fuzzy dijkstra algorithm in shortest path problem under uncertain environment

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FuzzyDijkstraalgorithmforshortestpathproblemunderuncertainenvironment

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YuxinChen

MINESParisTech

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SEEPROFILE

AvailablefromYuxinChen

Retrievedon07September2015

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Applied Soft Computing 12 (2012) 1231ndash1237

Contents lists available at SciVerse ScienceDirect

Applied Soft Computing

j ourna l ho mepage wwwelsev ier com locate asoc

hort communication

uzzy Dijkstra algorithm for shortest path problem under uncertain environment

ong Dengabclowast Yuxin Chenb Yajuan Zhanga Sankaran Mahadevanc

School of Computer and Information Sciences Southwest University Chongqing 400715 ChinaSchool of Electronics and Information Technology Shanghai Jiao Tong University Shanghai 200240 ChinaSchool of Engineering Vanderbilt University Nashville TN 37235 USA

r t i c l e i n f o

rticle historyeceived 7 June 2011eceived in revised form 18 October 2011ccepted 1 November 2011

a b s t r a c t

A common algorithm to solve the shortest path problem (SPP) is the Dijkstra algorithm In thispaper a generalized Dijkstra algorithm is proposed to handle SPP in an uncertain environment Twokey issues need to be addressed in SPP with fuzzy parameters One is how to determine the addi-tion of two edges The other is how to compare the distance between two different paths with

vailable online 20 November 2011

eywordshortest path problemijkstra algorithmuzzy numbersraded mean representation

their edge lengths represented by fuzzy numbers To solve these problems the graded mean inte-gration representation of fuzzy numbers is adopted to improve the classical Dijkstra algorithm Anumerical example of a transportation network is used to illustrate the efficiency of the proposedmethod

copy 2011 Elsevier BV All rights reserved

Introduction

In a transportation management system it is necessary torovide the shortest path from a specified origin node to otherestination nodes [12] In a network the lengths of the arcs aressumed to represent transportation time or cost rather than theeographical distances Consider an acyclic directed network G(N) consisting of a set of nodes N = 1 2 n and m directed arcs

sube N times N Each arc is denoted by an ordered pair (i j) where i j isin Nt is assumed that there is only one directed arc (i j) from i to jlthough in classical graph theory the weights of the edges in ahortest path finding are represented as real numbers most practi-al applications however have parameters that are not precise (egosts capacities demands time etc) [3] Fuzzy set theory providehe ability to handle vague information and is widely used in manyelds such as environmental assessment [45] pattern recognition6ndash8] decision making [9ndash12]

In finding the shortest path under uncertain environment anppropriate modelling approach is to make use of fuzzy numberss a result many researchers have paid attention to the fuzzy short-st path problem (FSPP) [13ndash22]

One of the most used methods to solve the shortest path prob-em is the Dijkstra algorithm In the case of crisp number to modelrc lengths the Dijkstra algorithm can be easily to implemented

lowast Corresponding author at School of Computer and Information Sciences South-est University Chongqing 400715 China Fax +86 023 68254555

E-mail addresses ydengswueducn professordenghotmailcom (Y Deng)

568-4946$ ndash see front matter copy 2011 Elsevier BV All rights reservedoi101016jasoc201111011

However due to the reason that many optimization methods forcrisp numbers cannot be applied directly to fuzzy numbers somemodifications are needed before using the classical methods Forexample one straight forward approach is to transform the fuzzynumber into a crisp one A typical work [23] transforms the trape-zoidal number into crisp number by the defuzzification functionalso called Yagerrsquos ranking index [24] Generally speaking there aretwo main issues that need to be solved when applying the Dijkstraalgorithm in a fuzzy environment One is the summing operationof fuzzy numbers the other is the ranking and comparison of fuzzynumbers which is still an open issue in fuzzy set theory researchfields

The canonical representation of operations on triangular fuzzynumbers that are based on the graded mean integration represen-tation method leads to the result that multiplication and addition oftwo fuzzy numbers can be represented as a crisp number [25] Thismethod is widely used in many applications such as multi-criteriadecision making [26ndash36] risk evaluation [37] portfolio selection[38] evaluation of enterprise knowledge management capabilities[39] product adoption [40] evaluation of airline service quality[41] and efficient network selection in heterogeneous wireless net-works [42]

In this paper the classical Dijkstra algorithm is generalizedbased on the canonical representation of operations on triangu-lar fuzzy numbers to handle the fuzzy shortest path problem

Compared with existing methods the proposed method is moreefficient due to the fact that the summing operation and the rank-ing of fuzzy numbers can be done is a easy and straight manner Thepaper is organized as follows Section 2 gives a brief introduction to

