NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

Fuzzy Shortest Path Problem with Finite FuzzyQuantities

S.MoazeniMS. Student of Applied Math

Department of Mathematics and Computer ScienceAmirkabir University of Technology (Tehran Polytechnic)Hafez Ave, No 424, P.O.BOX 15875-4413, Tehran ,Jran

moazenis@aut.ac.ir Phone:+989133163791

Abstract - We discuss the problem of finding the shortestpaths from a fixed origin to all nodes on a network notnecessarily acyclic, with each arc length represented as positivefuzzy quantity with finite support. At first we show that the onlyexisting paper on this problem, Klein's algorithm, in some caseswill lead to a dominated path in the sense of extension principleand then we introduce a new algorithm for the problem. Theproposed algorithm is on the basis of the multiple labelingmethod of Hansen and Dijkstra's shortest path algorithm.Keywords: Fuzzy Network, Shortest Path, Fuzzy quantity.

I. INTRODUCTION

The shortest path problem (SPP) is one of the mostfundamental and well-known combinatorial optimizationproblems that appears in many applications as a sub-problem.The classical problem seeks to select, from a finite set ofpaths, a path with minimum length. The lengths of arcs in thenetwork represent traveling time, cost, distance or othervariables. In the real life applications, the arc length could beuncertain and determining the exact value of these lengths isdifficult or sometimes impossible for the decision maker. Insuch positions a fuzzy shortest path problem (FSPP) seems tobe more realistic. While proposing an algorithm for solvingFSPP, we are faced with ranking methods between fizzy sets[1,2] and determining a path from source to a specialdestination whose length is minimum in most of the cases isimpossible. So FSPP has received researchers' attention in tworecent decades and several approaches have been proposed tosolve FSPP with fuzzy numbers as arc lengths which arenormal and semi-continuous. The best of the Refs[3,4,5,6,7,8,9,10,11,12] can be found related to this problem.

Although most efforts on FSPP concentrated on networkswith fuzzy numbers as the arc lengths, the problem onnetworks with finite fizzy quantities as arc lengths have notbeen treated intensively in the specialized literature despite itspotential applications. In fact, dealing with somegeneralizations of SPP in fuizzy environment, such as the time-constrained SPPs or SPP in a time-window network, we needto find fuizzy shortest path with finite fuizzy quantities as arclengths.

Literature on this topic is scarce. Only one approach forthe problem, Klein's algorithm, is available in the existingpapers. His approach which works just on acyclic networks,

in some cases, will result in a dominated path in the sense ofextension principle.

In this paper, a different algorithm was proposed whichtakes advantage of the multiple labeling method ofHansen[14] for the bi-criterion path problems and Dijkstra'sshortest path algorithm. Our approach is much more efficientbecause it not only works on both cyclic and acyclic networksbut also finds all non-dominated paths in the sense ofextension principle.

This paper is organized as follows; In section 2, someelementary concepts and definitions in fuwzzy set theory usedthrough out this paper are described. In section 3, someterminology is provided. In section 4, Klein's approach ispresented and is shown that executing this algorithm, in somecases, leads to a dominated path in the sense of extensionprinciple. In section 5, the proposed algorithm is presented andthe numerical example is illustrated in section 6. Possibleimprovements to the proposed algorithm is discussed insection 7. This paper is concluded in section 8.

II. PRELIMINARIESThe membership function AA (-) of a fuzzy set A defined

on the universe U, is a function AA: U [pl]. The fuzysetAis completely determined by the set of 2-tuplesA = {(uu,U2(u))juE U}. The support of a fuzzy set A is defined bySUpp(i) = {u E Uup(u) >q. A fuzzy set A defined on the set ofreal numbers, U = 91, is called a fuzzy quantity.

The height of a fuzzy set Ai on u, denoted by hgt(A), isdefined as hgt(;i) = Sup..uu, (u). A fuwzzy set i is called normal,if hgt(A) = I . A fuzzy set A is convex if and only ifVx, yE U VAi [0,11 y Ax+ (I-A)y) >min(u (x),4gu (y)). A fuzzynumber A is an upper semi-continuous, normal and convexfuzzy subset of the real line 9i.

