Hindawi Publishing CorporationApplied Computational Intelligence and Soft ComputingVolume 2013 Article ID 970197 5 pageshttpdxdoiorg1011552013970197

Research ArticleAn Algorithm for Extracting Intuitionistic FuzzyShortest Path in a Graph

Siddhartha Sankar Biswas Bashir Alam and M N Doja

Department of Computer Engineering Faculty of Engineering amp Technology Jamia Millia Islamia New Delhi 110025 India

Correspondence should be addressed to Siddhartha Sankar Biswas ssbiswas1984gmailcom

Received 15 September 2013 Accepted 5 October 2013

Academic Editor Baoding Liu

Copyright copy 2013 Siddhartha Sankar Biswas et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We consider an intuitionistic fuzzy shortest path problem (IFSPP) in a directed graphwhere theweights of the links are intuitionisticfuzzy numbers We develop a method to search for an intuitionistic fuzzy shortest path from a source node to a destination nodeWe coin the concept of classical Dijkstrarsquos algorithm which is applicable to graphs with crisp weights and then extend this conceptto graphs where the weights of the arcs are intuitionistic fuzzy numbers It is claimed that the method may play a major role inmany application areas of computer science communication network transportation systems and so forth in particular to thosenetworks for which the link weights (costs) are ill defined

1 Introduction

Graphs [1ndash4] are a very important model of networks Thereare many real-life problems of network of transportationcommunication circuit systems and so forth which aremodeled into graphs and hence solved Graph theory haswide varieties of applications in several branches of engi-neering science social science medical science economicsand so forth to list a few only out of many Many real-life situations of communication network transportationnetwork and so forth cannot be modeled into crisp graphsbecause of the reason that few or all of the arcslinks have thecostweight which is ill defined The weights of the arcs arenot always crisp but intuitionistic fuzzy (or fuzzy)

One of the first studies on fuzzy shortest path problem(FSPP) in graphs was done by Dubois and Prade [5] andthen by Klein [6] However few more solutions to FSPPproposed in [7ndash10] are also interesting Though the work ofDubois and Prade [5] was a major breakthrough that paperlacked any practical interpretation even if fuzzy shortest pathis found but still this may not actually be any of the pathin the corresponding network for which it was found Thereare very few works reported in the literature on finding anintuitionistic fuzzy shortest path in a graph Mukherjee [11]used a heuristic methodology for solving the IF shortest

path problem using the intuitionistic fuzzy hybrid geometric(IFHG) operator with the philosophy of Dijkstrarsquos algorithmIn [12] Karunambigai et al in a team work with Atanassovpresent a model based on dynamic programming to find theshortest paths in intuitionistic fuzzy graphs NagoorGani andMohammed Jabarulla in [13] also developed a method onsearching intuitionistic fuzzy shortest path in a network Butall these algorithms have both merits and demerits (none isabsolutely the best) as all these are greedy algorithms In thispaper we solve the intuitionistic fuzzy shortest path problem(IFSPP) for a graph where the arc weights are intuitionisticfuzzy numbers and then we reduce the method to the caseof finding fuzzy shortest path in a graph Our work here doesalso have the same kind of demerits (as in [12 13]) but themajor significance lies in the fact that we follow the conceptof classical Dijkstrarsquos algorithm which is applicable to graphswith crisp weights and then extend this concept to graphswhere theweights of the arcs are intuitionistic fuzzy numbers

2 Preliminaries

A graph119866 is an ordered pair (119881 119864) which consists of two sets119881 and 119864 where 119881 or 119881(119866) is the set of vertices (or nodes)and 119864 or 119864(119866) is the set of edges (or arcslinks) Throughout

2 Applied Computational Intelligence and Soft Computing

in our work here we consider those graphs which are withoutloops Graphs may be of two types undirected graphs anddirected graphs In an undirected graph the edge (119894 119895) andthe edge (119895 119894) if exist are obviously identical unlike that inthe case of directed graph For a fast visit on the theory ofgraphs one could see [1ndash4]

The intuitionistic fuzzy set theory of Atanassov [14ndash16] is now a well-known powerful soft computing tool tothe world scientists If 119883 is a universe of discourse anintuitionistic fuzzy set 119860 in 119883 is a set of ordered triplets119860 = ⟨119909 120583

119860(119909) ]119860(119909)⟩ 119909 isin 119883 where 120583

119860 ]119860 119883 rarr [0 1]

are functions such that 0 le 120583119860(119909) + ]

119860(119909) le 1 for all

119909 isin 119883 For each 119909 isin 119883 the values 120583119860(119909) and ]

119860(119909) represent

the degree of membership and degree of nonmembershipof the element 119909 to 119860 sub 119883 respectively and the amount120587119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) is called the hesitation part Of

course a fuzzy set is a particular case of the intuitionisticfuzzy set if 120587

119860(119909) = 0 for all 119909 isin 119883 For details of the

classical notion of intuitionistic fuzzy set theory one couldsee the book authored by Atanassov [15] The concept of anintuitionistic fuzzy number is of importance for quantifyingan ill-known quantity Intuitionistic fuzzy numbers are themore generalized form of fuzzy numbers involving twoindependently estimated degrees degree of acceptance anddegree of rejection In our work here throughout we usethe notion of triangular intuitionistic fuzzy numbers andthe basic operations like IF addition oplus IF subtraction ⊝ldquorankingrdquo of intuitionistic fuzzy numbers and so forthTrivially any crisp real number can be viewed as a fuzzynumber or as an intuitionistic fuzzy number There is nounique method for ranking the number of intuitionisticfuzzy numbers because all the existing methods [17ndash19] aresoft computing methods Each method has got merits anddemerits depending upon the properties of the applicationdomains and the problem under consideration However if1198601 1198602 1198603 119860

119899are 119899 intuitionistic fuzzy numbers sorted

in IF ascending order (in fact it is a kind of nonascendingorder assuming that the IF equal intuitionistic fuzzy numberstake corresponding positions at random if there is no lossof generality) by any good predecided method that is if1198601≺ 1198602≺ 1198603≺ sdot sdot sdot ≺ 119860

