> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 1 Reformulation an Solution Algorithms for Absolute an Percentile Robust Shortest Path Problems Xuesong Zhou, Member, IEEE an Tao Xing Abstract To moel river route choice behavior uner inherent traffic system stochasticity, an further provie better route guiance with travel time reliability guarantees, this paper examines two moels to evaluate the travel time robustness: absolute an α-percentile robust shortest path problems. A Lagrangian relaxation approach an a scenario-base representation scheme are integrate to reformulate the mini an percentile criteria uner ay-epenent ranom travel times. The complex problem structure is ecompose into several subproblems that can be solve efficiently as the stanar shortest path problems or univariate linear programming problems. Large-scale numerical experiments with real-worl ata are provie to emonstrate the efficiency of the propose algorithms. Inex Terms Route guiance, traffic information systems, algorithms, traffic planning. T I. INTROUCTION RAVEL time reliability has been wiely recognize as an important element of a traveler s route an eparture time scheuling, especially for ris-averse commuters. In recent years, transportation planning an management agencies have begun to shift their focus more towar monitoring an improving the reliability of transportation systems through traffic information provisions an integrate corrior management. In aition to proviing preicte travel times, the new generation of personal navigation systems nees to estimate the potential uncertainty an variability of origin-toestination (O) trip times, an further suggest the route that can imize the trip time reliability or on-time performance with respect to uncertainties incluing inherent traffic ynamics, measurement errors an possible poor forecasts. A wie range of efinitions an formulations have been propose to quantify travel time reliability, incluing (1) percentile travel time an absolute robust travel time, (2) ontime arrival probability, an (3) travel time variation expresse in terms of stanar eviation or coefficient of variation. From a trip planning point of view, the first class of criteria highlights travel time guarantees over uncertain traffic Manuscript receive October 3, 2011. X. Zhou an T. Xing are with the epartment of Civil & Environmental Engineering, University of Utah, Salt Lae City, UT 84112 USA (e-mail: zhou@eng.utah.eu; tao.xing@utah.eu). situations, while the secon an thir efinitions emphasize the probability of later arrivals for a given preferre arrival time or a given buffer time inex. Fig. 1, aapte from a recent FHWA report [1], shows a istribution of travel times on eastboun State Route 520 in Seattle, base on 3096 observation samples taen on weeays between 4:00 to 7:00 pm. In the heavy-taile travel time istribution along this 11.5-mile corrior, the mean an stanar eviation statistics (i.e. 15.9 min an 5.5 min) are insufficient to fully measure the extreme elay uring the aily commutes, where contributing factors may inclue traffic crashes or severe weather conitions. In particular, the worst or absolute robust travel time is about 31.5 min (uring the survey perio of four months), while the 95% percentile travel time is aroun 22.5 min. In this stuy, we will focus on the absolute robust shortest path (ARSP) an percentile robust shortest path (PRSP) problems, an a finite number of lin-base travel time samples (e.g., from ifferent ays) are use to escribe the travel time istribution. The absolute robust shortest path problem uner consieration aims to fin the path that minimizes the imum path travel time over all samples. Similarly, the a-percentile robust shortest path problem is efine as the path that minimizes the travel time within which a-percentile trips in all samples are complete. ARSP ten to emphasize the extreme tail of the travel time istribution, which might be unliely to occur. PRSP is able to systematically balance the trae-off between the overall ris an uncertainty, an also provies a better statistical measure to avoi possible outliers in the real-worl ata sample set. Aitionally, PRSP can meet the nees for travelers with ifferent egrees of ris-avoiance preferences. The concept of the percentile robust path has also attracte increasing attention recently in the area of transportation networ analysis. A traffic assignment moel propose by Nie [2] consiers a percentile user equilibrium where travelers follow PRSPs between each O pair, an the PRSP problem is solve by assuming inepenent probability istributions of lin travel times as a result of stochastic service rates. Oronez an Stier-Moses [3] examine a user equilibrium moel for ris-averse users, where the robust shortest path moel propose by Bertsimas an Sim [4] is embee to capture the trae-off between the normal an worst cases of lin travel times.
