Research ArticleNovel Applications of Intuitionistic Fuzzy Digraphs inDecision Support Systems

Muhammad Akram1 Ather Ashraf2 and Mansoor Sarwar2

1 Department of Mathematics University of the Punjab New Campus Lahore Pakistan2 Punjab University College of Information Technology University of the Punjab Old Campus Lahore 54000 Pakistan

Correspondence should be addressed to Muhammad Akram makrampucitedupk

Received 20 April 2014 Accepted 21 May 2014 Published 16 June 2014

Academic Editor Feng Feng

Copyright copy 2014 Muhammad Akram et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Many problems of practical interest can bemodeled and solved by using graph algorithms In general graph theory has a wide rangeof applications in diverse fields In this paper the intuitionistic fuzzy organizational and neural networkmodels intuitionistic fuzzyneurons in medical diagnosis intuitionistic fuzzy digraphs in vulnerability assessment of gas pipeline networks and intuitionisticfuzzy digraphs in travel time are presented as examples of intuitionistic fuzzy digraphs in decision support system We have alsodesigned and implemented the algorithms for these decision support systems

1 Introduction

Graph theory is an extremely useful tool in solving combina-torial problems in different areas including geometry algebranumber theory topology operations research optimizationcomputer science engineering and physical biological andsocial systems Point-to-point interconnection networks forparallel and distributed systems are usually modeled bydirected graphs (or digraphs) A digraph is a graph whoseedges have directions and are called arcs (edges) Arrows onthe arcs are used to encode the directional information an arcfrom vertex (node) 119909 to vertex 119910 indicates that onemaymovefrom 119909 to 119910 but not from 119910 to 119909

Presently science and technology are featured with com-plex processes and phenomena for which complete informa-tion is not always available For such cases mathematicalmodels are developed to handle types of systems containingelements of uncertainty A large number of these models arebased on an extension of the ordinary set theory namelyfuzzy sets The notion of fuzzy sets was introduced by Zadeh[1] as a method of representing uncertainty and vaguenessSince then the theory of fuzzy sets has become a vigorousarea of research in different disciplines including medicaland life sciences management sciences social sciencesengineering statistics graph theory artificial intelligence

signal processing multiagent systems pattern recognitionrobotics computer networks expert systems decision mak-ing and automata theory

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems Kauffmanrsquos initialdefinition of a fuzzy graph [2] was based on Zadehrsquos fuzzyrelations [3] Rosenfeld [4] introduced the fuzzy analogueof several basic graph-theoretic concepts and Bhattacharya[5] gave some remarks on fuzzy graphs Mordeson andNair [6] defined the concept of complement of fuzzy graphand studied some operations on fuzzy graphs In [7] thedefinition of complement of a fuzzy graph was modified sothat the complement of the complement is the original fuzzygraph which agrees with the crisp graph case Atanassov [8]introduced the concept of intuitionistic fuzzy relations andintuitionistic fuzzy graphs Akram et al [9ndash11] introducedmany new concepts including strong intuitionistic fuzzygraphs intuitionistic fuzzy hypergraphs intuitionistic fuzzycycles and intuitionistic fuzzy trees Wu [12] discussed fuzzydigraphs In this paper the intuitionistic fuzzy organizational

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 904606 11 pageshttpdxdoiorg1011552014904606

2 The Scientific World Journal

neural network models intuitionistic fuzzy neurons in med-ical diagnosis intuitionistic fuzzy digraphs in vulnerabilityassessment of gas pipeline networks and intuitionistic fuzzydigraphs in travel time are presented as examples of intuition-istic fuzzy digraphs in decision support systems Algorithmsof these decision support systems are also designed andimplemented

2 Preliminaries

A digraph is a pair 119866lowast = (119881 119864) where 119881 is a finite set and119864 sube 119881 times 119881 Let 119866lowast

1= (1198811 1198641) and 119866lowast

2= (1198812 1198642) be two

digraphsTheCartesian product of119866lowast1and119866lowast

2gives a digraph

119866lowast

1times 119866lowast

2= (119881 119864) with 119881 = 119881

1times 1198812and

119864 = (119909 1199092) 997888rarr (119909 119910

2) | 119909 isin 119881

1 1199092997888rarr 1199102isin 1198642

cup (1199091 119911) 997888rarr (119910

1 119911) | 119909

1997888rarr 1199101isin 1198641 119911 isin 119881

2

(1)

In this paper we will write 119909119910 isin 119864 to mean 119909 rarr 119910 isin 119864 andif 119890 = 119909119910 isin 119864 we say 119909 and 119910 are adjacent such that 119909 is astarting node and 119910 is an ending node

Definition 1 (see [1 3]) A fuzzy subset 120583 on a set 119883 is a map120583 119883 rarr [0 1] A fuzzy binary relation on119883 is a fuzzy subset120583 on 119883 times 119883 By a fuzzy relation we mean a fuzzy binaryrelation given by 120583 119883 times 119883 rarr [0 1]

Definition 2 (see [12]) Let 119881 be a finite set 119860 = ⟨119881 120583119860⟩ a

fuzzy set of119881 and 119861 = ⟨119881times119881 120583119861⟩ a fuzzy relation on119881 then

the ordered pair (119860 119861) is called a fuzzy digraph

In 1983 Atanassov [13] introduced the concept of intu-itionistic fuzzy sets as a generalization of fuzzy sets [1]Atanassov added a new component (which determines thedegree of nonmembership) in the definition of fuzzy setThe fuzzy sets give the degree of membership of an elementin a given set (and the nonmembership degree equals oneminus the degree of membership) while intuitionistic fuzzysets give both a degree of membership and a degree ofnonmembership which are more or less independent fromeach other the only requirement is that the sum of these twodegrees is not greater than 1

Definition 3 (see [8]) An intuitionistic fuzzy set (IFS) on auniverse119883 is an object of the form

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ 119909 isin 119883 (2)

where 120583119860(119909)(isin [0 1]) is called degree of membership of 119909

in 119860 ]119860(119909)(isin [0 1]) is called degree of nonmembership of

119909 in 119860 and 120583119860and ]119860satisfy the following condition for all

119909 isin 119883 120583119860(119909) + ]

119860(119909) le 1

Definition 4 An intuitionistic fuzzy relation 119877 =

(120583119877(119909 119910) ]

119877(119909 119910)) in a universe 119883 times 119884 (119877(119883 rarr 119884))

is an intuitionistic fuzzy set of the form119877 = ⟨(119909 119910) 120583

119860(119909 119910) ]

119860(119909 119910)⟩ | (119909 119910) isin 119883 times 119884 (3)

where 120583119860 119883 times 119884 rarr [0 1] and ]

119860 119883 times 119884 rarr [0 1] The

intuitionistic fuzzy relation119877 satisfies 120583119877(119909 119910)+]

119877(119909 119910) le 1

for all 119909 119910 isin 119883

Definition 5 Let 119877 be an intuitionistic fuzzy relation on uni-verse 119883 Then 119877 is called an intuitionistic fuzzy equivalencerelation on119883 if it satisfies the following conditions

(a) 119877 is intuitionistic fuzzy reflexive that is 119877(119909 119909) =(1 0) for each 119909 isin 119883

(b) 119877 is intuitionistic fuzzy symmetric that is 119877(119909 119910) =119877(119910 119909) for any 119909 119910 isin 119883

(c) 119877 is intuitionistic fuzzy transitive that is 119877(119909 119911) ge⋁119910(119877(119909 119910) and 119877(119910 119911))

Definition 6 Let 119876 (119883 rarr 119884) and 119877 (119884 rarr 119885) be twointuitionistic fuzzy relationsThemax-min-max composition119877 ∘ 119876 (119883 rarr 119885) is the intuitionistic fuzzy relation defined bythe membership function

120583119877∘119876(119909 119911) = ⋁

119910

(120583119876(119909 119910) and 120583

119877(119910 119911)) (4)

and the nonmembership function

]119877∘119876(119909 119911) = ⋀

119910

(]119876(119909 119910) or ]

119877(119910 119911)) (5)

for all (119909 119911) isin 119883 times 119885 and for all 119910 isin 119884

Definition 7 Let 119876 (119883 rarr 119884) and 119877 (119884 rarr 119885) be twointuitionistic fuzzy relations The max-product-min-productcomposition119877∘119876 (119883 rarr 119885) is the intuitionistic fuzzy relationdefined by the membership function

120583119877∘119876(119909 119911) = ⋁

119910

(120583119876(119909 119910) sdot 120583

119877(119910 119911)) (6)

and the nonmembership function

]119877∘119876(119909 119911) = ⋀

119910

(]119876(119909 119910) sdot ]

119877(119910 119911)) (7)

for all (119909 119911) isin 119883 times 119885 and for all 119910 isin 119884

Throughout this paper we denote 119866lowast a crisp simpledigraph and 119866 an intuitionistic fuzzy digraph

3 Applications of Intuitionistic FuzzyDigraphs in Decision Support Systems

Definition 8 An intuitionistic fuzzy digraph of a digraph 119866lowastis a pair 119866 = (119860 119861) where 119860 = ⟨119881 120583

119860 ]119860⟩ is an intuitionistic

fuzzy set in119881 and119861 = ⟨119881times119881 120583119861 ]119861⟩ is an intuitionistic fuzzy

relation on 119881 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(8)

and 0 le 120583119861(119909119910) + ]

119861(119909119910) le 1 for all 119909 119910 isin 119881 We note that 119861

may not be symmetric relation

Example 9 Consider a graph 119866lowast = (119881 119864) such that 119881 =

V1 V2 V3 V4 and 119864 = V

1V2 V2V3 V3V4 V4V1 sube 119881 times 119881 Let 119860

The Scientific World Journal 3

1 2

34

(04 01)

(01 01)

(04 01)(02 01)

(03 01)

(06 02)

(05 02) (05 01)

Figure 1 Intuitionistic fuzzy digraph

be an intuitionistic fuzzy set of119881 and let 119861 be an intuitionisticfuzzy relation on 119881 defined by

V1

V2

V3

V4

12058311986004 06 05 05

]11986001 02 01 02

V1V2V2V3V3V4V4V1

12058311986101 02 03 04

]11986101 01 01 01

(9)

By routine computations it is easy to see from Figure 1that 119866 = (119860 119861) is an intuitionistic fuzzy digraph of 119866lowastThe intuitionistic fuzzy digraph119866 is represented by adjacencymatrix given below

119860 =

[

[

[

[

(00 10) (01 01) (00 10) (00 10)

(00 10) (00 10) (02 01) (00 10)

(00 10) (00 10) (00 10) (03 01)

(04 01) (00 10) (00 10) (00 10)

]

]

]

]

(10)

We now present several applications of intuitionistic fuzzydigraphs in decision support systems in the areas of manage-ment marketing medical diagnosis gas pipeline networksand transportation

31 Intuitionistic Fuzzy Organizational Model In this subsec-tion we explore an intuitionistic fuzzy graph model to findout themost influential personwithin an organization whichis called influence graph In an influence graph verticesrepresent an employee and edges represent the influence ofan employee on another employee of a company Such graphshave applications in modeling social structures communica-tion and distributed computing

We consider an organization having employees and theirdesignation as shown in Table 1 For this organization the setof employee is 119864 = BODMQTMMZAKRB St

Upon some investigation we discover the following(i) Mujeeb Qayyum has worked with Munib Zia for

over 10 years and he values his input on strategicinitiatives

(ii) The board of directors is chaired by a long timeassociate of Munib Zia Like Mujeeb the chair ofboard also values Munib

Table 1 Names of employees in an organization and their designa-tions

Name DesignationBoard of Directors (BOD) Board of DirectorsMujeeb Qayyum (MQ) CEOTahir Mahmood (TM) CTOMunib Zia (MZ) Director of MarketingArif Kaleem (AK) Director of Product DevelopmentRizwan Bashir (RB) Director of Human ResourcesStaff (St) Staff

Table 2 Power of employees in terms of membership degree andnonmembership degree

BOD MQ TM MZ AK RB St120583119860

09 09 08 07 06 06 05]119860

00 00 01 01 03 03 03

(iii) For reorganization the entire marketing and HRteam will be very important Rizwan Bashir will beespecially important

