Dynamic Vehicle Routing with Moving Demands Part I: Low speed demands and high arrival rates

Transcription

1 Dynamic Vehicle Routing with Moing Demands Part I: Low speed demands and high arrial rates Shaunak D. Bopardikar Stephen L. Smith Francesco Bullo João P. Hespanha Abstract We introduce a dynamic ehicle routing problem in which demands arrie uniformly on a segment and ia a temporal Poisson process. Upon arrial, the demands translate perpendicular to the segment in a gien direction and with a fixed speed. A serice ehicle, with speed greater than that of the demands, seeks to sere these translating demands. For the existence of any stabilizing policy, we determine a necessary condition on the arrial rate of the demands in terms of the problem parameters: (i) the speed ratio between the demand and serice ehicle, and (ii) the length of the segment on which demands arrie. Next, we propose a noel policy for the ehicle that inoles sericing the outstanding demands as per a translational minimum Hamiltonian path (TMHP) through the moing demands. We derie a sufficient condition on the arrial rate of the demands for stability of the TMHP-based policy, in terms of the problem parameters. We show that in the limiting case in which the demands moe much slower than the serice ehicle, the necessary and the sufficient conditions on the arrial rate are within a constant factor. I. INTRODUCTION Vehicle routing is concerned with planning optimal ehicle routes for proiding serice to a gien set of customers. In contrast, Dynamic Vehicle Routing (DVR) considers scenarios in which not all customer information is known a priori, and thus routes must be re-planned as new customer information becomes aailable. An early DVR problem is the Dynamic Traeling Repairperson Problem (DTRP) [1], in which serice requests, or demands arrie sequentially in a region and a serice ehicle seeks to sere them by reaching each demand location. In this two-part paper, we introduce a DVR problem in which the demands moe with a specified elocity upon arrial. This problem has applications in areas such as perimeter defense, wherein the demands are moing targets trying to cross a region under sureillance by a UAV. Another application is in the automation industry where the demands are objects arriing on a coneyor belt and a robotic arm seeks to perform a pick-and-place operation on them. The goal in the DTRP [1] is to minimize the expected time spent by each demand before being sered. In [1], the authors propose a policy that is optimal in the case of low arrial rate, and seeral policies within a constant factor of the optimal in the case of high arrial rate. In [2], they study multiple serice ehicles, and ehicles with finite serice capacity. This material is based upon work supported in part by ARO-MURI Award W911NF , ONR Award N and by the Institute for Collaboratie Biotechnologies through the grant DAAD19-3-D-4 from the U.S. Army Research Office. The authors are with the Center for Control, Dynamical Systems and Computation, Uniersity of California at Santa Barbara, Santa Barbara, CA 9316, USA, {shaunak,stephen,bullo}@engineering.ucsb. edu, hespanha@ece.ucsb.edu In [3], a single policy is proposed which is optimal for the case of low arrial rate and performs within a constant factor of the best known policy for the case of high arrial rate. In [4], decentralized policies are deeloped for the multiple serice ehicle ersions of the DTRP. The Euclidean Traeling Salesperson Problem (ETSP) consists of determining the minimum length tour through a gien set of static points in a region [5]. Vehicle routing with objects moing on straight lines was introduced in [6], in which a fixed number of objects moe in the negatie y-direction with fixed speed, and the motion of the serice ehicle is constrained to be parallel to either the x- or the y-axis. For a ersion of this problem wherein the ehicle has arbitrary motion, termed as the translational Traeling Salesperson Problem, a polynomial-time approximation scheme was presented in [7] to catch all objects in minimum time. [8] and [9] address other ersions of ETSP with moing objects. We introduce a dynamic ehicle routing problem in which demands arrie uniformly on a segment of length W, ia a temporal Poisson process with rate λ. Upon arrial, the demands translate in a fixed direction perpendicular to the line and with a fixed speed < 1. A serice ehicle, modeled as a first-order integrator with unit speed, seeks to sere these mobile demands. Our contributions are as follows. First, we derie a necessary condition on the arrial rate for the existence of a stabilizing policy, i.e., a finite expected time spent by a demand in the enironment. Second, we propose a noel policy which inoles sericing all of the outstanding demands as per a translational minimum Hamiltonian path (TMHP) through them. We derie a sufficient condition for stability of this TMHP-based policy, and also obtain an upper bound on the steady-state