Computers and Mathematics with Applications. Computing a fuzzy shortest path in a network with mixed fuzzy arc lengths using α-cuts

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1 Computers and Mathematcs wth Applcatons 60 (00) Contents lsts avalable at ScenceDrect Computers and Mathematcs wth Applcatons journal homepage: Computng a fuzzy shortest path n a network wth mxed fuzzy arc lengths usng α-cuts Al Tajdn a, Iraj Mahdav a,, Nezam Mahdav-Amr b, Bahram Sadeghpour-Gldeh c a Department of Industral Engneerng, Mazandaran Unversty of Scence and Technology, Babol, Iran b Faculty of Mathematcal Scences, Sharf Unversty of Technology, Tehran, Iran c Department of Mathematcs and Statstcs, Mazandaran Unversty, Babolsar, Iran a r t c l e n f o a b s t r a c t Keywords: Fuzzy numbers α-cut Dstance functon Shortest path Lnear least squares Regresson Dynamc programmng We are concerned wth the desgn of a model and an algorthm for computng a shortest path n a network havng varous types of fuzzy arc lengths. Frst, we develop a new technque for the addton of varous fuzzy numbers n a path usng α-cuts by proposng a lnear least squares model to obtan membershp functons for the consdered addtons. Then, usng a recently proposed dstance functon for comparson of fuzzy numbers, we present a dynamc programmng method for fndng a shortest path n the network. Examples are worked out to llustrate the applcablty of the proposed model. 00 Elsever Ltd. All rghts reserved.. Introducton The problem of fndng a shortest path from a specfed source node to any other node s fundamental n graph theory, and s of contnung nterest [,]. Ths problem arses from many applcatons ncludng transportaton, routng, communcatons, supply chan management or models nvolvng agents. Let G = (V, E) be a graph, where V s the set of vertces (nodes) and E s the set of edges (arcs). A path between two nodes s an alternatng sequence of vertces and edges begnnng wth a startng node and endng wth an endng node. The dstance (cost) of a path s the sum of the weghts (arc lengths) of the edges on the path. However, snce there can be more than one path between any two vertces, the problem of fndng a path wth a mnmal cost between two specfed vertces of nterest s the so-called shortest path problem (SPP). Although n conventonal graph theory, the weghts of the edges n an SPP are assumed to be precse real numbers, for most practcal applcatons, these parameters (.e., costs, capactes, demands, tme, etc.) are naturally mprecse. In such cases, an approprate modelng approach may justfably make use of fuzzy numbers, and so does the name fuzzy shortest path problem (FSPP) appear n the lterature [,3,4]. The FSPP, nvolvng addton and comparson of fuzzy numbers, s qute dfferent from the conventonal SPP, whch only nvolves crsp numbers. In an FSPP, the costs beng fuzzy numbers, the task of fndng a path beng smaller than all the others s not straghtforward, as the comparson of fuzzy numbers as an operaton can be defned n a wde varety of ways. Recently, several results have been publshed on the FSPP [5]. The work of Dubos and Prade [6] s one of the frst on ths subject and consders extensons of the classcal Floyd and Ford Moore Bellman (FMB) algorthms. However, t was verfed that the algorthm would compute the shortest path dstance wthout dentfyng an exstng path (see [7]) as was outlned by Klen [8] wth a fuzzy domnance set. Ln and Chern [3] defned the denomnaton of vtal arcs as beng those whose removal from the path resulted n an ncrease of the cost. Another algorthm for ths problem, presented by Okada and Gen [9,0], s a generalzaton of the Djkstra algorthm. In ths algorthm, the weghts of the arcs are consdered to be nterval numbers, and are defned usng a partal order between nterval numbers. Okada and Soper [5] characterzed a Correspondng author. Tel.: ; fax: E-mal address: rajarash@redffmal.com (I. Mahdav) /$ see front matter 00 Elsever Ltd. All rghts reserved. do:0.06/j.camwa