(iv) Tahir Mahmood and Rizwan Bashir have a history ofconflict

(v) Tahir Mahmood has great influence on the develop-ment team

Considering the above points an influence graph can bedeveloped but such a graph cannot represent the power ofemployees within an organization and the degree of influenceof employees on each other As the powers and influencehave no defined boundaries it is desired to represent themin the form of fuzzy set The fuzzy digraph representsthe influence of employees on each other but there is afair chance of the existence of nonnull hesitation part ateach moment of evaluation of influence We apply here theconcept of intuitionistic fuzzy set which is more preciseabout the influence and conflicts between the employeesTheintuitionistic fuzzy set of the employees is as follows

We represent the influence in the intuitionistic fuzzydigraph by an edgeThe resultant intuitionistic fuzzy digraphis shown in Figure 2 and corresponding adjacency matrix isshown in Table 3

The nodes of intuitionistic fuzzy digraph in Figure 2represent the employee and its power in terms of degree ofmembership and nonmembership which can be interpretedas percentage for example MQ possesses 90 power withinthe organization (Table 2) Similarly the edges of an intu-itionistic fuzzy digraph represent the influence of one personon another person that is end nodes of edges The degree ofmembership and nonmembership can be interpreted as thepercentage of positive and negative influence for example60 of the time BOD works on MZrsquos opinion but 10 of thetime they do not follow his opinion

In Figure 2 it is clear that MZ has influence both onBOD and on MQ He can influence both of them equallyas the degree of membership in both cases is 06 that is60 But in case of MQ the degree of hesitation is 04 thatis (120587 = 1ndash06ndash00) and in case of BOQ it is 03 that is

4 The Scientific World Journal

Table 3 Adjacency matrix corresponding to Figure 2

BOD MQ TM MZ AK RB StBOD (00 10) (08 00) (00 10) (00 10) (00 10) (00 10) (00 10)MQ (00 10) (00 10) (07 00) (00 10) (00 10) (00 10) (02 03)TM (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (04 03)MZ (06 01) (06 00) (00 10) (00 10) (00 10) (00 10) (03 03)AK (00 10) (00 10) (00 10) (00 10) (00 10) (05 03) (03 03)RB (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (02 03)St (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

BOD (09 00) MZ (07 01) St (05 03) RB (06 03)

MQ (09 00) TM (08 01) AK (06 03)

(02 03)

(06 00)(08 00)

(02 03)

(03 03)(06 01)

(04 03)

(03 03)

(05 03)

(07 00)

Figure 2 Influence intuitionistic fuzzy digraph

(120587 = 1ndash06ndash01) which means that the hesitation in case ofBOQ is more than that of MQ But it is quite obvious thatMZ is themost influential employee in the organization Alsonote that no other employee has influence on both BOD andMQ each possessing 90 power within the organization

32 Intuitionistic Fuzzy Neurons in Medical Diagnosis Thefield of medicine is one of the most fruitful and interestingareas of application for intuitionistic fuzzy set theory Inthe discrimination analysis for diagnosis of an illness thesymptoms are ranked according to the grade of discrimi-nation of each disease by a particular symptom A properbase knowledge is required in the medical diagnosis of asymptom In this section we use intuitionistic fuzzy elementfor knowledge base

Consider the following set of diseasesdiagnoses 119863 andset of symptoms 119878

119863 = DiabetesDengueTuberculosis

119878 = Temperature InsulinBlood pressure

Blood plateletsCough

(11)

The intuitionistic fuzzy relation 119876 (119863 rarr 119878) is shown inTable 4

Consider the set of patients as119875 = Fayyaz Amir AslamThe intuitionistic fuzzy relation 119877 (119878 rarr 119875) is given inTable 5

The max product composition 119879 = 119877 ∘ 119876 is shown inTable 6

By applying Algorithm 1 on Table 6 it is identified thatFayyaz is suffering from Dengue Amir is suffering fromDiabetes and Aslam is a patient of Tuberculosis

In the first portion of Algorithm 1 from lines 1ndash5 theparameters are set and max product composition is calcu-lated Each disease is ranked using 119878

119879 that is 119878

119879= 120583119894 minus ]119894 lowast

120587119894 in a loop The patient is suffering from the disease havingmaximum 119878

119879 If two diseases have the same 119878

119879 then their

hesitation function is tested that is 120587 = 1 minus ]119894 The diseasehaving less hesitation is selected

33 Architecture of Intuitionistic Fuzzy Neurons inMarketabil-ity The marketability of a book can be studied based onthree criteria that is pictures 119875 cost 119862 and examples 119864 Itis known that if a book has more examples low cost anda large numbers of pictures the sale of the book improves

The Scientific World Journal 5

(1) void diagnoseDisease() (2) 119877 = Patient symptoms(3) 119876 = Relation between disease and symptoms(4) 119879 = 119876 ∘ 119877(5) 119878max = 0(6) 119888119900119906119899119905119863119894119904119890119886119904119890 = 119879height(7) for (int 119894 = 0 119894 le 119888119900119906119899119905119863119894119904119890119886119904119890 119894++) (8) 119878

119894= 120583119894 minus ]119894 lowast 120587119894

(9) if (119878max == 119878119894) (10) if (degree of hesitation of 119878max ⩽ degree of hesitation of 119878

119894)

(11) 119878max = 119878119894(12) Diagnose = disease(13)

(14) else if (119878max ⩽ 119878119894) (15) 119878max = 119878119894(16) Diagnose = disease(17)

(18)

(19) print diagnose(20) if (diagnose is not accepted by the doctor) (21) Modify 119876(22) diagnoseDisease()(23)

(24)

Algorithm 1 IF neurons in medical diagnosis

Table 4 Intuitionistic fuzzy relation 119876(119863 rarr 119878)

119876 Temperature Insulin Blood pressure Blood platelets CoughDiabetes (02 08) (09 01) (01 08) (01 08) (01 08)Dengue (09 01) (00 08) (08 01) (09 01) (01 08)Tuberculosis (06 02) (00 09) (04 04) (00 08) (09 01)

Table 5 Intuitionistic fuzzy relation 119877(119878 rarr 119875)

119877 Fayyaz Amir AslamTemperature (08 01) (06 02) (04 04)Insulin (02 06) (09 01) (02 07)Blood pressure (04 04) (01 08) (01 07)Blood platelets (08 01) (02 07) (03 06)Cough (03 04) (05 04) (08 02)

Suppose that by the ldquobetter SALErdquo we mean a sale of 60percent of books and the pattern of the set of criteria Cr thatis examples cost pictures in intuitionistic fuzzy set is

Cr = [(06 03) (01 08) (06 03)] (12)

This set can be interpreted as about 60 percent of the bookscontain examples and pictures but the cost is not very low asits degree of membership is 01 To determine the better salewe present it in intuitionistic fuzzy digraph given in Figure 3and apply Algorithm 2 to it

The digraph in Figure 3 shows a typical three-layeredarchitecture of intuitionistic fuzzy neuron that is input

Table 6 Composition 119879 = 119877 ∘ 119876(119863 rarr 119875)

119879 Fayyaz Amir AslamDiabetes (018 006) (081 001) (018 007)Dengue (072 001) (054 002) (036 004)Tuberculosis (048 002) (045 004) (072 002)

hidden and output layer In an intuitionistic fuzzy neuronthe input hidden and output weights are defined in termsof degree of membership and degree of nonmembershipTheaggregation or activation of a neuron involves the degrees ofboth membership and nonmembership A node in the inputlayer represents the criteria 119862 of sales A node in the hiddenlayer shows the aggregationactivation of the neuron and theoutput layer shows the expected sales The relation betweenthe input and hidden layers is

IH =[

[

(05 05) (05 03) (00 10)

(01 08) (00 10) (01 08)

(00 10) (05 03) (00 10)

]

]

(13)

6 The Scientific World Journal

(01 08)

(05 03)(01 08)

(05 05)

(05 03)

(05 05)

(05 05)

(01 05)

Input layer Hidden layer

Output layer

Pictures (06 03)

Example (06 03) O1998400(03 064)

O2998400(03 009)

O3998400(002 064)

O1998400998400(015 032)Cost (0108)

Figure 3 Intuitionistic fuzzy digraph of marketability

and the relation between the hidden and output layers is

HO = [

[

(03 064)

(03 009)

(002 064)

]

]

(14)

The output on the hidden layer can be computed by takingcomposition between IH and 119862 that is 1198741015840 = 119862 ∘ IH

1198741015840

= [(05 05) (05 03) (01 08)] (15)

Similarly the final output is calculated by taking compo-sition between 1198741015840 and HO that is 11987410158401015840 = 1198741015840 ∘HO

11987410158401015840

= [(015 032)] (16)

The fuzzy digraph output layer in Figure 3 shows that the saleis about 15 with 53 hesitation Algorithm 2 describes theoverall scheme

The first three lines of Algorithm 2 set the required inputAt line 4 output on hidden layer is calculated by taking thecomposition between 119862 and IH relation Final output is cal-culated on line 5 by taking the composition between outputof hidden layer and HO relation Finally lines 7ndash10 checkwhether the results are in the desirable limits or not If theyare not within limits the membership and nonmembershipfunctions are modified using back propagation

(1) void marketability() (2) 119862 = Criteria of sale(3) 119868119867 = Relation between input and hidden layer(4) 119867119874 = Relation between hidden and output layer(5) 119874

1015840

= 119862 ∘ 119868119867(6) 119874

10158401015840

= 1198741015840

∘ 119867119874(7) if (11987410158401015840 is not expected) (8) Modify 119862 using back propagation(9) marketability()(10)

(11) print 11987410158401015840(12)

Algorithm 2 Architecture of IF neurons in marketability

34 Intuitionistic Fuzzy Digraph in Vulnerability Assessmentof Gas Pipeline Networks Vulnerability assessment of gasnetwork can be categorized into structural componentsreliability connectivity reliability flow performance relia-bility andor interdependent reliability These reliabilitiesdepended on the type of pipe and fittings used their agingand the connection between fitting and pipe In most caseswe do not know the exact age and condition of connectivityWe can present these factors as an intuitionistic fuzzy setAny gas network can be represented as an intuitionistic fuzzydigraph119866(119865 119875) where 119865 is the intuitionistic fuzzy set of pipefittings presenting their ages and connectivity conditions as

The Scientific World Journal 7

C1 C2

C3

C4

C5

C6

(0503)

(07 01)

(06 03)

(07 02)

(05 04)

(05 03)

(06 03)

(05 04)

(05 04)

(0602)

(05 03)

(05 03)

(05 03)

(0303)

Figure 4 Intuitionistic fuzzy digraph of a gas pipeline network

degrees of membership 120583119865(119909) and nonmembership ]

119865(119909)

and 119875 is an intuitionistic fuzzy set of pipelines betweenfittings In graph theoretic terms 119875 is a set of edges (iepipelines) between two vertices (ie fittings) The degrees ofmembership 120583

119875(119909119910)and nonmembership ]

119875(119909119910)are calculated

as120583119875(119909119910)

le min (120583119865(119909) 120583119865(119910))

]119875(119909119910)

le max (]119865(119909) ]119865(119910))

(17)

Consider the intuitionistic fuzzy set of pipe fittings

1198621 1198622 1198623 1198624 1198625 1198626

120583119865(119909)

07 05 06 07 05 05

]119865(119910)

01 03 03 02 04 03

(18)

The intuitionistic fuzzy digraph 119866(119865 119875) of the gas pipelinenetwork shown in Figure 4 is represented by the followingadjacency matrix

119866 =

[

[

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (05 04) (03 03)

(06 02) (00 10) (00 10) (00 10) (05 04) (05 03)

(06 03) (00 10) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

]

]

]

]

]

]

]

]

(19)

The final weighted digraph WG that can be used fordifferent kind of vulnerabilities can be calculated by findingthe ranks of edges as 119878

119894= 120583119875119894 minus ]

119875119894 lowast 120587

119875119894 The final

adjacency matrix and weighted digraph shown in Figure 5are developed based on these weights

WG =

[

[

[

[

[

[

[

[

0 044 0 0 0 0

0 0 044 0 0 0

0 0 0 0 046 018

056 0 0 0 046 044

051 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

(20)

