Modified Banker s algorithm for scheduling in multi-agv systems

Transcription

1 2011 IEEE International Conference on Automation Science and Engineering Trieste, Italy - August 24-27, 2011 ThC3.1 Modified Banker s algorithm for scheduling in multi-agv systems Luka Kalinovcic, Tamara Petrovic, Stjepan Bogdan, Vedran Bobanac Abstract In today s highly complex multi-agv systems key research objective is finding a scheduling and routing policy that avoids deadlock while assuring that vehicle utilization is as high as possible. It is well known that finding such an optimal policy is a NP-hard task in general case. Therefore, big part of the research is oriented towards finding various suboptimal policies that can be applied to real world plants. In this paper we propose modified Banker s algorithm for scheduling in multi-agv systems. A predetermined mission s path is executed in a way that some non-safe states are allowed in order to achieve better utilization of vehicles. A graph-based method of polynomial complexity for verification of these states is given. Algorithm is tested on a layout of a real plant for packing and warehousing palettes. Results shown at the of the paper demonstrate advantages of the proposed method compared with other methods based on Banker s algorithm. I. INTRODUCTION Introduction of automated guided vehicles (AGVs) into automated materials handling systems, flexible manufacturing systems and containers handling systems has brought many notable advantages, but it also requires the use of effective on-line supervisory control strategies able to solve potential problems that arise in such a multi-agv system layout. Control of AGVs includes mission assignment, routing and scheduling and it can be performed either on-line or off-line. Off-line planning requires the knowledge of all tasks prior to execution of an control algorithm, while on-line (sometimes referred to as dynamic) planning allows new tasks to appear although the planning has been already done. AGV routing is focused on the spatial dimension of the problem, that is, determination of paths in the space domain, while scheduling provides deadlock free execution of the path. Which strategy will be used deps on various elements. For example, missions can be assigned to AGVs according to given priority. The notion of preemption is another issue that should be considered; once a mission is assigned to an AGV, question is whether it can be interrupted before it is finished. Also, positioning of idle AGVs, in most cases ignored by the researchers, in practice plays a very important role. Various methods for AGV routing and scheduling are currently in use [1-4], but new methods which employ faster and computationally more efficient algorithms are still the subject of intensive research. Very good reviews of the field can be L. Kalinovcic, T. Petrovic and S. Bogdan are with the Laboratory for Robotics and Intelligent Control Systems, Department of Control and Computer Engineering, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia stjepan.bogdan@fer.hr, tamara.petrovic@fer.hr V. Bobanac is with the Laboratory for Renewable Energy Sources, Department of Control and Computer Engineering, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia vedran.bobanac@fer.hr found in [5, 6, 7]. Many modern AGVs are free-ranging, i.e. instead of using a predefined path, the layout vehicles can travel with no restrictions as long as the collisions with other vehicles and obstacles are avoided. Usually such approach is based on heuristic-behavior and fuzzy algorithms [8, 9, 10]. However, due to very restrictive security issues, traditional centralized decision-making structures still dominate in the industrial implementations. In [11] authors propose a deadlock resolution by using a technique frequently adopted for vehicle management in AGV systems, so called the zone control. Authors allow the route of an AGV to change dynamically, as long as any path assigned to a vehicle s in the docking station. The control algorithm performs a simple one-step look-ahead policy to check if the configuration resulting from the vehicle movement corresponds to a deadlock situation or not. In [12] two types of AGVs are modeled by Petri nets; centralized - an AGV is exclusively assigned to a job until the job is completed, and decentralized - an AGV is shared by multiple jobs. In [13] authors use Colored Resource - Oriented Petri nets for the lower level logical control in an AGV system. There proposed method achieved better results on the typical case study system