In automated logistic systems Automated Guided Vehicles (AGVs) are used for transportation tasks. To deal with the interaction in such an AGV system one needs efficient and intelligent routing on the one hand and collision avoidance on the other. Obviously, AGV routing is an online routing problem (nothing is known about future requests) and even a real-time problem, because fast answers are required. A route should be computed in less than a second

Gawrilow, Ewgenij

Köhler, Ekkehard

Möhring, Rolf H.

Stenzel, Björn

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Conflict-free Real-time AGV RoutingRolf H. Möhring, Ekkehard Köhler, Ewgenij Gawrilow, andBjörn StenzelTechnische Universität Berlin, Institut für Mathematik, MA 6-1,Straße des 17. Juni 136, 10623 Berlin, Germany{moehring,ekoehler,gawrilow,stenzel}@math.tu-berlin.deAbstract. We present an algorithm for the problem of routing Automated GuidedVehicles (AGVs) in an automated logistic system. The algorithm avoids collisions,deadlocks and livelocks already at the time of route computation (conflict-free rout-ing). After a preprocessing step the real-time computation for each request consistsof the determination of a shortest path with time-windows and a following readjust-ment of these time-windows. Both is done in polynomial-time. Using goal-orientedsearch we get computation times which are appropriate for real-time routing. Ad-ditionally, in comparison to a static routing approach, used in Container TerminalAltenwerder (CTA) at Hamburg Harbour, our algorithm had an explicit advantage.1 IntroductionNowadays automation in logistic systems is very popular. In such an au-tomated logistic system Automated Guided Vehicles (AGVs) are used fortransportation tasks and the control of these AGVs is the key to an efficienttransportation system. Usually, the aim is to maximize the throughput.Here, control means computation of routes (routing) on the one hand andcollision avoidance on the other hand. The assignment of transportation tasksto AGVs is not part of this work.Various aspects of the considered problem are of importance. The routesmust be computed in appropriate time (for a real-time computation) withrespect to the physical properties of the AGVs. The collision avoidance par-ticularly has to deal with the dimensions of the AGVs. In our approachcollisions are avoided already at the time of route computaion. This is calledconflict-free routing.One of the first paper on routing (free-ranging) AGVs without causingcollisions was done by Broadbent et al. [3] in 1987. Krishnamatury, Battaand Karwan [9] discussed the problem for the special case when all requestsare known right from the beginning.We consider the online problem where requests appear sequentially andone must answer each request without having any information about later ar-riving requests. We extend the approaches of Huang, Palekar and Kapoor [7]and Kim and Tanchoco [8], respectively. In particular, we take physical prop-erties of the AGVs into consideration in a more exact and flexible way andpresent an efficient algorithm for the problem.2 Rolf H. Möhring et al.2 The ModelLet G = (V, A) be a directed graph which represents the lanes of the auto-mated transportation system and let τ(a) be the transit time on arc a. Thenan online routing algorithm has to deal with a sequence σ = r1, . . . , rn ofrequests. Each request rj = (sj , tj , θj) represents a task with start node sjand end node tj while θj denotes the desired starting time.2.1 Problems with static routing approachesIn static approaches for this problem one computes a route without takingtime dependencies into consideration. This can be done by standard shortestpath algorithms, i.e., the Dijkstra algorithm (see [4]). In such an approachcongestion can be considered by additionally using load dependent arc costs(see [1]). The arising routes are not conflict-free and therefore one needs anadditional conflict management at real-time. To guarantee that there are nocollisions the moving AGVs must allocate the area they want to use next.The advantage of this approach is clear: it is easy to implement a fastrouting algorithm. But the disadvantages of the described collision avoid-ance, namely the appearance of deadlocks and livelocks (see Fig. 1), have anenormous effect on the performance of the system.Deadlock LivelockFig. 