An Intuitionistic Fuzzy Shortest Path Problem Based on Level λ – LR Type Representation

I L (A) a

b

l

2

2 2

2(p q) (s r)

where

and

3(q r) 2(s r)

o

c m n 4 4

a b l

2

If the

i th

fuzzy path Length is

n

c m

Li (pi , qi , ri , si ; )LRand the fuzzy

I R (A) a b

l

shortest lengthis L (p, q, r, s; )

Then the

2

2 2

min LR

a b l n c m

shortest Mean and StandardDeviation Index

2

between LMin and Li is

The signed distance of level

LR Type

I(L

, L ) ( ) (i i )

Representation of Fuzzy interval

Min i

o i

Let

A p , q

, r , s ;

LR

Where

2(p q) (s r)

and

3(q r) 2(s r)

o

be

The

level LR type representation of fuzzy interval.

4 4

Introduces new definition are here.

cut of A is

Minimum Operation on

2 shaped

A A L ()A R ( , 0

and

(01]

Intuitionistic fuzzy numbers:

Where

A L ( )

p r

, A R ( )

q s

Since

For two 2

shaped Intuitionistic fuzzy numbers

L lw

, lp

rp , rw

lw

, lp

rp , rw ,

2

the function is continuous over the interval 0

1 1 1 1 1

1 1 1 1

integration is used for obtaining the mean value of signed distance as follows: d(A.0) = ¼(2(p+q)+(s-r))

L 2 lw , lp2

rp 2

, rw

lw , lp rp , rw

2

2

2 2 2

max(lw 1, lw2 ), min(lp1, lp2 ),

A p , q , r , s

p , q , r , s

is A L U

L min (L1, L 2 )

LR

A , A

min(rp 1, rp 2 ), min(rw 1, rw 2 )

min(lw 1 , lw 2 ), max(lp 1 , lp2 ),

cut intervalis embershipfunctionand

.The above

nonmembershipfunctionas follows

Lmax (L1, L2 )

0,1

max(rp 1 , rp 2 ), max(rw 1 , rw 2 )

Weighted Average index for level 2 Membership function

AL c (a c ) ; AU (b d ) d and

A

A

L c (a c ) ; U (b d ) d

i i i , i i

Let L lw , lp rp , rw ; and

cut interval for LR type representation

Lmin

(lw , lp , rp , rw ; ) be two

2 .

of Intuitionistic fuzzy interval

membershipfunctions

If lp lpi , rp rpi

then theweightedaverage

cut interval for LR type representation of fuzzy

interval A a, b is given by

index between

A L U .The above representation cut

A , A .

L i and L min can be calculated as

A A

interval is obtained for membership function and

nonmember ship function as follows 0,1

2

WAI(Lmin , Li ) i

0 1

A L b c A U d b

WhereA

lp rp 2

and Ai

lp i rp i 2

and

L

A

b

• c

U d

• b

A

i 1,2,….n.

Weighted Average index for level 2 non- membership function

Convex Index for Intuitionistic fuzzy number

Let A be an LR trapezoidal Intuitionistic fuzzy number, then

CoI(A ) A L 1 A U ; CoI(A ) (1 ) A L

A U

Let Li

and Lmax

lw i , lp i , rp i , rw i ;

(lw , lp , rp , rw ; )

be two non

,

Where A

A

L U

and A L

A

If

U are the

cut

membership functions, If

inter al of A p , q , r , s

p , q , r , s

A

A

A

A

l lpi , rp rp i then the weightedaverage index

betweenLi and Lmax can be calculatedas

A L 1 A U

min

L , U ;

A

A

(1 ) A L A U

max

L , U

A

A

A

A

WAI(L

, L ) A Ai

0 1

for all

max

WhereA

i

lp rp 2

2

andAi

lp i rp i 2

i 1,2,….n.