compared to others that are currently in use. However, its application is in general case of exponential complexity. A different approach to routing and scheduling is used in [14]. Author models the system as a Resource Allocation system (RAS) and uses a scheduling policy that is based on standard Banker s algorithm. Deadlocks are efficiently avoided, while AGVs move on paths that are not priory fixed (as long as the scheduling policy allows it). Property of such a system, which is direct consequence of the applied Banker s algorithm, is that at any allowed state there must exist an ordering of vehicles (missions) that guarantees sequential, one vehicle at a time, termination of all active missions. Main drawback is unavailability to resolve some potential livelocks. The objective of our work was to develop a scheduling algorithm that would, when compared to previous algorithms, provide an increase in vehicle utilization. The proposed modified Banker s algorithm does that by exting the set of states that system is allowed to achieve, while traveling the shortest fixed distance and ensuring that all missions can be safely executed. It is important to emphasize that the method is of low computational complexity and very simple to implement in real-world plants. Work described here is an extension of research presented in [15]. The paper is organized in the following way. First, we define key terms used throughout the paper. Then, a thorough /11/$ IEEE 351

2 description of scheduling policy based on the modified Banker s algorithm for AGV systems is given together with formal proofs of system stability. The proposed policy is tested by simulation and compared to other strategies based on Banker s algorithm. Results are presented at the of the paper. II. A MULTI-AGV PLANT LAYOUT REPRESENTATION Usually, when it comes to the mathematical analysis and the design of supervisory control algorithm, a layout of a considered multi-agv plant can be represented by a graph whose intersections and s of paths are referred to as graph nodes, while paths themselves are represented by arcs. Let us use for this purpose an undirected graph H = (N, A), which contains a set of nodes, N = {n 1, n 2,..., n p } and a set of arcs, A = {a 1, a 2,..., a q }. Three different types of nodes are present in the system: idle nodes (battery charging) I i, workstation nodes (palletization cell, manufacturing cell, warehouse drop off) W i, and crossroads nodes C i. Set of available vehicles is denoted as R. We make the following assumptions: 1) a vehicle can reside only on arcs, 2) only one vehicle at the time is allowed to occupy an arc (if paths are lengthy or wide, they can be segmented into a set of arcs), 3) each vehicle has its idle node, 4) vehicles are not allowed to drive backwards nor make U-turns, except on workstation positioning arcs. In the proposed approach a mission, starting and ing on an idle positioning arc (an arc with dead- in idle node), is executed by a vehicle that visits workstation nodes in particular given order, I i W k W l... I i. Once mission is finished, the vehicle resides on the idle positioning arc, waiting for a new mission assignment. Hence, mission m i is defined as m i = (σ r, r), where σ r is a mission path, and r R is a vehicle assigned to mission m i. A path σ r = a p a q a s... a p of length σ r, is defined as a sequence of arcs that connect visited nodes, with σ r (j) representing j-th arc of σ r. For each vehicle r we define its arc index k r such that k r = i means that r occupies i-th arc of σ r. For example, in system depicted in Fig. 1 [14], we can define mission m 1 = (σ 2, 2) as I 2 W 2 W 1 I 2. The mission is executed by vehicle 2 with σ 2 = and σ 2 (6) = 13. In case vehicle is positioned on arc 13 for the second time, we have k 2 = 6. A set of active paths, Σ = {σ 1, σ 2,..., σ l }, is defined as the set of all paths currently assigned to missions, and set K = {k 1, k 2,..., k l }, is defined as the set of all arcs indexes. Thus, the state of the system is uniquely defined with the set of active paths and the set of occupied arcs, S = (Σ, K). The state of the system S 0 = (Σ, K) is called initial if k r K : k r = 1. For the final state of the system, S F = (Σ, K), one has k r K : k r = σ r. Fig. 1: An example of muli-agv system Definition 1: (allowed state) State S = (Σ, K) is allowed if p, q R, p q : σ p (k p ) σ q (k q ). In other words, state S is allowed if each vehicle resides on a different arc. It should be noted that by definitions the initial S 0 and the final S F states are allowed. Following