Applied Mathematical Sciences, Vol. 7, 2013, no. 54, 2661 - 2673 HIKARI Ltd, www.m-hikari.com

An Approach for Solving Fuzzy Transportation

Problem Using Octagonal Fuzzy Numbers

S. U. Malini

Research Scholar, Stella Maris College (Autonomous), Chennai

Felbin C. Kennedy

Research Guide, Associate Professor, Stella Maris College, Chennai.

Copyright © 2013 S. U. Malini and Felbin C. Kennedy. This is an open access article

distributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, a general fuzzy transportation problem model is discussed. There are

several approaches by different authors to solve such a problem viz., [1,2,3,6,7,8].

We introduce octagonal fuzzy numbers using which we develop a new model to

solve the problem. By defining a ranking to the octagonal fuzzy numbers, it is

possible to compare them and using this we convert the fuzzy valued

transportation problem (cost, supply and demand appearing as octagonal fuzzy

numbers) to a crisp valued transportation problem, which then can be solved using

the MODI Method. We have proved that the optimal value for a fuzzy

transportation problem, when solved using octagonal fuzzy number gives a much

more optimal value than when it is solved using trapezoidal fuzzy number as done

by Basirzadeh [3] which is illustrated through a numerical example.

Mathematics Subject Classification: 03B52, 68T27, 68T37, 94D05

Keywords: Octagonal Fuzzy numbers, Fuzzy Transportation Problem.

1 Introduction

The transportation problem is a special case of linear programming problem,

which enable us to determine the optimum shipping patterns between origins and

2662 S. U. Malini and Felbin C. Kennedy

destinations. Suppose that there are m origins and n destinations. The solution of

the problem will enable us to determine the number of units to be transported

from a particular origin to a particular destination so that the cost incurred is least

or the time taken is least or the profit obtained is maximum. Let ai be the number

of units of a product available at origin i, and bj be the number of units of the

product required at destination j. Let cij be the cost of transporting one unit from

origin i to destination j and let xij be the amount of quantity transported or shipped

from origin i to destination j. A fuzzy transportation problem is a transportation

problem in which the transportation costs, supply and demand quantities are fuzzy

quantities. Michael [11] has proposed an algorithm for solving transportation

problems with fuzzy constraints and has investigated the relationship between the

algebraic structure of the optimum solution of the deterministic problem and its

fuzzy equivalent. Chanas et al [4] deals with the transportation problem wherein

fuzzy supply values of the deliverers and the fuzzy demand values of the receivers

are analysed. For the solution of the problem the technique of parametric

programming is used. Chanas and Kuchta [5] have given a definition for the

optimal solution of a transportation problem and as also proposed an algorithm to

determine the optimal solution. Shiang-Tai Liu and Chiang Kao[14] have given a

procedure to derive the fuzzy objective value of the fuzzy transportation problem

based on the extension principle. Two different types of the fuzzy transportation

problem are discussed: one with inequality constraints and the other with equality

constraints. Nagoor Gani and Abdul Razack [12] obtained a fuzzy solution for a

two stage cost minimising fuzzy transportation problem in which supplies and

demands are trapezoidal fuzzy numbers. Pandian et al [13] proposed a method

namely fuzzy zero point method for finding fuzzy optimal solution for a fuzzy

transportation problem where all parameters are trapezoidal fuzzy numbers.

In a fuzzy transportation problem, all parameters are fuzzy numbers. Fuzzy

numbers may be normal or abnormal, triangular or trapezoidal or it can also be

octagonal. Thus, they cannot be compared directly. Several methods were

introduced for ranking of fuzzy numbers, so that it will be helpful in comparing

them. Basirzadeh et al [2] have also proposed a method for ranking fuzzy

numbers using α – cuts in which he has given a ranking for triangular and

trapezoidal fuzzy numbers.

A ranking using α-cut is introduced on octagonal fuzzy numbers. Using this

ranking the fuzzy transportation problem is converted to a crisp valued problem,

which can be solved using VAM for initial solution and MODI for optimal

solution. The optimal solution can be got either as a fuzzy number or as a crisp

number.

2. Octagonal fuzzy numbers

Two relevant classes of fuzzy numbers, which are frequently used in

practical purposes so far, are “triangular and trapezoidal fuzzy numbers”. In this

paper we introduce octagonal fuzzy numbers which is much useful in solving

Solving fuzzy transportation problem 2663

fuzzy transportation problem (FTP).

