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ON INITIAL BASIC FEASIBLE SOLUTION (IBFS) OF FUZZY TRANSPORTATION
PROBLEM BASED ON RANKING OF FUZZY NUMBERS USING CENTROID OF
INCENTERS
aP. Maheswari* & bM. Vijaya aResearch Scholar & bAssistant Professor
P.G. & Research Department of Mathematics
Marudupandiyar College, Thanjavur – 613 403.
Abstract:
This paper develops a methodology for finding initial basic feasible solution of fuzzy
transportation problem based on proposed ranking technique of trapezoidal fuzzy number using
centroid of incenters. Moreover, the paper is purely depends on interval based transportation
problem (IBTP). Firstly IBTP is converted to fuzzy transportation problem and then a proposed
ranking technique based centroid of incenters is applied for the conversion of crisp number to
find it initial basic feasible solution. Finally, numerical example is given and compared with
traditional methods for finding IBFS through proposed ranking method of trapezoidal fuzzy
number.
Keywords: Fuzzy Transportation Problem, Initial Basic Feasible Solution, Interval Number,
Trapezoidal Fuzzy Number, Fuzzy Ranking Method.
1. Introduction:
In the present competitive world, different ranking methods are being introduced in
variety of forms and everyone is used in an effective manner to find the optimum solution of
transportation problem under uncertain environment.
In 2011, Amarpreet Kaur and Amit Kumar [1] proposed a new method for solving fuzzy
transportation problems using ranking function for non – normal fuzzy numbers and Hadi
Basirzadeh [6] introduced a systematic procedure for finding fuzzy optimal solution of fuzzy
transportation problem based on centroid based defuzzification technique in the same year. A
new algorithm based on proposed ranking function for finding an optimal solution of fully fuzzy
transportation problem introduced by Iden Hasan Hussein and Anfal Hasan Dheyab [7]
introduced in 2015. In 2016, Malini and Anathanarayanan [8] introduced a new ranking method
to solve the fuzzy transportation problem by converting it to a crisp valued problem.
Subsequently, they proposed a method for ranking of octagonal fuzzy numbers in order to find
the optimal solution of fuzzy transportation problem in the same year [9].
In the year 2017 many authors have taken their effort for solving the fuzzy transportation
problem based on various defuzzification techniques. Elumalai [5] and others proposed a new
algorithm by applying zero simplex method to find the optimal solution of fuzzy transportation
problem based on hexagonal fuzzy numbers using Robust ranking method. Using the same
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Robust’s ranking technique, Darunee Hunwisai and Poom Kumam [3] introduced a method for
solving fuzzy transportation problem. Moreover they used allocation table method to find its
initial basic feasible solution. Purushothkumar and Ananathanarayanan [11] introduced an
approach for solving fuzzy transportation problem using a centroid based ranking method [16]
which was proposed by wang et. al. in the year 2006. Anitha Kumari [2] and others developed a
fuzzy version of Vogel’s algorithm for finding fuzzy optimal solution of fuzzy transportation
problem based on centroid of triangular fuzzy numbers. Mohamed Ali and Danish Faraz [10]
proposed a fuzzy least cost method for solving fuzzy triangular transportation problem based on
signed distance ranking method. Uthra and others [15] proposed a method for obtaining the
optimal solution of fuzzy transportation problem based on ranking of symmetric triangular fuzzy
numbers. Recently in 2018, Purushothkumar [12] and others proposed a new ranking technique
based on centroid to solve the fuzzy transportation problem using traditional method of crisp
transportation problem. Subsequently Ramesh Kumar and Subramanian [13] also using Robust
ranking method in order to convert the fuzzy transportation problem into crisp one to its optimal
solution based on their proposed method in the same year.
In this paper, an interval data based fuzzy transportation algorithm is proposed to find Initial
Basic Feasible Solution (IBFS) based on proposed ranking method of generalized trapezoidal
fuzzy number. To illustrate the proposed method a numerical example of Interval based
Transportation Problem is solved and the obtained results are analyzed and compared with the
help of traditional methods of finding IBFS of Transportation Problem.
2. Preliminaries:
Definition: (Fuzzy Set) 1.1 [17]
Fuzzy set is a set of objects which has elements with degree of membership of belonging in it.
