Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment

Applied Soft Computing 12 (2012) 12311237

Contents lists available at SciVerse ScienceDirect

Applied Soft Computing

j ourna l ho mepage: www.elsev ier .co

Short communication

Fuzzy D ble

Yong Den ahaa School of Comb School of Elec Chinac School of Eng

a r t i c l

Article history:Received 7 JunReceived in reAccepted 1 NoAvailable onlin

Keywords:Shortest path Dijkstra algoriFuzzy numberGraded mean

shortm isSPP ww touzzy bers

ion n

1. Introdu

In a transportation management system, it is necessary toprovide the shortest path from a specied origin node to otherdestination nodes [1,2]. In a network, the lengths of the arcs areassumed to represent transportation time or cost, rather than thegeographicaA), consistinA N N. EIt is assumeAlthough inshortest patcal applicatcosts, capacthe ability telds such a[68], decis

In ndinappropriateAs a result, mest path pro

One of thlem is the Darc lengths

Corresponwest Universit

E-mail add

er, dcrisp numbers cannot be applied directly to fuzzy numbers, somemodications 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 defuzzication function,

1568-4946/$ doi:10.1016/j.l distances. Consider an acyclic directed network G(N,g of a set of nodes N = {1, 2, . . ., n}, and m directed arcsach arc is denoted by an ordered pair (i, j), where i, j N.d that there is only one directed arc (i, j) from i to j.

classical graph theory, the weights of the edges in ah nding are represented as real numbers, most practi-ions, however, have parameters that are not precise (e.g.ities, demands, time, etc.) [3]. Fuzzy set theory provideo handle vague information and is widely used in manys environmental assessment [4,5], pattern recognitionion making [912].g the shortest path under uncertain environment, an

modelling approach is to make use of fuzzy numbers.any researchers have paid attention to the fuzzy short-blem (FSPP) [1322].e most used methods to solve the shortest path prob-ijkstra algorithm. In the case of crisp number to model, the Dijkstra algorithm can be easily to implemented.

ding author at: School of Computer and Information Sciences, South-y, Chongqing 400715, China. Fax: +86 023 68254555.resses: [email protected], [email protected] (Y. Deng).

also called Yagers 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 researchelds.

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 [2636], risk evaluation [37], portfolio selection[38], evaluation of enterprise knowledge management capabilities[39], product adoption [40], evaluation of airline service quality[41] and efcient 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 moreefcient 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

see front matter 2011 Elsevier B.V. All rights reserved.asoc.2011.11.011ijkstra algorithm for shortest path pro

ga,b,c,, Yuxin Chenb, Yajuan Zhanga, Sankaran Mputer and Information Sciences, Southwest University, Chongqing 400715, Chinatronics and Information Technology, Shanghai Jiao Tong University, Shanghai 200240, ineering, Vanderbilt University, Nashville, TN 37235, USA

e i n f o

e 2011vised form 18 October 2011vember 2011e 20 November 2011

problemthmsrepresentation

a b s t r a c t

A common algorithm to solve the paper, a generalized Dijkstra algorithkey issues need to be addressed in tion of two edges. The other is hotheir edge lengths represented by fgration representation of fuzzy numnumerical example of a transportatmethod.

ction Howevm/locate /asoc

m under uncertain environment

devanc

est path problem (SPP) is the Dijkstra algorithm. In this proposed to handle SPP in an uncertain environment. Twoith fuzzy parameters. One is how to determine the addi-

compare the distance between two different paths withnumbers. To solve these problems, the graded mean inte-

is adopted to improve the classical Dijkstra algorithm. Aetwork is used to illustrate the efciency of the proposed

2011 Elsevier B.V. All rights reserved.

ue to the reason that many optimization methods for

1232 Y. Deng et al. / Applied Soft Computing 12 (2012) 12311237

1

0

( )A x

the basic ththeory and method in under fuzzyproposed m

2. Prelimin

In this including fu

2.1. Fuzzy n

Denition 2A is a fuzzy [0, 1] whichA, and is cal

Denition 2convex fuzz

Here, n

x R, x

and conve

x1 X, x min(A

Denition a triplet (afollows.

