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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Recommended