Constraint Reformulation and a Lagrangian Relaxation based Solution Algorithm for a Least Expected Time Path Problem Abstract 1.

Transcription

1 Constraint Reformulation an a Lagrangian Relaation base Solution Algorithm for a Least Epecte Time Path Problem Liing Yang State Key Laboratory of Rail Traffic Control an Safety, Being Jiaotong University, Being, , China, lyang@bjtu.eu.cn Xuesong hou epartment of Civil an Environmental Engineering, University of Utah, Salt Lae City, UT , USA, zhou@eng.utah.eu Abstract Using a sample-base representation scheme to capture spatial an temporal travel time correlations, this article constructs an integer programming moel for fining the a priori least epecte time paths. We eplicitly consier the unique path constraint associate with the a priori path in a time-epenent an stochastic networ, an propose a number of reformulations to establish linear inequalities that can be easily ualize by a Lagrangian relaation solution approach. The relae moel is further ecompose into two sub-problems, which can be solve irectly by using a moifie label-correcting algorithm an a simple single-value linear programming metho. The numerical eperiments investigate the quality an computational efficiency of the propose solution approach. Key wors: a priori least epecte time path; time-epenent traffic networ; Lagrangian relaation History: 1. Introuction Fining the shortest path in a stochastic networ is among the core networ optimization problems in the fiels of both operations research an transportation engineering. To mitigate recurring an non-recurring congestion, efficient routing algorithms are critically neee to minimize the epecte path travel time in networ-wie traffic flow management an personalize route guiance applications. Three categories of stochastic programming formulations have been propose to represent the inherent travel time uncertainties in traffic networs: (1) iscrete probability istribution functions (e.g., Miller-Hoos 1997, 2001; Nie an Wu 2009a), (2) moment-base characterization for continuous lin travel times through variance an other statistics (e.g., Fu an Rilett 1998; Sun, Gu an Mahmassani 2011) an (3) a sample-base representation, which can be viewe as a special version of the Monte Carlo approimation metho. The thir representation metho has been use by Liu (2002) to approimate service time winow constraints for a vehicle routing problem, by Chen an Ji (2005) to simulate uncertain objectives for a stochastic path fining problem an, recently, by Xing an hou (2011) to capture spatial travel time correlations for the most reliable path problem. In a transportation networ with stochastic, time-varying lin travel times, the optimal routing strategies can be evaluate accoring to various objective functions, incluing (1) fining the least epecte travel time (Hall 1986; Miller-Hoos 2001; Fu 2001; Nielsen 2003; Yang an Miller-Hoos 2004; Gao an Chabini 2006; Hicman an Bernstein 1997; Sivaumar an Batta 1994), (2) maimizing the probability of arriving on time (Nie an Wu 2009a; Samaranayae et al. 2011) an (3) minimizing the maimum possible travel time (Nielsen 2003). In aition, the eisting routing algorithms are usually use to provie either an a priori optimal path (as a single solution) or aaptive en-route path trees (as a set of solutions). In the a priori path problem, base on (preicte) stochastic travel times, travelers are require to select a single route before eparting from the origin noe; this selecte path cannot be upate, even though only one instance of stochastic travel time will be realize. In aaptive en-route guiance applications, which can be viewe as a multi-stage recourse problem (Miller-Hoos 2001), travelers nee to select their route at each noe after new or upate travel time information is available. In other wors, the selection of the net physical lin will be eecute after the lin traversal times are reveale en route once the lin is traverse. To hanle travel time stochasticity an ynamics in route selection, accoring to Miller-Hoos (2001), many stuies use two major classes of solution strategies to choose the esirable paths, namely, aaptive selection an a priori optimization. The early wor by Hall (1986) presents the terminology an solution technique for aaptively upating path solutions when seeing the least epecte travel time path. Recently, Miller-Hoos (2001), Yang an Miller-Hoos (2004) an Gao an Chabini (2006) establishe more comprehensive framewors for fining optimal routing policies or a set of strategies (i.e., hyperpaths) in

2 stochastic time-epenent networs. A stuy by Nielsen (2003) consiers the problem of fining the K best strategies an a bicriterion route with time-aaptive strategies for route choice. In aition, focusing on a route travel time reliability measure, Samaranayae et al. (2011) esigne an aaptive routing policy to maimize the probability of arriving on time at a estination. Aiming to eten the single-value ominance rule in the stanar shortest path algorithm with constant lin istance, many researchers focus on how to construct effective ominance rules to solve a priori optimization for the shortest path problem uner ynamic an uncertain traffic conitions. For instance, an early stuy by Wellman, For an Larson (1995) enhance the path planning algorithm base on the stochastic consistency an stochastic ominance principle. Miller-Hoos an Mahmassani (2000) efine an optimal a priori solution as the path that can realize the least epecte travel time in the networ. Moreover, in their later stuies, these researchers further investigate a systematic approach to compare sub-path travel times (at intermeiate noes) an the optimality of solutions by constructing various ominance rules, incluing a eterministic ominance rule, a first-orer stochastic ominance rule an an epecte-value ominance rule (Miller, Mahmassani an iliasopoulos 1994; Miller-Hoos 1997; Miller-Hoos an Mahmassani 2003). Base on a probability istribution function (PF)-base representation, Miller-Hoos an Mahmassani (2000) presente a novel lower boun estimator by consiering the least reveale travel times on travele lins en route through time-aaptive routing selection rules. Focusing on an epecte-value ominance rule, with the travel time on each arc being treate as a continuous time stochastic process, Fu an Rilett (1998) also evelope probability-base methos to approimate the mean an variance of the travel time for a given path in a ynamic an stochastic networ. Using the first-orer stochastic ominance rule, Nie an Wu (2009a, b) an Wu an Nie (2009) propose various methos of fining reliable a priori shortest paths to guarantee a given lielihoo of arriving on time at the estination. Pretolani (2000) presente several criteria, such as the least epecte travel time an minimization of the maimum possible travel time, to ran hyperpaths in a time-epane irecte hypergraph. Aitionally, a variety of ominance rules have been embee in a label-correcting algorithmic framewor by Miller, Mahmassani an iliasopoulos (1994), Miller-Hoos (1997), Nie an Wu (2009a), an Wu an Nie (2009). A route generation-oriente solution scheme, which is typically built on the -shortest path algorithm, has been aapte by Fu an Rilett (1998), Wu et al. (2005) an Nielsen (2003). In this research, we focus on the least epecte travel time (LET) criterion, which can be easily estimate an valiate within a traitional utility-maimization framewor. We are particularly intereste in how to fin the optimal a priori path using a sample-base representation for time-epenent an stochastic travel times. This paper aims to offer the following contributions to the growing boy of wor on optimum path algorithms in ynamic networ analysis. (1) A rigorous mathematical programming moel is formulate for fining the optimal route with the least epecte travel time problem, in which scenario-base time-epenent lin travel times are use to capture possible spatial an temporal correlations. The resulting unique path constraint (for the a priori path) across ifferent scenarios is then reformulate by ifferent equivalent forms to allow efficient problem ecomposition an constraint relaation. Note that formulating or reformulating the stochastic routing problem has not receive sufficient attention in the literature; this is because the wiely stuie aaptive routing problem is essentially a comple multi-stage ecision process, an the PF-base representation is ifficult in its own right to epress in a stanar integer programming moel. (2) Another emphasis of this paper is how to quantify the quality of the solution an provie benchmars for evaluating various heuristic algorithms. A novel Lagrangian relaation-base lower boun approach is evelope to hanle multiple stochastic scenarios an to iteratively approimate the optimal solution space, in which the ualize moel can be further ecompose into two sub-problems that are easily solve by efficient label-correcting algorithms an simple rules for a univariate linear program. In comparison with several relate important stuies on this topic, the main features of our paper are summarize in Table 1.