Mathematically, the Fuzzy subset A~
of a Universal set X is defined by its membership function
as an ordered pair. It is denoted as follows:
AxXxxAAA
~each ],1,0[: )(,
~~~
where the value of )(~ xA
at x shows the grade of membership of x in A~
.
Definition: (Fuzzy Number) 1.2 [14]
A fuzzy set A~
, defined on the universal set of real number R, is said to be a fuzzy number if it
possess at least the following properties:
i. A~
is convex.
ii. A~
is normal Rx 0 such that 1)( 0~ xA
.
iii. )(~ xA
is piecewise continuous.
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iv. A~
must be closed interval for every ]1 ,0[
v. The support of A~
, must be bounded.
Definition: (Trapezoidal Fuzzy Number) 1.3 [4]
Trapezoidal fuzzy number dcbaA ,,,~ is defined as:
therwise ,0
,
,
,
)(~
o
dxcdc
dx
cxbw
bxaab
ax
xA
Here w is any real number satisfying 10 w . If 1w then the trapezoidal fuzzy number is
said to be normal. It becomes a triangular fuzzy number if b = c.
Definition: (Fuzzificaiton) 1.4 [4]
Let us consider an interval number [L, U]. One – third length of the interval is taken as
3
)( LUd
. As per the definition of arithmetic progression, the required trapezoidal fuzzy
number is expressed as
(1) ---------------- ),2,,( UdLdLL
3. Proposed Ranking Method based on centroid of incenters
This section proposes a new area method for ranking generalized trapezoidal fuzzy number
based on centroid of incenters. Centroid is the balancing point of any type of plane figure. The
below plane figure is considered as the graphical representation of Generalized Trapezoidal
Fuzzy Number. To determine the centroid of the trapezoidal fuzzy number );,,,( wdcbaA
geometrically, the trapezoid is divided into three triangles AEH, EHF and HFD. In this work, the
centroid of the incenters of the three plane figures is considered as the reference point of
generalized trapezoidal fuzzy number to define its ranking function. The reason for selecting this
point as a reference point is that each incenter points (G1 =
2
))(2(,
abbdab of AEH, G2 =
4
4)2)(2(,
2
2bdacdadaof EHF, and G3 =
2
))(2(,
dccdac of HFD) are
balancing points of three triangles. Therefore, the centroid of these points would be a much more
balancing point rather than centroid of trapezoid.
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E ,b
F ,c
A(a,0) B(b,0) H
0,
2
da C(c,0) D(d,0)
Figure 1: Centroid of Incenters
Consider the generalized fuzzy number );,,,( wdcbaA . The incenters of the three triangles are
(2)------------------- ,2
,111
1
111
111
1 11
w
daba
yxIC II
Where 2
2
12
wda
b
,
2
12
daa
,
22
1 wba
(3)------------------- ,
2,
222
22
222
222
2 22
w
dacb
yxIC II
Where 2
2
22
wda
c
,
2
2
12
wda
b
,
21 cb
-(4)------------------- ,2
,333
3
333
333
3 33
w
dadc
yxIC II
1IC 3IC
2IC
G
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Where
2
32
dad
,
2
2
12
wda
c
,
22
1 wdc
The point IC3 does not line in the line ______
21ICIC . Therefore, IC1, IC2 and IC3 are non collinear and
they form a triangle.
We define the centroid of incenters IC1, IC2 and IC3 of the generalized trapezoidal fuzzy number
);,,,( wdcbaA as
-(5)----------------- 3
,3
)~
(),~
( 321321
IIIIII yyyxxxAyAxG
As a special case,, for triangular fuzzy number );,,( wcbaA , i.e., c = b the centroid of
incenters is given by
(6)------------------- 3
,3
)~
(),~
( 321321
IIIIII yyyxxxAyAxG
The ranking function of the generalized trapezoidal fuzzy number );,,,( wdcbaA which maps
the set of all fuzzy numbers to a set of real numbers is defined as:
(7)--------------------- )A
~()A
~()A
~R(
22
yx
This is the distance between the centroid of incenters as defined in (1) and the original point.
Definition 3.1
Let F(R) be the set of all generalized trapezoidal fuzzy numbers. One feasible way for solving
the transportation problem is based on the concept of comparison of fuzzy unit cost, fuzzy supply
and fuzzy demand by using ranking function. An effective approach for comparison of such
fuzzy parameter is to define a ranking function : F(R) → R which maps each fuzzy parameters
into crisp one.