Frequency > REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 2 Average 95% percentile Worst 350 300 250 200 150 100 50 0 11.5 14.5 17.5 20.5 23.5 26.5 31.5 Travel Time (min) Fig. 1. Travel time istribution for the eastboun lanes of State Route 520, Seattle, aapte from FHWA report [1]. The ARSP problem an its variants have been extensively stuie in the last few ecaes. Murty an Her [5] propose a relaxation base label-correcting proceure to provie exact solutions for the ARSP problem. Specifically, two pruning techniques, namely one-row an Lagrangian-base relaxation, were use to improve the algorithm efficiency. Their approach was later enhance by Bruni an Guerriero [6] by using heuristic rules an evaluation functions to better guie the solution search proceure. Yu an Yang [7] stuie both ARSP an robust eviation shortest path (RSP) problems by using a set of scenarios to capture the uncertainty of travel time. They first prove that both ARSP an RSP problems are NP-complete uner limite scenarios an NP-har for an unboune number of scenarios, an then propose ynamic programming algorithms with a pseuo-polynomial computational time an a few heuristic methos. Mainly focusing on the RSP problem, Karasan et al. [8] propose a simple ARSP approximation algorithm by setting each lin travel time to its upper boun over all scenarios/samples. Montemanni an Gambarella [9] evelope two algorithms for the ARSP problem, namely a Beners ecompositionbase algorithm an a solution metho by generating uality reformulation an solving through mixe integer linear programming techniques. In a recent stuy by Fan et al. [10], a path fining algorithm was propose to minimize the probability of arriving at the estination later than a specifie arrival time. Nie an Wu [11] evelope solution algorithms with first-orer stochastic ominance rules for the routing problem with a given on-time arrival reliability. Many stuies [12]-[14] have been evote to calibrating the travel time reliability measure through stanar eviation or variance. There are also a number of other efinitions relate to robust shortest paths. For example, Yu an Yang [7] consiere the robust eviation shortest path problem that minimizes the imum eviation of the path length from the optimal path length of the corresponing scenario, an Sigal et al. [15] suggeste using the probability of being the shortest path as an optimality inex. Using a sampling-base representation scheme, this research utilizes historical travel time recors from multiple ays of traffic measurements to capture ay-by-ay traffic ynamics an the complex spatial networ correlations. Specifically, a scenario (corresponing to travel time samples on a ay) is consiere as a realization of ranom travel time istributions. Recognizing the complexity in on-line path computation for path reliability measures, Chen et al. [16] propose a ynamic upating metho using off-line precalculate caniate paths base on historical travel time samples. In this research, we focus on how to efficiently fin approximate solutions for the ARSP an PRSP problems, an a Lagrangian relaxation base algorithm is use to generate satisfactory feasible solutions an provie the corresponing quality evaluation on large-scale real-worl networs. In particular, we aopt a variable splitting approach to reformulate the mini objective function of the ARSP problem an the percentile efinitional constraint of the PRSP problem. The variable splitting approach was propose by Joernsten an Naesberg [17]. To reformulate a complex objective function, auxiliary variables an aitional constraints are introuce so that easy-to-solve subproblems can be constructe in a Lagrangian relaxation solution framewor. This approach was aapte by Larsson et al. [18] to solve a minimum cost networ flow problem with a concave objective function, an recently by Xing an Zhou [19] to hanle the nonlinear an nonaitive cost function associate with the quaratic forms of the stanar eviation term in a reliable path problem. To fin paths that can imize travel time reliability on a networ with normally istribute an correlate lin travel times, Seshari an Srinivasan [20] propose bouns-base optimality conitions an an efficient path generation proceure. The remainer of this paper is structure as follows. The next two sections provie formal problem statements, theoretical erivations an algorithmic evelopment for both absolute an percentile robust shortest path problems. Section 4 evaluates the performance of propose algorithms through numerical experiments on a large-scale networ with realworl observation ata. II. FINING ABSOLUTE ROBUST SHORTEST PATH A. Problem Statement an Assumptions Consier a irecte, connecte transportation networ G(N, A) consisting of a set of noes N an a set of lins A. In this stuy, we assume a set of lin travel time samples or measurements is available for the same time perio of ays, for example, for the pea hour from 8am to 9am over = 1,, weeays uring a three-month perio ( = 60), where is the inex of ranom scenarios (in a stochastic optimization framewor) or the inex of ata collection ays (from a traffic ata mining perspective). With a sufficiently large sample set, the calculate path travel time measure is able to capture the inherent spatial an temporal correlation of lin travel times. Intereste reaers are referre to iscussions by Seshari an Srinivasan [20], an Xing an Zhou [19] on ifferent moels for representing spatial correlation for lin travel times.
> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 3 For notational simplicity, this stuy consiers timeinvariant travel times uring the analysis time perio of iniviual ays, but the presente solution framewor can be extene to a space-time expane networ by aing mapping constraints between physical lins an space-time arcs, as illustrate in a recent stuy [21]. We further enote the lin from noe i to noe j as a paire inex of, an accoringly the travel time of each lin at the sample ay is expresse as, s), a set of binary variables X x A c. For a given O pair (r, represents the selection of lins on a path (i.e. a path solution). The travel time for a path X at sample ay is then written as: T c x (1), A subject to a flow balance constraint x x b ji (2) j: A j: jia 1 i r where b 0 i N { r, s} represents the flow status 1 i s for each noe i in the networ. Given a ay-epenent sample set, the robust shortest path problem aims to fin a single path solution X that satisfies a certain robustness criterion over all realize samples/scenarios from ifferent ays. Specifically, two types of criteria are consiere for the evaluation of the path travel time robustness: absolute robust shortest path an α-percentile robust shortest path. The ARSP problem is mathematically expresse as Problem P0: z0 min c, x (3) x A subject to the flow balance constraint (2). B. Variable Splitting-base Moel Reformulation To reformulate the propose mini objective function (3), a variable splitting approach is aopte in this stuy. Expressly, we first introuce an auxiliary variable y into the objective function, y c x (4), A so the mini problem is transforme to a stanar minimization problem format as z = min y. The auxiliary variable y is efine as the imum path travel time for the path X from r to s over all samples, an y also correspons to the absolute robust travel time for a path solution X. Further, the imization sub-problem in (4) can also be equivalently expresse as a set of inequality constraints for y over ifferent ays =1,, : y c, x, (5) A Consequently, the ARSP problem P0 is formulate as P1 Problem P1: z1 min y (6) subject to constraints (2) an (5). To further iteratively fin solutions for P1 efficiently, a Lagrangian relaxation base approach is implemente. That is, we introuce a set of non-negative Lagrangian multipliers to relax an ualize the inequality constraint set (5) into the objective function (6). y c x y, (7) min A By re-grouping variables in (7), we now consier a Lagrangian problem: Problem L1: L(,,..., ) 1 2 min c, x 1 y A subject to constraint (2). For any feasible (non-negative) value set of the Lagrangian multipliers, the objective function value of the Lagrangian ual problem 1 2 the optimal value z 1 * of the original problem P1. By iteratively ajusting the Lagrangian multiplier set for L1, we want to imize the ual objective function in (8) an therefore improve the lower boun estimate of the primal problem. Aitionally, we will use paths generate through solving the ual problem to iscover better solutions, which can also reuce the upper boun to the optimal value z 1 * of the primal problem. (8) L(,,..., ) provies a lower boun to C. Lagrangian ecomposition In the ual problem L1, (8) can further be ecompose into an solve by two inepenent sub-problems for primal variables x an auxiliary variable y, respectively. L(,,..., ) 1 2 L (,,..., ) L (,,..., ) x 1 2 y 1 2 The subproblem Lx( 1, 2,..., ) is a binary integer programming problem for the primal variable set x, an it can be solve efficiently using stanar label correcting or label setting algorithms [22] for new lin cost values of c,. Subproblem SP1: Lx ( 1, 2,..., ) min c, x A (10) : x x ji b j: A j: jia The secon part of the ual problem in (9) is a linear (9)
> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 4 minimization problem for the single variable y. As a linear function, the optimal value of L y is achieve at one extreme point of the feasible range of y. Subproblem SP2: Ly ( 1, 2,..., ) min 1 y (11) To fin the optimal solution for subproblem SP2, we nee to consier a feasible range[y LB, y UB ] for y an propose a solution proceure for L y : Proposition 1: epening on the value of 1, the variable y in subproblem SP2 is selecte at one extreme point of its feasible range for the optimal value of L y, e.g.: LB y 1 0 y (12) UB y 1 0 Proof: The above proposition can be easily erive as subproblem SP2 is a univariate linear program with 1 as the cost coefficient. By referring bac to the efinitional equation (4) for variable y, for any feasible path solution X of the primal problem, it correspons to an upper boun on the optimal objective function. For example, we can fin the shortest path (with a cost function as the path istance), an use the corresponing imum ay-specific travel time as the upper boun y UB. In the iterative search process to be presente below, the upper boun y LB can be upate once a new path is iscovere with