The overall algorithm is explained in Algorithm 3It takes an intuitionistic fuzzy set of pipeline fittings as

an input Lines 3ndash6 calculate the degrees of membershipand nonmembership for edges and line 7 assigns them tointuitionistic fuzzy set of edges and adjacency matrix isprepared in line 8 Finally a weighted adjacency matrix iscalculated in lines 9ndash12 using rank techniques based on thedegrees of membership and nonmembership This weightedmatrix is printed in line 13 and is used for calculatingvulnerability in line 14

35 Intuitionistic Fuzzy Digraph in Travel Time In many net-work models such as transportation communication graphs

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

neural network models intuitionistic fuzzy neurons in med-ical diagnosis intuitionistic fuzzy digraphs in vulnerabilityassessment of gas pipeline networks and intuitionistic fuzzydigraphs in travel time are presented as examples of intuition-istic fuzzy digraphs in decision support systems Algorithmsof these decision support systems are also designed andimplemented

2 Preliminaries

A digraph is a pair 119866lowast = (119881 119864) where 119881 is a finite set and119864 sube 119881 times 119881 Let 119866lowast

1= (1198811 1198641) and 119866lowast

2= (1198812 1198642) be two

digraphsTheCartesian product of119866lowast1and119866lowast

2gives a digraph

119866lowast

1times 119866lowast

2= (119881 119864) with 119881 = 119881

1times 1198812and

119864 = (119909 1199092) 997888rarr (119909 119910

2) | 119909 isin 119881

1 1199092997888rarr 1199102isin 1198642

cup (1199091 119911) 997888rarr (119910

1 119911) | 119909

1997888rarr 1199101isin 1198641 119911 isin 119881

2

(1)

In this paper we will write 119909119910 isin 119864 to mean 119909 rarr 119910 isin 119864 andif 119890 = 119909119910 isin 119864 we say 119909 and 119910 are adjacent such that 119909 is astarting node and 119910 is an ending node

Definition 1 (see [1 3]) A fuzzy subset 120583 on a set 119883 is a map120583 119883 rarr [0 1] A fuzzy binary relation on119883 is a fuzzy subset120583 on 119883 times 119883 By a fuzzy relation we mean a fuzzy binaryrelation given by 120583 119883 times 119883 rarr [0 1]

Definition 2 (see [12]) Let 119881 be a finite set 119860 = ⟨119881 120583119860⟩ a

fuzzy set of119881 and 119861 = ⟨119881times119881 120583119861⟩ a fuzzy relation on119881 then

the ordered pair (119860 119861) is called a fuzzy digraph

In 1983 Atanassov [13] introduced the concept of intu-itionistic fuzzy sets as a generalization of fuzzy sets [1]Atanassov added a new component (which determines thedegree of nonmembership) in the definition of fuzzy setThe fuzzy sets give the degree of membership of an elementin a given set (and the nonmembership degree equals oneminus the degree of membership) while intuitionistic fuzzysets give both a degree of membership and a degree ofnonmembership which are more or less independent fromeach other the only requirement is that the sum of these twodegrees is not greater than 1

Definition 3 (see [8]) An intuitionistic fuzzy set (IFS) on auniverse119883 is an object of the form

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ 119909 isin 119883 (2)

where 120583119860(119909)(isin [0 1]) is called degree of membership of 119909

in 119860 ]119860(119909)(isin [0 1]) is called degree of nonmembership of

119909 in 119860 and 120583119860and ]119860satisfy the following condition for all

119909 isin 119883 120583119860(119909) + ]

119860(119909) le 1

Definition 4 An intuitionistic fuzzy relation 119877 =

(120583119877(119909 119910) ]

119877(119909 119910)) in a universe 119883 times 119884 (119877(119883 rarr 119884))

is an intuitionistic fuzzy set of the form119877 = ⟨(119909 119910) 120583

119860(119909 119910) ]

119860(119909 119910)⟩ | (119909 119910) isin 119883 times 119884 (3)

where 120583119860 119883 times 119884 rarr [0 1] and ]

119860 119883 times 119884 rarr [0 1] The

intuitionistic fuzzy relation119877 satisfies 120583119877(119909 119910)+]

119877(119909 119910) le 1

for all 119909 119910 isin 119883

Definition 5 Let 119877 be an intuitionistic fuzzy relation on uni-verse 119883 Then 119877 is called an intuitionistic fuzzy equivalencerelation on119883 if it satisfies the following conditions

(a) 119877 is intuitionistic fuzzy reflexive that is 119877(119909 119909) =(1 0) for each 119909 isin 119883

(b) 119877 is intuitionistic fuzzy symmetric that is 119877(119909 119910) =119877(119910 119909) for any 119909 119910 isin 119883

(c) 119877 is intuitionistic fuzzy transitive that is 119877(119909 119911) ge⋁119910(119877(119909 119910) and 119877(119910 119911))

Definition 6 Let 119876 (119883 rarr 119884) and 119877 (119884 rarr 119885) be twointuitionistic fuzzy relationsThemax-min-max composition119877 ∘ 119876 (119883 rarr 119885) is the intuitionistic fuzzy relation defined bythe membership function

120583119877∘119876(119909 119911) = ⋁

119910

(120583119876(119909 119910) and 120583

119877(119910 119911)) (4)

and the nonmembership function

]119877∘119876(119909 119911) = ⋀

119910

(]119876(119909 119910) or ]

119877(119910 119911)) (5)

for all (119909 119911) isin 119883 times 119885 and for all 119910 isin 119884

Definition 7 Let 119876 (119883 rarr 119884) and 119877 (119884 rarr 119885) be twointuitionistic fuzzy relations The max-product-min-productcomposition119877∘119876 (119883 rarr 119885) is the intuitionistic fuzzy relationdefined by the membership function

120583119877∘119876(119909 119911) = ⋁

119910

(120583119876(119909 119910) sdot 120583

119877(119910 119911)) (6)

and the nonmembership function

]119877∘119876(119909 119911) = ⋀

119910

(]119876(119909 119910) sdot ]

119877(119910 119911)) (7)

for all (119909 119911) isin 119883 times 119885 and for all 119910 isin 119884

Throughout this paper we denote 119866lowast a crisp simpledigraph and 119866 an intuitionistic fuzzy digraph

3 Applications of Intuitionistic FuzzyDigraphs in Decision Support Systems

Definition 8 An intuitionistic fuzzy digraph of a digraph 119866lowastis a pair 119866 = (119860 119861) where 119860 = ⟨119881 120583

119860 ]119860⟩ is an intuitionistic

fuzzy set in119881 and119861 = ⟨119881times119881 120583119861 ]119861⟩ is an intuitionistic fuzzy

relation on 119881 such that

120583119861(119909119910) le min (120583

119860(119909) 120583

119860(119910))

]119861(119909119910) le max (]

119860(119909) ]

119860(119910))

(8)

and 0 le 120583119861(119909119910) + ]

119861(119909119910) le 1 for all 119909 119910 isin 119881 We note that 119861

may not be symmetric relation

Example 9 Consider a graph 119866lowast = (119881 119864) such that 119881 =

V1 V2 V3 V4 and 119864 = V

1V2 V2V3 V3V4 V4V1 sube 119881 times 119881 Let 119860

The Scientific World Journal 3

1 2

34

(04 01)

(01 01)

(04 01)(02 01)

(03 01)

(06 02)

(05 02) (05 01)

Figure 1 Intuitionistic fuzzy digraph

be an intuitionistic fuzzy set of119881 and let 119861 be an intuitionisticfuzzy relation on 119881 defined by

V1

V2

V3

V4

12058311986004 06 05 05

]11986001 02 01 02

V1V2V2V3V3V4V4V1

12058311986101 02 03 04

]11986101 01 01 01

(9)

By routine computations it is easy to see from Figure 1that 119866 = (119860 119861) is an intuitionistic fuzzy digraph of 119866lowastThe intuitionistic fuzzy digraph119866 is represented by adjacencymatrix given below

119860 =

[

[

[

[

(00 10) (01 01) (00 10) (00 10)

(00 10) (00 10) (02 01) (00 10)

(00 10) (00 10) (00 10) (03 01)

(04 01) (00 10) (00 10) (00 10)

]

]

]

]

(10)

We now present several applications of intuitionistic fuzzydigraphs in decision support systems in the areas of manage-ment marketing medical diagnosis gas pipeline networksand transportation

31 Intuitionistic Fuzzy Organizational Model In this subsec-tion we explore an intuitionistic fuzzy graph model to findout themost influential personwithin an organization whichis called influence graph In an influence graph verticesrepresent an employee and edges represent the influence ofan employee on another employee of a company Such graphshave applications in modeling social structures communica-tion and distributed computing

We consider an organization having employees and theirdesignation as shown in Table 1 For this organization the setof employee is 119864 = BODMQTMMZAKRB St

Upon some investigation we discover the following(i) Mujeeb Qayyum has worked with Munib Zia for

over 10 years and he values his input on strategicinitiatives

(ii) The board of directors is chaired by a long timeassociate of Munib Zia Like Mujeeb the chair ofboard also values Munib

Table 1 Names of employees in an organization and their designa-tions

Name DesignationBoard of Directors (BOD) Board of DirectorsMujeeb Qayyum (MQ) CEOTahir Mahmood (TM) CTOMunib Zia (MZ) Director of MarketingArif Kaleem (AK) Director of Product DevelopmentRizwan Bashir (RB) Director of Human ResourcesStaff (St) Staff

Table 2 Power of employees in terms of membership degree andnonmembership degree

BOD MQ TM MZ AK RB St120583119860

09 09 08 07 06 06 05]119860

00 00 01 01 03 03 03

(iii) For reorganization the entire marketing and HRteam will be very important Rizwan Bashir will beespecially important

(iv) Tahir Mahmood and Rizwan Bashir have a history ofconflict

(v) Tahir Mahmood has great influence on the develop-ment team

Considering the above points an influence graph can bedeveloped but such a graph cannot represent the power ofemployees within an organization and the degree of influenceof employees on each other As the powers and influencehave no defined boundaries it is desired to represent themin the form of fuzzy set The fuzzy digraph representsthe influence of employees on each other but there is afair chance of the existence of nonnull hesitation part ateach moment of evaluation of influence We apply here theconcept of intuitionistic fuzzy set which is more preciseabout the influence and conflicts between the employeesTheintuitionistic fuzzy set of the employees is as follows

We represent the influence in the intuitionistic fuzzydigraph by an edgeThe resultant intuitionistic fuzzy digraphis shown in Figure 2 and corresponding adjacency matrix isshown in Table 3

The nodes of intuitionistic fuzzy digraph in Figure 2represent the employee and its power in terms of degree ofmembership and nonmembership which can be interpretedas percentage for example MQ possesses 90 power withinthe organization (Table 2) Similarly the edges of an intu-itionistic fuzzy digraph represent the influence of one personon another person that is end nodes of edges The degree ofmembership and nonmembership can be interpreted as thepercentage of positive and negative influence for example60 of the time BOD works on MZrsquos opinion but 10 of thetime they do not follow his opinion

In Figure 2 it is clear that MZ has influence both onBOD and on MQ He can influence both of them equallyas the degree of membership in both cases is 06 that is60 But in case of MQ the degree of hesitation is 04 thatis (120587 = 1ndash06ndash00) and in case of BOQ it is 03 that is

4 The Scientific World Journal

Table 3 Adjacency matrix corresponding to Figure 2

BOD MQ TM MZ AK RB StBOD (00 10) (08 00) (00 10) (00 10) (00 10) (00 10) (00 10)MQ (00 10) (00 10) (07 00) (00 10) (00 10) (00 10) (02 03)TM (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (04 03)MZ (06 01) (06 00) (00 10) (00 10) (00 10) (00 10) (03 03)AK (00 10) (00 10) (00 10) (00 10) (00 10) (05 03) (03 03)RB (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (02 03)St (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

BOD (09 00) MZ (07 01) St (05 03) RB (06 03)

MQ (09 00) TM (08 01) AK (06 03)

(02 03)

(06 00)(08 00)

(02 03)

(03 03)(06 01)

(04 03)

(03 03)

(05 03)

(07 00)

Figure 2 Influence intuitionistic fuzzy digraph

(120587 = 1ndash06ndash01) which means that the hesitation in case ofBOQ is more than that of MQ But it is quite obvious thatMZ is themost influential employee in the organization Alsonote that no other employee has influence on both BOD andMQ each possessing 90 power within the organization

