Application of Classical Transportation Methods for Solving Fuzzy Transportation Problems

JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 11, Issue 5, October 2011 Online English edition of the Chinese language journal RESEARCH PAPER

Cite this article as: J Transpn Sys Eng & IT, 2011, 11(5), 68í80.

Application of Classical Transportation Methods for Solving Fuzzy Transportation Problems KUMAR Amit*, KAUR Amarpreet School of Mathematics and Computer Applications, Thapar University, Patiala-147 004, India

Abstract: There are several methods, in the literature, for finding a fuzzy optimal solution to fully fuzzy transportation problems (transportation problems in which all the parameters are represented by fuzzy numbers). In this study, the shortcomings of some existing methods are pointed out, and to overcome these shortcomings, two new methods (based on fuzzy linear programming formulation and classical transportation methods) are proposed to find a fuzzy optimal solution to fuzzy transportation problems with a new representation of trapezoidal fuzzy numbers. The advantages of the proposed methods over existing methods are discussed. Also, it is shown that it is better to use the proposed representation of trapezoidal fuzzy numbers instead of the existing representation of trapezoidal fuzzy numbers for finding a fuzzy optimal solution to fuzzy transportation problems. To illustrate the proposed methods, a fuzzy transportation problem (FTP) is solved using the proposed methods and the obtained results are discussed. The proposed methods are easy to understand and to apply for finding a fuzzy optimal solution to fuzzy transportation problems occurring in real-life situations. Key Words: fuzzy transportation problem; ranking function; trapezoidal fuzzy number

1

Introduction

In today’s highly competitive market, the pressure on organizations to find better ways to create and deliver value to customers becomes stronger. How and when to send the products to the customers in the quantities, they want in a cost-effective manner, becomes more challenging. Transportation models provide a powerful framework to meet this challenge. They ensure efficient movement and timely availability of raw materials and finished goods. The basic transportation problem was originally developed by Hitchcock[1]. The transportation problems can be modeled as a standard linear programming problem, which can then be solved by the simplex method. However, because of its very special mathematical structure, it was recognized early that the simplex method applied to the transportation problem can be made quite efficient in terms of how to evaluate the necessary simplex method information (variable to enter the basis, variable to leave the basis and optimality conditions). Charnes and Cooper[2] developed the stepping stone method, which provides an alternative way of determining the simplex

method information. Dantzig and Thapa[3] used the simplex method to the transportation problem as the primal simplex transportation method. An initial basic feasible solution to the transportation problem can be obtained using the north-west corner rule, row minima, column minima, matrix minima (least-cost), or the vogel’s approximation method. The modified distribution method is useful for finding the optimal solution to the transportation problem. In conventional transportation problems, it is assumed that the decision-maker is sure about the precise values of transportation cost, availability and demand of the product. In real-world applications, all the parameters of the transportation problems may not be known precisely because of uncontrollable factors. These types of imprecise data are not always well represented by s random variable selected from a probability distribution. Fuzzy numbers introduced by Zadeh[4] may represent these data. Therefore, fuzzy decision making method is needed here. Zimmermann[5] showed that solutions obtained by fuzzy linear programming are always efficient. Subsequently, Zimmermann’s fuzzy linear programming has developed into

Received date: May 25, 2010; Revised date: Jul 5, 2010; Accepted date: Apr 18, 2011 *Corresponding author. E-mail: [email protected] Copyright © 2011, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(10)60141-9

KUMAR Amit et al. / J Transpn Sys Eng & IT, 2011, 11(5), 68í80

several fuzzy optimization methods for solving transportation problems. Oheigeartaigh[6] proposed an algorithm for solving transportation problems where the capacities and requirements are fuzzy sets with linear or triangular membership functions. Chanas et al.[7] presented a fuzzy linear programming model for solving transportation problems with crisp cost coefficients and fuzzy supply and demand values. Chanas and Kuchta[8] proposed concept of the optimal solution for the transportation problem with fuzzy coefficients expressed as fuzzy numbers, and developed an algorithm for obtaining the optimal solution. Saad and Abbas[9] discussed the solution algorithm for solving transportation problems in fuzzy environment. Liu and Kao[10] described a method of solving fuzzy transportation problems based on the extension principle. Gani and Razak[11] presented a two-stage cost-minimizing fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy numbers. A parametric approach is used to obtain a fuzzy solution and the aim is to minimize the sum of the transportation costs in the two stages. To deal with uncertainties of supply and demand parameters, Gupta and Mehlawat[12] transformed the past data pertaining to the amount of supply of the ith supply point and the amount of demand of the jth demand point using level (Ȝ, ȡ) interval-valued fuzzy numbers. Dinagar and Palanivel[13] investigated fuzzy transportation problems, with the aid of trapezoidal fuzzy numbers and proposed fuzzy modified distribution method to find the optimal solution in terms of fuzzy numbers. Pandian and Natarajan[14] proposed a new algorithm, namely, fuzzy zero point method of finding a fuzzy optimal solution to a fuzzy transportation problems, where the transportation cost, supply and demand are represented by trapezoidal fuzzy numbers. In this study, the shortcomings of some existing methods are pointed out and to overcome these shortcomings, two new methods (based on fuzzy linear programming formulation and classical transportation methods) are proposed to find a fuzzy optimal solution to fuzzy transportation problems by representing all the parameters as trapezoidal fuzzy numbers. The advantages of the proposed methods over existing methods are also discussed. To illustrate the proposed methods, a FTP is solved by the proposed methods and the obtained results are discussed.Proposed methods are easy to understand and to apply for finding a fuzzy optimal solution to fuzzy transportation problems occurring in real-life situations. This paper is organized as follows: In Section 2, formulation of FTP is presented and the shortcomings of the existing methods are pointed out. Application of ranking function for solving FTP is also presented. In Section 3, a new representation of trapezoidal fuzzy numbers, named as JMD (Jai Mata(Mehar) Di) representation of trapezoidal fuzzy numbers, is proposed and the existing methods[15], are modified with JMD representation of trapezoidal fuzzy