1232 Y Deng et al Applied Soft Computing 12 (2012) 1231ndash1237

a b c

1

0X

( )A x

ttmup

2

i

2

DA[A

Dc

exist

a

Daf

Fig 1 A triangular fuzzy number

he basic theory used in our proposed method including fuzzy setheory and the Dijkstra algorithm Section 3 develops the proposed

ethod in detail In Section 4 a numerical shortest path examplender fuzzy environment is used to illustrate the efficiency of theroposed method Section 5 concludes the paper

Preliminaries

In this section some basic concepts are briefly introducedncluding fuzzy sets theory and dynamic programming

1 Fuzzy numbers

efinition 21 ldquoFuzzy setrdquo Let X be a universe of discourse Where˜ is a fuzzy subset of X and for all x isin X there is a number A(x) isin0 1] which is assigned to represent the membership degree of x in˜ and is called the membership function of A [13]

efinition 22 ldquoFuzzy numberrdquo A fuzzy number A is a normal andonvex fuzzy subset of X [13]

Here ldquonormalityrdquo implies that

x isin R orxA(x) = 1 (1)

nd ldquoconvexrdquo means that

forallx1 isin X x2 isin X forall isin [0 1] A(˛x1 + (1 minus ˛)x2)

ge min(A(x1) A(x2)) (2)

efinition 23 A triangular fuzzy number A can be defined by triplet (a b c) where the membership can be determined asollows

A triangular fuzzy number A = (a b c) can be shown in Fig 1

A(x) =

⎧⎪⎪⎪⎪⎪⎨0 x lt a

x minus a

b minus a a le x le b

c minus x(3)

⎪⎪⎪⎪⎪⎩ c minus b

b le x le c

0 x gt c

Fig 2 A trapezoidal fuzzy number a

Definition 24 A trapezoidal fuzzy number A can be defined asA = (a1 a2 a3 a4) where the membership can be determined asfollows and shown in Fig 2

A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 x le a1

x minus a1

a2 minus a1a1 le x le a2

1 a2 le x le a3

a4 minus x


0 a4 le x

(4)

22 Canonical representation of operations on fuzzy numbers

In this paper the canonical representation of operations ontriangular fuzzy numbers which are based on the graded mean inte-gration representation method [25] is used to obtain a simple fuzzyshortest path algorithm

Definition 25 Given a triangular fuzzy number A = (a1 a2 a3)the graded mean integration representation of triangular fuzzynumber A is defined as

P(A) = 16

(a1 + 4 times a2 + a3) (5)

Let A = (a1 a2 a3) and B = (b1 b2 b3) be two triangular fuzzynumbers By applying Eq (5) the graded mean integration rep-resentation of triangular fuzzy numbers A and B can be obtainedrespectively as follows

P(A) = 16

(a1 + 4 times a2 + a3)

P(B) = 16

(b1 + 4 times b2 + b3)

The representation of the addition operation oplus on triangular fuzzynumbers A and B can be defined as

P(A oplus B) = P(A) + P(B) = 16

(a1 + 4 times a2 + a3) + 16

(b1 + 4 times b2 + b3)

(6)

The canonical representation of the multiplication operation ontriangular fuzzy numbers A and B is defined as

P(A otimes B) = P(A) times P(B) = 16

(a1 + 4 times a2 + a3) times 16

(b1 + 4 times b2 + b3)

(7)

Y Deng et al Applied Soft Computing 12 (2012) 1231ndash1237 1233

Table 1The arc lengths of the network

Arc Membership function Arc Membership function Arc Membership function

(12) (12131517) (710) (9101213) (1518) (891113)(13) (9111315) (711) (6789) (1519) (571012)(14) (8101213) (812) (58910) (1620) (9121416)(15) (78910) (813) (35810) (1720) (7101112)(26) (5101516) (916) (67910) (1721) (67810)(27) (6111113) (1016) (12131617) (1821) (15171819)(38) (10111617) (1017) (15192021) (1822) (3579)(47) (17202224) (1114) (891113) (1823) (57911)(411) (6101314) (1117) (691113) (1922) (15161719)(58) (691113) (1214) (13141618) (2023) (13141617)(511) (7101314) (1215) (12141516) (2123) (12151718)(512) (10131517) (1315) (10

(17(11

D(t

P

Stati

(69) (681011) (1319)

(610) (10111415) (1421)

efinition 26 Given a trapezoidal fuzzy number A =a1 a2 a3 a4) the graded mean integration representation ofriangular fuzzy number A is defined as

(A) = 16

(a1 + 2 times a2 + 2 times a3 + a4) (8)

imilar to the triangular fuzzy numbers the graded mean integra-

ion representation of operations on trapezoidal fuzzy numbers canlso obtained Let A = (a1 a2 a3 a4) and B = (b1 b2 b3 b4) be tworapezoidal fuzzy numbers By applying Eq (8) the graded meanntegration representation of the addition operation of trapezoidal

