A Novel Approach for Finding a Shortest Path in a Mixed Fuzzy Network

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1 Wireless Sensor Network doi:0.436/wsn Published Online Februry 00 ( A Novel Approch for Finding Shortest Pth in Mixed Fuzzy Network Abstrct Ali Tjdin Irj Mhdvi * Nezm Mhdvi-Amiri Bhrm Sdeghpour-Gildeh 3 Rez Hssnzdeh Deprtment of Industril Engineering Mzndrn University of Science nd Technology Bbol Irn Fculty of Mthemticl Sciences Shrif University of Technology Tehrn Irn 3 Deprtment of Mthemtics nd Sttistics Mzndrn University Bbolsr Irn E-mil: irjrsh@rediffmil.com Received October 009; revised November 3 009; ccepted November We present novel pproch for computing shortest pth in mixed fuzzy network network hving vrious fuzzy rc. First we develop new technique for the ddition of vrious fuzzy numbers in pth using -cuts. Then we present dynmic progrmming method for finding shortest pth in the network. For this we pply recently proposed distnce function for comprison of fuzzy numbers. Four exmples re worked out to illustrte the pplicbility of the proposed pproch s compred to two other methods in the literture s well s demonstrte the novel feture offered by our lgorithm to find fuzzy shortest pth in mixed fuzzy networks with vrious settings for the fuzzy rc. Keywords: Fuzzy Numbers -Cut; Shortest Pth Dynmic Progrmming. Introduction Determintion of shortest distnce nd shortest pth between two vertices is one of the most fundmentl problems in grph theory. Let G = (V E) be grph with V s the set of vertices nd E s the set of edges. A pth between two vertices is n lternting sequence of vertices nd edges strting nd ending with vertices nd no vertices or no edges pper more thn once in the sequence. The length of pth is the sum of the weights of the edges on the pth. There my exist more thn one pth between pir of vertices. The shortest pth problem is to find the pth with minimum length between specified pir of vertices. In clssicl grph theory the weight of ech edge is tken s crisp rel number. But prcticlly weight of ech edge of the network my not be fixed rel number nd it my well be imprecise. The shortest pth problem involves ddition nd comprison of the edge weights. Since the ddition nd comprison between two intervl numbers or between two tringulr fuzzy numbers re not like those between two precise rel numbers it is necessry to discuss them t first. Intervl rithmetic ws developed in Moore []. The cse of optimiztion with intervl vlued nd fuzzy constrints were discussed in Delgdo et l. Ishibuchi nd Tnk Sengupt nd Shocheng [ 5]. Vrious rnking methods for intervl numbers were introduced by severl reserchers. An extensive survey of the order reltions long with new proposl re given by Sengupt nd Pl [6]. There re lso rnking methods for fuzzy numbers vilble in the literture. Dubois nd Prde [7] introduced rnking of fuzzy numbers in the setting of possibility theory nd Chen [8] rnked fuzzy numbers using mximizing nd minimizing sets. Rnking of fuzzy numbers ws lso studied by Bortoln nd Degni Cheng nd Delgdo et l. [90]. Fuzzy grph problems were ddressed in Blue et l. nd Koczy Klein Li et l. Lin nd Chen Okd nd Gen [ 7] pid specil ttention to fuzzy shortest pths. In recent development Okd nd Soper [8] proposed n lgorithm to find the shortest pth in network with fuzzy edge weights. The lgorithm gives fmily of non-dominted shortest pths for specified stisfction level but it does provide ny guideline to the decisionmker to choose the best mongst the pths ccording to his/her view; i.e. optimistic pessimistic etc. The shortest pth (SP) problem hs received lots of ttention from reserchers in the pst decdes becuse it is importnt to mny pplictions such s communiction trnsporttion scheduling nd routing. In network the