numbers. It is shown that it is better to use the JMD representation of trapezoidal fuzzy numbers instead of existing representation of trapezoidal fuzzy numbers for finding a fuzzy optimal solution to fuzzy transportation problems. The advantages of the proposed methods are discussed in Section 4 . In Section 5, to illustrate the proposed methods, a numerical example is solved and obtained results are discussed. The conclusions are discussed in Section 6.

2

Fuzzy transportation problem

In conventional transportation problems, it is assumed that the decision-maker is sure about the precise values of transportation cost, availability and demand of the product. In real-world applications, all these parameters of the transportation problems may not be known precisely because of uncontrollable factors. For example, in real-life problems, the following situations may occur: (a) Let a product be transported first time at a destination and no expert has knowledge about the transportation cost, then there exists uncertainty about the transportation cost. (b) If a new product is launched in the market, then there always exists uncertainty about the demand of that particular product. (c) In daily-life problems, it can be seen that whenever a customer asks a supplier that the particular product is available or not, sometimes the supplier answers “yes it is available”, but after a few seconds the supplier answers “sorry” at this time “this product is not available”. Sometimes a supplier does not have any uncertainty about the statement that the product is available or not. When a customer demands for a particular product and the supplier answers “yes, the product is available”, but if the demand of product is large, then again the supplier says, “I check that so much, quantity is available or not i.e.”, there exists uncertainty about the availability of product. To deal with such situations, fuzzy set theory is applied in literature to solve the transportation problems. Several authors[10,13,14] have proposed different methods for solving balanced fuzzy transportation problems by representing the transportation cost, availability and demand as normal fuzzy numbers. The balanced fuzzy transportation problems, in which a decision-maker is uncertain about the precise values of transportation cost, availability and demand, may be formulated as follows: p

Minimize

q

¦¦c~

ij

…~ xij

i =1 j =1 q

subject to

¦~x

= a~i ,

ij

i = 1,2,3,, p

j =1

p

¦~x

ij

i =1 p

~ = bj , q

~

¦a~ = ¦b i

i =1

j = 1,2,3, , q j

j =1

~ xij is a non-negative trapezoidal fuzzy number.

KUMAR Amit et al. / J Transpn Sys Eng & IT, 2011, 11(5), 68í80

where, p=the total number of sources; q=the total number of destinations; a~i =the fuzzy availability of the product at the ith source; ~ b j =the fuzzy demand of the product at the jth destination; c~ij =the fuzzy transportation cost for unit quantity of the product from the ith source to the jth destination; ~ xij =the fuzzy quantity of the product that should be transported from the ith source to the jth destination (or fuzzy decision variables) to minimize the total fuzzy transportation cost; p

¦a~ =the total fuzzy availability of the product; i

i =1

q

~

¦b

j

transportation problem p

q

ij

q

i =1

j =1

~ Remark 2.1: If ¦a~i = ¦b j , then the FTP is said to be a

balanced fuzzy transportation problem; otherwise, it is called an unbalanced fuzzy transportation problem. 2.1 Shortcomings of the existing methods There are several methods to find optimal solutions to fuzzy transportation problems. However, there are shortcomings in some existing methods, which are as follows: (i) In the existing methods[6,7,10], fuzzy transportation problems are first converted into equivalent crisp transportation problem (CTP) by using Į-cut method, which are then solved by standard methods. The final results of a FTP are thus real numbers, which represents a compromise in terms of fuzzy numbers. (ii) It is very difficult to apply existing methods for finding a fuzzy optimal solution to fuzzy transportation problems. (iii) Dinager and Palanivel[13] proposed a method of solving fuzzy transportation problems, and they obtained the following fuzzy optimal solution by applying the proposed method for solving a FTP: (a) The fuzzy optimal quantity of the product that should be transported, from the second source to the third destination, the third source to the first destination and the third source to the third destination are trapezoidal fuzzy numbers (–5, –1, 6, 12), (–5, –1, 3, 7), and (–11, –3, 6, 12), respectively. (b) The minimum total fuzzy transportation cost is (–122, –2, 139, 257). It is clear that there exists a negative part in all the obtained trapezoidal fuzzy numbers, which depicts that quantity of the product and transportation cost may be negative. However, the negative quantity of the product and negative transportation cost have no physical meaning. Similarly the results obtained using the existing methods[14] have no physical meaning. 2.2 Application of ranking function for solving fuzzy transportation problems ~ The fuzzy optimal solution {xij } to the balanced fuzzy

