Journal of Theoretics Vol.4-3

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1 Journal of heoretics Vol.4-3 Development of a Model and Algorithm for ime-dependent Shortest Path Nazar M. Zai Faculty of Computer Science & Information System, University echnology Malalysia, Malaysia nazar@siswa.utm.my Abstract: In this study, we analyzed the complexity of the ime-dependent Shortest Paths Problem (D-2SP) and its variants, and developed model and algorithm to solve this problem as well as more general versions of it in which the pair of paths is required to be only partially disjoint with respect to certain ey arcs in the networ. Keywords: time-dependent, shortest pair, disjointed paths, linear programming. 1.0 INRODUCION he general time-dependent shortest pair of disjoint paths problem (D-2SP) is when we are given a graph having m nodes and n arcs along with a designated pair of nodes O and D. Each arc (i, j) has a time-dependent travel delay d (t i ) that varies with the time of arrival t i at the tail node i of the arc during some horizon interval [0, H]. For values of t i H, we assume that the delay is static. he problem then is to find a pair of arc-disjointed paths between O and D such that the total travel delay (cost) is minimized. 2.0 A 0-1 LINEAR PROGRAMMING MODEL FOR D-2SP PROBLEM Consider a digraph G (N, A ), where N and A are the sets of nodes and arcs of G, respectively. Furthermore, suppose that we have a designated pair of origin (start) and destination (final) nodes s and f, respectively. Let the start time at the origin node s be 0, and suppose that we are interested in finding a D- 2SP from s to f, where the travel delays are defined on a discrete set of times S = {0, d, 2d,..., Md} {0, d, 2d,..., M d}, for some integer M and for some suitably large value of M. (Note that Md merely puts a practical limit on time beyond which the characterization of the delay function is not of interest.) Hence, the nodes of G are visited along any path at the discrete points in time specified by S. Observe that this delay structure could be viewed as a discrete approximation to some general specified set of arc delay functions. Without loss of generality, assume that the networ has been preprocessed so that FS(f) = Æ and RS(s) = Æ, where FS( ) and RS( ) respectively denote the forward and reverse stars of any node ( ). Additionally, suppose that we find all the nodes that are reachable from s by successively scanning FS( ), starting at s, and that we find all nodes that can reach f by successively scanning RS( ), starting at f. Let N Í N be the set of nodes in the intersection of these two resulting node sets, and let G(N, A) be the subgraph of G induced by this node set N. Clearly, we only need to focus on this subgraph G in order to solve the stated D-2SP problem from s to f. Now, define: UB = some upper bound on the length (delay) of any acceptable path in the solution to D-2SP, BP = delay function breapoint such that d(ti) is a constant for ti ³ BP, for each (i, j) Î A, and let = minimum UB, BP}. (2.1) We will now use this parameter to govern the degree with which each node in G is replicated in a time-space representation of the given networ. Note that for static problems, we would have = 0 by virtue of the second term in the minimum in (2.1), and hence, no replicates would be necessary. Otherwise, either an upper bound on the path length, or the time beyond which the delay step function is flat for all arcs will influence the extent of replications. 1