A triang

A(x) =

tion , a2,s and

0 x a2

1 a4a4

0

noni

his plar f

reprst pa

tion adedr A i

16(a1

A = rs. Bationtively, as follows:

1a b cX

Fig. 1. A triangular fuzzy number.

eory used in our proposed method, including fuzzy setthe Dijkstra algorithm. Section 3 develops the proposeddetail. In Section 4, a numerical shortest path example

environment is used to illustrate the efciency of theethod. Section 5 concludes the paper.

aries

section, some basic concepts are briey introducedzzy sets theory and dynamic programming.

umbers

.1. Fuzzy set: Let X be a universe of discourse. Wheresubset of X; and for all x X, there is a number A(x)

is assigned to represent the membership degree of x inled the membership function of A [13].

.2. Fuzzy number: A fuzzy number A is a normal andy subset of X [13].ormality implies that:

A(x) = 1 (1)

x means that:

2 X, [0, 1], A(x1 + (1 )x2)(x1), A(x2)) (2)

DeniA = (a1follow

A =

2.2. Ca

In ttriangugrationshorte

Denithe grnumbe

P(A) =

Letnumberesentrespec2.3. A triangular fuzzy number A can be dened by, b, c), where the membership can be determined as

ular fuzzy number A = (a, b, c) can be shown in Fig. 1.

0, x < ax ab a , a x bc xc b , b x c

0, x > c

(3)

P(A) =6(a

P(B) = 16(b

The represenumbers A

P(A B) = P

The canonitriangular f

P(A B) = PFig. 2. A trapezoidal fuzzy number a.

2.4. A trapezoidal fuzzy number A can be dened as a3, a4), where the membership can be determined as

shown in Fig. 2:

x a1 a1

a1a1 x a2

a2 x a3 x a3

a3 x a4

a4 x

(4)

cal representation of operations on fuzzy numbers

aper, the canonical representation of operations onuzzy numbers which are based on the graded mean inte-esentation method [25], is used to obtain a simple fuzzyth algorithm.

2.5. Given a triangular fuzzy number A = (a1, a2, a3), mean integration representation of triangular fuzzys dened as:

+ 4 a2 + a3) (5)

(a1, a2, a3) and B = (b1, b2, b3) be two triangular fuzzyy applying Eq. (5), the graded mean integration rep-

of triangular fuzzy numbers A and B can be obtained,1 + 4 a2 + a3)

1 + 4 b2 + b3)

ntation of the addition operation on triangular fuzzyand B can be dened as:

(A) + P(B) = 16(a1 + 4 a2 + a3) +

16(b1 + 4 b2 + b3)

(6)

cal representation of the multiplication operation onuzzy numbers A and B is dened as:

(A) P(B) = 16(a1 + 4 a2 + a3)

16(b1 + 4 b2 + b3)

(7)

Y. Deng et al. / Applied Soft Computing 12 (2012) 12311237 1233

Table 1The arc lengths of the network.

Arc Membership function Arc Membership function Arc Membership function

(1,2) (12,13,15,17) (7,10) (9,10,12,13) (15,18) (8,9,11,13)(1,3) (9,11,13,15) (7,11) (6,7,8,9) (15,19) (5,7,10,12)(1,4) (8,10,12,13) (8,12) (5,8,9,10) (16,20) (9,12,14,16)(1,5) (7,8,9,10) (8,13) (3,5,8,10) (17,20) (7,10,11,12)(2,6) (5,10,15,16) (9,16) (6,7,9,10) (17,21) (6,7,8,10)(2,7) (6,11,11,13) (10,16) (12,13,16,17) (18,21) (15,17,18,19)(3,8) (10,11,16,17) (10,17) (15,19,20,21) (18,22) (3,5,7,9)(4,7) (17,20,22,24) (11,14) (8,9,11,13) (18,23) (5,7,9,11)(4,11) (6,10,13,14) (11,17) (6,9,11,13) (19,22) (15,16,17,19)(5,8) (6,9,11,13) (12,14) (13,14,16,18) (20,23) (13,14,16,17)(5,11) (7,10,13,14) (12,15) (12,14,15,16) (21,23) (12,15,17,18)(5,12) (10,13,15,17) (13,15) (10,12,14,15) (22,23) (4,5,6,8)(6,9) (6,8,10,11) (13,19) (17,18,19,20)(6,10) (10,11,14,15) (14,21) (11,12,13,14)

