International Journal of Mathematics Research. ISSN 0976-5840 Volume 4, Number 4 (2012), pp. 411-420 © International Research Publication House http://www.irphouse.com

The Transportation Problem in an Intuitionistic Fuzzy Environment R. Jahir Hussain1 and P. Senthil Kumar2 1

Associate Professor and 2Research Scholar PG and Research Department of Mathematics, Jamal Mohamed College, Tiruchirappalli: 620 020 India E-mail: hssn_jhr@yahoo.com, senthilsoft_5760@yahoo.com

Abstract In this paper, we investigate transportation problem in which supplies and demands are intuitionistic fuzzy numbers. Intuitionistic Fuzzy Vogel’s Approximation Method is proposed to find an initial basic feasible solution. Intuitionistic Fuzzy Modified Distribution Method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The solution procedure is illustrated with suitable numerical example. Keywords: Triangular intuitionistic fuzzy numbers, intuitionistic fuzzy transportation problem, intuitionistic fuzzy vogel’s approximation method, intuitionistic fuzzy modified distribution method, initial basic feasible solution, optimal solution.

Introduction The theory of fuzzy set introduced by Zadeh[8] in 1965 has achieved successful applications in various fields. The concept of Intuitionistic Fuzzy Sets (IFSs) proposed by Atanassov[1] in 1986 is found to be highly useful to deal with vagueness. The major advantage of IFS over fuzzy set is that IFSs separate the degree of membership (belongingness) and the degree of non membership (non belongingness) of an element in the set .The concept of fuzzy mathematical programming was introduced by Tanaka et al in 1947 the frame work of fuzzy decision of Bellman and Zadeh[2]. In [4], Nagoor Gani et al presented a two stage cost minimizing fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy number. In [7], Stephen Dinager et al investigated fuzzy transportation problem with the aid of trapezoidal fuzzy numbers. In[6], Pandian.P and Natarajan.G presented a new

412

R. Jahir Hussain and P. Senthil Kumar

algorithm for finding a fuzzy optimal solution for fuzzy transportation problem. In [3], Ismail Mohideen .S and Senthil Kumar .P investigated a comparative study on transportation problem in fuzzy environment. In this paper , a new ranking procedure which can be found in [5] and is used to obtain a basic feasible solution and optimal solution in an intuitionistic fuzzy transportation problem[IFTP]. The paper is organized as follows: section 2 deals with some terminology, section 3 provides the mathematical formulation of intuitionistic fuzzy transportation problem, section 4 deals with solution procedure, section 5 consists of numerical example, finally conclusion is given.

Terminology Definition 2.1: Let A be a classical set, set with the membership function ; ,

be a function from A to [0,1]. A fuzzy is defined by 0,1 .

Definition 2.2: Let X be denote a universe of discourse, then an intuitionistic fuzzy set A in X is given by a set of ordered triples, , , ; Where , : 0,1 , are functions such that 0 1, . For each x the membership represent the degree of membership and the degree of non – membership of the element to respectively. Definition 2.3: An Intuitionistic fuzzy subset A = {<x, µA(x), υA(x)> : x X } of the real line R is called an intuitionistic fuzzy number (IFN) if the following holds: i. There exist m R, µA(m) = 1 and υA(m) = 0, (m is called the mean value of A). ii. µA is a continuous mapping from R to the closed interval [0,1] and x x 1 holds. , the relation 0 The membership and non: membership function of A is of the following form 0 ∞ , 1 x , 0 ∞ Where f1(x) and h1(x) are strictly increasing and decreasing function in , and , respectively

413

The Transportation Problem in an Intuitionistic Fuzzy Environment 1

∞ ,

;0

1

;0

1

0

x ,

∞

1

Here m is the mean value of A. α and β are called left and right spreads of membership function x , respectively. α′ β′ represents left and right spreads of non membership function x , respectively. Symbolically, the intuitionistic fuzzy number is represented as AIFN =(m; , ; α′ , β′). Definition 2.4: A Triangular Intuitionistic Fuzzy Number (ÃI is an intuitionistic fuzzy set in R with the following membership function x and non membership function x :) x 0

x 1 Where x

x ,

and . This TrIFN is denoted by = , ,

,

x

0.5

,

Membership and non membership functions of TrIFN

for

x

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R. Jahir Hussain and P. Senthil Kumar

