APPLYING DIJKSTRA ALGORITHM FOR

SOLVING NEUTROSOPHIC SHORTEST PATH

PROBLEM

Luige Vladareanu

Institute of Solid Mechanics Romanian Academy Bucharest Romania

luigiv2007yahoocomsg luigivimsarbueduro

Said Broumi Assia Bakali Mohamed Talea Florentin

Smarandache Luige Vladareanu

Laboratory of Information processing Faculty of Science

Ben MrsquoSik University Hassan II BP 7955 Sidi

Othman Casablanca Morocco

Ecole Royale Navale Boulevard Sour Jdid BP 16303

Casablanca Morocco

Department of Mathematics University of New

Mexico705 Gurley Avenue Gallup NM 87301 USA

1

AbstractmdashThe selection of shortest path problem is

one the classic problems in graph theory In

literature many algorithms have been developed to

provide a solution for shortest path problem in a

network One of common algorithms in solving

shortest path problem is Dijkstrarsquos algorithm In

this paper Dijkstrarsquos algorithm has been redesigned

to handle the case in which most of parameters of a

network are uncertain and given in terms of

neutrosophic numbers Finally a numerical example

is given to explain the proposed algorithm

Keywords mdash Dijkstrarsquos algorithm Single valued

neutrosophic number Shortest path problem

Network2

IINTRODUCTION

To express indeterminate and inconsistent information which exist in

real world Smarandache [1] originally proposed the concept of a

neutrosophic set from a philosophical point of view The concept of the

neutrosophic set (NS for short) is powerful mathematical tool which

generalizes the concept of classical sets fuzzy sets [3] intuitionistic

fuzzy sets [4] interval-valued fuzzy sets [5] and interval-valued

intuitionistic fuzzy sets [6] The concept of the neutrosophic has three

basic components such that a truth-membership (T) indeterminacy-

membership (I) and a falsity membership (F) which are defined

independently of one another But a neutrosophic set So will be more

difficult to apply it in real scientific and engineering areas Thus

Wang et al [7] proposed the concept of single valued neutrosophic set

(for short SVNS) which is an instance of a neutrosophic set whose

functions of truth indeterminacy and falsity lie in [0 1] and provided

the set theoretic operators and various properties of SVNSs Some of

the recent research works on neutrosophic set theory and its

applications in various fields can be found in [8] In addition

Thamaraiselvi and Santhi [9] introduced a mathematical

representation of a transportation problems in neutrosophic

environment based on single valued trapezoidal neutrosophic

numbers and also provided the solution method The operations on

neutrosophic sets and the ranking methods are presented in [10]

3

4

The shortest path problem (SPP) is one of the most fundamental

and well-known combinatorial problems that appear in various

fields of science and engineering eg road networks application

transportation and other applications In a network the shortest

path problem aims at finding the path from one source node to

destination node with minimum weight where some weight is

attached to each edge connecting a pair of nodes The edge length of

the network may represent the real life quantities such as time

cost etc In conventional shortest path problem it is assumed that

decision maker is certain about the parameters (distance time etc)

between different nodes But in real life situations there always

exist uncertainty about the parameters between different nodes For

this purpose many algorithms have been developed to find the

shortest path under different types of input data including fuzzy

set intuitionistic fuzzy sets vague sets [11-15] One of the most

used methods to solve the shortest path problem is the Dijkstrarsquos

algorithm [16] Dijkstrarsquos algorithm solves the problem of finding

the shortest path from a point in a graph (the source) to a

destination

5

Recently numerous papers have been published on neutrosophic

graph theory [17-23] In addition Broumi et al [24-26] proposed

some algorithms to find the shortest path of a network (graph) where

edge weights are characterized by a neutrosophic numbers including

single valued neutrosophic numbers bipolar neutrosophic numbers

and interval valued neutrosophic numbers

The main purpose of this paper is to propose a new version of

Dijkstra algorithm for solving shortest path problem on a network

where the edge weights are characterized by a single valued

neutrosophic numbers The proposed method is more efficient due to

the fact that the summing operation and the ranking of SVNNs can

be done in a easy and straight manner

The rest of the article is organized as follows Section 2 introduces

some basic concepts of neutrosophic sets single valued neutrosophic

sets In Section 3 a network terminology is presented In section 4

we propose the new version of Dijkstrarsquoalgorithm for solving the

shortest path with connected edges in neutrosophic data Section 5

illustrates a practical example which is solved by the proposed

algorithm Conclusions and further research are given in section 6

6

Preliminaries

In this section some basic concepts and definitions on neutrosophic sets and single valued