3 Table 1 Comparative Summary of Shortest Path Problems with Stochastic Travel Times Characteristics Miller-Hoo an Mahmassani (1997, 2000, 2003) Nie an Wu (2009a) Xing an hou (2011) Yang an hou (2011) Representation of stochastic input ata Time-varying PF Time-varying PF Samples with spatial correlation, static stochastic travel times Scenario-base time-epenent lin travel time with both spatial an temporal correlations Objective function Pareto optimal solutions Local-reliable paths Mean + stanar eviation of path travel time Epecte path travel time Solution methoology Three ominance rules First-orer stochastic ominance rules Variable substitution for non-aitive objective function, Lagrangian relaation Unique path constraint reformulation, Lagrangian relaation with multiple scenarios Characteristic of solution Upper boun, lower boun through aaptive route choice Upper boun of objective function Upper boun, lower boun through Lagrangian relaation Upper boun, lower boun though Lagrangian relaation The remainer of this paper is structure as follows. Section 2 provies a formal problem statement, followe by three versions of reformulate moels for the original problem in Section 3. In Section 4, a Lagrangian relaation metho is aapte for seeing a tight lower boun of the objective function, an the corresponing relae moel is then ecompose into two classes of sub-problems associate with each scenario an each physical lin, respectively. Section 5 presents a esign of a sub-graient algorithm to iteratively improve the solution quality an reuce the optimality gap. Finally, several numerical eperiments are conucte in Section 6 to emonstrate the computational effectiveness of the propose algorithms. Table 2 Subscripts an Parameters Use in Mathematical Formulations Symbol efinition S = set of noes in the physical traffic networ. V = set of traffic lins. T = set of iscrete time stamps. t = ine of ifferent time stamps, t { t0, t0, t0 2,, t0 M}. = ine of scenarios/ays, {1, 2,, }. i, j = inices of noes, i, j S. ( i, j ) = ine of traffic lin between ajacent noes i an j, ( i, j) V. c = travel time on traffic lin ( i, j ) at the entering time t on scenario/ay. t C = vector representation for sequence t,(, ), c i j V t T on scenario/ay.

4 C = vector representation for sequence c,( i, j) V, t T, 1,2,,. B = set of binary vectors. t 2. Problem Statements Consier a irecte, connecte traffic networ ( SV, ), where S is a finite set of noes, an V is a finite set of traffic lins between ifferent ajacent noes. The planning time horizon is iscretize into a set of small time slots, enote by T { t0, t0, t0 2,, t0 M}. Symbol t 0 specifies the given eparture time from the origin noe O, an represents a short time interval (e.g. 6 secons) uring which no perceptible changes of travel times are assume to tae place in a transportation networ. M is a sufficiently large positive integer so that the time perio from t 0 to t M 0 covers the entire planning horizon. Corresponing to each physical lin (i, j), a time-variant lin travel time ct is given for traveling along the lin when a vehicle eparts from noe i at each time stamp tt uner scenario. Table 2 lists the notations for the least epecte travel time shortest path problem uner consieration ecision Variables Two types of ecision variables in Table 3 will be use to show the selection of physical lins an time-epenent arcs. Table 3 ecision Variables Use in Mathematical Formulations Symbol efinition = 1 if traffic lin ( i, j ) is selecte; 0 otherwise. y = 1 if traffic lin ( i, j ) is use at entering time t on scenario/ay ; 0 otherwise. t X = lin selection vector containing,( i, j) V. Y = time-epenent arc vector containing y,( i, j) V, t T on scenario/ay. Y = vector representation for t t y on all scenarios/ays, ( i, j) V, t T, 1, 2,, Constraints on Networs In the a priori LET path problem, the main tas is to see a unique physical route in the traffic networ such that the epecte total travel time over ifferent ranom scenarios can be minimize. The physical networ flow balance constraint is first introuce for the LET path selection process in the physical networ (S, V). 1, i O ji 1, i 0, otherwise. (1) ( i, j) V ( j, i) V The set of constraints ensures that all of the selecte physical lins can constitute a path from the origin O to the estination on each scenario. Constraint (1) will be abbreviate using the linear form AX b, as below.