= a~i ,

ij

~ = bj ,

p

i =1 p

q

i = 1,2,3,, p

i

i =1

j = 1,2,3,  , q

~

¦a~ = ¦b

j

j =1

~ xij is a non-negative fuzzy number, is a set of ~ xij number which satisfies the following

fuzzy characteristics: ~ (i) x ij is a non-negative fuzzy number; q

(ii)

¦~x

ij

= a~i , i = 1,2,3,, p and

p

¦~x

ij

i =1

~ = b j , j = 1,2,3, , q ;

~ (iii) If there exists any non-negative fuzzy number x 'ij q

p

ij

¦~x

…~ xij =the total fuzzy transportation cost;

i =1 j =1

…~ xij

j =1

j =1

q

¦~x

subject to

j =1 p

ij

i =1 j =1

=the total fuzzy demand of the product;

¦¦c~

q

¦¦c~

Minimize

¦~x '

such that

ij

j =1

p

p

¦~x '

= a~i , i = 1,2,3,, p and

q

p

i =1

ij

~ = b j , j = 1,2,3, , q ,

q

then ƒ(¦¦c~ij … ~xij ) < ƒ(¦¦c~ij … ~x 'ij ) . i =1 j =1

~

i =1 j =1

Remark 2.2: Let x ij be a fuzzy optimal solution to FTP ~ and there exist one or more x'ij such that ~ (i) x 'ij is a non-negative fuzzy number; q

(ii)

¦~x ' j =1

p

ij

= a~i , i = 1,2,3,, p and

p

¦~x ' i =1

q

p

ij

~ = b j , j = 1,2,3,  , q ;

q

(iii) ƒ(¦¦c~ij … ~xij ) = ƒ(¦¦c~ij … ~x 'ij ) ; i =1 j =1

i =1 j =1

~ then x 'ij is said to be an alternative fuzzy optimal solution to FTP.

3

Proposed methods with new representation of trapezoidal fuzzy numbers

In this section, a new representation of trapezoidal fuzzy numbers, named JMD representation of trapezoidal fuzzy numbers, is proposed and the existing methods[15], are modified with JMD representation of trapezoidal fuzzy numbers. Also, it is shown that it is better to use the JMD representation of trapezoidal fuzzy numbers instead of existing representation of trapezoidal fuzzy numbers for finding a fuzzy optimal solution to fuzzy transportation problems. 3.1 JMD representation of trapezoidal fuzzy numbers In this section, some new definitions are introduced. Definition 3.1: Let (m, n, Į, ȕ) be a trapezoidal fuzzy number; then, its JMD representation is (x, Į, Ȗ, ȕ)JMD, where x=m–Į, Į=Į•0, Ȗ=n–m•0, ȕ=ȕ•0. Definition 3.2: A trapezoidal fuzzy number ~ A = ( x, D , J , E )JMD is said to be a zero trapezoidal fuzzy number if and only if x=0, Į=0, Ȗ=0, ȕ=0. Definition 3.3: A trapezoidal fuzzy number ~ A = ( x, D , J , E ) JMD is said to be a non-negative trapezoidal fuzzy number if and only if x•0.

KUMAR Amit et al. / J Transpn Sys Eng & IT, 2011, 11(5), 68í80

Definition

3.4:

Two

trapezoidal

fuzzy

numbers

~ ~ A = ( x1 , D 1 , J 1 , E1 ) JMD and B = ( x 2 , D 2 , J 1 , E 2 ) JMD are said to ~ ~ be equal i.e., A = B if and only if x1 = x2 , D1 = D 2 , J 1 = J 2 , E1 = E 2 . [16]