expected time a demand spends in the enironment. As the arrial rate λ +, the necessary stability condition implies that the demands must hae +. This regime of low demand speed is the focus of this paper. In this regime, we show that the sufficient stability condition for the TMHP-based policy is within a constant factor of the necessary condition for stability. In companion paper [1], we analyze a first-come-firstsered (FCFS) policy in which the demands are sered in the order of their arrial. We show that in the regime of λ +, the FCFS policy minimizes the expected time spent by a demand before being sered; while in the regime of 1, we show that eery stable policy must sere demands in the FCFS order, and hence FCFS is optimal. Thus, for low demand speeds the TMHP-based policy can stabilize higher arrial rates, while for high demand speeds the FCFS can stabilize higher arrial rates. This is summarized in Fig. 1.

2 4 Necessary condition for stability of any policy Sufficient condition for stability of the TMHP based policy Sufficient condition for stability of the FCFS policy p = (X,Y ) C = (x,y + T) 2 (, ) q = (x, y) 14 TMHP TMHP and FCFS FCFS W Fig. 2. Constant bearing control. The ehicle motion towards the point C := (x, y + T) minimizes the time taken to reach the demand q Fig. 1. A summary of stability regions for the TMHP-based policy and the FCFS policy. Stable serice policies exist only for the region under the solid black cure. In the top figure, the solid black cure is due to Theorem IV.1, the dashed blue cure is due to part (i) of Theorem V.1, and the red cure is described in [1]. In the asymptotic regime shown in the bottom left, the dashed blue cure is due to Theorem V.2, and is different than the one in the top figure. In the asymptotic regime shown in the bottom right, the solid black cure is described in [1], and is different from the solid black cure in the top figure. minimum Hamiltonian path (EMHP) is a Euclidean Hamiltonian path that has minimum length. In this paper, we also consider the problem of determining a constrained EMHP which starts at a specified start point, isits a set of points and terminates at a specified end point. More specifically, the EMHP problem is as follows. Gien n static points placed in R 2, determine the length of the shortest path which isits each point exactly once. An upper bound on the length of such a path for points in a unit square was gien by Few [13]. By mimicking the technique of Few, we can extend the bound to the case of points in a rectangular region, which is described in the following lemma (cf. [11] for proof). This paper is organized as follows: we begin with background results on the traeling salesperson problems in Section II. The problem formulation is presented in Section III. The necessary condition for stability is deried in Section IV. The TMHP-based policy and its stability results are presented in Section V. Simulation results are presented in Section VI. Due to space constraints, we include only a sketch of proofs for some intermediate results. The complete proofs are presented in [11]. II. PRELIMINARY RESULTS We use the following motion to reach a demand. Proposition II.1 (Constant bearing control, [12]) Gien the locations p := (X, Y ) E and q := (x, y) E at time t of the ehicle and a demand, respectiely, then the motion of the ehicle towards the point (x, y + T), where (1 2 )(X x) T(p,q) := 2 + (Y y) 2 (Y y) , minimizes the time taken by the ehicle to reach the demand. Constant bearing control is illustrated in Fig. 2. We now reiew seeral results on determining shortest paths through sets of points. A. The Euclidean Minimum Hamiltonian Path (EMHP) Gien a set of points in the plane, a Euclidean Hamiltonian path is a path that isits each point exactly once. A Euclidean Lemma II.2 (EMHP length) Gien n points in a 1 h rectangle in the plane, where h R >, there exists a path that starts from a unit length edge of the rectangle, passes through each of the n points exactly once, and terminates on the opposite unit length edge, haing length upper bounded by 2hn + h + 5/2. We will also require the following result on the length of a path through a large number of points. Gien a set Q of n points in R 2, the Euclidean Traeling Salesperson Problem (ETSP) is to determine the shortest tour, i.e., a closed path that isits each point exactly once. Let ETSP(Q) denote the length of the ETSP tour through Q. The following is the classic result by Beardwood, Halton, and Hammersly [14]. Theorem II.3 (Asymptotic ETSP length, [14]) If a set Q of n points are distributed independently and uniformly in a compact region of area A, then there exists a constant β TSP such that, almost surely, ETSP(Q) lim = β TSP A. n + n The constant β TSP has been estimated numerically as β TSP.712 ±.2, [15]. B. The Translational Minimum Hamiltonian Path (TMHP) Next, we describe the TMHP problem which was proposed and soled in [7]. This problem is posed as follows.