Definition 2.1: An octagonal fuzzy number denoted by ω is defined to be the

ordered quadruple ω , for , and t

where

(i) is a bounded left continuous non decreasing function over [0, ω1],

[0 ω1 k]

(ii) is a bounded left continuous non decreasing function over [k, 2],

[k ω2 ]

(iii) is bounded left continuous non increasing function over [k, ω2],

[k ω2 ]

(iv) is bounded left continuous non increasing function over [0,ω1].

[0 ω1 k]

Remark 2.1: If ω=1, then the above-defined number is called a normal octagonal

fuzzy number.

The octagonal numbers we consider for our study is a subclass of the octagonal

fuzzy numbers (Definition 2.1) defined as follows:

Definition 2.2: A fuzzy number is a normal octagonal fuzzy number denoted

by (a1,a2,a3,a4,a5,a6,a7,a8) where a1, a2, a3, a4, a5, a6, a7, a8 are real numbers and

its membership function (x) is given below

μÃ(x) =

where 0 < k < 1.

Remark 2.2: If k = 0, the octagonal fuzzy number reduces to the trapezoidal

number (a3,a4,a5,a6) and if k=1, it reduces to the trapezoidal number (a1,a4,a5,a8).

Remark 2.3: According to the above mentioned definition, octagonal fuzzy

number ω is the ordered quadruple , for , and t where

=

, =

,

and

2664 S. U. Malini and Felbin C. Kennedy

Remark 2.4: Membership functions are continuous functions.

Remark 2.5: Here ω represents a fuzzy number in which “ω” is the maximum

membership value that a fuzzy number takes on. Whenever a normal fuzzy

number is meant, the fuzzy number is shown by , for convenience.

Definition 2.3: If ω be an octagonal fuzzy number, then the α-cut of ω is

[ ω]α = ω

= α

α α α ω

Remark 2.6: The octagonal fuzzy number is convex as their α-cuts are convex

sets in the classical sense.

Remark 2.7: The collection of all octagonal fuzzy real numbers from R to I is

denoted as Rω(I) and if ω=1, then the collection of normal octagonal fuzzy

numbers is denoted by R(I).

Graphical representation of a normal octagonal fuzzy number for k=0.5 is

Working Rule I:

Using interval arithmetic given by Kaufmann A, [10] we obtain α-cuts, α ∊ (0, 1],

addition, subtraction and multiplication of two octagonal fuzzy numbers as

follows:

a) α-cut of an octagonal fuzzy number: The α-cut of a normal octagonal fuzzy

number =n (a1, a2, a3, a4, a5, a6, a7, a8) given by Definition 2.3 (i.e. ), for

α ∊ (0, 1] is:

=

∊

–

∊

0

0.2

0.4

0.6

0.8

1

1.2

a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈

s1(t) s2(t)

l1(r) l2(r)

Solving fuzzy transportation problem 2665

b) Addition of octagonal fuzzy Numbers: Let =(a1, a2, a3, a4, a5, a6, a7, a8)

and = (b1, b2, b3, b4, b5, b6, b7, b8) be two octagonal fuzzy numbers. To calculate

addition of fuzzy numbers and we first add the α–cuts of and using

interval arithmetic.

]1()](1

)(1

[)](1

)(1

[

]0[)]()([)]()([

566343566343

788121788121

,kforbbk

kb,bb

k

kbaa

k

ka,aa

k

ka

k,forbbk

b,bbk

baak

a,aak

a

B~

A~

]1()](1

)(1

[

]0[)]()([

565666343433

787888121211

,kforbbaak

kba,bbaa

k

kba

k,forbbaak

ba,bbaak

ba

B~

A~

c) Subtraction of two octagonal fuzzy numbers: Let = (a1, a2, a3, a4, a5, a6, a7,

a8) and = (b1, b2, b3, b4, b5, b6, b7, b8) be two octagonal fuzzy numbers. To

calculate subtraction of fuzzy numbers and we first subtract the α–cuts of

and using interval arithmetic.