Fig 3 Pseudocode of the propose

121415) (2223) (4568)181920)121314)

fuzzy numbers A and B can be defined as

P(A oplus B) = 16

(a1 + 2 times a2 + 2 times a3 + a4)

+16

(b1 + 2 times b2 + 2 times b3 + b4) (9)

The multiplication operation on trapezoidal fuzzy numbers A andB is defined as

P(A otimes B) = 16


times16

(b1 + 2 times b2 + 2 times b3 + b4) (10)

d fuzzy Dijkstra algorithm

1 Computing 12 (2012) 1231ndash1237

Ft1tcl

trwmi

2

e

Fig 4 Pseudocode for shortest path from source to target

234 Y Deng et al Applied Soft

or example see Fig 5 and Table 1 From node 17 there arewo routes to node 23 One is 17 rarr 20 rarr 23 and the other is7 rarr 21 rarr 23 The use of the canonical representation of the addi-ion operation on fuzzy numbers in shortest path finding probleman be illustrated as follows For the first route 17 rarr 20 rarr 23 theength can be obtained as

Arc1(17 23)= Arc(17 20) oplus Arc(20 23)= (7 10 11 12) oplus (13 14 16 17)

= 16

(7 + 2 times 10 + 2 times 11 + 12) + 16

(13 + 2 times 14 + 2 times 16 + 17)

= 1516

For the second route the length can be obtained as


= 16

(6 + 2 times 7 + 2 times 8 + 10) + 16

(12 + 2 times 15 + 2 times 17 + 18)

= 1406

The result shows that the former is worse than the latter since151

6 gt 1406 As can be seen from the example above one merit of

he canonical representation of the addition operation is that itsesult is a crisp number The decision making can be easily obtainedithout the process of ranking fuzzy numbers commonly used inany other fuzzy shortest path problems This is very advantageous

n the Dijkstra algorithm under fuzzy environment

3 Dijkstra algorithm

Dijkstrarsquos algorithm was conceived by the Dutch computer sci-ntist Edsger Dijkstra in 1956 and published in 1959 [43]

Fig 6 The first three steps of fuzzy Dijkst

Fig 5 A transportation network

For a given source vertex (node) in the graph the algorithm findsthe path with lowest cost (ie the shortest path) between that ver-

tex and every other vertex It can also be used for finding costs ofshortest paths from a single vertex to a single destination vertexby stopping the algorithm once the shortest path to the destina-tion vertex has been determined For example if the vertices of the

ra algorithm in numerical example


ra algo

gtaa

ngv

1

23

4

5

[

Fig 7 Step 22 of the fuzzy Dijkst

raph represent cities and edge path costs represent driving dis-ances between pairs of cities connected by a direct road Dijkstrarsquoslgorithm can be used to find the shortest route between one citynd any other city

Let the node at the beginning of the path be called the originode Let the distance of node Y be the distance from the ori-in node to Y Dijkstrarsquos algorithm will assign some initial distancealues and will try to improve them step by step

Assign to every node a distance value set it to zero for the originnode and to infinity for all other nodes

Mark all nodes as unvisited Set initial node as current

For current node consider all its unvisited neighbors and calcu-late their tentative distance For example if current node A hasdistance of 6 and an edge connecting it with another node Bhas length 2 the distance to B through A will be 6 + 2 = 8 If thisdistance is less than the previously recorded distance overwritethe distance

Considering all neighbors of the current node mark it as visitedA visited node will not be checked again its distance recordedis final and minimal

If all nodes have been visited stop Otherwise set the unvisitednode with the smallest distance (from the initial node consid-ering all nodes in the graph) as the next ldquocurrent noderdquo and

continue from step 3

The details information of Dijkstrarsquos algorithm can be found in44]

rithm in the numerical example

3 Proposed method

In a fuzzy environment two issues namely the addition of fuzzynumbers and ranking of numbers should be solved Based on thecanonic representation [25] the classical Dijkstra algorithm can beeasily generalized to a fuzzy Dijkstra algorithm as follows

In the proposed method it is necessary to clarify some basicnotation Q is the set of all unvisited vertex (nodes) that are tobe removed In set Q u represents the vertex with the smallestdistance from the source node Let v be one variant (cf Fig 3)dist[v] denote the current shortest distance between v and thesource node distminusbetween(u v) the distance between two verticesu v previous[v] the nearest vertex to the current v along the shortestpath from the source node The sign lsquoaltrsquo is an alternative vari-ant of distance used for comparison At the initial state for eachvertex v in graph (excluding the source node) dist[v]=infinity andprevious[v]=0 and Q is the set of all nodes in graph For the sourcenode dist[source]=0 Then the following three steps are repeatedtill completion

1 Check if Q is empty While Q is not empty seek u in Q If dist[u]is not infinity then from Q u is removed (this means that allremaining vertices are inaccessible from source and the analysisis completed)