5 Figure 1 An Illustrative Simple Networ Figure 2 Space-Time Networ Representations along ifferent Paths This research aims to use a sampling-base approach to capture comple temporal an spatial travel time correlations in a traffic networ; in particular, historical travel times over ifferent ays are use to construct scenarios. Note that the travel time correlations can be contribute by physical bottlenecs an a number of elay resources such as incients, roa construction, severe weather conitions an special events. On the other han, in a pre-trip routing application involving an avance travel time forecasting engine, each scenario can also correspon to a stochastic instance of preicte travel times uner a certain incient uration or capacity reuction level. Consier an illustrative networ consisting of three noes an three lins in Figure 1. We nee to select an a priori LET path from noe 1 to noe 3 uner two scenarios of time-varying lin travel times ( = 1, 2), in which the eparture time at the origin noe is set as 1 minute. There are two potential paths, namely, P1: 12 3, an P2: 1 3 for the given O pair. Figure 2 isplays the corresponing scheules along those two paths, where the y-ais represents noes on each path an the -ais represents the planning time horizon. Then, the evaluation value of P1 (or P2) is the average travel time in cases of = 1, 2 on path 12 3 (or 1 3 ). Table 4 isplays the epecte travel time on each route, in which no waiting is permissible at intermeiate noes. The epecte travel times on P1 an P2 over two scenarios are 7 min an 7.5 min, respectively; thus, the least epecte time path is foun to be P1: Table 4 Path Travel Time ata Travel Time Route (=1) Travel Time (=2) 12 3 (optimal) Epecte Travel Time For each scenario, one can construct the corresponing space-time networ, enote by ( S, V ), which is epane from the physical networ ( SV, ) an time-varying lin travel time. In more etail, S { it is, t T} represents the set of time-epenent noes, where it S inicates the state of noe i at time stamp t, an each state will be treate as a separate noe. The set of time-epenent arcs is represente by V {( it, jt ') ( i, j) V, t ct t ', t0 t t ' t0 M}, in which time-epenent arc ( i, j ) occurs in the space-time networ when one can travel from physical noe i at time stamp t an t t' arrive at physical noe j at stamp t', where t ct t ', t 0 t t ' t 0 M.

6 For each scenario-base space-time networ, yt is the binary variable inicating whether a time-epenent arc from noe i to noe j at starting time t is selecte in the least travel time path uner scenario. For the same eparture time t0 in a space-time networ, there are ifferent time-epenent paths with ifferent arrival times at the estination. In orer to construct stanar flow balance constraints, ' a ummy estination noe can be ae to the space-time networ to represent the completion of t0 M the journey, in which the corresponing lin travel times are set to zero for incoming lins from the original ' estination to the ummy estination noe, where t' belongs to the feasible arrival times at the t ' t0 M estination. Figure 3 gives an illustration for the ae ummy estination with M 3. O t0 t 0 t 0 2 t 0 3 ' t 3 0 Figure 3 An Illustration of Space-Time Networ an ummy estination To ensure the feasibility of a time-epenent path, a space-time flow balance constraint shoul be formulate as follows: t ( it, jt ') V ( jt ', it ) V 1, i O, t t 0 jit ' 1, ', 0. (2) y y i t t M 0, otherwise Equation (2) will be enote below by an abbreviate form, B Y h. Essentially, the goal of the a priori LET path problem is to fin a path in the physical networ that can correspon to multiple paths in iniviual scenario-epenent space-time networs with possible ifferent arrival times at the estination an intermeiate noes. To characterize this critical linage between the physical an space-time networ representation, we introuce (space-time) arc-to-lin mapping constraints to capture the relationship between the selection of a physical lin an potential time-epenent arcs as follows: y, ( i, j) V, 1, 2,,, t T. (3) t This if-then conitional constraint (wiely use in integer programming formulations) ensures that a time-epenent arc ( i, j ) can be selecte if an only if a physical lin ( i, j) lies on the final path. t t' Constraint (3) will be referre to as Y X. Finally, a binary variable constraint shoul be impose on ecision variables Constraint (4) will be abbreviate as X, Y B. an y t. That is,, y B, ( i, j) V, 1, 2,,, t T. (4) t 2.3. Objective Function For any solution metho, a natural choice for a robust a priori path is to select a solution X so as to minimize the epecte total travel time in the scenario-base time-epenent travel ata. Then the objective is

7 F( X, Y) CY 1 ct y 1 ( i, j) V t T. Because the number of scenarios is a constant, without loss of generality, this equation is simplifie as an equivalent objective in the following iscussion. That is, F( X, Y)= CY. (5) 2.4. One-Stage Mathematical Moel By further introucing an objective that minimizes the epecte value of least time-epenent travel times over all scenarios in equation (5), the integer programming moel of the a priori LET problem can be constructe below: min epecte total travel time (5) F( X, Y)= CY (6) s.t. physical networ flow balance constraint (1) AX b space-time flow balance constraint for each scenario(2) B Y h, 1, 2,, space-time arc-to-lin mapping constraint (3) Y X, 1, 2,,. binary variable constraint (4) X, Y B t 2.5. Alternative Two-Stage Mathematical Moel Aaptive routing is recognize as an alternative metho to search for the least epecte time path in a stochastic an time-epenent networ, which can be treate using a two-stage optimization problem with recourse. Specifically, the process of a routing selection can be ivie into two steps. The first step is to choose a sub-path from the origin to some intermeiate noe in a sub-area of the networ. Then, the rest of the sub-path, which connects the aforementione intermeiate noe an the estination, is etermine by the least epecte time criterion. Meanwhile, the corresponing least epecte time will be use to evaluate this path selection. Note that in this paper we focus the first stage on entire routing selection; sub-area-base route selection at the first stage will be iscusse in future papers. The stochastic two-stage optimization moel with recourse can be formulate as follows: Min E[ Q( X, )] (7) where C Y c y t t ( i, j) V tt s.t. AX b X B where Q( X, ) min C Y s.t. B Y Y Y h X B,. In the above moel, the ecision variable X at the first stage is use for selecting physical routes in the transportation networ. At the secon stage, the least total travel time associate with the space-time networ on each scenario nees to be calculate. The recourse objective function of the first stage, which is essentially the same as the objective in moel (6), represents the least epecte time-epenent travel time over ifferent scenarios. Note that, although moel (7) can well aress the problem mathematically, some inherent ifficulties remain in solving multi-stage moels. For this reason, the emphasis of this paper will be on moel (6). 3. Reformulating Comple Mapping Constraints Within a Lagrangian relaation framewor, the flow balance constraints (1) an (2) can be viewe as easy constraints because they can be hanle eplicitly in stanar shortest path algorithms. The space-time arc-to-lin mapping constraint (3) then becomes a set of har constraints. The following iscussion focuses