= [ ,

where

1 2 1 1 2 1 1 2 1 8 8 7

8 8 7 1 2 1 8 8 7 8 8 7

min{( ( )) ( ( )), ( ( )) ( ( )),

( ( )) ( ( )), ( ( )) ( ( ))}

Lq a a a b b b a a a b b bk k k k

a a a b b b a a a b b bk k k k

1 2 1 1 2 1 1 2 1 8 8 7

8 8 7 1 2 1 8 8 7 8 8 7

max{( ( )) ( ( )), ( ( )) ( ( )),

( ( )) ( ( )), ( ( )) ( ( ))}

Rq a a a b b b a a a b b bk k k k

a a a b b b a a a b b bk k k k

[0, ]for k

and

3 4 3 3 4 3 3 4 3 6 6 5

6 6 5 3 4 3 6 6 5 6

min{( ( )) ( ( )), ( ( )) ( ( )),1 1 1 1

( ( )) ( ( )), ( ( )) ( (1 1 1 1

L k k k kq a a a b b b a a a b b b

k k k k

k k k ka a a b b b a a a b

k k k k

6 5 ))}b b

3 4 3 3 4 3 3 4 3 6 6 5

6 6 5 3 4 3 6 6 5 6

max{( ( )) ( ( )), ( ( )) ( ( )),1 1 1 1

( ( )) ( ( )), ( ( )) ( (1 1 1 1

R k k k kq a a a b b b a a a b b b

k k k k

k k k ka a a b b b a a a b

k k k k

6 5))}b b

( ,1]for k

d) Multiplication of two octagonal fuzzy numbers: Let = (a1, a2, a3, a4, a5, a6,

a7, a8) and = (b1, b2, b3, b4, b5, b6, b7, b8) be two octagonal fuzzy numbers. To

calculate multiplication of fuzzy numbers and we first multiply the α–cuts

of and using interval arithmetic.

= [ ,

where

2666 S. U. Malini and Felbin C. Kennedy

1 2 1 1 2 1 1 2 1 8 8 7

8 8 7 1 2 1 8 8 7 8 8 7

min{( ( ))( ( )), ( ( ))( ( )),

( ( ))( ( )), ( ( ))( ( ))}

Lq a a a b b b a a a b b bk k k k

a a a b b b a a a b b bk k k k

1 2 1 1 2 1 1 2 1 8 8 7

8 8 7 1 2 1 8 8 7 8 8 7

max{( ( ))( ( )), ( ( ))( ( )),

( ( ))( ( )), ( ( ))( ( ))}

Rq a a a b b b a a a b b bk k k k

a a a b b b a a a b b bk k k k

[0, ]for k

=[ ,

where

3 4 3 3 4 3 3 4 3 6 6 5

6 6 5 3 4 3 6 6 5 6 6

min{( ( ))( ( )), ( ( ))( ( )),1 1 1 1

( ( ))( ( )), ( ( ))( (1 1 1 1

L k k k kq a a a b b b a a a b b b

k k k k

k k k ka a a b b b a a a b b b

k k k k

5))}

3 4 3 3 4 3 3 4 3 6 6 5

6 6 5 3 4 3 6 6 5 6 6

max{( ( ))( ( )), ( ( ))( ( )),1 1 1 1

( ( ))( ( )), ( ( ))( (1 1 1 1

R k k k kq a a a b b b a a a b b b

k k k k

k k k ka a a b b b a a a b b b

k k k k

5))}

( ,1]for k

3 Ranking of octagonal fuzzy numbers [3]

The parametric methods of comparing fuzzy numbers, especially in fuzzy

decision making theory are more efficient than non-parametric methods. Cheng’s

centroid point method[6], Chu and Tsao’s method[7], Abbasbandy and

Assady’s [1] sign-distance method was all non-parametric and was applicable

only for normal fuzzy numbers. The non-parametric methods for comparing fuzzy

numbers have some drawbacks in practice.

Definition 3.1: A measure of fuzzy number is a function Mα: Rω(I) → R+

which assigns a non-negative real number Mα( ) that expresses the measure of

.

(Ãω) =

+

where 0

Definition 3.2: The measure of an octagonal fuzzy number is obtained by the

average of the two fuzzy side areas, left side area and right side area, from

membership function to α axis.

