Volume 3, Issue 11, November 2016
SK International Journal of Multidisciplinary Research Hub
Journal for all Subjects
Research Article / Survey Paper / Case Study Published By: SK Publisher (www.skpublisher.com)
Optimization of Transportation Problem with Fuzzy Parameters Ranking Method
V. Rajadurai1 Assistant Professor Dept. of Mathematics
Govt. Arts College Coimbatore – India
S. Kumara Ghuru2 Assistant Professor Dept. of Mathematics
Govt. Arts College Coimbatore – India
Abstract:In this paper, we are presenting a ranking technique with
-cut optimal solution for solving Fuzzy Transportation Problem, where demand quantities, supply quantities are in the form of Triangular fuzzy numbers and the capacity and costs of edges are represented by fuzzy numbers. Many researchers have discussed arithmetic operations, alpha level and simple ranking operations; Pandian [5] has presented some methods for fuzzy Transportation problems.Key Words: fuzzy transportation problem, triangular fuzzy number, robust ranking method, α- optimal solution.
I. INTRODUCTION
1. PRELIMINARIES
Fuzzy Set: Let X be a non-empty set A fuzzy set A in X is characterized by its membership function A : X [0,1], where A (x) is interpreted as the degree of membership of element x in fuzzy set A for all x X.
1.2 INTERNAL NUMBER
Let R be the set of all real numbers. Then closed interval [a, b] is said to be an internal number, where a, b R, a b.
1.3 Fuzzy Number
A fuzzy set A of the real line R with membership function
A( ) X
: R [0,1] is called fuzzy number if (i) A must be normal and convex fuzzy set.(ii) The support of A
must be bounded.
(iii) A must be closed internal for all [0,1].
1.4 Triangular Fuzzy Number
For a Triangular Fuzzy Numbers A(x), it can be represented by A (x1, x2, x3 : 1) with three parameters x1, x2, x3 with membership function is given by
Volume 3, Issue 11, November 2016 pg. 5-9
1 1
1 2
2 1
3
2 3
3 2
0
x x x x
A x x x
x x
x x
x x x x
x x
2.0 Robust Ranking Technique
To find the performance measures in-terms of crop values, the fuzzy numbers are defuzzified into crop ones by a fuzzy number ranking method. Robust ranking Technique which satisfies compensation, linearity and additive properties provides results which are consistent with human intuitions. Given a convex fuzzy number x, the Robust Ranking index its defined by,
1
0
( ) 1 { ) where ( )
2
L U L U
R x x
x
d x
x
is a -level cut of the Fuzzy number x. Here this method for ranking the fuzzy numbers is used. The Robust ranking Index Rx gives the representative value of the fuzzy number x. It satisfies the linearity and additive properly.
2.1Numerical Example
A company has 3 sources S1, S2, S3 and 3 destinations D1, D2, D3; the fuzzy Transportation cost for unit quantity of the product from ith source to jth destination is Cij where Cij is the 3 x 3 square matrix. [Cij] 3 x 3
3 3
(10,15, 20) (10,15,25) (10,20,25) ( . )[ ) (10,15, 25) (10,20,25) (10,15,20) (10,15, 25) (15,20,25) (15,20,25)
ij x
i e C
Fuzzy availability of the product at sources are (15,20,25), (25,35,45) and the Fuzzy demand of the product at destinations are (30,35,40), (15,20,25), (10,20,25) respectively.
The fuzzy transportation problems are
Table 1
x11 = (10,15,.20) x12 = (10,15,25) x13 = (10,20,25) Sources S1 = (15,20,25) x21 = (10,15,.25) x22 = (10,20,25) x23 = (10,15,20) S2 = (10,15,20) x31 = (10,15,.25) x32 = (10,15,25) x33 = (10,20,25) S3 = (25,35,45) D1 = (25,30,.40) D2 = (15,20,25) D3 = (10,25,25)
Destination D1 D2 D3
2.2 Solution procedure
The fuzzy transportation problem in Table 1 can be formulated in the following mathematical programming form.
= i = 1, 2 ...m j = 1, 2, ...m
X
ij
, i j
11 12 13 21 22 23 31 32 33
Minimise = Rx Rx Rx Rx Rx Rx Rx Rx Rx
= R(10,15,20)x11 + R(10,15,25)x12 + R(10,20,25)x13
= R(10,15,25)x21 + R(10,20,25)x22 + R(10,15,20)x23
= R(10,15,25)x31 + R(10,20,25)x32 + R(15,20,25)x33
By the Robust Ranking method, we have,
1 0
( ) 1 ,
2
L U
R x x
x
dx
Where
2 1 1 3 3 2
( x
L, x
U) ( x x ) x x , ( x x )
(ie)
(10,15, 20)
011 15 10) 10, 20 (20 15)
R 2 x dx
1
0
(.5) 5 10, 20 5 dx
1
0
(.5) 5 20, 10 5 dx
R (10,15,20) = 15
Similarly, we have the following values.