8 on how to convert this har constraint set for each scenario Introucing Scenario-epenent Physical Lin Inicators Recall that the path solution to the original LET moel shoul be unique across ifferent scenarios, an this requirement is enforce by constraints (1) an (3). We further introuce a unique path constraint as the first version of moel reformulation. To this en, an alternative set of ecision variables is neee, as shown in Table 5. Table 5 New ecision Variables Use in Reformulations Symbol efinition = 1 if traffic lin ( i, j ) is selecte on scenario/ay ; 0 otherwise. X = vector representation for sequence,( i, j) V on scenario/ay. X = vector representation for sequence,( i, j) V, 1,2,,. It is worth noting that, without a specific statement, the notation X mentione below always has the meaning shown in Table 5. With new ecision variable, the physical networ flow balance constraint (1) an space-time arc-to-lin mapping constraint (3) will be reformulate as follows: ( i, j) V ( j, i) V multi-equalities-base unique path constraint: 1, i O ji 1, i, 1, 2,, 0, otherwise, (8), ( i, j) V, (9) 1 2 y, ( i, j) V, 1, 2,,, t T. (10) t For notational convenience, constraints (8), (9) an (10) are abbreviate as vector-base formulations by 1 2 AX b, 1, 2,,, X X X an Y X, 1, 2,,, respectively. Substituting constraints (1) an (3) in moel (6) by constraints (8) through (10), we can generate an alternative version of the original problem Transforming Multiple Equalities to Nonlinear Variance Equality Although the moel mentione in Subsection 3.1 is linear integer programming, an a Lagrangian relaation metho can be use to ualize these equality constraints, the multi-equalities-base unique path constraint (9) consists of a large number of V ( 1) equality relationships, where V is the 2 carinality of the set V of physical lins. To capture the essential moeling requirement in a more convenient way, we begin to eamine its equivalent forms. Lemma 1 The multi-equalities-base unique path constraint (9) is equivalent to 2 ( ) 0 1 ( i, j) V, (11) 1 1, ( i, j) V. (12) Proof. If the unique path constraint (9) hols, then the set of lins on the physical networ can be ivie into two subsets, enote by E an F, respectively, satisfying: (i) 0 for all ( i, j) E, 1, 2,, for lins not being selecte in the LET solution; (ii) 1 for all ( i, j) F, 1, 2,, for lins being selecte in the LET solution.

9 From the efinitional constraint (12), case (i) yiels 0 for any lin ( i, j) E, 1, 2,,, resulting in 2 ( ) 0 1 ( i, j) E as a single equation across all lins. In case (ii), it is easy to verify that 1 for any lin ( i, j) F, which leas to 0 for each lin ( i, j) F, 1, 2,,. In other wors, we have can be ecompose as 2 ( ) 0 1 ( i, j) F. Alternatively, constraint (11) ( ) ( ) ( ) 0 1 ( i, j) V 1 ( i, j) E 1 ( i, j) F. If equation (11) is true, we have 0 for all ( i, j) E, 1, 2,,, resulting in, ( i, j) V. 1 2 Obviously, a large number of equalities in constraint (9) can be ramatically simplifie by Lemma 1 to a small constraint set with a size of V 1 efinitional constraints associate with an. In real-worl large-scale networ applications, constraint (11) may remain ifficult to solve because of its form of equality restriction. A further relae alternative constraint, referre to as a quaratic variance-base unique path constraint, is introuce in this stuy as the secon moel reformulation. A 1 positive threshol value 2 number of scenarios. Lemma 2 If X is use to set up the following inequality constraint, where is the total 1 2 ( ). (13) 2 1 ( i, j) V B, constraint (11) is equivalent to constraint (13). That is, 2 2 ( ) 0 ( ). 2 1 ( i, j) V 1 ( i, j) V Proof. Constraint (11) yiels constraint (13) because the former can be viewe as a special case with a smaller feasible region. Net, we nee to prove that constraint (13) leas to constraint (11) by contraiction. 0 1 If there eist two ays 0 an 1 with ifferent lin selection inicators such that for a given lin ( i, j) V, from the efinition of, we can erive {0,1} 1, namely, 1 2 1,,,. Without the loss of generality, we consier two particular ays 1 1 an among all ays. Then, it follows that ( ) an ( 2 1 ) (1 ), so the sum over all ays is ( ) 2. In other wors, if 2 conition that is, 1 1 ( i, j) V 2 ( ) hols, then there are no two ays with ifferent selection inicators; 2 1 ( i, j) V, ( i, j) V, satisfying equation (11). 1 2 In the following iscussion, the formulation 2 ( ) will be abbreviate by a vector-base 1 ( i, j) V

