Multi-Robot Task Scheduling

Transcription

1 Proc of IEEE Internationa Conference on Robotics and Automation, Karsruhe, Germany, 013 Muti-Robot Tas Scheduing Yu Zhang and Lynne E Parer Abstract The scheduing probem has been studied extensivey in the iterature Many agorithms have been deveoped to operate with different types of processors and tass In the robotics domain, when considering each robot as a processor, some of these agorithms can be directy adapted However, most of the existing agorithms can ony hande singe-robot tass, or muti-robot tass that can be divided into singerobot tass As the tas requirements may ony be partiay nown, and the avaiabe (heterogeneous) robots can dynamicay change, robots may be required to cooperate tighty to share different capabiities (ie, sensors and motors) In such cases, considering scheduing for individua robots is no onger sufficient, since the robots need to wor at the coaition eve Athough there exist a few agorithms that aso support these more compex cases, they do not represent efficient soutions that can be adapted by various muti-robot systems in a convenient manner In this paper, we propose heuristics to address the muti-robot tas scheduing probem at the coaition eve, which hides the detais of robot specifications, thus aowing these heuristics to be incorporated straightforwardy These heuristics are easy to impement and efficient enough to run in rea time We provide forma anayses and simuation resuts to demonstrate and compare their performances I INTRODUCTION The genera scheduing probem is often formuated as the probem of arranging a set of tass on a set of processors Here, processor represents a generic term that can assume any form, such as a CPU, an airpane, or a robot Aong with the objective function, a scheduing probem is often specified as three components separated by These three components, written together as P T func, can be used to specify the type of processors, tass, and the objective function For more detaied discussions of the different scheduing probems that have been studied, refer to [1], [3] To study the scheduing probems in robotic systems, one must first reaize that these probems represent specia cases of the genera scheduing probem In particuar, the scheduing probems in robotic systems are often characterized by the foowing restrictions: 1) Execution is assumed to be non-preemptive; ) Robots (and capabiities) are assumed to be non-divisibe, such that the same robot can ony wor on a singe tas at any time point (thus restricting the robots to be singe-tas robots [4]) The first restriction hods especiay in muti-robot tass, since it is often costy to regroup robots and bring them into new coaitions [15]; the second restriction is often due to geographica distances between different tas execution ocations [13] Consequenty, This materia is based upon wor supported by the Nationa Science Foundation under Grant No Yu Zhang and Lynne Parer are with the Distributed Inteigence Laboratory in the Department of Eectrica Engineering and Computer Science, University of Tennessee, Knoxvie, TN 37996, USA, {yzhang51,parer}@eecsutedu agorithms in this paper are deveoped under these two restrictions, athough preemptive tas execution and mutitas robots can sometimes be beneficia Furthermore, tas dependencies (eg, precedence orders) are not considered However, note that the agorithms proposed in this paper can be extended when tas dependencies are present, for exampe, using simiar approaches as in [10], [16] The soution bounds in these situations need to be re-estabished Foowing the above discussion, a simpe approach to sove the scheduing probems in robotic systems is to use existing agorithms for the restricted probem instances There are two cases to consider The simper case is scheduing with singe-robot tass, which can be specified as R func This probem with the objective function e, can be soved in poynomia time using a soution for the assignment probem [1] Here, R represents unreated machines, since the times required for robots to accompish the same tas can be unreated (or independent of their capabiities); is the index for tass and e denotes the finishing time of tas t The more compex case is with muti-robot tass Sometimes, when the muti-robot tass can be divided into singe-robot tass, the scheduing probem reduces to R e For situations in which this is infeasibe, the probem