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www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476

A METHOD FOR SOLVING INTUITIONISTIC FUZZY ASSIGNMENT PROBLEM USING BRANCH AND BOUND METHOD B.Srinivas Department of Mathematics ISTS Women’s Engineering College, Rajanagaram, Andhra Pradesh, India

G.Ganesan Department of Mathematics Adikavi Nannaya University, Rajahmundry, Andhra Pradesh, India

Abstract In this paper, we investigate an assignment problem in which cost coefficients are triangular intuitionistic fuzzy numbers. In conventional assignment problem, cost is always certain. This paper develops an approach to solve an intuitionistic fuzzy assignment problem where cost is not deterministic numbers but imprecise ones. Here, the elements of the costs (profits) matrix of the assignment problem are triangular intuitionistic fuzzy numbers. Then its triangular shaped membership and non-membership functions are defined. A new ranking procedure which can be found in [2] and is used to compare the intuitionistic fuzzy numbers so that an Intuitionistic Fuzzy Branch Bound method may be applied to solve the intuitionistic fuzzy assignment problem. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem. Keywords: Intuitionistic Fuzzy Set, Triangular Fuzzy Number, Triangular Intuitionistic Fuzzy Number, Intuitionistic Fuzzy Assignment Problem, Optimal Solution.

1. Introduction Assignment Problem(AP) is used worldwide in solving real world problems. An assignment problem plays an important role in an assigning of persons to jobs, or classes to rooms, operators to machines, drivers to trucks, trucks to routes, or problems to research teams, etc. The assignment problem is a special type of linear programming problem (LPP) in which our objective is to assign n number of jobs to n number of machines (persons) at a minimum cost. To find solution to assignment problems, various algorithm such as linear programming, Hungarian algorithm, neural network, genetic algorithm have been developed However, in real life situations, the parameters of assignment problem are imprecise numbers instead of fixed real numbers because time/cost for doing a job by a facility (machine/persion) might vary due to different reasons. The theory of fuzzy set introduced by Zadeh[14\] in 1965 has achieved successful applications in various fields. In 1970, Belmann and Zadeh[3] introduce the concepts of fuzzy set theory into the decision-making problems involving uncertainty and imprecision. Bai et.al[4] investigated Fuzzy generalized assignment problem with credibility Constraints. Fortemps and.Roubens[8] Ranking and defuzzification methods based area compensation fuzzy sets and systems. Das et.al[17] presented Ranking of trapezoidal intuitionistic fuzzy numbers. Kuhn[12], The Hungarian method for the assignment and transportation problems. Kalaiarasi et.al [13] Optimization of fuzzy assignment model with triangular fuzzy numbers. Different kinds of fuzzy assignment problems are solved in the papers [2,5,6,9].

227

B.Srinivas, S Ganesan

International Journal of Engineering Technology, Management and Applied Sciences

www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 The concept of Intuitionistic Fuzzy Sets (IFSs) proposed by Angelov [1] is found to be Optimization in an intuitionistic fuzzy environment. Jahir Hussian et al[11] presented .Algorithm approach for solving intuitionistic fuzzy transportation problem. Here we investigate a more realistic problem, namely intuitionistic fuzzy assignm of assigning the jth job to the ith machine. We assume that one machine can be assigned exactly one job also each machine can do at most one job. The problem is to find an optimal assignment so that the total intuitionistic fuzzy cost of performing all jobs is minimum or the total intuitionistic fuzzy profit is maximum. In this paper, ranking procedure of Annie Varghese and Sunny Kuriakose [2] is used to compare the intuitionistic fuzzy numbers. Finally an Intuitionistic Fuzzy Branch Bound method may be applied to solve an IFAP

RESEARCH ARTICLE OPEN ACCESS This paper is organized as follows: Section 2 deals with some basic terminology and ranking of triangular intuitionistic fuzzy numbers, In section 3, provides not only the definition of intuitionistic fuzzy assignment problem but also Assignment Problem. Section 4 describes the solution procedure of an intuitionistic fuzzy assignment problem, In section 5 to illustrate the proposed method a numerical example with results and discussion is discussed and followed by the conclusions are given in Section 6

II. Preliminaries Definition 2.1 Let A be a classical set, membership function

A

A

( ) is defined by

( ) be a function from A to [0,1]. A fuzzy set *

= {( ,

A

( ) ; ∈

A

*

with the

( ) ∈ [0,1]

Definition 2.2 Let X be denote a universe of discourse, then an intuitionistic fuzzy set A in X is ~ given by a set of ordered tripl A I ={˂x, A ( ),ʋ A (x)˃:x∈X} Where

