On Intuitionistic Fuzzy Transportation Problem Using Hexagonal Intuitionistic Fuzzy Numbers by IJFLS

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

On Intuitionistic Fuzzy Transportation Problem Using Hexagonal Intuitionistic Fuzzy Numbers A.Thamaraiselvi1 and R.Santhi2 1

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

DOI : 10.5121/ijfls.2015.5102

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

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])

of X is defined as A = 〈x, μ x 〉/ x ∈ X where Let be a nonempty set. A fuzzy set A 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

− '() ≤ ≤ % − , # 1 μ = − , '() ≤ ≤ $ − # " 0, (+ℎ-)./0-

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

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

− '() ≤ ≤ % − , # 1, '() ≤ ≤ 1 μ = − $ 2 , '() ≤ ≤ 2 # 2 − " 0, (+ℎ-)./0-

2.5. Definition (Hexagonal fuzzy number [HFN]) A hexagon fuzzy number [10] AH 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, 1 − 8 , '() ≤ ≤ % 7 2 − # # 1 1 − # 2 + 2 7 − 8 , '() ≤ ≤ # 1, '() ≤ ≤ 2 1 μ = $ 1 − 1 7 − 2 8 , '() ≤ ≤ 2 4 # 2 4 − 2 # 1 − # 7 5 8 , '() 4 ≤ ≤ 5 2 5 − 4 # " 0, (+ℎ-)./0-

2.6. Definition (Intuitionistic Fuzzy set [IFS])

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

x ,

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

2.7. Definition (Intuitionistic Fuzzy Number [IFN]) : = D〈x, μ An Intuitionistic fuzzy subset F of the real line is called an ; x , ϑ ; x 〉/ x ∈ 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. : are in the following form: (iii) The membership and non membership functions of 17

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

µ ;

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

Where ', ' ′ , J, J′ are functions from ℝ → [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

x ν : ; − '() ≤ ≤ % − , # 1 ν µ = − '() ≤ ≤ $ − , # " 0 , (+ℎ-)./0 − '() ≤ ≤ % − ′ , # 1 = x − '() ≤ ≤ $ ′ − , # " 1 , (+ℎ-)./0-

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

A trapezoidal Intuitionistic fuzzy number is denoted by : = , , , 2 , K , , , 2 K where K ≤ ≤ ≤ ≤ 2 ≤ 2 K with membership and non membership functions are defined as follows − − '() ≤ ≤ '() ≤ ≤ % − , % − , # # 1, '() ≤ ≤ 1 1, '() ≤ ≤ 1 ν = − μ = − $ 2 $ 2 , '() ≤ ≤ 2 , '() ≤ ≤ 2 # 2 − # 2 − " " 0, (+ℎ-)./00, (+ℎ-)./0-

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

3. HEXAGONAL INTUITIONISTIC FUZZY NUMBER 3.1. Definition (Hexagonal Intuitionistic fuzzy number [HIFN]) A hexagonal intuitionistic fuzzy number is specified by : K K K K 3 = , , , 2 , 4 , 5 , , , , 2 , 4 , 5 where , , , 2 , 4 , 5 , K , K , 4 K 5 K are real numbers such that K ≤ ≤ K ≤ ≤ ≤ 2 ≤ 4 ≤ 4 K ≤ 5 ≤ 5 K and its membership and non membership functions are given below, MNOP % LOQ NOP R , '() ≤ ≤ # MNOQ # + LOS NOQR , '() ≤ ≤ # 1, '() ≤ ≤ 2 1 μ = and $ 1 − L MNOT R , '() 2 ≤ ≤ 4 OU NOT # # L OV NM R , '() ≤ ≤ 4 5 # O NO "

0, (+ℎ-)./01 − ′ % 1 − W Y , '() ′ ≤ ≤ ′ 2 ′ − ′ # # 1 − # 2 7 − ′8 , '() ′ ≤ ≤ # 0, '() ≤ ≤ 2 1 ν = $ 1 7 − 2 8 , '() ≤ ≤ ′ 2 4 # 2 4 ′ − 2 # 1 1 − ′ 4 # + W Y , '() 4 ′ ≤ ≤ 5 ′ # 2 2 5 ′ − 4 ′ " 1, (+ℎ-)./0-