119899 then 119860

1and 119860

119899are called

respectively the IF-min and IF-max of these n intuitionisticfuzzy numbers Almost all the existing methods [17ndash19] ofranking intuitionistic fuzzy numbers were developed inde-pendently that is not as extensions of the existing methods[20ndash26] of ranking of fuzzy numbers [27] Although severalauthors [20ndash26] have reported several ranking methods offuzzy numbers all are having limitations too that is notan absolute method suitable for every application domainHowever if119860

1 1198602 1198603 119860

119899are 119899 fuzzy numbers sorted in

fuzzy ascending order by a predecided method that is 1198601≺

1198602≺ 1198603≺ sdot sdot sdot ≺ 119860

119899 then 119860

1and 119860

119899are called respectively

the fuzzy-min and fuzzy-max of these 119899 fuzzy numbers

3 Graphs with IF Weighted Arcs

In most of the real-life problems of networks be it in acommunication model or transportation model the weights

Approx 46km Approx 37km

Approx 65km

Approx 19km

Approx 15km

B

C D

A

Figure 1 A graph 119866 with IF weights of arcs

of the arcs are not always crisp but intuitionistic fuzzynumbers (or at best fuzzy numbers) For example Figure 3shows a public road transportationmodel for a traveler wherethe cost parameter for traveling each arc has been available tohim as an intuitionistic fuzzy number

But for such type of ill graph there is no attempt madeso far in the literature for searching an IF shortest path Inour method here we solve this intuitionistic fuzzy shortestpath problem (IFSPP) for graphswherewe also use the notionof Dijkstrarsquos algorithm but with simple soft-computationswithout using any hybrid geometric operators using onlybasics of Atanassovrsquos operators [15]

4 Intuitionistic Fuzzy ShortestPath in a Graph

In this section we solve the IFSPP for graphs where we usethe philosophy of Dijkstrarsquos algorithm but with simple soft-computations with IF data Consider a directed graph 119866

where the arcs are of intuitionistic fuzzy weights (intuition-istic fuzzy numbers) as shown in Figure 1 Suppose that thesubalgorithm IF119882(119866) returns the intuitionistic fuzzy weightset119882 corresponding to each arc

41 IF Shortest Path Estimate 119889[V] of a Vertex V in a DirectedGraph As shown in Figure 2 suppose that 119904 is the sourcevertex and the currently traversed vertex is 119906 The IF estimate119889[V] in graph 119866 is computed using IF addition as follows

(IF shortest path estimate of vertex V)

= (IF shortest path estimate of vertex 119906)

oplus (intuitionistic fuzzy number weight

corresponding to the arc from

the vertex 119906 to the vertex V)

(1)

or

119889 [V] = 119889 [119906] oplus 119908119906V (2)

42 Intuitionistic Fuzzy Relaxation of an Arc in a DirectedGraph We extend the classical notion of relaxation to the

Applied Computational Intelligence and Soft Computing 3

Intermediary vertices betweenArc between

Fuzzy shortestdistance between

Fuzzy weight

vertex s and vertex u

vertex s and vertex u

vertex s and vertex u

of the arc u

s u v

Figure 2 IF estimation procedure for 119889[V] in a graph 119866

IFISS (119866 119904)(1) For each vertex V isin 119881[119866](2) 119889[V] = infin(3) V sdot 120587 = NIL(4) 119889[119904] = 0

Algorithm 1

Intermediary vertices

If shortestdistance between

If sh

ortes

t

distan

ce be

twee

n

vertex s and vertex u

verte

x san

d vert

exu

s u

v

Inter

med

iary n

odes IFN

Wuv

betw

een s

and

between s and

Arc between vertices betweenuand

Figure 3 Diagram showing how the IF-RELAX algorithm works ina graph

case here with intuitionistic fuzzy number weights We callit ldquoIF relaxationrdquo For this first of all we initialize the graphalong with its starting vertex and IF shortest path estimatefor each vertices of the graph 119866 The ldquoINTUITIONISTIC-FUZZY-INITIALIZATION-SINGLE-SOURCErdquo algorithmIFISS will do what is shown in Algorithm 1

After the IF initialization the process of IF relaxationof each arc begins as shown in Figure 3 The subalgorithmIF-RELAX plays the vital role to update 119889[V] that is the IFshortest distance value between the starting vertex 119904 and thevertex V which is a neighbour of the current traversed vertex119906 (see Algorithm 2) where119908

119906V isin 119882 is the IF weight of the arcfrom the vertex 119906 to the vertex V and V sdot 120587 denotes the parentnode of vertex V

IF-RELAX (119906 119907119882)

(1) IF 119889[V] ≻ 119889[119906] oplus 119908119906V

(2) THEN 119889[V] larr 119889 [119906] oplus 119908119906V(3) V sdot 120587 larr 119906

Algorithm 2

IFSP (119866 119904)(1) IFISS (119866 119904)(2) 119882larr IF119882(119866)

(3) 119878 larr 0

(4) 119876 larr 119881[119866]

(5) WHILE 119876 = 0

(6) DO 119906 larr EXTRACT-IF-MIN (119876)

(7) 119878 larr 119878 cup 119906

(8) FOR each rn vertex V isin Adj[119906](9) DO IF-RELAX (119906 V119882)

Algorithm 3

43 IF Shortest Path Algorithm (IFSP Algorithm) in a GraphWe now present our main algorithm to find single source IFshortest path in a graph We name this ldquointuitionistic fuzzyshortest path algorithmrdquo that is in short by the title IFSPalgorithm In this algorithm we use the previously designedabove subalgorithms and also the subalgorithm EXTRACT-IF-MIN (119876) which extracts the node 119906 with minimum keyusing a predecided IF ranking method and updates 119876 (seeAlgorithm 3)