2 990 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) soluton not as the shortest path, but as a fuzzy set soluton, where each element of the set s a no domnated path or a Pareto optmal path wth fuzzy edge weghts. However, ths algorthm does not provde decson-makers wth any gudelnes for choosng a best path accordng to ther own vewponts (optmstc/pessmstc, rsky/conservatve) []. Blue et al. [] presented an algorthm whch would fnd a cut value to lmt the number of analyzed paths, and then appled a modfed verson of the k-shortest path (crsp) algorthm proposed by Eppsten [7]. Followng the dea of fndng a fuzzy set soluton, Okada [3] ntroduced the concept of the degree of possblty of an arc beng on a shortest path. Among the most recent work s the one by Nayeem and Pal [4] that proposes an algorthm based on the acceptance ndex ntroduced by Sengupta and Pal [4] and whch gves a sngle fuzzy shortest path or a gudelne for choosng a best fuzzy shortest path accordng to the decson-maker vewpont [5]. Here, we propose a new approach and an algorthm to fnd a shortest path n a network wth varous fuzzy arc lengths. The remander of the paper s organzed as follows. In Secton, basc concepts and defntons are gven. Secton 3 explans ways of computng α-cuts for fuzzy numbers. We present our fuzzy sum operator by use of a lnear least squares model n Secton 4. In Secton 5, usng a recently proposed dstance functon, we present a dynamc programmng algorthm for fndng fuzzy shortest path n a mxed fuzzy network. We conclude n Secton 6.. Defntons We start wth basc defntons of some well-known fuzzy numbers. Defnton. An LR fuzzy number s represented by ã = (m, a, b) LR, wth the membershp functon, µã(x), defned by [ ] m x L x m a µã(x) = [ ] x m R x m, b where L and R are non-ncreasng functons from R + to [0, ], L(0) = R(0) =, m s the center, a s the left spread and b s the rght spread. Note that f L(x) = R(x) = x wth 0 < x <, then x s a trangular fuzzy number and s represented by the trplet ã = (m, a, b), wth the membershp functon, µã(x), defned by ( ) m x x m a µã(x) = ( ) x m x m. b Defnton. A trapezodal fuzzy number ã s shown by ã = (a, a, a 3, a 4 ), wth the membershp functon as follows: 0 x a x a a x a a a µã(x) = a x a 3 a 4 x a 3 x a 4 a 4 a 3 0 a 4 x. A general trapezodal fuzzy number, along wth a cut (to be explaned later n Defnton 4), s shown n Fg.. It s apparent that a trangular fuzzy number s a specal trapezodal fuzzy number wth a = a 3. Defnton 3. If L(x) = R(x) = e x, wth x R, then x s a normal fuzzy number that s shown by (m, σ ) and the correspondng membershp functon s defned to be: x m µã(x) = e ( σ ), x R, where m s the mean and σ s the standard devaton. A normal fuzzy number, along wth a cut (to be explaned later n Defnton 4), s shown n Fg.. Defnton 4. The α-cut and strong α-cut for a fuzzy number ã are shown by ã α and ã + α, respectvely, and for α [0, ] are defned to be: ã α = { x µã(x) α, x X }, ã + α = { x µã(x) > α, x X }, where X s the unversal set.

3 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) Fg.. A trapezodal fuzzy number wth an α-cut. Fg.. A normal fuzzy number wth an α-cut. Note that the upper and lower bounds for the α-cut set (ã α ) are shown by sup ã α and nf ã α, respectvely. Here, we assume that the upper and lower bounds of α-cuts are fnte values and for smplcty we show sup ã α by ã R α and nf ã α by ã L α (see Fgs. and ). 3. Computng α-cuts for fuzzy numbers For the LR fuzzy numbers wth L and R nvertble functons, the α-cuts are: [ ] m x α = L m x = L (α) ã L α a a = x = m al (α), [ ] x m α = R x m = R (α) ã R α b b = x = m + br (α). For specfc L and R functons, the followng cases are dscussed. 3.. α-cuts for trapezodal fuzzy numbers Let ã = (a, a, a 3, a 4 ) be a trapezodal fuzzy number. An α-cut for ã, ã α, s computed as: α = x a a a ã L α = x = (a a )α + a, α = a 4 x a 4 a 3 ã R α = x = a 4 (a 4 a 3 )α, ã α = {ãl α = (a a )α + a ã R α = a 4 (a 4 a 3 )α, 0 α, () where ã = α [ãl α, ãr α ] s the correspondng α-cut. The α-cuts for trangular fuzzy numbers are obtaned by usng the above equatons consderng a = a α-cuts for normal fuzzy numbers If ã = (m, σ ) s a normal fuzzy number, then ã α s computed as: m x α = e ( σ ) ln(α) = m x ã L α σ = x = m σ ln α