Ranking of Triangular intuitionistic Fuzzy Numbers The Ranking of a triangular intuitionistic fuzzy number is completely defined by its membership and non- membership as follows [5]: Let ÃI = (a,b,c) (e,b,f) 1 1 2 3 3 2 6 6 2 6

1

3

2

1

6

2

3

2 1 3 2 3 Rank (A) = (Sqrt ((xµ(A))2+ (yµ(A))2), Sqrt ((xυ(A))2 + (yυ(A))2 )) be two TrIFNs. The ranking of Definition 2.5: Let and E, the set of TrIFNs is defined as follows: i. R( )>R( ) iff ii. R( )<R( ) iff iii. R( )=R( ) iff Definition 2.6: The ordering as follows i. ii.

and

between any two TrIFNs iff iff

Definition 2.7: Let , 1,2, … , then the TrIFN is the minimum of

and

or

by the R(.) on

and

are defined

and or

be a set of TrIFNs. If , 1,2, … , .

for all i,

Definition 2.8: Let , 1,2, … , be a set of TrIFNs. If then the TrIFN is the maximum of , 1,2, … , .

for all i,

Arithmetic Operations Addition: = I I Subtraction: Ã Θ B̃ = Multiplication

, , A

Where,

, , B

l ,l ,l

, , l ,l ,l

, ,

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The Transportation Problem in an Intuitionistic Fuzzy Environment min

,

,

,

= max { min {

,

,

,

,

,

max { Scalar multiplication i. ii.

, ,

, ,

,

,

, , ,

} ,

}

, ,

, ,

0 0

Intuitionistic fuzzy transportation problem and its mathematical formulation Consider a transportation with m IF origins (rows) and n IF destinations (columns). Let be the cost of transporting one unit of the product from ith IF (Intuitionistic Fuzzy) origin to jth IF destination. , , , , be the quantity of commodity available at IF origin i. , , , , the quantity of commodity needed at intuitionistic fuzzy destination j. , , , , is the quantity transported from ith IF origin to jth IF destination, so as to minimize the IF transportation cost. ∑ ∑ (IFTP) Minimize Subject to

0, Where

,

1,2, … ,

,

1,2, … , 1,2, … , 1,2, … ,

m = the number of supply points n = the number of demand points , , , , is the number of units shipped from ith IF origin to jth IF destination. = the cost of shipping one unit from IF supply point i to IF demand point j. , , , , is the intuitionistic fuzzy supply at supply point i and , , , , is the IF demand at demand point j.

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R. Jahir Hussain and P. Senthil Kumar

The above IFTP can be stated in the below tabular form

Definition 3.1: Any set of intuitionistic fuzzy non negative allocations >(2δ,0,2δ)(-3δ,0,3δ) where δ is small positive number, which satisfies the row and column sum is a IF feasible solution. Definition 3.2: Any feasible solution is an intuitionistic fuzzy basic feasible solution if the number of non negative allocations is at most (m+n-1) where m is the number of rows and n is the number of columns in the transportation table. Definition 3.3: Any intuitionist fuzzy feasible solution to a transportation problem containing m origins and n destinations is said to be intuitionist fuzzy non degenerate, if it contains exactly (m+n-1) occupied cells. Definition 3.4: If an intuitionistic fuzzy basic feasible solution contains less than (m+n-1) non negative allocations, it is said to be degenerate.

Solution of an intuitionistic fuzzy transportation problem The solution of an IFTP can be solved two stages, namely initial solution and optimal solution. Finding an initial solution of an IFTP there are numerous methods but intuitionistic fuzzy vogel’s approximation method (IFVAM) is preferred over the other methods, since the initial intuitionistic fuzzy basic feasible solution obtained by this method is either optimal or very close to the optimal solution. We are going to discuss IFVAM. 4.1 Intuitionistic fuzzy vogel’s approximation method Step 1: Find the penalty cost, namely the difference between the smallest and nextto-smallest costs for each row and display them to the right of the corresponding row. If there are more than one least cost, the difference is