Example 1 (an example) Consider the following directedgraph 119866 where the IF weights (here they are intuitionisticfuzzy numbers) are shown against each link as shown inFigure 4 We want to solve the single-source IF shortest pathproblem (IFSPP) taking the vertex119860 as the source vertex andthe vertex119863 as the destination vertex

Our algorithm computes the following results

(1) 119882 = 119908119860119861

= 15 119908119860119862

= 3 119908119861119862

= 1 119908119862119861

= 4 119908119862119863

=

6 119908119861119863

= 2 and then

4 Applied Computational Intelligence and Soft Computing

C

B

A D

15

1 4

2

3 6

AB

BD

CD

AC CB

BC

Figure 4 A graph 119866 with IF weights of arcs

(2) 119878 = 119860 119862 119861119863 that is the IF shortest path from thesource vertex 119860 is

119860 997888rarr 119862 997888rarr 119861 997888rarr 119863 (3)

(3) with 119889-values that is IF shortest distance estimatevalues of each vertex from the starting vertex 119860 are

119889 [119860] = 0 119889 [119862] = TFN 3 119889 [119861] = TFN 7

119889 [119863] = TFN 9

(4)

5 Conclusion

There are many real-life problems in the networks of trans-portation communication circuit systems and so forthwhich are initially modeled into graphs and hence solved Inmany of these directed graphs in reality the weights of thearcs are not always crisp but fuzzy numbers In this paperwe develop a new method to solve the intuitionistic fuzzyshortest path problem (IFSPP) from a source vertex to adestination vertex in a directed graphThe importance of ourmethod lies in its potential to give solution in intuitionisticfuzzy environment unlike any of the existing algorithms ofIFSPP Obviously our algorithm does also work in case fewor all of the weights are fuzzy numbers or crisp numbers as aspecial case of IF numbers

References

[1] V K BalakrishnanGraphTheory McGraw-Hill NewYork NYUSA 1997

[2] B Bollobas Modern Graph Theory Springer New York NYUSA 2002

[3] R Diestel GraphTheory Springer 2000[4] F Harary Graph Theory Addison Wesley Boston Mass USA

1995[5] D Dubois and H Prade Fuzzy Sets and Systems Academic

Press New York NY USA 1980[6] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol

39 no 1 pp 27ndash41 1991[7] S Okada and T Soper ldquoA shortest path problem on a network

with fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[8] L Sujatha and S Elizabeth ldquoFuzzy shortest path problem basedon similarity degreerdquo Applied Mathematical Sciences vol 5 no66 pp 3263ndash3276 2011

[9] J-S Yao and F-T Lin ldquoFuzzy shortest-path network problemswith uncertain edgeweightsrdquo Journal of Information Science andEngineering vol 19 no 2 pp 329ndash351 2003

[10] J-R Yu and T-HWei ldquoSolving the fuzzy shortest path problemby using a linear multiple objective programmingrdquo Journal ofthe Chinese Institute of Industrial Engineers vol 24 no 5 pp360ndash365 2007

[11] S Mukherjee ldquoDijkstrarsquos algorithm for solving the shortest pathproblem on networks under intuitionistic fuzzy environmentrdquoJournal of Mathematical Modelling and Algorithms vol 11 no 4pp 345ndash359 2012

[12] M G Karunambigai P Rangasamy K Atanassov and NPalaniappan ldquoAn intuitionistic fuzzy graph method for findingthe shortest paths in networksrdquoAdvances in SoftComputing vol42 pp 3ndash10 2007

[13] A Nagoor Gani and M Mohammed Jabarulla ldquoOn searchingintuitionistic fuzzy shortest path in a networkrdquo Applied Mathe-matical Sciences vol 4 no 69 pp 3447ndash3454 2010

[14] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[15] K Atanassov Intuitionistic Fuzzy SetsTheory and ApplicationsPhysica 1999

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] D-F Li ldquoA ratio ranking method of triangular intuitionisticfuzzy numbers and its application to MADM problemsrdquo Com-puters and Mathematics with Applications vol 60 no 6 pp1557ndash1570 2010

[18] D F Li J X Nan andM J Zhang ldquoA ranking method of trian-gular intuitionistic fuzzy numbers and application to decisionmakingrdquo International Journal of Computational IntelligenceSystems vol 3 no 5 pp 522ndash530 2010

[19] H B Mitchell ldquoRanking-intuitionistic fuzzy numbersrdquo Inter-national Journal of Uncertainty Fuzziness and Knowlege-BasedSystems vol 12 no 3 pp 377ndash386 2004

[20] S Abbasbandy ldquoRanking of fuzzy numbers some recent andnew formulasrdquo in Proceedings of the IFSA-EUSFLAT pp 642ndash646 Lisbon Portugal July 2009

[21] T Allahviranloo S Abbasbandy and R Saneifard ldquoA methodfor ranking of fuzzy numbers using new weighted distancerdquoMathematical and Computational Applications vol 16 no 2 pp359ndash369 2011

[22] L Q Dat V F Yu and S Y Chou ldquoAn improved rankingmethod for fuzzy numbers using left and right indicesrdquo inProceedings of the 2nd International Conference on ComputerDesign and Engineering (IPCSIT rsquo12) vol 49 pp 89ndash94 2012

[23] B Farhadinia ldquoRanking fuzzy numbers on lexicographicalorderingrdquo International Journal of Applied Mathematics andComputer Sciences vol 5 no 4 pp 248ndash251 2009

[24] N Parandin Araghi and M A Fariborzi ldquoRanking of fuzzynumbers by distance methodrdquo Journal of Applied Mathematicsvol 5 no 19 pp 47ndash55 2008

[25] N R Shankar and P P B Rao ldquoRanking fuzzy numbers with adistance method using circumcenter of centroids and an indexof modalityrdquo Advances in Fuzzy Systems vol 2011 Article ID178308 7 pages 2011

Applied Computational Intelligence and Soft Computing 5

[26] R Saneifard and R Ezzati ldquoA new approach for ranking fuzzynumbers with continuous weighted quasi-arithmatic meansrdquoMathematical Sciences vol 4 no 2 pp 143ndash158 2010

