Advanced Topics in AI

Transcription

1 Advanced Topics in AI - Task Allocation - Alexander Felfernig and Gerald Steinbauer Institute for Software Technology Inffeldgasse 16b/2 A-8010 Graz Austria

2 Agenda Motivation Examples Formal Problem Description Taxonomy of Task Allocation Solutions Application Example -2-

3 Motivating Movie 3

4 Rescue Simulation League Agent Competition large scale disaster simulation simulators for earthquake, fire, civilians, and traffic the task is to develop software agents with different roles, that make roads passable (police) extinguish the fires (fire brigades) rescue all civilians (ambulances) difference to Soccer Simulation: a challenging Multi Agent Problem since Agents must cooperate simulator components are developed within the Infrastructure Competition the more civilians alive and fires put out, the higher the score

5 RoboCup Rescue Scenario exemplar application: earthquake in a given city Foligno dense building layout Fire Brigade Police Ambulance (Team) Unhurt Civilian Hurt Civilian Dead Civilian Road Block

6 Platoon Agents Fire Brigades put out fires have limited water tank capacity can collaborate to extinguish more quickly Ambulances dig out civilians who are buried under rubble (takes a number of cycles) carry victims to a refuge can collaborate to dig out more quickly Police Force can remove obstructions bt ti on roads cannot collaborate to remove obstructions faster

7 Centre Agents Ambulance Centre talks to ambulances and to other centres Fire Centre talks to fire brigades and other centres Police Centre talks to police force agents and other centres located in a building which may or may not collapse or get burnt usually act as a synchronisation point/computation centre 7

8 Kobe (1/4) long distances, loose building layout, sparse bits

9 Random Maps with different building/road densities

10 Exploration Attacking Fires -10-

11 Challenges heterogeneous agents a large fixed number of agents a variable ibl number of tasks dynamic environment incomplete knowledge limited communication touches a number of research areas team exploration (victim search) multi-agent path planning (removal of road blockage) scheduling (treatment of victims) -11-

12 Support System for Officer in Charge Rosenbauer International AG

13 Multi Agent System (MAS) a MAS can be defined as a loosely coupled network of problem solvers that interact to solve problems that are beyond the individual capability or knowledge of each problem solver these problem solver, often called agents, are autonomous an can be heterogeneous in nature -13-

14 -14- Multi Agent System (MAS)

15 Task Allocation a (basic) task allocation problem is given by 1. a set of tasks T, <t 1,,t n > 2. a set of robots R, <r 1,,rr m > 3. a assignment matrix A, a ij =1 if task i is assigned to robot j otherwise 0 4. a utility function U:A defining the value or reward of executing tasks by particular robots 5. maximize the reward argmax A U -15-

16 Utility Function has to reflect two criteria expected quality, e.g., map generated by a particular robot or number of victims rescued expected costs, e.g., energy used by the robot given agent r and task t a utility function that reflects these criteria is: if r is capable to execute t Qrt Crt Ut Urt and Qrt Crt 0 otherwise Q rt is are the quality achievable by executing t by r C rt are the costs to execute t by r -16-

17 Taxonomy for TA to describe the nature of a task allocation problem three axis are used [Gerky & Mataric 2004] 1. robot: single-task robots (ST) versus multi-task robots (MT) 2. task: single-robot tasks (SR) versus multi-robot tasks (MR) 3. events: instantaneous assignment (IA) versus timeextended assignment (TA) TA problems are classified by three 2-letter abbreviation, e.g., ST-MR-IA -17-

18 Taxonomy for TA -18- single-task robot: a robot is able to perform one task at the time multi-task task robot: a robot is able to perform multiple tasks at the time single-robot task: a task can be completed by a single robot multi-robot task: a group of robots is necessary to complete the task instantaneous assignment: the available information allows only an instantaneous assignments time-extended assignment: information are changing and knowledge over potential future tasks is availableailable

19 Optimal Assignment Problem ST-SR-IA is an instance of Optimal Assignment Problem (OAP) [Gale 1960] simple case studied in context of game theory, operation research, personnel scheduling assign m workers to n prioritized jobs maximize i the overall expected performance -19-

20 Optimal Assignment Problem find mxn non negative integers to maximize find mxn non-negative integers ij to maximize U U m n n j U U m i j j ij ij m i n j n i ij m i j ij j are the weights for the prioritization of tasks -20-