The Transportation Problem in an Intuitionistic Fuzzy Environment

417

zero. Similarly, compute the difference for each column. Step 2: Among the penalties as found in step1, choose the maximum penalty. If this maximum penalty is more than one, choose arbitrarily. Step 3: In the selected row or column as by step2, allocate the maximum possible amount to the cell with the least cost in the selected row or column i.e., = min { , } by ranking procedure. If = , then delete the ith row and adjust the amount of IF demand. If = , then delete the jth column and adjust the amount of IF supply. If = = , then delete either ith row or jth column, but not both. Step 4: Repeat step1 to step3 until all the intuitionistic fuzzy supply points are fully used and all the intuitionistic fuzzy demand points are fully received. 4.2 Intuitionistic Fuzzy Modified Distribution Method This proposed method is used for finding the optimal solution in an intuitionistic fuzzy environment and the following step by step procedure is utilized to find out the same. 1. Find out a set of numbers and for each row and column satisfying for each occupied cell. To start with we assign an intuitionistic fuzzy zero to any row or column having maximum number of allocations. If this maximum number of allocation is more than one, select any one arbitrarily. 2. For each empty (un occupied) cell, we find intuitionistic fuzzy sum and . 3. Find out for each empty cell the net evaluation value, = , this step gives the optimality conclusion. i. If all 2δ, 0,2δ 3δ, 0,3δ the solution is IF optimal and a unique solution exists. 2δ, 0,2δ 3δ, 0,3δ then the solution is IF optimal, but an ii. If alternate optimal solution exists. iii. If at least one 2δ, 0,2δ 3δ, 0,3δ the solution is not IF optimal. In this case we go to next step, to improve the total IF transportation cost. 4

5

from this cell we Select the empty cell having the most negative value of draw a closed path drawing horizontal and vertical lines with corner cell occupied. Assign sign + and – alternately and find the IF minimum allocation from the cell having negative sign. This allocation having negative sign. The above step yield a better solution by making one (or more) occupied cell as empty and one empty cell as occupied. For this new set of intuitionistic fuzzy basic feasible allocation repeat from the step1, till an intuitionistic fuzzy optimal solution is desired.

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Numerical Example Consider the 4× 4 IFTP IFD1 IFD2 IFD3 IFD4 IF supply IFO1 16 1 8 13 (2,4,5)(1,4,6) IFO2 11 4 7 10 (4,6,8)(3,6,9) IFO3 8 15 9 2 (3,7,12)(2,7,13) IFO4 6 12 5 14 (8,10,13)(5,10,16) IF demand (3,4,6)(1,4,8) (2,5,7)(1,5,8) (10,15,20)(8,15,22) (2,3,5)(1,3,6)

∑ = (17, 27, 38) (11, 27, 44), the problem is balanced Since ∑ IFTP. There exists an IF initial basic feasible solution. IFD1 IFO1 IFO2 IFO3

(-2,4,10) 8 (-4,4,12)

IFO4

(-13,0,14) 6 (-21,0,22)

IF demand

(3,4,6) (1,4,8)

IFD2 (2,4,5) 1 (1,4,6) (-3,1,5) 4 (-5,1,7)

IFD3

(-1,5,11) 7 (-4,5,14)

IFD4

IF supply (2,4,5) (1,4,6) (4,6,8) (3,6,9) (2,3,5) 2 (3,7,12) (1,3,6) (2,7,13)

(-1,10,21) 5 (-6,10,26) (2,5,7) (1,5,8)

(10,15,20) (8,15,22)

(8,10,13) (5,10,16) (2,3,5) (1,3,6)

Since the number of occupied cell having m+n-1 and are also independent, there exist a non degenerate IF basic feasible solution. Therefore, the initial IF transportation minimum cost is, Min I =(-112,131,381)(-233,131,502) To find the optimal solution Applying the intuitionistic fuzzy modified distribution method, we determine a set of numbers , , , , , , , , , each nrow and column such that , , , , + , , , , for each occupied cell. Since maximum number of allocations in row and column are same, so we give intuitionistic fuzzy number , , , , (-2,0,2)(-3,0,3). The remaining numbers can be obtained as given below. = , , , , + , , , , , , , , =(6,8,10)(5,8,11)

419

The Transportation Problem in an Intuitionistic Fuzzy Environment ,

=

, , ,

,

=

,

,

, ,

,

=

, ,

, ,

=

, ,

, ,

=

, , , , , =(4,6,8)(3,6,9) , + , , , , , , =(-6,-4,-2)(-7,-4,-1) , , + , , , , , , =(6,8,10)(5,8,11) , , + , , , , , , =(-3,-1,1)(-4,-1,2) , , + , , , , , , =(-8,-6,-4)(-9,-6,-3) , , + , , , , , , =(3,5,7)(2,5,8)