becomes the MPM MPT e probem (muti-purpose machine with mutiprocessor tas) As a specia case of MPM MPT e, the MPT probem is shown in [6] to be N P-hard, even with ony two machines, ie, MPT e Unfortunatey, very few agorithms are provided to address this very difficut probem As muti-robot systems deveop, we need efficient agorithms to generate acceptabe soutions fast in dynamic environments In this paper, we propose four heuristics to address the MPM MPT e probem To the best of our nowedge, this is the first wor that provides efficient heuristics for this probem with provabe soution bounds Furthermore, since these heuristics directy operate at the coaition eve, they can be easiy incorporated in a centraized or distributed manner using a ayering technique [1] The coaition information is provided by the underying ayer, in which robots search for coaitions and compute their costs based on the tas requirements and robot capabiities (see [14] for an exampe) This information is then submitted to the upper ayer (eg, our heuristics) for tas scheduing This paper is organized as foows After a brief discussion of the reated wor in Section II, we formuate the probem in Section III We introduce and anayze two simpe heuristics in Section IV, and two more compex ones in Section V The compexity anayses are provided in Section VI Finay, we present resuts in Section VII and concude in Section VIII

2 II RELATED WORK It is first noted in [4] that the muti-robot tas scheduing probem is equivaent to the MPM MPT e probem, athough [4] does not assume singe-tas robots This probem is often addressed by considering it as composed of two probems: an assignment probem and a MPT scheduing probem The assignment probem determines the assignment of tass to coaitions; the MPT scheduing probem then determines the tas execution schedues Whie the assignment probem can be soved optimay using the Hungarian agorithm [7], as we have discussed, the MPT probem is N P- hard To reduce the compexity, simpifying assumptions are often made, such as requiring the same processing times for tass, restricting the number of processors, or aowing preemptive scheduing However, the soution quaity is often difficut to anayze after combining soutions of the assignment and the MPT probems On the other hand, the MPT probem is often formuated more generay as the RCPSP (resource-constrained project scheduing probem) probem [3] The RCPSP probem reduces to the MPT probem when each robot is considered as one unique resource, such that neither the robots nor their capabiities are divisibe (ie, singe-tas robots) The corresponding MPM MPT e probem is then referred to as the muti-mode RCPSP probem Many agorithms have been provided to address the RCPSP probem Reviews of different methods can be found in [], [5] For exampe, mixed integer inear programming [8], as we as branch and bound agorithms [9], have been appied Some of these agorithms have been extended to the muti-mode case [11] However, none of these extensions represent efficient soutions with provabe soution bounds Athough the number of coaitions can be exponentia in the number of robots, in muti-robot systems, the number of executabe coaitions [15] is often imited These coaitions are associated with assignments that do not have any unsatisfied preconditions Consequenty, the number of coaitions that need to be considered for any given tas is aso imited This observation motivates the design of some of the heuristics in this paper Furthermore, a desirabe heuristics must be convenient to be incorporated One soution is to use a ayering technique [1], which requires the agorithms to directy operate at the coaition eve III PROBLEM FORMULATION We now present the formuation of the muti-robot tas scheduing probem, or MPM MPT e Given: A set of tass T = {t 1, t, }, and a set of robots R; A set of distinct coaitions C = {c 1, c, }, c j R; A function f : C T R +, which returns the time, p j, required for a coaition c j to accompish a tas t ; The scheduing probem we consider in this paper is: Minimize: e (1) subject to the foowing restrictions: 1) Non-preemptive scheduing; ) Non-divisibe robot resources and capabiities; Note that in this formuation, a coaition may be abe to accompish mutipe tass and a tas may be accompished by mutipe coaitions Again, e denotes the finishing time of tas t The reason that we are interested in e is because we often