definitions describe how the system state can be changed. Definition 2: (state change by vehicle movement) Let S = (Σ, K) and S = (Σ, K ) be two states and let there exists r such that { 1 r R, k i ki, i r = k i + 1, i = r then we say that state S is changed to state S movement of vehicle r, denoted as S r S. Definition 3: (state change by new mission assignment) Let S = (Σ, K) and S = (Σ, K ) be two states and let there exists r such that { 1 r R, k r = σ r, k i ki, i r = 1, i = r i r : σ i = σ i then we say that state S is changed to state S by new mission assignment to vehicle r, denoted as S r S. Now we can define properties of the system states. Definition 4: (deadlock state) State S = (Σ, K) is deadlock state if there is no vehicle r such that S r S and S = (Σ, K ) is allowed state. Evidently, no allowed movements of vehicles are possible once system gets in deadlock state. As a consequence, missions cannot be completed and system is not able to reach final state. Definition 5: (safe state) State S = (Σ, K) is safe state if there exists a sequence of vehicles movements, r 1 r 2... r M, such that S r 1 S r 2 S... r M S F. by 352

3 In other words, state is safe if there is at least one scenario of vehicle moves such that all active missions can be successfully completed. Except for safe and deadlock states there is a third, remaining group of states from which system is inevitably going to up in deadlock, regardless of the control policy. These states are commonly referred to as higher level deadlocks [11]. III. SCHEDULING IN MULTI-AGV SYSTEM Prior to execution of scheduling policy, routing algorithm has to assign a path to each newly requested mission. Usually the shortest distance path is assigned to the mission, although, sometimes the shortest distance does not necessarily mean the shortest traveling time as on some parts of the floor shop vehicles are required to travel very slow due to curves or security reasons. There are many known methods for calculation of paths of the graph representing the system layout [16,17]. Some approaches allow the path to change dynamically [14]. As the routing in multi-agv systems is out of the scope of this paper, in a case study presented at the of the paper set Σ is comprised of predetermined shortest distance paths. A problem of scheduling in multi-agv system that we are interested to solve is formalized as follows: having defined initial state of the system S 0 = (Σ, K) and final state of the system S F = (Σ, K) find a scheduling algorithm such that transition from S 0 to S F is free of deadlock states. There exists a trivial solution to this problem; a scheduling algorithm which executes one mission at a time, i.e. one vehicle is moving while all other vehicles remain in their idle positioning arcs. Once the first vehicle finishes its mission second mission is started and so on. Clearly, performance of such an algorithm, in sense of vehicle utilization, is very poor. However, direct consequence of this algorithm is the following corollary. Corollary 1: The initial state S 0 = (Σ, K) is safe. According to Definition 3. state transition can be initiated by a new mission assignment. As the proposed method allows dynamic assignment of mission (once a vehicle is positioned on idle positioning arc), in the next lemma we show that such action does not influence state safety. Lemma 1: Let S = (Σ, K) is a safe state and let S = (Σ, K ) is such that S r S. Then S is a safe state. Lemma 1 is important as it states that under any scheduling policy new mission assignment does not influence the system stability as long as each vehicle has its idle node. System in safe state shall remain in safe state upon new mission assignment. Conclusion that can be drawn from Corollary 1 and Lemma 1 is that system can up in a deadlock state only by vehicle movement. Hence, each time a vehicle request allocation of next arc on its mission s path, scheduling policy should allow it if and only if the resulting state is safe. Fig. 2: An example However, in the general case, detection of state safety is NP-complete problem [18]. Therefore, minimally restrictive control policy, where only deadlock and higher level deadlock states are forbidden and only safe permitted, is computationally intractable and inapplicable in real world. On the price of sacrificing optimality, today s scheduling algorithms allow only those safe states that can be verified in polynomial time. One of such commonly used scheduling algorithms is standard Banker s algorithm, which is described in the following