A

( ),ʋ A (x): →[0,1], are functions such that 0≤

membership

A

( )

A

( ),+ ʋ A (x): ≤1,∀ ∈ . For each x the

ʋ A (x) represent the degree of membership and the degree of non –

membership of the element ∈ to ⊂ respectively Definition 2.3 A fuzzy number A is defined to be a triangular fuzzy number if its membership functions A ( ) :ℝ→ [0, 1] is equal to.

 x  a1 a  a 1 if  2  a3  x u A ( x)   if a  a 2  3 otherwise  0  228 B.Srinivas, S Ganesan

x  [a1 , a 2 ] x  [a 2 , a3 ]

International Journal of Engineering Technology, Management and Applied Sciences

www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 where

1

≤

≤

2

3

. This fuzzy number is denoted by (

1

,

2

,

3

).

~ Definition 2.4 A Triangular Intuitionistic Fuzzy Number ( A I is an intuitionistic fuzzy set in R with the following membership function A ( ) and non membership function ʋ A (x):)

0  xa 1   a 2  a1  u A ( x)  1 a x  3  a3  a 2 0 

for

x  a1

for

a1  x  a 2

for

x  a2

for

a 2  x  a3

for

x  a3

1 a x  2  a 2  a1   A ( x)  0  xa 2  a  a  3 2 1 

for

x  a1

for

a1  x  a 2

for

x  a2

for

a 2  x  a3

for

x  a3

′≤ 1 ≤ 2 ≤ 3 ≤ 3 ′and ~ denoted by A I = ( 1 , 2 , 3 ) ( Where

1

~ Particular Cases Let A I =(

Case 1: If,

1

′═ 1 ,

=(

3

).

1

,

2

Case 2: If

, 1

′=

1

=

3

2

1

( ),ʋ A (x ) ≤0.5 for

′,

2

,

3

A

( )

= ʋ A (x) ∀ ∈ . This TrIFN is

′)

′) be a TrIFN. Then the following cases arise ~I ~I 3 ′then represent A Tringular Fuzzy Number(TFN). It is denoted by A

═

=

A

3

=

1

,

3

′=

2

,

3

)(

1

′,

2

,

3

~ then A I represent a real number

.

~ ~ Definition 2.5 Let and be two TrIFNs. The ranking of A I and B I by the R (.) on E, the set of TrIFNs is defined as follows: ~ ~ ~ ~ i. R ( A I )> R ( B I ) iff A I ≻ B I ~ ~ ~ ~ ii. R ( A I )< R ( B I ) iff A I ≺ B I

229

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www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 ~ ~ ~ ~ iii. R ( A I )= R ( B I ) iff A I ≈ B I ~

~

~

~

iv. R ( A I + B I )= R ( A I )+ R ( B I )

~ ~ ~ ~ v. R( A I − B I )= R ( A I ) −R ( B I ) Arithmetic Operations ~ Let A I =(

1

,

2

,

3

)(

1

′,

2

,

arithmetic operations as follows: ~ ~ Addition: A I ⊕ B I =( 1 + 1 ,

~ ~ Subtraction: A I - B I = (

1

−

~ ′) and B I =(

3

+

2

3

,

2

,

−

2

3

2

+

,

3

,

2

,

3

)(

1

′,+

−

1

1

)(

3

1

)(

1

1

′,

2

′, 2+

′− 3′,

2

,

2

′)be any two TrIFNs then the

3

3

′+ 3′)

2

,

,

,−

3

′−

1

′)

′,

2

Ranking of triangular intuitionistic fuzzy numbers ~ The Ranking of a triangular intuitionistic fuzzy number A I =(

1

,

2

,

3

)(

1

,

3

′) is defined

by [4] 2

2

1 (a '  a1' )(a 2  2a3'  2a1' )  (a3  a1 )(a1  a 2  a3 )  3(a3'  a1' ) R( A )  [ 3 ]’ 3 a3'  a1'  a3  a1 ` I

The ranking technique [4] is: ~ ~ ~ ~ ~ ~ ~ If R( A I ) ≤ R( B I ), then A I ≤ B I i.e., min { A I , B I }= A I ~ ~ Example: Let A I =( 8,10,12 ;6,10,14) and B I = (3,5,8 ;1,5,10) be any two TrIFN, then its rank is ~ ~ ~ ~ defined by R A I =10, R( B I )=5.33 this implies A I ≻ B I III. Intuitionistic Fuzzy Assignment Problem