2 It is well nown that there exists a time-space static equivalent networ representation for this problem (Kaufman and Smith, 1990). In this representation, each node is replicated as i, t} "t Î S. Furthermore, for each (i, j) Î A, we construct an arc {(i, t), (j, t+ d (t))} having a fixed delay of d(t),"t Î S such that t+ d (t) Md, where note that t + d (t) then also belongs to S by our assumption. Hence, each node and arc gets replicated (up to) S times. Remove (p,t) From NOW Scan FS(p) in G. For each arc (p, q) Î A corresponding to q Î FS(p), DO: NOW = {(S,0)} NEX = Æ (SE º NOW È NEX) l(s,0) = 0 NOW = Æ N Y NEX = Æ Y Pic(p,t) NOW Put NOW = NEX N NEX = Æ (a) Compute t q = l (p,t) + d pq [ l (p,t) ]. If t q >, or if FS (q) = Æ, go to (c). Else, proceed to (b). (b) i. Create a node (q, t q ) for G if it doesn t already exist. Put λ q, t ) =t q. ( q ii. Create an arc (p, t) (q, t q ) having a delay equal to d pq [t]. iii. If (q, t q ) Ï SE and it has been newly created at this visit to (b)i., put it in NEX. Return to the top of this loop. (c) i. Create a node (q, ) for G, if it doesn t already exist, and in this case, let λ ( q, ) = temporarily. ii. Create an arc (p, t) (q, ) having a delay = d pq [t], if it doesn t already exist. iii. If t q < l(q,), then revise the label l(q,) to tq, and then, if (q, ) Ï SE and FS(q) ¹ Æ, put (q, ) in NEX. Return to the top of this loop. SOP Comment: λ ( q, t ) º t for t <, and so, λ( q, t) is required only for t =. Figure 2.1 Generation of G (N, A) and the solution of D-1SP on G. We will now describe a procedure that dynamically generates a reduced-size time-space networ using a minimal number of time-based node replications, while simultaneously finding the D-1SP from s to f. his is done within the framewor of a dynamic programming routine for finding the shortest timedependent path from s to all the nodes in G, and therefore also to f, implemented using an extension of the partitioned shortest path (PSP) algorithm for the static case due to Glover, et al. (1985). Figure 2.1 describes the proposed networ generation procedure. Here, time-expanded replicates (p, t) of each node p Î N are automatically created only for specific, necessary values of t, based on possible visitation times. Let G (N, A) denote the time-space networ that is generated by this procedure. Note that we obtain: Minimum l(p, t) = delay for the D-1SP from node s to node p, " p Î N, ( 2.2) where the minimum in (2.2) is taen over all the nodes (p, t) created in this process. Moreover, the actual shortest path that yields the delay (2.2) can be traced by maintaining the appropriate predecessor labels while revising the l-labels. Consider the Graph G depicted in Figure 2.2, where delays are constants for all the arcs except for arc (4, f), for which the corresponding delay function is as depicted in Figure 2.2(b). Our aim is to find the D-2SP from s to f, starting from s at time t = 0. 2

3 s d 4j (t) d 4j (t) f 0 3 t (a) (b) Figure 2.2 Networ G having ime-dependent Lin Delays. Let us first consider the procedure of Figure 2.1 to generate the time-space networ G. We can use = 4. he resultant graph G generated is shown in Figure 2.3. S,0 1 3, ,1 3,2 4,3 2, f, 4, ,2 2, , Figure 2.3 ime-space networ G corresponding to the graph G of Figure 2.2. Note that by the comment given in Figure 2.1, the labels λ(p, t) are equal to t nodes (p, t) having t <, while for the nodes (p, ), these labels are shown against the corresponding nodes. Also, by (2.2), the D-1SP to nodes 2, 3, 4 and f from s are respectively of lengths 1, 1, 2 and 5. In particular, observe that the D-1SP to node f is given by {(s, 0)-(3, 1)-(4, 2)-(2, 3)-(3, )-(4, )-(f, )}. his corresponds to the nonsimple path {s f} in G. Consider now the D-2SP problem. Permitting non-simple paths, the shortest pair of disjoint paths is given by {s f} and {s-3-f}, having a total length of 14. However, if we restrict the paths to be simple, the optimal pair of disjoint paths becomes {s f} and {s- 3-f}, or {s-2-3-f} and {s-3-4-f}. Both these pairs of paths have a total length of 21 > 14. We now present models to determine the time-dependent shortest pair of disjoint paths under the various restrictions discussed above. First, let us consider the case where non-simple paths are allowed. Because of the permissibility for arcs to repeat for any given path, but not between paths, we need to carry the two path flows separately in the model. hese path flows are respectively identified by variables x and y as follows. Given G as generated by Figure 2.1, let and 1, ifarc ( i, j ) ofg x = 0, otherwise i sin graph #1 (2.3a) 1, ifarc ( i, j ) ofg i sin path #2 y =. (2.3b) 0, Otherwise 3