3 Gien initial coordinates; s of a start point, Q := {q 1,...,q n } of a set of points, and f of a finish point, all translating with the same constant speed and in the same direction, determine a path that starts at time zero from point s, isits all points in the set Q exactly once and ends at the finish point, and the length L T, (s, Q,f) of which is minimum. In what follows, we wish to determine the TMHP through points which translate in the positie y direction. We also assume the speed of the serice ehicle to be normalized to unity, and hence consider the speed of the points ], 1[. A solution for the TMHP problem is: (i) for ], 1[, define the map g : R 2 R 2 by ( x g (x, y) =, y ) (ii) Compute the EMHP that starts at g (s), passes through the set of points gien by {g (q 1 ),..., g (q n )} =: g (Q) and ends at g (f). (iii) Moe between any two demands using constant bearing control. The following result is established. Lemma II.4 (TMHP length, [7]) Let the initial coordinates s = (x s, y s ) and f = (x f, y f ), and the speed of the points ], 1[. The length of the TMHP is L T, (s, Q,f) = L E (g (s), g (Q), g (f)) + (y f y s ) 1 2, where L E (g (s), g (Q), g (f)) denotes the length of the EMHP with starting point g (s), passing through the set of points {g (q 1 ),..., g (q n )}, and ending at g (f). III. PROBLEM FORMULATION We consider a single serice ehicle that seeks to serice mobile demands that arrie ia a spatio-temporal process on a segment with length W along the x-axis, termed the generator. The ehicle is modeled as a first-order integrator with speed upper bounded by one. The demands arrie uniformly distributed on the generator ia a temporal Poisson process with intensity λ >, and translate with constant speed ], 1[ along the positie y-axis, as shown in Figure 3. We assume that once the ehicle reaches a demand, the demand is sered instantaneously. The ehicle is assumed to hae unlimited fuel and demand sericing capacity. (, ) (X(t), Y (t)) Fig. 3. The problem setup. The thick line segment is the generator of mobile demands. The dark circle denotes a demand and the square denotes the serice ehicle. W We define the enironment as E := [, W] R R 2, and let p(t) = [X(t), Y (t)] T E denote the position of the serice ehicle at time t. Let Q(t) E denote the set of all demand locations at time t, and n(t) the cardinality of Q(t). Sericing of a demand q i Q and remoing it from the set Q occurs when the serice ehicle reaches the location of the demand. A static feedback control policy for the system is a map P : E FIN(E) R 2, where FIN(E) is the set of finite subsets of E, assigning a commanded elocity to the serice ehicle as a function of the current state of the system: ṗ(t) = P(p(t), Q(t)). Let D i denote the time that the ith demand spends within the set Q, i.e., the delay between the generation of the ith demand and the time it is sericed. The policy P is stable if under its action, lim sup E[D i ] < +, i + i.e., the steady state expected delay is finite. Equialently, the policy P is stable if under its action, limsup t + E[n(t)] < +, that is, if the ehicle is able to serice demands at a rate that is on aerage at least as fast as the rate at which new demands arrie. In what follows, our goal is to design stable policies for the system. IV. A NECESSARY CONDITION FOR STABILITY In this section, we proide a necessary condition on the arrial rate for the existence of a stabilizing policy. We begin by stating the main result of the section, with the remainder of the section dedicated to its proof. Theorem IV.1 (Necessary