3.2. Graphical representation of Hexagonal Intuitionistic fuzzy numbers

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

Figure 1.Hexagonal

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

3.3. Arithmetic operations on Hexagonal Intuitionistic fuzzy numbers

• • •

•

: Let 3 = , , , 2 , 4 , 5 , K , K , , 2 , 4 K , 5 K and : Z 3 = [ , [ , [ , [2 , [4 , [5 , [ K , [ K , [ , [2 , [4 K , [5 K be two HIFNs. Then : : 3 + Z 3 = + [ , + [ , + [ , 2 + [2 , 4 + [4 , 5 + [5 , K + [ K , K + [ K , + [ , 2 + [2 , 4 K + [4 K , 5 K + [5 K . : : 3 − Z 3 = − [5 , − [4 , − [2 , 2 − [ , 4 − [ , 5 − [ , K − [5 K , K − [4 K , − [2 , 2 − [ , 4 K − [ K , 5 K − [ K .

: : 3 ∗ Z 3 = ] , ] , ] , ]2 , ]4 , ]5 where

: ` 3

] = ^/ /^_^ [ , [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

` , ` , ` , ` 2 , ` 4 , ` 5 , ` K , ` K , ` , ` 2 , ` 4 K , ` 5 K /' ` > 01 =a

` 5 , ` 4 , ` 2 , ` , ` , ` , ` 5 K , ` 4 K , ` 2 , ` , ` K , ` K /' ` < 0

3.4. Ranking of Hexagonal Intuitionistic fuzzy numbers

The ranking function[11] of a HIFN 3 = , , , 2 , 4 , 5 , K , K , , 2 , 4 K , 5 K :

maps the set of all fuzzy numbers to a set of real numbers defined as : : : : O g O g2O g2O g O g O 3 = cd Je L 3 R , d Jν 3 f where d Je L 3 R = P Q S h T U V and

: O d Jν L 3 R = P

i g O i g2O g2O g O i g O i Q S T U V

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 : : 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

Example:

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

: Z 3 .

: : : : : Since d Je L 3 R > d Je LZ 3 R and d Jν L 3 R > d Jν LZ 3 R, 3 >

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

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

o : : : Minimize j : = ∑r lmn s̃mn subject to ∑onpq lmn = lm: , / = 1,2, … , ^ mpq ∑npq : : ∑r lmn = [ n: , v = 1,2, … , and lmn ≥ 0 ∀/, v. mpq : : : In the above model the transportation costs s̃mn , 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 jth 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 kth row and lth column. Now choose the zero with maximum sum and allocate the maximum possible quantity to that cell. 21

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

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.

Origins

Destinations D3

IF Supply

(7,9,11,13,16,2 0) (5,7,11,13,19,2 3) (6,8,11,14,19,2 5) (4,7,11,14,21,2 7) (9,11,13,15 ,18,20) (8,10,13,15,19, 22)

O 1

O 2

O 3

(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)

IF Dema nd 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

Origins

D1 O 1

O 2

Destinations D2 D3 6

D4 9

IF Supply

Di ff

(7,9,11,13,16, 20) (5,7,11,13,19, 23) (6,8,11,14,19, 25) (4,7,11,14,21, 27)

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

O 3

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)

5 ▲

Diff.

(9,11,13,15 ,18,20) (8,10,13,15,19 ,22)

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

Origins

O 1 O 2 O 3

Destinations D3

IF Supply

Diff .

(7,9,11,13,16,20) (5,7,11,13,19,23)

(9,11,13,15 ,18,20) (8,10,13,15,19,2 2)

(6,8,11,14,19,2 5) (4,7,11,14,21,2 7)

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.

8 ▲

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

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015 Table 3

Destinations D2 D3 (-9,(6,7,9,11,13,16 4,0,4,9,14) ) (-13,(5,6,9,11,16,18 9,0,4,13,18 ) )

Origins

O 1

O 2 O 3

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 )

Diff.

IF Supply

Diff .

(6,8,11,14,19,25 ) (4,7,11,14,21,27 )

(9,11,13,15 ,18,20) (8,10,13,15,19,22 )

(-15,-7,0,5,12,18) (-19,-11,0,5,13,21)

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

Origins

Table 4

Destinations D3

IF Supply

(-9,4,0,4,9,14) (-13,9,0,4,13,18)

(6,7,9,11,13,16) (5,6,9,11,16,18)

(6,8,11,14,19,25) (4,7,11,14,21,27)

(-15,-7,0,5,12,18) (-19,-11,0,5,13,21)

(-11,-

(3,4,5,6,8,10) 4,3,9,16,24) O3 (2,4,5,6,10,12) (-19,-

9,3,9,22,30)

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 , 24

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

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

Origins

The optimum solution is illustrated by the following example. Destinations D1 D2 D3