21 Linear Programming OAP can be formalized as linear program a linear program comprises: 1. a vector of non-negative variables x,x i 0 2. a linear objective function to be maximized c T x 3. constraints equations Axb 4. non-negative right hand side constant b i algorithms to find the optimal solution Simplex Hungarian Method O(mn 2 )

22 Auction Algorithms tasks are sold to agents by a broker during an auction for optimal auction algorithms construct a price-based task market the broker assign a value c j on each task j each robot i also assign a value h ij on task j in order to be sold by the broker there must be a price p j greater as c j a robot is elect to buy the task with the most profit t i =argmax j {h ij -pp j } a robot is not happy if the above condition is not satisfied -22-

23 Auction Algorithm the auction mechanism is conducted in rounds and works as follows: 1. start with any assignment and any prices for t i 2. if all robots are happy then terminate otherwise select an unhappy robot i and find the best task j i 3. exchange the task with the robot assigned before 4. set the price of task j i so that is indifferent to the second best task: p ji + i, i =v i -w i, v i =argmax j {h ij -p j }, w i =argmax jji {h ij -pp j } 5. repeat until all robots are happy -23-

24 Auction Algorithm problem: the algorithm may not terminate if more than one object offers maximum value to robot i, i =0 to break the cycle i have to be a positive increment a robot is almost happy with task j i if h iji -p ji max j=1,,n, {h ij -p j}- if we chose i =v i -w i + robot j is almost happy after each round this guarantees that the process terminates after a finite number of rounds (n rounds) and it reaches almost the equilibrium if all h ij are integer and <1/n a reached assignment is optimal -24-

25 Communication the approaches differs in their nature: centralized versus distributed centralized: needs n 2 messages to distribute all utilities works fine with n<200 distributed: needs fewer messages, i.e., less than n in some larger systems communication latency can not be ignored -25-

26 Iterative Assignment -26- consider the Cooperative Multi-robot Observation of Multiple Moving Targets Problem (CMOMMT) given: S: a bounded enclosed region R: a set of robots with limited sensors O(t): a set of n targets where In(o j (t),s) is true In(o j (t),s): target o j (t) is within S at time t A(t): a ij (t)=1 if r i is observing o j and 0 otherwise goal is to maximize: O T m r t i a t 1 j 1 1 ij( ) t m

27 Iterative Assignment [Kleiner 2010] -27-

28 Broadcast Local Eligibility behavior-based algorithm [Werger & Mataric 2001] coordination without explicit discussion on tasks broadcast local l eligibilities ibiliti to inhibit other robots behavior use subsumption style controller -28-

29 Broadcast Local Eligibility it resembles a Greedy algorithm the task is not matroid BLE is 2-competitive robust against wrong or suboptimal assignments is able to react on dynamic environments -29-

30 Role Assignment another instantiation of OAP RoboCup robot soccer robots or agent are interchangeable (expect the goalie) player take a role based on the current situation player broadcast their utility usually Greedy or ad-hoc algorithms occurs on higher frequency, e.g., 10 Hz dynamic environment simply pyalgorithms are sufficent -30-

31 On-line Assignment in some domains the set of tasks is not known in advance tasks are introduced one at the time is a variant of SRST SR-ST-IA if robots can be reassigned the problem can be solved optimally otherwise use again a greedy algorithm 1. when a new task arrive assign it to the most fit robot available -31- all greedy for OA are 3-competitive models of how task arrive can help to perform better

32 Time-Extended Assignment if there are more tasks than robots, a model of how task arrive is available, or the robot s future utility can be predicted d building a time-extended schedule of tasks while minimizing the costs: R w j C i problem is NP-hard solution: treat SR-ST-TA as an instance of on-line assignment -32-

33 Time-Extended Assignment given: m robots, n tasks, n > m the following approximation algorithm can be used: 1. optimally solve the initial m n assignment problem 2. use the greedy algorithm to assign tasks in an on-line fashion, as the robots become available the performance is limited by the Greedy approach: 3- competitive -33-

34 Time-Extended Assignment if the difference between #robots and #tasks assigned initially decrease, e.g., (n-m) 0, it reaches optimality works good in practice, in particular on short time horizons another solution uses price-based market and sub-contracting -34-