condition for stability) A necessary condition for the existence of a stabilizing policy is that λ 4 W. To proe Theorem IV.1, we begin by looking at the distribution of demands in the serice region. Lemma IV.2 (Poisson point process) Suppose the generation of demands commences at time and no demands are sericed in the interal [, t]. Let Q denote the set of all demands in [, W] [, t] at time t. Then, gien a compact region R of area A contained in [, W] [, t], e λa ( λa) n P[ R Q = n] =, where n! λ := λ/(w). Proof: [Sketch] This proof requires the calculation of the probability that at time t, R Q = n, where R = [l, l+ l] [h, h+ h] (that is, the probability that the region R contains n points in Q). Since the generation process is temporally Poisson and spatially uniform, this is equal to the probability that the region [, l] [, h] contains n points in Q. After some simplifications, we obtain that the result is true for the aboe considered rectangular region. Finally, since eery measurable, compact region can be written as a countable union of rectangles, the result is established. Remark IV.3 (Uniformly distributed demands) Lemma IV.2 shows us that the number of demands in an unsericed region is Poisson distributed with rate

4 λ/(w), and conditioned on this number, the demands are distributed uniformly. We now establish a result on the expected time to trael from a demand to its nearest neighbor. Lemma IV.4 (Trael time bound) Consider the set Q of demands in E at time t. Let T d be a random ariable giing the minimum amount of time required to trael to a demand in Q from a ehicle position (X, Y ), selected a priori. Then E[T d ] 1 W 2 λ. Proof: Using Proposition II.1, we can write the trael time T from an a priori ehicle position p := (X, Y ) to a demand location q := (x, y) implicitly as T(p,q) 2 = (X x) 2 + ((Y y) T(p,q)) 2. (1) Thus, any demand in S T, where S T := {(x, y) E : (X x) 2 + ((Y T) y) 2 T 2 }, can be reached from (X, Y ) in T time units. In general, the area of S T satisfies S T πt 2. By Lemma IV.2 the demands in an unsericed region are uniformly randomly distributed with density λ = λ/(w). Thus, for eery ehicle position p chosen before the generation of demands, P[T d > T] = P[ S T Q = ] e λ S T e λπt2 /(W). Therefore, E[T d ] = + P[T d > T]dT + π 2 λπ/(w) = 1 W 2 λ. e λπt2 /(W) dt We can now proe Theorem IV.1. Proof: [Proof of Theorem IV.1] A necessary condition for the stability of any policy (see, for example [1]) is that λe[t] 1, where E[T] is the steady-state expected trael time between demands i and i + 1. For eery policy E[T] E[T d ] 1 W 2 λ. Thus a necessary condition for stability is that λ 1 W 2 λ 1 λ 4 W. V. THE TMHP-BASED POLICY AND ITS STABILITY In this section, we present the TMHP-based policy for the ehicle along with the sufficient condition for its stability. The TMHP-based policy is summarized in Algorithm 1, and an instance of the policy is illustrated in Fig. 4. The TMHP-based policy gies the following result. Algorithm 1: TMHP-based policy Assumes: Serice ehicle has initial position (X, Y ), and all demands hae lower y-coordinates. 1 if no outstanding demands in the enironment then 2 Moe towards the generating line for a time interal of Y/(1 + ). 3 else 4 Let V be a irtual demand located at (X, ) translating with speed in the positie y-direction. 5 Serice all the outstanding demands by following a TMHP starting from (X, Y ), and terminating at irtual demand V. Use the constant bearing control to moe between demands. 