Denition 2.6. Given a trapezoidal fuzzy number A =(a1, a2, a3, a4), the graded mean integration representation oftriangular fuzzy number A is dened as:

P(A) = 16(a1 + 2 a2 + 2 a3 + a4) (8)

Similar to the triangular fuzzy numbers, the graded mean integra-tion representation of operations on trapezoidal fuzzy numbers canalso obtained. Let A = (a1, a2, a3, a4) and B = (b1, b2, b3, b4) be twotrapezoidal fuzzy numbers. By applying Eq. (8), the graded meanintegration representation of the addition operation of trapezoidal

fuzzy numbers A and B can be dened as

P(A B) = 16(a1 + 2 a2 + 2 a3 + a4)

+16(b1 + 2 b2 + 2 b3 + b4) (9)

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

P(A B) = 16(a1 + 2 a2 + 2 a3 + a4)

16(b1 + 2 b2 + 2 b3 + b4) (10)Fig. 3. Pseudocode of the proposed fuzzy Dijkstra algorithm.


For example, see Fig. 5 and Table 1. From node 17, there aretwo routes to node 23. One is 17 20 23 and the other is17 21 23. The use of the canonical representation of the addi-tion operation on fuzzy numbers in shortest path nding problemcan be illustrated as follows. For the rst route 17 20 23, thelength can be obtained as

Arc1(17, 23)= Arc(17, 20) Arc(20, 23)= (7, 10, 11, 12) (13, 14, 16, 17)= 1

6(7 + 2 10 + 2 11 + 12) + 1

6(13 + 2 14 + 2 16 + 17)

= 1516

For the second route, the length can be obtained as

Arc2(17, 23)= Arc(17, 21) Arc(21, 23)= (6, 7, 8, 10) (12, 15, 17, 18)= 1

6(6 + 2 7 + 2 8 + 10) + 1

6(12 + 2 15 + 2 17 + 18)

= 1406

The result shows that the former is worse than the latter since1516 >

1406 . As can be seen from the example above, one merit of

the canonical representation of the addition operation is that itsresult is a crisp number. The decision making can be easily obtainedwithout the process of ranking fuzzy numbers, commonly used inmany other fuzzy shortest path problems. This is very advantageousin the Dijkstra algorithm under fuzzy environment.

2.3. Dijkstra algorithm

Dijkstras algorithm, was conceived by the Dutch computer sci-entist Edsger Dijkstra in 1956 and published in 1959 [43].

Fig. 4. Pseudocode for shortest path from source to target.

Fig. 5. A transportation network.

For a given source vertex (node) in the graph, the algorithm ndsthe path with lowest cost (i.e. the shortest path) between that ver-tex and every other vertex. It can also be used for nding 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

ithm in numerical example.Fig. 6. The rst three steps of fuzzy Dijkstra algor


graph repretances betwalgorithm cand any oth

Let the nnode. Let tgin node tovalues and

1. Assign tonode and

2. Mark all 3. For curre

late theirdistance has lengtdistance the dista

4. ConsiderA visitedis nal an

5. If all nodnode witering allcontinue

The deta[44].Fig. 7. Step 22 of the fuzzy Dijkstra algorithm in the n

sent cities and edge path costs represent driving dis-een pairs of cities connected by a direct road, Dijkstrasan be used to nd the shortest route between one cityer city.ode at the beginning of the path be called the originhe distance of node Y be the distance from the ori-

