, �5 , ��K, ��K, ��, �2, �4K, �5K maps the set of all fuzzy numbers to a set of real numbers defined as ��3: = cd�Je L��3:R ,d�Jν��3: f where d�Je L��3:R = �OPg�OQg2OSg2OTg�OUg�OV�h and

d�Jν L��3:R = �OPig�OQig2OSg2OTg�OUig�OVi�h .

Note:

If ��3:and Z�3:are any two HIFNs. Then

1. ��3: < Z�3:if d�Je L��3:R < d�Je LZ� 3:R and d�Jν L��3:R < d�Jν LZ� 3:R 2. ��3: > Z�3:if d�Je L��3:R > d�Je LZ� 3:R and d�Jν L��3:R > d�Jν LZ� 3:R 3. ��3: = Z�3:if d�Je L��3:R = d�Je LZ�3:R and d�Jν L��3:R = d�Jν LZ�3:R

Example: ��3: =(3,5,7,9,12,15),(2,4,7,9,13,17) and Z�3: =(3,4,5,6,8,10)(2,4,5,6,10,12) d�Je L��3:R =8.38 and d�Jν L��3:R =8.5

d�Je LZ� 3:R = 5.89 and d�Jν LZ� 3:R = 6.33

Since d�Je L��3:R > d�Je LZ� 3:Rand d�Jν L��3:R > d�Jν LZ� 3:R, ��3: >Z�3:.


21

4. HEXAGONAL INTUITIONISTIC FUZZY TRANSPORTATION PROBLEM

(HIFTP)

4.1. Mathematical formulation

Consider a transportation problem with ‘m’ sources and ‘n’ destinations. The mathematical

formulation of the HIFTP whose parameters are HIFNs under the case that the total supply is

equivalent to the total demand is given by

Minimize j: = ∑ ∑ �lmn:onpqrmpq s̃mn: subject to∑ �lmn:onpq = �lm:, / = 1,2,… ,^

∑ �lmn:rmpq = [�n:, v = 1,2, … , � and �lmn: ≥ 0∀/, v. In the above model the transportation costss̃mn: , supplies �lm: and the demands [�n: are HIFNs.

4.2. Intuitionistic Fuzzy Initial Basic Feasible Solution (IFIBFS)

A feasible solution to a ‘m’ sources and ‘n’ destinations transportation problem is said to be basic

feasible solution if the number of positive allocations are ‘m+n-1’.Here the IFIBFS is obtained by

Vogel’s Approximation method(VAM). The method proceeds as follows.

Step 1: Calculate the magnitude of difference between the minimum and next to minimum

transportation cost in each row and column and write it as “Diff.” along the side of the

table against the corresponding row/column.

Step 2: In the row /column corresponding to maximum “Diff.”, make the maximum allotment at

the box having minimum transportation cost in that row/ column.

Step 3: If the maximum “Diff.” corresponding to two or more rows or columns are equal, select

the top most row and the extreme left column.

Repeat the above procedure until all the HIF supplies are fully used and IF demands are fully

received.

4.3. Intuitionistic Fuzzy Optimal Solution (IFOS)

The optimality algorithm [7] is as follows:

Step 1: construct the transportation table for the given problem.

Step 2: Subtract each row entries of the table from the row minimum.

Step 3: In the table obtained from step 1, subtract each column entries from the column

minimum. Now there will be atleast one zero in each row and column in the resultant

table.

Step 4: In the above resultant table, for every zero, count the total number of zeros in the

corresponding row and column. Suppose (i,j)th zero is selected , count the total number of zeros

in the ith row and j

th column.

Step 5: Now select a zero for which the number of zeros counted in step 4 is minimum. And

allocate the maximum possible hexagonal Intuitionistic Fuzzy quantity to that cell. If tie

occurs for some zeros in step 4 then find the sum of all the elements in the corresponding

row and column. For example, for (k,l)th zero, find the sum of all the elements in the k

th row

and lth column. Now choose the zero with maximum sum and allocate the maximum possible

quantity to that cell.


22

Step 6: After every allocation, delete the row or column for which the demand fulfilled and the

supply is depleted.

Step 7: Repeat step 3 to step 6 until all the demands are satisfied and all the supplies are

exhausted.