a lower value of the imum ay-specific travel time compare to the current y UB. Essentially, y LB shoul provie a lower boun to the imum ay-specific travel time on the optimal path. In this stuy, we first compute min c min c, as the least travel time of lin () across ifferent ays, an then use min c x as the objective function to fin the least travel A time path an the corresponing path travel time T min. As min c is the least possible travel time of each lin, T min z 1 * for sure, for both ARSP an PRSP problems.. Subgraient Metho Let us enote L* to be the imum value of L(,,..., ) over ifferent Lagrangian multiplier sets: 1 2 L* L(,,..., ) (13) 1, 2,..., 0 1 2 In orer to fin a tighter lower boun for the primal problem, we aopt a subgraient approach to iteratively search the Lagrangian multiplier set an the corresponing values of x an y. The search irections of µ are typically calculate as the subgraient of L: L(,,..., ) 1 2 (14) c,1 x y, c,2 x y,..., c, x y A A A Let us use to enote the number of iterations. Starting from any feasible initial value set, we first fin solutions an y for subproblems SP1 an SP2, respectively. Then the values of the Lagrangian multipliers 1 x at iteration +1are upate using the following subgraient equation: 1 ( c, x y ) (15) A where the step-size set can be calculate by using the following heuristic algorithm: UB 1 ( 1, 2,..., z L ) (16) z UB In (16), 1 is the current best objective function value for feasible solutions in the primal problem an can be upate when a tighter upper boun is foun. A scalar chosen between 0 an 2 is use in this stuy to ajust the step-size of the search process an ensure non-negativity of Lagrangian multipliers. E. Solution Proceure The overall algorithm for solving the absolute robust shortest path problem is escribe below. Algorithm 1: Step 1: Initialization Set iteration number = 0; Choose positive values to initialize the set of Lagrangian multipliers ; Select initial values for y UB an y LB. The upper bouns an lower bouns of the original problem P1 are 1 z LB 1 y LB z UB y UB an Step 2: Solve ecompose ual problems Solve subproblem SP1 using a stanar shortest path algorithm an fin a path solution x; Solve subproblem SP2 with (12) in Proposition 1 an fin a value for y; Calculate primal, ual an gap values, an upate the upper an lower boun of y. Step 3: Upate Lagrangian multipliers If K or the gap is smaller than a preefine toleration gap, terminate the algorithm, otherwise go bac to Step 2. K is a preetermine imum iteration value. F. Solution Quality Measurement To measure the path solution quality, we efine ε= z UB z LB as the uality gap between the lower boun z LB an the upper boun z UB of the optimal solution. As a result, the gap between the optimal value z * an the objective function value of the current best solution z UB is no larger than the gap ε. To
> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 5 normalize the uality gap for comparison purposes, we can efine a relative optimality measure as UB LB * * z z z L ' (17) UB * z z With a reasonably small relative gap, we provie a satisfie solution quality guarantee on the suggeste absolute robust path. It is important to notice that, ue to the approximate nature of the Lagrangian relaxation estimator, there coul still be a positive gap even if the optimal solution of the primal problem has been achieve. The above propose algorithm has a complexity of O( A K) where K is the number of iteration (e.g., 10-20 for our experiments in Section IV), while the complexity of Yu an Yang s heuristic solution algorithm is O( A ) with being the number of scenarios/ays. Our algorithm is comparatively more efficient when a large number of scenarios (say =50) is require to achieve a low sampling error in capturing the networ travel time stochasticity an ynamics. III. FINING α -PERCENTILE ROBUST SHORTEST PATH An α-percentile robust shortest path problem aims to minimize the α-percentile path travel time among all feasible paths. For instance, given α = 0.9, each feasible path solution x of the given O pair has a path reliability measure y(x) corresponing to 90 th -percentile travel time. This measure ensures that, over all sample ays, 90% of those ays have ay-specific path travel times less than y(x). Among all feasible paths, the path solution x * with a minimum 90 th - percentile travel time is then consiere as the 90 th -percentile robust shortest path. As a special case, the absolute robust shortest path problem can be viewe as the 100 th -percentile robust shortest path, where the path reliability measure y(x) is minimize an y( x) c, x A A. Problem Formulation The α-percentile represents the value below which α percent of the observations (in ascening orer) may be foun. To represent the α-percentile travel time over a finite number of ays =1,,, let us first enote the sorte path travel times of a path on ifferent ays as T T T T. The 1 2 3 percentage α then