4 99 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) x m α = e ( σ ) ln(α) = x m ã R α σ = x = m + σ ln α {ãl ã α = α = m σ ln α ã R α = m + σ, 0 < α. () ln α 4. Fuzzy approxmate sum operators Here, we propose an approach for summng varous fuzzy numbers approxmately usng α-cuts. The approxmaton s based on fttng an approprate model for the sum usng α-cuts of the addton as the ftness data. Let us dvde the α-nterval [0, ] nto n equal subntervals by lettng α 0 = 0, α = α + α, α =, =,..., n. Ths way, we have a set of n + n equdstant ponts. For the normal fuzzy numbers x (, + ), t s mproper to assume α beng equal to zero. Therefore, n ths case we consder α (0, ], and thus use the nonzero α, n. Here, we ntend to show how to add up a trapezodal fuzzy number wth a normal one. We present a numercal approach to approxmate the sum and ts correspondng membershp functon. 4.. α-cut sum Let ã = (a, a, a 3, a 4 ) and b = (m, σ ) be the trapezodal and normal fuzzy numbers, respectvely. Gven α (0, ], n, the α-cut sum of these fuzzy numbers usng Eqs. () and () s obtaned as follows: L { c c α = ã α + bα c α = α = (ã L α + bl α ) c α R = (ãr α +, n, br α ), where, [ c α L = (a a )α + a + m σ ] ln α, [ c α R = a 4 (a 4 a 3 )α + m + σ ] ln α. (3) Usng Eq. (3), correspondng to α, n, n ponts are obtaned for c (n ponts for the c α L and n ponts for the c α R ). Usng these ponts, t s possble to approxmate the sum of the two fuzzy numbers. An approxmate membershp functon of the sum s computed by fttng an approprate functon usng the α-cut ponts. For the addton of normal and trapezodal fuzzy numbers, the case beng consdered n our examples later on, we propose an exponental membershp functon for approxmatng the sum as follows (later, we wll see that ths choce would ndeed provde a good model for the approxmatng sum of trapezodal and normal fuzzy numbers). Let x = c α R and y = µ( c α R ), and usng the n ponts ( ) (x, y ), n, consder the fttng model as y = e x λ β. The unknown parameters λ and β appear nonlnearly. We lnearze the model, by notng that for any x > λ (as s the case here for the rght hand model), we must have: ( ) x λ ln y =. (4) β Snce 0 < y, then ln(y ) 0, and thus we can wrte, ( ) x λ ln y =, β and hence, β ln y + λ = x. Therefore, we defne a lnear of least squares model for the mnmzaton of error as follows: n ( mn E = β ) ln y + λ x, = where x = c R α s gven by (3). To solve (7), we need to have: E β = E λ = [( ) ln y (β )] ln y + λ x = 0, (8) [ (β )] ln y + λ x = 0. (5) (6) (7)

5 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) Hence, we need to solve the followng so-called normal equatons for the unknown parameters β and λ: x ln y, (9) β β ln y + λ ln y + nλ = ln y = x. (0) Usng Cramer s rule to solve (9) and (0), β and λ are explctly determned to be: x ln y ln y x n n x ln y ln y x β = β = ln y ln y n ln y ln y, () ln y ln y n ( ) ln y ln y x ln y x ln y x ( ) x ln y ln y λ = λ = ln y ln y n ln y ln y. () ln y ln y n ( Now, smlarly let x = c α L and ȳ = µ( c α L ), and consder the model y = e λ ) x β. Usng the above approach, we have: Or, ( λ ) x ln y =. β β ln y + λ = x. (3) Thus, solvng mn E = n = smlarly leads to: ( β ln y + λ x ), ln y β = n n x ln y + ln y x ln y ln y, (4) ( x ) ln y ln y λ = ln y n x ln y ln y ln y. (5) Thus, the membershp functon s determned to be: ) β x < λ µ c (x) = λ x λ ( ) e x λ β x > λ, e ( λ x (6) wth λ, β, λ and β as defned by (), (), (4) and (5), respectvely.