6 Repeat. (X, Y ) W Fig. 4. The TMHP-based policy. The ehicle seres all outstanding demands inside the shaded rectangular region R(X, Y ) as per the TMHP that begins at (X, Y ) and terminates at the irtual demand V. Theorem V.1 (Stability of TMHP-based policy) (i) TMHP-based policy is stable if λ < (1 2 ) 3/2 2W(1 + ) 2, and, V The (ii) assuming that the TMHP-based policy is stable, the steady state expected time spent by a demand in the enironment is upper bounded by ( ) 5W /(1 + ). 2Wλ/(1 2 ) 3/2 Proof: Let R(X, Y ) denote the region [, W] [, Y ] defined by the position (X, Y ) of the serice ehicle, as shown in Fig. 4. Obsere that at the end of eery iteration of this policy, all outstanding demands hae their y-coordinates less than or equal to that of the ehicle, and hence would be contained in R(X, Y ). Let the ehicle be located at p(t i ) = (X(t i ), Y (t i )) at time instant t i. If there are no outstanding demands in R(X(t i ), Y (t i )), then Y (ti) 1+ is the distance that the ehicle moes towards the generator. Thus, we hae Y (t i+1 ) = Y (t i ) Y (t i) 1 + = Y (t i) 1 +, if there are no unsericed demands in R(X(t i ), Y (t i )) at time t i. Otherwise, for any unsericed demands {q 1,...,q ni } where n i 1, in R(X(t i ), Y (t i )), Y (t i+1 ) = L T, (p(t i ), {q 1,...,q ni }, V (t i )),

5 where L T, (p(t i ), {q 1,...,q ni }, V (t i )) is the time taken for the ehicle as per the TMHP that begins at p(t i ), seres all n i demands and ends at the irtual demand V (t i ). Since the distribution of the demands inside R(X(t i ), Y (t i )) is spatially Poisson (cf. Lemma IV.2 from Section IV), Y (t i+1 ) = Y (t i) 1 +, w.p. e λa, = L T, (p(t i ), {q 1 }, V (t i )), w.p. ( λa)e λa, = L T, (p(t i ), {q 1,q 2 }, V (t i )), w.p. ( λa) 2 e λa, 2! and so on, where A = WY (t i ) is the area of R(X(t i ), Y (t i )). We now seek an upper bound for the length L T, (p(t i ), {q 1,...,q ni }, V (t i )) of the TMHP for which we use the method from Section II-B. For n i = k 1, inoking Lemma II.4 and writing Y i := Y (t i ) and Q k := {q 1,...,q k }, L T, (p, Q k, V ) = L E (g (p), g (Q k ), g (V )) Y i 1 2 2WY i k (1 2 ) 3/2 + Y i W 2 1 2, where the second equality is due to y V (ti) =, and the inequality is obtained using Lemma II.2. Thus, we hae E[Y i+1 Y i ] Y i e λa ( 2WY i k (1 2 ) + Y i 3/ W ) ( λa) k 2 e λa, 1 2 k! k=1 where λ = λ/w from Lemma IV.2. Collecting the terms with Y i /(1 + ) together, we obtain E[Y i+1 Y i ] Y i 1 + ( λa) k e λa + k! k= ( 2 2 WY i k 5W ) ( λa) + (1 2 ) 3/2 2 k 1 k=1 2 k! Y i W Yi E [ ] 5W ni Y (1 2 ) 3/2 i Y i = Y i = Y i e λa 2 2 W Yi E[ni Y (1 2 ) 3/2 i ] + 5W W λwyi Yi (1 2 ) 3/2 W + 5W W λ (1 2 ) 3/2 Y i + 5W 2 1, 2 where the inequality in the third step follows by applying Jensen s inequality to the conditional expectation and the equality in the fourth step is due to the arrial process is Poisson with rate λ and for a time interal Y i /. Using the law of iterated expectation, we hae E[Y i+1 ] = E[E[Y i+1 Y i ]] 1 + E[Y 2λW i] + (1 2 ) E[Y 5W i] + 3/2 2 1, (2) 2 which is a linear recurrence in E[Y i ]. Thus, lim i + E[Y i ] is finite if Wλ (1 2 ) < 1 λ < (1 2 ) 3/2 3/2 2W(1 + ) 2. Thus, if λ satisfies the condition aboe, then expected number of demands in the enironment is finite and the TMHP-based policy is stable. The upper bound in part (ii) follows since the recurrence Eq. (2) is linear. A. Limiting Case of Low Speed Demands In this section, we focus on the case when λ + and, by the necessary stability condition in Theorem IV.1, +. Recall that for this case, the sufficient stability condition for the TMHP-based policy is that λ < 1/(2W). This differs by