neutrosophic sets are reviewed from the literature

7

8

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

IINTRODUCTION

To express indeterminate and inconsistent information which exist in

real world Smarandache [1] originally proposed the concept of a

neutrosophic set from a philosophical point of view The concept of the

neutrosophic set (NS for short) is powerful mathematical tool which

generalizes the concept of classical sets fuzzy sets [3] intuitionistic

fuzzy sets [4] interval-valued fuzzy sets [5] and interval-valued

intuitionistic fuzzy sets [6] The concept of the neutrosophic has three

basic components such that a truth-membership (T) indeterminacy-

membership (I) and a falsity membership (F) which are defined

independently of one another But a neutrosophic set So will be more

difficult to apply it in real scientific and engineering areas Thus

Wang et al [7] proposed the concept of single valued neutrosophic set

(for short SVNS) which is an instance of a neutrosophic set whose

functions of truth indeterminacy and falsity lie in [0 1] and provided

the set theoretic operators and various properties of SVNSs Some of

the recent research works on neutrosophic set theory and its

applications in various fields can be found in [8] In addition

Thamaraiselvi and Santhi [9] introduced a mathematical

representation of a transportation problems in neutrosophic

environment based on single valued trapezoidal neutrosophic

numbers and also provided the solution method The operations on

neutrosophic sets and the ranking methods are presented in [10]

3

4

The shortest path problem (SPP) is one of the most fundamental

and well-known combinatorial problems that appear in various

fields of science and engineering eg road networks application

transportation and other applications In a network the shortest

path problem aims at finding the path from one source node to

destination node with minimum weight where some weight is

attached to each edge connecting a pair of nodes The edge length of

the network may represent the real life quantities such as time

cost etc In conventional shortest path problem it is assumed that

decision maker is certain about the parameters (distance time etc)

between different nodes But in real life situations there always

exist uncertainty about the parameters between different nodes For

this purpose many algorithms have been developed to find the

shortest path under different types of input data including fuzzy

set intuitionistic fuzzy sets vague sets [11-15] One of the most

used methods to solve the shortest path problem is the Dijkstrarsquos

algorithm [16] Dijkstrarsquos algorithm solves the problem of finding

the shortest path from a point in a graph (the source) to a

destination

5

Recently numerous papers have been published on neutrosophic

graph theory [17-23] In addition Broumi et al [24-26] proposed

some algorithms to find the shortest path of a network (graph) where

edge weights are characterized by a neutrosophic numbers including

single valued neutrosophic numbers bipolar neutrosophic numbers

and interval valued neutrosophic numbers

The main purpose of this paper is to propose a new version of

Dijkstra algorithm for solving shortest path problem on a network

where the edge weights are characterized by a single valued

neutrosophic numbers The proposed method is more efficient due to

the fact that the summing operation and the ranking of SVNNs can

be done in a easy and straight manner

The rest of the article is organized as follows Section 2 introduces

some basic concepts of neutrosophic sets single valued neutrosophic

sets In Section 3 a network terminology is presented In section 4

we propose the new version of Dijkstrarsquoalgorithm for solving the

shortest path with connected edges in neutrosophic data Section 5

illustrates a practical example which is solved by the proposed

algorithm Conclusions and further research are given in section 6

6

Preliminaries

In this section some basic concepts and definitions on neutrosophic sets and single valued

neutrosophic sets are reviewed from the literature

7

8

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

4

The shortest path problem (SPP) is one of the most fundamental

and well-known combinatorial problems that appear in various

fields of science and engineering eg road networks application

transportation and other applications In a network the shortest

path problem aims at finding the path from one source node to

destination node with minimum weight where some weight is

attached to each edge connecting a pair of nodes The edge length of

the network may represent the real life quantities such as time

cost etc In conventional shortest path problem it is assumed that

decision maker is certain about the parameters (distance time etc)