paragraph. In the remainder of the paper term Banker s algorithm will be used instead of standard Banker s algorithm. A. Banker s algorithm in multi-agv systems First thing in defining Banker s algorithm is to identify system resources and processes, which use a certain number of resources. In multi-agv systems one associates resources with arcs and processes with missions (vehicles). In such scenario a vehicle, that executes particular mission, requests allocation of next arc on its path, solely if that arc is not occupied by some other vehicle. Banker s algorithm then checks if this allocation will allow all vehicles to finish their missions in a sequential manner, one by one. If the new state is proved to be safe, requested arc shall be granted to the vehicle and vehicle continues with execution of the assigned mission. In case state is not proved to be safe, movement will be forbidden. Let us examine a system shown in Fig. 2 There are 4 vehicles, 6 workstation nodes and 4 idle position nodes. Layout comprises 24 arcs (it should be noted that numbers in Fig. 2 represent arcs IDs, not weights). Vehicles 1, 3 and 4 are returning to their idle positioning arcs, while vehicle 2 is moving towards workstation W 1 and then back to its idle node I 2. Vehicle 1, as the vehicle that is closest to an intersection, requests arc 13. Using Banker s algorithm we will check the safety of that consequent state, with vehicle 1 positioned on arc 13. Formally, we first define all resources that a particular process already holds (allocates). For system in Fig.2 vehicle 1 holds arc 13, vehicle 2 holds arc 15, vehicle 3 holds arc 8 and vehicle 4 holds arc 17. Further, we define resources 353

4 needed by processes to be finished. Let us suppose that vehicles are using shortest paths, then vehicle 1 needs arcs 1, 5, 15,vehicle 2 arcs 2, 5, 7, 15, vehicle 3 arcs 3, 6, 8, 14, 17 and vehicle 4 needs arcs 4, 6. The outcome of Banker s algorithm, is a sequence describing order in which missions can be carried out, one by one. That sequence is not necessary unique. In our case sequences are, for example (m i here denotes mission of vehicle i), m 2 m 1 m 4 m 3 and m 4 m 2 m 1 m 3, hence, current state is safe. For example, m 1 m 2 m 4 m 3 is not a valid sequence for Banker s algorithm since mission m 1, that is first in the sequence, cannot be finished with no movement of other vehicles in the system (vehicle 2 in particular). Here should be noted that vehicles are not going to necessarily execute missions as given in the output sequence. From the scheduling point of the view it is only important to know that such an option exist. In the remainder of the paper this kind of missions sequence will be called BA-sequence and the associated state BA-safe state (in the literature these states are sometimes referred to as ordered [14]). Formal procedure for determination of system BA-safety is as follows. First, one has to form directed wait-for graph [19]: G = (N g, A g ), N g = {n 1, n 2,..., n n }, and a ij = (n i, n j ) A g iff vehicle j occupies an arc that is on the path needed by vehicle i to finish its mission. As an example of graph formation let us consider situation depicted in Fig. 2. Vehicle 2 occupies arc 15 which is part of the path required for vehicle 1 to complete its mission, hence, in the graph there is an edge from node n 1 to node n 2. Vehicle 4 occupies arc 17. Since this arc belongs to the path executed by vehicle 3, there is an edge from node n 3 to node n 4. Vehicles 2 and 4 have open paths and as such they have no output arcs. Fig. 3: Wait-for graph for system in Fig. 2 If a node has no output arcs, its outdegree is equal to zero. That means that its mission path is not occupied by any other vehicle and the vehicle corresponding to the node can advance to its final arc. The correlation between wait-for graph of state S and its BA-safety is given as: Theorem 1: State S = (Σ, K) is BA-safe if and only if wait-for graph G = (N g, A g ) is acyclic. Proof : Proof of biconditional (P Q) is split into two stages: i)(q P ): If G is acyclic, then state S is BA-safe. Let us assume that graph G is acyclic. Since every acyclic directed graph has at least one node with outdegree zero, the vehicle corresponding to that node can safely finish its mission with no movement of other vehicles. Hence, its node can be removed from the graph. The graph remains acyclic, still having at least one node with outdegree zero. If we continue with this procedure, all nodes will be removed. These nodes, in the exact order of their removal, form a BA