Definition 3.5: The existing ranking formula , is converted into the following ranking formula: Let ( x, D , J , E ) JMD be a trapezoidal fuzzy number then ƒ( x, D , J , E ) = ( 4 x  3D  2J  E ) / 4 . 3.2 Arithmetic operations of JMD type trapezoidal fuzzy numbers The arithmetic operations of JMD type trapezoidal fuzzy numbers can be easily defined by using the following steps: Step 1: Convert the given JMD type trapezoidal fuzzy numbers into (m, n, Į, ȕ) type trapezoidal fuzzy numbers by using the Definition 3.1. Step 2: Now apply the arithmetic operations of (m, n, Į, ȕ) type trapezoidal fuzzy numbers, discussed in Section 2.2[15]. Step 3: Convert the (m, n, Į, ȕ) type trapezoidal fuzzy number, obtained from Step 2, into a JMD type trapezoidal fuzzy number. ~ A = (6, 2, 3, 11)JMD Example 3.1: Let and ~ B = (7, 4, 10, 16 ) JMD be two JMD type trapezoidal fuzzy ~ and ~ can be obtained by the numbers then addition of A B following steps: ~ and ~ is (8, 11, Step 1: (m, n, Į, ȕ) representation of A B 2, 11) and (11, 21, 4, 16), respectively. ~ ~ Step 2: A † B =(19, 32, 6, 27). Step 3: JMD representation of (19, 32, 6, 27) is (13, 6, 13, 27)JMD, i.e., (6, 2, 3, 11) JMD † (7, 4, 10, 16) JMD = (13, 6, 13, 27) JMD . ~ ~ Example 3.2: Let A = (6, 2, 3, 11)JMD , B = (7, 4, 10, 16)JMD be two JMD type trapezoidal fuzzy numbers then ~ and ~ can be obtained by the multiplication of A B following steps: ~ and ~ is (8, 11, 2, Step 1: (m, n, Į, ȕ) representation of A B 11) and (11, 21, 4, 16), respectively. ~ ~ ; (88, 231, 46, 583). Step 2: A … B Step 3: JMD representation of (88, 231, 46, 583) is (42, 46, 143, 583)JMD. i.e., (6,2,3,11) JMD … (7,4,10,16 ) JMD , (42, 46, 143, 583)JMD. 3.3 Method based on fuzzy linear programming formulation (FLPF) In this section the existing method[15], to find the fuzzy optimal solution of FTP occurring in real life situations is modified by representing all the parameters as JMD type trapezoidal fuzzy numbers. The steps of the proposed method are as follows: Step 1: Formulate the balanced fuzzy transportation problem into the following fuzzy linear programming problem: m

n

Minimize ¦¦c~ij … ~xij i =1 j =1

n

subject to

¦~xij = a~i , j =1

i = 1,2,3,, m; m = p or p+1

m

¦~x

ij

~ = bj ,

j = 1,2,3,, n; n = q or q+1

i =1

~ xij is a non-negative JMD trapezoidal fuzzy number. where, m=total number of sources; n=total number of destinations; c~ij = ( xijc , D ijc , J ijc , E ijc ) JMD =fuzzy transportation cost for unit quantity of the product from ith source to jth destination; a~i = ( xi , D i , J i , E i ) JMD =the fuzzy availability of the product at ith source; ~ b j = ( xcj , D ' j , J ' j , E ' j ) JMD =the fuzzy demand of the product at jth destination; ~ xij = ( xij , D ij , J ij , E ij ) JMD =the fuzzy quantity of the product that should be transported from the ith source to jth destination (or fuzzy decision variables) to minimize the total fuzzy transportation cost; m

¦a~ =total fuzzy availability of the product; i

i =1

m

n

¦¦c~

ij

…~ xij =total fuzzy transportation cost.

i =1 j =1

~ Step 2: Now our objective is to find xij which satisfies the following properties: ~ (i) x ij is a non-negative trapezoidal fuzzy number. n

(ii)

¦~x

ij

j =1

= a~i , i = 1,2,3,  , m

m

¦~x

and

ij

i =1

~ = b j , j = 1,2,3,, n .

(iii) If there exists any non-negative trapezoidal fuzzy ~ number x 'ij such that n

¦~x '

ij

= a~i , i = 1,2,3,  , m

m

ij

~ = b j , j = 1,2,3,, n

j =1

and

¦~x ' i =1

then m

n

ƒ(¦ ¦c~ij … ~ xij ) i =1 j =1

m

n

< ƒ(¦¦c~ij … ~x 'ij ) i =1 j =1

i.e., m

n

~ ~ Minimize ƒ(¦¦cij … xij ) i =1 j =1

n

¦~xij = a~i ,

subject to

i = 1,2,3,, m

j =1

m

¦~x

ij

~ = bj ,

j = 1,2,3,, n

i =1 m

n

~

¦a~ = ¦b i

i =1

j

j =1

~ xij a mnon-negative JMD trapezoidal fuzzy number. n Step 3: Let ¦¦c~ij … ~x ij = ( x 0 , D 0 , J 0 , E 0 ) JMD , then the fuzzy i =1 j =1

linear programming problem (FLPP), obtained in Step 2, can be written as: Minimize ƒ( x0 , D 0 , J 0 , E 0 ) subject to n

n

n

n

j =1

j =1

j =1

j =1

(¦xij , ¦D ij , ¦J ij , ¦E ij ) JMD = ( xi , D i , J i , E i ) JMD , i = 1,2,3,  , m