Definition 3.3: Let be a normal octagonal fuzzy number. The value ,

called the measure of is calculated as follows:

M0Oct

( ) =

where 0<k<1

=

[(a1+ a2+a7 + a8)k + (a3 +a4+a5+a6)(1-k)] ---------------(3.1)

Solving fuzzy transportation problem 2667

Remark 3.1: Consider the trapezoidal number which is

got from the above octagonal number by equating , for

k=0.5 and ω=1

The measure of the normal fuzzy trapezoidal number is given by

M0tra

( ) =

------- (3.2)

Remark 3.2: If k=0.5, M0Oct

( ) =

(a1+ a2+ a3 +a4+a5+a6+a7 + a8)

When a2 coincides with a3 and a6 coincides with a7 it reduces to trapezoidal fuzzy

number, which is given by Equation (3.2).

Remark 3.3: If

a1+ a2+a7 + a8 = a3 +a4+a5+a6 ---------------(3.3)

then the measure of an octagonal number is the same for any value of k (0<k<1).

Remark 3.4: If ω=(a1 +

(a2-a1), a3 +

(a4-a3), a6 –

(a6-a5),

a8 -

(a8-a7)) for , and t be an arbitrary octagonal fuzzy

number at decision level higher than “α” and α, ω ∊ [0,1], the value α ct(A ω),

assigned to A ω may be calculated as follows:

Working Rule II:

If ω>α, then

(Ãω) =

+

;

=

{ [ +

] + +

(ω+k-2 ω)] (ω-k)}

Obviously if ω α then the above quantity will be zero.

If ω = it becomes a normal octagonal number, then

α ct(A ) =

+

; α ∊

=

{ [ +

] + +

(1-k)] (1-k)} α ∊ [0,1)

0

0.5

1

1.5

a₁ b a₄ a₅ c a₈

2668 S. U. Malini and Felbin C. Kennedy

Remark 3.5: If ω and ω’ are two octagonal fuzzy numbers and ω, ω’ [0, 1],

then we have:

1. ω ω’ ∀α [0, 1] ( ω)

( ω’) 2. ω = ω’ ∀α [0, 1]

( ω) = ( ω’)

3. ω ω’ ∀α [0, 1] ( ω)

( ω’) Remark 3.6: If α is close to one, the pertaining decision is called a “high level

decision”, in which case only parts of the two fuzzy numbers, with membership

values between “α” and “1”, will be compared. Likewise, if “α” is close to zero,

the pertaining decision is referred to as a “low level decision”, since members

with membership values lower than both the fuzzy numbers are involved in the

comparison.

4 Mathematical formulation of a Fuzzy Transportation Problem

Consider the following fuzzy transportation problem (FTP) having fuzzy

costs, fuzzy sources and fuzzy demands,

(FTP) Minimize z =

Subject to

≈ ãi, for i= ,2,…m (4.1)

≈ , for j= , 2,…n (4.2)

≽ for i= , 2, …m and j= , 2,…n (4.3)

where m = the number of supply points;

n = the number of demand points;

≈

is the uncertain number of units shipped

from supply point i to demand point j;

≈

is the uncertain cost of shipping one unit

from supply point i to the demand point j;

ãi ≈

is the uncertain supply at supply point i and

≈

is the uncertain demand at demand point j.

The necessary and sufficient condition for the linear programming problem given

above to have a solution is that

≈

The above problem can be put in table namely fuzzy transportation table given

below: Supply

Demand …………….

………………

.

.

.

.

.

.

.

.

.

………………

Solving fuzzy transportation problem 2669

5. Procedure for Solving Fuzzy Transportation Problem

We shall present a solution to fuzzy transportation problem involving

shipping cost, customer demand and availability of products from the producers

using octagonal fuzzy numbers.

Step 1: First convert the cost, demand and supply values which are all octagonal

fuzzy numbers into crisp values by using the measure defined by (Definition 3.3)

in Section 3.

Step 2: [9] we solve the transportation problem with crisp values by using the

VAM procedure to get the initial solution and then the MODI Method to get the

optimal solution and obtain the allotment table.

Remark 5.1: A solution to any transportation problem will contain exactly

(m+n-1) basic feasible solutions. The allotted value should be some positive

integer or zero, but the solution obtained may be an integer or non-integer,

because the original problem involves fuzzy numbers whose values are real

numbers. If crisp solution is enough the solution is complete but if fuzzy

solution is required go to next step.