[27] R Biswas ldquoFuzzy numbers redefinedrdquo Information vol 15 no4 pp 1369ndash1380 2012

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Human-ComputerInteraction

Advances in

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2 Applied Computational Intelligence and Soft Computing

in our work here we consider those graphs which are withoutloops Graphs may be of two types undirected graphs anddirected graphs In an undirected graph the edge (119894 119895) andthe edge (119895 119894) if exist are obviously identical unlike that inthe case of directed graph For a fast visit on the theory ofgraphs one could see [1ndash4]

The intuitionistic fuzzy set theory of Atanassov [14ndash16] is now a well-known powerful soft computing tool tothe world scientists If 119883 is a universe of discourse anintuitionistic fuzzy set 119860 in 119883 is a set of ordered triplets119860 = ⟨119909 120583

119860(119909) ]119860(119909)⟩ 119909 isin 119883 where 120583

119860 ]119860 119883 rarr [0 1]

are functions such that 0 le 120583119860(119909) + ]

119860(119909) le 1 for all

119909 isin 119883 For each 119909 isin 119883 the values 120583119860(119909) and ]

119860(119909) represent

the degree of membership and degree of nonmembershipof the element 119909 to 119860 sub 119883 respectively and the amount120587119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) is called the hesitation part Of

course a fuzzy set is a particular case of the intuitionisticfuzzy set if 120587

119860(119909) = 0 for all 119909 isin 119883 For details of the

classical notion of intuitionistic fuzzy set theory one couldsee the book authored by Atanassov [15] The concept of anintuitionistic fuzzy number is of importance for quantifyingan ill-known quantity Intuitionistic fuzzy numbers are themore generalized form of fuzzy numbers involving twoindependently estimated degrees degree of acceptance anddegree of rejection In our work here throughout we usethe notion of triangular intuitionistic fuzzy numbers andthe basic operations like IF addition oplus IF subtraction ⊝ldquorankingrdquo of intuitionistic fuzzy numbers and so forthTrivially any crisp real number can be viewed as a fuzzynumber or as an intuitionistic fuzzy number There is nounique method for ranking the number of intuitionisticfuzzy numbers because all the existing methods [17ndash19] aresoft computing methods Each method has got merits anddemerits depending upon the properties of the applicationdomains and the problem under consideration However if1198601 1198602 1198603 119860

119899are 119899 intuitionistic fuzzy numbers sorted

in IF ascending order (in fact it is a kind of nonascendingorder assuming that the IF equal intuitionistic fuzzy numberstake corresponding positions at random if there is no lossof generality) by any good predecided method that is if1198601≺ 1198602≺ 1198603≺ sdot sdot sdot ≺ 119860

119899 then 119860

1and 119860

119899are called

respectively the IF-min and IF-max of these n intuitionisticfuzzy numbers Almost all the existing methods [17ndash19] ofranking intuitionistic fuzzy numbers were developed inde-pendently that is not as extensions of the existing methods[20ndash26] of ranking of fuzzy numbers [27] Although severalauthors [20ndash26] have reported several ranking methods offuzzy numbers all are having limitations too that is notan absolute method suitable for every application domainHowever if119860

1 1198602 1198603 119860

119899are 119899 fuzzy numbers sorted in

fuzzy ascending order by a predecided method that is 1198601≺

1198602≺ 1198603≺ sdot sdot sdot ≺ 119860

119899 then 119860

1and 119860

119899are called respectively

the fuzzy-min and fuzzy-max of these 119899 fuzzy numbers

3 Graphs with IF Weighted Arcs

In most of the real-life problems of networks be it in acommunication model or transportation model the weights

Approx 46km Approx 37km

Approx 65km

Approx 19km

Approx 15km

B

C D

A

Figure 1 A graph 119866 with IF weights of arcs

of the arcs are not always crisp but intuitionistic fuzzynumbers (or at best fuzzy numbers) For example Figure 3shows a public road transportationmodel for a traveler wherethe cost parameter for traveling each arc has been available tohim as an intuitionistic fuzzy number

But for such type of ill graph there is no attempt madeso far in the literature for searching an IF shortest path Inour method here we solve this intuitionistic fuzzy shortestpath problem (IFSPP) for graphswherewe also use the notionof Dijkstrarsquos algorithm but with simple soft-computationswithout using any hybrid geometric operators using onlybasics of Atanassovrsquos operators [15]

4 Intuitionistic Fuzzy ShortestPath in a Graph

In this section we solve the IFSPP for graphs where we usethe philosophy of Dijkstrarsquos algorithm but with simple soft-computations with IF data Consider a directed graph 119866

where the arcs are of intuitionistic fuzzy weights (intuition-istic fuzzy numbers) as shown in Figure 1 Suppose that thesubalgorithm IF119882(119866) returns the intuitionistic fuzzy weightset119882 corresponding to each arc

41 IF Shortest Path Estimate 119889[V] of a Vertex V in a DirectedGraph As shown in Figure 2 suppose that 119904 is the sourcevertex and the currently traversed vertex is 119906 The IF estimate119889[V] in graph 119866 is computed using IF addition as follows

(IF shortest path estimate of vertex V)

= (IF shortest path estimate of vertex 119906)

oplus (intuitionistic fuzzy number weight

corresponding to the arc from

the vertex 119906 to the vertex V)

(1)

or

119889 [V] = 119889 [119906] oplus 119908119906V (2)

42 Intuitionistic Fuzzy Relaxation of an Arc in a DirectedGraph We extend the classical notion of relaxation to the

Applied Computational Intelligence and Soft Computing 3

Intermediary vertices betweenArc between

Fuzzy shortestdistance between

Fuzzy weight

vertex s and vertex u

vertex s and vertex u

vertex s and vertex u

of the arc u

s u v

Figure 2 IF estimation procedure for 119889[V] in a graph 119866

IFISS (119866 119904)(1) For each vertex V isin 119881[119866](2) 119889[V] = infin(3) V sdot 120587 = NIL(4) 119889[119904] = 0