,

,

=

,

, , ,

,

, , ,

+

We find, for each empty cell of the sum , , , , , . Next we find the net evaluation is given by

IFO1 IFO2 IFO3

IFD1 *(7,11,15)16 (5,11,17) *(-1,3,7) 11 (-3,3,9) (-2,4,10) 8 (-4,4,12)

IFO4

(-13,0,14) 6 (-21,0,22)

IF demand

(3,4,6) (1,4,8)

IFD2 (2,4,5) 1 (1,4,6) (-3,1,5) 4 (-5,1,7) *(7,11,15)15 (5,11,17)

IFD3 *(0,4,8) 8 (-2,4,10) (-1,5,11) 7 (-4,5,14) *(-2,2,6) 9 (-4,2,8) (-1,10,21)5 *(6,10,14)12 (-6,10,26) (4,10,16) (2,5,7) (1,5,8)

,

, ,

,

and

,

,

,

IFD4 *(10,14,18) 13 (8,14,20) *(4,8,12) 10 (2,8,14) (2,3,5) 2 (1,3,6)

IF supply (2,4,5) (1,4,6) (4,6,8) (3,6,9) (3,7,12) (2,7,13)

*(10,14,18) 14 (8,14,20)

(8,10,13) (5,10,16)

(10,15,20) (8,15,22)

(2,3,5) (1,3,6)

Where ,

= ,

,

,

,

,

, =

-[

, ,

, ,

,

= ,

, ,

+

,

,

,

,

,

,

]

Since all , , , , >0 the solution is intuitionistic fuzzy optimal and unique.The intuitionistic fuzzy optimal solution in terms of triangular intuitionistic fuzzy numbers =(2,4,5)(1,4,6), =(-3,1,5)(-5,1,7), =(-1,5,11)(-4,5,14), =(-2,4,10)(4,4,12), =(2,3,5)(1,3,6), =(-13,0,14)(-21,0,22), =(-1,10,21)(-6,10,26)

420

R. Jahir Hussain and P. Senthil Kumar Hence, the total intuitionistic fuzzy transportation minimum cost is Min I =(-112,131,381)(-233,131,502)

Conclusion Mathematical formulation of intuitionistic fuzzy transportation problem and procedure for finding an intuitionistic fuzzy optimal solution in two stages are discussed with suitable numerical example. In the first stage, initial basic intuitionistic fuzzy feasible solution using intuitionistic fuzzy vogelâ€™s approximation method is determined. In second stage, intuitionistic fuzzy optimal solution using intuitionistic fuzzy modified distribution method is calculated. The new arithmetic operations of triangular intuitionistic fuzzy numbers are employed to get the optimal solution in terms of triangular intuitionistic fuzzy numbers. This method is a systematic procedure, both easy to understand and to apply also; it can serve as an important tool for the decision makers when they are handling various types of logistic problems having intuitionistic fuzzy parameters.

References [1] K.T.Atanassov, Intuitionistic fuzzy sets, fuzzy sets and systems, vol.20, no.1.pp.87- 96,1986. [2] R.Bellman,L.A.Zadeh,Decision making in a fuzzy environment, management sci.17(B)(1970)141-164. [3] S.Ismail Mohideen, P.Senthil Kumar, A Comparative Study on Transportation Problem in fuzzy environment. International Journal of Mathematics Research,Vol.2Number.1 (2010),pp. 151-158. [4] A.Nagoor gani, K.Abdul Razak, Two stage fuzzy transportation problem, journal of physical sciences,vol.10,2006,63-69. [5] A.Nagoor Gani, Abbas., Intuitionistic Fuzzy Transportation problem, proceedings of the heber international conference pp.445-451. [6] P.Pandian and G.Natarajan., A new algorithm for finding a fuzzy optimal solution for fuzzy Transportation problems. Applied mathematics sciences, Vol. 4,2010, no.2, 79-90. [7] D.Stephen Dinager, K.Palanivel,The Transportation problem in fuzzy environment, int.journal of Algorithm, computing and mathematics , vol2, no3, 2009. [8] L.A. Zadeh, Fuzzy sets ,information and computation, vol.8,pp.338-353,1965