Step 3: Determine the locations of nonzero basic feasible solutions in

transportation table. There must be atleast one basic cell in each row and one in

each column of the transportation table. Also the m+n-1 basic cells should not

contain a cycle. Therefore, there exist some rows and columns which have only

one basic cell. By starting from these cells, we calculate the fuzzy basic

solutions, and continue until (m+n-1) basic solutions are obtained.

6. Numerical Example

Consider the following fuzzy transportation problem.

According to the definition of an octagonal fuzzy number Ã, the measure of à is

calculated as

M0Oct

(A ) =

where 0 k 1

2670 S. U. Malini and Felbin C. Kennedy

=

[(a1+a2+a7 + a8)k + (a3 +a4+a5+a6)(1-k)] where 0 k 1

Step 1: Convert the given fuzzy problem into a crisp value problem by using the

measure given by Definition 3.3 in Section 3.

This problem is done by taking the value of k as 0.4, we obtain the values of

M0Oct

( ), M0Oct

( ) and M0Oct

( ) as

c (-1,0,1,2,3,4,5,6) M0Oct

(c )=

2

(-1+0+5+6)+

(1+2+3+4)] = 2.5

c 2 (0,1,2,3,4,5,6,7) M0Oct

(c 2)=

2

(0+1+6+7)+

(2+3+4+5)] = 3.5

c (8,9,10, 11,12,13,14,15) M0Oct

(c )=

2

(8+9+14+15)+

(10+11+12+13)] = 11.5

c (4, 5,6,7,8,9,10,11) M0Oct

(c )=

[2

(4+5+10+11)+

(6+7+8+9)] = 7.5

c 2 (-2,-1,0,1,2,3,4,5) M0Oct

(c 2 )=

2

(-2-1+4+5)+

(0+1+2+3)] = 1.5

c 22 = (-3,-2,-1,0,1,2,3,4) M0Oct

(c 22)=

2

(-3-2+3+4)+

(-1+0+1+2)] = 0.5

c 2 = (2,4,5,6,7,8,9,11) M0Oct

(c 2 )=

2

(2+4+9+11)+

(5+6+7+8)] = 6.5

c 2 = (-3,-1,0,1,2,4,5,6) M0Oct

(c 2 )=

2

(-3-1+5+6)+

(0+1+2+4)] = 1.75

c (2,3,4,5,6,7,8,9) M0Oct

(c )=

2

(2+3+8+9)+

(4+5+6+7)] = 5.5

c 2 (3,6,7,8,9,10,12,13) M0Oct

(c 2)=

2

(3+6+12+13)+

(7+8+9+10)] = 8.5

c = (11,12,14,15,16,17,18,21) M0Oct

(c )=

2

(11+12+18+21)+

(14+15+16+17)]=15.5

c (5,6,8,9,10,11,12,15) M0Oct

(c )=

2

(5+6+12+15)+

(8+9+10+11)] = 9.5

And the fuzzy supplies are

= (1,3,5,6,7,8,10,12) M0Oct

( ) =

(1+3+10+12)+

(5+6+7+8)] = 6.5

= (-2,-1,0,1,2,3,4,5) M0Oct

( ) =

(-2-1+4+5)+

(0+1+2+3)] = 1.5

= (5,6,8,10,12,13,15,17) M0Oct

( ) =

(5+6+15+17)+

(8+10+12+13)] = 10.75

And the fuzzy demands are

(4,5,6,7, 8,9,10,11) M0Oct

( ) =

[

(4+5+10+11)+

(6+7+8+9)] = 7.5

(1,2,3,5,6,7,8,10) M0Oct

( ) =

(1+2+8+10)+

(3+5+6+7)] = 5.25

(0,1,2,3,4,5,6,7) M0Oct

( ) =

(0+1+6+7)+

(2+3+4+5)] = 3.5

(-1,0,1,2,3,4,5,6) M0Oct

( ) =

(-1+0+5+6)+

(1+2+3+4)] = 2.5

Solving fuzzy transportation problem 2671

Remark 6.1: In the above problem since condition 3.3 (Equation 3.3) is satisfied

by all the octagonal numbers (cost, supply and demand), for any value of k we

will get the same table as below.