2 After finding v each neighbor node of u and calculate alt =dist[u] + distbetween(u v) by using the canonical representationoperation introduced in Section 2 If there exists alt lt dist[v]then replace dist[v] with alt and record the previous [v] = u (if


kstra a

3

iNisp

4

ppFti

nn

Ut

Fig 8 The final step of fuzzy Dij

not the previous record of each neighbor vertex v remains thesame)

Choose the one among each v with shortest dist[v] and replaceit with u and go to Step 1 The pseudocode of the proposed fuzzyDijkstra algorithm is shown in Fig 3

If only the shortest path between a single source and target pairs of interest the search can be terminated at line 13 if u = targetow the shortest path from origin to target can be found as shown

n Fig 4 Sequence S is the list of vertices constituting one of thehortest paths from source to target or the empty sequence if noath exists

Transportation network application

In this section a numerical example of the fuzzy shortest pathroblem illustrated in [1718] is used to show the efficiency of theroposed method Consider the transportation network shown inig 5 with 23 nodes and 40 arcs It is assumed that all arc lengths arerapezoidal fuzzy numbers with membership functions as shownn Table 1

The first three steps are detailed in Fig 6 Let S be the set of visitedodes representing the shortest path and U the set of unvisited

odes which are to be settled

In Step 1 node (1) which is the source node is moved from to S and the shortest distance from (1) to (1) is zero while

he distance from source node to accessible neighbor nodes are

lgorithm in numerical example

calculated Among them the shortest one (1)rarr(5) is picked outalong with distance value 85

In Step 2 we move (5) from U to S this time the seeking ofshortest path should be begun from (5) The shortest distance from(1) rarr (5) to its neighbors are calculated None of them is shorterthan the path (1)rarr(4) by comparing all the records in the columnSet U

So in Step 3 (4) is moved and all the data starts with (4) and (1)(3) with distance 12000 is the result

Due to the limitation of space the other steps are not illus-trated Only Steps 22 and 23 the final two steps in this numericalexample are shown in Figs 7 and 8 respectively At the endof Step 21 the shortest path is (1)rarr(5)rarr(11)rarr(17)rarr(21)rarr(23)However there are no neighbor nodes accessible from (23) andonly (1)rarr(5)rarr(12)rarr(15)rarr(18)rarr(22) is found from the previ-ous record of column Set U As (22) is drawn to S U becomes anempty set and the search is complete The shortest path whoselength is 5283 based on the proposed fuzzy Dijkstra algorithmis (1)rarr(5)rarr(11)rarr(17)rarr(21)rarr(23) which is the same in Refs[1718]

5 Conclusion

This paper extended the Dijkstra algorithm to solve the shortest

path problem with fuzzy arc lengths Two key issues are addressedOne is how to determine the addition of two edges The other ishow to compare the distance between two different paths whentheir edges length are represented by fuzzy numbers The proposed

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Computing Journal 11 (2011) 3734ndash3743

Y Deng et al Applied Soft

ethod to find the shortest path under fuzzy arc lengths is basedn the graded mean integration representation of fuzzy numbers

numerical example was used to illustrate the efficiency of theroposed method The proposed method can be applied to realpplications in transportation systems logistics management andany other network optimization problem that can be formulated

s shortest path problemIt should be pointed out that the uncertainty in the shortest path

roblem is not limited to the geometric distance For example dueo the weather and other unexpected factors the travel time fromne city to another city may be represented as a fuzzy variable evenf the geometric distance is fixed

cknowledgements

The authors thank the anonymous reviews for their valuableuggestions The PhD students of the first author in Shanghaiiao Tong University XY Su and PD Xu provided valuable dis-ussions The work is partially supported by National Naturalcience Foundation of China Grant Nos 60874105 61174022rogram for New Century Excellent Talents in University Granto NCET-08-0345 Chongqing Natural Science Foundation Granto CSCT 2010BA2003 Aviation Science Foundation Grant No0090557004 the Fundamental Research Funds for the Centralniversities Grant No XDJK2010C030 the Fundamental Researchunds for the Central Universities Grant No XDJK2011D002 theouthwest University Scientific amp Technological Innovation Fundor Graduates Grant No ky2011011 and Doctor Funding of South-est University Grant No SWU110021

eferences

[1] M Xu Y Liu Q Huang Y Zhang G Luan An improved Dijkstra shortestpath algorithm for sparse network Applied Mathematics and Computation 185(2007) 247ndash254

[2] X Lu M Camitz Finding the shortest paths by node combination AppliedMathematics and Computation 217 (2011) 6401ndash6408

[3] TN Chuang JY Kung The fuzzy shortest path length and the correspond-ing shortest path in a network Computers amp Operations Research 32 (2005)1409ndash1428

