International Journal of Mathematics Research. ISSN 0976-5840 Volume 2, Number 1 (2010), pp. 151--158 © International Research Publication House http://www.irphouse.com

A Comparative Study on Transportation Problem in Fuzzy Environment S. Ismail Mohideen1* and P. Senthil Kumar2 Associate Professor, 2Research Scholar, PG and Research, Department of Mathematics, Jamal Mohamed College, Tiruchirappalli – 620 020. India. E-mail: simohideen@yahoo.co.in, senthilsoft_5760@yahoo.com 1*

Abstract In this paper, transportation problem in fuzzy environment using trapezoidal fuzzy number is discussed. A new algorithm called the fuzzy zero point method for finding a fuzzy optimal solution of fuzzy transportation problem in single stage with the multiplication used by Stephen Dinegar.D & Palanivel.K [5] is discussed. The solution procedure is illustrated with the existing Stephen Dinegar.D & Palanivel.K [5] algorithm with numerical example. Finally the comparative result is given. Keywords: Trapezoidal fuzzy numbers, Fuzzy transportation, Fuzzy Vogel’s Approximation Method, Fuzzy Modified Distribution Method, Fuzzy Optimal Solution, Fuzzy Zero Point Method.

Introduction The fuzzy transportation problem (FTP) is one of the special kinds of fuzzy linear programming problems. A fuzzy transportation problem is a transportation problem in which the transportation costs, supply and demand quantities are fuzzy quantities. To deal quantitatively with imprecise information in making decisions, Bellman and Zadeh [1] and Zadeh [7] introduced the notion of fuzziness. Fuzzy transportation is the transportation of fuzzy quantity from the fuzzy origin to fuzzy destination in such a way that the total fuzzy transportation cost is minimum. The objective of the fuzzy transportation problem is to determine the shipping schedule that minimizes the total fuzzy transportation cost, while satisfying fuzzy supply and demand limits. Here we use the existing fuzzy modified distribution method(FMDM) [5] and fuzzy zero point method(FZPM) [4] with the multiplication used by Stephen Dinegar.D & Palanivel.K [5].The paper is organized as follows:

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Section 2 deals with some basic terminology, section 3 provides the mathematical formulation of Fuzzy transportation problem, section 4 deals with the numerical example based on fuzzy modified distribution method and its rank, section 4.2 deals with the same numerical example based on Fuzzy zero point method and its rank with new multiplication operation, finally conclusion is given.

Terminology Definition Let A be a classical set μA(x) be a function from A to [0, 1]. A fuzzy set A* with the membership function μA(x) is defined by, A* = {(x, μA(x)) : x ∈ A and μA(x) ∈ [0, 1]}. Definition A real fuzzy number ã = (a1, a2, a3, a4) is a fuzzy subset from the real line R with the membership function μã(a) satisfying the following conditions. μã(a) is a continuous mapping from R to the closed interval [0, 1] μã(a) = 0 for every a ∈ (-∞, a1] μã(a) is strictly increasing and continuous on [a1, a2]. μã(a) = 1 for every a ∈ [a2, a3] μã(a) is strictly decreasing and continuous on [a3, a4] μã(a) = 0 for every a ∈ [a4, + ∞) Arithmetic Operations Let ã = [a1, a2, a3, a4] and b = [b1, b2, b3, b4] are two trapezoidal fuzzy numbers then the arithmetic operations [3,6] on ã and b are as follows. Addition : ã + b = (a1 + b1, a2 + b2, a3 + b3, a4 + b4) Subtraction : ã - b = (a1 - b4, a2 - b3, a3 - b2, a4 - b1) Multiplication used by Stephen Dinegar.D & Palanivel.K [5]: a a a a ã · b = ( 1 (b1+b2+ b3+b4), 2 (b1+b2+b3 + b4), 3 (b1+b2+ b3+ b4), 4 (b1+ b2+ 4 4 4 4 b3+ b4)), if R (ã) > 0. a a a a = ( 4 (b1+b2+b3+ b4), 3 (b1+ b2+ b3+ b4), 2 (b1+ b2+ b3+ b4), 1 (b1+b2+ b3+ b4)), 4 4 4 4 if R (ã) < 0.