IF Supply

O 1

(7,9,11,13,16,20) (5,7,11,13,19,23)

O 2

(6,8,11,14,19,25) (4,7,11,14,21,27)

O 3 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)

(9,11,13,15 ,18,20) (8,10,13,15,19,22 )

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

Origins

Table 5

Destinations D3

(7,9,11,13,16,20) (5,7,11,13,19,23)

(6,8,11,14,19,25) (4,7,11,14,21,27)

IF Deman d

IF Supply

(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)

(9,11,13,15 ,18,20) (8,10,13,15,19,22 )

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

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

(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 )

O1 Origins

Destinations D3 (6,7,9,11,13,1 6) (5,6,9,11,16,1 8)

IF Deman d

D4 2

IF Supply (-9,-4,0,4,9,14) (-13,-9,0.4,13,18) (6,8,11,14,19,25) (4,7,11,14,21,27) (9,11,13,15 ,18,20) (8,10,13,15,19,22)

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

Origins

O 1

O 2 O 3

IF Deman d

(-9,4,0,4,9,14) (-13,9,0.4,13,18 )

(3,5,7,9,12,15 ) (2,4,7,9,13,17 )

Destinations D3 (6,7,9,11,13,16 ) (5,6,9,11,16,18 ) -

D4 2

IF Supply

(-9,4,2,7,14,22) (-13,6,2,7,17,25) (9,11,13,15 ,18,20) (8,10,13,15,19, 22)

(-11,5,1,6,12,19 ) (-16,9,1,6,19,25 )

(10,12,14,16,20,24 ) (8,10,14,16,20,25) 26

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

Now allocate the remaining to fulfill the demand and supply we the following allocation table. Table 8

Destinations D1

(6,7,9,11,13,16) (5,6,9,11,16,18)

(-9,4,2,7,14,22) (-13,6,2,7,17,25)

(-10,1,7,14,23,31) (-17,9,7,14,28,38)

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

O1 (-13,Origins

9,0.4,13,18)

1 (-11,5,1,6,12,19) (-16,9,1,6,19,25)

IF Demand

IF Supply

(3,5,7,9,12,15) (2,4,7,9,13,17)

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,

Minimize

j : = 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.

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

REFERENCES [1] [2]

Atanassov.K.T., (1986) “Intuitionistic fuzzy sets”, Fuzzy sets and systems, vol.20, No.1, pp. 87-96. Atanassov, K.T.,(1999) “Intuitionistic Fuzzy Sets: Theory and Applications”, Physica Verlag , Heidelberg , New York. [3] Bellmann.R.E. and Zadeh.L.A., (1970) “Decision making in fuzzy environment”, Management sciences, Vol.17, pp.141-164. [4] Chanas.S and Kutcha.D., (1996) “A concept of the optimal solution of the transportation problem with fuzzy cost coefficients”, Fuzzy Sets and Systems, 82, pp.299-305. [5] Hussain.R.J and Senthil Kumar.P., (2012) “Algorithm approach for solving intuitionistic fuzzy transportation problem”, Applied mathematical sciences, Vol. 6, No.80, pp.3981- 3989. [6] Nagoor Gani.A, and Abbas.S., (2012) “A New method for solving Intuitionistic Fuzzy Transportation problem using zero suffix algorithm”, International Journal of Mathematical Science and Engineering , Vol.6, No.3, pp.73- 82. [7] Nagoor Gani.A, Abbas.S., (2013) “A New method for solving Intuitionistic Fuzzy Transportation problem”, Applied Mathematical Sciences, Vol.7, No.28, pp. 1357-1365. [8] O’heigeartaigh.H.,(1982) “A fuzzy transportation algorithm”, Fuzzy Sets and Systems, pp. 235-243. [9] Pandian.P and Natarajan.G., (2010) “A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problem”, Applied Mathematical Sciences, Vol. 4, No. 2, pp.79-90. [10] Rajarajeswari.P., Sahaya Sudha.A. and Karthika.R., (2013) “A New Operation on Hexagonal Fuzzy Number”, International Journal of Fuzzy Logic Systems ,Vol. 3, No. 3, pp.15-26. [11] Rajarajeswari.P. and Sahaya Sudha.A., (2014) “Ranking of Hexagonal Fuzzy Numbers using Centroid”, AARJMD , Vol. 1, No. 17 , pp. 265-277. [12] Zadeh.L.A., (1965) “Fuzzy sets”, Information and Control, 8, pp. 338-353.