Y. Dijkstras algorithm will assign some initial distancewill try to improve them step by step.

every node a distance value: set it to zero for the origin to innity for all other nodes.nodes as unvisited. Set initial node as current.nt node, consider all its unvisited neighbors and calcu-

tentative distance. For example, if current node A hasof 6, and an edge connecting it with another node Bh 2, the distance to B through A will be 6 + 2 = 8. If thisis less than the previously recorded distance, overwritence.ing all neighbors of the current node, mark it as visited.

node will not be checked again; its distance recordedd minimal.es have been visited, stop. Otherwise, set the unvisitedh the smallest distance (from the initial node, consid-

nodes in the graph) as the next current node and from step 3.

ils information of Dijkstras algorithm can be found in

3. Propose

In a fuzznumbers ancanonic repeasily gene

In the pnotation. Qbe removeddistance frodist[v] denosource nodu, v, previoupath from ant of distavertex v in previous[v]=node, dist[still complet

1. Check if is not inremaininis compl

2. After ndist[u] +operatiothen repumerical example.

d method

y environment, two issues, namely the addition of fuzzyd ranking of numbers should be solved. Based on theresentation [25], the classical Dijkstra algorithm can beralized to a fuzzy Dijkstra algorithm as follows.roposed method, it is necessary to clarify some basic

is the set of all unvisited vertex (nodes) that are to. In set Q, u represents the vertex with the smallestm the source node. Let v be one variant (cf. Fig. 3),te the current shortest distance between v and thee, distbetween(u, v) the distance between two verticess[v] the nearest vertex to the current v along the shortestthe source node. The sign alt is an alternative vari-nce used for comparison. At the initial state, for eachgraph (excluding the source node), dist[v]=innity and0 and Q is the set of all nodes in graph. For the sourceource]=0. Then the following three steps are repeatedion.

Q is empty. While Q is not empty, seek u in Q. If dist[u]nity, then from Q, u is removed (this means that allg vertices are inaccessible from source and the analysiseted).ding v, each neighbor node of u, and calculate alt =

distbetween(u, v) by using the canonical representationn introduced in Section 2. If there exists alt < dist[v],lace dist[v] with alt and record the previous [v] = u (if


not, the same).

3. Choose tit with u Dijkstra

If only this of interesNow the shin Fig. 4. Seshortest papath exists.

4. Transpo

In this sproblem, illproposed mFig. 5, with 2trapezoidalin Table 1.

The rstnodes reprnodes whic

In Step U to S, andthe distancFig. 8. The nal step of fuzzy Dijkstra algorithm in n

previous record of each neighbor vertex v remains the

he one among each v with shortest dist[v], and replaceand go to Step 1. The pseudocode of the proposed fuzzyalgorithm is shown in Fig. 3.

e shortest path between a single source and target pairt, the search can be terminated at line 13 if u = target.ortest path from origin to target can be found, as shownquence S is the list of vertices constituting one of theths from source to target, or the empty sequence if no

rtation network application

ection, a numerical example of the fuzzy shortest pathustrated in [17,18], is used to show the efciency of theethod. Consider the transportation network shown in3 nodes and 40 arcs. It is assumed that all arc lengths are

fuzzy numbers with membership functions as shown

three steps are detailed in Fig. 6. Let S be the set of visitedesenting the shortest path and U the set of unvisitedh are to be settled.1, node (1), which is the source node, is moved from

the shortest distance from (1) to (1) is zero, whilee from source node to accessible neighbor nodes are

calculated. along with

In Step shortest pa(1) (5) tothan the paSet U.

So in Ste(3) with dis

Due to trated. Onlyexample, aof Step 21, However, tonly (1)(ous record empty set length is 5is (1)(5)[17,18].