4.4. Numerical example

4.4.1. Hexagonal IFIBFS

Consider the following IFTP with hexagonal intuitionistic fuzzy demands and supplies. Destinations

IF Supply D1 D2 D3 D4

Ori

gin

s

O

1 5 6 12 9

(7,9,11,13,16,2

0)

(5,7,11,13,19,2

3)

O

2 3 2 8 4

(6,8,11,14,19,2

5)

(4,7,11,14,21,2

7)

O

3 7 11 20 9

(9,11,13,15

,18,20)

(8,10,13,15,19,

22)

IF

Dema

nd

(3,4,5,6,8,1

0)

(2,4,5,6,10,

12)

(3,5,7,9,12,

15)

(2,4,7,9,13,

17)

(6,7,9,11,13,

16)

(5,6,9,11,16,

18)

(10,12,14,16,20

,24)

(8,10,14,16,20,

25)

Solution:

The Intuitionistic Fuzzy IBFS of the above IFTP can be obtained by VAM as follows:

Now using Step 1 of the VAM calculate the value “Diff” for each row and column as mentioned

in the last row and column the following table. Table 1

Destinations IF Supply

Di

ff D1 D2 D3 D4

Ori

gin

s

O

1 5 6 12 9

(7,9,11,13,16,

20)

(5,7,11,13,19,

23)

1

O

2 3 2 8 4

(6,8,11,14,19,

25)

(4,7,11,14,21,

27)

1


23

O

3 7 11 20 9

(9,11,13,15

,18,20)

(8,10,13,15,19

,22)

2

IF Dem

and

(3,4,5,6,8,1

0)

(2,4,5,6,10,

12)

(3,5,7,9,12,1

5)

(2,4,7,9,13,1

7)

(6,7,9,11,13,1

6)

(5,6,9,11,16,1

8)

(10,12,14,16,2

0,24)

(8,10,14,16,20,

25)

Diff. 2 4 4 5

▲

Using the step 2 identify the row/column corresponding to the highest value of “Diff”. In this

case it occurs at column 4. In this column minimum cost cell is (2,4). And the corresponding

demand and supply are (10,12,14,16,20,24)(8,10,14,16,20,25) and (6,8,11,14,19,25)

(4,7,11,14,21,27) respectively. Now allocate the (minimum of the above demand and supply)

maximum possible units (6,8,11,14,19,25) (4,7,11,14,21,27) to the minimum cost position (2, 4).

And write the remaining in column 4. After removing the second row repeats the step 1, we

obtain the table Table 2


Diff

. D1 D2 D3 D4

Ori

gin

s

O

1 5 6 12 9

(7,9,11,13,16,20)

(5,7,11,13,19,23) 1

O

2 - - -

(6,8,11,14,19,2

5)

(4,7,11,14,21,2

7)

- -

O

3 7 11 20 9

(9,11,13,15

,18,20)

(8,10,13,15,19,2

2)

2

IF

Deman

d

(3,4,5,6,8,10)

(2,4,5,6,10,1

2)

(3,5,7,9,12,1

5)

(2,4,7,9,13,1

7)

(6,7,9,11,13,1

6)

(5,6,9,11,16,1

8)

(-15,-7,0,5,12,18)

(-19,-

11,0,5,13,21)

Diff. 2 5 8

▲

0

In the above Table 2 the highest value of “Diff” occurs at third column. Now allocate the

maximum possible units (6,7,9,11,13,16) (5,6,9,11,16,18) to the minimum cost position (1,3).And

write the remaining in first row. After removing the third column repeat the steps 1 to 3. Now

highest value of “Diff” occurs at second column. Now allocate the maximum possible units (-9,-

4,0,4,9,14) (-13,-9,0,4,13,18) to the minimum cost position (1,2).After writing the remaining in

column 2, remove the first row and repeats the step 1, we obtain the table


24

Table 3


Diff

. D1 D2 D3 D4

Ori

gin

s

O

1 -

(-9,-

4,0,4,9,14)

(-13,-

9,0,4,13,18

)

(6,7,9,11,13,16

)

(5,6,9,11,16,18

)

- - -

O

2 - - -

(6,8,11,14,19,25

)

(4,7,11,14,21,27

)

- -

O

3 7 11 - 9

(9,11,13,15

,18,20)