Algorithm 1

Intermediary vertices

If shortestdistance between

If sh

ortes

t

distan

ce be

twee

n

vertex s and vertex u

verte

x san

d vert

exu

s u

v

Inter

med

iary n

odes IFN

Wuv

betw

een s

and

between s and

Arc between vertices betweenuand

Figure 3 Diagram showing how the IF-RELAX algorithm works ina graph

case here with intuitionistic fuzzy number weights We callit ldquoIF relaxationrdquo For this first of all we initialize the graphalong with its starting vertex and IF shortest path estimatefor each vertices of the graph 119866 The ldquoINTUITIONISTIC-FUZZY-INITIALIZATION-SINGLE-SOURCErdquo algorithmIFISS will do what is shown in Algorithm 1

After the IF initialization the process of IF relaxationof each arc begins as shown in Figure 3 The subalgorithmIF-RELAX plays the vital role to update 119889[V] that is the IFshortest distance value between the starting vertex 119904 and thevertex V which is a neighbour of the current traversed vertex119906 (see Algorithm 2) where119908

119906V isin 119882 is the IF weight of the arcfrom the vertex 119906 to the vertex V and V sdot 120587 denotes the parentnode of vertex V

IF-RELAX (119906 119907119882)

(1) IF 119889[V] ≻ 119889[119906] oplus 119908119906V

(2) THEN 119889[V] larr 119889 [119906] oplus 119908119906V(3) V sdot 120587 larr 119906

Algorithm 2

IFSP (119866 119904)(1) IFISS (119866 119904)(2) 119882larr IF119882(119866)

(3) 119878 larr 0

(4) 119876 larr 119881[119866]

(5) WHILE 119876 = 0

(6) DO 119906 larr EXTRACT-IF-MIN (119876)

(7) 119878 larr 119878 cup 119906

(8) FOR each rn vertex V isin Adj[119906](9) DO IF-RELAX (119906 V119882)

Algorithm 3

43 IF Shortest Path Algorithm (IFSP Algorithm) in a GraphWe now present our main algorithm to find single source IFshortest path in a graph We name this ldquointuitionistic fuzzyshortest path algorithmrdquo that is in short by the title IFSPalgorithm In this algorithm we use the previously designedabove subalgorithms and also the subalgorithm EXTRACT-IF-MIN (119876) which extracts the node 119906 with minimum keyusing a predecided IF ranking method and updates 119876 (seeAlgorithm 3)

Example 1 (an example) Consider the following directedgraph 119866 where the IF weights (here they are intuitionisticfuzzy numbers) are shown against each link as shown inFigure 4 We want to solve the single-source IF shortest pathproblem (IFSPP) taking the vertex119860 as the source vertex andthe vertex119863 as the destination vertex

Our algorithm computes the following results

(1) 119882 = 119908119860119861

= 15 119908119860119862

= 3 119908119861119862

= 1 119908119862119861

= 4 119908119862119863

=

6 119908119861119863

= 2 and then

4 Applied Computational Intelligence and Soft Computing

C

B

A D

15

1 4

2

3 6

AB

BD

CD

AC CB

BC

Figure 4 A graph 119866 with IF weights of arcs

(2) 119878 = 119860 119862 119861119863 that is the IF shortest path from thesource vertex 119860 is

119860 997888rarr 119862 997888rarr 119861 997888rarr 119863 (3)

(3) with 119889-values that is IF shortest distance estimatevalues of each vertex from the starting vertex 119860 are

119889 [119860] = 0 119889 [119862] = TFN 3 119889 [119861] = TFN 7

119889 [119863] = TFN 9

(4)

5 Conclusion

There are many real-life problems in the networks of trans-portation communication circuit systems and so forthwhich are initially modeled into graphs and hence solved Inmany of these directed graphs in reality the weights of thearcs are not always crisp but fuzzy numbers In this paperwe develop a new method to solve the intuitionistic fuzzyshortest path problem (IFSPP) from a source vertex to adestination vertex in a directed graphThe importance of ourmethod lies in its potential to give solution in intuitionisticfuzzy environment unlike any of the existing algorithms ofIFSPP Obviously our algorithm does also work in case fewor all of the weights are fuzzy numbers or crisp numbers as aspecial case of IF numbers

References

[1] V K BalakrishnanGraphTheory McGraw-Hill NewYork NYUSA 1997

[2] B Bollobas Modern Graph Theory Springer New York NYUSA 2002

[3] R Diestel GraphTheory Springer 2000[4] F Harary Graph Theory Addison Wesley Boston Mass USA

1995[5] D Dubois and H Prade Fuzzy Sets and Systems Academic

Press New York NY USA 1980[6] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol

39 no 1 pp 27ndash41 1991[7] S Okada and T Soper ldquoA shortest path problem on a network

with fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[8] L Sujatha and S Elizabeth ldquoFuzzy shortest path problem basedon similarity degreerdquo Applied Mathematical Sciences vol 5 no66 pp 3263ndash3276 2011

[9] J-S Yao and F-T Lin ldquoFuzzy shortest-path network problemswith uncertain edgeweightsrdquo Journal of Information Science andEngineering vol 19 no 2 pp 329ndash351 2003

[10] J-R Yu and T-HWei ldquoSolving the fuzzy shortest path problemby using a linear multiple objective programmingrdquo Journal ofthe Chinese Institute of Industrial Engineers vol 24 no 5 pp360ndash365 2007

[11] S Mukherjee ldquoDijkstrarsquos algorithm for solving the shortest pathproblem on networks under intuitionistic fuzzy environmentrdquoJournal of Mathematical Modelling and Algorithms vol 11 no 4pp 345ndash359 2012

[12] M G Karunambigai P Rangasamy K Atanassov and NPalaniappan ldquoAn intuitionistic fuzzy graph method for findingthe shortest paths in networksrdquoAdvances in SoftComputing vol42 pp 3ndash10 2007