Consider the situation of assigning n machines to n jobs and each machine is capable of doing any job at different costs. Let ij be an intuitionistic fuzzy cost of assigning the jth job to the ith machine. Let x ij be the decision variable denoting the assignment of the machine i to the job j. The objective is to minimize the total intuitionistic fuzzy cost of assigning all the jobs to the available machines (one machine per job) at the least total cost. This situation is known as balanced intuitionistic fuzzy assignment problem. n ~ (IFAP)Minimize A I cij =  j 1

230

B.Srinivas, S Ganesan

n

c i 1

ij

x ij

International Journal of Engineering Technology, Management and Applied Sciences

www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 Subject to n

c i 1

ij

=1 for

ij

=1 for

i=1,2,.. . . . . . n

n

c j 1

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

to 1, if the i th machine is assigned th not assigned 0, if the i machine is

Where xij  

j th job th to j

job

x ij ∈{0,1} ` I

'

C ij  (Cij1 , Cij2 , Cij3 )(Cij1 , Cij2 , Cij3 ) '

Job1

Job2

Job3

.Jobj

JobN

Person1

a~11

a~12

a~13

.. a~ij

a~1n

Person 2

a~21

a~22

a~23

... a~ij

a~2 n

.

-

-

-

-

-

.

a~

-

-

-

-

.. a~ij .

-

Person i

i1

a~i 2

a~i 3

. PersonN

-

a~n1

a~n 2

a~n 3

.. a~nj ..

a~nn

3.1Fundamental Theorems of an Intuitionistic Fuzzy Assignment Problem The solution of an intuitionistic fuzzy assignment problem is fundamentally based on the following two theorems Theorem 1: In an intuitionistic fuzzy assignment problem, if we add or subtract an intuitionistic fuzzy number to every element of any row(or column) of the intuitionistic fuzzy cost matrix [ cij ], then an assignment that minimizes the total intuitionistic fuzzy cost on one matrix also minimizes the total intuitionistic fuzzy cost on the other matrix. In other words if x ij = * x ij Minimizes

231

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International Journal of Engineering Technology, Management and Applied Sciences

www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 ` I

Z  i 1 ` I*

Z

` I



 j 1 C ij xij with

n

n

 i 1

xij  1,  j 1 xij  1, xij  0 or 1 then xij* also minimizes

n

n

i 1

` I*

` I*

` I

`I

`I

`I

`I

 j 1 C ij xij where C ij = C ij  u i  v j for all i,j= 1,2,…n and u i , v j are some real

n

n

triangular intuitionistic fuzzy numbers. ` I*

Proof: Now, Z

 i 1 n

=

 

=



n

i 1

j 1



`I



n

i 1

`I

( C ij  u i  v j ) xij ` I

 j 1 C ij xij = n

i 1

=Z  Since

` I

n

n

` I

` I*

 j 1 C ij xij n

`I



`I

n

ui

i 1



n



xij =

j 1

n

i 1

xij



n j 1

`I

vj

`I

u i -  j 1 v j n

xij =  j 1 xij =1. This shows that the minimization of the new objective function

n

n

i 1

` I*

` I

Z yields the same solution as the minimization of original objective function Z because and





n

i 1

`I

n

vj .

j 1

Theorem 2: ` I

` I

`I

In an intuitionistic fuzzy assignment problem with cost [ C ij ], if all [ C ij ]  0 then a feasible solution xij which satisfies ` I

  n

n

i 1

j 1

`I

` I

C ij xij = 0 is optimal for the problem.

`I

Proof: Since all [ C ij ]  0 and all [ xij ]  0 , ` I

The objective function Z  i 1 ` I

n

` I

 j 1 C ij xij cannot be negative. The minimum possible n

`I

value that Z can attain 0 ` I

Thus any feasible solution [ xij ] that satisfies Z  i 1

232

B.Srinivas, S Ganesan

n



n j 1

` I

`I

C ij xij = 0 , will be an optimal.

`I

ui

International Journal of Engineering Technology, Management and Applied Sciences

www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 IV. The Computational Procedure for Intuitionistic Fuzzy Assignment Problem. BRANCH BOUND TECHNIQE AND BOUND TECHNIQUE FOR ASSIGNMENT PROBLEM; The assignment problem can also be solved using a branch and bound algorithm: It is a curtailed enumeration technique. The terminologies of branch and bound technique applied to the assignment problem are presented below. 1. Let k be the level number in the branch tree (for root node it is 0) Be an assignment in the current node of a branching tree. 2. Pk be an assignment at level k of the branching tree. A is the set of assigned cells (partial assignment) up to the node Pk from the root node ( set of I and j values with respect to the assigned cells up to the node Pk from the root node), and V be the lower bound of the partial assignment up to Pk such that,