6.5

1.5

10.75

7.5 5.25 3.5 2.5

Step 2: Using VAM procedure we obtain the initial solution as

1.25 5.25

1.5

6.25 3.5 1

which is not an optimal solution.

Hence by using the MODI method we shall improve the solution and get the

optimal solution as

5.25 1.25

1.5

7.5 0.75 2.5

Step 3: Now using the allotment rules, the solution of the problem can be

obtained in the form of octagonal fuzzy numbers

Therefore the fuzzy optimal solution for the given transportation problem is

(1,2,3,5,6,7,8,10), = (-9,-5,-2,0,2,5,8,11) = (-2,-1,0,1,2,3,4,5),

=(4,5,6,7,8,9,10,11), = (-12,-9,-5,-1,3,6,10,14), = (-1,0,1,2,3,4,5,6)

and the fuzzy optimal value of z = (-416,-224,-73,58,188,333,516,773).

And the crisp solution to the problem is Minimum cost = 119.125. Also for

different values of k (0 k 1) we obtain the same solution. Hence the solution is

(-9,-5,-2,0,2,5,8,11) (1,3,5,6,7,8,10,12)

(-2,-1,0,1,2,3,4,5)

(-12,-9,-5,-1,3,6,10,14) (5,6,8,10,12,13,15,17)

(4,5,6,7,8,9,10,11) (1,2,3,5,6,7,8,10) (0,1,2,3,4,5,6,7) (-1,0,1,2,3,4,5,6)DEMAND

SUPPLYDESTINATION

(4,5,6,7,8,9,10,11)

(1,2,3,5,6,7,8,10)

(-2,-1,0,1,2,3,4,5)

(-1,0,1,2,3,4,5,6)

2.5 3.5 11.5 7.5

1.5 0.5 6.5 1.75

5.5 8.5 15.5 9.5

2672 S. U. Malini and Felbin C. Kennedy

independent of k.

Remark 6.2: When we convert the octagonal fuzzy transportation problem into

trapezoidal fuzzy transportation problem we get

Supply

(1,6,7,12)

(-2,1,2,5)

(5,10,12,17)

Demand (4,7,8,11) (1,5,6,10) (0,3,4,7) (-1,2,3,6)

When this problem is solved as in [13] we would get the fuzzy optimal solution

for the given transportation problem as (1,5,6,10), = (-9,0,2,11) =

(-2,-1,2,5), =(4,7,8,11), = (-12,-1,3,14), = (-1,2,3,6), the fuzzy

optimal value of z as z = (-416,59,185,773) and the crisp value of the optimum

fuzzy transportation cost for the problem is z = 140.83. If it is solved as in [3], we

will get the same value for the variables but the optimal cost will be 121.375. On

the other hand if the problem is solved using octagonal fuzzy numbers we get the

optimum cost as 119.125

Remark 6.3: If the octagonal numbers are slightly modified so that the condition

(Equation (3.3)) is not satisfied, i.e. a1+ a2+a7 + a8 ≠ a3 +a4+a5+a6, then for such

a problem the optimal solution for different values of k (0 k 1) can be easily

checked to lie in a finite interval.

7. Conclusion

In this paper a simple method of solving fuzzy transportation problem

(supply, demand, and cost are all octagonal fuzzy numbers) were introduced by

using ranking of fuzzy numbers. The shipping cost, availability at the origins and

requirements at the destinations are all octagonal fuzzy numbers and the solution

to the problem is given both as a fuzzy number and also as a ranked fuzzy number.

It also gives us the optimum cost which is much lower than, when it is done using

trapezoidal fuzzy numbers.

Acknowledgement.

The authors wish to thank Professor M.S. Rangachari, Former Director

and Head, Ramanujan Institute for Advanced Study in Mathematics, University of

Madras, Chennai and Professor P.V. Subramanyam, Department of Mathematics,

IIT Madras, Chennai for their valuable suggestions in the preparation of this

paper.

(-1,2,3,6) (0,3,4,7) (8,11,12,15) (4,7,8,11)

(-2,1,2,5) (-3,0,1,4) (2,6,7,11) (-3,1,2,6)

(2,5,6,9) (3,8,9,13) (11,15,16,21) (5,9,10,15)

Solving fuzzy transportation problem 2673

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Received: February 25, 2013