want to reduce the overa system operating time, incuding robot waiting or ide times, as they too consume resources The second restriction adds interference between different coaitions, which prohibits coaitions sharing the same robots to wor on different tass at the same time A MinProcTime IV SIMPLE HEURISTICS MinProcTime chooses at every step the assignment (ie, coaition-tas pair) that has the shortest processing time This process creates an ordering of tas assignments; we assume that ties are broen in a deterministic manner For scheduing assignments, MinProcTime aways assigns the eariest starting times The intuition behind MinProcTime is to process tass with shorter processing times earier Theorem 41: The MinProcTime heuristic yieds a soution quaity bounded by Proof: MinProcTime chooses an assignment for one remaining tas in T at each step unti a tass are assigned Given the greedy choice, when choosing a tas t, the heuristic necessariy chooses the coaition that can accompish t in the shortest time Denote this processing time as We further suppose that tass are ordered non-decreasingy based on, which is aso the order for MinProcTime to choose tass Then, the soution produced by MinProcTime must satisfy: S g (C, T ) ( T + 1) () in which S g is used throughout to represent the soutions of our heuristics (g stands for greedy, since there are aways some greedy criteria invoved) This inequaity hods because, in the worst case, tass woud be accompished in a sequentia manner by their respective coaitions On the other hand, the soution of the optima soution ceary satisfies: S (C, T ) (3) which happens ony when a tass are processed in parae Given the non-decreasing ordering, we now that when Combining Equations () and (3) yieds: S g (C, T ) T + 1 ( T + 1) T + 1 S (C, T ) (4) Hence, the concusion hods Athough the reaxations in the proof seem to be pessimistic, the bound is actuay tight; it is not difficut to create probem instances with the worst case soution quaity

3 B MinStepSum Simiar to MinProcTime, MinStepSum aways assigns the eariest starting time when considering an assignment It then chooses at every step the assignment that contributes the east to the objective function We again assume that ties are broen in a deterministic manner Note that whie the chosen assignments and their ordering are pre-determined in MinProcTime given the probem instance, they are determined incrementay (ie, one assignment at each greedy step) in MinStepSum Athough the two greedy heuristics operate differenty, interestingy, they share the same soution bound We introduce the foowing emma before proving this resut, in which we sti assume that the tass are ordered non-decreasingy based on Lemma 4: Denoting the finishing time of the th tas chosen by MinStepSum as e, we have that: max e (5) =1 =1 Proof: We prove this resut by an induction on Suppose that the inequaity hods up to At step + 1, MinStepSum chooses an assignment that contributes the east to the objective function There are two cases Case 1, when a tass in {t 1,, t } are aready assigned: since t is not assigned in this case, the east we can do is to have it accompished by the coaition with processing time In such a case, we have: max e max e + =1 =1 pmin =1 (6) Case, when not a tass in {t 1,, t } are assigned: since tass are ordered non-decreasingy based on, the east we can do is to have any unassigned tas t h (h ) accompished In such a case, we have: max e max e + =1 =1 pmin h max e + =1 pmin (7) Finay, since the induction ceary hods for = 1, we have that the concusion hods Theorem 43: The MinStepSum heuristic yieds a soution quaity bounded by Proof: Since we now that S g (C, T ) = e, the resut foows directy from Lemma 4 and Equation (4) Simiary, it can be shown that the bound in Theorem 43 is aso tight Athough the number of tass can be arge, as we see in the simuation resuts section, both MinProcTime and MinStepSum produce soutions with quaity much better than this bound on average =1 V COMPLEX HEURISTICS One imitation of MinProcTime and MinStepSum is that they do not consider the interference between coaitions Hence, they cannot mae an informed decision when two assignments have simiar processing times but different infuences on other coaitions However, a coaition can directy or indirecty infuence other coaitions in compex ways The heuristics in this section, nevertheess, assume that a coaition