6 994 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) Normal Fuzzy Number Trapezod Fuzzy Number x 8 Fg. 3a. The α-cut ponts of fuzzy numbers. Remarks. An equvalent approach for fttng the least squares model s to consder the bass functons { ln y, }, for the fttng relaton (6). Ths would lead to the mnmzaton model, wth mn θ Aθ x, ln y A =.., ln yn θ = [ β λ ], x = x.. x n. The soluton of (7) s obtaned by solvng the normal equatons, A T Aθ = A T x, yeldng the same results as () and (). Convenently, ths general data fttng approach can be used to consder other membershp functons n cases usng varous types of fuzzy numbers. The key element s thus to decde the approprate basc functons. Now, we provde a numercal llustraton for our proposed model for addton of trapezodal and normal fuzzy numbers. Example. Consder the followng two fuzzy numbers, one beng normal and the other beng trapezodal: ã = (3, 5), b = (4, 0, 7, 6). The dagrams of the numbers at the α-cuts and ther sum, (ã + b), usng Eq. (3), are shown n Fgs. 3a and 3b, respectvely. As ndcated n the dagram, the result s nether trapezodal nor normal. Usng the ponts obtaned from the α-cuts consderng n = 00, the values of (λ, β) and (λ, β ) are obtaned by (9) (5). For ths example, we obtan: ( ) e 3 x x < 3 µ c (x) = 3 x 30 ( ) e x x > 30. (7) 5. An algorthm for fuzzy shortest path n a network 5.. Dstance between fuzzy numbers Knowng that we can obtan a good approxmaton for the addton of varous fuzzy numbers by use of α-cuts, we compute the dstance between two fuzzy numbers usng the resultng ponts from the α-cuts. Assume that ã and b are two fuzzy

7 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) Trapezod + Normal < α Output x 50 Fg. 3b. The sum of mxed fuzzy numbers usng α-cuts. numbers. We apply a fuzzy rankng method for fuzzy numbers. We have used ths rankng method effectvely n a recent work [6]. Let us consder fuzzy mn operatons as follows: Mn value (ã, b) = (mn(a, b ), mn(a, b ), mn(a 3, b 3 ), mn(a 4, b 4 )). (8) It s evdent that, for non-comparable fuzzy numbers ã and b, the fuzzy mn operaton results n a fuzzy number dfferent from both of them. For example, for ã = (5, 0, 3, 9) and b = (6, 9, 5, 0), we get from (8) a fuzzy MṼ = Mn value(ã, b) = (5, 9, 3, 9), whch s dfferent from both ã and b. To allevate ths drawback, we use a method based on the dstance between fuzzy numbers. We use the dstance functon ntroduced n [7]. The man advantages of ths dstance functon are the generalty of ts usage on varous fuzzy numbers, and ts relablty n dstngushng unequal fuzzy numbers. Indeed, the usage of the dstance functon below worked out to be qute approprate for our approach. The D p,q -dstance, ndexed by parameters < p < and 0 < q <, between two fuzzy numbers ã and b s a nonnegatve functon gven by: [ ] ( q) a α b p α dα + q a + α b + p p α dα, p <, D p,q (ã, b) = 0 0 (9) ( q) sup a α b α + q nf a + α b + α, p =. 0<α 0<α The analytcal propertes of D p,q depend on the frst parameter p, whle the second parameter q of D p,q characterzes the subjectve weght attrbuted to the end ponts of the support;.e., the a + α and a α of the fuzzy numbers. If there s no reason for dstngushng any sde of the fuzzy numbers, then D p, s recommended. Havng q close to results n consderng the rght sde of the support of the fuzzy numbers more favorably. Snce the sgnfcance of the end ponts of the support of the fuzzy numbers s assumed to be the same, then we consder q =. For two fuzzy numbers ã and b wth correspondng α -cuts, the D p,q dstance s approxmately proportonal to: D p,q (ã, b) = [( q) n a α b p α + q = n a + α b + p α = ] p. (0) If q =, p =, then the above equaton turns nto: D, (ã, b) = n (a α b α ) + n (a + α b + α ). () = =