5. Conclus

This pappath probleOne is howhow to comtheir edgesumerical example.

Among them, the shortest one (1)(5) is picked outdistance value 8.5.2, we move (5) from U to S, this time the seeking ofth should be begun from (5). The shortest distance from

its neighbors are calculated. None of them is shorterth (1)(4) by comparing all the records in the column

p 3, (4) is moved and all the data starts with (4) and (1)tance 12.000 is the result.the limitation of space, the other steps are not illus-

Steps 22 and 23, the nal two steps in this numericalre shown in Figs. 7 and 8, respectively. At the endthe shortest path is (1)(5)(11)(17)(21)(23).here are no neighbor nodes accessible from (23), and5)(12)(15)(18)(22) is found from the previ-of column Set U. As (22) is drawn to S, U becomes anand the search is complete. The shortest path, whose2.83 based on the proposed fuzzy Dijkstra algorithm,

(11)(17)(21)(23), which is the same in Refs.

ion

er extended the Dijkstra algorithm to solve the shortestm with fuzzy arc lengths. Two key issues are addressed.

to determine the addition of two edges. The other ispare the distance between two different paths when

length are represented by fuzzy numbers. The proposed


method to nd the shortest path under fuzzy arc lengths is basedon the graded mean integration representation of fuzzy numbers.A numerical example was used to illustrate the efciency of theproposed method. The proposed method can be applied to realapplications in transportation systems, logistics management, andmany other network optimization problem that can be formulatedas shortest path problem.

It shouldproblem is to the weatone city to aif the geom

Acknowled

The authsuggestionsJiao Tong Ucussions. TScience FouProgram foNo. NCET-0No. CSCT, 2200905570UniversitiesFunds for tSouthwest for Graduatwest Unive

References

[1] M. Xu, Ypath algo(2007) 24

[2] X. Lu, M.Mathema

[3] T.N. Chuaing short1409142

[4] R. Sadiq,ordered wResearch

[5] Y. Deng, bution neApplicati

[6] Y. Deng, Wnumbers 25 (2004)

[7] H.W. Liubetween

[8] J. Ye, Cositions, Ma

[9] Y. Deng, PAdvanced

[10] Y. Deng, Fin supplie

[11] Y. Deng, on multip9854986

[12] Y. Deng, ronment:Applicati

[13] D. DuboisPress, Ne

[14] C. Lin, M.SSets and S

[15] A. Boulmashortest p(2004) 26

[16] T.N. Chuaproblem 1660166

[17] X. Ji, K. Iwamura, Z. Shao, New models for shortest path problem with fuzzy arclengths, Applied Mathematical Modelling 31 (2007) 259269.

[18] I. Mahdavi, R. Nourifar, A. Heidarzade, N.M. Amiri, A dynamic programmingapproach for nding shortest chains in a fuzzy network, Applied Soft Comput-ing 9 (2009) 503511.

[19] M. Ghatee, S.M. Hashemi, M. Zarepisheh, E. Khorram, Preemptive priority-based algorithms for fuzzy minimal cost ow problem: an application inhazardous materials transportation, Computers & Industrial Engineering 57

09) 341354.Ghatenetwo332eshavrnal oajdinrtest p

and M. Liu, CSystem. Yage

Scien. Choufuzzy 116. Chouctingspecties in A148. Chourine trpute

. Chouixed

empt110. Chouombinmminputin

. Chouices, S. Chou

the oovativhamody of ibuternal 1Tsen

nt in:10.10TsengompleTsengwledg013Tseng

the enlicatioan, H

of ae on M1.Chen,E Inte. Qi, wledgnagemim, K.rket wlicati. Choulity urnal 1hamocient putin. Dijkthema. Cormss and be pointed out that the uncertainty in the shortest pathnot limited to the geometric distance. For example, dueher and other unexpected factors, the travel time fromnother city may be represented as a fuzzy variable, evenetric distance is xed.