sequence, that is, S is BA-safe. ii)( Q P ): If G has at least one cycle, then state S is not BA-safe. Let us assume that graph G contains a cycle of nodes: n 1 n 2... n k n 1. None of the nodes belonging to that cycle has outdegree equal to zero. That means none of vehicles that correspond to the cycle nodes will be able to finish their missions with no movement of other vehicles in the system, that is, BA-sequence cannot be formed. In the text that follows we define an algorithm for checking of BA-safety based on Theorem 1. Algorithm sequentially removes nodes with outdegree zero from wait-for graph. Algorithm terminates either if all nodes have been removed (state is BA-safe) or if there are no nodes with outdegree zero (graph has a cycle and state is not BA-safe). For graph G with n nodes algorithm complexity is O(n 2 ). Algorithm 1: BA-SAFETY CHECK (G = (N g, A g)) output; % state is BA-safe while ( n i N g : outdegree(n i ) = 0) N g = N g\n i if N g = output := true else output := false B. Modified banker s algorithm in multi-agv systems In general, set of BA-safe states is a subset of a wider set of safe states that can eventually reach final state. Scheduling policy that allows for system to be only in a BA-safe state often results in low vehicle utilization and long vehicle waiting times. For example, let us examine the situation depicted in Fig. 4. Vehicle 1 has just visited workstation W 3 and it is returning to idle node I 1 by using arcs 14, 13, 15, 5 and 1. In the same time vehicle 2 has a mission m 2 (visit workstations W 1 and W 3 ) such that σ 2 = {2, 5, 15, 7, 13, 14, 9, 14, 13, 15, 5, 2}. In its current position vehicle 2 requests arc 5. It can be shown, by execution of Banker s algorithm, that a new state obtained by allocation of arc 5 to vehicle 2 is not BA-safe since vehicle 1 cannot finish its mission. Arc 5 is allocated by vehicle 2, and vehicle 2 is not able to proceed due to allocation of arc 14 by vehicle 1 - deadlock. Hence, Banker s algorithm would prevent vehicle 2 to proceed to arc 5. Let us, nevertheless, continue to observe what happens if vehicle 2 proceeds to arc 5 and then requests allocation of arc 15. As with the previous arc, result is not BA-safe state. We continue with the previous reasoning and now vehicle 2 requests arc 354

5 Fig. 4: An example of temporarily BA-unsafe state 7 which is next on the path. This time state of the system is BA-safe. We can summarize described scenario as follows: although allocation of arc 5 to vehicle 2 leads system to temporarily BA-unsafe state, since there exist free pathway from arc 5 to arc 7 and since the system is BA-safe when vehicle 2 resides on arc 7, granting of arc 5 to vehicle 2 is allowed and would not cause the system deadlock. Let us denote this kind of temporarily BA-unsafe state as Modified Banker s safe state. Formal definitions are given as follows: Definition 6: (Modified Banker s (MBA) sequence) Sequence r 1 r 2... r M is called MBA sequence if there exist k, 1 k M such that: i) (i, j), 1 i < j k : r i = r j and ii) (i, j), k < i < j M : r i = r j r i = r i+1 =... = r j For example, sequence is MBA sequence (k = 2), while is not (k = 2, property ii) does not hold for i = 3 and j = 7). Definition 7: State S = (Σ, K) is MBA-safe state if there exists MBA sequence r 1 r 2... r M, such that S r 1 S r 2 S... r M S F. In other words, state S is MBA-safe if by moving of a single vehicle on its path, system can reach BA-safe state and thus can reach final state. Proposition 1: Every BA-safe state is also a MBA-safe state. Algorithm for checking of MBA-safety uses previously defined wait-for graph and Algorithm 1, with certain modifications. Let us assume that vehicle r requests allocation of next arc on its mission s path. First, we associate to vehicle r an integer value that is equal to the maximum number of arcs that r can transverse on its path without being intercepted by any other vehicle. Let us denote this value as e max. For system in Fig. 4, when vehicle 2 requests arc 5, e max = 4, since vehicle 2 can move freely along arcs 5, 15, 7 and 13. Further, for each j-th position of the vehicle, 1 j e max, we construct wait-for graph and using Algorithm 1 check whether that particular state is BA-safe. If during this process BA-safe state is reached, we conclude that moving the vehicle on the requested arc (j = 1) results in MBAsafe state and the procedure is terminated. Otherwise, state is not MBA nor BA safe. For system with n active missions (vehicles) algorithm complexity is O(e max n 2 ). It should be noted here