8 996 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) To compare two fuzzy arc lengths ã and b wth α -cuts as ther approxmatons, snce they are supposed to represent postve values, we compare them wth MṼ = (0, 0,..., 0). In fact, we use formula () to compute D, (ã, MṼ) and D, ( b, MṼ) and then use these values for comparson of the two numbers. Here, we consder ã = (6, 9, 5, 0) and b = (5, 0, 3, 9) wth n = 0. Then, the α-cuts for ã are obtaned to be a + α = {9.5, 9, 8.5, 8, 7.5, 7, 6.5, 6, 5.5, 5}. a α = {9, 8.7, 8.4, 8., 7.8, 7.5, 7., 6.9, 6.6, 6.3} and for b we have b + α = {8.4, 7.8, 7., 6.6, 6, 5.4, 4.8, 4., 3.6, 3}. b α = {0, 9.5, 9, 8.5, 8, 7.5, 7, 6.5, 6, 5.5}. As a result, the dstances are D, (ã, MṼ) = D, ( b, MṼ) = Therefore, ã > b. 5.. An algorthm for computng a shortest path The followng dynamc programmng algorthm s for computng the shortest path n a network. The algorthm s based on Floyd s dynamc programmng method to fnd a shortest path, f t exsts, between every par of nodes and j n the network [8]. We make use of the followng optmal value functon f k (, j) and the correspondng labelng functon P k (, j): f k (, j) = length of the shortest path from node to node j when the path s consdered to use only the nodes from the set of nodes {,..., k}, P k (, j) = the last ntermedate node on the shortest path from node to node j usng {,..., k} as ntermedate node, where, s a source node, j s the end node and k refers to an ntermedate node. The dynamc updatng for the optmal path length and ts correspondng labelng are: f k (, j) = mn {f k (, j), f k (, k) + f k (k, j)}, { Pk (, j) f k s not on shortest path from to j usng {,..., k} P k (, j) = P k (k, j) otherwse. We are now ready to gve the steps of the algorthm. Algorthm. A dynamc programmng method for computng a shortest path n a fuzzy network G = (V, A), where V s the set of nodes wth V = N, and A s the set of arcs. The value dj s the fuzzy arc dstance for arc (, j), f t exsts. Below, fk (, j) the shortest path length s set to, when there s no arc. Step : Let k = 0 and fk (, j) = dj, for all (, j) A, fk = (, j) =, for all (, j) A. If an arc exsts from node to node j then let P k (, j) =. Step : Let k = k +. Do the followng steps for =,, 3,..., N, j =,, 3,..., N, j. ]. Compute the value of fk (, j) = mn [ fk (, j), fk (, k) + fk (k, j) (for the addton, our proposed method dscussed n Secton 4. and for comparson of fuzzy numbers the D p,q dstance functon () of Secton 5. are appled).. If node k s not on the shortest path usng the nodes {,,..., k} as ntermedate nodes, then let P k (, j) = P k (, j) else let P k (, j) = P k (k, j). Step 3: If k < N then go to Step. Step 4: Obtan the shortest path usng the P k (, j) values. If f N (, j) =, then there s no path between and j. The shortest path from node to j, f t exsts, s dentfed backwards and read by the nodes: j, P N (, j) = k followed by P N (, k),..., P N (, l) =, where l s the node mmedately after n the path.