4 Furthermore, for each arc in G, = 1, 2,..., K, say, define S = { Arcs of G that represent copies of arc in G }, = 1, 2,..., K. (3.4) he following 0-1 model D-2SP(NS) then represents the problem of finding non-simple (NS) shortest pair of disjoint paths. Here, FS ( ) and RS ( ) respectively denote the forward and reverse stars of nodes ( ) in G. Note that constraints (2.5b) and (2.5c) represent the shortest path networ flow constraints for each of the individual paths, while the side-constraints (2.5d) restrict these paths to be disjoint. Because of these side-constraints, extreme points of (2.5b)-(2.5e) are not necessarily binary valued, and hence, (2.5f) is not superfluous. D-2SP(NS): Minimize j) A d ( x + y ) (2.5a) Subject to: 1, fori = ( s,0) x - x ji = 1, fori = ( f, ) j FS j RS 0, Otherwise (2.5b) 1, fori = ( s,0) y - y = 1, fori = ( f, ) j FS j FS 0, Otherwise (2,5c) x + y pq 1 (i,j) S, (p,q) S, and = 1, 2,, (2.5d) 0 x 1 and 0 y 1 (I,j) A (2.5e) x, y binary. (2.5f) Remar 2.1: Note that there are S 2 constraints in (2.5d). Also, the upper bounds in (3.5e) are implied by = 1 (3.5d) and the non-negativity restrictions in (2.5e), but are explicitly stated here only for clarity. he LP relaxation of this problem is defined by (2.5a) - (2.5e). 2.1 ALERNAIVE YPES OF SIDE-CONSRAINS In many practical cases, the user may be interested in dealing with only arc-simple paths, even if they are non-optimal with respect to D-2SP(NS) as illustrated previously. Hence, it would be desirable to reformulate the side-constraints (2.5d) such that the solution yields optimal arc-simple paths. his might be more practical, especially when combined with different starting times in order to prescribe waitinginduced departure times at the origin node. o model this case, we only need to define one set of binary flow variables X, where 1, if j) iswithedineitherpath X = (3.6) 0, Otherwise. Observe that now, the X-variables must represent a flow of two units from s to f, one along each identified path, where the corresponding two paths jointly use any lin of G at most once. his can be modeled as follows. 4

5 D-2SP(S): Minimize j FS j ) S j ) A d X (2.7a) Subject to: 2 fori = ( s,0) X - X ji = 2 fori = ( f, ) (2.7c) j RS 0, Otherwise X 1 = 1,2,, (2.7c) 0 X 1, (i,j) A, X binary (2.7d) Let us now simplify the form of (2.7) for the special case in which the shortest path in G from s to f corresponds to a shortest simple path in G for problem D-1SP. Hence, this path in G satisfies the sideconstraints (2.7c). Let X =1 if (i, j) A belongs to this shortest path and X = 0 otherwise. Also, using the shortest path labels λi i N as obtained in Figure 2.1, let us denote the reduced costs d as: Furthermore, define d = d + λ i -λ j (i, j) A. (2.8) Now, let us use the transformation: P = {(i, j) A : X = 1}. (2.9) Y = X (i, j) A - P, Yji = 1 - X (i, j) P. (2.10) Note that arc (i, j) P is flipped in the transformed problem. hen, (2.7) gets transformed to the following problem. D-2SP(S) : Minimize d ' Y (2.11a) Subject to: j FS j) S 1, fori = ( s,0) Y Y ji = 1, fori = ( f, ) (2.11b) ( i) j RS ( i) 0, Otherwise Y j) S { p, q} K 1 P S (2,11c) Y Y p S, where p S = {( p, q)} (2,11d) qp 0 Y 1 (i,j), Y binary. (2,11e) Note that problem (2.11) effectively sees a shortest path from s to f subject to the side-constraints (2.11c) and (2.11d). Also, note that P S 1 = 1, 2,..., K. Due to its modified structure, in the presence of alternative optimal solutions to the LP relaxation, the fractionally of variables resulting at optimality for the relaxation of (2.11), as opposed to that obtained for (2.7), might differ. Hence, whenever 5