Two fuzzy sets are equal, i.e. Ai= if and only if forevery ueU ,u, (u) = u, (u)

0-7803-91 87-X105/$20.00 02005 IEEE. 664

The extended sum and the extended minimum of fuizzysets A and B, on the basis of extension principle, denoted as

iB and mlin(A,B) respectively, are defined as follows:=(U) Supu=.,y minm (x),yu (y)}

= Sup, min{, (x),,Uy (u - x)}

min(A B)(u) = Supu=mP X y) minl#,(x),#lB(y)}Corollary 1. Let A and B are two fuizzy quantities with

finite support, the following equation holds:SUmin(A,B)(z) = max{ min(y4 (z), max,,>: , (v)),

min(max U>z,f (U)"UNW)IZ)Proof: Since Supp(A) and Supp(B) are finite crisp sets,

the extended minimum of A and B can be written as:itmin(A,B)(z) = maxz=minuy) min{Ju,(u), 1 (v)} =max {max,> u=mili {8A (U),, UBj (V))} maxu>,,v=z min {1A (U),,uh (V)}}

Now inequality 9Uk(v.) < maxv2, phB(V), V v. E 91 results in

min(p;i (z),puh(v)) < min(1u;(z),max,,z1fl(v)). Thus maxv,2 (mm(Xu1 (z),a (v.)) . min(A (z),max,>7 'UB(v)) . On the other hand,since Supp(B) is a finite set then there exists v' > z such that

max,> pu(v) = B ) Therefore min(#uA(z),maxv2z,uB(v)) is oneof the elements of the statement maxV :>(min(#u (z),l,g(v.)).

Consequently we have max, ,(min(#u (z),f,lu(v)) = min

(/12 (z),max j/ (v)) or after renaming, we obtainmax, j;(mn(uA (z),#u (v)) = min(u (z),max,2uff (v))

Similarly, we can obtainmaxu,Z (mun(,u (u),p (z))) = min(max,,;>L (u),upU (z)) whichalong with the equation, n(A )(z) = max{maxv,> min(u; (u),

,, (v)), maxu>z min(,uA (u), #,u (v))} ,the conclusion is proved.ov=z

III. TERMINOLOGY

Consider a directed network G(N,A), consisting of afinite set of nodes N = {l,...,n} and a set of m directed arcsA c Nx N. Each arc is denoted by an ordered pair (i,j),where i, je N and i. j. It is supposed that there is only onedirected arc (i,]) from i to j whose fuzzy length is denotedby Tu. If 7y is a fuzzy quantity with finite support, then

=-{(1,J(1)),(2,,u, (2))S..(k,u(k))} nm which k = maxSupp(7.).The source of the network is numbered by 1 . The SPP is todetennine for every non-source node is N a shortest lengthdirected path from node 1 to node i in which the length of anypath P denoted by 7e is defined as2p=-

The fuzzy counter part of the problem is the SPP on thenetwork with each arc length represented as fuzzy sets. Theproblem can be formulated as the following linearprogramming form:

P: mn f(x)= X(i,j)A

SI x-(D.ji =n-1 if i=ls. ex J- jji 1_ if i#lV(i, j) E A xa ?E }

Where tff s are fuizzy sets (quantities or numbers)corresponding to the arc lengths and E is the extended sum offuzzy sets. The meaning of "min" in the objective function isambiguous and is undefined [8]. An optimal solution to thisproblem in the normal sense cannot be obtained because sucha problem is "ill-posed" and the shortest path length is notdeterministic [11].

We can not use the extended minimum in the objectivefunction because the minimum of two fuzzy sets A and B isnot necessarily one of A or B. In fact for fuzzy sets A and Bdefined on 91, the statement (A4 < B * m i n(A, B) = A ), isjust a partial order on fuzzy sets defined on 91. But in thesense of extension principle, we will define the concept ofdominated path (dominated solution) as follows.