9 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) (,3,4,5) (4,) (5,4) 4 (4,8,,6) 3 (5,) Fg. 4. A small szed network havng mxed fuzzy arc lengths. Table The f0 (, j) matrx for k = 0. /j 3 4 (, 3, 4, 5) (4, 8,, 6) (4, ) (5, 4) 3 (5, ) Table The P 0 (, j) matrx for k = 0. /j Table 3 The f (, j) matrx for k =. /j 3 4 (, 3, 4, 5) (4, 8,, 6) (4, ) (5, 4) 3 (5, ) Table 4 The P (, j) matrx for k =. /j Termnaton and complexty of the algorthm The proposed algorthm termnates after N outer teratons correspondng to k. A total of N(N ) addtons and comparsons are needed for every k. For each addton, n fuzzy addtons for the α -cuts should be performed resultng n n(n)(n ) addtons. For comparsons, we have (n + )N(N ) addtons and (n + ) N(N ) multplcatons usng (7). Therefore, the total needed operatons are (6n + ) N(N ) addtons and multplcatons, wth N(N ) comparsons. Example. Consder the mxed fuzzy network n Fg. 4 wth four nodes and fve arcs havng two trapezodal and three normal arc lengths as specfed n Table. Step : We set fk (, j) = dj, for k = 0, as specfed n Table. Therefore, wth P k (, j) =, Table s obtaned. Step : Here, we consder k = and compute the value of f k (, j) = mn [f k (, j), f k (, k) + f k (k, j)]. The result s shown n Table 3. Therefore, for P k (, j) =, Table 4 s obtaned. If node k s not on the shortest path usng {,,..., k} as ntermedate nodes, then we consder P k (, j) = P k (, j), otherwse we let P k (, j) = P k (, k). We now report the results obtaned for other values of k n Tables 5 0. Note that, the

10 998 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) Table 5 The f (, j) matrx for k =. /j 3 4 (, 3, 4, 5) V W (4, ) (5, 4) 3 (5, ) Table 6 The P (, j) matrx for k =. /j Table 7 The f3 (, j) matrx for k = 3. /j 3 4 (, 3, 4, 5) V W (4, ) (9, ) 3 (5, ) Table 8 The P 3 (, j) matrx for k = 3. /j Table 9 The f4 (, j) matrx for k = 4. /j 3 4 (, 3, 4, 5) V 3 W 3 (4, ) (9, ) 3 (5, ) Table 0 The P 4 (, j) matrx for k = 4. /j sets V and W are the ponts obtaned by α-cut addtons, where the V and W values are obtaned by the α -cuts consderng n = 0. It ncludes 0 ponts for the a α and 0 ponts for the a + α : V = {(4.5857, 0.474), (4.9336, ), (5.074, ), (5.4477, ), ( , ), (5.8858, 9.47), (6.078, 8.897), (6.376, ), (6.5754, ), (7, 8)} W = {(.0303, ), (.55, ), (.9, 4.089), (3.57, 3.489), (4.698,.830), (4.74,.589), (5.3,.6889), (5.905,.0895), (6.606, ), (8, 9)} V = {(4.5857, 0.474), (4.9336, ), (5.074, ), (5.4477, ), ( , ), (5.8858, 9.47), (6.078, 8.897), (6.376, ), (6.5754, ), (7, 8)} W = {(8.0655, ), (8.6673, ), (9.0549, ), ( , 5.545), ( , 5.65), (0.706, 4.894), (0.5056, ), (0.855, 4.448), (.508, 3.749), (, 3)}

11 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) x Fg. 5. The membershp functon. Table The analyss of parameter varaton. n Path β β λ λ Table The parameters of non-optmal path wth ther dstance values for Example. n Path β β λ λ D, V 3 = {(4.5857, 0.474), (4.9336, ), (5.074, ), (5.4477, ), ( , ), (5.8858, 9.47), (6.078, 8.897), (6.376, ), (6.5754, ), (7, 8)} W 3 = {(8.0655, ), (8.6673, ), (9.0549, ), ( , 5.545), ( , 5.65), (0.706, 4.894), (0.5056, ), (0.855, 4.448), (.508, 3.749), (, 3)}. Fnally, when k = N, we dentfy the shortest path as follows: Shortest path from to 4: 3 4. Shortest path length from to 4: (8.0655, ), (8.6673, ), (9.0549, ), ( , 5.545), ( , 5.65), (0.706, 4.894), (0.5056, ), (0.855, 4.448), (.508, 3.749), (, 3). Here, we obtan the membershp functon as shown n Fg. 5. To nvestgate the varaton of β, β, λ, and λ wth respect to n, for the membershp functon (6), we solve the least squares problem for dfferent values of n and obtan the left and rght membershp functon parameters. The results are reported n Table. Note that, the path does not change for dfferent szes of n and after n = 00, the varatons n membershp functon parameters are neglgble. Also, for ths optmal path D, = Moreover, the costs of other paths wth n = 00 are obtaned as shown n Table. Next, we consder a problem wth a larger number of nodes to specfy the general nput and output structures usng the proposed algorthm. Example 3. We consder a larger network as shown n Fg. 6. We consder the network havng mxed arc lengths (a combnaton of normal and trapezodal fuzzy numbers) and use our dynamc algorthm to fnd the shortest paths. The arc lengths are specfed n Table 3. Usng the dstance functon D p,q (for q = / and p = ), the shortest path from the source node to the destnaton node 3 s determned to be: To fnd an optmal path, Table 4 s used. For nstance, to fnd an optmal path from node to node 3, accordng to P 3 (, j) matrx, we have P 3 (, 3) and P 3 (, 3) = 8. From P 3 (, 3) = 8, the path 8 3 s obtaned.