Fuzzy Transportation Problem and its Mathematical Formulation Consider transportation with m fuzzy origins (rows) and n fuzzy destinations (columns). Let Cij = ⎡⎣Cij(1) , Cij(2) , Cij(3) , Cij(4) ⎤⎦ be the cost of transporting one unit of the product

from ith

fuzzy origin

to

jth

fuzzy destination.

Let

ai

=

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153

⎡⎣ ai(1) , ai(2) , ai(3) , ai(4) ⎤⎦ be the quantity of commodity available at fuzzy origin i. Let ⎤ bj = ⎡⎣b (1) b (2) b (3) b (4) j , j , j , j ⎦ be the quantity of commodity needed at fuzzy destination j. Let xij = ⎡⎣ xij(1) , xij(2) , xij(3) , xij(4) ⎤⎦ be the quantity transported from ith

fuzzy origin to jth fuzzy destination, so as to minimize the fuzzy transportation cost. The linear programming model representing the fuzzy transportation problem is given by. m

Min Z = ∑

i=1

n

∑

j=1

⎡⎣Cij(1) , Cij(2) , Cij(3) , Cij(4) ⎤⎦ ⎡⎣ xij(1) , xij(2) , xij(3) ,

xij(4) ⎤⎦

Subject to the constraints n

∑

j=1

⎡⎣ xij(1) , xij(2) , xij(3) ,

xij(4) ⎤⎦ = ⎡⎣ ai(1) , ai(2) , ai(3) , ai(4) ⎤⎦ for i = 1, 2, …,m

(Row sum) m

∑

i=1

⎡⎣ xij(1) , xij(2) , xij(3) ,

⎤ xij(4) ⎤⎦ = ⎡⎣b (1) b (2) b (3) b (4) j , j , j , j ⎦ for j = 1, 2, …,n

(Column sum) ⎡⎣ xij(1) , xij(2) , xij(3) , xij(4) ⎤⎦ ≥ 0

Numerical Example Ranking Technique - I Here the algorithm of Pandian.P & Natarajan.G [4] and Stephen Dinegar.D & Palanivel.K [5] is used with the ranking technique

.

The Fuzzy Transportation Problem FD1 FO1 [1, 2,3, 4] FO2 [0, 1,2, 4] FO3 [3, 5,6, 8] Fuzzy Demand [5,7,8,10]

FD2 [1,3,4,6] [-1,0,1,2] [5,8,9,12] [1,5,6,10]

FD3 [9,11,12,14] [5, 6, 7, 8] [12,15,16,19] [1, 3, 4, 6]

FD4 [5,7,8,11] [0,1, 2, 3] [7,9,10,12] [1, 2, 3, 4]

Fuzzy capacity [1,6,7,12] [0,1, 2, 3] [5,10,12,17]

Two Stage Method Stage I The fuzzy initial basic feasible solution using Fuzzy Vagol’s Approximation Method(FVAM) is, (1) (2) (3) (4) ⎤⎦ =[-6,5, 8,19], ⎡⎣ x11(1) , x11(2) , x11(3) , x11(4) ⎤⎦ =[-9,0,2,11], ⎡⎣ x31 , x31 , x31 , x31 (1) (2) (3) (4) ⎤⎦ =[1,3,4,6], ⎡⎣ x12(1) , x12(2) , x12(3) , x12(4) ⎤⎦ =[1,5,6,10], ⎡⎣ x33 , x33 , x33 , x33

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S. Ismail Mohideen and P. Senthil Kumar (1) (2) (3) ⎡⎣ x24 , x24 , x24 ,

(4) (1) (2) (3) ⎤⎦ =[0,1,2,3], ⎡⎣ x34 x24 , x34 , x34 ,

(4) ⎤⎦ =[-2,0, 2,4] x34

The initial fuzzy transportation cost is Z = [-55.5, 93, 154, 302.5] and the crisp value of Z = (-55.5+93+154+302.5)/4 = 123.5 Stage II The fuzzy optimal solution using FMDM in terms of trapezoidal fuzzy (1) (2) (3) (4) ⎤⎦ = [-15,5,10,30], numbers, ⎡⎣ x12(1) , x12(2) , x12(3) , x12(4) ⎤⎦ =[1,5,6,10], ⎡⎣ x31 , x31 , x31 , x31 ⎡⎣ x13(1) , x13(2) , x13(3) , (1) (2) (3) ⎡⎣ x23 , x23 , x23 ,