[13] A Nagoor Gani and M Mohammed Jabarulla ldquoOn searchingintuitionistic fuzzy shortest path in a networkrdquo Applied Mathe-matical Sciences vol 4 no 69 pp 3447ndash3454 2010

[14] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[15] K Atanassov Intuitionistic Fuzzy SetsTheory and ApplicationsPhysica 1999

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] D-F Li ldquoA ratio ranking method of triangular intuitionisticfuzzy numbers and its application to MADM problemsrdquo Com-puters and Mathematics with Applications vol 60 no 6 pp1557ndash1570 2010

[18] D F Li J X Nan andM J Zhang ldquoA ranking method of trian-gular intuitionistic fuzzy numbers and application to decisionmakingrdquo International Journal of Computational IntelligenceSystems vol 3 no 5 pp 522ndash530 2010

[19] H B Mitchell ldquoRanking-intuitionistic fuzzy numbersrdquo Inter-national Journal of Uncertainty Fuzziness and Knowlege-BasedSystems vol 12 no 3 pp 377ndash386 2004

[20] S Abbasbandy ldquoRanking of fuzzy numbers some recent andnew formulasrdquo in Proceedings of the IFSA-EUSFLAT pp 642ndash646 Lisbon Portugal July 2009

[21] T Allahviranloo S Abbasbandy and R Saneifard ldquoA methodfor ranking of fuzzy numbers using new weighted distancerdquoMathematical and Computational Applications vol 16 no 2 pp359ndash369 2011

[22] L Q Dat V F Yu and S Y Chou ldquoAn improved rankingmethod for fuzzy numbers using left and right indicesrdquo inProceedings of the 2nd International Conference on ComputerDesign and Engineering (IPCSIT rsquo12) vol 49 pp 89ndash94 2012

[23] B Farhadinia ldquoRanking fuzzy numbers on lexicographicalorderingrdquo International Journal of Applied Mathematics andComputer Sciences vol 5 no 4 pp 248ndash251 2009

[24] N Parandin Araghi and M A Fariborzi ldquoRanking of fuzzynumbers by distance methodrdquo Journal of Applied Mathematicsvol 5 no 19 pp 47ndash55 2008

[25] N R Shankar and P P B Rao ldquoRanking fuzzy numbers with adistance method using circumcenter of centroids and an indexof modalityrdquo Advances in Fuzzy Systems vol 2011 Article ID178308 7 pages 2011

Applied Computational Intelligence and Soft Computing 5

[26] R Saneifard and R Ezzati ldquoA new approach for ranking fuzzynumbers with continuous weighted quasi-arithmatic meansrdquoMathematical Sciences vol 4 no 2 pp 143ndash158 2010

[27] R Biswas ldquoFuzzy numbers redefinedrdquo Information vol 15 no4 pp 1369ndash1380 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 3

Intermediary vertices betweenArc between

Fuzzy shortestdistance between

Fuzzy weight

vertex s and vertex u

vertex s and vertex u

vertex s and vertex u

of the arc u

s u v

Figure 2 IF estimation procedure for 119889[V] in a graph 119866

IFISS (119866 119904)(1) For each vertex V isin 119881[119866](2) 119889[V] = infin(3) V sdot 120587 = NIL(4) 119889[119904] = 0

Algorithm 1

Intermediary vertices

If shortestdistance between

If sh

ortes

t

distan

ce be

twee

n

vertex s and vertex u

verte

x san

d vert

exu

s u

v

Inter

med

iary n

odes IFN

Wuv

betw

een s

and

between s and

Arc between vertices betweenuand

Figure 3 Diagram showing how the IF-RELAX algorithm works ina graph

case here with intuitionistic fuzzy number weights We callit ldquoIF relaxationrdquo For this first of all we initialize the graphalong with its starting vertex and IF shortest path estimatefor each vertices of the graph 119866 The ldquoINTUITIONISTIC-FUZZY-INITIALIZATION-SINGLE-SOURCErdquo algorithmIFISS will do what is shown in Algorithm 1

After the IF initialization the process of IF relaxationof each arc begins as shown in Figure 3 The subalgorithmIF-RELAX plays the vital role to update 119889[V] that is the IFshortest distance value between the starting vertex 119904 and thevertex V which is a neighbour of the current traversed vertex119906 (see Algorithm 2) where119908

119906V isin 119882 is the IF weight of the arcfrom the vertex 119906 to the vertex V and V sdot 120587 denotes the parentnode of vertex V

IF-RELAX (119906 119907119882)

(1) IF 119889[V] ≻ 119889[119906] oplus 119908119906V

(2) THEN 119889[V] larr 119889 [119906] oplus 119908119906V(3) V sdot 120587 larr 119906

Algorithm 2

IFSP (119866 119904)(1) IFISS (119866 119904)(2) 119882larr IF119882(119866)

(3) 119878 larr 0

(4) 119876 larr 119881[119866]

(5) WHILE 119876 = 0

(6) DO 119906 larr EXTRACT-IF-MIN (119876)

(7) 119878 larr 119878 cup 119906

(8) FOR each rn vertex V isin Adj[119906](9) DO IF-RELAX (119906 V119882)

Algorithm 3

43 IF Shortest Path Algorithm (IFSP Algorithm) in a GraphWe now present our main algorithm to find single source IFshortest path in a graph We name this ldquointuitionistic fuzzyshortest path algorithmrdquo that is in short by the title IFSPalgorithm In this algorithm we use the previously designedabove subalgorithms and also the subalgorithm EXTRACT-IF-MIN (119876) which extracts the node 119906 with minimum keyusing a predecided IF ranking method and updates 119876 (seeAlgorithm 3)

Example 1 (an example) Consider the following directedgraph 119866 where the IF weights (here they are intuitionisticfuzzy numbers) are shown against each link as shown inFigure 4 We want to solve the single-source IF shortest pathproblem (IFSPP) taking the vertex119860 as the source vertex andthe vertex119863 as the destination vertex