can ony infuence a imited number of coaitions, given the observations in [13], [15] A InterfereAssign The idea that motivates InterfereAssign is the fact that the scheduing probem with singe-robot tass can be recast as an assignment probem, and hence can be soved optimay To add in the effects of the interference, we first define the notion of interference: Definition 51: Coaition Interference For any two coaitions c j and c j (j j ), c j interferes (or conficts) with c j if and ony if c j c j This definition appies simiary to assignments Given this definition, we denote F j as the set of coaitions that interfere with c j Observe that an assignment c j t can increase the objective function by the foowing three factors: 1) The assignment s processing time p j ; ) The tass that are schedued on c j after t ; 3) The tass that are schedued on any coaitions in F j 1 Denoting the scheduing position for t on c j as I j, the first and second factors above together contribute I j p j The third factor can be estimated by the number of tass that are schedued on F j Athough this number is not nown a priori, we now that for each coaition c, the number of tass that are schedued on c must be no greater than N c, in which N c represents the set of tass that c can accompish Hence, the third factor must be no greater than c Fj N c Then, we can formuate the new assignment probem: Create a tas node for each tas t ; Create a coaition-position node for each coaition c j and position pair, with positions ranging from 1 to N cj for coaition c j ; If a coaition c j can accompish a tas t, connect t with a coaition-position nodes for c j, and set the weights to be ( c Fj N c + I j ) p j, respectivey, based on I j The above assignment probem can be soved optimay, which gives the positions of tass to be executed on the coaitions This soution satisfies the foowing property: Lemma 51: There exists a schedue that is no worse than the soution of the assignment probem Proof: We prove this resut by constructing a schedue from the soution of the assignment probem This soution specifies the coaitions to execute the tass and the sequences of these executions The basic idea for the construction is to move from the beginning to the end of the entire execution and resove any conficts aong the path The resoution ordering guarantees that the process does not introduce new conficts before where the current confict 1 Athough these tass (and the corresponding assignments) can recursivey infuence the objective function, such infuences are too compicated to mode and hence are not considered Scheduing positions are numbered from the atest to the eariest tas, such that the ast tas to be executed on a coaition is at position 1

4 occurs This process ends when no conficts exist and the resuting schedue is vaid A we need to show now is how to resove a confict When there are conficts with an assignment c j t, given the resoution ordering, we now that no conficts exist before the starting of its execution Since conficts can ony occur when a coaition conficts with c j, we ony need to adjust tas executions schedued on F j In the worst case, the number of assignments that we need to move to start time p j ater is c Fj N c In such a way, the confict is resoved and the process can move forward to the next assignment in the entire execution Theorem 5: The schedue that is constructed in Lemma 51 from the soution of the assignment probem yieds a soution quaity bounded by max j c Fj N c + 1 Proof: In the assignment probem, setting the connection weight to be I j p j ceary estabishes a ower bound on the objective function for the scheduing probem, since it assumes that there are no conficts between the coaitions Based on this observation and Lemma 51, the soution for the schedue that is constructed in Lemma 51 satisfies: S g (C, T ) S assign (T, {C, I}) c Fj N c + I j max S (C, T ) j I j (max c Fj N c + 1) S (C, T ) (8) j Hence, the concusion hods Breaing Ties: Note that from the construction in Lemma 51, we now that when there are ties in resoving the conficts (ie, when mutipe conficting assignments are schedued to start at the same time), we can choose any one of these assignments to perform the resoution process In InterfereAssign, we choose the one that minimizes its contribution to e with consideration of the interference (ie, computed as e in addition to the necessary moves for