32 Intuitionistic Fuzzy Neurons in Medical Diagnosis Thefield of medicine is one of the most fruitful and interestingareas of application for intuitionistic fuzzy set theory Inthe discrimination analysis for diagnosis of an illness thesymptoms are ranked according to the grade of discrimi-nation of each disease by a particular symptom A properbase knowledge is required in the medical diagnosis of asymptom In this section we use intuitionistic fuzzy elementfor knowledge base

Consider the following set of diseasesdiagnoses 119863 andset of symptoms 119878

119863 = DiabetesDengueTuberculosis

119878 = Temperature InsulinBlood pressure

Blood plateletsCough

(11)

The intuitionistic fuzzy relation 119876 (119863 rarr 119878) is shown inTable 4

Consider the set of patients as119875 = Fayyaz Amir AslamThe intuitionistic fuzzy relation 119877 (119878 rarr 119875) is given inTable 5

The max product composition 119879 = 119877 ∘ 119876 is shown inTable 6

By applying Algorithm 1 on Table 6 it is identified thatFayyaz is suffering from Dengue Amir is suffering fromDiabetes and Aslam is a patient of Tuberculosis

In the first portion of Algorithm 1 from lines 1ndash5 theparameters are set and max product composition is calcu-lated Each disease is ranked using 119878

119879 that is 119878

119879= 120583119894 minus ]119894 lowast

120587119894 in a loop The patient is suffering from the disease havingmaximum 119878

119879 If two diseases have the same 119878

119879 then their

hesitation function is tested that is 120587 = 1 minus ]119894 The diseasehaving less hesitation is selected

33 Architecture of Intuitionistic Fuzzy Neurons inMarketabil-ity The marketability of a book can be studied based onthree criteria that is pictures 119875 cost 119862 and examples 119864 Itis known that if a book has more examples low cost anda large numbers of pictures the sale of the book improves

The Scientific World Journal 5

(1) void diagnoseDisease() (2) 119877 = Patient symptoms(3) 119876 = Relation between disease and symptoms(4) 119879 = 119876 ∘ 119877(5) 119878max = 0(6) 119888119900119906119899119905119863119894119904119890119886119904119890 = 119879height(7) for (int 119894 = 0 119894 le 119888119900119906119899119905119863119894119904119890119886119904119890 119894++) (8) 119878

119894= 120583119894 minus ]119894 lowast 120587119894

(9) if (119878max == 119878119894) (10) if (degree of hesitation of 119878max ⩽ degree of hesitation of 119878

119894)

(11) 119878max = 119878119894(12) Diagnose = disease(13)

(14) else if (119878max ⩽ 119878119894) (15) 119878max = 119878119894(16) Diagnose = disease(17)

(18)

(19) print diagnose(20) if (diagnose is not accepted by the doctor) (21) Modify 119876(22) diagnoseDisease()(23)

(24)

Algorithm 1 IF neurons in medical diagnosis

Table 4 Intuitionistic fuzzy relation 119876(119863 rarr 119878)

119876 Temperature Insulin Blood pressure Blood platelets CoughDiabetes (02 08) (09 01) (01 08) (01 08) (01 08)Dengue (09 01) (00 08) (08 01) (09 01) (01 08)Tuberculosis (06 02) (00 09) (04 04) (00 08) (09 01)

Table 5 Intuitionistic fuzzy relation 119877(119878 rarr 119875)

119877 Fayyaz Amir AslamTemperature (08 01) (06 02) (04 04)Insulin (02 06) (09 01) (02 07)Blood pressure (04 04) (01 08) (01 07)Blood platelets (08 01) (02 07) (03 06)Cough (03 04) (05 04) (08 02)

Suppose that by the ldquobetter SALErdquo we mean a sale of 60percent of books and the pattern of the set of criteria Cr thatis examples cost pictures in intuitionistic fuzzy set is

Cr = [(06 03) (01 08) (06 03)] (12)

This set can be interpreted as about 60 percent of the bookscontain examples and pictures but the cost is not very low asits degree of membership is 01 To determine the better salewe present it in intuitionistic fuzzy digraph given in Figure 3and apply Algorithm 2 to it

The digraph in Figure 3 shows a typical three-layeredarchitecture of intuitionistic fuzzy neuron that is input

Table 6 Composition 119879 = 119877 ∘ 119876(119863 rarr 119875)

119879 Fayyaz Amir AslamDiabetes (018 006) (081 001) (018 007)Dengue (072 001) (054 002) (036 004)Tuberculosis (048 002) (045 004) (072 002)

hidden and output layer In an intuitionistic fuzzy neuronthe input hidden and output weights are defined in termsof degree of membership and degree of nonmembershipTheaggregation or activation of a neuron involves the degrees ofboth membership and nonmembership A node in the inputlayer represents the criteria 119862 of sales A node in the hiddenlayer shows the aggregationactivation of the neuron and theoutput layer shows the expected sales The relation betweenthe input and hidden layers is

IH =[

[

(05 05) (05 03) (00 10)

(01 08) (00 10) (01 08)

(00 10) (05 03) (00 10)

]

]

(13)

6 The Scientific World Journal

(01 08)

(05 03)(01 08)

(05 05)

(05 03)

(05 05)

(05 05)

(01 05)

Input layer Hidden layer

Output layer

Pictures (06 03)

Example (06 03) O1998400(03 064)

O2998400(03 009)

O3998400(002 064)

O1998400998400(015 032)Cost (0108)

Figure 3 Intuitionistic fuzzy digraph of marketability

and the relation between the hidden and output layers is

HO = [

[

(03 064)

(03 009)

(002 064)

]

]

(14)

The output on the hidden layer can be computed by takingcomposition between IH and 119862 that is 1198741015840 = 119862 ∘ IH

1198741015840

= [(05 05) (05 03) (01 08)] (15)

Similarly the final output is calculated by taking compo-sition between 1198741015840 and HO that is 11987410158401015840 = 1198741015840 ∘HO

11987410158401015840

= [(015 032)] (16)

The fuzzy digraph output layer in Figure 3 shows that the saleis about 15 with 53 hesitation Algorithm 2 describes theoverall scheme

The first three lines of Algorithm 2 set the required inputAt line 4 output on hidden layer is calculated by taking thecomposition between 119862 and IH relation Final output is cal-culated on line 5 by taking the composition between outputof hidden layer and HO relation Finally lines 7ndash10 checkwhether the results are in the desirable limits or not If theyare not within limits the membership and nonmembershipfunctions are modified using back propagation

(1) void marketability() (2) 119862 = Criteria of sale(3) 119868119867 = Relation between input and hidden layer(4) 119867119874 = Relation between hidden and output layer(5) 119874

1015840

= 119862 ∘ 119868119867(6) 119874

10158401015840

= 1198741015840

∘ 119867119874(7) if (11987410158401015840 is not expected) (8) Modify 119862 using back propagation(9) marketability()(10)

(11) print 11987410158401015840(12)

Algorithm 2 Architecture of IF neurons in marketability

34 Intuitionistic Fuzzy Digraph in Vulnerability Assessmentof Gas Pipeline Networks Vulnerability assessment of gasnetwork can be categorized into structural componentsreliability connectivity reliability flow performance relia-bility andor interdependent reliability These reliabilitiesdepended on the type of pipe and fittings used their agingand the connection between fitting and pipe In most caseswe do not know the exact age and condition of connectivityWe can present these factors as an intuitionistic fuzzy setAny gas network can be represented as an intuitionistic fuzzydigraph119866(119865 119875) where 119865 is the intuitionistic fuzzy set of pipefittings presenting their ages and connectivity conditions as

The Scientific World Journal 7

C1 C2

C3

C4

C5

C6

(0503)

(07 01)

(06 03)

(07 02)

(05 04)

(05 03)

(06 03)

(05 04)

(05 04)

(0602)

(05 03)

(05 03)

(05 03)

(0303)

Figure 4 Intuitionistic fuzzy digraph of a gas pipeline network

degrees of membership 120583119865(119909) and nonmembership ]

119865(119909)

and 119875 is an intuitionistic fuzzy set of pipelines betweenfittings In graph theoretic terms 119875 is a set of edges (iepipelines) between two vertices (ie fittings) The degrees ofmembership 120583

119875(119909119910)and nonmembership ]

119875(119909119910)are calculated

as120583119875(119909119910)

le min (120583119865(119909) 120583119865(119910))

]119875(119909119910)

le max (]119865(119909) ]119865(119910))

(17)

Consider the intuitionistic fuzzy set of pipe fittings

1198621 1198622 1198623 1198624 1198625 1198626

120583119865(119909)

07 05 06 07 05 05

]119865(119910)

01 03 03 02 04 03

(18)

The intuitionistic fuzzy digraph 119866(119865 119875) of the gas pipelinenetwork shown in Figure 4 is represented by the followingadjacency matrix

119866 =

[

[

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (05 04) (03 03)

(06 02) (00 10) (00 10) (00 10) (05 04) (05 03)

(06 03) (00 10) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

]

]

]

]

]

]

]

]

(19)

The final weighted digraph WG that can be used fordifferent kind of vulnerabilities can be calculated by findingthe ranks of edges as 119878

119894= 120583119875119894 minus ]

119875119894 lowast 120587

119875119894 The final

adjacency matrix and weighted digraph shown in Figure 5are developed based on these weights

WG =

[

[

[

[

[

[

[

[

0 044 0 0 0 0

0 0 044 0 0 0

0 0 0 0 046 018

056 0 0 0 046 044

051 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

(20)

The overall algorithm is explained in Algorithm 3It takes an intuitionistic fuzzy set of pipeline fittings as

an input Lines 3ndash6 calculate the degrees of membershipand nonmembership for edges and line 7 assigns them tointuitionistic fuzzy set of edges and adjacency matrix isprepared in line 8 Finally a weighted adjacency matrix iscalculated in lines 9ndash12 using rank techniques based on thedegrees of membership and nonmembership This weightedmatrix is printed in line 13 and is used for calculatingvulnerability in line 14

35 Intuitionistic Fuzzy Digraph in Travel Time In many net-work models such as transportation communication graphs

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

1 2

34

(04 01)

(01 01)

(04 01)(02 01)

(03 01)

(06 02)

(05 02) (05 01)

Figure 1 Intuitionistic fuzzy digraph

be an intuitionistic fuzzy set of119881 and let 119861 be an intuitionisticfuzzy relation on 119881 defined by

V1

V2

V3

V4

12058311986004 06 05 05

]11986001 02 01 02

V1V2V2V3V3V4V4V1

12058311986101 02 03 04

]11986101 01 01 01

(9)

By routine computations it is easy to see from Figure 1that 119866 = (119860 119861) is an intuitionistic fuzzy digraph of 119866lowastThe intuitionistic fuzzy digraph119866 is represented by adjacencymatrix given below

119860 =

[

[

[

[

(00 10) (01 01) (00 10) (00 10)

(00 10) (00 10) (02 01) (00 10)

(00 10) (00 10) (00 10) (03 01)

(04 01) (00 10) (00 10) (00 10)

]

]

]

]

(10)

We now present several applications of intuitionistic fuzzydigraphs in decision support systems in the areas of manage-ment marketing medical diagnosis gas pipeline networksand transportation

31 Intuitionistic Fuzzy Organizational Model In this subsec-tion we explore an intuitionistic fuzzy graph model to findout themost influential personwithin an organization whichis called influence graph In an influence graph verticesrepresent an employee and edges represent the influence ofan employee on another employee of a company Such graphshave applications in modeling social structures communica-tion and distributed computing

We consider an organization having employees and theirdesignation as shown in Table 1 For this organization the setof employee is 119864 = BODMQTMMZAKRB St

Upon some investigation we discover the following(i) Mujeeb Qayyum has worked with Munib Zia for

over 10 years and he values his input on strategicinitiatives

(ii) The board of directors is chaired by a long timeassociate of Munib Zia Like Mujeeb the chair ofboard also values Munib

Table 1 Names of employees in an organization and their designa-tions

Name DesignationBoard of Directors (BOD) Board of DirectorsMujeeb Qayyum (MQ) CEOTahir Mahmood (TM) CTOMunib Zia (MZ) Director of MarketingArif Kaleem (AK) Director of Product DevelopmentRizwan Bashir (RB) Director of Human ResourcesStaff (St) Staff