35 Sub-Contracting Robot 1: costs 100, reward 120, profit 20 Robot 2: costs 150, reward 180, profit 30 Robot 2 sub-contracts task B to Robot 1 for 130 Robot 1: costs 210, reward , profit 40 Robot 2: 0, reward , profit

36 Sub-Contracting agents exchange tasks opportunistically frequently modify their schedule bidding robots negotiate until price is mutually beneficial this moves the global solution towards optimum robots can negotiate several deals at once deals can potentially be multi-party prices determined by supply and demand example: if there are a lot of haulers, they won t be able to command a high h price this helps distribute robots among occupations -36-

37 Sub-Contracting based on economical principles provides not necessarily the optimal solution cost and revenue functions are difficult to obtain for some domains robots are self-interested maximize i the individual id profit the team revenue function has to be defined according to this -37-

38 Coalition Formation some task require combined effort of robots heterogeneous skills, e.g., special sensors combine same skills, e.g., push a heavy boy instance of SR-MT-IA application domains e-commerce surveillance search and rescue -38-

39 -39- Coalition Formation

40 -40- Coalition Formation

41 Coalition Formation Set Partitioning Problem (SPP) given a set of robots E a family X is a partition of E iff elements of X are disjoint: y z the union of elements of X are E: 0 y, z X, xx E given a finite i set E and F are acceptable subsets of E a utility function u:f + find a family X of F that maximizes the utility and X is a partition of E y z -41-

42 Coalition Formation for n robots the number of possible coalitions is 2 n -1 but the number of partitions is n n/2-42-

43 Coalition Formation general problem is intractable NP-hard many heuristic algorithms are proposed Simple Heuristic 1. search the bottom two levels of the graph, the two bottom levels comprise any possible coalition 2. continue breath-first search from top as long as there is time left 3. return the partition with the highest utility so far -43-

44 Add Scheduling ST-MR-TA includes scheduling and coalition formation example: robots have to deliver packages of different size to different places sizes and destination are known in advance is NP-hard again go back to STMR ST-MR-IA and use a Greedy algorithm other solution: market-place techniques assign a leader for groups the leader may arrange local groups the leader may opportunistically bd bid for tasks -44-

45 Criteria for TA -45- Computational Requirements how much computational burden for the agents and a potential center how does the burden increase with the number of agents and tasks Communication Requirements how many massages are necessary to achieve a solution unicasts versus multicasts Solution Quality is the optimal utility achieved what minimal utility is guaranteed -competitive

46 ResQ Freiburg the RoboCup rescue agent simulation team of the University of Freiburg won the RoboCup competition in n ambulances bl have to rescue m civilians iili after an earthquake civilians are characterized by buriedness, damage and hit- points costs to recue a victim comprise the joint travel time of ambulances bl and the time to rescue the victim overall utility is the number of rescued civilians some assumptions: same travel costs, super-additive

47 ResQ Freiburg -47- the problem reduces to assign a sequence R of rescue tasks to the entire set of agents A R=<r 1,rr 2,.,rr n > where r i denotes a recue task and i the position in the sequence U(R) ) denoted the predicted utility ty (number of survivors) vo s) when execute R optimize the sequence in respect to the utility: R*=argmax R U(R) exact Solution is impossible to obtain: many sequences, limited time Greedy solution prefer victims that can be rescued fast (buriedness) Prefer urgent victims (demage)

48 ResQ Freiburg -48- non-allocated agents (e.g., police and fire brigades) continuously search unexplored location and update information o (e.g., buriedness, ess, health) about known victims the ambulance station (agent) predicts for each known victim the lifetime and costs for rescue simulates rescue sequences, genetic algorithm when a better sequence is found, the sequence in the field is altered life time prediction learning a decision tree to classify victims (die or survive) linear regression to learn life time prediction calculation of confidence values in respect of the age of data

49 Literature Michael Woolridge: An Introduction to Multi-Agent- Systems, John Wiley & Sons, 2 nd Edition, Brian Gerkey and Maja Mataric: A Formal Analysis and Taxonomy of Task Allocation in Multi-Robot Systems, International Journal of Robotics Research, 23(9), , September2004. Bernhardine Dias, Robert Zlot, Nidhi Kalra and Anthony Stentz: Market-Based Multirobot Coordination: A Survey and Analysis, Proceedings of the IEEE, 94(7), ,,July

50 -50- Thank You!