Our algorithm computes the following results

(1) 119882 = 119908119860119861

= 15 119908119860119862

= 3 119908119861119862

= 1 119908119862119861

= 4 119908119862119863

=

6 119908119861119863

= 2 and then

4 Applied Computational Intelligence and Soft Computing

C

B

A D

15

1 4

2

3 6

AB

BD

CD

AC CB

BC

Figure 4 A graph 119866 with IF weights of arcs

(2) 119878 = 119860 119862 119861119863 that is the IF shortest path from thesource vertex 119860 is

119860 997888rarr 119862 997888rarr 119861 997888rarr 119863 (3)

(3) with 119889-values that is IF shortest distance estimatevalues of each vertex from the starting vertex 119860 are

119889 [119860] = 0 119889 [119862] = TFN 3 119889 [119861] = TFN 7

119889 [119863] = TFN 9

(4)

5 Conclusion

There are many real-life problems in the networks of trans-portation communication circuit systems and so forthwhich are initially modeled into graphs and hence solved Inmany of these directed graphs in reality the weights of thearcs are not always crisp but fuzzy numbers In this paperwe develop a new method to solve the intuitionistic fuzzyshortest path problem (IFSPP) from a source vertex to adestination vertex in a directed graphThe importance of ourmethod lies in its potential to give solution in intuitionisticfuzzy environment unlike any of the existing algorithms ofIFSPP Obviously our algorithm does also work in case fewor all of the weights are fuzzy numbers or crisp numbers as aspecial case of IF numbers

References

[1] V K BalakrishnanGraphTheory McGraw-Hill NewYork NYUSA 1997

[2] B Bollobas Modern Graph Theory Springer New York NYUSA 2002

[3] R Diestel GraphTheory Springer 2000[4] F Harary Graph Theory Addison Wesley Boston Mass USA

1995[5] D Dubois and H Prade Fuzzy Sets and Systems Academic

Press New York NY USA 1980[6] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol

39 no 1 pp 27ndash41 1991[7] S Okada and T Soper ldquoA shortest path problem on a network

with fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[8] L Sujatha and S Elizabeth ldquoFuzzy shortest path problem basedon similarity degreerdquo Applied Mathematical Sciences vol 5 no66 pp 3263ndash3276 2011

[9] J-S Yao and F-T Lin ldquoFuzzy shortest-path network problemswith uncertain edgeweightsrdquo Journal of Information Science andEngineering vol 19 no 2 pp 329ndash351 2003

[10] J-R Yu and T-HWei ldquoSolving the fuzzy shortest path problemby using a linear multiple objective programmingrdquo Journal ofthe Chinese Institute of Industrial Engineers vol 24 no 5 pp360ndash365 2007

[11] S Mukherjee ldquoDijkstrarsquos algorithm for solving the shortest pathproblem on networks under intuitionistic fuzzy environmentrdquoJournal of Mathematical Modelling and Algorithms vol 11 no 4pp 345ndash359 2012

[12] M G Karunambigai P Rangasamy K Atanassov and NPalaniappan ldquoAn intuitionistic fuzzy graph method for findingthe shortest paths in networksrdquoAdvances in SoftComputing vol42 pp 3ndash10 2007

[13] A Nagoor Gani and M Mohammed Jabarulla ldquoOn searchingintuitionistic fuzzy shortest path in a networkrdquo Applied Mathe-matical Sciences vol 4 no 69 pp 3447ndash3454 2010

[14] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[15] K Atanassov Intuitionistic Fuzzy SetsTheory and ApplicationsPhysica 1999

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] D-F Li ldquoA ratio ranking method of triangular intuitionisticfuzzy numbers and its application to MADM problemsrdquo Com-puters and Mathematics with Applications vol 60 no 6 pp1557ndash1570 2010

[18] D F Li J X Nan andM J Zhang ldquoA ranking method of trian-gular intuitionistic fuzzy numbers and application to decisionmakingrdquo International Journal of Computational IntelligenceSystems vol 3 no 5 pp 522ndash530 2010

[19] H B Mitchell ldquoRanking-intuitionistic fuzzy numbersrdquo Inter-national Journal of Uncertainty Fuzziness and Knowlege-BasedSystems vol 12 no 3 pp 377ndash386 2004

[20] S Abbasbandy ldquoRanking of fuzzy numbers some recent andnew formulasrdquo in Proceedings of the IFSA-EUSFLAT pp 642ndash646 Lisbon Portugal July 2009

[21] T Allahviranloo S Abbasbandy and R Saneifard ldquoA methodfor ranking of fuzzy numbers using new weighted distancerdquoMathematical and Computational Applications vol 16 no 2 pp359ndash369 2011

[22] L Q Dat V F Yu and S Y Chou ldquoAn improved rankingmethod for fuzzy numbers using left and right indicesrdquo inProceedings of the 2nd International Conference on ComputerDesign and Engineering (IPCSIT rsquo12) vol 49 pp 89ndash94 2012

[23] B Farhadinia ldquoRanking fuzzy numbers on lexicographicalorderingrdquo International Journal of Applied Mathematics andComputer Sciences vol 5 no 4 pp 248ndash251 2009

[24] N Parandin Araghi and M A Fariborzi ldquoRanking of fuzzynumbers by distance methodrdquo Journal of Applied Mathematicsvol 5 no 19 pp 47ndash55 2008

[25] N R Shankar and P P B Rao ldquoRanking fuzzy numbers with adistance method using circumcenter of centroids and an indexof modalityrdquo Advances in Fuzzy Systems vol 2011 Article ID178308 7 pages 2011

Applied Computational Intelligence and Soft Computing 5

[26] R Saneifard and R Ezzati ldquoA new approach for ranking fuzzynumbers with continuous weighted quasi-arithmatic meansrdquoMathematical Sciences vol 4 no 2 pp 143ndash158 2010

[27] R Biswas ldquoFuzzy numbers redefinedrdquo Information vol 15 no4 pp 1369ndash1380 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Applied Computational Intelligence and Soft Computing