(8,10,13,15,19,22

)

2

IF

Demand

(3,4,5,6,8,10)

(2,4,5,6,10,12

)

(-11,-

4,3,9,16,24

)

(-19,-

9,3,9,22,30

)

- (-15,-7,0,5,12,18)

(-19,-11,0,5,13,21)

Diff. 2 - - -

Now allocate the remaining demands and supplies, we get the following complete allocation

table. Table 4


D1 D2 D3 D4

Ori

gin

s

O1 -

(-9,-

4,0,4,9,14)

(-13,-

9,0,4,13,18)

(6,7,9,11,13,16)

(5,6,9,11,16,18) - -

O2 - - - (6,8,11,14,19,25)

(4,7,11,14,21,27) -

O3 (3,4,5,6,8,10)

(2,4,5,6,10,12)

(-11,-

4,3,9,16,24)

(-19,-

9,3,9,22,30)

- (-15,-7,0,5,12,18)

(-19,-11,0,5,13,21) -

IF

Demand - - - -

Therefore, the intuitionistic fuzzy IBFS in terms of HIFNs for the given IFTP is,

�� = �−9,−4,0,4,9,14 �−13,−9,0,4,13,18 , �� = �6,7,9,11,13,16 �5,6,9,11,16,18 ��2 = �6,8,11,14,19,25 �4,7,11,14,21,27 , �� = �3,4,5,6,8,10 �2,4,5,6,10,12 �� = �−11,−4,3,9,16,24 �−19,−9,3,9,22,30 , ��= �−15,−7,0,5,12,18 �−19,−11,0,5,13,21 And the minimum total fuzzy transportation cost is given by ,


25

Minimize j: = 6�−9,−4,0,4,9,14 �−13,−9,0,4,13,18 + 12�6,7,9,11,13,16 �5,6,9,11,16,18 +

4�6,8,11,14,19,25 �4,7,11,14,21,27 + 7�3,4,5,6,8,10 �2,4,5,6,10,12 +

11�−11,−4,3,9,16,24 �−19,−9,3,9,22,30 +

9�−15,−7,0,5,12,18 �−19,−11,0,5,13,21 = �−193,13,220,398,626,872 �−335,−124,220,398,783,1035

4.4.2. Hexagonal Intuitionistic Fuzzy Optimal Solution

The optimum solution is illustrated by the following example.


D1 D2 D3 D4

Ori

gin

s

O

1 5 6 12 9

(7,9,11,13,16,20)

(5,7,11,13,19,23)

O

2 3 2 8 4

(6,8,11,14,19,25)

(4,7,11,14,21,27)

O

3 7 11 20 9

(9,11,13,15

,18,20)

(8,10,13,15,19,22

)

IF

Deman

d

(3,4,5,6,8,10

)

(2,4,5,6,10,1

2)

(3,5,7,9,12,1

5)

(2,4,7,9,13,1

7)

(6,7,9,11,13,

16)

(5,6,9,11,16,

18)

(10,12,14,16,20,

24)

(8,10,14,16,20,2

5)

Solution:

Now using Step 2 and step3 of the optimality algorithm [7] we get the following table in which

there will be at least one zero in each row and column Table 5


D1 D2 D3 D4

Ori

gin

s

O1 0 1 0* 2

(7,9,11,13,16,20)

(5,7,11,13,19,23)

O2 1 0 1 0

(6,8,11,14,19,25)

(4,7,11,14,21,27)

O3 0 4 8 0

(9,11,13,15

,18,20)

(8,10,13,15,19,22

)

IF

Deman

d

(3,4,5,6,8,10

)

(2,4,5,6,10,1

2)

(3,5,7,9,12,1

5)

(2,4,7,9,13,1

7)

(6,7,9,11,13,1

6)

(5,6,9,11,16,1

8)

(10,12,14,16,20,

24)

(8,10,14,16,20,2

5)


26

Now using the step 4 and step 5 for the above table. ie For every zero count the total number of

zeros in the corresponding row and column. Since the zero in the cell (1,3) has minimum

number(2) of zeros with maximum sum(12) of elements in the first row and third column. Now

allocate the maximum possible units (6,7,9,11,13,16)(5,6,9,11,16,18) to the position (1,3).And

write the remaining in row 1. After removing the third column we obtain the following table.