[4] R Sadiq S Tesfamariam Developing environmental using fuzzy numbersordered weighted averaging (FN-OWA) operators Stochastic EnvironmentalResearch and Risk Assessment 22 (2008) 495ndash505

[5] Y Deng W Jiang R Sadiq Modeling contaminant intrusion in water distri-bution networks a new similarity-based DST method Expert Systems withApplications 38 (2011) 571ndash578

[6] Y Deng WK Shi F Du Q Liu A new similarity measure of generalized fuzzynumbers and its application to pattern recognition Pattern Recognition Letters25 (2004) 875ndash883

[7] HW Liu New similarity measures between intuitionistic fuzzy sets andbetween elements Mathematical and Computer Modelling 42 (2005) 61ndash70

[8] J Ye Cosine similarity measures for intuitionistic fuzzy sets and their applica-tions Mathematical and Computer Modelling 53 (2011) 91ndash97

[9] Y Deng Plant location selection based on fuzzy topsis International Journal ofAdvanced Manufacturing Technology 28 (2006) 839ndash844

10] Y Deng FTS Chan A new fuzzy Dempster MCDM method and its applicationin supplier selection Expert Systems with Applications 38 (2011) 6985ndash6993

11] Y Deng FTS Chan Y Wu D Wang A new linguistic MCDM method basedon multiple-criterion data fusion Expert Systems with Applications 38 (2011)9854ndash9861

12] Y Deng R Sadiq W Jiang S Tesfamariam Risk analysis in a linguistic envi-ronment a fuzzy evidential reasoning-based approach Expert Systems withApplications (2011) doi101016jeswa201106018

13] D Dubois H Prade Fuzzy Sets and Systems Theory and Applications AcademicPress New York 1980

14] C Lin MS Chern The fuzzy shortest path problem and its most vital arcs FuzzySets and Systems 58 (1993) 343ndash353

15] A Boulmakoul Generalized path-finding algorithms on semirings and the fuzzy

shortest path problem Journal of Computational and Applied Mathematics 162(2004) 263ndash272

16] TN Chuang JY Kung A new algorithm for the discrete fuzzy shortest pathproblem in a network Applied Mathematics and Computation 174 (2006)1660ndash1668

[

[

ting 12 (2012) 1231ndash1237 1237

17] X Ji K Iwamura Z Shao New models for shortest path problem with fuzzy arclengths Applied Mathematical Modelling 31 (2007) 259ndash269

18] I Mahdavi R Nourifar A Heidarzade NM Amiri A dynamic programmingapproach for finding shortest chains in a fuzzy network Applied Soft Comput-ing 9 (2009) 503ndash511

19] M Ghatee SM Hashemi M Zarepisheh E Khorram Preemptive priority-based algorithms for fuzzy minimal cost flow problem an application inhazardous materials transportation Computers amp Industrial Engineering 57(2009) 341ndash354

20] M Ghatee SM Hashemi Application of fuzzy minimum cost flow problemsto network design under uncertainty Fuzzy Sets and Systems 160 (2009)3263ndash3289

21] E Keshavarz E Khorram A fuzzy shortest path with the highest reliabilityJournal of Computational and Applied Mathematics 230 (2009) 204ndash212

22] A Tajdin I Mahdavi NM Amiri B Sadeghpour-Gildeh Computing a fuzzyshortest path in a network with mixed fuzzy arc lengths using a-cuts Comput-ers and Mathematics with Applications 60 (2010) 989ndash1002

23] ST Liu C Kao Network flow problems with fuzzy arc lengths IEEE Transactionon Systems Man and Cybernetics Part B Cybernetics 34 (2004) 765ndash769

24] RR Yager A procedure for ordering fuzzy subsets of the unit interval Informa-tion Sciences 24 (1981) 143ndash151

25] CC Chou The canonical representation of multiplication operation on triangu-lar fuzzy numbers Computers and Mathematics with Applications 45 (2003)1601ndash1610

26] CC Chou PC Chang A fuzzy multiple criteria decision making model forselecting the distribution center location in china a Taiwanese manufacturerrsquosperspective Lecture Notes in Computer Science (including subseries LectureNotes in Artificial Intelligence and Lecture Notes in Bioinformatics) 5618 (2009)140ndash148

27] CC Chou Application of FMCDM model to selecting the hub location in themarine transportation A case study in southeastern Asia Mathematical andComputer Modelling 51 (2010) 791ndash801

28] CC Chou RH Gou CL Tsai MC Tsou CP Wong HL Yu Application ofa mixed fuzzy decision making and optimization programming model tothe empty container allocation Applied Soft Computing Journal 10 (2010)1071ndash1079

29] CC Chou FT Kuo RH Gou CL Tsai CP Wong MC Tsou Application ofa combined fuzzy multiple criteria decision making and optimization pro-gramming model to the container transportation demand split Applied SoftComputing Journal 10 (2010) 1080ndash1086