gements

ors thank the anonymous reviews for their valuable. The Ph.D. students of the rst author in Shanghainiversity, X.Y. Su and P.D. Xu, provided valuable dis-he work is partially supported by National Naturalndation of China, Grant Nos. 60874105, 61174022,r New Century Excellent Talents in University, Grant8-0345, Chongqing Natural Science Foundation, Grant010BA2003, Aviation Science Foundation, Grant No.04, the Fundamental Research Funds for the Central, Grant No. XDJK2010C030, the Fundamental Researchhe Central Universities, Grant No. XDJK2011D002, theUniversity Scientic & Technological Innovation Fundes, Grant No. ky2011011, and Doctor Funding of South-rsity, Grant No. SWU110021.

. Liu, Q. Huang, Y. Zhang, G. Luan, An improved Dijkstra shortestrithm for sparse network, Applied Mathematics and Computation 1857254.

Camitz, Finding the shortest paths by node combination, Appliedtics and Computation 217 (2011) 64016408.ng, J.Y. Kung, The fuzzy shortest path length and the correspond-est path in a network, Computers & Operations Research 32 (2005)8.

S. Tesfamariam, Developing environmental using fuzzy numberseighted averaging (FN-OWA) operators, Stochastic Environmental

and Risk Assessment 22 (2008) 495505.W. Jiang, R. Sadiq, Modeling contaminant intrusion in water distri-tworks: a new similarity-based DST method, Expert Systems withons 38 (2011) 571578.

.K. Shi, F. Du, Q. Liu, A new similarity measure of generalized fuzzyand its application to pattern recognition, Pattern Recognition Letters

875883., New similarity measures between intuitionistic fuzzy sets andelements, Mathematical and Computer Modelling 42 (2005) 6170.ne similarity measures for intuitionistic fuzzy sets and their applica-thematical and Computer Modelling 53 (2011) 9197.lant location selection based on fuzzy topsis, International Journal of

Manufacturing Technology 28 (2006) 839844..T.S. Chan, A new fuzzy Dempster MCDM method and its applicationr selection, Expert Systems with Applications 38 (2011) 69856993.F.T.S. Chan, Y. Wu, D. Wang, A new linguistic MCDM method basedle-criterion data fusion, Expert Systems with Applications 38 (2011)1.

R. Sadiq, W. Jiang, S. Tesfamariam, Risk analysis in a linguistic envi- a fuzzy evidential reasoning-based approach, Expert Systems withons (2011), doi:10.1016/j.eswa.2011.06.018., H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academicw York, 1980.. Chern, The fuzzy shortest path problem and its most vital arcs, Fuzzyystems 58 (1993) 343353.koul, Generalized path-nding algorithms on semirings and the fuzzyath problem, Journal of Computational and Applied Mathematics 1623272.ng, J.Y. Kung, A new algorithm for the discrete fuzzy shortest pathin a network, Applied Mathematics and Computation 174 (2006)8.

(20[20] M.

to 326

[21] E. KJou

[22] A. Tshoers

[23] S.Ton

[24] R.Rtion

[25] C.Clar 160

[26] C.CseleperNot140

[27] C.CmaCom

[28] C.Ca mthe107

[29] C.Ca cgraCom

[30] C.Ccho

[31] C.CingInn

[32] I. CstuattrJou

[33] M. medoi

[34] M. inc

[35] M. kno134

[36] M. ateapp

[37] X. Priskenc111

[38] W. IEE

[39] R.GknoMa

[40] S. KmaApp

[41] C.CquaJou

[42] I. CefCom

[43] E.WMa

[44] T.HPree, S.M. Hashemi, Application of fuzzy minimum cost ow problemsrk design under uncertainty, Fuzzy Sets and Systems 160 (2009)89.arz, E. Khorram, A fuzzy shortest path with the highest reliability,f Computational and Applied Mathematics 230 (2009) 204212., I. Mahdavi, N.M. Amiri, B. Sadeghpour-Gildeh, Computing a fuzzyath in a network with mixed fuzzy arc lengths using a-cuts, Comput-athematics with Applications 60 (2010) 9891002.. Kao, Network ow problems with fuzzy arc lengths, IEEE Transactions, Man, and Cybernetics. Part B: Cybernetics 34 (2004) 765769.