2 A. TAJDIN ET AL. 49 rc length my represent time or cost. Conventionlly it is ssumed to be crisp. However it is difficult for decision mkers to specify the rc. For exmple using the sme modem to trnsmit the dt from node to b in network the dt trnsmission time my not be the sme every time. Therefore in rel world the rc length could be uncertin. Fuzzy set theory s proposed by Zdeh [9] is frequently utilized to del with uncertinty. Zdeh presented the possibility theory using membership functions to describe uncertinties. Considering directed network tht is composed of finite set of nodes nd set of directed rcs we denote ech rc by n ordered pir (i where i nd j re two different nodes. The rc length represents the distnce needed to trverse (i from node i to j. We denote it by l(i or L(i when it is crisp or fuzzy respectively. We cll L(i s fuzzy rc length. The shortest pth problem is the following: given weighted directed grph nd two specil vertices s nd t compute the weight of the shortest pth between s nd t. The shortest pth problem is one of the most fundmentl network optimiztion problems. This problem comes up in prctice nd rises s subproblem in mny network optimiztion lgorithms. Algorithms for this problem hve been studied for long time [0 ]. However dvnces in the method nd theory of shortest pth lgorithms re still being mde [3 5]. In the network we consider here the of the rcs re ssumed to represent trnsporttion times or costs rther thn geogrphicl distnces. As time or cost fluctute with trffic conditions pylod nd so on it is not prcticl to represent the rcs s crisp vlues. Thus it is pproprite to utilize fuzzy numbers bsed on fuzzy set theory. In proposing n lgorithm for solving the problem we re first fced with the comprison or ordering of fuzzy numbers tsk not considered to be routine. For this reson fuzzy shortest pth problems hve rrely been studied despite their potentil ppliction to mny problems [86]. The problem turns out to be even more complicted in our more generl cse of llowing vrious fuzzy rc. Here we propose new pproch nd n lgorithm to find shortest pth in mixed network hving vrious fuzzy rc. The reminder of the pper is orgnized s follows. In Section bsic concepts nd definitions re given. A dynmic progrmming lgorithm for finding fuzzy shortest pth in network is presented in Section 3. There we mke use of -cuts for computing pproximtions for the ddition of two different types of fuzzy numbers nd pply distnce function for the comprison of fuzzy numbers. Comprtive exmples re given in Section 4. Section 5 works out n exmple to show the novel feture of our lgorithm to find fuzzy shortest pths in mixed fuzzy networks with vrious settings for the fuzzy rc. We conclude in Section 6.. Concepts nd Definitions We strt with bsic definitions of some well-known fuzzy numbers. Definition. An LR fuzzy number is represented by A = ( m β ) LR with the membership function µ ( x) defined by the expression m x L x m µ ( x) = x m R x m