12 000 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) Fg. 6. A network. Table 3 The arc lengths. Arc Fuzzy number Arc Fuzzy number Arc Fuzzy number (, ) (, 3, 5, 7) (, 3) (40, ) (, 4) (8, 0,, 3) (, 5) (7, 8, 9, 0) (, 6) (35, 0) (, 7) (6,,, 3) (3, 8) (40, ) (4, 7) (7, 0,, 4) (4, ) (6, 0, 3, 4) (5, 8) (9, 9) (5, ) (7, 0, 3, 4) (5, ) (0, 3, 5, 7) (6, 9) (6, 8, 0, ) (6, 0) (35, ) (7, 0) (9, 0,, 3) (7, ) (6, 7, 8, 9) (8, ) (5, 8, 9, 0) (8, 3) (0, 5) (9, 6) (6, 7, 9, 0) (0, 6) (40, 3) (0, 7) (5, 9, 0, ) (, 4) (8, 9,, 3) (, 7) (8, 9) (, 4) (3, 4, 6, 8) (, 5) (, 4, 5, 6) (3, 5) (37, ) (3, 9) (7, 8, 9, 0) (4, ) (,, 3, 4) (5, 8) (8, 9,, 3) (5, 9) (5, 7) (6, 0) (38, ) (7, 0) (7, 0,, ) (7, ) (6, 7, 8, 0) (8, ) (5, 7, 8, 9) (8, ) (6, 5) (8, 3) (5,5) (9, ) (5, 6, 7, 9) (0, 3) (3, 4, 6, 7) (, 3) (, 5, 7, 8) (, 3) (0, 5) Table 4 P 3 (, j) matrx. k/j Also, P 3 (, 8) = 5, P 3 (, 5) =, and P 3 (, ) = 5. Snce P 3 (, 5) =, then we Stop. Therefore, the shortest path s: Usng the least squares model, we regress the estmated ponts and obtan the followng equaton as membershp functon for the shortest path. By addton of varous fuzzy numbers on the correspondng path, the membershp functon s obtaned

13 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) α-cut value the regressed model x Fg. 7. The obtaned membershp functon usng the least squares model. Table 5 The results of parameter varaton. n Path β β λ λ Fg. 8. The network. Table 6 The arc lengths of network (both crsp and fuzzy). Arc Fuzzy lengths Crsp lengths Arc Fuzzy lengths Crsp lengths Arc Fuzzy lengths Crsp lengths (, ) (,, 7) (, 3) (, 3, 4) 3 (, 3) (3, 5, 8) 5 (, 4) (,, 6) (, 5) (, 5, 6) 5 (3, 4) (, 3, 4) 3 (4, 5) (, 4, 5) 4 (4, 6) (,, 5) (5, 6) (4, 5, 8) 5 by () (4) as follows: ( ) e x x < 59 µ c (x) = 59 < x < 65 ( ) e x x > 65. The regressed membershp functon s presented n Fg. 7. To analyze the varatons of β, β, λ, and λ wth respect to n, we use dfferent szes of n and obtan the left and rght membershp functon parameters. The results are reported n Table 5. Note that the path does not change for dfferent values of n bgger than 00, and the varatons of membershp functon parameters are neglgble. 6. Dscusson Our approach can easly be used when the dstances are crsp values. Ths can serve as a tool for comparng crsp versus fuzzy arc length for a partcular case. Consder the example shown by Fg. 8. Both the fuzzy and crsp dstances of arcs are gven n Table 6. As an example, we consdered the crsp value of each arc to be the correspondng maxmum membershp value. The obtaned results for the shortest paths correspondng to the

14 00 A. Tajdn et al. / Computers and Mathematcs wth Applcatons 60 (00) two cases show the dfference of the optmal paths. For the fuzzy case, the optmal path s 3 4 6, whle for the crsp case, the optmal path s 4 6. The dstance value of optmal path n the fuzzy network, usng our proposed algorthm s 5.93, whle for the crsp stuaton s 6 (for the path 4 6). Moreover, for the optmal path n the fuzzy network, we wll obtan the value of 8 for the crsp case, showng the dfference of =.07 n favor of the fuzzy case. Ths example shows the possble effectveness of consderng fuzzy arc lengths n a network. 