a factor of 8 from the policy independent necessary stability condition of λ < 4/(W). By utilizing the tight asymptotic expression for the length of the TSP path, gien in Theorem II.3, in place of the bound in Lemma II.2, we can reduce this factor to approximately 2. To begin, consider an iteration i of the TMHP-based policy, and let Y i > be the position of the serice ehicle. In the limit as +, the length of the TMHP at the ith iteration equals the length of the corresponding static EMHP as described in Lemma II.4. Further, in the limit as λ +, the number of outstanding demands in that iteration n i +, and hence the length of the EMHP tends to the length of the ETSP through all outstanding demands at the end of the iteration. Thus, applying Theorem II.3, the position of the ehicle at the end of the iteration is gien by Y i+1 = β TSP ni A = β TSP ni Y i W, where A := Y i W is the area of the region below the ehicle at the ith iteration. Conditioned on Y i, E[Y i+1 ] = β TSP E [ Wn i Y i ] βtsp WYi E[n i ], where we hae applied Jensen s inequality. Using Lemma IV.2, E[n i ] = WY i λ/(w) and thus E[Y i+1 Y i ] β TSP W 2 Y 2 i Thus, we arrie at the following result. λ W = β TSP λwyi. Theorem V.2 (TMHP stability for low speeds) In the limit as +, a sufficient condition for stability of the TMHP-based policy is λ < β 2 TSP W W.

6 VI. SIMULATIONS In this section, we present a numerical study to determine stability of the TMHP-based policy. We numerically determine the region of stability of the TMHP-based policy, and compare it with the theoretical results from the preious sections. The linkern 1 soler was used to generate approximations to the TMHP at eery iteration of the policy. The linkern soler takes as an input an instance of the Euclidean Traeling salesperson problem. To transform the constrained EMHP problem into an ETSP, we replaced the distance between the start and end points with a large negatie number, ensuring that this edge was included in the linkern output. For a gien alue of (, λ), we begin with 1 demands in the enironment and determine the ehicle s aerage y coordinate at the end of the iteration. If it exceeds the y coordinate at the beginning of the iteration, then that particular data point of (, λ) is classified as being unstable; otherwise, it is stable. The results of this numerical experiment are presented in Figure 5. For the purpose of comparison, we oerlay the plots for the theoretical cures, which were presented in Figure 1. We obsere that the numerically obtained stability boundary for the TMHP-based policy falls between the necessary and the sufficient conditions for stability, and is well approximated by the cure established by Theorem V.2 for almost the entire interal of ], 1[ Fig. 5. Numerically determined region of stability for the TMHP-based policy. A lightly shaded (green-coloured) dot represents stability while a darkly shaded (blue-coloured) dot represents instability. The uppermost (thick solid) cure is the necessary condition for stability for any policy as deried in Theorem IV.1. The lowest (dashed) cure is the sufficient condition for stability of the TMHP-based policy as established by part (i) of Theorem V.1. The broken cure between the two cures is the sufficient stability condition of the TMHP-based policy in the low speed regime as established in Theorem V.2. The enironment width is W = 1. VII. CONCLUSIONS AND FUTURE DIRECTIONS We introduced a ehicle routing problem in which a serice ehicle seeks to sere demands that arrie ia a Poisson process on a line segment and that translate