10 representation X X 2, an the efinitional constraint (12) for the mean lin selection rate will be represente by X l( X ), where l() enotes a linear transformation Constructing Equivalent Linear Inequality The unique a priori path constraint is now reformulate an epresse through equations (11) an (13). Because the stanar shortest path algorithm accepts only linear formulations, the nonlinear function constraints (11) an (13) might not be esirable. To avoi comple linearization-relate computation, this stuy offers a linear equivalent as below. Lemma 3 If X B, equalities, ( i, j) V hol if an only if 1 2 1, ( i, j ) V, 1, 2,,. (14) Proof. Because the necessary conition is obvious, we nee to prove the sufficient conition by contraiction. Consier that equation (14) hols, but the unique path constraint is not true. Then, there must eist a lin ( i, j) V an two of the scenarios 0 an 1 from ays 1, 2,, such that 0 1 than 1 an, which implies that one of two variables 0 an 1. Without the loss of generality, we set has a value of 0, is strictly less 1 1 an 0. Then, it is easy to obtain. Thus, a contraiction proves the sufficient conition. Compare with constraint (9), equations (14) an (12) are also simplifie formulations because the total number of constraints in (14) an (12) is V ( 1), an both sets of constraints have linear forms. To 1 further simplify the notation, we introuce a smaller positive threshol value v, where 1 0, an enote constraint (14) by X X v. Then, the fourth version of the original problem is formulate by min epecte total travel time (5) F( X, Y)= CY (15) s.t. physical networ flow balance constraint (8) AX b, 1, 2,, linear ifference-base unique path constraint (14) X X v efinitional constraint for mean lin selection rate (12) X l( X ) space-time flow balance constraint (2) B Y h, 1, 2,, space-time arc-to-lin mapping constraint (10) Y X, 1, 2,, binary variable constraint (4) X, Y B 0 4. Lagrangian Relaation To ualize har constraints an search for tight lower bouns of moel (15), this section will present a Lagrangian relaation approach for solving the least epecte time shortest path problem ualizing Har Constraints. In moel (15), constraints (14) an (12) are consiere comple constraints to be ualize. In particular, an auiliary variable will be introuce for the efinitional constraint (12). Two sets of multipliers are also neee for ualizing constraints (14) an (12), i.e., unique path constraint multiplier 0 an mean

11 lin selection rate constraint multiplier R, where ( i, j) V, 1, 2,,. Although the omain of 1 2 is a finite iscrete set of 0,,,,1, we herein still rela the value of in a continuous interval [0, 1], enote by X [0,1] for simplicity. At this point, moel (14) can be reformulate as: 1 ct yt v 1 ( i, j) V tt 1 ( i, j) V ( i, j) V 1 (16) min + ( ) + s.t. physical networ flow balance constraint (8) AX b, 1, 2,, space-time flow balance constraint (2) B Y h, 1, 2,, space-time arc-to-lin mapping constraint (10) Y X, 1, 2,, binary variable constraint (4) X, Y B auiliary variable constraint X [0,1]. By regrouping the variables, we finally have a more systematic view of the components of the ual problem: min + - ct y t v 1 ( i, j) V t T 1 ( i, j) V ( i, j) V 1 1 ( i, j) V s.t. physical networ flow balance constraint (8) AX b, 1, 2,, space-time flow balance constraint (2) B Y h, 1, 2,, space-time arc-to-lin mapping constraint (10) Y X, 1, 2,, binary variable constraint (4) X, Y B auiliary variable constraint X [0,1] Problem ecomposition It is easy to prove that the optimal objective of moel (16) is a lower boun of the optimal objective to the primal moel. To solve moel (16) for any given Lagrangian multiplier vector (, ), we ecompose this moel into two sub-problems. Sub-problem 1 SP1(, ) : time-epenent shortest path problem with generalize costs ct yt 1 ( i, j) V tt 1 ( i, j) V min + (17) s.t. physical networ flow balance constraint (8) AX b, 1, 2,, space-time flow balance constraint (2) B Y h, 1, 2,, space-time arc-to-lin mapping constraint (10) Y X, 1, 2,, binary variable constraint (4) X, Y B. Moel (17) involves ecision variables yt an, an it can be further separate into sub-problems, enote by SP1(,, ), 1, 2,,, which are referre to as the scenario-base time-epenent shortest path problems with generalize costs. That is, min ct yt ( i, j) V tt ( i, j) V (18)

12 s.t. AX b an B Y h an Y X an X, Y B In fact, for each scenario {1, 2,, }, the sub-problem SP1(,, ) can be solve by efficient label-setting or label-correcting algorithms. The optimal objective of moel (18) will be enote by (,, ) SP1. Base on Bellman s principle of optimality, moifie label-correcting algorithms (iliasopoulos an Mahmassani 1992; Pallottino an Scutella 1997) can be aopte to search for the optimal time-epenent least-cost paths of moel (18). In this iterative algorithm, the generalize lin cost consists of two parts: (1) the time-epenent lin travel time c, whose value is epenent on both time stamps t T an physical lins ( i, j) t t V, an (2) the penalty an time-invariant cost selecte physical lins, which are correlate to each physical lin ( i, j) time-epenent lin cost, enote by g t, can be calculate by. associate with the V. Then, the generalize gt ct. (19) In the searching proceure, an M-imensional vector-base label, enote by j ( j ( t0), j ( t0 ),, j ( t0 M)), can be use to show the least travel cost from origin O to current noe j at each time stamp t' T, where j ( t ') is compute by ( t ' t c t ): min j ( t '), g t i ( t), j V / i, t ' T j ( t '). (20) 0, j O, t ' T After solving sub-problems SP1(,, ), 1, 2,,, the sum of corresponing least travel costs (across ifferent scenarios) becomes the optimal objective of moel SP1(, ). Sub-problem 2 SP2(, ) : auiliary-variable optimization problem SP2(, ) : ma : X [0,1] ( i, j) V 1. (21) Moel ( SP2(, ) ) can be further re-epresse in terms of the following lin-base auiliary-variable optimization problem for each lin ( i, j) V, namely, SP2(,, i, j) : ma : [0,1] 1. (22) Equation (22) is a simple univariant linear program with a boun, in which unique path constraint multiplier an efinitional mean lin selection rate constraint multiplier are input parameters preetermine in moel (16). For each moel SP2(,, i, j), we can calculate its optimal objective as follows: SP2 i j 1 1 (,,, )), if 0. (23) 0, otherwise After solving SP2(,, i, j) for each lin ( i, j) V, the sum of corresponing optimal objective values equals the optimal objective value of moel SP2(, ). For any Lagrangian multipliers (, ), the optimal objective value of moel (16), enote by LR(, ), can be compute by the formulation below:

13 SP1 SP2 1 ( i, j) V 1 ( i, j) V. (24) LR(, ) (,, ) (,, i, j) v 4.3. Comparison with a Wait an See Solution After we present the Lagrangian relaation moel, it woul be interesting to compare the corresponing tightest lower boun with the well-nown Wait an See (WAS) boun for stochastic programming, which aims to choose the optimal paths base on the realize lin travel time of each scenario. In etail, for each scenario, we first search for a time-epenent shortest path associate with the realize sample ata. The summation, i.e., 1 TX [ ], offers a lower boun of the optimal objective value to the primal stochastic programming problem, where T(X ) enotes the least travel time on scenario. Lemma 4 If 0, 0 for any ( i, j) V, 1, 2,,, then the Lagrangian relaation-base moel (16) is egenerate to a WAS moel. That is, compare with our propose Lagrangian relaation-base approach, a WAS solution oes not impose the unique a priori path constraint, which leas to a larger feasible region an then prouces a relatively loose lower boun compare with the original moel (15). 5. Solution Algorithm 5.1. Solution Strategies Because the primal moel is to minimize the total travel time over ifferent scenario-base ata, the path travel time of each feasible path solution in moel (15) can be regare as an upper boun of the optimal objective to the original problem. To fin an approimate optimal solution with the guarantee quality, the propose iterative solution algorithm is intene to minimize the gap between the upper boun an the lower boun. Particularly, if the minimize gap is equal to 0, the eact optimal solution will be obtaine. In this section, a sub-graient algorithm will be esigne to improve the lower boun. This solution technique aims to ecrease the ual gap iteratively by upating the Lagrangian multipliers (, ) along the sub-graient irection of the objective function in moel (16). For fie values of multipliers (, ), the optimal objective LR(, ) of moel (16) is a lower boun of the optimal objective of the primal moel (15), an the following Lagrangian ual problem nees to be solve: LR(, ) ma { LR(, )}. (25) LR 0, R Meanwhile, the upper boun will be upate using newly available better feasible path solutions (with a lower objective function (5)) to minimize the ual gap. To iteratively upate the parameter vector (, ) for reaching tighter lower bouns, at each iteration, the vector components of sub-graient will be calculate by: 1 LR (, ) v, LR (, ) (26) 1 for any ( i, j) V, 1, 2,,. Consequently, the sub-graient vector LR(, ), consisting of LR (, ) an LR (, ), ( i, j) V, 1, 2,,, will be treate as the searching irection for the net iteration. In the searching process, the algorithm starts with a preetermine initial solution, an notation is use as the ine of the iterative number. enote the Lagrangian multiplier vector by (, ) at iteration. To obtain the upate multipliers for the net iteration, first we nee to solve moel (16) at the current iteration, where variables of the optimal solution are enote by y,, t an,, ( i, j) V, t T, 1, 2,,, respectively. Then, the following equations will be use to upate the Lagrangian multipliers for iteration +1:

14 1, 1,, 1, ( v), 1 for any ( i, j) V, 1, 2,,, where the parameter is etermine as follows:, f (, ) (27) ( UB LR(, )), (28) 1 f (, ) ( v),, 2, 1 ( i, j) V ( i, j) V In equation (28), notation UB represents the best objective value encountere so far in the primal problem an can be upate iteratively to spee up the optimization process. The parameter is a scalar in interval [0, 2], the purpose of which is to ajust the step size of the process an guarantee that no negative cost appears in the objective function. In the propose algorithm, UB can be iteratively upate by solving sub-problems SP1(,, ), {1, 2,, } at each iteration. Specifically, let UB enote the objective of moel (15) corresponing to the optimal routing plan of SP1(,, ) ; then, the upper boun at iteration will be upate by the following formulation: UB min UB, min UB. 1 {1, 2,, } Once the ual gap UB LR(, ) becomes less than a preetermine toleration gap, or the iteration is larger than a preefine maimum iteration K ma, the algorithm will terminate, an the UB will be use to evaluate the solution quality in terms of gaps Solution Proceure We now present the complete solution proceure as follows: Step 1. (Initialization) Let = 1 (referre to as inner loop iteration ine in numerical eperiments); initialize the multipliers (, ), set UB as a sufficiently number or an objective value of a feasible solution to the primal moel. Step 2. (Solve the Relae Moel) This part is ivie into two types of sub-problems, where parameters (, ) are use. (1) Solve the moel (18) for each 1, 2,, by a moifie label-correcting algorithm; (2) Compute the equality (23) for each ( i, j) V., Step 3. (Compute the Gap) enote the optimal solution of the above problem by y, ( i, j) V, 1, 2,,, t T. Base on the optimal solutions, calculate primal, ual an gap values. Step 4. (Upate Lagrangian Multipliers) Compute the multipliers for the net iteration by equation (27). Step 5. (Termination Conitions) If Kma (a preetermine maimum number of iterations) or the gap is less than the preetermine value, then the algorithm will be terminate. Otherwise, let 1, go to Step 2. To clarify the proceure of the solution methoologies, a flow chart illustrating each step of the Lagrangian relaation approach is shown in Figure 4. t, an,

15 Initialize (, ) at iteration =1 1 Lagrangian relaation moel with given multipliers (, ) Scenario-base timeepenent shortest path problem with generalize costs Lin-base auiliaryvariable optimization problem Upate Lagrangian multipliers (, ) SP1(,, ) for 1,2,, SP2(,, i, j) for lin ( i, j) V Compute sub-graient LR (, ), LR (, ) no Upate lower boun Upate upper boun Compute optimality gap En conition? yes Output optimal solution Figure 4 Proceure of the Solution Methoology 6. Numerical Eperiments In this section, the effectiveness an computational efficiency of the Lagrangian relaation metho is teste by using a simplifie networ an two real-worl networs. The algorithm is implemente in C++ on the Winows 7.0 platform an evaluate on a personal computer with an Intel(R) Core(TM) i5 CPU an 4 GB memory Illustration of the Solution Approach with a Simplifie Networ In the first set of numerical eperiments, we consier a simple transportation networ, as shown in Figure 1, in which the travel time on each lin is assume to be time-invariant, an the travel time information is given in Table 6 with two scenarios ( 1, 2 ). For this special case, ecision variables in moel (15) will be egenerate to a single ecision variable will be reuce to a time-invariant parameter c., an time-epenent lin travel time Table 6 Travel Time on Each Lin for ifferent Scenarios (Unit: min) Scenarios Lin (1,2) Lin (1,3) Lin (2,3) = = c t First, we apply the optimization moel (15) with an O pair from noe 1 to noe 3, where v is set as a small value < 1 an where 2 inicates the number of scenarios. Because there are only 2 paths 2 between this O pair, by enumeration, one can simply obtain the optimal solution 12 3 with an objective function LET = 14, where LET is the optimal objective value of the primal moel. As analyze in Subsection 4.3, the WAS moel in this case leas to the two shortest paths corresponing to ifferent scenarios; that is, path 1 3 on scenario 1 an path 12 3 on scenario 2, which leas to a WAS-base lower boun of 12 min. The etaile results are shown in Table 7. Table 7 Solution Provie by Wait an See Moel Scenarios =