7. Conclusons A novel practcal approach was proposed for computng a shortest path n a fuzzy network havng mxed fuzzy arc lengths. In dong ths, an α-cut method was presented to compute the addton of varous fuzzy numbers as arc lengths. To obtan an approxmaton of the correspondng membershp functon for the addton, we proposed a lnear least squares model. Fnally, usng a recently proposed dstance functon, we showed how to decde dstances for comparson of fuzzy arc lengths to be used n our proposed dynamc programmng algorthm for fndng an optmal (shortest) path. The effectveness of our approach was shown by workng out llustratng examples. The proposed model, whle beng practcally smple, has the flexblty to consder a mxture of varous types of fuzzy arc lengths n a general network. We also gave a comparatve case of fuzzy and crsp fuzzy cases to pont out a possble effectveness of a fuzzy network. References [] T.-N. Chuang, J.-Y. Kung, The fuzzy shortest path length and the correspondng shortest path n a network, Comput. Oper. Res. 3 (005) [] M.T. Takahash, Contrbuções ao estudo de grafos fuzzy: teora e algortmos, Ph.D. Thess, Faculdade de Engenhara Elétrca e de Computação, UNICAMP, 004. [3] C. Ln, M.S. Chern, The fuzzy shortest path problem and ts most vtal arcs, Fuzzy Sets and Systems 58 (993) [4] S.M.A. Nayeem, M. Pal, Shortest path problem on a network wth mprecse edge weght, Fuzzy Optm. Decs. Mak. 4 (005) [5] S. Okada, T. Soper, A shortest path problem on a network wth fuzzy arc lengths, Fuzzy Sets and Systems 09 (000) [6] D. Dubos, H. Prade, Fuzzy Sets and Systems: Theory and Applcatons, Academc Press, New York, 980. [7] D. Eppsten, Fndng the k-shortest paths, n: Proc. IEEE Symp. on Foundatons of Computer Scence, 994, pp [8] C.M. Klen, Fuzzy shortest paths, Fuzzy Sets and Systems 39 (99) 7 4. [9] S. Okada, M. Gen, Order relaton between ntervals and ts applcatons to shortest path problem, n: Proc. 5th Annu. Conf. on Computers and Industral Engneerng, vol. 5, 993, pp [0] S. Okada, M. Gen, Fuzzy shortest path problem, n: Proc. 6th Ann. Conf. on Computers and Industral Engneerng, vol. 7, 994, pp [] D. Dubos, H. Prade, Rankng fuzzy numbers n the settng of possblty theory, Inform. Sc. 30 (983) [] M. Blue, B. Bush, J. Puckett, Unfed approach to fuzzy graph problems, Fuzzy Sets and Systems 5 (00) [3] S. Okada, Fuzzy shortest path problems ncorporatng nteractvty among paths, Fuzzy Sets and Systems 4 (3) (004) [4] A. Sengupta, T.K. Pal, On comparng nterval numbers, European J. Oper. Res. 7 (000) [5] J.A. Moreno, J.M. Moreno, J.L. Verdegay, Fuzzy locaton problems on networks, Fuzzy Sets and Systems 4 (004) [6] I. Mahdav, R. Nourfar, A. Hedarzade, N. Mahdav Amr, A dynamc programmng approach for fndng shortest chans n a fuzzy network, Appl. Soft Comput. 9 (009) [7] B. Sadeghpour Gldeh, D. Gen, La dstance-dp, q et le coffcent de corrélaton entre deux varables aléatores floues, Actes de LFA 00, Monse, Belgum, 00, pp [8] R.W. Floyd, Algorthm 97, shortest path, Commun. ACM 5 (96) 345.