correspons to the n th value an T n, where n. For example, when consiering α = 0.9 an = 100 sample ays, n=90. If turns out to be a floating point number, especially when the sample size is small, then the ran n can be obtaine by rouning to the nearest integer of the value of. To moel the α-percentile robust shortest path problem, we introuce the following formulation: Problem P2: z2 min y subject to c, x y Mw, (18) A w (1 ) (19) where M is a sufficiently large number an w is a binary variable for sample. When w is 0, then (18) reuces to A, y c x (20) an this active inequality shoul be hel for all ay-specific path travel times T less than the α-percentile travel time. When w =1, (18) leas to an always-feasible an inactive constraint c, x y M (21) A To mae sure (21) is vali for those sample ays on which T is greater than the final y * (i.e. the optimal α-percentile travel time), the parameter M shoul be sufficiently large. To ensure the robust path travel time measure y is larger than for a certain percentage of ays, the variable set w is constraine by (19). For example, for α = 0.9 an = 100, w (1 0.9) 100 10, so there are a total of 10 inactive constraints, an 90 active constraints. Because problem P2 nees to minimize the variable of y, the optimization result nees to select the n active constraints for travel time values on ifferent ays raning from smallest to largest. B. Illustrative Example Consier a single origin-estination pair with two parallel paths, as shown in Fig. 2. In this illustrative networ, both paths share a common lin A. Table I shows the lin an path travel times (min) over =4 sample ays. As calculate in this table, for the ARSP problem, path AB (along lins A an B) has a mini travel time of 12 min over four sample ays. On the other han, for a 75 th -percentile robust shortest path problem, each path can only have (1 75%) 4 1 ay of sample travel time greater than the variable of y. For path AB, only ay 4 has a w =1, leaing to its 75 th -percentile travel time as 11 min. Path AC nees to set w =1 on ay 3, leaing to its 75 th -percentile travel time as 10 min an therefore the optimal solution to the PRSP problem. Origin A Fig. 2. Networ of the illustrative example B C estimation
> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 6 TABLE I TRAVEL TIME CALCULATION OF THE ILLUSTRATIVE EXAMPLE (MIN) w for Lin Lin Lin Path Path Path A B C AB AC AB ay1 3 5 6 8 0 9 0 ay2 4 7 6 11 0 10 0 ay3 5 6 8 11 0 13 1 ay4 4 8 6 12 1 10 0 100 th percentile 75 th percentile 12 (ARSP solution) 11 13 10 (PRSP solution) w for Path AC C. Lagrangian Relaxation an ecomposition Following the same Lagrangian relaxation moeling framewor for the ARSP problem, we introuce a set of nonnegative Lagrangian multipliers µ an v to relax the inequality constraint set (18) an (19) into the objective function: min y c, x y Mw A w (1 ) (22) By re-grouping variables in (22), a Lagrangian ual problem is constructe with three sets of inepenent variables: Problem L2: L( 1,...,, ) min c, x A (23) 1 y M w v1 We then ivie the ual function into three inepenent subproblems. For notational convenience, we enote a constant variable 1 h v. L(,...,, ) L (,..., ) 1 x 1 L (,..., ) L (,...,, ) h y 1 w 1 (24) The first two subproblems in the ual function are ientical to subproblems SP1 an SP2 in the ARSP problem, an both can be solve efficiently. The thir part of the ual problem in (24) is a number of binary integer problems, each corresponing to a ay an a single variable w. Subproblem SP3: w 1 L (,...,, ) M w, (25) Proposition 2: For each univariate linear programming problem in subproblem SP3, the optimal value of variable w is etermine accoring to the given values of µ an v, i.e.: w 0 M 0 1 M 0 (26) Proof: The above proposition can be easily erive as subproblem SP3 is a univariate linear program with M as the cost coefficient, an variable w is boune by an interval of [0, 1].. Subgraient Metho Similar to the ARSP problem, we nee to improve the upper an lower bouns of the primal problem by iteratively imizing the ual problem in (23). The subgraient metho is implemente here with two sets of Lagrangian multipliers µ an ν. L( 1,...,, v) c,1x y Mw1, A (27)..., c, x y Mw, w 1 A an 1 c, x y Mw, A 1 w 1 (28) (29) A heuristic algorithm is use to upate the step-size set UB y L( 1,...,, ), (30) UB ( 1,...,, y L ) (31) E. Solution Proceure The overall algorithm for solving the α-percentile robust shortest path problem is escribe below. Algorithm 2: Step 1: Initialization Set iteration number = 0; Choose positive values to initialize the set of Lagrangian multipliers, µ an ν; Select initial values for M, y UB an y LB. Step 2: Solve ecompose ual problems Solve Subproblem SP1 using a stanar shortest path algorithm an fin a solution x; Solve Subproblem SP2 with (12) in Proposition 1 an fin a value for y; Solve Subproblem SP3 with (26) in Proposition 2 an fin values for w ; Calculate primal, ual an gap values, an upate the upper an lower bouns of the optimization problem P2. Step 3: Upate Lagrangian multipliers Upate Lagrangian multipliers with (27-31) Step 4: Termination conition test If K or the gaps are smaller than the preefine toleration gap, terminate the algorithm, otherwise go bac to Step 2.
> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 7 F. Illustrative Numerical Examples Now we apply the propose Lagrangian relaxation approach in Algorithms 1 an 2 to fin the ARSP an 75 th -percentile PRSP in the sample networ (Fig. 2). Tables II an III show some ey intermeiate computational results in the first few iterations of the search proceure. TABLE II RESULTS OF FIRST FEW ITERATIONS FOR ALGORITHM 1 Iteration μ1 μ2 μ3 μ4 y Lx1 Lx2 Lx Ly L LB UB gap Relative Gap 1 0.25 0.25 0.25 0.25 8 10.5 10.5 10.5 0 10.5 10.5 13 2.5 19% 2 0.5 0.75 1.5 0.75 13 37.75 39 37.75-32.5 5.25 10.5 12 1.5 12.5% 3 0.01 0.45 1.2 0.6 12 25.43 26.19 25.43-15.12 10.31 10.5 12 1.5 12.5% 4 0.01 0.34 1.09 0.6 12 22.96 23.6 22.96-12.42 10.54 10.54 12 1.465 12% 5 0.01 0.25 0.99 0.6 12 21.02 21.58 21.02-10.31 10.71 10.71 12 1.2892 11% 6 0.01 0.19 0.94 0.6 12 19.6 20.1 19.6-8.76 10.84 10.84 12 1.16 10% 7 0.01 0.14 0.89 0.6 12 18.51 18.95 18.51-7.57 10.94 10.94 12 1.06 9% TABLE III RESULTS OF FIRST FEW ITERATIONS FOR ALGORITHM 2 Iteration μ1 μ2 μ3 μ4 v M y Lx1 Lx2 Lx Ly w1 w2 w3 w4 Lw L LB UB gap Relative Gap 1 0.25 0.25 0.25 0.25 1.5 4 8 10.5 10.5 10.5 0 0 0 0 0-1.5 9 9 11 2 18% 2 0.25 0.85 0.85 1.05 1.3 4 11 33.3 32.3 32.3-22 0 1 1 1-8.4 1.9 9 10 1 10% 3 0.05 0.35 0.65 0.55 1.5 4 10 18 17.9 17.9-6 0 0 1 1-3.3 8.6 9 10 1 10% 4 0.01 0.35 0.575 0.25 1.575 4 10 13.26 13.57 13.26-1.85 0 0 1 0-2.3 9.11 9.11 10 0.89 9% 5 0.01 0.40 0.41 0.36 1.575 4 10 13.36 13.08 13.08-1.85 0 1 1 0-1.7 9.54 9.54 10 0.46 5% 6 0.01 0.31 0.39 0.36 1.6 4 10 12.09 11.85 11.85-0.69 0 0 0 0-1.6 9.56 9.56 10 0.44 4% 7 0.01 0.31 0.45 0.36 1.58 4 10 12.7 12.58 12.58-1.25 0 0 1 0-1.79 9.54 9.56 10 0.44 4% Aitional notations in Tables II & III: Lx1, Lx2: objective function values of subproblem SP1 for paths AB an AC, respectively. Lx: the cost of the shortest path foun in subproblem SP1. Ly: optimal value of subproblem SP2 at each iteration. Lw: optimal value of subproblem SP3 at each iteration. L: the value of the ual problem for each iteration. LB: lower boun of the solution, obtaine from the best ual value L. UB: upper boun of the solution, generate from the best primal value among the paths uncovere up to the current search iteration. Gap: the ifference between UB an LB. Relative Gap: as efine in (17). In the above two examples, by iteratively configuring weights μ on ifferent samples, the propose approach successfully uncovere the optimal solutions (Path AB for ARSP an Path AC for PRSP). Specifically, starting with uniform istribute values (1/4 = 0.25), the Lagrangian multipliers are ajuste to improve the lower boun of the optimal solution even after the optimal upper bouns have been achieve. It shoul be remare that, although the optimal solution was foun in both problems, a relative gap still exists ue to the approximation nature of the Lagrangian relaxation metho. In the propose subproblem SP3, the parameter M is inclue in the ualize objective function (25). Given its corresponing negative sign, a larger value of M coul lea to a lower value in the final optimal function for (25) an therefore a looser lower boun estimator. Thus, we nee to select a value for parameter M which is not only sufficiently large enough to mae inequality (21) vali, but also small enough to construct a tight lower boun for function (25). In the illustrative example, M is selecte to be 4 minutes so that it is larger than the imum value of the gap between 100% an 75% travel times, which is 3 minutes for Path AC. IV. NUMERICAL EXPERIMENTS In this section, numerical experiments are conucte on a large-scale real-worl transportation networ for the Bay Area, California, which is comprise of 53,124 noes an 93,900 lins. Specifically, 8,511 lins (9.1% of lins) of the entire networ are freeways with a total length of 1,774.8 miles (i.e., 15.8% of the total mileage), while 85,389 lins (90.9%) are arterial roas with a total length of 9,431.8 miles (84.2%). The algorithm is implemente in C# on the Winows Vista platform an evaluate on a personal computer with an Intel Core uo 1.8GHz CPU an 2 GB memory. The samples of lin travel time are calculate base on available historical recors from the NAVTEQ traffic atabase. In particular, 73 ays of travel time measurements between November 2009 an February 2010 are collecte for the time interval of 9:00 AM to 9:15 AM of each sample ay. As the observation ata use in this stuy (mainly from freeway segments) cover about 4.1% of the total mileage in the Bay Area, ranom sample travel times are generate for lins without ata coverage. For simplicity, this research oes not remove traffic ata from weeen ays an holiays. Fig.