Table 2 Power of employees in terms of membership degree andnonmembership degree

BOD MQ TM MZ AK RB St120583119860

09 09 08 07 06 06 05]119860

00 00 01 01 03 03 03

(iii) For reorganization the entire marketing and HRteam will be very important Rizwan Bashir will beespecially important

(iv) Tahir Mahmood and Rizwan Bashir have a history ofconflict

(v) Tahir Mahmood has great influence on the develop-ment team

Considering the above points an influence graph can bedeveloped but such a graph cannot represent the power ofemployees within an organization and the degree of influenceof employees on each other As the powers and influencehave no defined boundaries it is desired to represent themin the form of fuzzy set The fuzzy digraph representsthe influence of employees on each other but there is afair chance of the existence of nonnull hesitation part ateach moment of evaluation of influence We apply here theconcept of intuitionistic fuzzy set which is more preciseabout the influence and conflicts between the employeesTheintuitionistic fuzzy set of the employees is as follows

We represent the influence in the intuitionistic fuzzydigraph by an edgeThe resultant intuitionistic fuzzy digraphis shown in Figure 2 and corresponding adjacency matrix isshown in Table 3

The nodes of intuitionistic fuzzy digraph in Figure 2represent the employee and its power in terms of degree ofmembership and nonmembership which can be interpretedas percentage for example MQ possesses 90 power withinthe organization (Table 2) Similarly the edges of an intu-itionistic fuzzy digraph represent the influence of one personon another person that is end nodes of edges The degree ofmembership and nonmembership can be interpreted as thepercentage of positive and negative influence for example60 of the time BOD works on MZrsquos opinion but 10 of thetime they do not follow his opinion

In Figure 2 it is clear that MZ has influence both onBOD and on MQ He can influence both of them equallyas the degree of membership in both cases is 06 that is60 But in case of MQ the degree of hesitation is 04 thatis (120587 = 1ndash06ndash00) and in case of BOQ it is 03 that is

4 The Scientific World Journal

Table 3 Adjacency matrix corresponding to Figure 2

BOD MQ TM MZ AK RB StBOD (00 10) (08 00) (00 10) (00 10) (00 10) (00 10) (00 10)MQ (00 10) (00 10) (07 00) (00 10) (00 10) (00 10) (02 03)TM (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (04 03)MZ (06 01) (06 00) (00 10) (00 10) (00 10) (00 10) (03 03)AK (00 10) (00 10) (00 10) (00 10) (00 10) (05 03) (03 03)RB (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (02 03)St (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

BOD (09 00) MZ (07 01) St (05 03) RB (06 03)

MQ (09 00) TM (08 01) AK (06 03)

(02 03)

(06 00)(08 00)

(02 03)

(03 03)(06 01)

(04 03)

(03 03)

(05 03)

(07 00)

Figure 2 Influence intuitionistic fuzzy digraph

(120587 = 1ndash06ndash01) which means that the hesitation in case ofBOQ is more than that of MQ But it is quite obvious thatMZ is themost influential employee in the organization Alsonote that no other employee has influence on both BOD andMQ each possessing 90 power within the organization

32 Intuitionistic Fuzzy Neurons in Medical Diagnosis Thefield of medicine is one of the most fruitful and interestingareas of application for intuitionistic fuzzy set theory Inthe discrimination analysis for diagnosis of an illness thesymptoms are ranked according to the grade of discrimi-nation of each disease by a particular symptom A properbase knowledge is required in the medical diagnosis of asymptom In this section we use intuitionistic fuzzy elementfor knowledge base

Consider the following set of diseasesdiagnoses 119863 andset of symptoms 119878

119863 = DiabetesDengueTuberculosis

119878 = Temperature InsulinBlood pressure

Blood plateletsCough

(11)

The intuitionistic fuzzy relation 119876 (119863 rarr 119878) is shown inTable 4

Consider the set of patients as119875 = Fayyaz Amir AslamThe intuitionistic fuzzy relation 119877 (119878 rarr 119875) is given inTable 5

The max product composition 119879 = 119877 ∘ 119876 is shown inTable 6

By applying Algorithm 1 on Table 6 it is identified thatFayyaz is suffering from Dengue Amir is suffering fromDiabetes and Aslam is a patient of Tuberculosis

In the first portion of Algorithm 1 from lines 1ndash5 theparameters are set and max product composition is calcu-lated Each disease is ranked using 119878

119879 that is 119878

119879= 120583119894 minus ]119894 lowast

120587119894 in a loop The patient is suffering from the disease havingmaximum 119878

119879 If two diseases have the same 119878

119879 then their

hesitation function is tested that is 120587 = 1 minus ]119894 The diseasehaving less hesitation is selected

33 Architecture of Intuitionistic Fuzzy Neurons inMarketabil-ity The marketability of a book can be studied based onthree criteria that is pictures 119875 cost 119862 and examples 119864 Itis known that if a book has more examples low cost anda large numbers of pictures the sale of the book improves

The Scientific World Journal 5

(1) void diagnoseDisease() (2) 119877 = Patient symptoms(3) 119876 = Relation between disease and symptoms(4) 119879 = 119876 ∘ 119877(5) 119878max = 0(6) 119888119900119906119899119905119863119894119904119890119886119904119890 = 119879height(7) for (int 119894 = 0 119894 le 119888119900119906119899119905119863119894119904119890119886119904119890 119894++) (8) 119878

119894= 120583119894 minus ]119894 lowast 120587119894

(9) if (119878max == 119878119894) (10) if (degree of hesitation of 119878max ⩽ degree of hesitation of 119878

119894)

(11) 119878max = 119878119894(12) Diagnose = disease(13)

(14) else if (119878max ⩽ 119878119894) (15) 119878max = 119878119894(16) Diagnose = disease(17)

(18)

(19) print diagnose(20) if (diagnose is not accepted by the doctor) (21) Modify 119876(22) diagnoseDisease()(23)

(24)

Algorithm 1 IF neurons in medical diagnosis

Table 4 Intuitionistic fuzzy relation 119876(119863 rarr 119878)

119876 Temperature Insulin Blood pressure Blood platelets CoughDiabetes (02 08) (09 01) (01 08) (01 08) (01 08)Dengue (09 01) (00 08) (08 01) (09 01) (01 08)Tuberculosis (06 02) (00 09) (04 04) (00 08) (09 01)

Table 5 Intuitionistic fuzzy relation 119877(119878 rarr 119875)

119877 Fayyaz Amir AslamTemperature (08 01) (06 02) (04 04)Insulin (02 06) (09 01) (02 07)Blood pressure (04 04) (01 08) (01 07)Blood platelets (08 01) (02 07) (03 06)Cough (03 04) (05 04) (08 02)

Suppose that by the ldquobetter SALErdquo we mean a sale of 60percent of books and the pattern of the set of criteria Cr thatis examples cost pictures in intuitionistic fuzzy set is

Cr = [(06 03) (01 08) (06 03)] (12)

This set can be interpreted as about 60 percent of the bookscontain examples and pictures but the cost is not very low asits degree of membership is 01 To determine the better salewe present it in intuitionistic fuzzy digraph given in Figure 3and apply Algorithm 2 to it

The digraph in Figure 3 shows a typical three-layeredarchitecture of intuitionistic fuzzy neuron that is input

Table 6 Composition 119879 = 119877 ∘ 119876(119863 rarr 119875)

119879 Fayyaz Amir AslamDiabetes (018 006) (081 001) (018 007)Dengue (072 001) (054 002) (036 004)Tuberculosis (048 002) (045 004) (072 002)

hidden and output layer In an intuitionistic fuzzy neuronthe input hidden and output weights are defined in termsof degree of membership and degree of nonmembershipTheaggregation or activation of a neuron involves the degrees ofboth membership and nonmembership A node in the inputlayer represents the criteria 119862 of sales A node in the hiddenlayer shows the aggregationactivation of the neuron and theoutput layer shows the expected sales The relation betweenthe input and hidden layers is

IH =[

[

(05 05) (05 03) (00 10)

(01 08) (00 10) (01 08)

(00 10) (05 03) (00 10)

]

]

(13)

6 The Scientific World Journal

(01 08)

(05 03)(01 08)

(05 05)

(05 03)

(05 05)

(05 05)

(01 05)

Input layer Hidden layer

Output layer

Pictures (06 03)

Example (06 03) O1998400(03 064)

O2998400(03 009)

O3998400(002 064)

O1998400998400(015 032)Cost (0108)

Figure 3 Intuitionistic fuzzy digraph of marketability

and the relation between the hidden and output layers is

HO = [

[

(03 064)

(03 009)

(002 064)

]

]

(14)

The output on the hidden layer can be computed by takingcomposition between IH and 119862 that is 1198741015840 = 119862 ∘ IH

1198741015840

= [(05 05) (05 03) (01 08)] (15)

Similarly the final output is calculated by taking compo-sition between 1198741015840 and HO that is 11987410158401015840 = 1198741015840 ∘HO

11987410158401015840

= [(015 032)] (16)

The fuzzy digraph output layer in Figure 3 shows that the saleis about 15 with 53 hesitation Algorithm 2 describes theoverall scheme

The first three lines of Algorithm 2 set the required inputAt line 4 output on hidden layer is calculated by taking thecomposition between 119862 and IH relation Final output is cal-culated on line 5 by taking the composition between outputof hidden layer and HO relation Finally lines 7ndash10 checkwhether the results are in the desirable limits or not If theyare not within limits the membership and nonmembershipfunctions are modified using back propagation

(1) void marketability() (2) 119862 = Criteria of sale(3) 119868119867 = Relation between input and hidden layer(4) 119867119874 = Relation between hidden and output layer(5) 119874

1015840

= 119862 ∘ 119868119867(6) 119874

10158401015840

= 1198741015840

∘ 119867119874(7) if (11987410158401015840 is not expected) (8) Modify 119862 using back propagation(9) marketability()(10)

(11) print 11987410158401015840(12)

Algorithm 2 Architecture of IF neurons in marketability

34 Intuitionistic Fuzzy Digraph in Vulnerability Assessmentof Gas Pipeline Networks Vulnerability assessment of gasnetwork can be categorized into structural componentsreliability connectivity reliability flow performance relia-bility andor interdependent reliability These reliabilitiesdepended on the type of pipe and fittings used their agingand the connection between fitting and pipe In most caseswe do not know the exact age and condition of connectivityWe can present these factors as an intuitionistic fuzzy setAny gas network can be represented as an intuitionistic fuzzydigraph119866(119865 119875) where 119865 is the intuitionistic fuzzy set of pipefittings presenting their ages and connectivity conditions as

The Scientific World Journal 7

C1 C2

C3

C4

C5

C6

(0503)

(07 01)

(06 03)

(07 02)

(05 04)

(05 03)

(06 03)

(05 04)

(05 04)

(0602)

(05 03)

(05 03)

(05 03)

(0303)

Figure 4 Intuitionistic fuzzy digraph of a gas pipeline network

degrees of membership 120583119865(119909) and nonmembership ]

119865(119909)

and 119875 is an intuitionistic fuzzy set of pipelines betweenfittings In graph theoretic terms 119875 is a set of edges (iepipelines) between two vertices (ie fittings) The degrees ofmembership 120583

119875(119909119910)and nonmembership ]

119875(119909119910)are calculated

as120583119875(119909119910)

le min (120583119865(119909) 120583119865(119910))

]119875(119909119910)

le max (]119865(119909) ]119865(119910))

(17)

Consider the intuitionistic fuzzy set of pipe fittings

1198621 1198622 1198623 1198624 1198625 1198626

120583119865(119909)

07 05 06 07 05 05

]119865(119910)

01 03 03 02 04 03

(18)

The intuitionistic fuzzy digraph 119866(119865 119875) of the gas pipelinenetwork shown in Figure 4 is represented by the followingadjacency matrix

119866 =

[

[

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (05 04) (03 03)

(06 02) (00 10) (00 10) (00 10) (05 04) (05 03)

(06 03) (00 10) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

]

]

]

]

]

]

]

]

(19)

The final weighted digraph WG that can be used fordifferent kind of vulnerabilities can be calculated by findingthe ranks of edges as 119878