between different nodes But in real life situations there always

exist uncertainty about the parameters between different nodes For

this purpose many algorithms have been developed to find the

shortest path under different types of input data including fuzzy

set intuitionistic fuzzy sets vague sets [11-15] One of the most

used methods to solve the shortest path problem is the Dijkstrarsquos

algorithm [16] Dijkstrarsquos algorithm solves the problem of finding

the shortest path from a point in a graph (the source) to a

destination

5

Recently numerous papers have been published on neutrosophic

graph theory [17-23] In addition Broumi et al [24-26] proposed

some algorithms to find the shortest path of a network (graph) where

edge weights are characterized by a neutrosophic numbers including

single valued neutrosophic numbers bipolar neutrosophic numbers

and interval valued neutrosophic numbers

The main purpose of this paper is to propose a new version of

Dijkstra algorithm for solving shortest path problem on a network

where the edge weights are characterized by a single valued

neutrosophic numbers The proposed method is more efficient due to

the fact that the summing operation and the ranking of SVNNs can

be done in a easy and straight manner

The rest of the article is organized as follows Section 2 introduces

some basic concepts of neutrosophic sets single valued neutrosophic

sets In Section 3 a network terminology is presented In section 4

we propose the new version of Dijkstrarsquoalgorithm for solving the

shortest path with connected edges in neutrosophic data Section 5

illustrates a practical example which is solved by the proposed

algorithm Conclusions and further research are given in section 6

6

Preliminaries

In this section some basic concepts and definitions on neutrosophic sets and single valued

neutrosophic sets are reviewed from the literature

7

8

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

5

Recently numerous papers have been published on neutrosophic

graph theory [17-23] In addition Broumi et al [24-26] proposed

some algorithms to find the shortest path of a network (graph) where

edge weights are characterized by a neutrosophic numbers including

single valued neutrosophic numbers bipolar neutrosophic numbers

and interval valued neutrosophic numbers

The main purpose of this paper is to propose a new version of

Dijkstra algorithm for solving shortest path problem on a network

where the edge weights are characterized by a single valued

neutrosophic numbers The proposed method is more efficient due to

the fact that the summing operation and the ranking of SVNNs can

be done in a easy and straight manner

The rest of the article is organized as follows Section 2 introduces

some basic concepts of neutrosophic sets single valued neutrosophic

sets In Section 3 a network terminology is presented In section 4

we propose the new version of Dijkstrarsquoalgorithm for solving the

shortest path with connected edges in neutrosophic data Section 5

illustrates a practical example which is solved by the proposed

algorithm Conclusions and further research are given in section 6

6

Preliminaries

In this section some basic concepts and definitions on neutrosophic sets and single valued

neutrosophic sets are reviewed from the literature

7

8

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

6

Preliminaries

In this section some basic concepts and definitions on neutrosophic sets and single valued

neutrosophic sets are reviewed from the literature

7

8

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

7

8

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

8

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

9

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

10

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

11

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

12

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

13

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

14

IVILLUSTRATIVE EXAMPLE

Now we solve an hypothetical example to verify the proposed

approach Consider the network shown in figure1 we want to

obtain the shortest path from node 1 to node 6 where edges have

a single valued neutrosophic numbers Let us now apply the

extended Dijkstra algorithm to the network given in figure 1

Fig 1 A network with single valued neutrosophic weights

(0406 07)

(0503 01)

(0104 06) (0304 07)

(0706 08)

(0605 03)

(0302 06)

(0203 04)

1

3

4

2 5

6

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

15

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

16

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

17

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

18

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

19

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

20

1

34

2 5

6

(0503 01)

(0605 03)

(0302 06)

(0406 07)

[(06009 004) 3][(0203 04) 1]

[(093018 017) 5]

[(0406 07) 1]

(0304 07)

(0706 08)[(082036 056) 2]

(0203 04)

(0104 06)

[(01 1) -]

Fig 2 Network with single valued neutrosophic shortest distance of each node from

node 1

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

VCONCLUSIONS

This paper extended the fuzzy Dijkstrarsquos algorithm to

find the shortest path of a network with single valued

neutrosophic edge weights The use of neutrosophic

numbers as weights in the graph express more

uncertainty than fuzzy numbers The proposed

algorithm proposes solution to one issue this issue is

addressed by identification of shortest path in

neutrosophic environment A numerical example was

used to illustrate the efficiency of the proposed

method In future we will research the application of

this algorithm

21

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

22

REFERENCES

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

1 F Smarandache A unifying field in logic Neutrosophy neutrosophic probability set and logic American Research Press Rehoboth 1998