C

B

A D

15

1 4

2

3 6

AB

BD

CD

AC CB

BC

Figure 4 A graph 119866 with IF weights of arcs

(2) 119878 = 119860 119862 119861119863 that is the IF shortest path from thesource vertex 119860 is

119860 997888rarr 119862 997888rarr 119861 997888rarr 119863 (3)

(3) with 119889-values that is IF shortest distance estimatevalues of each vertex from the starting vertex 119860 are

119889 [119860] = 0 119889 [119862] = TFN 3 119889 [119861] = TFN 7

119889 [119863] = TFN 9

(4)

5 Conclusion

There are many real-life problems in the networks of trans-portation communication circuit systems and so forthwhich are initially modeled into graphs and hence solved Inmany of these directed graphs in reality the weights of thearcs are not always crisp but fuzzy numbers In this paperwe develop a new method to solve the intuitionistic fuzzyshortest path problem (IFSPP) from a source vertex to adestination vertex in a directed graphThe importance of ourmethod lies in its potential to give solution in intuitionisticfuzzy environment unlike any of the existing algorithms ofIFSPP Obviously our algorithm does also work in case fewor all of the weights are fuzzy numbers or crisp numbers as aspecial case of IF numbers

References

[1] V K BalakrishnanGraphTheory McGraw-Hill NewYork NYUSA 1997

[2] B Bollobas Modern Graph Theory Springer New York NYUSA 2002

[3] R Diestel GraphTheory Springer 2000[4] F Harary Graph Theory Addison Wesley Boston Mass USA

1995[5] D Dubois and H Prade Fuzzy Sets and Systems Academic

Press New York NY USA 1980[6] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol

39 no 1 pp 27ndash41 1991[7] S Okada and T Soper ldquoA shortest path problem on a network

with fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[8] L Sujatha and S Elizabeth ldquoFuzzy shortest path problem basedon similarity degreerdquo Applied Mathematical Sciences vol 5 no66 pp 3263ndash3276 2011

[9] J-S Yao and F-T Lin ldquoFuzzy shortest-path network problemswith uncertain edgeweightsrdquo Journal of Information Science andEngineering vol 19 no 2 pp 329ndash351 2003

[10] J-R Yu and T-HWei ldquoSolving the fuzzy shortest path problemby using a linear multiple objective programmingrdquo Journal ofthe Chinese Institute of Industrial Engineers vol 24 no 5 pp360ndash365 2007

[11] S Mukherjee ldquoDijkstrarsquos algorithm for solving the shortest pathproblem on networks under intuitionistic fuzzy environmentrdquoJournal of Mathematical Modelling and Algorithms vol 11 no 4pp 345ndash359 2012

[12] M G Karunambigai P Rangasamy K Atanassov and NPalaniappan ldquoAn intuitionistic fuzzy graph method for findingthe shortest paths in networksrdquoAdvances in SoftComputing vol42 pp 3ndash10 2007

[13] A Nagoor Gani and M Mohammed Jabarulla ldquoOn searchingintuitionistic fuzzy shortest path in a networkrdquo Applied Mathe-matical Sciences vol 4 no 69 pp 3447ndash3454 2010

[14] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[15] K Atanassov Intuitionistic Fuzzy SetsTheory and ApplicationsPhysica 1999

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] D-F Li ldquoA ratio ranking method of triangular intuitionisticfuzzy numbers and its application to MADM problemsrdquo Com-puters and Mathematics with Applications vol 60 no 6 pp1557ndash1570 2010

[18] D F Li J X Nan andM J Zhang ldquoA ranking method of trian-gular intuitionistic fuzzy numbers and application to decisionmakingrdquo International Journal of Computational IntelligenceSystems vol 3 no 5 pp 522ndash530 2010

[19] H B Mitchell ldquoRanking-intuitionistic fuzzy numbersrdquo Inter-national Journal of Uncertainty Fuzziness and Knowlege-BasedSystems vol 12 no 3 pp 377ndash386 2004

[20] S Abbasbandy ldquoRanking of fuzzy numbers some recent andnew formulasrdquo in Proceedings of the IFSA-EUSFLAT pp 642ndash646 Lisbon Portugal July 2009

[21] T Allahviranloo S Abbasbandy and R Saneifard ldquoA methodfor ranking of fuzzy numbers using new weighted distancerdquoMathematical and Computational Applications vol 16 no 2 pp359ndash369 2011

[22] L Q Dat V F Yu and S Y Chou ldquoAn improved rankingmethod for fuzzy numbers using left and right indicesrdquo inProceedings of the 2nd International Conference on ComputerDesign and Engineering (IPCSIT rsquo12) vol 49 pp 89ndash94 2012

[23] B Farhadinia ldquoRanking fuzzy numbers on lexicographicalorderingrdquo International Journal of Applied Mathematics andComputer Sciences vol 5 no 4 pp 248ndash251 2009

[24] N Parandin Araghi and M A Fariborzi ldquoRanking of fuzzynumbers by distance methodrdquo Journal of Applied Mathematicsvol 5 no 19 pp 47ndash55 2008

[25] N R Shankar and P P B Rao ldquoRanking fuzzy numbers with adistance method using circumcenter of centroids and an indexof modalityrdquo Advances in Fuzzy Systems vol 2011 Article ID178308 7 pages 2011

Applied Computational Intelligence and Soft Computing 5

[26] R Saneifard and R Ezzati ldquoA new approach for ranking fuzzynumbers with continuous weighted quasi-arithmatic meansrdquoMathematical Sciences vol 4 no 2 pp 143ndash158 2010

[27] R Biswas ldquoFuzzy numbers redefinedrdquo Information vol 15 no4 pp 1369ndash1380 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 5

[26] R Saneifard and R Ezzati ldquoA new approach for ranking fuzzynumbers with continuous weighted quasi-arithmatic meansrdquoMathematical Sciences vol 4 no 2 pp 143ndash158 2010

[27] R Biswas ldquoFuzzy numbers redefinedrdquo Information vol 15 no4 pp 1369ndash1380 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014