119894= 120583119875119894 minus ]

119875119894 lowast 120587

119875119894 The final

adjacency matrix and weighted digraph shown in Figure 5are developed based on these weights

WG =

[

[

[

[

[

[

[

[

0 044 0 0 0 0

0 0 044 0 0 0

0 0 0 0 046 018

056 0 0 0 046 044

051 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

(20)

The overall algorithm is explained in Algorithm 3It takes an intuitionistic fuzzy set of pipeline fittings as

an input Lines 3ndash6 calculate the degrees of membershipand nonmembership for edges and line 7 assigns them tointuitionistic fuzzy set of edges and adjacency matrix isprepared in line 8 Finally a weighted adjacency matrix iscalculated in lines 9ndash12 using rank techniques based on thedegrees of membership and nonmembership This weightedmatrix is printed in line 13 and is used for calculatingvulnerability in line 14

35 Intuitionistic Fuzzy Digraph in Travel Time In many net-work models such as transportation communication graphs

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

Table 3 Adjacency matrix corresponding to Figure 2

BOD MQ TM MZ AK RB StBOD (00 10) (08 00) (00 10) (00 10) (00 10) (00 10) (00 10)MQ (00 10) (00 10) (07 00) (00 10) (00 10) (00 10) (02 03)TM (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (04 03)MZ (06 01) (06 00) (00 10) (00 10) (00 10) (00 10) (03 03)AK (00 10) (00 10) (00 10) (00 10) (00 10) (05 03) (03 03)RB (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (02 03)St (00 10) (00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

BOD (09 00) MZ (07 01) St (05 03) RB (06 03)

MQ (09 00) TM (08 01) AK (06 03)

(02 03)

(06 00)(08 00)

(02 03)

(03 03)(06 01)

(04 03)

(03 03)

(05 03)

(07 00)

Figure 2 Influence intuitionistic fuzzy digraph

(120587 = 1ndash06ndash01) which means that the hesitation in case ofBOQ is more than that of MQ But it is quite obvious thatMZ is themost influential employee in the organization Alsonote that no other employee has influence on both BOD andMQ each possessing 90 power within the organization

32 Intuitionistic Fuzzy Neurons in Medical Diagnosis Thefield of medicine is one of the most fruitful and interestingareas of application for intuitionistic fuzzy set theory Inthe discrimination analysis for diagnosis of an illness thesymptoms are ranked according to the grade of discrimi-nation of each disease by a particular symptom A properbase knowledge is required in the medical diagnosis of asymptom In this section we use intuitionistic fuzzy elementfor knowledge base

Consider the following set of diseasesdiagnoses 119863 andset of symptoms 119878

119863 = DiabetesDengueTuberculosis

119878 = Temperature InsulinBlood pressure

Blood plateletsCough

(11)

The intuitionistic fuzzy relation 119876 (119863 rarr 119878) is shown inTable 4

Consider the set of patients as119875 = Fayyaz Amir AslamThe intuitionistic fuzzy relation 119877 (119878 rarr 119875) is given inTable 5

The max product composition 119879 = 119877 ∘ 119876 is shown inTable 6

By applying Algorithm 1 on Table 6 it is identified thatFayyaz is suffering from Dengue Amir is suffering fromDiabetes and Aslam is a patient of Tuberculosis

In the first portion of Algorithm 1 from lines 1ndash5 theparameters are set and max product composition is calcu-lated Each disease is ranked using 119878

119879 that is 119878

119879= 120583119894 minus ]119894 lowast

120587119894 in a loop The patient is suffering from the disease havingmaximum 119878

119879 If two diseases have the same 119878

119879 then their

hesitation function is tested that is 120587 = 1 minus ]119894 The diseasehaving less hesitation is selected

33 Architecture of Intuitionistic Fuzzy Neurons inMarketabil-ity The marketability of a book can be studied based onthree criteria that is pictures 119875 cost 119862 and examples 119864 Itis known that if a book has more examples low cost anda large numbers of pictures the sale of the book improves

The Scientific World Journal 5

(1) void diagnoseDisease() (2) 119877 = Patient symptoms(3) 119876 = Relation between disease and symptoms(4) 119879 = 119876 ∘ 119877(5) 119878max = 0(6) 119888119900119906119899119905119863119894119904119890119886119904119890 = 119879height(7) for (int 119894 = 0 119894 le 119888119900119906119899119905119863119894119904119890119886119904119890 119894++) (8) 119878

119894= 120583119894 minus ]119894 lowast 120587119894

(9) if (119878max == 119878119894) (10) if (degree of hesitation of 119878max ⩽ degree of hesitation of 119878

119894)

(11) 119878max = 119878119894(12) Diagnose = disease(13)

(14) else if (119878max ⩽ 119878119894) (15) 119878max = 119878119894(16) Diagnose = disease(17)

(18)

(19) print diagnose(20) if (diagnose is not accepted by the doctor) (21) Modify 119876(22) diagnoseDisease()(23)

(24)

Algorithm 1 IF neurons in medical diagnosis

Table 4 Intuitionistic fuzzy relation 119876(119863 rarr 119878)

119876 Temperature Insulin Blood pressure Blood platelets CoughDiabetes (02 08) (09 01) (01 08) (01 08) (01 08)Dengue (09 01) (00 08) (08 01) (09 01) (01 08)Tuberculosis (06 02) (00 09) (04 04) (00 08) (09 01)

Table 5 Intuitionistic fuzzy relation 119877(119878 rarr 119875)

119877 Fayyaz Amir AslamTemperature (08 01) (06 02) (04 04)Insulin (02 06) (09 01) (02 07)Blood pressure (04 04) (01 08) (01 07)Blood platelets (08 01) (02 07) (03 06)Cough (03 04) (05 04) (08 02)

Suppose that by the ldquobetter SALErdquo we mean a sale of 60percent of books and the pattern of the set of criteria Cr thatis examples cost pictures in intuitionistic fuzzy set is

Cr = [(06 03) (01 08) (06 03)] (12)

This set can be interpreted as about 60 percent of the bookscontain examples and pictures but the cost is not very low asits degree of membership is 01 To determine the better salewe present it in intuitionistic fuzzy digraph given in Figure 3and apply Algorithm 2 to it

The digraph in Figure 3 shows a typical three-layeredarchitecture of intuitionistic fuzzy neuron that is input

Table 6 Composition 119879 = 119877 ∘ 119876(119863 rarr 119875)

119879 Fayyaz Amir AslamDiabetes (018 006) (081 001) (018 007)Dengue (072 001) (054 002) (036 004)Tuberculosis (048 002) (045 004) (072 002)

hidden and output layer In an intuitionistic fuzzy neuronthe input hidden and output weights are defined in termsof degree of membership and degree of nonmembershipTheaggregation or activation of a neuron involves the degrees ofboth membership and nonmembership A node in the inputlayer represents the criteria 119862 of sales A node in the hiddenlayer shows the aggregationactivation of the neuron and theoutput layer shows the expected sales The relation betweenthe input and hidden layers is

IH =[

[

(05 05) (05 03) (00 10)

(01 08) (00 10) (01 08)

(00 10) (05 03) (00 10)

]

]

(13)

6 The Scientific World Journal

(01 08)

(05 03)(01 08)

(05 05)

(05 03)

(05 05)

(05 05)

(01 05)

Input layer Hidden layer

Output layer

Pictures (06 03)

Example (06 03) O1998400(03 064)

O2998400(03 009)

O3998400(002 064)

O1998400998400(015 032)Cost (0108)

Figure 3 Intuitionistic fuzzy digraph of marketability

and the relation between the hidden and output layers is

HO = [

[

(03 064)

(03 009)

(002 064)

]

]

(14)

The output on the hidden layer can be computed by takingcomposition between IH and 119862 that is 1198741015840 = 119862 ∘ IH

1198741015840

= [(05 05) (05 03) (01 08)] (15)

Similarly the final output is calculated by taking compo-sition between 1198741015840 and HO that is 11987410158401015840 = 1198741015840 ∘HO

11987410158401015840

= [(015 032)] (16)

The fuzzy digraph output layer in Figure 3 shows that the saleis about 15 with 53 hesitation Algorithm 2 describes theoverall scheme

The first three lines of Algorithm 2 set the required inputAt line 4 output on hidden layer is calculated by taking thecomposition between 119862 and IH relation Final output is cal-culated on line 5 by taking the composition between outputof hidden layer and HO relation Finally lines 7ndash10 checkwhether the results are in the desirable limits or not If theyare not within limits the membership and nonmembershipfunctions are modified using back propagation

(1) void marketability() (2) 119862 = Criteria of sale(3) 119868119867 = Relation between input and hidden layer(4) 119867119874 = Relation between hidden and output layer(5) 119874

1015840

= 119862 ∘ 119868119867(6) 119874

10158401015840

= 1198741015840

∘ 119867119874(7) if (11987410158401015840 is not expected) (8) Modify 119862 using back propagation(9) marketability()(10)

(11) print 11987410158401015840(12)

Algorithm 2 Architecture of IF neurons in marketability

34 Intuitionistic Fuzzy Digraph in Vulnerability Assessmentof Gas Pipeline Networks Vulnerability assessment of gasnetwork can be categorized into structural componentsreliability connectivity reliability flow performance relia-bility andor interdependent reliability These reliabilitiesdepended on the type of pipe and fittings used their agingand the connection between fitting and pipe In most caseswe do not know the exact age and condition of connectivityWe can present these factors as an intuitionistic fuzzy setAny gas network can be represented as an intuitionistic fuzzydigraph119866(119865 119875) where 119865 is the intuitionistic fuzzy set of pipefittings presenting their ages and connectivity conditions as

The Scientific World Journal 7

C1 C2

C3

C4

C5

C6

(0503)

(07 01)

(06 03)

(07 02)

(05 04)

(05 03)

(06 03)

(05 04)

(05 04)

(0602)

(05 03)

(05 03)

(05 03)

(0303)

Figure 4 Intuitionistic fuzzy digraph of a gas pipeline network

degrees of membership 120583119865(119909) and nonmembership ]

119865(119909)

and 119875 is an intuitionistic fuzzy set of pipelines betweenfittings In graph theoretic terms 119875 is a set of edges (iepipelines) between two vertices (ie fittings) The degrees ofmembership 120583

119875(119909119910)and nonmembership ]

119875(119909119910)are calculated

as120583119875(119909119910)

le min (120583119865(119909) 120583119865(119910))

]119875(119909119910)

le max (]119865(119909) ]119865(119910))

(17)

Consider the intuitionistic fuzzy set of pipe fittings

1198621 1198622 1198623 1198624 1198625 1198626

120583119865(119909)

07 05 06 07 05 05

]119865(119910)

01 03 03 02 04 03

(18)

The intuitionistic fuzzy digraph 119866(119865 119875) of the gas pipelinenetwork shown in Figure 4 is represented by the followingadjacency matrix

119866 =

[

[

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (05 04) (03 03)

(06 02) (00 10) (00 10) (00 10) (05 04) (05 03)

(06 03) (00 10) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

]

]

]

]

]

]

]

]

(19)

The final weighted digraph WG that can be used fordifferent kind of vulnerabilities can be calculated by findingthe ranks of edges as 119878

119894= 120583119875119894 minus ]

119875119894 lowast 120587

119875119894 The final

adjacency matrix and weighted digraph shown in Figure 5are developed based on these weights

WG =

[

[

[

[

[

[

[

[

0 044 0 0 0 0

0 0 044 0 0 0

0 0 0 0 046 018

056 0 0 0 046 044

051 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

(20)

The overall algorithm is explained in Algorithm 3It takes an intuitionistic fuzzy set of pipeline fittings as

an input Lines 3ndash6 calculate the degrees of membershipand nonmembership for edges and line 7 assigns them tointuitionistic fuzzy set of edges and adjacency matrix isprepared in line 8 Finally a weighted adjacency matrix iscalculated in lines 9ndash12 using rank techniques based on thedegrees of membership and nonmembership This weightedmatrix is printed in line 13 and is used for calculatingvulnerability in line 14