that, if the procedure terminates for exactly j = 1, then state is BA (and MBA) - safe. Described procedure is formally given as Algorithm 2. Algorithm 2: MBA-SAFETY CHECK (r, e max ) using: BA-SAFETY CHECK if e max = 0 return(false) else for j = 1 to e max construct G j = (N g, A g ) %r moves forward for j arcs if G j G j 1 if BA-SAFETY CHECK (G j ) return (true) return (false) The scheduling policy is given in the following Theorem: Theorem 2: Let multi-agv system be in state S = (Σ, K) and let vehicle r requests arc a. The system is deadlock free if allocating a to r leads to a new state, S = (Σ, K ), S r S, which is MBA-safe. Proof: Initial state of the system is MBA-safe according to Corollary 1. Further, state can be changed only in to ways: if a new mission is assigned to vehicle, and if vehicle (upon request for arc allocation) advances to the next arc on path. New mission assignment results in MBA-safe state (Lemma 1). Let us assume that control policy is such that enables only those arc allocation requests that result in MBA-safe state. Under such policy, all reachable system states are MBA-safe. Since definition of MBA-state implies that system can always reach its final state, we conclude that under the given control policy system is deadlock free. IV. SIMULATION RESULTS The proposed multi-agv scheduling algorithm has been extensively tested on various layouts of factory floors. Here we present results obtained for the layout depicted in Fig. 5; a layout has 4 idle nodes, 8 workstations and 34 arcs. It replicates layout from a real plant for packing and warehousing palettes. Tests were done in the following way. A set of

6 Fig. 5: A case study layout predefined missions consisting of three to eight workstations for each vehicle was defined and executed in the same order for standard Banker s algorithm (BA), for Reveliotis variation of Banker s algorithm (RBA) [14] and for here described modified Banker s algorithm (MBA). For RBA, which determines vehicle s path dynamically, performance based policy was designed to prefer deadlock-free path with least traveling distance to next workstation on mission s path. Obtained results are shown in Table I. For each vehicle average traveled distance, average mission execution time and average vehicle waiting time in seconds have been calculated. TABLE I: CASE STUDY RESULTS Vh.1 Vh.2 Vh.3 Vh.4 Avg. BA traveling RBA distance [m] MBA Avg. miss. BA execution RBA time [s] MBA Avg. miss. BA waiting RBA time [s] MBA It can be seen that average travel distances for BA and MBA are the same (since both algorithms execute the shortest path) and lower than in RBA. Moreover, total traveling times for MBA system are lower than those for BA and RBA. In summary, obtained results show that MBA algorithm, when compared to other two, provides faster mission execution with least traveled distance, that is, least energy consumption. Since RBA algorithm allows a vehicle to chose an alternative (longer) path if the shortest one is not safe, RBA results with the lowest vehicle waiting time. Although results show clear advantages of MBA algorithm over RBA, finding mathematical framework that would allow exact comparison of these two methods for general system layout is still an open question. V. CONCLUSION A new scheduling algorithm for multi-agv systems has been proposed. Algorithm is based on Banker s algorithm, which is modified to allow unsafe system states under specific circumstances in order to decrease unnecessary vehicle waiting times and mission execution times. Algorithms for verification of system safety are based on graphs and are of polynomial complexity with respect to number of missions. Results obtained by simulation of four vehicles working in a factory layout demonstrate significant improvement of resource utilization compared with the other variations of Banker s algorithm for AGV systems. Currently, authors are working on the extension of proposed method to the systems with mission priorities. REFERENCES [1] A. J. Broadbent, C. B. Besant, S. K. Premi, and S. P. Walker, Free ranging AGV Systems: Promises, Problems and Pathways, Proc. of the 2nd Int l Conf. on Aut.Mat.Hand, pp , May 1985, Birmingham. [2] S. C. Daniels, Real-time Conflict Resolution in Automated Guided Vehicle Scheduling, Ph.D. thesis, 1988, Dept. of Industrial Eng., Penn. State University, USA. [3] G. Desaulniers, A. Langevin, and D. Riopel, Dispatching and conflictfree routing of automated guided vehicles: an exact approach, Int l J. of Flex. Man. 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