with a fixed speed in a direction perpendicular to the line. For the existence of a stabilizing policy, we first deried a necessary condition on the arrial rate of the demands as a function of the speed ratio between the demands and the ehicle, and the length of the line segment. Then, we proposed a noel serice policy for the ehicle which inoles sericing all the outstanding demands as per a TMHP through the translating demands. We deried a sufficient condition on the arrial rate of the demands for stability of the TMHP-based policy. In the limiting case of the relatie speed tending to zero, we showed that the necessary and the sufficient conditions for stability are within a constant factor. In the companion paper [1], we analyze the first-come-first-sered (FCFS) policy and show that in the regimes of high demand speeds, the policy is stable for higher arrial rates than the TMHP-based policy. Further, we show that in the high demand speed regime, the FCFS is an optimal policy. In future, we plan to address ersions of the present problem inoling multiple serice ehicles, and also with non-uniform spatial arrial of the demands. This extension has been completed for the placement problem. REFERENCES [1] D. J. Bertsimas and G. J. an Ryzin, A stochastic and dynamic ehicle routing problem in the Euclidean plane, Operations Research, ol. 39, pp , [2], Stochastic and dynamic ehicle routing in the Euclidean plane with multiple capacitated ehicles, Operations Research, ol. 41, no. 1, pp. 6 76, [3] J. D. Papastarou, A stochastic and dynamic routing policy using branching processes with state dependent immigration, European Journal of Operational Research, ol. 95, pp , [4] E. Frazzoli and F. Bullo, Decentralized algorithms for ehicle routing in a stochastic time-arying enironment, in IEEE Conf. on Decision and Control, Paradise Island, Bahamas, Dec. 24, pp [5] B. Korte and J. Vygen, Combinatorial Optimization: Theory and Algorithms, 3rd ed., ser. Algorithmics and Combinatorics. Springer, 25, ol. 21. [6] P. Chalasani and R. Motwani, Approximating capacitated routing and deliery problems, SIAM Journal on Computing, ol. 28, no. 6, pp , [7] M. Hammar and B. J. Nilsson, Approximation results for kinetic ariants of TSP, Discrete and Computational Geometry, ol. 27, no. 4, pp , 22. [8] C. S. Helig, G. Robins, and A. Zelikosky, The moing-target traeling salesman problem, Journal of Algorithms, ol. 49, no. 1, pp , 23. [9] Y. Asahiro, E. Miyano, and S. Shimoirisa, Grasp and deliery for moing objects on broken lines, Theory of Computing Systems, ol. 42, no. 3, pp , 28. [1] S. L. Smith, S. D. Bopardikar, F. Bullo, and J. P. Hespanha, Dynamic ehicle routing with moing demands Part II: High speed demands or low arrial rates, in American Control Conference, St. Louis, MO, Jun. 29, to appear. [11] S. D. Bopardikar, S. L. Smith, F. Bullo, and J. P. Hespanha, Dynamic ehicle routing for translating demands: Stability analysis and receding-horizon policies, IEEE Transactions on Automatic Control, Mar. 29, submitted. [12] R. Isaacs, Differential games. John Wiley, [13] L. Few, The shortest path and the shortest road through n points in a region, Mathematika, ol. 2, pp , [14] J. Beardwood, J. Halton, and J. Hammersly, The shortest path through many points, in Proceedings of the Cambridge Philosophy Society, ol. 55, 1959, pp [15] G. Percus and O. C. Martin, Finite size and dimensional dependence of the Euclidean traeling salesman problem, Physical Reiew Letters, ol. 76, no. 8, pp , The TSP soler linkern is freely aailable for academic research use at