Definition 1. Let PI and p2 are two paths from sourcenode to a special node i, path j, dominates path Pj ifm in(p,,fP, ) = ep . For a given path p,, if there is no path p,in the network such that min(- ,-{ ) = , we say p1 is anon-dominated path.

Consequently although finding optimal solution in thesense of extension principle for FSPP in most of the cases isimpossible but the set of non-dominated solutions or pareto-optimal solutions can be defined for the problem. This conceptcan be a criteria for comparing algorithms for the problembecause if there is a path which is the extended minimum ofall paths in the network, both optimistic and pessimisticdecision makers will choose it. In the regard of the concept ofnon-domination, the existing approaches can be considered intwo categories.

The first category consists of the algorithms that find theset of all non-dominated solutions and offer to the decisionmaker such as [8] by Okada and Soper, [11 ] by Okada and [9]by Blue and Bush and Puckett that obtains the set of all non-dominated paths along with some dominated paths.

The second category are the algorithms which choose oneof non-dominated paths that has special useful property fordecision maker as a shortest path in fuzzy network. [6] byOkada and Gen, [10] by Yao and Lin, [12] by Chuang andKung and all other algorithms based on ranking functions areconsidered in this category.

The only existing approach dealing with FSPP in which afuizzy quantity with finite support is assigned to each arclength is Klein's approach [5]. We will show in section 4 thatthis algorithm, in some cases, results in a dominated solution.

As stated previously, our algorithm will find all non-dominated paths in a network with finite fuizzy quantities.

IV. RELATED WORKA. An overview ofKlein 's algorithm

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He proposed an algorithm based on dynamicprogramming to solve FSPP with positive fuzzy quantities asarc lengths on acyclic and layered networks (indeed anynetwork without negative cycles can be transformed into alayered network by the Bellman-Ford method[15].). Each arccan take an integer value for length between 1 and a fixedintegerR. Let the integer value 1 through R be fuzzy sets andlet the grade of membership of each arc (i, j) in each frizzy setI, 1< I < R, be denoted by p, (i, j) . Therefore each arc hasassociated with it an R -tuples of membership values.

His proposed recursion algorithm for solving FSPP withfuizzy quantities as arc lengths is:

f(n) = (, lf(i) = dom(Q,)EA (eY + f(j))

where ey is an R -tuples associated with arc (i,j). This R -tuples consists of the membership grades of arc (i,j) or themembership grades of the paths from i to j in the respectivefuzzy sets associated with the possible lengths, 1 through R.Hence eq =(j(ij)-...flPR (i,j)). The operator + represents thecombinatorial sum, and dom(.,.) is the domination operatorbased on more is better:

dom((a1,...,a),(b,,..., b)) = (max(a,,b1),..., (a ,bj))The combinatorial sum for fuwzzy shortest paths is defined

as follows. Recall that it is assumed that, there are M layersand the possible lengths of an arc are 1 through R. Therefore,the shortest a path could be in length is M -1 and the longesta path could be in length is (M-l).R To find the paths ofpossible lengths, combination of the possible lengths must beconsidered. To find the possible length I of a path, the lengthsthat can be used in combination are l2,...,(I -M + 1). Thecombinatorial sum oftwo tuples is then defined as follows:Let z = min{ (j, k),px (k, q)} . Then ejq = eIk + ek, where thei -th element of the R -tuple ejq is given by (ejq)i

maxx+.=i (z) = Hi (i q) -

The recursion in f(i) = dom(A (e, 4 f(j)) will yield theset of non-dominated paths from source 1 to n. Finally, thefuizzy shortest path length is defined asFSP = {(l,max i (al (1, n))),..., (k, maxi,( (l,n))),..., (R,max (AR (ln)))}

where u4 (Jn) represents the membership grade in the fuzzyset k of the path from node 1 to node n given by the i -th non-dominated R -tuples.