r, A procedure for ordering fuzzy subsets of the unit interval, Informa-ces 24 (1981) 143151., The canonical representation of multiplication operation on triangu-numbers, Computers and Mathematics with Applications 45 (2003)10., P.C. Chang, A fuzzy multiple criteria decision making model for

the distribution center location in china: a Taiwanese manufacturersve, Lecture Notes in Computer Science (including subseries Lecturerticial Intelligence and Lecture Notes in Bioinformatics) 5618 (2009)

., Application of FMCDM model to selecting the hub location in theansportation: A case study in southeastern Asia, Mathematical andr Modelling 51 (2010) 791801., R.H. Gou, C.L. Tsai, M.C. Tsou, C.P. Wong, H.L. Yu, Application offuzzy decision making and optimization programming model toy container allocation, Applied Soft Computing Journal 10 (2010)79., F.T. Kuo, R.H. Gou, C.L. Tsai, C.P. Wong, M.C. Tsou, Application ofed fuzzy multiple criteria decision making and optimization pro-g model to the container transportation demand split, Applied Softg Journal 10 (2010) 10801086., An integrated quantitative and qualitative fmcdm model for locationoft Computing 14 (2010) 757771., A fuzzy backorder inventory model and application to determin-ptimal empty-container quantity at a port, International Journal ofe Computing, Information and Control 5 (2009) 48254834.drakas, I. Leftheriotis, D. Martakos, In-depth analysis and simulationan innovative fuzzy approach for ranking alternatives in multiple

decision making problems based on topsis, Applied Soft Computing1 (2011) 900907.g, A. Chiu, Evaluating rms green supply chain manage-

linguistic preferences, Journal of Cleaner Production (2010),16/j.jclepro.2010.08.007., Green supply chain management with linguistic preferences andte information, Applied Soft Computing. 11 (2011) 48944903., Using a hybrid mcdm model to evaluate rm environmentale management in uncertainty, Applied Soft Computing 11 (2011)

52., Using linguistic preferences and grey relational analysis to evalu-vironmental knowledge management capacity, Expert systems withns 37 (2010) 7081.. Wang, W. Chang, A fuzzy synthetic evaluation method for failureviation product R&D project, in: 5th IEEE International Confer-anagement of Innovation and Technology, ICMIT, 2010, pp. 1106

S. Tan, Fuzzy portfolio selection problem under uncertain exit time,rnational Conference on Fuzzy Systems 5 (2009) 550554.S.G. Liu, Research on comprehensive evaluation of enterprisese management capabilities, in: 2010 International Conference onent Science and Engineering, ICMSE, 2010, pp. 10361301.

Lee, J.K. Cho, C.O. Kim, Agent-based diffusion model for an automobileith fuzzy topsis-based product adoption process, Expert Systems withons 38 (2011) 72707276., L.J. Liu, S.F. Huang, J.M. Yih, T.C. Han, An evaluation of airline servicesing the fuzzy weighted SERVQUAL method, Applied Soft Computing1 (2011) 21172128.drakas, D. Martakos, A utility-based fuzzy topsis method for energynetwork selection in heterogeneous wireless networks, Applied Softg Journal 11 (2011) 37343743.stra, A note on two problems in connexion with graphs, Numerischetik 1 (1959) 269271.en, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, MIT

McGraw-Hill, 2001.

Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment1 Introduction2 Preliminaries2.1 Fuzzy numbers2.2 Canonical representation of operations on fuzzy numbers2.3 Dijkstra algorithm

3 Proposed method4 Transportation network application5 ConclusionAcknowledgementsReferences

Documents

Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment