A Polyomial Iterval Shortest-Route Algorithm for Acyclic Network Hossai M Akter Key words: Iterval; iterval shortest-route problem; iterval algorithm; ucertaity Abstract A method ad algorithm is preseted for solvig the shortest-route problem New algorithm is applicable to the case whe the geeralized legth (distace, cost, time, etc) associated with each arc is oegative, iterval or real A iterval algorithm is developed o the base of midpoit ad half-width represetatio of itervals ad the ew algorithm is more efficiet tha the iterval algorithm that could be proposed by usig traditioal iterval descriptio The complexity of the ew algorithm is evaluated Itroductio There are may reasos why etwork models, methods ad algorithms are widely used, for istace, they exactly represet the real world systems, they facilitate extremely efficiet solutio to large real problems, they ca solve problems with sigificatly more variables ad costraits tha ca be solved by other optimizatio techiques, etc [] Network cosists of special poits called odes ad liks coectig pairs of odes called arc (or brach or edge or lik) A etwork is called acyclic etwork, if it does ot have ay loop The acyclic algorithm is easier tha the cyclic algorithm, because it yields fewer computatios [0,5,6] Cosider a coected etwork G = (N, A), where N = {,, } is the set of the odes ad A = {(i, j), (k, l),, (y, z)} is the fiite set of arcs joiig odes i N The cardiality of N ad A are deoted by N ad A respectively, ad N =, A = m Let d(i) is the distace label ad i N; d(i, j) is the geeralized legth (distace, time, cost, etc) of arc (i, j) A; P is the directed route from source ode s to destiatio ode ad s, N; the legth d(p ) of the route P is give by d( P ) = d( i, j) such that (i, j) belogs to P, where by covetio d(p s ) = 0; the predecessor of ode j that is deoted by p(j), is started from ode i of the sigle arc (i, j) A i the tree termiatig at j [7,8,5] The Shortest-Route Problem (SRP) is cocered with determiig the shortest route from a origi to a destiatio through a coectig etwork, give oegative distaces associated with the respective arcs of the etwork [3,9,,3,5] The SRP is a classical etwork problem, ad it is the most popular problem/model amog all etwork problems [3,9,3] I literature, Dkstra algorithm [] is cosidered a classical algorithm for SRP Last five decades may variats of Dkstra algorithm have bee developed, for example, a alterative method for SRPs is proposed i [], which reduces the upper boud of ruig time, ad makes empirical comparisos for a certai class of etworks Reachig, Pruig, ad Buckets are the three cocepts that are used i these methods Reachig is a label settig scheme, reachig allows a etwork to be prued durig computatio of some of its odes ad/or braches, ad bucket is a list of odes whose labels fall withi a give rage I [], the author assumed odes, ad the existece of at least oe route betwee ay two odes Two fudametal problems were cosidered: to obtai the tree of miimum total legth betwee the odes, ad to fid the route of miimum total legth betwee two give odes The iterval SRP is cocered with determiig the iterval shortest route from a origi to a destiatio through a coectig etwork, give the iterval geeralized legth betwee odes i ad j is a oegative, iterval ad iterval umbers are represeted by D, D = [ d, d ] [-6,,] The aim of this paper is to develop simple ad effective method ad algorithm for solvig the SRP for acyclic etwork uder parametric ucertaities The aalysis of the complexity of the iterval algorithm will be discussed Related Work A iterval algorithm is proposed for solvig SRP uder parametric ucertaity i [] The exact values of the parameters of a give etwork are ukow, but upper ad lower limits withi which the values are expected to fall are cosidered The iterval algorithm is developed o the base of midpoit ad half-width represetatio of itervals Cosiderable uificatio ad simplificatio are obtaied by usig the mea-value lemma This iterval algorithm is applicable whe the parameters of a give etwork are iterval ad real The iterval algorithm is applicable whe the give etwork is acyclic Updated versio of this algorithm is preseted i [6,] A iterval algorithm for cyclic etwork is preseted i [5,] Fial versios of iterval methods ad algorithms are give for solvig the well-kow SRP for acyclic ad cyclic etworks i [] The formulatio of the iterval shortest-route algorithm for cyclic etwork is a iterval extesio of Dktra algorithm The author cosidered the 0
iterval geeralized legth betwee odes i ad j is a oegative, iterval umber The ew methods ad algorithms are developed o the base of midpoit ad halfwidth represetatio of itervals These iterval algorithms are applicable whe the parameters are iterval ad real The complexity of these algorithms is evaluated Both method ad algorithm are more efficiet tha the method ad algorithm that could be obtaied by usig traditioal iterval descriptio ad compariso, ad the complexity of such a algorithm will be too high from the poit of view of computatio ad practical applicatios A method to fid the most reliable route i a give etwork is give i [5] The probability of a arc is certai The author coverts probability to log probability The the shortest-route algorithm is used to fid the shortest distace (log) Fially, this log probability is coverted back to o-log probability There are some limitatios of this method If the probabilities of a give etwork are with higher degree of ucertaity, this method ca ot be used to solve the problem The author has ot cosidered the complexity aalysis of this method To covert probability to log probability, the log probability to probability, he eeds more operatios So, the complexity of the method will be higher Five algorithms are proposed for solvig the most reliable route problem i fiite fuzzy acyclic ad cyclic etworks i [,3] The ucertaity about the reliability of a route is represeted i a possibilistic settig The plausibility of ot beig stopped o a segmet of the route is described usig the correspodig possibility The cocept of iterval possibility is itroduced to icrease the degree of ucertaity These algorithms maximize the possibility of ot beig stopped o the route betwee a origi ode ad a destiatio ode The complexity of these algorithms is evaluated Brief descriptio of the algorithms is give below: The first ad secod algorithms are based o the usage of ad ad product operators to determie the strogest route, that is, the most reliable route i a fiite fuzzy acyclic etwork The first algorithm takes less time for computatios tha the secod algorithm So, the first algorithm is better suited for large etwork The third algorithm uses multiplicatio of iterval possibilities ad yields directly the largest iterval possibility of ot beig stopped o the route 3 The fourth ad fifth algorithms are based o the cocept of iterval possibility for acyclic etwork ad cyclic etwork, respectively Oly oce at the begiig, the trasformatio of the iitial represetatio of itervals possibilities ito logarithmic form is accomplished, ad the the simple midpoit algorithm for solvig iterval acyclic algorithm ad iterval cyclic algorithm is applied, respectively A variat of SRP has cosidered i [7,8,9] Cosider a directed etwork G = (N, A), where N is the set of odes ad A is the set of arcs, ad s, N The costs (travel times) of each arc is give by a iterval Itervals represet rages of possible costs A iterval [d, d ] is associated with each arc (i, j) A, ad 0 d d A route H from source to destiatio is said to be a Robust Shortest Route (RSR) if it has the smallest (amog all routes from source to destiatio) maximum (amog all possible scearios) robust deviatio I [9], the authors proposed a brach ad boud algorithm for the RSR problem with iterval data The ew algorithm is based o a lower boud ad o some reductio rules which work by exploitig some properties of the particular brachig strategy The algorithm starts by iitializig the structures of r, the root of the searchtree, which is the iserted ito the set of odes to be examied A iterative statemet is the repeated util the search-tree has bee completely examied The authors tested their methods o differet etworks: radom etworks, real etworks, etc I [7], the authors preseted a exact algorithm for RSR problem with iterval data The algorithm is based o the cojecture that a RSR is oe of the first routes i a shortest route rakig i a simple directed etwork, where the cost o each arc (i, j) is equal to d They adopted the algorithm which is based o the cocept of route deletio, ad also implemeted the Dkstra algorithm to evaluatio of the robustess cost of a give route The algorithm works i the followig way: a procedure raks routes i the simple directed etwork For each route retrieved, the respective robustess cost is calculated The algorithm stops whe a lower value for the robustess costs of the routes ot yet examied matches a upper boud for the same routes The limitatios of the proposed algorithm are as follows: a) if the robust route from s to t is log, all the routes from s to t will ted to be log, ad the shortest route algorithm will be slower, b) if the robust route from s to t is log, more alterative routes will exist betwee s ad t, ad the algorithm eed more iteratios to coverge, c) the ew method obtais poor results o problems based o Karasa, etworks The mai advatage is that the algorithm gives the optimal solutio of some etwork problems I [8], two versios of ovel exact algorithms are give for the RSR problem with iterval data, ad these algorithms are based o Beders decompositio The Beders decompositio approach is the best oe for etworks with low arc desity, ad the brach ad boud method give i [9], is the most promisig while the arc desity icreases The authors made a experimet o real road etworks that showed that the Beders decompositio approach is the most appropriate for this type of etworks Moreover, the choice of the most appropriate approach is strictly coected with the characteristics of the problem to be solved I [], the authors examied a specific SRP i acyclic etwork, i which arc costs are ukow fuctios of certai eviromet variables at etwork odes, ad each of these variables evolve accordig to a idepedet 0 3
Markov process The vehicle ca wait at a ode (at a cost) i aticipatio of more favorable arc costs First, the authors developed two recursive procedures for the idividual arc case, based o successive approximatios, ad policy iteratio Several procedures have bee used to determie which of the eviromet states at each ode are gree (the vehicle departs immediately) ad which are red (the vehicle waits), based o successive approximatios, policy iteratio, ad parametric liear programmig methods The 3 complexity of this method is O ( K + K ), where is the umber of odes ad K is the umber of Markov states at each ode Sometimes sigle objective fuctio may ot be sufficiet to characterize may practical problems completely I a real trasportatio etwork several objectives, ie, time, cost, distace, etc ca be assiged to each arc If oly oe objective is give o each arc, the solutio of the problem ca be obtaied by classical shortest-route algorithm, give i [] Whe more tha oe objective is give o each arc, the solutio of the problem ca ot be obtaied by classical shortest-route algorithm The shortest route may be ot wise to use because it could be expesive To deal with a real problem with more tha oe objective, ew variats of classical shortest-route algorithm have bee developed, which are called the bicriterio or multi-criteria shortest-route algorithms [,-6,7] I [7], the authors proposed a method to solve the fuzzy SRP The weighted additive method is itroduced to solve a multiple objective iteger programmig problem, which met the requiremets of the Network LPs costraits Weights i the weighted additive model show the relative importace of the goals For simplicity, the authors assumed that the importace of the four objectives is the same Therefore, all objective fuctios were reformulated as a sigle objective fuctio, ad oe eed ot to add the costraits of iteger programmig The fuzzy shortest route was obtaied whe the model met the requiremets of the Network LPs costraits This ew approach reduced the complexity of solvig the basic fuzzy shortest route formulatio The author assumed that the importace of all objective fuctios is same Mixed Iteger Liear Programmig (ILP) approach is proposed i [,,5] to solve the bicriterio etwork problem The method is based o the approach, proposed i [8], for solvig multicriterio cotiuous problems, which itroduces fuzzy sets of the values ear to the optimal values for each criterio Cosider a etwork G = (N, A), where N = {,, } is the set of the odes ad A = {(i, j), (k, l),, (y, z)} is fiite set of directed arcs joiig odes i N Assume we have A = m arcs Each arc (i, j) A has two attributes, for example, d = (d', d'' ) d' is the distace betwee ode i ad ode j, d'' is the travel time betwee ode i ad ode j I [5], Mixed ILP approach is proposed to solve the multicriterio etwork problem The method also is based o the approach, proposed i [8], for solvig multicriterio cotiuous problems, which itroduces fuzzy sets of the values ear to the optimal values for each criterio Cosider a etwork G = (N, A), where N = {,, } is the set of the odes ad A = {(i, j), (k, l),, (y, z)} is fiite set of directed arcs joiig odes i N Assume we have A = m arcs Each arc (i, j) A has three attributes, for example, d = (d ', d'',d''' ) d' is the distace betwee ode i ad ode j, d '' is the travel time betwee ode i ad ode j, ad d ''' is the travel cost betwee ode i ad ode j 3 Theoretical Prelimiaries A iterval umber is a pair of real umbers ( r, r), with r r The iterval aalysis cocepts are itroduced i [0,] Let R be a iterval We will deote its lower (left) edpoit by r ad its upper (right) edpoit by r, so that R = [ r, r] The set of all itervals will be deoted by I (R) Let R, S I(R), ad let deote ay of the iterval arithmetic operatios, = +,,, / The the set theory defiitio of the iterval arithmetic operatios is as follows: () R S = { r s r R, It follows that the sum of R = [r, r], S = [s, s] deoted by R + S, is the iterval R + S = [r, r] + [s, s] = [r + s, r + s] The product R S is agai a iterval R S = [mi{rs, rs, rs, rs}, max{rs, rs, rs, rs}] For R, S > 0 the defiitio reduces to () R S = [rs, rs] The half-width of a iter R = [r, r] is the real umber, w (R) = (r r), ad the midpoit of R is the real umber, m(r) = ( r + r)/ Usig the set iclusio relatio ad the relatio, we ca defie the supremum-like (sup) ad ifimum-like (if) elemets: (3) sup(r, S) = [sup(r, s), sup(r, s)] () if(r, S) = [if(r, s), if(r, s)] To compare itervals the cocept of metric ρ is itroduced For each R ad S i I(R) the distace ρ is defied by (5) ρ(r, S) = {r s + r s} Now the itervals R ad S ca be compared The followig importat results hold i [] R S iff (if ad oly if) 0
(6) ρ (R, if(r, S)) ρ (S, if(r, S)) I a similar way, R S iff (7) ρ (R, sup(r, S)) ρ (S, sup(r, S)) Two itervals R ad S are said to be equivalet R ~ S if the followig coditio holds: (8) ρ (R, sup(r, S)) = ρ (S, sup(r, S)) (9) ρ (R, if(r, S)) = ρ (S, if(r, S)) It meas that r - s = s - r, ie, the midpoits of R ad S coicide I practical cases whe R ~ S ad oe have to make a choice i the sese of, the coditio (6) should be modified We say that R S if (0) ρ (R, if (R, S)) = ρ (S, if (R, S)) ad r s or () ρ (R, if (R, S)) = ρ (S, if (R, S)) ad r s We use, further, the otatio R S i the usual sese, whe r s ad r s, ad i the case of iclusio, R S, whe ρ(r,if(r, S)) ρ (S,if(R, S)) The coditios (6) ad (7) lead to the followig result, as prove i [] Let m(p) deote the midpoit of P, m(p) = ( p + p )/ The () R S iff m(r) m (S) Let [m(r), Δ(R)] deote the iterval R, R = [ r, r ], where m(r) = ( r + r)/ is the midpoit of R, ad Δ(R) = ( r - r)/ is the half-width of R, so that (3a) R = [m(r) - Δ(R), m(r) + Δ(R)] (3b) R = [m(r), Δ(R)] The followig result is easily show: Let R, S, ad T I(R) The T = R + S iff () m(t) = m(r) + m(s) (5) Δ(T) = Δ(R) + Δ(S) Method ad Algorithm I may practical cases, the parameters of the etwork models are ot exactly kow, they are ucertai A typical way to express these ucertaities i the edge weights is to utilize tools based o probability theory, iterval mathematics, fuzzy sets theory, etc The aim is to fid the shortest route betwee a source ode ad ay destiatio ode t i a etwork with odes, t A iterval extesio of the well-kow acyclic algorithm ca be obtaied, usig the iterval operatio +, the metric ρ as defied i (5), ad the coditios (6) or (0), () Let D ad U j deote the iterval distace betwee odes i ad j, ad the shortest iterval distace from the source ode (ode ) to ode j, correspodigly The destiatio ode is ode The iterval values of U j = [ u, u ], j =, may be j j computed recursively usig the iterval formula (6) U j = mi {U i + D } i where U i + D = [ u i + d, u i + d ], ad U = [0, 0] The operator mi{} is performed o the basis of the metric (5) ad the coditios (6) or (0), () This way a iterval extesio of the well-kow shortest-route algorithm for acyclic etwork is obtaied We preset a more effective algorithm, usig the midpoit ad half-width otatio, (3b) ad the coditios (), () ad (5) Let u j deote the real shortest distace from to ode j The real values u j, j =, are computed usig the recursive oiterval formula (7) u j = mi {u i + d } i where d is the midpoit of D, u = 0 To obtai the optimal solutio of the SRP, it is importat to idetify the odes ecoutered alog the route ad the correspodig iterval widths The followig labelig of ode j is used (8) ode j Label = [u j, k, Δ ] kj where k is the ode immediately precedig j that leads to the shortest distace u j, ad Δ kj is the half-width of D kj Further it is assumed that the etwork is described usig iterval otatio with midpoit ad half-width (3b) It is also assumed a atural cosecutive umberig of odes from to, such that the umber of ay ode i, i N is greater tha the umber of ay immediately precedig ode k, k N, ad where N is the set of odes, ad N i is the set of all precedig ode The geeralized steps of the iterval acyclic algorithm are summarized as follows: Step Assig the label [0,, 0] to the source ode Set j = Step Set j = j + Compute the shortest distace from source ode to ode j, by usig recursive formula (7) Label ode j by usig (8) If j < t repeat step Step 3 Obtai the optimum route H * betwee odes ad ode, startig from ode ad tracig backward through the odes usig the label s iformatio Step Obtai the half-width Δ(U ) of the iterval solutio U, addig the correspodig Δ ecoutered alog the optimum route H * Δ(U ) = Σ Δ (i, j) H* 0 5
Step 5 Obtai the iterval solutio U, U = [u Δ(U ), u + Δ(U )] The algorithm provides the shortest route betwee ode ad ay ode j, j i the etwork Aalysis of the complexity of the iterval shortest route algorithm for acyclic etwork Cosider the etwork i figure The cardiality N j of the set of eterig arcs N j ito ode j is (j ), N j = j To calculate (shortest distace from to ode j, j =, ), we eed the followig additio(s) ad compariso(s): u = u + d We eed oly additio to determie u u 3 = mi {(u + d 3 ), (u + d 3 )} We eed oly additios ad compariso to determie u 3 u = mi {( d u + ), ( u + d ), ( u 3 + d3 )} We eed oly 3 additios ad comparisos to determie u u 5 = mi {( u + d5 ), ( u + d 5 ), ( u 3 + d35 ), ( u + d 5 )} We eed oly additios ad 3 comparisos to determie u 5 u (-) = mi {(u + d (-) ), (u + d (-) ), (u 3 + d 3(-) ),, (u - +d (-)(-) )} We eed oly ( ) additios ad ( 3) comparisos to determie u (-) u = mi {( ( + ( )( ) u d + ), ( u + d ), ( u + 3 d3 ),, u d ), ( u ( ) + d( ) )} We eed oly ( - ) additios ad ( - ) comparisos to determie Hece, to obtai u j we eed oly (j ) additios ad (j ) comparisos The umber of additios is ( j ) j= The umber of comparisos is ( j ) j= We set χ = j ad δ = j, ad we obtai: χ = Σδ= χ = ( ) ( ) ( ) δ = The total umber of additios is ϑ, ϑ = ( ) + ( ) (to get total half-width) + additios to obtai the traditioal iterval represetatio The total umber of comparisos is ε, ( ) ( ) ε = So, the ruig time of the algorithm is bouded by O(addi = ϑ, comp = ε) Note that if the iterval formula (6) were used, the each compariso of two itervals V = [v, v] ad W = [w, w] icludes the followig comparisos ad additios: Step if v w set Lab = V, go to the ext iterval compariso comparisos; Step if w v set Lab = W, go to the ext iterval compariso comparisos; Step 3 if V = W (v = w ad v = w) set Lab = V, W, go to the ext iterval compariso comparisos; Step if v w ad v w set Lab = V, go to the ext iterval compariso comparisos; Step 5 if w v ad w v set Lab = W, go to the ext iterval compariso comparisos; Step 6 if v w set ρ = ρ (V, if) = v w; set ρ = ρ (W, if) = w v comparisos, additios; if ρ < ρ set Lab = V else set Lab = W go to the ext iterval compariso compariso; Step 7 if w v set et ρ = ρ (V, if) = v w; set ρ = ρ (W, if) = w v comparisos, additios; if ρ ρ set Lab = V else set Lab = W compariso To compare two itervals to comparisos ad 0 to additios are eeded So, if we develop iterval shortestroute algorithm based o traditioal iterval represetatio, the complexity of the algorithm will be very high 5 3 - Figure Acyclic etwork ode Commet The aalysis of the complexity is very importat for two reasos: practical reasos ad theoretical reasos The first reaso ca be summarized as a eed to obtai the executio-time that are eeded i the implemetatio of algorithm The secod reaso for the complexity aalysis of the algorithm is the desirability of quatitative stadards that would allow the compariso of more tha oe algorithm desiged to solve the same problem The aalysis of the complexity of the iterval shortest route algorithm is to provide upper bouds o the amout of computatioal work (comparisos (comp) ad additios 6 0
Table The computatios of all iteratios for figure Node j Computatio of u j Coected from Label u = 0 - [0, -, 0] u = 0 + 8 = 8 ode [8,, ] 3 u 3 = 0 + 9 = 9 ode [9,, ] u = mi {0 + 0, 8 +, 9 + 5} = 0 ode [0,, ] 5 u 5 = mi {8 + 5, 0 + 7} = 7 ode [7,, ] 6 u 6 = mi {9 + 5, 0 + } = ode [,, ] 7 u 7 = mi {7 + 9, + 8} = 6 ode 5 [6, 5, ] (addi)) ivolved i the applicatio of iterval shortest route algorithm for acyclic etwork It is assumed that a compariso ad a additio require approximately the same uit of time The worst-case coditios for the executio of a algorithm meas that the required umber of elemetary operatios to termiate the algorithm is maximum [] Numerical Example Cosider the etwork i figure The geeralized legth of the arcs are ucertai ad give by itervals, i the form (3b) Usig the algorithm, as described i sectio, we obtai the results for odes,,, 7, that are put o table The computatios for all iteratios are summarized directly o figure The optimal solutio is obtaied tracig backward from ode 7 ad usig the label s iformatio 7 [6,5,] 5 [7,, ] [0,,] The half-width of the optimal solutio is as follows: Δ 7 = Δ57 + Δ 5 + Δ = + + = 3 Hece, U 7 = [ 6 3, 6 + 3] = [3, 9] The algorithm provides the shortest iterval distace betwee ode ad ay other ode I figure, the solid lies show the obtaied the iterval shortest route (the desired route) betwee the source ad the destiatio ode amely 5 7 [0, -, 0] [8, ] [9, ] [8,, ] [7,, ] [5, ] 5 [0, ] 3 [9,, ] [, ] [5, ] [5, ] [0,, ] [7, ] [, ] 6 [,, ] [9, ] [8, ] 7 [6, 5, ] Figure Acyclic etwork with midpoit ad half-width otatio Note that if the iterval formula (6) were used, at ode 7, for example, we would have to compare two itervals: [5, 9] + [8, 0] = [3, 9] ad [9, 5] + [7, 9] = [6, 3] 5 Coclusios I this paper, we proposed a method ad a algorithm for solvig shortest-route problem for acyclic etwork based o midpoit ad half-width represetatio of itervals (3b), ad the coditios (), (), ad (5) The ew iterval algorithm is applicable whe the parameters are real or iterval valued This approach yields simple ad computatioally effective algorithm, whe the exact values of the parameters are ukow, but upper ad lower limits withi which the values are expected to fall are give Istead of comparig itervals usig the distace (5) ad ifimum-like itervals (), the coditios (0) ad (), the algorithm compares real values, ie, the midpoits of the itervals The complexity of the iterval acyclic algorithm is evaluated, it is a polyomial algorithm Ackowledgemets The author expresses his gratitude to Prof Dr Geo I Gatev for beig his PhD supervisor ad cotiuous guidace over the years Refereces Deardo, E V, B L Fox, Shortest-Route Methods: Reachig, Pruig ad buckets, Operatios Research, 7, 979, 6-86 Dkstra, E W, A Note o Two Problems i Coexio with Graphs, Joural of Numerische Mathematik,, 959, 69-7 3 Dreyfus, S E, A appraisal of some shortest-path algorithms, Joural of Operatios Research, 7, 969, 395- Gatev, G, Iterval aalysis approach ad algorithms for solvig etwork ad dyamic programmig problems uder parametric ucertaity, Proceedigs of the Techical Uiversity of Sofia, 8, 995, 35-350 0 7
5 Gatev, G, A Hossai, Iterval Shortest Route Algorithms, Proceedigs of the Iteratioal Coferece, Automatio & Iformatio, Sofia, Bulgaria, 000, 68-7 6 Gatev, G, A Hossai, Iterval Algorithms for Solvig Miimal Spaig Tree ad Shortest-Route Models, Joural of Iformatio Techologies ad Cotrol,, 007, 37-6 7 Glover, F, D Kligma, New Sharpess Properties, Algorithms ad Complexity boud for Partitioig Shortest Path Procedures, Joural of Operatios Research, 37, 989, 5-56 8 Glover, F, D Kligma, N V Phillips, R F Scheider, New Polyomial Shortest Path Algorithms ad their Computatioal Attributes, Joural of Maagemet Sciece, 3, 985 06-8 9 Golde, B, Shortest-Path Algorithm: A Compariso, Operatios Research,, 976, 6-68 0 Hillier, S F, G J Lieberma, Itroductio to Operatios Research, Fifth editio McGraw-Hill Publishig Compay, New York, 990 Hossai, A, Iterval Algorithms for solvig Network ad Dyamic programmig uder parametric ucertaity, Master s Thesis, Techical Uiversity of Sofia, 999 Hossai, A, Network Models uder Parametric Ucertaity, PhD Thesis, Techical Uiversity of Sofia, 009 3 Hossai, A, G Gatev, Polyomial Algorithms for solvig the Most Reliable Route Problem, Joural of Iformatio Techologies ad Cotrol, Ed K Boyaov,, 009, 7-6 Hossai, A, Method ad Algorithm for solvig the Bicriterio Network Problem, Proceedig of the 00 Iteratioal Coferece o Idustrial Egieerig ad Operatios Maagemet, Dhaka, Bagladesh, 00 5 Hossai, A, Methods ad Algorithms for solvig the Bicriterio ad Multicriterio Network Problem, Joural of Iteratioal Joural of Logistics ad Trasportatio Research,, 00, 39-7 6 Martis, E Q V, O a multicriteria shortest path problem, Europea Joural of Operatioal Research, 6, 98, 36-5 7 Motemai, R, L M Gambardella, A exact algorithm for the robust shortest path problem with iterval data, Computers & Operatios Research, 3, 00, 667-680 8 Motemai, R, L M Gambardella, The robust shortest path problem with iterval data via Beders decompositio, A Quarterly Joural of Operatios Research, 3, 005, 35-38 9 Motemai, R, L M Gambardella, A V Doati, A brach ad boud algorithm for the robust shortest path problem with iterval data, Operatio Research Letters, 3, 00, 5-3 0 Moore, R E, Iterval Aalysis, Pretice Hall, Eglewood Cliffs, NJ, 966 Moore, R E, Methods ad Applicatios of Iterval Aalysis, SIAM, Philadelphia, 979 Phillips, Do T, A Garcia-Diaz, Fudametals of Network Aalysis, Pretice-Hall Ic, Eglewood Cliffs, NJ, 98 3 Pollack, M, W Wiebeso, Solutio of the Shortest-Route Problem - A Review, Joural of Operatios Research, 8, 960, - 30 Psaraftis, H N, J N Tsitsiklis, Dyamic Shortest Paths i Acyclic Networks with Markovia Arc Costs, Operatios Research,, 993 9-0 5 Taha, H A, Operatios Research, A Itroductio, Eighth Editio, Pearso Educatio Ic, Delhi, 007 6 Wager, D K, Shortest Paths is almost acyclic graphs, Operatios Research Letters, 7, 000, 3-7 7 Yu, J-R, T-H Wei, Solvig the Fuzzy Shortest Path Problem by usig a Liear Multiple Objective Programmig, Joural of the Chiese Istitute of Idustrial Egieers,, 007, 360-365 H-J Zimmerma, Fuzzy Set Theory - ad its Applicatios, Secod Revised Editio, Kluwer Academic Publishers, Bosto, 99 Mauscript received o 0 Akter Hossai was bor i Dhaka, Bagladesh He received his academic degrees i BSc i Idustrial Egieerig (997), MSc i Commuicatio Egieerig (999), MPhill (00) ad PhD (009) At the preset, he has bee workig as Associate Professor & Head, Departmet of Computer Sciece & Egieerig, IBAIS Uiversity, Dhaka, Bagladesh His research ad teachig iterests are i the field of E-Commerce, Decisio-Makig, Maagemet, Productio Operatio Maagemet, Operatios Research, etc Dr Hossai has 3 Iteratioal Publicatios published i Frace, Bulgaria, Idia ad Bagladesh Cotacts: e-mail: akter_h@yahoocom 8 0