Table 6


D1 D2 D3 D4

Ori

gin

s

O1 0 1

(6,7,9,11,13,1

6)

(5,6,9,11,16,1

8)

2 (-9,-4,0,4,9,14)

(-13,-9,0.4,13,18)

O2 1 0* - 0

(6,8,11,14,19,25)

(4,7,11,14,21,27)

O3 0 4 - 0

(9,11,13,15

,18,20)

(8,10,13,15,19,22)

IF

Deman

d

(3,4,5,6,8,10

)

(2,4,5,6,10,1

2)

(3,5,7,9,12,1

5)

(2,4,7,9,13,1

7)

-

(10,12,14,16,20,2

4)

(8,10,14,16,20,25

)

Now repeating the steps 4 and 5 , allocate the maximum possible units

(3,5,7,9,12,15)(2,4,7,9,13,17) to the position (2,2).And write the remaining in second row. After

removing the second column again apply the steps 4 to 6 we obtain the following table.

Table 7


D1 D2 D3 D4

Ori

gin

s

O

1

(-9,-

4,0,4,9,14)

(-13,-

9,0.4,13,18

)

-

(6,7,9,11,13,16

)

(5,6,9,11,16,18

)

2 -

O

2 1

(3,5,7,9,12,15

)

(2,4,7,9,13,17

)

- 0

(-9,-

4,2,7,14,22)

(-13,-

6,2,7,17,25)

O

3 0 - - 0

(9,11,13,15

,18,20)

(8,10,13,15,19,

22)

IF

Deman

d

(-11,-

5,1,6,12,19

)

(-16,-

9,1,6,19,25

)

- -

(10,12,14,16,20,24

)

(8,10,14,16,20,25)


27

Now allocate the remaining to fulfill the demand and supply we the following allocation table.

Table 8


D1 D2 D3 D4

Ori

gin

s

O1 (-9,-,0,4,9,14)

(-13,-

9,0.4,13,18)

- (6,7,9,11,13,16)

(5,6,9,11,16,18) 2 -

O2 1 (3,5,7,9,12,15)

(2,4,7,9,13,17) -

(-9,-

4,2,7,14,22)

(-13,-

6,2,7,17,25)

-

O3

(-11,-

5,1,6,12,19)

(-16,-

9,1,6,19,25)

- -

(-10,-

1,7,14,23,31)

(-17,-

9,7,14,28,38)

-

IF

Demand - - - -

Therefore, the intuitionistic fuzzy optimal solution in terms of HIFNs for the given IFTP is, �� = �−9,−4,0,4,9,14 �−13,−9,0,4,13,18 ,�� = �6,7,9,11,13,16 �5,6,9,11,16,18 �� = �3,5,7,9,12,15 �2,4,7,9,13,17 ,��2 = �−9,−4,2,7,14,22 �−13,−6,2,7,17,25 �� = �−11,−5,1,6,12,19 �−16,−9,1,6,19,25 , ��2= �−10,−1,7,14,23,31 �−17,−9,7,14,28,38 And the total minimum fuzzy transportation cost is given by, Minimizej: = 5�−9,−4,0,4,9,14 �−13,−9,0,4,13,18 + 12�6,7,9,11,13,16 �5,6,9,11,16,18 +2�3,5,7,9,12,15 �2,4,7,9,13,17 + 4�−9,−4,2,7,14,22 �−13,−6,2,7,17,25 +7�−11,−5,1,6,12,19 �−16,−9,1,6,19,25 +9�−10,−1,7,14,23,31 �−17,−9,7,14,28,38 = �−170,−6,200,366,572,792 �−318,−133,200,366,736,957

5. CONCLUSION

Usually the transportation problems are discussed with triangular intuitionistic fuzzy numbers or

trapezoidal intuitionistic fuzzy numbers. In the present paper Hexagonal Intuitionistic Fuzzy

number has been newly introduced to deal IFTP. The arithmetic operations on hexagon

intuitionistic fuzzy numbers are employed to find the solutions. Intuitionistic fuzzy problems with

six parameters can be solved by introducing HIFNs. For future research we propose generalized

Hexagonal Intuitionistic fuzzy numbers to deal problems in intuitionistic fuzzy environment.


28

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