, 0,1

Total Integral Index

For an intuitionsiticfuzzy numberA p

, q

, r

, s

p , q

, r

, s

cut interval for LR trapezoidal

Intuitionistic fuzzy number

cut interval for LR trapezoidal Intuitionistic fuzzy number

The totalintegralindexIT (A)can be constructed from theleft integralsvaluesIL (A) and therightintervalvalue

Mean and standard deviation of level LR trapezoidal Intuitionistic Fuzzy Number

IR (A).The totalintegralvalue IT (A) with index

A p , q , r , s p , q , r , s

LR

of optimism where 0 1.

is intuitiontsictrapezoidal fuzzy number

Is then defined for both membership and non membership function.

If the i th fuzzy path Length for membershipfunction

IT A IL A (1 )I R A ;

is L i (p i , q i , r i , s i ; )LRand the

IT A (1 ) IL A I R A

nonmembershipfunction L i (p i , q i , r i , s i ; )LR

c m n

then fuzzy shortestlengthsare Lmin (p , q , r , s )

I L (A ) a b l

2

and Lmax (p , q , r , s ) Then the shortestMean

n d m

a b

and StandardDeviation Index between LMin andLi is

IR (A ) a b l

n l a , m b l

l ,

2 2

LMax andLi ,

I(LMin , Li )

( ) (i i ) , i

c m n

I(L

, L ) ( ) (i i )

where

IL (A ) a b l ;

Max i

o

2 i

I (A ) a b l n d m

2(p q) (s r)

R

2

and

4

Signed distance of level LR Intuitionistic

3(q p) 2(s r)

o

4

for membershipfunctionand

Fuzzy number(Membership Function)

2(p q) (r s)

A p , q , r , s p , q , r , s is A L U

and

4

LR

The mean valueof signeddistanceis

A , A

3(p q) 2(s r)

o

4

for

NonmembershipFunction

d(A,0)

1

(2(p q ) (s r ))

4

Proposed Algorithm

Algorithm for Intuitionistic fuzzy shortest path problem based on convex index

Signed distance of level LR Intuitionistic Fuzzy number(Non Membership Function)

A p ,q ,r ,s p ,q ,r ,s is A L U

In this section a new algorithm is proposed for finding the Intuitionistic fuzzy shortest path based on convex index.

Step 1:Construct a network G= (V, E) Where V is the

set of vertices and E is the set of edges.

LR

The mean valueof signeddistanceis

A , A

Step 2:Form he possible path

Pi from source node to

destination node and compute the corresponding path

1

d ( A,0)

(2( p q ) (r s ))

4

length Li ii=1,2,m for possible m path and set

Li p i , q i , r i , s i

p i , q i , r i , s i

LR

Step3:Calculate cut interval for LR trapezoidal Intuitionistic fuzzy number ( or LR Type representation of fuzzy interval) for all possible path length, Li I

=1,2,m and set

Step5:Rank the shortest path with the highest weighted index.

Algorithm For Intuitionistic Fuzzy Shortest path problem based on signed distance

Li ( LL

, LU

i 1,2,…m ;

A new algorithm is proposed for finding the

i ( i (

L LL , LU

i 1,2,…m

Intuitionistic fuzzy shortest path based on signed

i ( i (

i (

distance in this paper introduces a new method Signed

Step 4:Calculate Convex Index

distance

CoI(Li ) LL 1 A U ;

i (

L

i (

U

Let Li p i , q i , r i , s i

p i , q i , r i , s i

CoI(Li ) (1 ) Li ( Li (

LR

i

and i=1,2,n where

for all possiblepathlengths

Step 5:Ranking the shortest path with the minimum

L denotes the level -LR Intuitionistic fuzzy path

convex index and the corresponding path length the Intuitionistic fuzzy shortest path.

Li is

length.

Step 1: Construct a network G=(V,E) Where V is the set of vertices and E is the set of edges. Here G is an

Algorithm for Intuitionistic fuzzy shortest path problem based on weighted average

acyclic digraph and the arc length takes the level –

LR intuitionistic fuzzy numbers.

index

Step 2: Calculate all possible paths

Pi from source

A new algorithm is proposed for finding the

node to destination node and compute the

Intuitionistic fuzzy shortest path based on weighted average index

corresponding path length

L i =1,2,m for possible

i

Step1:Construct a network G =(V,E) Where V is the set of vertices and E is the set of edges. Here G is an

m paths using definition and set

and

acyclic digraph.

Step 2:Calculate all possible paths Pi from source node

Li

p i , q i , r i , s i

p i , q i , r i , s

i LR

to destination node and compute the corresponding

i 1 to n and 0 1 .

path length Li i =1,2,m for possible m path and set

Step 3: calculate signed distance of level – LR type representation of fuzzy intervals for each possible path

Li p i , q i , r i , s i

p i , q i , r i , s i

LR

using definition 3.8&3.9

Step 4: The path having the minimum signed distance

Step3:Calculate the Intuitionistic fuzzy shortest length.

is identified as the shortest path.

Lmin

using definition and set

Algorithm for Intuitionistic Fuzzy Shortest path

Lmin

p

, q

, r , s ;

Lmax

using definition

problem based on Mean and standard deviation index

A new algorithm is proposed for finding the Intuitionistic fuzzy shortest path based on Mean and

and set

Lmax

p , q , r

, s

standard deviation index in this paper introduces a new method called Mean and Standard Deviation index

Step4:Calculate weighted average index using

which acts as the measurement tool between

Lmin

and

definition and the rank to the path based on weighted index.

L L

, L and

i max i

Let L

p , q

, r , s

p , q

, r , s

P1 :1 2 4

L1 (7,34,39,9;1)

i i

i i

i i

i i

i LR

and L

(7,36,34,9;1)

i

and i=1,2,n where L denotes the level -LR Intuitionistic fuzzy path length.

1 P2 :1 4

L2 (6,28,30,7;1)

Step1:Construct a network G =(V,E) Where V is the

and

L2 (3,30,31,6;1)

set of vertices and E is the set of edges. here G is an acyclic digraph.

Step 2:Calculate all possible paths Pi from source node to destination node and compute the corresponding

P3 :1 3 4 L3 (9,41,49,14;1)

and L3 (13,46,467, ;1)

path length Li ,

i =1,2,m for possible m path and set

Step 3:

Lmin 9,28,30,7;1

and Lmax 3,46,46,9;1

Li p i , q i , r i , s i

p i , q i , r i , s i

LR

Step 4: Let distance index.

1

for weighted index, signed

Step 3:Calculate the Intuitionistc fuzzy shortest length.

Table 1 Results of the Network Based on Level

Lmin

using definitionand set

Lmin

p , q , r , s ;

Lamda TLR Weighted Index

Lmax

using definition and set

Lmax

p , q , r , s

Step 4:Calculate Mean and standard deviation index

between

definition 3.10.

Lmin ,

and

L Lmax and

i

,

L using

i

Step 5:The path having the maximum Mean and Standard deviation index is identified as the shortest path.

ILLUSTRATIVE EXAMPLE

Step1:Construct a network with 4 vertices and 5 edges as follows: Fig (1)

Table 2 Results of the Network Based on Level Lamda TLR Signed Distance Index

Step 2: The possible paths and the corresponding path lengths are as follows

Let .3

and .5 for convex index

Table 3 Results of the Network Based on Convex Index

Step 5:Path P2 :1 4 is the Intuitionistic fuzzy shortest path since it has the highest level Trapezoidal LR weighted average index and the corresponding shortest path length is

L2 (6,28,30,7;1)LR

and

L2 (3,30,31,6;1)LR . he

Total integral index for different values of

solution obtained for Intuitionistic fuzzy shortest path problem in this paper coincides with the solution of the existing algorithm.

Conclusion

This paper developed three algorithms for solving the shortest path problem on a network with intuitionistic fuzzy arc length. The ranking given to the paths is helpful for the decision-makers as they make decisions

(i)

Let .3 then,

in choosing the best of all the possible path alternatives.

(i)

Table 4 Results of the Network Based on Total Integral Index

1 then,

Table 5 Results of the Network Based on Total Integral Index

Verification is also done with the existing methods, which helps to conclude that the algorithms developed in the current paper are an alternative and improved form of previous methods, to get the shortest path in intuitionistic fuzzy environments.

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Promblem Based on Level

LR type representation of

Fuzzy Interval; International conference on Mathematics in Engineering and Business management 978-81-8286-015-5 2012.

Journal of computer applications vol 63-No.20, Feb 2013.