35 Intuitionistic Fuzzy Digraph in Travel Time In many net-work models such as transportation communication graphs

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

(1) void diagnoseDisease() (2) 119877 = Patient symptoms(3) 119876 = Relation between disease and symptoms(4) 119879 = 119876 ∘ 119877(5) 119878max = 0(6) 119888119900119906119899119905119863119894119904119890119886119904119890 = 119879height(7) for (int 119894 = 0 119894 le 119888119900119906119899119905119863119894119904119890119886119904119890 119894++) (8) 119878

119894= 120583119894 minus ]119894 lowast 120587119894

(9) if (119878max == 119878119894) (10) if (degree of hesitation of 119878max ⩽ degree of hesitation of 119878

119894)

(11) 119878max = 119878119894(12) Diagnose = disease(13)

(14) else if (119878max ⩽ 119878119894) (15) 119878max = 119878119894(16) Diagnose = disease(17)

(18)

(19) print diagnose(20) if (diagnose is not accepted by the doctor) (21) Modify 119876(22) diagnoseDisease()(23)

(24)

Algorithm 1 IF neurons in medical diagnosis

Table 4 Intuitionistic fuzzy relation 119876(119863 rarr 119878)

119876 Temperature Insulin Blood pressure Blood platelets CoughDiabetes (02 08) (09 01) (01 08) (01 08) (01 08)Dengue (09 01) (00 08) (08 01) (09 01) (01 08)Tuberculosis (06 02) (00 09) (04 04) (00 08) (09 01)

Table 5 Intuitionistic fuzzy relation 119877(119878 rarr 119875)

119877 Fayyaz Amir AslamTemperature (08 01) (06 02) (04 04)Insulin (02 06) (09 01) (02 07)Blood pressure (04 04) (01 08) (01 07)Blood platelets (08 01) (02 07) (03 06)Cough (03 04) (05 04) (08 02)

Suppose that by the ldquobetter SALErdquo we mean a sale of 60percent of books and the pattern of the set of criteria Cr thatis examples cost pictures in intuitionistic fuzzy set is

Cr = [(06 03) (01 08) (06 03)] (12)

This set can be interpreted as about 60 percent of the bookscontain examples and pictures but the cost is not very low asits degree of membership is 01 To determine the better salewe present it in intuitionistic fuzzy digraph given in Figure 3and apply Algorithm 2 to it

The digraph in Figure 3 shows a typical three-layeredarchitecture of intuitionistic fuzzy neuron that is input

Table 6 Composition 119879 = 119877 ∘ 119876(119863 rarr 119875)

119879 Fayyaz Amir AslamDiabetes (018 006) (081 001) (018 007)Dengue (072 001) (054 002) (036 004)Tuberculosis (048 002) (045 004) (072 002)

hidden and output layer In an intuitionistic fuzzy neuronthe input hidden and output weights are defined in termsof degree of membership and degree of nonmembershipTheaggregation or activation of a neuron involves the degrees ofboth membership and nonmembership A node in the inputlayer represents the criteria 119862 of sales A node in the hiddenlayer shows the aggregationactivation of the neuron and theoutput layer shows the expected sales The relation betweenthe input and hidden layers is

IH =[

[

(05 05) (05 03) (00 10)

(01 08) (00 10) (01 08)

(00 10) (05 03) (00 10)

]

]

(13)

6 The Scientific World Journal

(01 08)

(05 03)(01 08)

(05 05)

(05 03)

(05 05)

(05 05)

(01 05)

Input layer Hidden layer

Output layer

Pictures (06 03)

Example (06 03) O1998400(03 064)

O2998400(03 009)

O3998400(002 064)

O1998400998400(015 032)Cost (0108)

Figure 3 Intuitionistic fuzzy digraph of marketability

and the relation between the hidden and output layers is

HO = [

[

(03 064)

(03 009)

(002 064)

]

]

(14)

The output on the hidden layer can be computed by takingcomposition between IH and 119862 that is 1198741015840 = 119862 ∘ IH

1198741015840

= [(05 05) (05 03) (01 08)] (15)

Similarly the final output is calculated by taking compo-sition between 1198741015840 and HO that is 11987410158401015840 = 1198741015840 ∘HO

11987410158401015840

= [(015 032)] (16)

The fuzzy digraph output layer in Figure 3 shows that the saleis about 15 with 53 hesitation Algorithm 2 describes theoverall scheme

The first three lines of Algorithm 2 set the required inputAt line 4 output on hidden layer is calculated by taking thecomposition between 119862 and IH relation Final output is cal-culated on line 5 by taking the composition between outputof hidden layer and HO relation Finally lines 7ndash10 checkwhether the results are in the desirable limits or not If theyare not within limits the membership and nonmembershipfunctions are modified using back propagation

(1) void marketability() (2) 119862 = Criteria of sale(3) 119868119867 = Relation between input and hidden layer(4) 119867119874 = Relation between hidden and output layer(5) 119874

1015840

= 119862 ∘ 119868119867(6) 119874

10158401015840

= 1198741015840

∘ 119867119874(7) if (11987410158401015840 is not expected) (8) Modify 119862 using back propagation(9) marketability()(10)

(11) print 11987410158401015840(12)

Algorithm 2 Architecture of IF neurons in marketability

34 Intuitionistic Fuzzy Digraph in Vulnerability Assessmentof Gas Pipeline Networks Vulnerability assessment of gasnetwork can be categorized into structural componentsreliability connectivity reliability flow performance relia-bility andor interdependent reliability These reliabilitiesdepended on the type of pipe and fittings used their agingand the connection between fitting and pipe In most caseswe do not know the exact age and condition of connectivityWe can present these factors as an intuitionistic fuzzy setAny gas network can be represented as an intuitionistic fuzzydigraph119866(119865 119875) where 119865 is the intuitionistic fuzzy set of pipefittings presenting their ages and connectivity conditions as

The Scientific World Journal 7

C1 C2

C3

C4

C5

C6

(0503)

(07 01)

(06 03)

(07 02)

(05 04)

(05 03)

(06 03)

(05 04)

(05 04)

(0602)

(05 03)

(05 03)

(05 03)

(0303)

Figure 4 Intuitionistic fuzzy digraph of a gas pipeline network

degrees of membership 120583119865(119909) and nonmembership ]

119865(119909)

and 119875 is an intuitionistic fuzzy set of pipelines betweenfittings In graph theoretic terms 119875 is a set of edges (iepipelines) between two vertices (ie fittings) The degrees ofmembership 120583

119875(119909119910)and nonmembership ]

119875(119909119910)are calculated

as120583119875(119909119910)

le min (120583119865(119909) 120583119865(119910))

]119875(119909119910)

le max (]119865(119909) ]119865(119910))

(17)

Consider the intuitionistic fuzzy set of pipe fittings

1198621 1198622 1198623 1198624 1198625 1198626

120583119865(119909)

07 05 06 07 05 05

]119865(119910)

01 03 03 02 04 03

(18)

The intuitionistic fuzzy digraph 119866(119865 119875) of the gas pipelinenetwork shown in Figure 4 is represented by the followingadjacency matrix

119866 =

[

[

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (05 04) (03 03)

(06 02) (00 10) (00 10) (00 10) (05 04) (05 03)

(06 03) (00 10) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

]

]

]

]

]

]

]

]

(19)

The final weighted digraph WG that can be used fordifferent kind of vulnerabilities can be calculated by findingthe ranks of edges as 119878

119894= 120583119875119894 minus ]

119875119894 lowast 120587

119875119894 The final

adjacency matrix and weighted digraph shown in Figure 5are developed based on these weights

WG =

[

[

[

[

[

[

[

[

0 044 0 0 0 0

0 0 044 0 0 0

0 0 0 0 046 018

056 0 0 0 046 044

051 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

(20)

The overall algorithm is explained in Algorithm 3It takes an intuitionistic fuzzy set of pipeline fittings as

an input Lines 3ndash6 calculate the degrees of membershipand nonmembership for edges and line 7 assigns them tointuitionistic fuzzy set of edges and adjacency matrix isprepared in line 8 Finally a weighted adjacency matrix iscalculated in lines 9ndash12 using rank techniques based on thedegrees of membership and nonmembership This weightedmatrix is printed in line 13 and is used for calculatingvulnerability in line 14

35 Intuitionistic Fuzzy Digraph in Travel Time In many net-work models such as transportation communication graphs

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

(01 08)

(05 03)(01 08)

(05 05)

(05 03)

(05 05)

(05 05)

(01 05)

Input layer Hidden layer

Output layer

Pictures (06 03)

Example (06 03) O1998400(03 064)

O2998400(03 009)

O3998400(002 064)

O1998400998400(015 032)Cost (0108)

Figure 3 Intuitionistic fuzzy digraph of marketability

and the relation between the hidden and output layers is

HO = [

[

(03 064)

(03 009)

(002 064)

]

]

(14)

The output on the hidden layer can be computed by takingcomposition between IH and 119862 that is 1198741015840 = 119862 ∘ IH

1198741015840

= [(05 05) (05 03) (01 08)] (15)

Similarly the final output is calculated by taking compo-sition between 1198741015840 and HO that is 11987410158401015840 = 1198741015840 ∘HO

11987410158401015840

= [(015 032)] (16)

The fuzzy digraph output layer in Figure 3 shows that the saleis about 15 with 53 hesitation Algorithm 2 describes theoverall scheme

The first three lines of Algorithm 2 set the required inputAt line 4 output on hidden layer is calculated by taking thecomposition between 119862 and IH relation Final output is cal-culated on line 5 by taking the composition between outputof hidden layer and HO relation Finally lines 7ndash10 checkwhether the results are in the desirable limits or not If theyare not within limits the membership and nonmembershipfunctions are modified using back propagation

(1) void marketability() (2) 119862 = Criteria of sale(3) 119868119867 = Relation between input and hidden layer(4) 119867119874 = Relation between hidden and output layer(5) 119874

1015840

= 119862 ∘ 119868119867(6) 119874

10158401015840

= 1198741015840

∘ 119867119874(7) if (11987410158401015840 is not expected) (8) Modify 119862 using back propagation(9) marketability()(10)

(11) print 11987410158401015840(12)

Algorithm 2 Architecture of IF neurons in marketability

34 Intuitionistic Fuzzy Digraph in Vulnerability Assessmentof Gas Pipeline Networks Vulnerability assessment of gasnetwork can be categorized into structural componentsreliability connectivity reliability flow performance relia-bility andor interdependent reliability These reliabilitiesdepended on the type of pipe and fittings used their agingand the connection between fitting and pipe In most caseswe do not know the exact age and condition of connectivityWe can present these factors as an intuitionistic fuzzy setAny gas network can be represented as an intuitionistic fuzzydigraph119866(119865 119875) where 119865 is the intuitionistic fuzzy set of pipefittings presenting their ages and connectivity conditions as

The Scientific World Journal 7

C1 C2

C3

C4

C5

C6

(0503)

(07 01)

(06 03)

(07 02)

(05 04)

(05 03)

(06 03)

(05 04)

(05 04)

(0602)

(05 03)

(05 03)

(05 03)

(0303)

Figure 4 Intuitionistic fuzzy digraph of a gas pipeline network

degrees of membership 120583119865(119909) and nonmembership ]

119865(119909)

and 119875 is an intuitionistic fuzzy set of pipelines betweenfittings In graph theoretic terms 119875 is a set of edges (iepipelines) between two vertices (ie fittings) The degrees ofmembership 120583

119875(119909119910)and nonmembership ]

119875(119909119910)are calculated

as120583119875(119909119910)

le min (120583119865(119909) 120583119865(119910))

]119875(119909119910)

le max (]119865(119909) ]119865(119910))

(17)

Consider the intuitionistic fuzzy set of pipe fittings

1198621 1198622 1198623 1198624 1198625 1198626

120583119865(119909)

07 05 06 07 05 05

]119865(119910)

01 03 03 02 04 03

(18)

The intuitionistic fuzzy digraph 119866(119865 119875) of the gas pipelinenetwork shown in Figure 4 is represented by the followingadjacency matrix

119866 =

[

[

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (05 04) (03 03)

(06 02) (00 10) (00 10) (00 10) (05 04) (05 03)

(06 03) (00 10) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

]

]

]

]

]

]

]

]

(19)

The final weighted digraph WG that can be used fordifferent kind of vulnerabilities can be calculated by findingthe ranks of edges as 119878

119894= 120583119875119894 minus ]