30] CC Chou An integrated quantitative and qualitative fmcdm model for locationchoices Soft Computing 14 (2010) 757ndash771

31] CC Chou A fuzzy backorder inventory model and application to determin-ing the optimal empty-container quantity at a port International Journal ofInnovative Computing Information and Control 5 (2009) 4825ndash4834

32] I Chamodrakas I Leftheriotis D Martakos In-depth analysis and simulationstudy of an innovative fuzzy approach for ranking alternatives in multipleattribute decision making problems based on topsis Applied Soft ComputingJournal 11 (2011) 900ndash907

33] M Tseng A Chiu Evaluating firmrsquos green supply chain manage-ment in linguistic preferences Journal of Cleaner Production (2010)doi101016jjclepro201008007

34] M Tseng Green supply chain management with linguistic preferences andincomplete information Applied Soft Computing 11 (2011) 4894ndash4903

35] M Tseng Using a hybrid mcdm model to evaluate firm environmentalknowledge management in uncertainty Applied Soft Computing 11 (2011)1340ndash1352

36] M Tseng Using linguistic preferences and grey relational analysis to evalu-ate the environmental knowledge management capacity Expert systems withapplications 37 (2010) 70ndash81

37] X Pan H Wang W Chang A fuzzy synthetic evaluation method for failurerisk of aviation product RampD project in 5th IEEE International Confer-ence on Management of Innovation and Technology ICMIT 2010 pp 1106ndash1111

38] W Chen S Tan Fuzzy portfolio selection problem under uncertain exit timeIEEE International Conference on Fuzzy Systems 5 (2009) 550ndash554

39] RG Qi SG Liu Research on comprehensive evaluation of enterprisesknowledge management capabilities in 2010 International Conference onManagement Science and Engineering ICMSE 2010 pp 1036ndash1301

40] S Kim K Lee JK Cho CO Kim Agent-based diffusion model for an automobilemarket with fuzzy topsis-based product adoption process Expert Systems withApplications 38 (2011) 7270ndash7276

41] CC Chou LJ Liu SF Huang JM Yih TC Han An evaluation of airline servicequality using the fuzzy weighted SERVQUAL method Applied Soft ComputingJournal 11 (2011) 2117ndash2128

42] I Chamodrakas D Martakos A utility-based fuzzy topsis method for energyefficient network selection in heterogeneous wireless networks Applied Soft

43] EW Dijkstra A note on two problems in connexion with graphs NumerischeMathematik 1 (1959) 269ndash271

44] TH Cormen CE Leiserson RL Rivest C Stein Introduction to Algorithms MITPress and McGraw-Hill 2001

S

F

Ya

b

c

a

ARRAA

KSDFG

1

pdagAAIAscctfi[

aAe

la

w

1d

Applied Soft Computing 12 (2012) 1231ndash1237

Contents lists available at SciVerse ScienceDirect

Applied Soft Computing

j ourna l ho mepage wwwelsev ier com locate asoc

hort communication

uzzy Dijkstra algorithm for shortest path problem under uncertain environment

ong Dengabclowast Yuxin Chenb Yajuan Zhanga Sankaran Mahadevanc

School of Computer and Information Sciences Southwest University Chongqing 400715 ChinaSchool of Electronics and Information Technology Shanghai Jiao Tong University Shanghai 200240 ChinaSchool of Engineering Vanderbilt University Nashville TN 37235 USA

r t i c l e i n f o

rticle historyeceived 7 June 2011eceived in revised form 18 October 2011ccepted 1 November 2011

a b s t r a c t

A common algorithm to solve the shortest path problem (SPP) is the Dijkstra algorithm In thispaper a generalized Dijkstra algorithm is proposed to handle SPP in an uncertain environment Twokey issues need to be addressed in SPP with fuzzy parameters One is how to determine the addi-tion of two edges The other is how to compare the distance between two different paths with

vailable online 20 November 2011

eywordshortest path problemijkstra algorithmuzzy numbersraded mean representation

their edge lengths represented by fuzzy numbers To solve these problems the graded mean inte-gration representation of fuzzy numbers is adopted to improve the classical Dijkstra algorithm Anumerical example of a transportation network is used to illustrate the efficiency of the proposedmethod

copy 2011 Elsevier BV All rights reserved

Introduction

In a transportation management system it is necessary torovide the shortest path from a specified origin node to otherestination nodes [12] In a network the lengths of the arcs aressumed to represent transportation time or cost rather than theeographical distances Consider an acyclic directed network G(N) consisting of a set of nodes N = 1 2 n and m directed arcs

sube N times N Each arc is denoted by an ordered pair (i j) where i j isin Nt is assumed that there is only one directed arc (i j) from i to jlthough in classical graph theory the weights of the edges in ahortest path finding are represented as real numbers most practi-al applications however have parameters that are not precise (egosts capacities demands time etc) [3] Fuzzy set theory providehe ability to handle vague information and is widely used in manyelds such as environmental assessment [45] pattern recognition6ndash8] decision making [9ndash12]