β where L nd R re non-incresing functions from R + to [0] L(0)=R(0)= m is the center is the left spred nd β is the right spred. Note tht if L(x)=R(x)=-x with 0<x< then x is tringulr fuzzy number nd is represented by the triplet = ( 3) with the membership function µ ( x) defined by 0 x x < x ( ) = µ x 3 x < x x > 3. A tringulr fuzzy number is shown in Figure. Definition. A trpezoidl fuzzy number is shown by = ( 3 4 ) with the membership function s follows: 0 x µ ( x) = 4 x x x x 4 x x. Figure. A tringulr fuzzy number. 3 4

3 50 A. TAJDIN ET AL. A generl trpezoidl fuzzy number is shown in Figure. It is pprent tht tringulr fuzzy number is specil trpezoidl fuzzy number with = 3. Definition 3. If L(x)=R(x)= e x with x R then x is norml fuzzy number tht is shown by ( m σ ) nd the corresponding membership function is defined to be: x m ( ) µ ( x) = e σ x R where m is the men nd σ is the stndrd devition. A norml fuzzy number is shown in Figure 3. Definition 4. The -cut nd strong -cut for fuzzy set A re shown by A nd nd for [0] re defined to be: A = { x µ ( x) x X } A + A = { x µ ( x) > x X } A + A respectively where X is the universl set. Upper nd lower bounds for ny -cut ( A ) re shown by sup A nd inf A respectively. Here we ssume tht the upper nd lower bounds of -cuts re Figure. A trpezoidl fuzzy number. Figure 3. A norml fuzzy number. finite vlues nd for simplicity we show + A nd A inf by A. sup A by.. Computing -Cuts for Fuzzy Numbers For n LR fuzzy number with L nd R s invertible functions the -cuts re: m x m x = L[ ] = L ( ) x = m βl ( ) β β x m = R[ ] γ x m = R γ ( ) x = m + γr ( ). For specific L nd R functions the following cses re discussed. Let = ( 3 4 ) be trpezoidl fuzzy number. The -cut for ( ) is computed s: 0 x x x µ % ( x) = x 3 4 x 3 x x x = x = ( ) + 4 x = x = 4 ( 4 3) = 4 ( 4 3) = 0. () = ( ) + Note tht the -cut for tringulr fuzzy numbers is simply obtined by using the bove equtions considering = 3 : + = 3 ( 3 ) = 0. () = ( ) + For = ( m σ ) norml fuzzy number is computed s: m x ( ) m x σ = e ln( ) = σ x m ( ) x m σ = e ln( ) = σ = L R = m σ = m + σ ln ln 0<. (3)

4 A. TAJDIN ET AL. 5.. Fuzzy Sum Opertors Here we propose n pproch for summing vrious fuzzy numbers pproximtely using -cuts. The pproximtion is bsed on fitting n pproprite model for the sum using -cuts of the ddition by set of i vlues. Let us divide the -intervl [0] into n equl subintervls by letting 0 = 0 i = i + i i= n with i = i= n. This wy we hve n set of n+ equidistnt i points. For ddition of two different fuzzy numbers we dd the set of corresponding i -cut points of the two numbers to yield the i -cuts of the sum s n pproximtion for the fuzzy ddition. 3. An Algorithm for Fuzzy Shortest Pth in Network 3.. Distnce between Fuzzy Numbers Knowing tht we cn obtin good pproximtion of the ddition of vrious fuzzy numbers by use of -cuts we compute the distnce between two fuzzy numbers using the resulting points from the -cuts. For nd b s two different fuzzy numbers we use new fuzzy rnking method for the fuzzy numbers. Let us consider the fuzzy min opertion to be defined s follows MV % = Min vlue( b % % ) = (min( b) min( b ) min( 3 b3)min( 4 b 4)). (4) It is evident tht for non-comprble fuzzy numbers nd b the fuzzy min opertion results in fuzzy number different from both of them. For exmple for = (5039) nd b = (6950) we get from (4) fuzzy MV = (5939 ) which is different from both nd b. To llevite this drwbck we use method bsed on the distnce between fuzzy numbers. We use the distnce function introduced by Sdeghpour-Gildeh nd Gien [7]. The min dvntges of this distnce function re the generlity of its usge on vrious fuzzy numbers nd its relibility in distinguishing unequl fuzzy numbers. Indeed the use of this distnce function worked out to be quite pproprite for our pproch here s well s in different context before where we considered the rc in the network to be ll of the sme type (see Mhdvi et l. [8]). The proposed D p q -distnce indexed by prmeters < p < nd 0< q < between two fuzzy numbers % nd b % is nonnegtive function given by: The nlyticl properties of depend on the first D p q D pq p p + + p ( q) b d + q b d p < ( b % %) = ( q)sup ( b ) + q inf ( b ) p =. 0< 0< (5) prmeter p while the second prmeter q of chrcterizes the subjective weight ttributed to D p q + the end points of the support; i.e. ( ) of the fuzzy numbers. If there is no reson for distinguishing ny side of the fuzzy numbers then D is recommended. p Hving q close to results in considering the right side of the support of the fuzzy numbers more fvorbly. Since the significnce of the end points of the support of the fuzzy numbers is ssumed to be the sme then we consider q =. For two fuzzy numbers nd b with corresponding i -cuts the D pq distnce is pproximtely proportionl to: n n p p + + Dp q ( b ) = ( q) b q b. i + i i i i= i= (6) If q = p = then the bove eqution turns into: D ( b ) = ( ) ( b + i i n i= i= n + i b (7) To compre two fuzzy rc nd b with i -cuts s their pproximtions since they re supposed to represent positive vlues we compre them with M V = (00...0). In fct we use (7) to compute D ( 0) nd D ( b 0) nd use these vlues for comprison of the two numbers. + i ) p

5 5 A. TAJDIN ET AL. 3.. An Algorithm for Computing s Shortest Fuzzy Pth The following dynmic progrmming lgorithm is for computing the shortest pth in network. The lgorithm is bsed on Floyd s dynmic progrmming method to find shortest pth if it exists between every pir of nodes i nd j in the network (see Floyd [9]). We mke use of the following optiml vlue function f k ( i nd the corresponding lbeling function P k ( i : f k ( i = length of the shortest pth from node i to node j when the pth is considered to use only the nodes...k. from the set of nodes { } P k ( i = the lst intermedite node on the shortest pth from node i to node j using {...k} s intermedite node. The dynmic updting for the optiml pth length nd its corresponding lbeling re: f i min f ( i f { ( i k) f ( k )} k ( = k k + k j P Pk ( i = P k k ( i if k is not on shortest pth from i to j using {...k} ( k otherwise. We re now redy to give the steps of the lgorithm. Algorithm: A dynmic progrmming method for computing shortest pth in fuzzy network G = ( V A) where V is the set of nodes with V = N nd A is the set of rcs. Step : Let k=0 nd fk ( i = dij for ll ( i A f k = i j = for ll ( i A. If n rc exists from ( ) node i to node j then let P k ( i = i. Step : Let k = k +. Do the following steps for i = 3... N j = 3... N i j.. Compute the vlue of fk ( i j ) = min f ( i f ( ik) f ( k [ ] k k k + (for the ddition our proposed method discussed in Subection. nd for comprison of fuzzy numbers the D method of Subection 3. re pplied).. If node k is not on the shortest pth using nodes...k { } s intermedite nodes then let Pk ( i j ) = P ( ) i j Pk ( i = Pk ( k j k else let ) Step 3: If k < N then go to Step. Step 4: Obtin the shortest pth using the ( i. If f N ( i = then there is no pth between i nd j. The shortest pth from node i to j if it exists is identified bckwrds nd red by the nodes: j P N ( i = k followed by PN ( i k)... PN ( i l) = i where l is the node immeditely fter i in the pth. p q 3.3. Termintion nd Complexity of the Algorithm The proposed lgorithm termintes fter N outer itertions corresponding to k. A totl of N(N-) dditions nd comprisons re needed for every k. For ech ddition n fuzzy dditions for the -cuts should be performed i P k resulting in n(n)(n-) dditions. For comprisons we hve (n+)n(n-) dditions nd (n+) N(N-) multiplictions using (7). Therefore the totl needed opertions re (6n+) N(N-) dditions nd multiplictions with N(N-) comprison 4. Comprtive Exmples Here we illustrte exmples for comprison of our proposed method nd two other pproches. Exmple : Consider the following network Figure 4 considered by Chung nd Kung [30]. The tringulr rc re presented in Tble. The results obtined by the pproch ( P ) of Chung nd Kung [30] re f nd 6 6 shown in Tbles nd 3. The shortest pth nd the corresponding length using the proposed pproch in Chung nd Kung [30] re reported below: () (4) (45) Shortest pth from to 6 : 4 6. Shortest pth length fromto 6 : ( ) Figure 4. The network for Exmple. Tble. The rc for exmple. (334550) (56587) (34046) (3) (5) (46) (4576) (57985) (88934) (3) (35) (56) (5056) (435560) (7504)

6 A. TAJDIN ET AL. 53 Tble. The f vlues obtined by the pproch of Chung nd Kung. 6 i/j (334550) (4576) (8903) (85) (779556) - - (5056) (56587) (57985) (445006) (435560) (86574) (34046) (88934) (7504) Tble 3. The P6 vlues obtined by the pproch of Chung nd Kung. i/j Here we solve the sme problem using our proposed lgorithm givenin Subection 3. using the rnking method of Sdeghpour-Gildeh nd Gien [7]. The results of the proposed pproch for f 6 nd P 6 re given in Tbles 4 nd 5. Here the shortest pth obtined nd the corresponding length re exctly the sme s the ones we obtined by the pproch of Chung nd Kung [30]. Exmple : Consider the following network Figure 5 considered Tble 4. The f 6 vlues obtined by our proposed lgorithm i/j (334550) (4576) (8903) (85) (779556) - - (5056) (56587) (57985) (445006) (435560) (86574) (34046) (88934) (7504) by Hernndes et l. [3]. The fuzzy tringulr rc re given in Tble 6. The results ( f 6 nd P 6 ) for the pproch of Hernndes et l. [3] re given in Tbles 7 nd 8. The shortest pth nd the corresponding length using the proposed pproch of Hernndes et l. [3] re reported below: Shortest pth fromto: 9 7. Shortest pth length from to: ( ) We solved the sme problem using our proposed lgorithm of Subsection 3. using the rnking method of Sdeghpour-Gildeh nd Gien [7]. The results of our proposed pproch ( f nd P ) re given in Tbles 9 nd 0. The shortest pth nd the corresponding length re reported below: Shortest pth fromto: 9 7. ( ) Shortest pth length from to:. Clerly the proposed lgorithm computes lmost the sme solution s obtined by Hernndes et l. [3]. Tble 5. The P 6 vlues obtined by our proposed lgorithm. i/j Figure 5. The network for Exmple.