He claimed that this model not only finds a fuizzy number(indeed a fuzzy quantity) representing the fuzzy shortest pathlength but also associates an actual path or paths to thatnumber. After finding f(l), if the decision maker has athreshold of membership of a, then the shortest path withmembership degree equal to a in f(l) is chosen.B. Klein 's algorithm can lead to a dominatedpath.

Now we present an example in which execution ofKlein's algorithm on a network which satisfies assumptions,

for every threshold of membership degree that is chosen bydecision maker, leads to a dominated path.

Example 1. The following network has three layers. Thearc lengths shown in Fig 1, are fuzzy quantities with finitesupport.

£13= {(l,0.4),(2,0.8)} = (0.4,0.8)f12 = {(l,l),(2,0.3)} = (1,0.3)\24 = {(1,0),(2,0.3)} = (0,0.3)

Fig. f34= {(1,0),(2,0.8)} = (0,0.8)Fig. 1

There are two paths P1:I-2-4 and P2:-3-4 in thenetwork whose lengths are 7,={(3,0.3),(4,0.3)}=(0,0,0.3,0.3)and p2 = {(3,0.4),(4,0.8)} = (0,0,0.4,0.8) . According to theextended minimum min(: ,p =p, e and the decision makeralways choose p, as the shortest path.

According to the Klein's algorithm, we have:R.(M-1) =2.(3 -1) =4f (2) = (Op.3Po)f (3) = P.8p)f(l) = dom(e12 T f(2),e13 T f(3))

= dom(0P.3P.3),(0M.4P.8)) = (0PO).4p.8)Consequently for every threshold of membership of a ,

the algorithm suggests the decision maker to choose p2 as theshortest path.o

V. PROPOSED ALGORITHM

A. The algorithmWe will show any fuzzy quantity A with finite support by

a k -tuple whose i th-element denoted by(j). is equal to yu (i)and (A)k is nonzero. i.e. we denote {(1,0.1),(2,0.3),(4,0.7),(5,0)} by (0.1, 0.3, 0, 0.7).

The algorithm is based on the Dijkstra's label-settingshortest path algorithm [13] and multiple labeling methodproposed by Hansen [14].

The multiple label used in this paper contains a fuzzyquantity (correspondence to the path length) and two pointers.Let jEN be a node of G(N, A), the k -th label associated withnode j, which is one of its labels, is denoted as [-k ,(i,ki)]k,

where i(i .1) is a predecessor node of the label, 4k is thefizzy distance along the path pj of the k -th label of node jand node k, indicates some label of i , for whichk k,=p, ek, i

Like Dijsktra's algorithm that divides the nodes into twogroups (Temporary nodes and permanent nodes), the presentalgorithm partitions the labels into two sets: The set oftemporary labels, T and the set ofpermanent labels, P.

Permanent labels of node j show that there are non-dominated paths from node 1 to node j. A temporary label

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may be deleted from T without changing to a pennanent one,or may be deleted from T and becomes permanent. So while atemporary label may be deleted or changed to a permanentlabel, a permanent label remains unchanged once a temporarylabel changes to a permanent one.

When a node gets a new temporary label, this label shouldbe compared with other previous labels of the node; If thislabel is dominated by another label of the node, it will bedeleted from T without becoming permanent. If this labeldominates another label of the node, the dominated label willbe deleted. Any way all the labels of a node will be non-dominated relative to each other.

In order to select a new permanent label (similar to nodeselection step in Dijkstra's algorithm), we define alexicographic order among finite fuzzy quantities as follows:

Definition 2. Lexicographic order for finite fuzzyquantities. For two fuzzy quantities with finite supportsA=(a ,a2,...,a ) and B=(b1,b2,...,b,) , we say A islexicographically smaller than B, if one of the following fourcases holds:1) {n<m} or2) {n= m and an < bn} or3) {n= m and an = b, and (3i= ,...,n-1 s.t

an-i .bn-i and Vk >n-i(ak =bk) and

min(ani ,maxk>n,i ak) > min(bn-i maxk>n-i bk ))} or

4) {n = m and an = bn and (3i = ,...,n - s.t

an-i < bn-i and Vk > n - i(ak = bk) andmin(a,n, maxk>ni ak ) = min(bn , max k>n-i bk ))}

Unlike Okada's lexicographic order [8] that is flexibleaccording to the decision maker's preference, the proposedorder in definition 2, can not be started from the left.