1. Deadlocks and LivelocksDeadlocks appear if two or more AGVs wish to allocate the same area.None is able to continue its route and the system is blocked. Livelock is thegeneric term for situations where an AGV is blocked repeatedly by otherAGVs without having the possibility of allocating the area which is next onits route.2.2 Conflict-free routingIn order to avoid the problems of the simple model given in Section 2.1, wecompute conflict-free routes because in a conflict-free approach there is noneed for an additional collision avoidance.There are two key ingredients which must be considered in that approach.On the one hand, one has to deal with the physical dimensions of the AGVsConflict-free Real-time AGV Routing 3because they usually have to claim several arcs in the directed graph atthe same time. On the other hand, the approach has to be time-dependent(dynamic).To avoid conflicts we use polygons P (a) for each arc a which describe theblocked area when an AGV (the center of an AGV) is on arc a. Thus, it isprohibited to use two arcs at the same time if the corresponding polygonsintersect each other. For each arc a, this leads to a set confl(a) of so calledgeographic-dependent arcs which should not be used at the same time.The time-dependent behavior is modelled by time intervals. If an AGVtravels over an arc a during the interval [t1, t2], all geographic-dependentarcs are blocked from t1 to t2. This leads to time-windows for each arc ofthe directed graph G. These time-windows represent the times when arc ais free. After routing a request one has to readjust these time-windows ac-cording to the used arcs and their geographic-dependent arcs. If this is doneconsequently, one does not have to take care of the AGV dimension in routecomputation since it is already considered.3 The AlgorithmThe algorithm consists of a preprocessing step and (for each request) a routecomputation followed by a readjustment of the appropriate time-windows.3.1 PreprocessingIn a preprocessing step all polygons P (a) are compared pairwise. If two poly-gons P (a) and P (b), for arcs a, b ∈ A(G), intersect, a is assigned to confl(b)and b is added to confl(a).Additionally, in this preprocessing step the behavior of the AGVs is mod-elled by a list OUT (a) of arcs, containing the set of arcs, which are allowedto be used after arc a respecting the physical properties of the AGV.3.2 Route computation: shortest paths with time-windowsAs pointed out in Section 2.2, the route computation can be done in anidealized model where the dimension of the AGV is not considered explicitly.One just has to compute a route for an infinitesimal point which representsthe center of the AGV.This problem is known as Shortest Path Problem with Time-Windows(SPPTW) [5,10]: Given are a graph G, a source node s, a destination nodet, a start time θ, transit times τa, costs ca and a set of time-windows Faon each arc a. The aim is to compute a shortest path (concerning costs c)respecting the given time-windows. Here ”respecting” means that AGVs waitand traverse the corresponding arc only during (open) time-windows.4 Rolf H. Möhring et al.The SPPTW is NP-hard. The hardness can be shown by reduction ofthe of the Constrained Shortest Path Problem (CSPP, see [2]).1Our algorithm for this problem is a generalized arc-based Dijkstra algo-rithm which deals with labels. A label L = (aL, dL, IL, predL) on an arc aLconsists of a distance value dL, a predecessor predL and a time interval IL.Each label L represents a path from start node s to the tail of aL, whereasdL contains the overall transit time (travelling and waiting together) and thelabel interval IL = (AL, BL) represents the possible arrival times at arc aL(at the tail of aL). predL is the predecessor of aL on that path.We define an order for these labels.Definition 1 A label L dominates a label L′ if and only ifdL ≤ dL′ and IL′ ⊆ IL.The labels are stored in a priority queue H (a binary heap for example).The generalized arc-based Dijkstra algorithm works as follows.• Initialization.Create a label L = (a, 0, (θ,∞), nil) for all outgoing arcs a of s and addthem to the priority queue H .• Loop.Take the label L with lowest distance value dL from H . If there is nolabel left in the queue, notify that there is no path from s to t. If t is thetail of aL, write the corresponding path to output.– For each time-window on aL.∗ Label Expansion.Try to expand the label interval along aL through the currenttime-window (new label interval should be as large as possible,see Fig. 2), add the costs c(aL) to the distance value and set thepredecessor. If there is no possible expansion, consider the nexttime-window.∗ Dominance Test.Add the new label to each outgoing arc a in OUT (aL), if it is notdominated by any other label on a.In general, the algorithm cannot be executed in polynomial-time, unlessP 6= NP . But in our setting we have special costs on the arcs: the transittimes of the paths including the corresponding waiting times. In that casewe get a polynomial-time algorithm, because costs (distance values) correlatewith the lower bounds of the label intervals.Theorem 2 In the case of transit times (including waiting times) as costfunction, the described generalized arc-based Dijkstra algorithm solves theSPPTW in polynomial-time.1 The instance of the SPPTW is constructed by placing time-windows [0, R] ateach arc while R denotes the resource constraint in the CSPP instance.Conflict-free Real-time AGV Routing 521 1 096581010 10210 10 10 10 100 0 0 0 021 1 092651081010 10 10 10210 100 0 0 0 0(a) (b)21 1 092651081054810 10 10 10210 100 0 0 0 021 1 092651010545810 10 10 10810 100 0 00082(c) (d)Fig. 2. Label Expansion. The label intervals are represented by grey bars (nodes).The blockings are colored black (arcs). The white intervals between these blockingsare the time-windows. The figures (a) to (d) show the successive expansion of thelabel intervals.Proof. The algorithm computes all required paths since the expansion of thelabel intervals is maximal and no required path (label) will be dominated.Consider the correlation between costs and transit times: they differ justby an constant additive, namely the starting time. Thus, for two labels thatare expanded via the same time-window the distance value controls the dom-inance relation. That means, that in this case a label dominates another ifand only if it has a lower distance value.Therefore, the number of labels on an arc a is bounded by the numberof time-windows on all ingoing arcs. Thus, the number of iterations in eachloop is either bounded by the number of arcs or bounded by the number oftime-windows. Hence, the algorithm terminates in polynomial-time with ashortest path (or the notification that there is no path). utWe can directly conclude for the SPPTW:6 Rolf H. Möhring et al.Corollar 3 The SPPTW with transit time (including waiting times) as costfunction can be solved in polynomial-time.For additional acceleration of the algorithm we use goal-oriented search [6].3.3 Readjustment of the time-windowsFor each arc a of a computed route we consider all geographic dependent arcsin confl(a) during the time interval [t1, t2] representing the transit over a.Then we verify for each time-window on an arc of confl(a), whether the time-window and [t1, t2] overlap. Depending on that, the time-window is shortened,erased or left unchanged.4 Computational ResultsTwo questions certainly arise concerning the presented conflict-free approach.Is the approach really better than the static one? Is the algorithm suitable forreal-time computation? Concerning these questions we present some resultsfor test instances (scenarios) on a graph with about 30.000 arcs.4.1 Comparison with a static approachIn order to measure the performance of the computed routes we consider thesum of all transit times (all requests). We compare these overall transit timeswith a static approach used in Container Terminal Altenwerder (CTA) atHamburg Harbour.The comparison shows that the conflict-free approach is superior to thestatic one. The strongest dominance of our algorithm is achieved for scenarioswith heavy traffic (many AGVs) which clearly indicates the potential of theconflict-free approach.4.2 Computation timesBesides obtaining a better solution in comparison with other approaches,a real-time computation requires fast answers. The computation times2 inTable 1 show that our algorithm is able to provide this. The table is dividedinto computation times for determining a shortest path with time-windows(search) and computation times for readjusting the time-windows (blocking).On average, the computation (in both cases) take not more than somehundredth of a second. The entire real-time computation (search and blockingtogether) in all tested cases takes less than half a second which is suitable forthis real-time computation.2 Hardware: AMD-Athlon 2100+ (1,7 Mhz) with 512 MB RAM.Conflict-free Real-time AGV Routing 7Table 1. Computational times (in sec.)Scenarios Search Blockingmaximal  maximal 1A (15 AGVs) 0.21 0.018 0.19 0.0262A (15 AGVs) 0.14 0.020 0.17 0.0383A (20 AGVs) 0.22 0.021 0.20 0.0304A (44 AGVs) 0.21 0.019 0.19 0.0221B (32 AGVs) 0.18 0.022 0.18 0.0242B (38 AGVs) 0.28 0.021 0.17 0.0251C (22 AGVs) 0.09 0.021 0.13 0.0422C (44 AGVs) 0.17 0.023 0.09 0.0293C (48 AGVs) 0.17 0.022 0.12 0.040References1. Aspnes, J. et al. (1997) On-line routing of virtual circuits with applications toload balancing and machine scheduling. Journal of the ACM 44, 486–5042. Beasley, J. E., Christofides, N. (1989) An algorithm for the resource constrainedshortest path problem. Networks 19, 379–3943. Broadbent, A. J. et al. (1987) Free-ranging agv and scheduling system. In Au-tomated Guided Vehicle Systems, 301–3094. Cormen, T. H., Leiserson, Ch. E., Rivest, R. L. (1990) Introduction to Algo-rithms. MIT Press, Cambridge, Massachusetts5. Desrosiers et al. (1986) Methods for routing with time windows. European Jour-nal of Operations Research 23, 236–2456. Hart, P., Nilsson, N., Raphael, B. (1968) A formal basis for the heuristic de-termination of minimum cost paths. In IEEE Transactions on Systems, Scienceand Cybernetics SCC-4, 100–1077. Huang, J., Palekar, U. S., Kapoor, S. (1993) A labeling algorithm for the nav-igation of automated guided vehicles. Journal of engineering for industry 115,315–3218. Kim, Ch. W., Tanchoco, J. M. A. (1991) Conflict-free shortest-time bidirectionalagv routeing. International Journal of Production Research 29(12), 2377–23919. Krishnamurthy, N., Batta, R., Karwan, M. (1993) Developing conflict-free routesfor automated guided vehicles. Operations Research 41, 1077–109010. Sancho, N. G. F. (1994) Shortest path problems with time windows on nodesand arcs. Journal of mathematical analysis and applications 186, 643–648

Conflict-free Real-time AGV Routing

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Conflict-free Real-time AGV Routing

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