7 54 A. TAJDIN ET AL. Tble 6. The rc for Exmple. () ( ) (35) ( ) (84) ( ) (3) ( ) (38) ( ) (87) (30455) (6) ( ) (45) (90990) (97) (03050) (9) ( ) (46) ( ) (98) (303745) (0) ( ) (4) ( ) (90) (30460) (3) (808693) (56) ( ) (07) ( ) (5) ( ) (6) (30460) (0) ( ) (9) ( ) (76) ( ) (34) ( ) (7) ( ) Tble 7. The f ( i ) vlues obtined by Hernndes et l. j i/j ( ) ( ) ( ) ( ) ( ) ( ) - - (808693) ( ) ( ) (05705) ( ) ( ) ( ) ( ) ( ) (90990) ( ) ( ) ( ) ( ) ( ) ( ) (30455) ( ) ( ) ( ) (03050) ( ) ( ) Tble 7. continued. i/j ( ) ( ) ( ) ( ) ( ) ( ) (3070) ( ) 3 ( ) - - (05570) ( ) ( ) (30460) ( ) ( ) 9 (303745) - (30460) ( ) ( ) Exmple 3: A wireless sensor network Consider mobile service compny which hndles 3 geogrphicl centers. A configurtion of telecommuniction network is presented in Figure 6. Assume tht the distnce between ny two centers is trpezoidl fuzzy number (the rc re given in Tble ). The compny wnts to find shortest pth for n effective messge flow mongst the centers. The results obtined by our pproch ( f3 nd P 3) re given in Tbles nd 3. The shortest pth nd the corresponding length re reported below: Shortest pth fromto 3:

8 A. TAJDIN ET AL. 55 ( ) Shortest pth length fromto 3:. 5. Discussion We cn lso pply the proposed lgorithm to networks hving possibly mixture of different fuzzy numbers s rc. To see how the steps of the proposed lgorithm re crried out on such networks smll sized Tble 8. The P ( i ) vlues obtined by Hernndes et l. j mixed fuzzy network with 4 nodes s shown in Figure 7 is considered where the rc re considered to be mixture of trpezoidl nd norml fuzzy numbers. Exmple 4: Consider the mixed fuzzy network in Figure 4 with four nodes nd five rcs hving two trpezoidl nd three norml rc s specified in Tble 4. Step : We gin the fk ( i = d ij for k=0 s specified in Tble 4. i/j Tble 9. The f ( i vlues obtined by our proposed lgorithm. i/j ( ) ( ) ( ) ( ) ( ) ( ) - - (808693) ( ) ( ) (05705) ( ) ( ) ( ) ( ) ( ) (90990) ( ) ( ) ( ) ( ) ( ) ( ) (30455) ( ) ( ) ( ) (03050) ( ) ( ) Tble 9. continued. i/j ( ) ( ) ( ) ( ) ( ) ( ) (3070) ( ) 3 ( ) - - (085370) ( ) ( ) (30460) ( ) ( ) 9 (303745) - (30460) ( ) ( ) Tble 0. The P ( i ) vlues obtined by our proposed lgorithm. j i/j

9 56 A. TAJDIN ET AL. () (5) (38) (58) (69) (7) (96) (4) (5) (4) (60) (8) (9) (3) (357) (7890) (067) (693) (680) (6789) (6790) (893) (456) (34) (946) (5789) (5679) (4568) Tble. The rc for Exmple 3. (3) (6) (47) (5) (60) (8) (06) (7) (35) (58) (70) (8) (03) (935) (5056) (704) (7034) (045) (5890) (367) (693) (045) (893) (70) (3579) (3467) (4) (7) (4) (5) (70) (83) (07) (4) (39) (59) (7) (83) (3) Tble. The f ( i ) vlues obtined by our lgorithm. 3 j (803) (63) (6034) (0357) (903) (3580) (590) (3468) (7890) (570) (6780) (579) (578) i/j (357) (935) (803) (7890) (733033) (84630) (3703) (5056) (63) (067) - - (704) (693) Tble. continued. i/j (334044) ( ) (484) (747) (6833) (73337) ( ) (857) (536) (89) - - (073035) (5957) (3647) (833445) (38384) 4 - ( ) (6034) - - (4947) (7034) (0357) (9493) (5947) (73033) 6 (680) (045) (903) (6789) - - (469) (5890) (3580) (858) (375) (893) (3468) (456) (045)

10 A. TAJDIN ET AL. 57 Tble. continued. i/j ( ) (073337) ( ) ( ) ( ) (634447) ( ) ( ) (753437) (873035) - - (537447) ( ) - ( ) ( ) ( ) - ( ) ( ) ( ) 4 ( ) (947) - - (993539) (86337) - ( ) 5 - (3947) ( ) (633843) (093539) (96337) ( ) (344955) 6 (59) ( ) - - (73337) (337446) - ( ) 7 (3830) (69) - - (963034) (8373) - ( ) (63338) (03730) - (934384) (434047) (633449) 9 (6790) (5936) - - ( ) 0 (367) (590) - - (53033) (683) - ( ) - (693) - - (395) (693) - (43364) - - (0369) (758) - (4693) (383338) ( ) (858) (7890) - ( ) (6337) (383439) - (34) - (37303) (893) (570) - (3693) (48) (3604) (946) - - (63033) (70) (6780) - (858) (5789) (3579) (579) (5679) (937) (3467) (578) (4568) Tble 3. The P ( i ) vlues obtined by our lgorithm. 3 j i/j