(1) (2) (3) x13(4) ⎤⎦ =[-9,0,2,11], ⎡⎣ x33 , x33 , x33 ,

(4) ⎤⎦ =[-13,-1,3,15], x33

(4) (1) (2) (3) (4) ⎤⎦ = [-2,1, 4,7] ⎤⎦ = [0,1,2,3], ⎡⎣ x34 x23 , x34 , x34 , x34 The corresponding total fuzzy transportation cost is Z = [-403, 45.5, 196.5, 645] and the crisp value of Z = (-403+45.5+196.5+645)/4 = 121

Fuzzy Zero Point Method The optimum solution by using FZPM with the multiplication used by Stephen Dinegar.D & Palanivel.K [5]. x12 = [1, 5, 6, 10], x31 = [5, 7, 8, 10], x13 = [-9, 0, 2, 11], x33 = [-9, -1, 3, 11], x23 = [0, 1, 2, 3], x34 = [1, 2, 3, 4] The total fuzzy transportation cost is Z = [-202.5, 66, 176, 444.5] and the crisp value of Z = (-202.5+66+176+444.5)/4 = 121 Defuzzified Transportation Problem The Defuzzified Transportation Problem of the Fuzzy Transportation Problem in section 4.1.1. is

O1 O2 O3 Demand

D1 2.5 1.75 5.5 7.5

D2 3.5 0.5 8.5 5.5

D3 11.5 6.5 15.5 3.5

D4 7.75 1.5 9.5 2.5

Supply 6.5 1.5 11 19

Two Stage Method Stage I The initial basic feasible solution using Vogel’s Approximation method is x11 = 1, x12 = 5.5, x24 = 1.5, x31 = 6.5, x33 = 3.5, x34 = 1 The total transportation Cost is Z = 123.5

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Stage II The optimum solution using modified distribution method is x12 = 5.5, x13 = 1, x23 = 1.5, x31 = 7.5, x33 = 1, x34 = 2.5 Total transportation Cost is Z = 121 Zero Point Method (ZPM) The optimum solution is x12 = 5.5, x13 = 1, x23 = 1.5, x31 = 7.5, x33 = 1, x34 = 2.5 The total transportation cost is Z = 121 Comparative Study Rank used

Fuzzy Transportation Problem(FTP) Stage - II Stage - I FZPM FMDM FVAM [-55.5, 93, 154, [-403, 45.5, 196.5, 645] 302.5] 123.5

121

Defuzzified TP Stage- Stage- ZPM I II VAM MODI [-202.5, 66, 176, 123.5 121 121 444.5] 121

Ranking Technique – II Here also the algorithm of Pandian.P & Natarajan.G [4] and Stephen Dinegar.D & Palanivel.K [5] is used with different ranking[2] technique

The Fuzzy Transportation Problem Consider the same fuzzy transportation problem in 4.1.1 Two Stage Method Stage I The fuzzy initial basic feasible solution using FVAM is, (1) (2) (3) (4) ⎤⎦ =[-6,5, 8,19], ⎡⎣ x11(1) , x11(2) , x11(3) , x11(4) ⎤⎦ =[-9,0,2,11], ⎡⎣ x31 , x31 , x31 , x31 (1) (2) (3) (4) ⎤⎦ =[1,3,4,6], ⎡⎣ x12(1) , x12(2) , x12(3) , x12(4) ⎤⎦ =[1,5,6,10], ⎡⎣ x33 , x33 , x33 , x33 (1) (2) (3) ⎡⎣ x24 , x24 , x24 ,

(4) (1) (2) (3) ⎤⎦ =[0,1,2,3], ⎡⎣ x34 x24 , x34 , x34 ,

(4) ⎤⎦ =[-2,0, 2,4] x34

The initial fuzzy transportation cost is Z = [-55.5, 93, 154, 302.5] and the value of Z = (-55.5+2(93)+2(154)+302.5)/6 = 123.5

crisp

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Stage II The fuzzy optimal solution using FMDM in terms of trapezoidal fuzzy (1) (2) (3) (4) ⎤⎦ =[-15,5,10,30], numbers, ⎡⎣ x12(1) , x12(2) , x12(3) , x12(4) ⎤⎦ =[1,5,6,10], ⎡⎣ x31 , x31 , x31 , x31 (1) (2) (3) (4) ⎤⎦ =[-13,-1,3,15], ⎡⎣ x13(1) , x13(2) , x13(3) , x13(4) ⎤⎦ =[-9,0,2,11], ⎡⎣ x33 , x33 , x33 , x33 (1) (2) (3) ⎡⎣ x23 , x23 , x23 ,