119875119894 lowast 120587

119875119894 The final

adjacency matrix and weighted digraph shown in Figure 5are developed based on these weights

WG =

[

[

[

[

[

[

[

[

0 044 0 0 0 0

0 0 044 0 0 0

0 0 0 0 046 018

056 0 0 0 046 044

051 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

(20)

The overall algorithm is explained in Algorithm 3It takes an intuitionistic fuzzy set of pipeline fittings as

an input Lines 3ndash6 calculate the degrees of membershipand nonmembership for edges and line 7 assigns them tointuitionistic fuzzy set of edges and adjacency matrix isprepared in line 8 Finally a weighted adjacency matrix iscalculated in lines 9ndash12 using rank techniques based on thedegrees of membership and nonmembership This weightedmatrix is printed in line 13 and is used for calculatingvulnerability in line 14

35 Intuitionistic Fuzzy Digraph in Travel Time In many net-work models such as transportation communication graphs

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

C1 C2

C3

C4

C5

C6

(0503)

(07 01)

(06 03)

(07 02)

(05 04)

(05 03)

(06 03)

(05 04)

(05 04)

(0602)

(05 03)

(05 03)

(05 03)

(0303)

Figure 4 Intuitionistic fuzzy digraph of a gas pipeline network

degrees of membership 120583119865(119909) and nonmembership ]

119865(119909)

and 119875 is an intuitionistic fuzzy set of pipelines betweenfittings In graph theoretic terms 119875 is a set of edges (iepipelines) between two vertices (ie fittings) The degrees ofmembership 120583

119875(119909119910)and nonmembership ]

119875(119909119910)are calculated

as120583119875(119909119910)

le min (120583119865(119909) 120583119865(119910))

]119875(119909119910)

le max (]119865(119909) ]119865(119910))

(17)

Consider the intuitionistic fuzzy set of pipe fittings

1198621 1198622 1198623 1198624 1198625 1198626

120583119865(119909)

07 05 06 07 05 05

]119865(119910)

01 03 03 02 04 03

(18)

The intuitionistic fuzzy digraph 119866(119865 119875) of the gas pipelinenetwork shown in Figure 4 is represented by the followingadjacency matrix

119866 =

[

[

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (05 04) (03 03)

(06 02) (00 10) (00 10) (00 10) (05 04) (05 03)

(06 03) (00 10) (00 10) (00 10) (00 10) (00 10)

(00 10) (00 10) (00 10) (00 10) (00 10) (00 10)

]

]

]

]

]

]

]

]

(19)

The final weighted digraph WG that can be used fordifferent kind of vulnerabilities can be calculated by findingthe ranks of edges as 119878

119894= 120583119875119894 minus ]

119875119894 lowast 120587

119875119894 The final

adjacency matrix and weighted digraph shown in Figure 5are developed based on these weights

WG =

[

[

[

[

[

[

[

[

0 044 0 0 0 0

0 0 044 0 0 0

0 0 0 0 046 018

056 0 0 0 046 044

051 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

(20)

The overall algorithm is explained in Algorithm 3It takes an intuitionistic fuzzy set of pipeline fittings as

an input Lines 3ndash6 calculate the degrees of membershipand nonmembership for edges and line 7 assigns them tointuitionistic fuzzy set of edges and adjacency matrix isprepared in line 8 Finally a weighted adjacency matrix iscalculated in lines 9ndash12 using rank techniques based on thedegrees of membership and nonmembership This weightedmatrix is printed in line 13 and is used for calculatingvulnerability in line 14

35 Intuitionistic Fuzzy Digraph in Travel Time In many net-work models such as transportation communication graphs

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

C1 C2

C3

C4

C5

C6

044

051

044

046

018

044

046

056

Figure 5 Weighted digraph of a gas pipeline network

(1) void fuzzyPipelineVulnerability() (2) 119865 = Intuitionistic fuzzy set of pipeline fitting(3) 119888119900119906119899119905119865119894119905119905 = count(119865)(4) 119875 = Empty intuitionistic fuzzy set(5) for (int 119909 = 0 119909 lt 119888119900119906119899119905119865119894119905119905 119909++) (6) for (int 119910 = 0 119910 lt 119888119900119906119899119905119865119894119905119905 119910++) (7) if (119865(119909) is adjacent to 119865(119910)) (8) 120583

119875(119909119910)= min(120583

119865(119909) 120583119865(119910))

(9) ]119875(119909119910)

= max(]119865(119909) ]119865(119910))

(10)

(11)

(12)

(13) 119875 = Intuitionistic fuzzy set of edges(14) 119866 = Intuitionistic fuzzy relation (adjacency matrix of 119865 times 119865)(15) 119882119866 =Weighted relation (adjacency matrix of 119865 times 119865(16) 119899119900119874119891119864119889119892119890119904 = count(119875)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119875119894minus ]119875119894lowast 120587119875119894

(19) 119909 = Adjacent from Node of 119875119894

(20) 119910 = Adjacent to Node of 119875119894

(21) 119882119866119909119910= 119878119894

(22)

(23) print119882119866(24) Calculate Vulnerability using119882119866(25)

Algorithm 3 IF in vulnerability assessment of gas pipeline networks

are used as a natural mathematical model to identify prob-lems and solve themMany of these networks can bemodeledusing communication graphs to find the shortestoptimalpaths between the endpoints that is vertices and nodes ofnetworksThe optimality criteria are often evaluated in termsof weights of arcsedges between two adjacent vertices in thenetwork In case of transportation and road networks thetravel time is mostly used as weight The travel time is a

function of the traffic density on the road andor the length ofthe roadThe length of a road is a crisp quantity but the trafficdensity is fuzzy In a road network we represent crossingsas nodes and roads as edges The traffic density is mostlycalculated on the road between adjacent crossings Thesenumbers can be represented as intuitionistic fuzzy numbersFigure 6 shows a model of a road network represented asan intuitionistic fuzzy graph 119877lowast = (119862 119871) where 119862 is an

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

(1) void fuzzyShortestPath()(2) 119862 = Intuitionistic fuzzy set of crossings(3) 119899119900119874119891119862119903119900119904119904119894119899119892 = count(119862)(4) 119871 = Empty intuitionistic fuzzy set of roads(5) for (int 119909 = 0 119909 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119909++) (6) for (int 119910 = 0 119910 lt 119899119900119874119891119862119903119900119904119904119894119899119892 119910++) (7) if (119862(119909) is adjacent to 119862(119910)) (8) 120583

119871(119909119910)= min(120583

119862(119909) 120583119862(119910))

(9) ]119871(119909119910)

= max(]119862(119909) ]119862(119910))

)(10)

(11)

(12)

(13) 119871 = Intuitionistic fuzzy set of edges(14) 119877 = Intuitionistic fuzzy relation (Adjacency matrix of 119862 times 119862)(15) 119882119877 =Weighted relation (adjacency matrix of 119862 times 119862(16) 119899119900119874119891119864119889119892119890119904 = count(119871)(17) for (int 119894 = 0 119894 lt 119899119900119874119891119864119889119892119890119904 119894++) (18) 119878

119894= 120583119871119894minus ]119871119894lowast 120587119871119894

(19) 119909 = Adjacent from Node of 119871119894

(20) 119910 = Adjacent to Node of 119871119894

(21) 119882119877119909119910= 119878119894

(22)

(23) print119882119877(24) Calculate Optimal path using119882119877(25)

Algorithm 4 IF digraph in travel time

intuitionistic fuzzy set of crossings at which the traffic densityis calculated

119862 = ⟨1198621 08 01⟩ ⟨1198622 05 03⟩ ⟨1198623 06 03⟩

⟨1198624 07 02⟩ ⟨1198625 05 03⟩

(21)

and 119871 is an intuitionistic fuzzy set of roads between twocrossings The degrees of membership 120583

119871(119909119910) and nonmem-

bership ]119871(119909119910)

are calculated as

120583119871(119909119910)

le min (120583119862(119909) 120583119862(119910))

]119871(119909119910)

le max (]119862(119909) ]119862(119910))

(22)

The intuitionistic fuzzy digraph 119877 of the road network isrepresented by the adjacency matrix given below

119877 =

[

[

[

[

[

[

(00 10) (05 03) (00 10) (00 10) (00 10)

(00 10) (00 10) (04 03) (00 10) (00 10)

(06 03) (00 10) (00 10) (05 03) (04 03)

(06 02) (00 10) (05 03) (00 10) (05 03)

(00 10) (04 03) (00 10) (00 10) (00 10)

]

]

]

]

]

]

(23)

The final weights on edges can be calculated by findingthe rank as 119878119871

119894= 120583119871119894 minus ]119871119894 lowast 120587119871119894 The final adjacency matrix

and graph are developed based on these weights as shown inFigure 7

WR =[

[

[

[

[

[

0 044 0 0 0

0 0 031 0 0

057 0 0 044 031

056 0 044 0 044

0 031 0 0 0

]

]

]

]

]

]

(24)

The above weighted adjacency matrix represents the finalweighted digraph WR which can be used for finding theshortestoptimal path between two vertices by any of theknown methods including Djkastra and A star Algorithm 4generates the weighted digraph WR for the given intuition-istic fuzzy graph 119877lowast and uses it to calculate the optimal pathfrom a source node

Algorithm 4 is quite similar to Algorithm 3 This algo-rithm initially sets an intuitionistic fuzzy set of crossingsLines 3ndash6 calculate the values of degrees of membership andnonmembership for roads which are assigned to intuitionis-tic fuzzy set of edges in line 7 and then the adjacency matrixis prepared in line 8 Finally a weighted adjacency matrixis calculated in lines 9ndash12 using rank techniques based ondegrees of membership and nonmembership This weightedmatrix that is printed on line 13 can be used for calculatingthe shortest path using any known algorithm like Djkastra orA star in line 14

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 The Scientific World Journal

C1 C2

(05 03)(08 01)

C3

(06 03)

C4

(07 02)C5

(05 03)

(05 03)

(05 03)

(05 03)

(04 03)

(04

03

)

(04

03

)

(06 03)

(06

02

)

Figure 6 Intuitionistic fuzzy digraph of a road network

C1 C2

C3

C4 C5

044

057

031

031

031

044

044

056

Figure 7 Weighted digraph of a road network

4 Conclusions

Fuzzy digraph theory has numerous applications in modernsciences and technology especially in the fields of operationsresearch neural networks artificial intelligence and decisionmaking An intuitionistic fuzzy set is a generalization of afuzzy set Intuitionistic fuzzy models give more precision

flexibility and compatibility to the system as compared tothe fuzzy models We have discussed several intuitionisticfuzzy intelligent systems in this paper The natural extensionof this research work is application of intuitionistic fuzzydigraphs in the area of soft computing including neuralnetworks decision making and geographical informationsystemsWe plan to extend our research of fuzzification to (1)

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 11

application of fuzzy soft graphs in decision support systems(2) application of rough graphs in decision support systemsand (3) application of bipolar fuzzy graphs in decisionsupport systems

Conflict of Interests

The authors declare that they do not have any conflict ofinterests regarding the publication of this paper

Acknowledgment

The authors are highly grateful to the anonymous referees fortheir insightful comments and valuable suggestions

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[2] A Kauffman Introduction a la Theorie des Sous-EmsemblesFlous vol 1 Masson et Cie 1973

[3] L A Zadeh ldquoSimilarity relations and fuzzy orderingsrdquo Informa-tion Sciences vol 3 no 2 pp 177ndash200 1971

[4] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[5] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 1998ndash2001

[7] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[8] K T Atanassov Intuitionistic Fuzzy Sets Theory and Appli-cations Studies in Fuzziness and Soft Computing PhysicaHeidelberg Germany 2012

[9] M Akram and B Davvaz ldquoStrong intuitionistic fuzzy graphsrdquoFilomat vol 26 no 1 pp 177ndash195 2012

[10] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013

[11] M Akram andNO Al-Shehrie ldquoIntuitionistic fuzzy cycles andintuitionistic fuzzy treesrdquoThe ScientificWorld Journal vol 2014Article ID 305836 11 pages 2014

[12] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[13] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo VII ITKRrsquos SessionDeposed in Central for Science-Technical Library of BulgarianAcademy of Sciences 169784 Sofia Bulgaria June 1983(Bulgarian)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of