11 Figure 6. The telecommuniction network proposed for Exmple 3. Figure 7. A smll sized network hving vrious fuzzy rc. f Tble 4. The 0 ( i mtrix for k=0. i/j (345) (486) - - (4) (54) (5) Therefore with P k ( i = i we hve Tble 5. Step : Here we consider k= nd compute the vlue of fk ( i = min[ fk ( i fk ( i k) + fk ( k ]. The result is shown in Tble 6. Therefore for P k ( i = i we hve Tble 7. 3 Tble 5. The P ( i ) mtrix for k=0. 0 j i/j Tble 6. The f ( i mtrix for k=. i/j (345) (486) (4) (54) (5) Tble 7. The P ( i ) mtrix for k=. j i/j A. TAJDIN ET AL. 3 s intermedite nodes then we consider Pk ( i = Pk ( i otherwise we let Pk ( i = Pk ( i k). We now report the results obtined for other vlues of k in Tbles 8-3. Note tht the sets V i nd W i re the points obtined by -cut dditions where the V nd W vlues re obtined by the If node k is not on the shortest pth using {...k} i -cuts considering n=0. It includes 0 points for the nd 0 points for the + : i V ={( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (78)} W ={( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (89)} V ={( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (78)} W ={( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3)} Tble 8. The f ( i ) mtrix for k=. i j i/j (345) V W - - (4) (54) (5) Tble 9. The P ( i mtrix for k=. i/j Tble 0. The f ( i ) mtrix for k= 3 3 j i/j (345) V W - - (4) (9) (5) Tble. The P ( i ) mtrix for k= 3. 3 j i/j

12 A. TAJDIN ET AL. 59 Tble. The f ( i ) mtrix for k= 4. 4 j i/j (345) V 3 W (4) (9) (5) Tble 3. The P ( i ) mtrix for k= 4. 4 j i/j V 3 ={( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (78)} W 3 ={( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3)} Finlly when k = N we identify the shortest pth s follows: Shortest pth from to 4: Shortest pth length from to 4: ( )( )( )( )( )( )( )( )( )(3) 6. Conclusions We presented novel pproch for computing shortest pth in mixed network hving vrious fuzzy rc. First we developed new technique for the ddition of vrious fuzzy numbers in pth using -cuts. Then we pplied dynmic progrmming method for finding shortest pth in the network using recently proposed distnce function to compre fuzzy numbers in the proposed lgorithm. Four comprtive exmples were worked out to illustrte the pplicbility of our proposed pproch s compred to two other methods in the literture s well s demonstrte the dditionl novel feture offered by our lgorithm to find fuzzy shortest pth in mixed fuzzy networks hving vrious settings for the fuzzy rc. 7. References [] R. E. Moore Method nd ppliction of intervl nlysis SIAM Phildelphi 997. M. Delgdo J. L. Verdegy nd M. A. Vil A procedure for rnking fuzzy numbers using fuzzy reltions Fuzzy Sets nd Systems Vol. 6 pp [] M. Delgdo J. L. Verdegy nd M. A. Vil A procedure for rnking fuzzy numbers using fuzzy reltions Fuzzy Sets nd Systems Vol. 6 pp [3] H. Ishibuchi nd H. Tnk Multi-objective progrmming in optimiztion of the intervl objective function Europen Journl of Opertionl Reserch Vol. 48 pp [4] J. K. Sengupt Optiml decision under uncertinty