KUMAR Amit et al. / J Transpn Sys Eng & IT, 2011, 11(5), 68í80 m

m

m

m

i =1

i =1

i =1

i =1

Table 2 First crisp transportation table

(¦x ij , ¦D ij , ¦J ij , ¦E ij ) JMD = ( x cj , D cj , J cj , E cj ) JMD , j = 1,2,3,  , n m

¦( x , D , J i

i

i =1

n

i

, E i ) JMD = ¦( x cj , D cj , J cj , E cj ) JMD

Destinations ĺ Sources Ļ

D1

D2

···

Dj

···

Dn

S1

c c11

cc12

···

c1c j

···

c1cn

x1

















Si

cic1

cic2

···

cijc

···

cinc

xi

















Sm

ccm1

cmc 2

···

ccmj

···

c cmn

j =1

( xij ,D ij , J ij , E ij ) JMD is a non-negative JMD trapezoidal fuzzy number. Step 4: The FLPP, obtained in Step 3, is converted into following crisp linear programming problem:

Minimize

1 (4 x0  3D 0  2J 0  E 0 ) 4 n

subject to

¦x

ij

= xi ,

i = 1,2,3,  , m

n

¦D

¦J ij = J i , i = 1,2,3,  , m j =1

n

ij

= E i , i = 1,2,3,  , m

¦x

ij

= xcj , j = 1,2,3,  , m

¦D

ij

= D cj , j = 1,2,3,  , m

ij

= J cj , j = 1,2,3,  , m

i =1

m

¦J i =1

m

¦E

ij

i =1

j

j =1

D2

···

Dj

···

Dn

S1

ȡ11

ȡ12

···

ȡ1j

···

ȡ1n

Į1

















Si

ȡi1

ȡi2

···

ȡij

···

ȡin

Įi

















Sm

ȡm1

ȡm2

···

ȡmj

···

ȡmn

= E cj , j = 1,2,3,  , m

D '1

xij , D ij , J ij , E ij t 0

D '2

D'j

···

Įm m

n

¦D = ¦D '

D 'n

···

i

i =1

 i, j

Step 5: Find the optimal solution xij, Įij, Ȗij, ȕij, by solving the crisp linear programming problem, obtained in Step 4. Step 6: Find the fuzzy optimal solution {~xij } by putting the ~ values of xij, Įij, Ȗij, ȕij, in xij = ( xij , D ij , J ij , E ij ) JMD . Step 7: Find the minimum total fuzzy transportation cost by m n ~ ~ …~ c x putting the values of ij in ¦¦ ij xij . i =1 j =1 3.4 Method based on classical transportation methods In this section, the existing method[15], to find a fuzzy optimal solution to FTP occurring in real-life situations is modified by representing all the parameters as JMD -type trapezoidal fuzzy numbers. The steps of the proposed method are as follows: Step 1: Represent the balanced fuzzy transportation problem in a tabular form as shown by Table 1.

j

j =1

Table 4 Third crisp transportation table Destinations ĺ Sources Ļ S1

D1

D2

···

Dj

···

Dn

į11

į12



į1j



į1n

Ȗ1



 



 







  



  



 Si

įi1



 Sm

įi2

įm1

įm2

J '1

J '2

įij

įmj

J 'j

įin

Ȗi





įmn

Ȗm

J 'n

m

n

¦J = ¦J ' i

i =1

Table 1 Tabular representation of balanced fuzzy transportation problem

j =1

Table 5 Fourth crisp transportation table Destinations

D1

D2

···

Dj

···

Dn

Availability

S1

c~11

c~12

···

c~1 j

···

c~1n

a~1

















Si

c~i1

c~i 2

···

c~ij

···









Sm

c~m1

 c~m 2

Demand

i

D1

i =1

Destinations ĺ Sources Ļ

n

¦a = ¦b

Destinations ĺ Sources Ļ

j =1

i =1 m

xnc

···

Table 3 Second crisp transportation table

n

m

xcj

···

= D i , i = 1,2,3,  , m

ij

j =1

¦E

xc2

xc1

j =1

xm m

~ b1

~ b2

 ··· ···

c~mj

~ bj

··· ···

c~in

a~i

ĺ Sources Ļ

D1

D2

···

Dj

···

Dn

S1

ȟ11

ȟ12

···

ȟ1j

···

ȟ1n

ȕ1

















Si

ȟi1

ȟi2

···

ȟij

···

ȟin

ȕi

c~mn





a~m















Sm

ȟm1

ȟm2

···

ȟmj

···

ȟmn

ȕm

~ bn

¦a~ = ¦b

E '1

E '2

···

E'j

···

E 'n



m

n

i

i =1

j =1

~ j

m

n

¦E = ¦E ' i

i =1

j =1

j

j

KUMAR Amit et al. / J Transpn Sys Eng & IT, 2011, 11(5), 68í80

Step 2: Split Table 1 into four crisp transportation tables, i.e., Tables 2, 3, 4 and 5, respectively. In Table 2, cijc =