In finding the shortest path under uncertain environment anppropriate modelling approach is to make use of fuzzy numberss a result many researchers have paid attention to the fuzzy short-st path problem (FSPP) [13ndash22]

One of the most used methods to solve the shortest path prob-em is the Dijkstra algorithm In the case of crisp number to modelrc lengths the Dijkstra algorithm can be easily to implemented

lowast Corresponding author at School of Computer and Information Sciences South-est University Chongqing 400715 China Fax +86 023 68254555

E-mail addresses ydengswueducn professordenghotmailcom (Y Deng)

568-4946$ ndash see front matter copy 2011 Elsevier BV All rights reservedoi101016jasoc201111011

However due to the reason that many optimization methods forcrisp numbers cannot be applied directly to fuzzy numbers somemodifications are needed before using the classical methods Forexample one straight forward approach is to transform the fuzzynumber into a crisp one A typical work [23] transforms the trape-zoidal number into crisp number by the defuzzification functionalso called Yagerrsquos ranking index [24] Generally speaking there aretwo main issues that need to be solved when applying the Dijkstraalgorithm in a fuzzy environment One is the summing operationof fuzzy numbers the other is the ranking and comparison of fuzzynumbers which is still an open issue in fuzzy set theory researchfields

The canonical representation of operations on triangular fuzzynumbers that are based on the graded mean integration represen-tation method leads to the result that multiplication and addition oftwo fuzzy numbers can be represented as a crisp number [25] Thismethod is widely used in many applications such as multi-criteriadecision making [26ndash36] risk evaluation [37] portfolio selection[38] evaluation of enterprise knowledge management capabilities[39] product adoption [40] evaluation of airline service quality[41] and efficient network selection in heterogeneous wireless net-works [42]

In this paper the classical Dijkstra algorithm is generalizedbased on the canonical representation of operations on triangu-lar fuzzy numbers to handle the fuzzy shortest path problem

Compared with existing methods the proposed method is moreefficient due to the fact that the summing operation and the rank-ing of fuzzy numbers can be done is a easy and straight manner Thepaper is organized as follows Section 2 gives a brief introduction to


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Preliminaries


1 Fuzzy numbers







ge min(A(x1) A(x2)) (2)



A(x) =

⎧⎪⎪⎪⎪⎪⎨0 x lt a

x minus a


c minus x(3)


b le x le c

0 x gt c



A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 x le a1

x minus a1


1 a2 le x le a3

a4 minus x


0 a4 le x

(4)




P(A) = 16

(a1 + 4 times a2 + a3) (5)


P(A) = 16

(a1 + 4 times a2 + a3)

P(B) = 16

(b1 + 4 times b2 + b3)


P(A oplus B) = P(A) + P(B) = 16

(a1 + 4 times a2 + a3) + 16

(b1 + 4 times b2 + b3)

(6)




(b1 + 4 times b2 + b3)

(7)




(12) (12131517) (710) (9101213) (1518) (891113)(13) (9111315) (711) (6789) (1519) (571012)(14) (8101213) (812) (58910) (1620) (9121416)(15) (78910) (813) (35810) (1720) (7101112)(26) (5101516) (916) (67910) (1721) (67810)(27) (6111113) (1016) (12131617) (1821) (15171819)(38) (10111617) (1017) (15192021) (1822) (3579)(47) (17202224) (1114) (891113) (1823) (57911)(411) (6101314) (1117) (691113) (1922) (15161719)(58) (691113) (1214) (13141618) (2023) (13141617)(511) (7101314) (1215) (12141516) (2123) (12151718)(512) (10131517) (1315) (10

(17(11

D(t

P

Stati

(69) (681011) (1319)

(610) (10111415) (1421)


(A) = 16

(a1 + 2 times a2 + 2 times a3 + a4) (8)




121415) (2223) (4568)181920)121314)


P(A oplus B) = 16


+16

(b1 + 2 times b2 + 2 times b3 + b4) (9)


P(A otimes B) = 16


times16

(b1 + 2 times b2 + 2 times b3 + b4) (10)



Ft1tcl

trwmi

2

e





= 16

(7 + 2 times 10 + 2 times 11 + 12) + 16

(13 + 2 times 14 + 2 times 16 + 17)

= 1516



= 16

(6 + 2 times 7 + 2 times 8 + 10) + 16

(12 + 2 times 15 + 2 times 17 + 18)

= 1406













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3 Proposed method






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5 Conclusion



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a b c

1

0X

( )A x

ttmup

2

i

2

DA[A

Dc

exist

a

Daf




Preliminaries


1 Fuzzy numbers







ge min(A(x1) A(x2)) (2)