Using the proposed ranking function ( ), we define ranking order for generalized trapezoidal
fuzzy numbers (fuzzy parameters) as follows:
(i) If )~
()~
( BRAR then BA~~
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(ii) If )~
()~
( BRAR then BA~~
(iii) If )~
()~
( BRAR then BA~~
4. Proposed Algorithm for finding IBFS of Transportation Problem:
Step 1: Tabular form of specified problem based on interval data to be constructed.
Step 2: The interval data to be fuzzified as per the definition given in Section 4.
Step 3: The proposed ranking function to be applied for the conversion of fuzzy
transportation problem as a crisp one then check whether it is a balanced
transportation problem.
Step 4: Classical Methods (North West Corner Rule or Least Cost Method or Vogal’s
Approximation Method) to be applied for finding Initial Basic Feasible Solution.
5. Numerical Example:
A firm has three workshops A, B, C, D and four warehouses P, Q, R, S. The number of units
available at the workshops is [110, 150], [130, 170], [150, 190] and the demand at P, Q, R, S are
[70,110], [80, 120], [120, 160], [100, 140] respectively. The unit costs of transportation is given
by the following table.
Table 1
Transportation Cost
P Q R S
A [7, 13] [9, 15] [12, 18] [5, 11]
B [11, 17] [8, 14] [6, 12] [7, 13]
C [17, 23] [2, 8] [4, 10] [15, 21]
Table 2
Tabular Form
P Q R S Supply
A [7, 13] [9, 15] [12, 18] [5, 11] [127, 133]
B [11, 17] [8, 14] [6, 12] [7, 13] [147, 153]
C [17, 23] [2, 8] [4, 10] [15, 21] [167, 173]
Demand [87, 93] [97, 103] [137, 143] [117, 123]
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Solution
The interval data in Table 2 using equation (1) as follows:
Table 3
Fuzzified Data
P Q R S Supply
A (7, 9, 11, 13) (9, 11, 13, 15) (12, 14, 16, 18) (5, 7, 9, 11) (127, 129, 131, 133)
B (11, 13, 15, 17) (8, 10, 12, 14) (6, 8, 10, 12) (7, 9, 11, 13) (147, 149, 151, 153)
C (17, 19, 21, 23) (2, 4, 6, 8) (4, 6, 8, 10) (15, 17, 19, 21) (167, 169, 171, 173)
Demand (87, 89, 91, 93) (97, 99, 101, 103) (137, 139, 141, 143) (117, 119, 121, 123)
Applying the proposed ranking technique in equation (7), the above Table becomes
Table 4
Defuzzified Data
P Q R S Supply
A 10.0123 12.0102 15.0082 8.0154 130.0009
B 14.0088 11.0112 9.0137 10.0123 150.0008
C 20.0061 5.0245 7.0176 18.0068 170.0007
Demand 90.0014 100.0012 140.0009 120.0100
As the total demand and total supply of the transportation problem obtained in the above table
are not equal, so it is not a balanced transportation problem. A dummy supplier is introduced
with supply 0.085 units.
Table 4
Balanced Table
P Q R S Supply
A 10.0123 12.0102 15.0082 8.0154 130.0009
B 14.0088 11.0112 9.0137 10.0123 150.0008
C 20.0061 5.0245 7.0176 18.0068 170.0007
Dummy 0 0 0 0 0.01110
Demand 90.0014 100.0012 140.0009 120.0100
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The IBFS for the above transportation Problem using North West Corner Rule is
Z = (10.01 × 90) + (12.01 × 40) + (11.01 × 60) + (9.01 × 90) + (7.02 × 50) + (18.01 × 120) + (0
× 0.01) = 5365.13
The IBFS for the same problem using Least Cost Method is
Z = (10.01 × 9.99) + (8.02 × 120.01) + (14.01 × 80) + (9.01 × 70) + (5.02 × 100) + (7.02 × 70) +
(0×0.01) = 3807.31
Using Vogal’s Approximation, the IBFS is
Z = (10.01 × 89.99) + (8.02 × 40.01) + (9.01 × 70) + (10.01 × 80) + (5.02 × 100) + (7.02 × 70) +
(0 × 0.01) = 3647.34
Figure 1: IBFS based on Proposed Ranking Methods
6. Conclusion:
This paper proposes an algorithm with the combination of two key ideas to find the IBFS of
Interval based Fuzzy Transportation Problem. Among the two key ideas, first one is fuzzy
ranking method based on centroid of incenters and the second one is to find the IBFS of Interval
based Transportation Problem. The algorithm proposed in this research article is found to be
effective to find the IBFS of Interval valued Transportation Problem, which gives the better
solution as per the traditional methods for finding IBFS of Transportation Problem with crisp
data.
