on initial basic feasible solution (ibfs) of fuzzy ... · problem (ibtp). firstly ibtp is converted...

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

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 4, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com

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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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on initial basic feasible solution (ibfs) of fuzzy ... · problem (ibtp). firstly ibtp is converted...

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