resoving the conficts) Note that the soution quaity of InterfereAssign is dependent on the probem instance For cases when the number and size of coaitions are imited (eg, due to communication and coordination costs, or restrictions from ocation constraints), such that the coaitions interfere with ony a imited number of other coaitions, InterfereAssign is iey to produce good quaity soutions In particuar, when ony singe-robot coaitions are aowed, the probem reduces to an assignment probem and InterfereAssign produces the optima soution B MinInterfere In InterfereAssign, c Fj N c is often too much of an overestimate of the number of tass that are schedued on F j However, one cannot mae a more precise estimation before some assignments are made MinInterfere is a greedy heuristic introduced to address this issue The idea is to find an upper bound on the number of tass schedued on F j that is as tight as possibe, so that the estimation of the interference becomes more accurate At step during the greedy process, MinInterfere has aready made 1 assignments Then, for every remaining tas t, for any coaition c j that can execute t, MinInterfere schedues the assignment at the eariest time possibe, ie, the eariest time foowing the ast chosen assignment on c j that incurs no conficts MinInterfere computes the foowing vaue as the greedy criterion to minimize: β j = e j + c Fj N c \ M j p j (9) in which M j incudes t, and the set of tass that are schedued before c j t in the greedy process MinInterfere then schedues an assignment according to arg min j, β j The forma anaysis of MinInterfere is eft as future wor To provide an exampe of how these heuristics wor, we provide the agorithm for MinInterfere as foows Agorithm 1 MinInterfere for muti-robot tas scheduing 1: whie there are tass remaining do : for a t tass remaining do 3: for a c j that can accompish t do 4: Compute the eariest starting time s j 5: Compute e j = s j + p j 6: Compute β j in Equation (9) 7: end for 8: end for 9: Seect j, according to arg min j, β j 10: Assign tas t to c j 11: end whie VI COMPLEXITY ANALYSIS To sove the MPM MPT e probem optimay, one needs to search a orderings of tass For each ordering, we must iterate through a possibe coaition assignments for each tas Since finding the eariest starting time taes O( T ), the compexity is O(( C T ) T T!) 3 Ceary, computing the optima soution is intractabe MinProcTime requires us to first sort the tass based on their minimum processing times This process taes O( C T + T og T ) using a set structure For scheduing each tas, since finding the starting time taes O( T ), the compexity of MinProcTime is O( C T + T ) At every greedy step, MinStepTime checs every assignment s contribution to the objective function This process taes O( C T ) Hence, the computationa compexity of MinStepTime is O( C T 3 ) InterfereAssign is a direct appication of the Hungarian agorithm Hence, the compexity is O( C 3 T 3 ), given that the number of coaition-position pairs is O( C T ) The compexity for breaing ties is absorbed Simiary, from Agorithm 1, it can be inferred that the computationa compexity of MinInterfere is O( C T 3 ) Tabe I provides a summary of the discussed heuristics, incuding soution bounds, and computationa compexities 3 Athough C can be exponentia in R, the number of coaitions that need to be considered is often much smaer in practice [10], [13], [15], and thus the probem can be soved quicy

5 TABLE I SUMMARY OF DISCUSSED HEURISTICS Name Soution Bound Compexity Optima 1 O(( C T ) T T!) MinProcTime O( C T 3 ) MinStepTime O( C T 3 ) InterfereAssign max j c Fj N c + 1 O( C 3 T 3 ) MinInterfere Not Determined O( C T 3 ) TABLE II TASKS FOR THE SCENARIO IN FIGURE 1 Tas Robots Required Process Time 1) Object 1 One gripper, one ocaizer 6 ) Object One gripper, one ocaizer 6 3) Large Object Two grippers 5 Fig Schedues created by the heuristics for the scenario in Tabe II Fig 1 A simpe scenario of the scheduing probem VII SIMULATION RESULTS Simuations are run on a 4GHz aptop with GB memory; wa-coc times are reported Statistics are coected over 100 runs; probem instances are randomy generated A An Object Coection Scenario We start with an object coection scenario, as shown in Figure 1, to demonstrate interference between assignments There are two types of robot, and two robots for each type The first type has a gripper but cannot ocaize; the