6 the shortest path in G from s to f yields a simple path in G, we can equivalently use the revised model (2.11). 3.0 COMPUAIONAL RESUL he proposed models were tested for two inds of networs: Random networs of different densities and having random delays. he networ instances that correspond to the partition problem. 3.1 RANDOM NEWORKS Random digraphs having m nodes and density r were generated using the procedure given in Siscim and Golden (1989) as follows. Let the number of arcs, n, be given by érm(m-1)ù. he values of the density r range from 0 to 1. First we generate random (static) delays for the complete graph having m(m-1) arcs and we find the shortest path tree rooted at a given origin. We then add the remaining é rm(m-1)ù-(m- 1)} arcs to this tree graph based on whether a random number generated for each of the remaining arcs in the complete graph exceeds a given threshold value. o model delays, piecewise-continuous non-negative polynomial functions of time were used, having random coefficients and degrees, and these were then put into step functions using d = 1. he delay functions generated can admit non-simple paths, since the consistency assumption was not enforced. he problems were run using the CPLEX-M pacage on a UNIX - Sun SPARC 1000 computer for each of the model (2.5), (2.7), and (2.11), where applicable. he computational time and the number of branch-and-bound tree nodes enumerated for each model are given in able 3.1. Here, m, n, and M are respectively the number of nodes and arcs in G, and the number of timeperiods for the delay function step breapoints. he next two columns give the linear programming (LP) effort (CPU seconds) required to solve the relaxation and its objective value, and the last three columns give the additional time required to solve the integer program (), the number of branch-and- bound nodes generated in the process, and the optimal objective value. he objective values reported for model (2.11) include the constants resulting from the transformation of model (2.7). able 3.1 (a) Model (2.5) Nodes (m) Arcs (n) ime Periods (M) ime for LP LP Opt ime for nodes Opt (b) Model (2.7) Nodes (m) Arcs (n) ime Periods (M) ime for LP LP Opt ime for nodes Opt

7 (C) Model (2.11) Nodes (m) Arcs (n) ime Periods (M) ime for LP LP Opt ime for nodes Opt for each path in able 3.1 highlights the tightness of the continuous relaxation to D-2SP(S) (and D- 2SP(S) ), in that for all the cases reported in able 3.1, no branching was required and the relaxed LP produced optimal 0-1 flow values. We can also see that in each test case, the optimal paths were arc-simple, since the optimal values for models (2.5) and (2.7) (and (2.11)) coincide. However, by using random delays within a highly restricted range [0, 4], one test case was found that gave both, fractional flows and non-simple optimal paths. he corresponding results for this case are shown below. able 3.2. (a) Model (2.5) Nodes Arcs ime ime for LP LP Opt ime for nodes Opt (m) (n) Periods (M) (b) Model (2.7) Nodes Arcs ime ime for LP LP Opt ime for nodes (m) (n) Periods (M) Opt (c ) Model (2.11) Nodes Arcs ime ime for LP Opt ime for nodes Opt (m) (n) Periods (M) LP While the optimal values obtained via model (2.5) and (2.7) are the same, model (2.5) actually produced a non-simple pair of disjoint paths. his implies that model (2.5) has multiple optimal solutions, since any optimal solution to model (2.7) is feasible to model (2.5). 3.2 NEWORK INSANCES OF HE POSIION PROBLEM he integers {ai, i = 1,..., n} were randomly generated. Let a i = 2S. he dummy arcs having zero delays were split into two serial arcs, each of zero delay. Hence, for n integers, the number of arc equals 3n+3 and the number of nodes equal 2n+3. he number of time periods, M, was set equal to 4S+1. able 3.2 presents the results obtained. Note that the networ for this problem does not include the possibility of non-simple paths. However, ignoring this fact in formulating model (2.5) results in a considerably weaer LP relaxation than that obtained for models (2.7) and (2.11), hence requiring a significantly greater solution effort for this model. Observe that for the models (2.7) and (2.8), in each of the test cases reported in able 3.2, the optimal objective function value of the LP relaxation was equal to the final optimal value. Hence, the enumeration required was to essentially find an alternative optimal LP solution that happened to binary. A comparison of the results in able 3.2(b) and 3.2(c) indicates the relative advantage of formulating the problem as one of finding a shortest path in the transformed networ n i= 1 7