1 0
(10,15, 25) 1 15 10) 10, 25 (25 15)
R 2 dx
1 0
1 5 10, 25 10
2 dx
= 15.25
1 0
(10, 20, 25) 1 20 10) 10, 25 (25 20)
R 2
dx
1 0
1 10 10, 25 5
2 dx
= 18.75
1 0
(10,15, 25) 1 15 10) 10, 25 (25 15)
R 2
dx
= 16.75
Volume 3, Issue 11, November 2016 pg. 5-9
1 0
(10, 20, 25) 1 20 10) 10, 25 (25 20)
R 2
dx
= 18.75
R(10, 15, 20) = 15
R(10, 15, 25) = 16.75
R(15, 20, 25) = 20
R(15, 20, 25) = 20
2.3 Rank of all sources
= R(15, 20, 25)=20; R(10, 15, 20) =15; R(25, 35, 45) = 35 2.4 Rank of all destinations
= R(25, 30, 40)=31.25; R(15, 20, 25) =20; R(10, 20, 25) = 18.75 Table 2
FD1 FD2 FD3 Fuzzy Capacity
FO1 13.75
15
6.25
15.25 18.75 20
FO2 16.75
16.25
12.50
18.75 15 15
F03 16.75 20 18.75 35
Fuzzy Demand 31.25 20 18.75 70
Optimal solution is given by
15 + 13.75 + 15.25 x 6.25 + 18.75 x 12.50
16.75 x 16.25 + 20 x 18.75 = 1181.55
Table 3
FD1 FD2 FD3 Fuzzy Capacity
FO1 (10,15,20) (10,15,25) (10,20,25) (15,20,25)
FO2 (10,15,25) (10,20,25) (10,15,20) (10,15,20)
F03 (25,35,45)
(10,15,25) (15,25,20) (15,20,25) (25,35,45)
Fuzzy Demand (25,30,40) (15,20,25) (10,20,25)
From the above table 2 and table 3 if has been show that the optimal cost is (1181.55) by using this method remains as that II. CONCLUSION
In this paper, we have discussed an optimal solution for fuzzy transportation problem using Ranking Technique Fuzzy numbers. The Transportation costs are considered as imprecise numbers described by fuzzy numbers which are more realistic.
method we can have the optimal solution. We can conclude that the solution of fuzzy Transportation problems can be obtained by Robust’s ranking method effectively. The empirical results show that the proposed method approaches out performs the other fuzzy method for solving the transporation problem with fuzzy demands and fuzzy supplies.
References
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2. H.BasirZadeh; R. Abbasi – A new Approach for ranking fuzzy numbers based -cuts. JAMI – Journal of Applied Mathematics & Informatic, 26 (2008) 767-778.
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661-674 (2004).
4. Liu, Feng, Tse and Tsai, Tzoung Ru: “A Two Staan aenotic algorithm for solve the transportation problem with fuzzyh supplies and fuzzy demands” – Int. J. of Innovation Computing Information and Control, Vol. 5(2009).
5. Pandian P and Natarajan G, “An optimal more for less solution to fuzzy transportation problem with mixed constraints” – Applied Mathematical Sciences, Vol. 4. No. 29, 1405-1415 (2010).
6. Pandian P and Natarajan G, “A New algorithm for finding a fuzzy optimal solution for fuzzy transportation problem” – Applied Mathematical Sciences, Vol. 4 (2), pp.79-90 (2010).
7. R.R. Yagger, “A procedure for ordering fuzzy subsets of the unit interval” – Information Sciences, 24, 143-161 (1981).
8. Ritha W and Vinotha J. Merline: “Multi-objective Two stage fuzzy transportation problem”- Journal of Physical Sciences, Vol. 13, pp. 107-1200 (2009).
9. R.Nagarajan and Solairaju: “Computing improved optimal Hungarian Assignment Problems with Fuzzy costs under Robust Ranking Techniques” – International Journal of Computer Application, Vol. 6, No. 4 (2010).
AUTHOR(S)PROFILE
V. Rajadurai, received the M.Sc degree in IMAthematics from Bharathiar University and M.Phil from Periyar University in 1999 and 2001, respectively. Currently he is Assistant professor in Governments arts and Science College, Coimbatore, India. His area of interest includes operation research, differential equations etc.,