second type can ocaize but does not have a gripper The goa is to move a three objects to the home area Whie the smaer objects can be moved by one gripper robot, they are reativey far from the home area so that a ocaization capabiity is required The arger object can ony be moved by two gripper robots cooperativey but is coser to the home area Note that athough the tas execution times are ony dependent on the distances between the home area and the objects in this scenario, they can aso be dependent on the current robot ocations and other dynamic factors This is reated to the issue of tas dependency and is not specificay considered in this wor See [16] for some discussions on this issue Tabe II shows the robots required for each object and the time required to accompish the move Figure iustrates the schedues created by the four proposed heuristics We can see that ony the heuristics that consider the interference between the assignments (ie, InterfereAssign and MinInterfere) produce the optima soution Whie this simuation presents ony a sma exampe, the impact can be arge as the probem size increases, which is shown next B Performance Comparison First, the parameters used in the foowing simuations are isted in Tabe III In the first simuation, we vary n c from 3 to 6 whie eeping the other parameters fixed Due to the compexity for computing the optima soution, we set n t = 9 Other parameters are: n f = (09 18) (based on n c ), n e = 7, n min = 5, and n max = 10 The resut is shown in Figure 3 We can see that the performance generay decreases as n c increases for a heuristics This may seem to be counter-intuitive However, note that as n c increases, n f aso increases, which compicates the scheduing We can aso see that MinStepSum, InterfereAssign and MinInterfere perform simiary whie InterfereAssign is sighty better (at the 5% significance eve) than MinStepSum and MinInterfere for smaer n c (ie, 3 4) Another observation is that the average soution quaity is better than the proven bounds In the next simuation, we vary n t from 8 to 11 The other parameters are: n c = 4, n f = 1, n e = (4 33) (based on n t ), n min = 5, and n max = 10 The resut is shown in Figure 4 We can see simiar concusions as in the previous simuation (ie, InterfereAssign performs significanty better than MinStepSum and MinInterfere when n t = (8 9)) Since n f stays as a constant this time, we can see that the curve formed by the soution quaity is smoother Next, we show how the performance changes whie we vary n f from 05 to 5 Other parameters are: n c = 5, n t = 10, n e = 3, n min = 5, and n max = 10 Figure 5 gives the resut Again, we can see that the performance decreases as the interference becomes more compex (or as n f increases) We are aso interested in how n max infuences the soution In this simuation, we vary n max from 5 to 10 Other Parameter n c n t n f n e n min, n max TABLE III PARAMETERS USED IN THE SIMULATIONS Description No of coaitions No of tass Average no of conficting coaitions per coaition Average no of executabe tass per coaition Minimum and maximum processing time

6 Fig 3 Scheduing with varying n c Top: Average soution quaity (the ower the better) Bottom: Separate detaied resuts with standard deviations The green data points in each subgraph represent the worst soution quaity Fig 5 Scheduing with varying n f Top: Average soution quaity Bottom: Separate and more detaied resuts with standard deviations Fig 4 Scheduing with varying n t Top: Average soution quaity Bottom: Separate and more detaied resuts with standard deviations Fig 6 Scheduing with varying n max Top: Average soution quaity Bottom: Separate and more detaied resuts with standard deviations parameters are: n c = 5, n t = 10, n f = 15, n e = 3, and n min = 5 The resut is shown in Figure 6 We can see from the figure that the performance decreases as n max increases, which is different from the previous simuations This is understandabe because it is more difficut to schedue assignments when their processing times are simiar but their interferences on other assignments differ Now, we want to see how the heuristics perform with arge