8 represented by model (2.11) in contrast with the direct use of model (2.7). Due to its structure, model (211) has a greater propensity toward producing discrete solutions via its LP relaxation, when such solutions exist, and is therefore recommended for use over model (2.7) whenever applicable. able 3.3 (a) Model (2.5) n 4S ime for LP LP Opt ime for Nodes () Opt (b) Model (2.7) n 4S ime for LP LP Opt ime for Nodes () Opt (c) Model (2.11) n 4S ime for LP LP Opt ime for Nodes () Opt CONCLUSION AND FUURE RESEARCH In this study, we have examined the time-dependent shortest pair of disjoint paths problem (D- 2SP). his problem, including its selectively disjoint path variants, finds applications in dynamic traffic routing and assignment procedures in the context of Intelligent ransportation Systems (IS). A computational complexity analysis was performed to exhibit that this (optimization) problem as well as several of its variants is all NP-Hard, even under mild departures from the corresponding time-dependent, polynomial solvable class of problems. Subsequently, a 0-1 linear programming model was developed to solve the D-2SP problem. his model can accommodate various degrees of disjointedness of the paths, and some simplified representations can be obtained when the optimal pair of paths is required to be arcsimple. Computational tests using randomly generated networs and networ instances of the partition problem reveal the efficacy of the proposed models. In particular, the computational requirements for the specialized models formulated to determine a time-dependent shortest pair of arc-simple disjoint paths are considerably less than that required by the general D-2SP model in this case. Hence, these specialized models are recommended for use whenever applicable. Future research possibilities include the study of various applications of D-2SP, and its minimal variant, in transportation and communication networ flow problems. he relative advantages of using D- 2SP for the minimum-ris routing of hazmat carriers and the routing of multiple emergency-response vehicles in comparison with currently used routing methods can be investigated. he efficacy of prescribing selectively disjoint paths as a routing tool in a dynamic traffic assignment procedure is another interesting subject of future study. 8

9 GENERAL REFERENCES: 1. Bazaraa, M. S., Jarvis, J. J., and Sherai, H. D. Linear Programming and Networ Flows, 2 nd Edition. John Wiley and Sons, New Yor, NY, Bellman, R. E. on a routing problem found in Quarterly of Applied Mathematics, Vol. 16 (1958), Brogan, J. and Cashwell, J. Routing Models for the ransportation of Hazardous Materials State Level Enhancements and Modifcations, ransportation Research Record, Vol (1985), pp Cai, X., Kios,., and Wong, C. K. ime-varying Shortest Path Problems with Constraints. Networs, Vol. 29, No. 3 (1997), pp Calamai, P. H., and A. R. Conn. A Projected Newton Method for l p Norm Location Problems. Mathematical Programming, Vol. 38, (1987), pp Coo, K. L. and Halsey, E. he Shortest Route hrough a Networ With ime-dependent Internodal ransit imes, Journal of Mathematical Analysis and Application. 14 (1966) Dreyfus, S. E An Appraisal of Some Shortest Path Algorithms, Operations Research. 17 (1969) pp Halpern, J. L. Shortest Route With ime-dependent Length of Edges and Limited Delay Possibilities in Nodes,. Zeitschrift fur Operations Research. Vol. 21 (1977) Journal Home Page Journal of heoretics, Inc