The next property will show that the lexicographic orderin definition 2, is a suitable choice for label selection step inthe algorithm.

Property 1. If = (a,..., ) is lexicographically smallerthan B=(b1,b2.bm),then min(A,B) .B -

Proof: Since A is lexicographically smaller than B, soone of four cases mentioned in definition 2, happens. In orderto complete the proof, we verify each part separately:Casel. If n < m then B has nonzero i -th element such thatn <mi. m . Thus i -th element of min(A,B) is equal to

SuPmi,(k,)=i min((A)k,(B),) = SUPk:,, min(O,b,) = O ; consequently

(min(A,B))i =0.(B)0.Case2. If n=m and an <b, , then

(min(A,B)), = SUPn=mini]j) min((A),,(B)1) = min(a,,b,) = an . (B)n . So (min(A,B))n . (B)n.Case 3. According to the corollary 1 and the hypothesis in thiscase, min(an-i ,maxk,n i ak) > min(bn-iE max k>,, bk) . Thus we

have (min(A,B))n E = min(an i,max>,,, ak)- On the other hand,by contradiction, if min(A, B) = B then (m i n(A, B)),,i = (B)n-i

= b_i or equivalently b-i = min(a,,_jImax k>,j ak) >min(b"-,Imaxk>,,i bk). Hence bn_i > maxk>" i bk . By using the fact thatak =b, for every k>n-i, we have b_i >maxk>ak and thusb_-i >maxk>iiak > min(a -. maxk>-iaak) which is incontradiction to the fact b. i = min(a., ,maxk> ak) So

(m i n(A, B)), . (B) E..Case 4. By assumptions we have (min(A,B)),,_ =mm

(an Imaxk>i ak) = min(b,,,maxk>,,-i bk), now we will show that(min(A,B)),, . b,,.i . Demonstrating by contradiction, if

(min(A, B))nis equal to bn-i then min(an i, maxk>ni ak) = b,,-i,or equally an-i . b_i which is a contradiction to the fact that

an-i < bn-i -

Consequently in all cases, min(A,B) and A are different in atleast one element and so the conclusion is proved.o

According to the above property, using lexicographicorder in selecting a temporary label to change to a permanentone, we always choose a non-dominated label.

The proposed algorithm is summarized as follows:The Algorithm:

beginP:=0;T:={[tIi,(l,l)]for node(i)IiE N and (l,i)e A};While T 0 do{beginLet[7-, ,(j,k)], E T be a label which islexicographically the smallest one in T;T := T - {[ ' ,(]j,k)],}P := PU {[el ,(j,k)],};For each node h E N such that (i, h) E A dobegin [Distance update]Determine tpr = tp 3 7h-Put[7; ,(i,l)], as a new temporary label ofj;T <- T U 4{Vl,, I(i,l)]r};[The dominance check step]For u -th label ofnode h such that u < (r -1)if m in(;-,7)= then Tv T{[TPlh ("1lrV [P, (-,-)]u E T, if mrin(7p. ) = 7p thenT <- T -{epCu I(-,-)]u}end;end;Find all non-dominated paths from node 1 to other nodesby using pointers of each label and tracking backward thelist of nodes of that path until reaching node 1.end;

B. The correctness ofthe algorithm

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In this section we will show that the algorithm will obtainall non-dominated paths. At first we prove the correctness of astatement for fuzzy quantities that its correctness for fuzzynumbers is almost trivial.