7 Fig 8 Time anaysis Left: Varying n c Right: Varying n t soution bounds can be further improved It is aso usefu to estabish bounds on the approximation ratios that any poynomia-time agorithms can achieve Finay, we aso oo forward to impementing these heuristics on muti-robot systems to address rea-word scheduing probems Fig 7 Scheduing with varying n max, with arge n c and n t Top: Average soution quaity Bottom: Separate detaied resuts with standard deviations n c and n t Due to the compexity for computing the optima soutions, we use MinInterfere as the standard to compare against This time, we vary n max from 5 to 160 We set: n c = 5, n t = 50, n f = 15, n e = 3, and n min = 5 Figure 7 shows the resut We can see the same trend in the soution quaity as in Figure 6 Another observation is that the change of n max does not seem to have a significant infuence on the comparative performances of these heuristics Finay, we perform time anayses on these heuristics For the first resut, we vary n c from 15 to 30; for the second, we vary n t from 40 to 55 Other parameters remain the same as in the previous simuation Figure 8 shows the resut, which aigns with Tabe I A simuations together show that: when the coaitions interfere with ony a imited number of other coaitions (ie, 0 1), InterfereAssign can be used to produce better soutions (eg, see Figure 4, or Figure 3 when n c = (3 4)), since they expicity consider this interference in their formuations; otherwise, when the interference becomes difficut to be modeed accuratey by our approach, InterfereAssign, MinInterfere, and MinStep- Sum perform simiary In such cases, we can run these three heuristics and choose the best schedue produced VIII CONCLUSIONS AND FUTURE WORK In this paper, we have introduced four efficient heuristics to address the MPM MPT e probem We have formay anayzed them and provided simuations to demonstrate and compare their performances Furthermore, it is convenient to incorporate them using a ayering technique For future wor, it is interesting to study the soution quaity of MinInterfere and other heuristics, to see if the REFERENCES [1] P Brucer Scheduing Agorithms Springer-Verag New Yor, Inc, Secaucus, NJ, USA, 001 [] P Brucer, A Drex, RH Mohring, K Neumann, and E Pesch Resource-constrained project scheduing: Notation, cassification, modes, and methods European Journa of Operationa Research, 11(1):3 41, 1999 [3] P Brucer and S Knust Compex Scheduing Gor-Pubications Springer, 006 [4] BP Gerey and MJ Mataric A forma anaysis and taxonomy of tas aocation in muti-robot systems The Internationa Journa of Robotics Research, 3(9): , September 004 [5] S Hartmann and R Koisch Experimenta evauation of state-of-theart heuristics for the resourc-constrained project scheduing probem European Journa of Operationa Research, 17: , 1998 [6] JA Hoogeveen, SL van de Vede, and B Vetman Compexity of scheduing mutiprocessor tass with prespecified processor aocations Discrete Appied Mathematics, 55(3):59 7, 1994 [7] H W Kuhn The Hungarian method for the assignment probem Nava Research Logistic Quartery, :83 97, 1955 [8] E Neron, C Artigues, P Baptiste, J Carier, J Damay, S Demassey, and P Laborie Lower bounds for resource constrained project scheduing probem In Perspectives in Modern Project Scheduing, voume 9, pages Springer US, 006 [9] JH Patterson, F Brian Tabot, R Sowinsi, and J Wegarz Computationa experience with a bactracing agorithm for soving a genera cass of precedence and resource-constrained scheduing probems European Journa of Operationa Research, 49(1):68 79, Nov 1990 [10] O Shehory and S Kraus Methods for tas aocation via agent coaition formation Artificia Inteigence, 101(1-):165 00, 1998 [11] A Sprecher and A Drex Muti-mode resource-constrained project scheduing by a simpe, genera and powerfu sequencing agorithm European Journa of Operationa Research, 107(): , 1998 [1] F Tang and LE Parer A compete methodoogy for generating muti-robot tas soutions using ASyMTRe-D and maret-based tas aocation In Proceedings of the IEEE Internationa Conference on Robotics and Automation, pages , Apr 007 [13] L Vig and JA Adams Muti-robot coaition formation IEEE Transactions on Robotics, (4): , 006 [14] Y Zhang and LE Parer IQ-ASyMTRe: Forming executabe coaitions for tighty couped mutirobot tass IEEE Transactions on Robotics, PP(99):1 17, 01 [15] Y Zhang and LE Parer Tas aocation with executabe coaitions in mutirobot tass In Proceedings of the IEEE Internationa Conference on Robotics and Automation, pages , May 01 [16] Y Zhang and LE Parer Considering inter-tas resource constraints in tas aocation Autonomous Agents and Muti-Agent Systems, 6(3): , 013