Average Travel Time for All O Pairs (min) Average Travel Time for All O Pairs (min) Average Relative Gap (%) > REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 8 3 shows the test networ an its sensor ata coverage. As short-istance O pairs might be covere by no or inaequate raw observations, an they typically have very limite alternative routes to examine, this stuy imposes the following rules to select O pairs to be teste: (1) the average path travel time is larger than 45 minutes, an (2) the measurement coverage on the least expecte travel time path is larger than 30% in istance. the absolute robust path problem, which can be explaine by the ifference in the constructe ual objective functions, an in particular the aitional complexity in turning the parameter M for subproblem SP3. 18% ARSP 95% PRSP 14% 10% 6% 2% 0 10 20 30 40 50 Iterations Fig. 4. Relative gap for ARSP an 95% PRSP. 90 88 Upper Boun Lower Boun 84 80 Measurement Coverage 76 Fig. 3. Sensor ata coverage for Bay Area, California. Lins with measurements are highlighte. As a result, an O-pair set U containing u = 246 ranom O pairs is generate from the Bay Area networ. The performance of propose algorithms are assesse using the average relative gap, which is calculate as the average value of the relative gaps for all 246 O pairs uner a preefine ' imum number of iterations K, e.g. ( r, s) U ( r, s), K u Aitionally, the average objective function value of primal an ual problems among all O pairs are also use to emonstrate the improvement of the solution quality over the iterative proceure. As shown in Fig. 4, the average gap ecreases along with the increase of the preefine imum number of iterations K. Figs. 5 an 6 illustrate that, for both moels, after about 5 iterations, the reuction of upper boun becomes very slow, while the lower boun eeps improving. The average gap of the 95% percentile robust path problem is larger than that of. 72 0 10 20 30 40 50 Iterations Fig. 5. Upper an lower bouns evolution for ARSP. 84 80 76 72 68 0 10 20 30 40 50 Iterations Fig. 6. Upper an lower bouns evolution for 95% PRSP. Upper Boun Lower Boun
> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 9 Overall, our experiments inicate that 20 iterations are sufficient for both moels to achieve relatively small gap values, an the solution quality improvement begins to iminish after 10 iterations. It shoul be mentione that a small uality gap can still exist even when an optimal solution is foun, mainly ue to the inherent limitation of Lagrangian lower boun estimation techniques. V. EXTENSIONS AN CONCLUSIONS As an emerging research topic on moeling river route choice behavior an proviing reliability oriente route guiance, the robust shortest path problem is stuie in this paper with two moeling criteria: absolute robust shortest path an percentile robust shortest path. To reformulate the complex mini objective function in the ARSP problem, we applie a variable splitting an relaxation technique to generate a ual problem that provies tight lower bouns for the optimal solution. Furthermore, a subgraient metho is aopte in the solution proceure algorithm to iteratively improve both upper an lower bouns of the original problem. Along this line, the α-percentile robust shortest path problem is reformulate as a set of easy-to-solve subproblems by introucing auxiliary variables an aitional efinitional constraints. The comprehensive experiment results on a largescale networ with real-worl travel time measurements emonstrate that 10-20 iterations of stanar shortest path algorithms for the reformulate moels can offer a very small relative uality gap of about 3-6%. The moel presente in this paper assumes ay-epenent but time-invariant travel times. In our future research, one challenging tas is how to consier both absolute an percentile robust shortest path problems in stochastic an time-epenent networs. To capture traffic ynamics, one nees to first buil space-time expane networs, an then establish a complex multi-stage stochastic integer programming moel. In this case, a large number of nonanticipativity constraints nee to be incorporate to moel the first-stage path choice ecisions across ifferent scenarios. Intereste reaers are referre to a paper by Rocafellar an Wets [23] on the use of various Lagrangian formations to hanle scenario-base stochastic integer programming problems. Recently, Yang an Zhou [21] examine some alternative reformulation schemes to relax the nonanticipativity constraints as linear objective functions, so that the unerlying time-epenent shortest path problem can be solve using an efficient label-correcting algorithm by Ziliasopoulos an Mahmassani [24] on each sample ay. Our future research irections also inclue the following extensions: (1) incorporate ARSP an PRSP moels (for iniviual commuters) into the route choice component for networ-wie ynamic traffic assignment an flow management problems; an (2) evelop effective istribute or parallel computing techniques to improve the overall computational efficiency, as the propose subproblems can be solve inepenently. ACKNOWLEGMENT The authors woul lie to than the California Partners for Avance Transit an Highways (PATH) in the University of California, Bereley an NAVTEQ Corporate for proviing traffic measurement an map ata of the Bay