16 = WAS 12 As epecte, 12 = WAS < LET = 14. In comparison, solving the Lagrangian relaation moel (16) in Subsection 4.1 yiels computational results in the first 10 iterations liste in Table 8. Table 8 Upating Process of Multipliers an Gap in the First 10 Iterations Iteration LB UB Gap Relative Gap % % % % % % % % % % In Table 8, LB an UB, respectively, enote the best lower boun an upper boun encountere up to the current iteration, whereas Gap an Relative Gap are calculate by the formulations below: Gap = UB-LB, UB - LB Relative Gap = 100% UB. A lower boun of is obtaine at iteration 10, which shows only that the propose Lagrangian relaation-base approach can provie a much tighter lower boun than the WAS metho, as = 12 < LR = LET = 14. For this simple networ, a total of 9 multipliers nee to be consiere in the propose Lagrangian ual problem, incluing 6 multipliers for unique path constraints an 3 multipliers associate with equality constraints for calculating the mean of lin selection inicators across ifferent scenarios. The 1 close-to-optimal values for multipliers 2 13, 2 12 an 23 (shown as shae in Table 8) are quicly iscovere at the thir iteration, an their values remain nearly unchange afterwar. The searching process continues to ajust the values of the other Lagrangian multipliers, an the relative solution quality gap graually reuces to 3.74%, as shown in Figure 5. It is noteworthy that before iteration 10, the lower boun solution still generates two ifferent paths, even with a relatively high lower boun value. After iteration 10, the scenario-epenent solutions will become reuce to a single unique path. WAS

17 Objective Function Inner Loop Iteration Figure 5 Variations of Lower Boun an Upper Boun in the First 20 Iterations 6.2. Numerical Eperiments for a Meium-Scale Networ The secon set of eperiments consiers a real-worl freeway networ with 127 noes an 284 lins etracte from the San iego region (see Figure 6, left: Google Maps, right: noe-lin networ structure). Figure 6 Teste Transportation Networ in San iego (Source: Google Maps) This eperiment consiers a one-hour planning time horizon with a 0.5-min travel time aggregation time interval. Ten ays (i.e., scenarios) of time-epenent travel time ata are ranomly simulate by using a ata-generation metho similar to that propose by Miller-Hoos (1997). Specifically, the free-flow travel time of each lin (i, j), enote by t, is first calculate. Then, we generate a ranom integer 2} for time interval t to construct q t q in set {1, t as the lin travel time on lin (i, j) at eparture time t on scenario. Consiering the O pair shown in Figure 6, a total of 3124 Lagrangian multipliers nee to be consiere in the propose Lagrangian ual moel. Thus, to avoi generating only local optimums, a global optimization searching metho is use in the eperiments, in which multiple initial solutions are prouce to restart the propose sub-graient algorithm for every 15 inner iterations (i.e., Kma 15 ). After eecuting the algorithm for 20 (ranomly generate) starting sees of Lagrangian multipliers, the obtaine best upper boun an tightest lower boun of the primal problem are = an = , where Relative Gap = 2.57%. In comparison, the WAS metho generate a lower boun of WAS = with a relative gap of 7.02% (see Table 9). The computational results show that the propose Lagrangian relaation approach can fin a high-quality solution an a much tighter lower boun for the primal problem. LET t LR

18 Objective Function Table 9 Optimal Solution an Lower Bouns LET LR WAS Value Gap Relative Gap % 7.02% To show the searching process clearly, Figure 7 isplays the variation of the lower bouns with respect to ifferent sees of initial multipliers. The -ais represents the iteration inices for generating initial multipliers in the searching process (i.e., outer loop iteration ine). With ifferent initial solutions, the lower boun encountere so far (Line LB(OI), Figure 7) will be upate in a non-ecreasing sequence. When the outer loop iteration ine equals 2, the tightest lower boun LR is obtaine with the smallest relative gap. Figure 8 shows the variation of relative gaps with the increase of the outer loop iteration. Clearly, in each iteration of the searching process, the most time-consuming step of the sub-graient algorithm is the time-epenent shortest path generation for ifferent scenarios. Thus, the computational compleity of the propose algorithm is epenent to a great etent on the scale of the networ an the number of scenarios. In the eperiments, for the computational process of the sub-graient algorithm with a maimum of 15 inner iterations, the general time consumption in each iteration is about 4 to 7 secons (incluing 10 time-epenent shortest path solving steps) LB(II) WAS UB LB(OI) Outer Loop Iteration l Figure 7 Variations of Lower Bouns an Upper Bouns LB(II): The best lower boun obtaine with respect to ifferent restarting sees of initial multipliers; LB(OI): The tightest lower boun obtaine up to outer loop iteration l; UB: The variation of upper bouns; WAS: The variation of lower bouns by the WAS.

19 Relative Gap 6.00% 5.00% RelGap1 RelGap2 4.00% 3.00% 2.00% Outer Loop Iteration l Figure 8 Variation of the Relative Gap Associate with ifferent Sets of Initial Lagrangian Multipliers RelGap1: The best relative gap associate with ifferent restarting sees of initial multipliers; RelGap2: The best relative gap obtaine up to outer loop iteration l Numerical Eperiments for a Large-scale Networ with Real-worl Sensor ata The thir set of numerical eperiments is performe on a part of the real-worl transportation networ near en Haag-Rotteram in the Netherlans with the recore time-epenent lin travel time ata obtaine by sensors. As shown in Figure 9 (left: Google Maps, right: noe-lin networ structure), an abstract networ, consisting of 299 noes an 736 lins, is etracte from the real networ for the eperiments; lins mare by squares are equippe with sensors for obtaining the real-time lin travel times.