V =  Ci , j +  (  min Cij ) i X

i , jA

jY

Where Cij ithe cell entry of the cost matrix with respect to the Ith row and jth column. X be the set of rows which are not deleted up to the node Pk from the root node in the branching node. Branching guidelines: 1. At Level k, the row marked as k of the assignmenet assigned problem, will be assigned with the best column of the assignment problem. 2. If there is tie on the lower bound . then the terminal node at the lower –most is to be considered for the further branching. 3. Stooping rule: If the minimum lower bound happiness to be at any of the terminal node at the (n-1) th level, the optimality is reached. Then the assignments on the path of the node to that node along with the missing pair of row-column combination from the optimum solution.

4.Numerical Examples: Example: Let us consider an intuitionistic fuzzy assignment problem with rows representing 3 machines

1

,

2

,

3

and columns representing the 3 jobs 1 ,

2

,

3

. The cost matrix [

ij

] is given

whose elements are TrIFN. The problem is to find the optimal assignment so that the total cost of job assignment becomes minimum J1

J2

3

4

M1

(8,,10,12)(6,10,14) (7,21,29)(2,21,34)

(7,20,27)(3,20,27)

(12,25,56)(8,25,56)

M2

(8,9,16)(2,9,22)

(8,10,21)(6,10,14)

(4,14,28)(3,14,31)

(4,9,12)(3,9,14)

M3

(5,9,22)(2,9,25)

(4,14,28)(3,14,31)

(4,16,19)(1,16,22)

(3,14,26)(3,14,27)

M4

(3,14,26)(3,14,27)

(5,9,22)(2,9,25)

(8,9,16)(2,9,22)

(7,8,12)(2,8,17)

233

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www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 Solution: The above intuitionistic fuzzy assignment problem can be formulated in the following mathematical programming form

Mim[(8,,10,12)(6,10,14)x11+(7,21,29)(2,21,34)x12+(7,20,27)(3,20,27)x13+(12,25,56)(8,25,56)x14(8,9 ,16)(2,9,22)x 21 +(8,10,21)(6,10,14)x22+(4,14,28)(3,14,31)x23+(4,9,12)(3,9,14)x24+ (5,9,22)(2,9,25)x31+(4,14,28)(3,14,31)x32+(4,16,19)(1,16,22)x33+ (3,14,26)(3,14,27)x34+ (3,14,26)(3,14,27)x41+(5,9,22)(2,9,25)x42+(8,9,16)(2,9,22)x43 +( (7,8,12)(2,8,17)x44] Subject to

x11+x12+x13+x14=1 x21+x22+x23+x24=1 x31+x32+x33+x34=1 x41+x42+x43+x44=1

x11+x21+x31+x41=1 x12+x22+x32+x42=1 x13+x23+x33+x43=1 x14+x24+x34+x44=1 where xij∈ [0,1]

1

~ R( C11 ) =  0.5( 7+14+29+8)d=10

calculated

0

~ ~ R(C13 ) = 31 Similarly, R( C12 ) = 17.33 ~ ~ ~ R( C 21 ) =11 R( C 22 ) = 10 R( C 23 ) = 15.69 ~ ~ ~ R( C31 ) = 12 R( C32 ) = 15.69 R( C33 ) = 13 ~ ~ ~ R( C 41 ) = 14.14 R( C 42 ) = 12 R( C 43 ) = 11

~

R( C14 ) =25.43

~

R( C 24 ) = 8.52

~

R( C34 ) = 14.14

~

R( C 44 ) = 9.1

We replace these values for their corresponding aij in which result in a convenient assignment problem in the linear programming problem Now, using the Step 1 of the intuitionistic fuzzy ones assignment method, we have the following assignment method to get the following optimal solution. J1 10

J2 17.33

31

J3 25.43

11 M

10

15.69

8.52

15.69

13

14.14

12

11

9.1

M1` M2 12 M M3 14.14

We solve it by Branch and Bound method to get the following optimal solution Step 1 Initially, no job is assigned to any operator, so the assignment(  ) at the root (level o) of the branching tree is a null set and the corresponding lower bound is also 0 for each Further branching: the four different sub problem under the root nodes are shown as in Fig lower bound the solution problems shown on its right hand side.

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P0

P111  40.62

P131  60.62

P121  46.95

P141  58.43

1

Compute the lower bound for P11 :

V =  Ci , j

+

i , jA

Where

 ={(11)}

min C     (

i X

A= {(11)}

V(11)=C11+



(

i {2 , 3, 4}

ij )

j Y

X= {2,3}

 min C

j {2 , 3}

ij

Y= {2,3} then

)

P111 =10+(8.52+13+9.1)=40.62

P121 =17.33+(8.52+12+9.1)=46.95

P131 =31+(8.52+12+9.1)=60.62

P141 =25.43+(10+12+11)=58.43

Further branching: further branching is done from the terminal node which has the least lower bound At this stage, the nodes P111 , P121 P131 P141 are the terminal nodes. The node P111 has the least lower bound. Hence, further branching from this node is shown as in Fig.

P111  40.62

P222  42.1

P121  46.95

P232  48.93

V(22)=C11+ C22+



i {, 3, 4}

235

(

 min C

j {3, 4}

B.Srinivas, S Ganesan

P131  60.62

P242  42.52

ij

)

P141  58.63

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P232 =10+15.69+(14.14+9.1)=48.93

P222 =10+10+(13+9.1)=42.1

P242 =10+8.52+(13+11)=42.52

Further branching: At this stage the nodes P111 , P222 , P232 P242 are the terminal node. Among these nodes

P222 , is the lower bound. Then these terminal nodes at lower-most is to be considering further branching. V(3,3)=C11+ C22+C33+  ( i {4}

min C  

ij

) P333 =10+10+13+9.1=42.1

j {4}

P343 =10+10+14.14+11=45.14

P111  40.62

P222  42.1

P  42.1

P121  46.95

P232  48.93

P343  45.47

3 33

P131  60.62

P141  58.63

P242  42.52

P323  45.21

P333  40.62

We replace these values for their corresponding the above table which results in Conventional assignment problem in the LPP form. Solving it, we get the solution as with the optimal objective value 23 which represents the optimal total cost. In other words the optimal assignment is (1,1), (2,2), (3,3), (4,4) The fuzzy optimal cost is calculated 10+10+13+9.1=42.1 6. Conclusion:

In this paper, the assignment cost has been considered as an intuitionistic fuzzy numbers described by fuzzy numbers which are more realistic and general in nature. Here, the fuzzy assignment problem has been converted into crisp assignment problem using ranking procedure of Annie Varghese and Sunny Kuriakose [2] and Branch bounded method has been applied to find an optimal solution. Numerical example has been shown that the total cost obtained is optimal. This method is systematic procedure, easy to apply and can be utilized for all type of intuitionistic fuzzy assignment problem whether maximize or minimize objective function.

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www.ijetmas.com February 2015, Volume 3 Issue 2, ISSN 2349-4476 References:

[1] Angelov. P Optimization in an intuitionistic fuzzy environment, Fuzzy Sets and Systems, 86, (1997),299-306 [2] Annie Varghese,M and Sunny Kuriakose,S compare the intuitionistic fuzzy numbers. Applied Mathematical Sciences, 6(47) (200) 2345 – 2355. [3] Bellmann B.E and Zadeh.L.A, Decision making in fuzzy environment management sciences, 17(1970), 141-164 [4] Bai. X.J. Liu Y.K. and Shen. S.Y. Fuzzy generalized assignment problem with credibility constraints, Proceedings of the Eighth International Conference on Machine Learning and Cybernetics, Baoding, (2009) 657-662. [5] Chi-Jen Lin, Ue-Pyng Wen, A Labelling algorithm for the fuzzy assignment problem, fuzzy Sets and system 142 (2004), 373-391. [6] Chen M.S, On a fuzzy assignment problem, Tamkang J, 22 (1985), 407-411. [7] De ,PK and Das,D .Ranking of trapezoidal intuitionistic fuzzy numbers, 12th International Conference on Intelligent Systems Design and Applications (ISDA), (2012), 184 – 188. [8] Fortemps P and.Roubens M “Ranking and defuzzification methods based area compensation” fuzzy sets and systems Vol.82,(1996 ),PP 319-330, [9] Geetha, S. and Nair, K.P.K.. A variation of the assignment problem, European Journal of Operation Research. 68(3), (1993), 422-426. [10] Hussain , RJ and Senthil Kumar,P .Algorithm approach for solving intuitionistic fuzzy transportation problem, Applied mathematical sciences, 6 (80), (2012), 3981- 3989. [12] Kuhn, H. W, The Hungarian method for assignment problem, Naval Research Logistics Quarterly. 2, (1955), 83-97. [13] Kalaiarasi. K , Sindhu.S, and Arunadevi.M, Optimization of fuzzy assignment model with triangular fuzzy numbers IJISET Vol. 1 ISSN, 3, ( 2014). 2348 - 7968 [14] Zadeh L.A., Fuzzy sets, Information and Control, 8 (1965), 338-353

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