Area, CA. REFERENCES [1] Cambrige Systematics, Traffic congestion an reliability: Trens an avance strategies for congestion mitigation. Technical report, Feeral Highway Aministration, Washington,.C., Sep. 2005. [2] Y. Nie, Multi-class percentile user equilibrium with flow-epenent stochasticity, Trans. Res. Part B, Article in Press. Available at OI: 10.1016/j.trb.2011.06.001. [3] F. Oronez an N. Stier-Moses, Warrop equilibria with ris-averse users, Trans. Sci. vol. 44, no.1, pp. 63-86, 2010. [4]. Bertsimas an M. Sim, Robust iscrete optimization an networ flows, Math. Program. Ser. B vol. 98, no. 1-3, pp. 49-71, 2003. [5] I. Murty an S. S. Her, Solving min- shortest path problems on a networ, Naval Res. Logistics vol. 39, pp. 669-683, 1992. [6] M. E. Bruni an F. Guerriero, An enhance exact proceure for the absolute robust shortest path problem, Intl. Trans. in Op. Res. Vol. 17, no. 2, pp. 207-220, 2010. [7] G. Yu an J. Yang, On the robust shortest path problem, Comput. Oper. Res, vol. 25, no. 6, pp. 457-468, 1998. [8] O. E. Karasan, M. C. Pinar an H. Yaman, The robust shortest path problem with interval ata, Woring paper. Available at http://www.bilent.eu.tr/~hyaman/robustsp.ps. [9] R. Montemanni an L. M. Gambarella, Robust shortest path problems with uncertain costs, ISIA / USI-SUPSI, Switzerlan, Report No. ISIA-03-08, 2008. [10] Y. Y. Fan, R. E. Kalaba anj. E. Moore, Arriving on time, J. Optimiz. Theory App. Vol. 127, no. 3, pp. 497-513, 2005. [11] Y. Nie an X. Wu, Shortest path problem consiering on-time arrival probability, Trans. Res. Part B, vol. 43, no. 6, pp 597-613, 2009. [12] K. A. Small, The scheuling of consumer activities: wor trips, Amer. Econ. Rev., vol. 72, no. 3, pp. 467-479, 1982. [13] R. B. Nolan, K. A. Small, P. M. Kosenoja an X. Chu, Simulating travel reliability. Reg. Sci. Urban Econ., vol. 28, no. 5, pp. 535-564, 1998. [14] R. B. Nolan an J. W. Pola, Travel time variability: a review of theoretical an empirical issues, Trans. Rev., vol. 122, no. 1, pp. 39-54, 2002. [15] C. E. Sigal, A. Alan, B. Pritser an J. J. Solberg, The stochastic shortest route problem, Oper. Res., vol. 28, no. 5, pp. 1122-1129, 1980. [16] Y. Chen, M. G. H. Bell, an K. Bogenberger, Reliable pretrip multipath planning an ynamic aaptation for a centralize roa navigation system, IEEE Trans. Intell. Transp. Syst, vol. 8, no. 1, pp. 14 19, 2007. [17] K. Joernsten an M. Naesberg, A new Lagrangian relaxation approach to the generalize assignment problem, Eur. J. Oper. Res., vol. 27, no. 3, pp. 313-323, 1986. [18] T. Larsson, A. Migalas an M. Ronnqvist, A Lagrangean heuristic for the capacitate concave minimum cost networ flow problem, Eur. J. Oper. Res., vol. 78, no. 1, pp. 116-129, 1994. [19] T. Xing an X. Zhou, Fining the most reliable path with an without lin travel time correlation: A Lagrangian substitution base approach, Trans. Res. Part B, Article in Press. Available at OI: 10.1016/j.trb.2011.06.004. [20] R. Seshari an K. K. Srinivasan, "Algorithm for etermining most reliable travel time path on networ with normally istribute an correlate lin travel times", Trans. Res. Rec., vol. 2196, pp. 83-92, 2010. [21] L. Yang an X. Zhou, Constraint reformulation an a Lagrangian relaxation-base solution algorithm for a least expecte time path problem, Submitte to Trans. Sci,. Available at http://www.civil.utah.eu/~zhou/least-expecte-time-path- Problem.pf. [22] R. K. Ahuja, T. L. Magnanti an J. B. Orlin, Shortest paths: labelsetting algorithms; Shortest paths: label-correcting algorithms, Networ Flow: Theory, Algorithms an Applications. N.J.: Prentice Hall, 1993, ch. 4 & 5, pp. 93-156.
> REPLACE THIS LINE WITH YOUR PAPER IENTIFICATION NUMBER (OUBLE-CLICK HERE TO EIT) < 10 [23] R. T. Rocafellar an R. J-B. Wets, Scenarios an policy aggregation in optimization uner uncertainty, Math. Oper. Res., vol. 16, no. 1, pp. 119-147, 1991. [24] A. K. Ziliasopoulos an H. S. Mahmassani, Time-epenent, shortestpath algorithm for real-time intelligent vehicle highway system applications, Trans. Res. Rec., vol. 1408, pp. 94-100, 1993. Tao Xing receive his B.Eng. egree in communication engineering from Being University of Posts & Telecommunications, Being, China, in 2005, an his M.Sc. egree in raio frequency communication systems from University of Southampton, Southampton, UK, in 2006. He is currently pursuing the Ph.. egree in transportation engineering with the epartment of Civil an Environmental Engineering at the University of Utah. His current research interests inclue real-time traveler information systems, multicriteria routing, real-time traffic estimation an preiction, an heterogeneous sensor location problems. Xuesong Zhou (M 07) receive his B.S. egree in Railroa Management from Being Jiaotong University, Being, China, in 1995, an his Ph.. egree in Civil Engineering from the University of Marylan, College Par, in 2004. He is currently an Assistant Professor in the epartment of Civil an Environmental Engineering at the University of Utah. His current research interests inclue analytical moeling of transportation systems, real-time ynamic traffic estimation, an large-scale routing an rail scheuling. He is the Co-Chair of the IEEE ITS Society Technical Committee on Traffic an Travel Management an serves as a Committee Member for TRB Committee on Transportation Networ Moeling (AB30).