(4) (1) (2) (3) ⎤⎦ = [0,1,2,3], ⎡⎣ x34 x23 , x34 , x34 ,

(4) ⎤⎦ = [-2,1, 4,7] x34

The corresponding total fuzzy transportation cost is Z = [-403, 45.5, 196.5, 645] and the crisp value of Z = (-403+2(45.5)+2(196.5)+645)/6 = 121 Fuzzy Zero Point Method The optimum solution by using FZPM with the multiplication used by Stephen Dinegar.D & Palanivel.K [5]. x12 = [1, 5, 6, 10], x31 = [5, 7, 8, 10], x13 = [-9, 0, 2, 11], x33 = [-9, -1, 3, 11], x23 = [0, 1, 2, 3], x34 = [1, 2, 3, 4] The total fuzzy transportation cost is Z = [-202.5, 66, 176, 444.5] and the crisp value of Z = (-202.5+2(66)+2(176)+444.5)/6 = 121 Defuzzified Transportation Problem The Defuzzified Transportation Problem of the Fuzzy Transportation Problem in section 4.1.1. is D1 O1 2.5 O2 1.67 O3 5.5 Demand 7.5

D2 3.5 0.5 8.5 5.5

D3 11.5 6.5 15.5 3.5

D4 7.67 1.5 9.5 2.5

Supply 6.5 1.5 11

Two Stage Method Stage I The initial basic feasible solution using Vogel’s Approximation method is x11 = 1, x12 = 5.5, x24 = 1.5, x31 = 6.5, x33 = 3.5, x34 = 1 Total transportation Cost is Z = 123.5 Stage II The optimum solution using modified distribution method is x12 = 5.5, x13 = 1, x23 = 1.5, x31 = 7.5, x33 = 1, x34 = 2.5 The Total Transportation Cost is Z = 121

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Zero Point Method The optimum solution is x12 = 5.5, x13 = 1, x23 = 1.5, x31 = 7.5, x33 = 1, x34 = 2.5 The total transportation cost is Z = 121 Comparative Study

Rank used

Fuzzy Transportation Problem(FTP) Defuzzified TP Stage - I Stage - II Stage- Stage- ZPM FZPM II I FVAM FMDM VAM MODI [-55.5, 93, [-403, 45.5, [-202.5, 154, 302.5] 196.5, 645] 66, 176, 444.5] 123.5 121 121

123.5 121

121

Conclusion Mathematical formulation of fuzzy transportation problem and procedure for finding a fuzzy optimal solution in two stages are discussed with the numerical example. In the first stage, initial basic fuzzy feasible solution using “Fuzzy Vogel’s Approximation method” is determined. In second stage, fuzzy optimal solution using “Fuzzy modified distribution method” is calculated. A new algorithm called the fuzzy zero point method for finding a fuzzy optimal solution of fuzzy transportation problem in single stage [4] with the multiplication operation due to Stephen Dinegar.D & Palanivel.K [5] is discussed, with Numerical example. The same optimum solution is got using both the methods. But fuzzy zero point method with new multiplication operation gives optimum solution in single stage. Hence we conclude that fuzzy zero point method with new multiplication operation is better when compared to fuzzy modified distribution method.

Reference [1] R.E. Bellman and L.A. Zadeh, Decision – making in a fuzzy environment, Management Science, 17 (1970), B 141 – B 164 [2] S. H. Chen and C. H. Hsieh, Graded mean integration representation of generalized fuzzy numbers, Journal of Chiness Fuzzy systems, 5 (1999), 1 – 7. [3] George J. Klir and Bo Yuan., Fuzzy sets and Fuzzy Logic: Theory and application (Fifth Edition).

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[4] 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. [5] Stephen Dinagar. D and Palanivel. K., The transportation problem in Fuzzy Environment. International Journal of Algorithms, Computing and Mathematics. Vol 2, Number 3, August 2009. [6] H.J. Zimmermann, Fuzzy set theory and its applications, Kluwer – Nijhoff, Boston, 1996. [7] L.A. Zadeh, fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1 (1978), 3 – 28.