Springer New York 98. [5] T. Shocheng Intervl number nd fuzzy number liner progrmming Fuzzy Sets nd Systems Vol. 66 pp [6] A. Sengupt T. K. Pl Theory nd methodology on compring intervl numbers Europen Journl of Opertionl Reserch Vol [7] D. Dubois nd H. Prde Rnking fuzy numbers in the setting of possibility theory Informtion Sciences Vol. 30 pp [8] S. H. Chen Rnking fuzzy numbers with mximizing set nd minimizing set Fuzzy Sets nd Systems Vol. 7 pp [9] G. Bortoln nd R. Degni A review of some methods for rnking fuzzy subsets Fuzzy Sets nd Systems Vol. 5 pp [0] C.-H. Cheng A new pproch for rnking fuzzy numbers by distnce method Fuzzy Sets nd Systems Vol. 95 pp [] M. Blue B. Bush nd J. Puckett Unified pproch to fuzzy grph problems Fuzzy Sets nd Systems Vol. 5 pp [] L. T. Koczy Fuzzy grphs in the evlution nd optimiztion of networks Fuzzy Sets nd Systems Vol. 46 pp [3] C. M. Klein Fuzzy Shortest Pths Fuzzy Sets nd Systems Vol. 39 pp [4] Y. Li M. Gen nd K. Id Solving fuzzy shortest pth problem by neurl networks Computers Industril Engineering Vol. 3 pp [5] K. Lin nd M. Chen The fuzzy shortest pth problem nd its most vitl rcs Fuzzy Sets nd Systems Vol. 58 pp [6] S. Okd nd M. Gen Order reltion between intervls nd its ppliction to shortest pth problem Computers & Industril Engineering Vol [7] S. Okd nd M. Gen Fuzzy shortest pth problem Computers & Industril Engineering Vol. 7 pp [8] S. Okd nd T. Soper A shortest pth problem on network with fuzzy rc Fuzzy Sets nd Systems Vol. 09 pp [9] L. A. Zdeh Fuzzy sets s bsis for theory of possibility Fuzzy Sets nd Systems Vol. pp [0] R. K. Ahuj K. Mehlhorn J. B. Orlin nd R. E. Trjn

13 60 A. TAJDIN ET AL. Fster lgorithms for the shortest pth problem Technicl Report CS-TR Deprtment of Computer Science Princeton University 988. [] A. Andersson T. Hgerup S. Nilsson nd R. Rmn Sorting in liner time? In Proceedings of 7th ACM Symposium on Theory of Computing pp [] A. V. Goldberg Scling lgorithms for the shortest pths problem In: Proceedings of the 4th ACM-SIAM Symposium on Discrete Algorithms pp [3] Y. Asno nd H. Imi Prcticl efficiency of the liner-time lgorithm for the single source shortest pth problem Journl of the Opertions Reserch Society of Jpn Vol. 43 pp [4] Y. Hn Improved lgorithm for ll pirs shortest pths Informtion Processing Letters Vol. 9 pp [5] S. Sunders T. Tkok Improved shortest pth lgorithms for nerly cyclic grphs Electronic Notes in Theoreticl Computer Science Vol. 4 pp [6] S. Chns Fuzzy optimiztion in networks In: J. Kcprzyk S. A. Orlovski (Eds.) Optimiztion Models Using Fuzzy Sets nd Possibility Theory D. Reidel Dordrecht pp [7] B. Sdeghpour-Gildeh D. Gien Q. L Distnce-Dp et l. Cofficient de Corréltion entre deux Vribles Alétoires floues Actes de LFA 0 pp Monse- Belgium 00. [8] I. Mhdvi R. Nourifr A. Heidrzde nd N. Mhdvi- Amiri A dynmic progrmming pproch for finding shortest chins in fuzzy network Applied Soft Computing Vol. 9 No. pp [9] R. W. Floyd Algorithm 97 Shortest pth CACM 5 pp [30] T.-N. Chung nd J.-Y. Kung The fuzzy shortest pth length nd the corresponding shortest pth in network Computers & Opertions Reserch Vol [3] M. L. Fredmn nd R. E. Trjn Fiboncci heps nd their uses in improved network optimiztion lgorithm J. ACM 34 Vol. 3 pp [3] F. Hernndes M. T. Lmt J. L. Verdegy nd A. Ymkmi The shortest pth problem on networks with fuzzy prmeters Fuzzy Sets nd Systems Vol. 58 pp