Theorem 1. If A and B and e are three fuzzy quantitieswith finite support, then the following statement is established:If min(A,B)=A then min(AEC,BEC)=AOC"

Proof: Let A=(ai,...,am),B=(bp.bn) and C=(cl,...cCk)are three fuzzy quantities with finite support. The statementmin(A,B) = A leads that m < n . On the other hand:

min{(fa,,a,, ,a.),(b,,b2l , ,bn)j- { max{min(a,,maxj=_.,1 bi), min(b,,maxj=l,._. ai)},

max{min(a2,maxi=2,...,n bi),min(b2, maXi=2., ai)) I

min(an,,maxj=_,. bi)}=(a,,...,a.)So V t = l...,m is max{min(a,,maxi=/,...,n b,),min(bt,max=t,..,m

a1)) = a, or equivalently max=t.,,b > at and min(b,,max,=,.a,) < a,t

Thus in general min(A, B) = A if and only if

maxi...In b, >a and min(b,,max,j=, m,ai)<a,. So in order to

prove min(A e, B'C)=) = , it suffices to show that

maxi=,_n+(Bk C) >(AeC), and min((B e C),,max,=,,...(Ae C).) <(C)JA . We prove these inequalities in two stages.

1) Proof of min((B C))t,maxi=t,..,+k(A C)e) < (A C),e -

min((B e C), ,maxi=1, n+k (A ® C),) = min{maxj=A_...,(min(b,j ,cI)), max,=,.n+k (max1..1. j(min(a,-j,cj)))}< min {maxj=lA.t (min(b,tj ,cj)), maxj=..,- (min(at,. j

cj)), max,=,+,,,,. (min(a,.j ,cI ))} < maxj..1.,-, (min(a,-j,j=l.i-l

cj)) = (A G C),. So the conclusion is proved.

2) Proof ofmax__f n E( e) >()(A EC),max ,..,.fl÷(B D C)j =maxi=,.,n+k(maxj=l,..,,.1(min(bi-j, cj))) =

maxj=1 ...,t-l (maxiO,..,n+k- (min(b,j+t, cj)))=maxj=1-l.1 min(maxi=O,.n+k-tb,-j+t1,cj) =maxj=1 .,-1 min(maxs=t-j,...,n+k-jb,, cj) >maxj=, t.'-l min(at- I,Cj) = (A e C)t

in which in the last inequality we used the statementmax,=,,, b, > a with t- j instead of t that is a result of

m i n(A,B) = A4. Hence the proof.oNow we will present a theorem which is the basis of

correctness of the algorithm.

Theorem 2. If a path p, from node 1 to node t is a non-

dominated path, then every sub-path j from node 1 to anintermediate node i is also a non-dominated path.

Proof: This can be proved by contradiction as follows;Let p,, be a non-dominated path from node I tot. Assume an

intermediate node i on the path ,, distinct from node 1 andt, such that the sub-path j,. on the path p,, is dominated. Thenwe can construct a path p1, disfinct from j,. , such that

min(7^,tp)= . Let p,be the sub path on p, and letpit = pliUPitu By theorem 1, min(ep G) C,Jp, t,p ,)= fpeSo P, is dominated by pl ® p,,, this is a contradiction with

the assumption that pl, is a non-dominated path. Hence theproof.o

Consequently a path which has some dominated sub path,will be dominated and the algorithm will not more consider it.Other paths which do not have dominated sub path will befound and all these paths are non-dominated.

VI. NUMERICAL EXAMPLE

Now we will execute the proposed algorithm on thefollowing network.