4 xijc  3D ijc  2J ijc  E ijc 4

, i = 1,2,3,  , m and

j = 1,2,3,  , n

In Table 3, Uij =

3xijc  3D ijc  2J ijc  Eijc 4

, i = 1,2,3,, m

and

j = 1, 2, 3,  , n

In Table 4, G ij =

2 x ijc  2D ijc  2J ijc  E ijc 4

, i = 1,2,3,  , m

and

j = 1,2,3,  , n

In Table 5, [ ij =

xijc  D ijc  J ijc  E ijc 4

,

i = 1,2,3,  , m

and

j = 1,2,3,  , n

Step 3: Find the optimal solution xij, Įij, Ȗij, ȕij by solving the crisp transportation problems shown in Tables 2, 3, 4 and 5, respectively. Step 4: Find the fuzzy optimal solution by putting the ~ values of xij, Įij, Ȗij, ȕij in xij = ( xij , D ij , J ij , E ij ) JMD . Step 5: Find the minimum total fuzzy transportation cost by ~ putting the values of xij in m

n

¦¦c~

ij

…~ xij

i =1 j =1

Remark 3.1: Because in the transportation problems negative parameters have no physical meaning. Therefore, in the proposed methods all the parameters may be assumed as non-negative trapezoidal fuzzy numbers.

4

Advantages of JMD representation over existing representation

In this section, the advantages of JMD representation over existing representation of trapezoidal fuzzy numbers are discussed: 4.1 Advantages of JMD representation in proposed method with fuzzy linear programming formulation By comparing the methods, proposed Section 3.3 and the existing method[15], the following results are obtained: (i) If all the parameters are represented by (m, n, Į, ȕ)-type trapezoidal fuzzy numbers and a balanced fuzzy transportation problem with m constraints and n variables is converted into a balanced crisp transportation problem, then the number of constraints in CTP=4×number of constraints in fuzzy transportation problem+2×number of fuzzy variables. (ii) If all the parameters of fuzzy transportation problems are represented by JMD-type trapezoidal fuzzy numbers and a balanced fuzzy transportation problem with m constraints and n variables is converted into a balanced crisp transportation problem, then the number of constraints in CTP=4×number of constraints in fuzzy transportation problem. On the basis of these results, it can be easily seen that if all the parameters are represented by JMD-type trapezoidal fuzzy numbers, then the number of constraints in the converted CTP will be less then the number of constraints obtained by the

existing representation of trapezoidal fuzzy numbers. Remark 4.1: Non-negative restrictions on fuzzy variables, ~ i.e., x ij is a non-negative trapezoidal fuzzy number, is also considered as a constraint. Similarly, non-negative restrictions on crisp variables i.e., xij•0, is also considered as a constraint. 4.2 Advantages of JMD representation in proposed method with classical transportation methods By comparing the methods, proposed in Section 3.4 and the existing method[15], the following results are obtained: (i) If in the existing method[15], all the parameters are represented by (m, n, Į, ȕ) trapezoidal fuzzy numbers, then the values of mij–Įij; Įij; nij–mij and ȕij will be obtained from Tables 2 to 5 respectively by the Step 4 of the existing method[15]. Now the values of mij, nij, Įij, and ȕij will be obtained by solving these equations and the fuzzy optimal solution will be (mij, nij, Įij, ȕij) with using these values. However, if all the parameters are represented by JMD type trapezoidal fuzzy numbers, then in the same step the value of xij, Įij, Ȗij, and ȕij will be obtained and using these values the fuzzy optimal solution will be (xij, Įij, Ȗij, ȕij)JMD which can be easily converted into (m, n, Į, ȕ)-type trapezoidal fuzzy number. On the basis of results, discussed in Sections 4.1 and 4.2, it can be concluded that it is better to use the JMD representation of trapezoidal fuzzy numbers instead of existing representation of trapezoidal fuzzy numbers[15] for finding the fuzzy optimal solution of fuzzy transportation problems. 4.3 Advantages of the proposed methods By using the proposed methods a decision maker has the following advantages: (i) The final results are non-negative trapezoidal fuzzy numbers i.e., there is no negative part in the obtained trapezoidal fuzzy numbers. (iii) It is easy to apply the proposed methods in comparison to the existing methods, to find the fuzzy optimal solution of FTP occurring in real life situations.

5

Numerical example

To illustrate the methods, proposed in Sections 3.3 and 3.4, a FTP is solved by using both the proposed methods and it is shown that the fuzzy optimal solution and minimum total fuzzy transportation cost, obtained using both the proposed methods, are the same. Also the obtained results are compared with the results obtained by existing method. Example 5.1: A company has two sources S1 and S2 and three destinations D1, D2 and D3; the fuzzy transportation cost for unit quantity of the product from ith source to jth destination is c~ij where, § (10,10,10,10) JMD (50,10,10,20) JMD (80,10,20,10) JMD · ¸¸ © (60,10,10,10) JMD (70,10,20,20) JMD (20,10,20,10) JMD ¹

[c~ij ]2u3 = ¨¨

The fuzzy availability of the product at first and second

KUMAR Amit et al. / J Transpn Sys Eng & IT, 2011, 11(5), 68í80

sources are (70, 20, 10, 30)JMD and (40, 20, 10, 30)JMD and the fuzzy demand of the product at first, second and third destinations are (30, 10, 10, 20)JMD, (20, 10, 10, 10)JMD and (60, 20, 0, 30)JMD, respectively. The company wants to determine the fuzzy quantity of the product that should be transported from each source to each destination so that the total fuzzy transportation cost is minimum. 5.1 Fuzzy optimal solution using method based on FLPF The fuzzy optimal solution of the chosen FTP by using the method based on FLPF, proposed in Section 3.3, can be obtained as follows: Step 1: Total fuzzy availability=(110, 40, 20, 60)JMD and total fuzzy demand = (110, 40, 20, 60)JMD. Because total fuzzy availability = total fuzzy demand, so it is a balanced fuzzy transportation problem. ~ Let xij = ( xij , D ij , J ij , E ij ) JMD be the fuzzy quantity of the product that should be transported from the ith source to the jth destination so that the total fuzzy transportation cost is minimum then the problem can be formulated into the following FLPP: Minimize( (10,10,10, 10) JMD … ~x11 † (50,10,10,20) JMD … ~x12 † (80,10,20, 10) JMD … ~ x13 † (60,10,10 ,10) JMD … ~ x21 ~ ~ † (70,10,20,20) … x † (20,10,20,10) …x ) JMD

subject to

22

JMD

23

~ x11 † ~ x12 † ~ x13 = (70,20,10, 30) JMD ~ ~ x 21 † x 22 † ~ x 23 = (40,20,10, 30) JMD ~ x †~ x = (30,10,10,20) 11

21

JMD

~ x12 † ~ x 22 = (20,10,10, 10) JMD ~ x13 † ~ x23 = (60,20,0,3 0) JMD ~ x11 , ~ x12 , ~ x13 , ~ x21 , ~ x22 , ~ x23 are non-negative JMD trapezoidal fuzzy

numbers. Step 2: Using Steps 2 to 4 of the proposed method, the formulated FLPP is converted into the following crisp LPP: Minimize (100x11+90Į11+70Ȗ11+40ȕ11+270 x12+220Į12 +160Ȗ12+90ȕ12+230Ȗ13+120ȕ13+300x21+240Į21 +170Ȗ21+90ȕ21+370x22+300Į22+220Ȗ22+120ȕ22 +160x23+140Į23+110Ȗ23+60ȕ23)/4 subject to x11+x12+x13+x14=70, Į11+Į12+Į13=20, Ȗ11+Ȗ12+Ȗ13=10, ȕ11+ȕ12+ȕ13=30, x21+x22+x23=40, Į21+Į22+Į23=20, Ȗ21+Ȗ22+Ȗ23=10, ȕ21+ȕ22+ȕ23=30, x11+x21=30, Į11+Į21=10, Ȗ11+Ȗ21=10, ȕ11+ȕ21=20, x12  x22 = 20 , D12  D 22 = 10 , Ȗ12+Ȗ22=10, ȕ12+ȕ22=10, x13+x23=60, Į13+Į23=20, Ȗ13+Ȗ23=0, ȕ13+ȕ23=30 x11, Į11, Ȗ11, ȕ11, x12, Į12, Ȗ12, ȕ12, x13, Į13, Ȗ13, ȕ13, x21, Į21, Ȗ21, ȕ21, x22, Į22, Ȗ22, ȕ22, x23, Į23, Ȗ23, ȕ23•0 Step 3: The optimal solution of crisp LPP, in Step 2, is

Step 4: Putting the values of xij, Įij, Ȗij, and ȕij in

~ x ij = ( x ij , D ij , J ij , E ij ) JMD , the fuzzy optimal solution is ~x11 =(30, 10, 10, 20)JMD, ~x12 =(20, 10, 10, 10)JMD, ~x13 =(20, 0, 0, 0)JMD, ~ x 22 =(0, 0, 10, 0)JMD, ~x 23 =(40, 20, 0, 30)JMD. Step 5: Putting the values of ~x11 , ~x12 , ~x13 , ~x21 , ~x22 , ~x23 in x11 † (50,10,10,20)JMD (10,10,10,10)JMD … ~ x † (80,10,20,10) … ~ x † (60,10,10,10) …~ 12

13

JMD

then the minimum total fuzzy transportation cost is (2 800, 3 400, 2 600, 6 600)JMD. Step 6: Using Definition 3.1, (m, n, Į, ȕ) representation of (2 800, 3 400, 2 600, 6 600)JMD is (6 200, 8 800, 3 400, 6 600). 5.2 Fuzzy optimal solution using classical transportation methods The fuzzy optimal solution of the chosen FTP by using the method based on classical transportation methods, proposed in Section 3.4, can be obtained as follows: Step 1: The tabular representation of the balanced FTP, obtained in Step 1 of Section 5.1, can be represented in tabular form as shown by Table 6. Table 6 Tabular representation of balanced fuzzy transportation problem D1

D2

D3

Availability

S1

(10, 10, 10, 10)JMD

(50, 10, 10, 20)JMD

(80, 10, 20, 10)JMD

(70, 20, 10, 30)JMD

S2

(60, 10, 10, 10)JMD

(70, 10, 20, 20)JMD

(20, 10, 20, 10)JMD

(40, 20, 10, 30)JMD

Demand

(30, 10, 10, 20)JMD

(20, 10, 10, 10)JMD

(60, 20, 0, 30)JMD

¦a~ = ¦b

2

3

i

i =1

j =1

Table 7 First crisp transportation table D1

D2

D3

S1

25

67.5

100

70

S2

75

92.5

40

40

30

20

60

Table 8 Second crisp transportation table D1

D2

S1

22.5

55

80

20

S2

60

75

35

20

10

10

20

D3

Table 9 Third crisp transportation table D1

D2

D3

S1

17.5

40

57.5

10

S2

42.5

55

27.5

10

10

10

0

Table 10 Fourth crisp transportation table

x11 = 30, D11 = 10, J 11 = 10, E11 = 20, x12 = 20,

D12 = 10, J 12 = 0, E12 = 10, x13 = 20, D13 = 0, J 13 = 0, d13 = 0, x21 = 0, D 21 = 0, J 21 = 10, E 21 = 0, x22 = 0, D 22 = 10, J 22 = 0, E 22 = 0, x23 = 40, D 23 = 20, J 23 = 0, E 23 = 30

JMD

…~ x21 † (70,10,20,20)JMD … ~ x22 † (20,10,20,10)JMD … ~ x23 )

D1

D2

D3

S1

10

22.5

30

30

S2

22.5

30

15

30

20

10

30

~ j

KUMAR Amit et al. / J Transpn Sys Eng & IT, 2011, 11(5), 68í80

Table 11 Results using existing and proposed representation of trapezoidal fuzzy numbers Minimum total Type of Number of Number of Number of fuzzy trapezoidal constraints in fuzzy variables constraints in transportation fuzzy number FTP in FTP converted CTP cost (4h11)+(2h (6200, 8800, 11 6 (m, n, Į, ȕ) 3400, 6600) 6)=56 (2800, 3400, 2600, 6600)JMD (x, Į, Ȗ, 11 2 =(6200, 8800, ȕ)JMD 3400, 6600)

Step 2: Using Step 2 of the proposed method, discussed in section 3.4, split the Table 6 into four crisp transportation tables shown by Tables 7 to 10. Step 3: The optimal solution of crisp transportation problems shown by Tables 7, 8, 9 and 10, are, x11 = 30, D11 = 10, J 11 = 10, E11 = 20, x12 = 20, D12 = 10, J 12 = 0, E12 = 10, x13 = 20, D13 = 0, J 13 = 0, E13 = 0, x22 = 0, D 22 = 0, J 22 = 10, E 22 = 0, x23 = 40, D 23 = 20, J 23 = 0, E 23 = 30 Step 4: Putting the values of xij, Įij, Ȗij, and ȕij in ~ xij = ( xij , D ij , J ij , E ij ) JMD , the fuzzy optimal solution is ~x11 =(30, 10, 10, 20)JMD, ~x12 =(20, 10, 10, 10)JMD, ~x13 =(20, 0, 0, 0)JMD, ~ x 22 =(0, 0, 10, 0)JMD, ~x 23 =(40, 20, 0, 30)JMD. x11 , ~ x12 , ~ x13 , ~ x21 , ~ x22 , ~ x23 in Step 5: Putting the values of ~ ~ ~ (10,10,10,10) JMD … x11 † (50,10,10,20) JMD … x12 † (80,10,20,10) … ~ x † (60,10,10,10) JMD

13

Acknowledgements The authors would like to thank to the Editor-in-Chief `Prof. Yang’ and the anonymous reviewers for their suggestions which have led to an improvement in both the quality and clarity of the paper. I, Dr. Amit Kumar, want to acknowledge the adolescent inner blessings of Mehar. I believe that Mehar is an angel for me and without Mehar’s blessing it was not possible to think the idea proposed in this paper. Mehar is a lovely daughter of Parampreet Kaur (Research Scholar under my supervision).

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6

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