5365.13
3807.31
3647.34
0
1000
2000
3000
4000
5000
6000
NWCM LCM VAM
IBFS
IBFS
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Reference:
1. Amarpreet Kaur & Amit Kumar, A New Method for Solving Fuzzy Transportation
Problems using Ranking Function, Applied Mathematical Modelling, 35, pp. 5652–5661,
2011.
2. Anitha Kumari, T., Fuzzy Transportation Problems with New Kind of Ranking Function,
The International Journal of Engineering and Science, 6(11), pp. 15-19, 2017.
3. Darunee Hunwisai & Poom Kumam, A Method for Solving a Fuzzy Transportation
Problem via Robust Ranking Technique and ATM, 4, pp. 1 – 11, 2017.
4. Dinesh, C.S. Bisht, & Pankaj Kumar Srivastava, Trisectional Fuzzy Trapezoidal
Approach to Optimize Interval Data based Transportation Problem, Journal of King Saud
University - Science, Article in Press, 2018.
5. Elumalai, P., et al., Fuzzy Transportation Problem using Hexogonal Fuzzy Numbers by
Robust Ranking Method, Emperor International Journal of Finance and Management
Research, 3(7), pp. 52 – 58, 2017.
6. Hadi Basirzadeh, An Approach for Solving Fuzzy Transportation Problem, Applied
Mathematical Sciences, 5(32), pp. 1549 – 1566, 2011.
7. Iden Hasan Hussein, Anfal Hasan Dheyab, A New Algorithm using Ranking Function to
find Solution for Fuzzy Transportation Problem, International Journal of Mathematics
and Statistics Studies, 3(3), pp. 21-26, 2015.
8. Malini, P. & Ananthanarayanan, M., Solving Fuzzy Transportation Problem using
Ranking of Trapezoidal Fuzzy Numbers, International Journal of Mathematics Research,
8(2), pp. 127-132, 2016.
9. Malini, P. & Ananthanarayanan, M.,, Solving Fuzzy Transportation Problem using
Ranking of Octagonal Fuzzy Numbers, International Journal of Pure and Applied
Mathematics, 110(2), pp. 275-282, 2016.
10. Mohamed Ali, A., & Danish Faraz, Solving Fuzzy Triangular Transportation Problem
using Fuzzy Least Cost Method with Ranking Approach, 3(7), pp. 15 – 20, 2017.
11. Purushothkumar, M.K., & Ananathanarayanan, M., Fuzzy Transportation Problem of
Trapezoidal Fuzzy Numbers with New Ranking Technique, IOSR Journal of
Mathematics, 13(6), pp. 6-12, 2017.
12. Purushothkumar, M.K., Solution to Transportation problem in fuzzy environment with
New Ranking Technique, International Journal of Scientific Research and Reviews, 7(3),
pp. 638-650, 2018.
13. Ramesh kumar, M., & Subramanian, S., Solution of Fuzzy Transportation Problems with
Trapezoidal Fuzzy Numbers using Robust Ranking Methodology, 119(16), pp. 3763-
3775, 2018.
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14. Sankar Prasad Mondal & Manimohan Mandal, Pentagonal fuzzy number, its properties
and application in fuzzy equation, Future Computing and Informatics Journal, 2, pp. 110
– 117, 2017.
15. Uthra, G., et al., An improved ranking for Fuzzy Transportation Problem using
Symmetric Triangular Fuzzy Number, Advances in Fuzzy Mathematics, 12(3), pp. 629-
638, 2017.
16. Ying-Ming Wang et al., On the centroids of fuzzy number .Fuzzy set and systems,
157(7), pp. 919 – 926, 2006.
17. Zadeh, L.A., Fuzzy sets, Inf. Control 8, pp. 338–356, 1965.
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