(0.30.2p

048 P (0.24 .38)\92)

g #~~~~~~~.320.).8,0.9,0.2)(O." 'A1)(0.T9.l)

In the network, there are five paths as follows:PI: 1-3 -5-6 -* e,, = (0,0,0.2,0.3,0.4,0.4,0.4,0.4,0.3,0.2,0.2,0. 1)P2 :1-3-5 -4-6 -e+ p = (0,0,0,0. 1,0.2,0.2,02,0.2,0,02,0. 1)P3 :1-2-4-6 -X* p = (0,0,0.1,02,03,0.2,0.2,0. 1)1

P4 :1 -2-5 -4-6 < p4 = (0,0,0,0.1,0.2,0.2,0.2,0.2,0.1)P5:1-2-5-6 - p = (0,0,0.2,0.3,0.3,0.3,0.3,0.2,0.2,0.1)Since min(CP,2)-=pe and msn(4,,o,)=PSE and P2are dominated paths respectively by p, and p4 . Verifyingother paths determine that they are non-dominated. Byexecution the algorithmT:={[f',,(l,l)]1 VjE N st (l,j)EA}

-{[(0.3,0.2,0.1),(l,l)],for(2),[(0.5,0.3,0.7,0.2),(11)],for(3)}Since3 <4, so the first label of node 2 is lexicographicallysmaller than the first label of node 3 and changes to apermanent label. Now labels [(0,0.3,0.2,0.2,0.1),(2,1)], for node4 and [(0,0.3,0.3,0.2,0.1),(2,1)]j, for node 5 are added to the set T.Now the first label of node 3 is lexicographically the smallestone in T and becomes permanent. After distance updating wehave

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T = {[(w.3p.2P.2P. 1), (2,1)], for(4),[(00.3P.3P.2P. 1), (2,1)], for(5),[(O)PA4P3PAPS.3P.2P). 1), (3,1)12fbor(5)j

but m i n{(0,0.3,0.3,0.2,0.1),(0,0.4,0.3,0.4,0.3,0.2,0.1)} = (0,0.3,0.3,0.2,0.1) leads that the second label of node 5 is a dominatedone and will be deleted from T.The first label of node 4 islexicographically the smallest in T and becomes permanent.The iterations are continued until the set of temporary labelsbecomes empty. The non-domianted path(s) from node 1 toeach node are denoted in the table 1..

1-3 1-2-41-2-5-4

1-2 1-2-5 1-2-5-61-2-4-61-2-5-4-6

Table. 1VII. SOME MODIFICATIONS

In order to present the result as a fuzzy set we can useindices based on possibility measure and necessity measurefor comparing two fuzzy sets[l,2]. Consequently, each arc canreceive its own degree of possibility defining the possibilitythis arc is on the shortest path.

Definition 3. [12] Let Aand B be two fiuzzy quantities,Poss(A < B) = Sup., min(plA(u),uB(v)) . Now based on

possibility theory, the degree of possibilityD, E [0,1] for an

arc (i, j) E A is defined as Du = maxPEP(Q){Dp)where P(Q, j) means a set of all non-dominated paths travellingfrom (i, j) . More over, the degree of possibility D for P isdefined as Dp = min pEP(1, n) Poss(fep < )-

However, till know, we could find all non-dominatedpaths, but the number of these paths in a large size practicalproblem would be too large and consequently makingdecision for decision maker would be difficult and thedecision maker may be confused. For solving this drawbackwe can use the same solution as Okada proposed [8] In thissituation, it is applicable to relax the proposed algorithm byreplacing non dominated paths with h-nondominated paths.That is defined as follows:

Definition 4. Let A and Bbe two fuzzy quantities withfinite support, we say A h-dominates B if min(A',B')=A'in which Ai'and B'contains all of elements of Supp(A) andsupp(B) respectively, whose membership is not smaller thanh.

We remark that the higher possibility level h is set, theless the number of h-non dominated paths is.

VII. CONCLUSION

In this paper, we presented an algorithm for finding all ofthe non-dominated paths on a network in which the length ofeach arc is a fiuzzy quantity with finite support.

This kind of networks highly appears in generalizingconstrained shortest path problem to fuzzy environment.

Future consideration in this research would involvedeveloping an algorithm for finding all non-dominated pathsin a network in which some of its arc lengths are finite fuzzyquantities and some of them are fuzzy numbers or a finitecrisp set of fuzzy numbers.

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