2 F Smarandache A geometric interpretation of the neutrosophic set mdashA generalization of the intuitionistic fuzzy set Granular Computing (GrC) 2011 IEEE International Conference 2011 pp602ndash 606

3 L Zadeh Fuzzy sets Inform and Control 8 1965 pp338-353

4 K Atanassov Intuitionistic fuzzy sets Fuzzy Sets and Systems vol 20 1986 pp 87-96

5 I Turksen ldquoInterval valued fuzzy sets based on normal formsrdquo Fuzzy Sets and Systems vol 201986 pp 191-210

6 K Atanassov and G Gargov Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems vol31 1989 pp343-349

7 H Wang F Smarandache Y Zhang and R Sunderraman ldquoSingle valued Neutrosophic Sets Multisspace and Multistructure 4 2010 pp 410-413

8 httpfsgallupunmeduNSS

9 A Thamaraiselvi and RSanthi A New Approach for Optimization of Real Life Transportation Problems in Neutrosophic Environment Mathematical Problems in Enginnering2016 9 pages

10 P Chi and P Liu An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic setNeutrosophic Set and Systems1 2013 pp 1-8

23

24

11 P Jayagowri and G Geetha Ramani Using Trapezoidal Intuitionistic Fuzzy Number to Find Optimized Path in a Network Volume 2014 Advances in Fuzzy Systems 2014 6 pages

12 A Kumar and M Kaur A New Algorithm for Solving Shortest Path Problem on a Network with Imprecise Edge Weight Applications and Applied Mathematics Vol 6 Issue 2 2011 pp 602 ndash 619

13 A Kumar and M Kaur Solution of fuzzy maximal flow problems using fuzzy linear programming World Academy of Science and Technology 87 28-31 (2011)

14 S Majumder and A Pal Shortest Path Problem on Intuitionistic Fuzzy Network Annals of Pure and Applied Mathematics Vol 5 No 1 November 2013 pp 26-36

15 S Okada and M Gen Fuzzy shortest path problems Computers and Industrial Engineering vol 27 N1-4 1994 pp465-468

16 EWDijikstra A note on two problems in connexion with graphsNumerishe Mathematik 11959 pp 269-271

17 S Broumi M Talea A Bakali F Smarandache Single Valued Neutrosophic Graphs Journal of New Theory N 10 2016 pp 86-101

18 S Broumi M Talea A Bakali F Smarandache On Bipolar Single Valued Neutrosophic Graphs Journal of New Theory N11 2016 pp84-102

19 S Broumi M Talea A Bakali F Smarandache Interval Valued Neutrosophic Graphs (2016) in press

20 S Broumi A Bakali M Talea and F Smarandache Isolated Single Valued Neutrosophic Graphs Neutrosophic Sets and Systems Vol 11 2016 pp74-78

21 S Broumi F Smarandache M Talea and A Bakali An Introduction to Bipolar Single Valued Neutrosophic Graph Theory Applied Mechanics and Materials vol841 2016 184-191

22 S Broumi M Talea F Smarandache and A Bakali Single Valued Neutrosophic Graphs Degree Order and Size 2016 IEEE International Conference on Fuzzy Systems (FUZZ) 2016 pp2445-2451

23 F Smarandache Symbolic Neutrosophic Theory Europanova asbl Brussels 2015 195p

25

24 S Broumi Bakali M Talea and F Smarandache A Computation

of Shortest Path Problem in a Network with Single Valued

Neutrosophic Number Based on Ranking Method 2016 (submitted)

25 S Broumi A Bakali M Talea and F Smarandache M Ali

Shortest Path Problem Under Bipolar Neutrosophic Setting 2016

(submitted)

26 S Broumi A Bakali M Talea F Smarandache and Shortest Path

Problem Under Interval Valued Neutrosophic Setting 2016

(submitted)

27 NBhatia Algorithms for Some Fuzzy Network Problems Using

Ranking Function ( master Thesis Thapar university Pataiala-

147004(PUNJAB) School of Mathematics and Computer

Applications 2009

THANKS

26

THANKS

26