A(x) =

⎧⎪⎪⎪⎪⎪⎨0 x lt a

x minus a


c minus x(3)


b le x le c

0 x gt c



A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 x le a1

x minus a1


1 a2 le x le a3

a4 minus x


0 a4 le x

(4)




P(A) = 16

(a1 + 4 times a2 + a3) (5)


P(A) = 16

(a1 + 4 times a2 + a3)

P(B) = 16

(b1 + 4 times b2 + b3)


P(A oplus B) = P(A) + P(B) = 16

(a1 + 4 times a2 + a3) + 16

(b1 + 4 times b2 + b3)

(6)




(b1 + 4 times b2 + b3)

(7)




(12) (12131517) (710) (9101213) (1518) (891113)(13) (9111315) (711) (6789) (1519) (571012)(14) (8101213) (812) (58910) (1620) (9121416)(15) (78910) (813) (35810) (1720) (7101112)(26) (5101516) (916) (67910) (1721) (67810)(27) (6111113) (1016) (12131617) (1821) (15171819)(38) (10111617) (1017) (15192021) (1822) (3579)(47) (17202224) (1114) (891113) (1823) (57911)(411) (6101314) (1117) (691113) (1922) (15161719)(58) (691113) (1214) (13141618) (2023) (13141617)(511) (7101314) (1215) (12141516) (2123) (12151718)(512) (10131517) (1315) (10

(17(11

D(t

P

Stati

(69) (681011) (1319)

(610) (10111415) (1421)


(A) = 16

(a1 + 2 times a2 + 2 times a3 + a4) (8)




121415) (2223) (4568)181920)121314)


P(A oplus B) = 16


+16

(b1 + 2 times b2 + 2 times b3 + b4) (9)


P(A otimes B) = 16


times16

(b1 + 2 times b2 + 2 times b3 + b4) (10)



Ft1tcl

trwmi

2

e





= 16

(7 + 2 times 10 + 2 times 11 + 12) + 16

(13 + 2 times 14 + 2 times 16 + 17)

= 1516



= 16

(6 + 2 times 7 + 2 times 8 + 10) + 16

(12 + 2 times 15 + 2 times 17 + 18)

= 1406













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ngv

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3 Proposed method






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nn

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5 Conclusion



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ting 12 (2012) 1231ndash1237 1237
































(12) (12131517) (710) (9101213) (1518) (891113)(13) (9111315) (711) (6789) (1519) (571012)(14) (8101213) (812) (58910) (1620) (9121416)(15) (78910) (813) (35810) (1720) (7101112)(26) (5101516) (916) (67910) (1721) (67810)(27) (6111113) (1016) (12131617) (1821) (15171819)(38) (10111617) (1017) (15192021) (1822) (3579)(47) (17202224) (1114) (891113) (1823) (57911)(411) (6101314) (1117) (691113) (1922) (15161719)(58) (691113) (1214) (13141618) (2023) (13141617)(511) (7101314) (1215) (12141516) (2123) (12151718)(512) (10131517) (1315) (10

(17(11

D(t

P

Stati

(69) (681011) (1319)

(610) (10111415) (1421)


(A) = 16

(a1 + 2 times a2 + 2 times a3 + a4) (8)




121415) (2223) (4568)181920)121314)


P(A oplus B) = 16


+16

(b1 + 2 times b2 + 2 times b3 + b4) (9)


P(A otimes B) = 16


times16

(b1 + 2 times b2 + 2 times b3 + b4) (10)



Ft1tcl

trwmi

2

e





= 16

(7 + 2 times 10 + 2 times 11 + 12) + 16

(13 + 2 times 14 + 2 times 16 + 17)

= 1516



= 16

(6 + 2 times 7 + 2 times 8 + 10) + 16

(12 + 2 times 15 + 2 times 17 + 18)

= 1406













ra algo

gtaa

ngv

1

23

4

5

[












3 Proposed method






kstra a

3

iNisp

4

ppFti

nn

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5 Conclusion



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ting 12 (2012) 1231ndash1237 1237






























Ft1tcl

trwmi

2

e





= 16

(7 + 2 times 10 + 2 times 11 + 12) + 16

(13 + 2 times 14 + 2 times 16 + 17)

= 1516



= 16

(6 + 2 times 7 + 2 times 8 + 10) + 16

(12 + 2 times 15 + 2 times 17 + 18)

= 1406













ra algo

gtaa

ngv

1

23

4

5

[












3 Proposed method






kstra a

3

iNisp

4

ppFti

nn

Ut

















5 Conclusion



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3 Proposed method






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5 Conclusion



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5 Conclusion



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fuzzy dijkstra algorithm in shortest path problem under uncertain environment

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