20 Figure 9 Teste Transportation Networ in the Netherlans (Source: Google Maps) In the eperiments, a set of ten ays of time-epenent lin travel times obtaine by 130 point etectors are use to search for the robust a priori LET routes. Specifically, the time horizon is consiere to be from 15:30 to 17:00 an is iscretize by 0.1-min time intervals. For the lins equippe with the sensors, the real-time epenent travel time ata are collecte from 15:30 to 17:00 on ten weeays. Meanwhile, for the lins without etectors, the region-wie lin travel time ine is aopte as the efault values. Aitionally, to simulate the impact of non-recurring elays, we further a incients on lin travel times on ifferent ays. Specifically, we assume the probability of incient occurrence is 0.2 for each ay. If an incient occurs on a particular ay, the lin travel times of the involve lin over the planning horizon will be increase by multiplying the actual travel time by a ranom integer in set {2, 3}. Consier an O pair with potential multiple alternative routes, as shown in Figure 9. Note that the Lagrangian ual problem has introuce a large number of variables, namely, 8096 multipliers. A global search metho is also use to see the tightest boun of the prime problem, in which for each see of initial multipliers, the inner loop of the sub-graient algorithm will be repeate for ten iterations (i.e., Kma 10 ). After eecuting the algorithm with 20 outer-loop iterations with ifferent initial sees of LR multipliers, the obtaine best upper boun an tightest lower boun of the primal problem, respectively, are LR = In comparison, the WAS metho generates a lower boun of gap of 7.36%. The gap an relative gap are liste in Table 10. Table 10 Optimal Solution an Lower Bouns LET LR WAS Value Gap Relative Gap % 7.36% WAS =135.35, LET = with a relative 6. Conclusions an Future Research To solve critical path-fining problems in a time-epenent, stochastic transportation networ, this paper formulate two equivalent optimization moels with a scenario-base representation for capturing correlations of spatial an temporal lin travel time. Aiming to obtain the a priori LET shortest paths, first, we investigate the unique path constraint across ifferent solutions corresponing to ranom scenarios. Then, we propose several equivalent reformulations to establish linear inequalities that can be easily ualize by a Lagrangian relaation-base approach. To search for high-quality solutions, the formulation was further ecompose into two simple sub-problems. Then, a sub-graient algorithm was esigne to see the optimal solution of the corresponing Lagrangian ual problem. The effectiveness of the propose approach was emonstrate by using a simplifie networ an two real-worl networs. The eperimental results on a meium-scale

21 networ etracte from the San iego region show that the propose algorithm can prouce a smaller uality gap (about 2.57%) than the WAS. Even for the large-scale networ in the Netherlans, the propose Lagrangian relaation-base approach generate reasonably tight lower bouns. Further research will focus on the following three major aspects. (1) The propose a priori optimal path moeling methoology will be further etene for a two-stage or multi-stage optimization moel for emerging real-time aaptive routing applications. (2) The propose moel for the single O shortest path problem in a time-epenent, stochastic transportation networ can be further etene to a networ-wie traffic assignment problem involving multiple O pairs with stochastic eman patterns or roa capacities. (3) To spee up the computational performance for large-scale real-worl networs, improve reformulations or solution methos are also critically neee to aggregate or reuce the large number of Lagrangian relaation multipliers for the propose reformation moels. Acnowlegments The research of the first author was supporte by the National Natural Science Founation of China (No ) an the Research Founation of State Key Laboratory of Rail Traffic Control an Safety, Being Jiaotong University (Nos. RCS2009T001, RCS ). We also want to than Yusen Chen for proviing the thir networ an sensor ata. The wor presente in this paper remains the sole responsibility of the authors. References Chen, A.,. Ji Path fining uner uncertainty, Journal of Avance Transportation. 39(1) Fu, L., L.R. Rilett Epecte shortest paths in ynamic an stochastic traffic networs, Transportation Research Part B. 32(7) Fu, L An aaptive routing algorithm for in-vehicle route guiance systems with real-time information. Transportation Research Part B. 35(8) Gao, S., I. Chabini Optimal routing policy problems in stochastic time-epenent networs. Transportation Research Part B. 40(2) Hall, R.W The fastest path through a networ with ranom time-epenent travel times. Transportation Science. 20(3) Hicman, M..,.H. Bernstein Transit service an path choice moels in stochastic an time-epenent networs. Transportation Science. 31(2) Liu, B Theory an Practice of Uncertain Programming. Physica-Verlag, Heielberg. Pages: Miller, E.., H.S. Mahmassani, A. iliasopoulos Path search techniques for transportation networs with time-epenent, stochastic arc costs, IEEE International Conference on Systems, Man an Cybernetics: Humans, Information an Technology 2, Miller-Hoos, E Optimal Routing in Time-Varying, Stochastic Networ: Algorithms an Implementations. Ph. issertation. University of Teas at Austin, USA. Miller-Hoos, E.., H.S. Mahmassani Least epecte time paths in stochastic, time-varying transportation networs. Transportation Science 34(2) Miller-Hoos, E Aaptive least-epecte time paths in stochastic, time-varying transportation an ata networs. Networs 37(1) Miller-Hoos, E., H. Mahmassani Path comparisons for a priori an time-aaptive ecisions in stochastic, time-varying networs. European Journal of Operational Research 146(1) Nie, Y., X. Wu. 2009a. Shortest path problem consiering on-time arrival probability, Transportation Research Part B 43(6) Nie, Y., X. Wu. 2009b. Reliable a priori shortest path problem with limite spatial an temporal epenencies. Proceeings of the 18 th International Symposium on Transportation an Traffic Theory, Hong Kong, China, Nielsen, L.R Route Choice in Stochastic Time-epenent Networs. Ph. issertation. University of Aarhus, enmar. Pallottino, S., M.G. Scutella Shortest path algorithms in transportation moels: classical an innovative aspects, In: Marcotte, P., Nguyen, S. (Es.), Equilibrium an Avance Transportation