15-382 C OLLECTIVE I NTELLIGENCE – S18 L ECTURE 23: T ASK A LLOCATION 2 I NSTRUCTOR : G IANNI A. D I C ARO
MT-SR-TA: VRP Robots can work in || on multiple tasks and have a time-extended schedule of tasks (quite uncommon in current MR literature) Vehicle routing problems with capacity constraints and pick-up and delivery fall in this category: § Multiple vehicles transporting multiple items (goods, people,…) and picking up items along the way § Between a pick-up and delivery location the vehicle is dealing with MT § Visiting multiple locations is equivalent to TA NP-hard! 2
ST-SR-TA: G ENERALIZED A SSIGNMENT If dependencies / constraints are included, “more” NP-Hard → If the utility is related to traveling distances the problem falls in the class of m TSP, VRP problems Multi-robot routing 3
T RAVELING S ALESMAN P ROBLEM (TSP) § How can we compute the paths? à Order of visiting locations / Order performing assigned tasks § Find the minimum cost Hamiltonian tour among n cities / tasks Find the ordering of minimal path cost in the set {1,2,… ,𝑜} § n n P P min z = d ij x ij i =1 j =1 n P s.t. x ij = 1 j = 1 , . . . n entering city j once and only once i =1 n P x ij = 1 i = 1 , . . . n exiting city i once and only once j =1 x ij ∈ { 0 , 1 } { Solution is a Hamiltonian tour of n cities } § For n = 4, solution: (x 13 = x 24 = x 31 = x 42 = 1) and all other x ij = 0 is a feasible solution for the Assignment, but not for the TSP since it contains sub-tours (1-3-1) and (2-4-2) 15781 Fall 2016: Lecture 13 § Assignment is a relaxation of the TSP 4
TSP COMPLEXITY n n z = P P d ij x ij min i =1 j =1 n P s.t. x ij = 1 j = 1 , . . . n i =1 n P x ij = 1 i = 1 , . . . n j =1 x ij ∈ { 0 , 1 } n P x ij ≤ | S | − 1 , ∀ S ⊂ V, 2 ≤ | S | ≤ n − 1 i,j ∈ S DFJ formulation, O(2 n ) constraints! 15781 Fall 2016: Lecture 13 World TSP, ~2 ) 10 + cities 5
V EHICLE R OUTING P ROBLEMS § One or more vehicles / agents with limited capacity § Customers / Task can have time windows § … Precedence relationships § … Conflicts § …. 6
(T EAM ) O RIENTEERING P ROBLEMS § Reward collection problem: § Select the sequence of places to visit such that the total reward is maximized and the time/distance budget is not exceeded § Select a subset of places + define the order of visiting them § Start and end points are given 7
(T EAM ) O RIENTEERING P ROBLEMS 8
S INGLE A GENT O RIENTEERING P ROBLEM Start from 1 and returning in 𝑜 Assignment, not all places need a visit Time budget cannot be exceeded MTZ subtour elimination constraints 9
O RIENTEERING P ROBLEMS Agent(s) à Tasks (subset + path) 10
S ET C OVERING Are given a set 𝐵 of 𝑙 “ activities” , and a set 𝑆 of 𝑛 “ requirements ” § Each activity 𝐵 1 can “cover” one or more requirements 𝑆 2 with cost 𝑑 1 . § § Select a subset of the activities such that all requirements are covered by at least one activity and the total cost is minimized § Duplication admitted: one requirement can be covered by multiple activities § 𝑏 21 = 1 if 𝐵 1 covers 𝑆 2 , 0 otherwise 𝑦 1 variable corresponds to selection of activity 𝑘 § A One linear constraint 1 2 3 4 5 6 min Z = c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 + c 6 x 6 per requirement x x x 1 s.t. x 1 + x 2 + x 5 > 1 x x x 1 + x 3 > 1 2 x 2 + x 4 > 1 R x x 3 x 3 + x 6 > 1 x x 4 x 2 + x 3 + 15781 Fall 2016: Lecture 13 x 6 > 1 x x x 5 x 1 , x 2 , x 3 , x 4 , x 5 x 6 ∈ { 0 , 1 } 11
S ET C OVERING § Agents à Tasks § Do all tasks at the minimum cost selecting from the agent set § Agents do not interfere (neither conflict nor collaborate), multiple agents can be on the same task (no advantage, just extra costs) § Personnel / Turns à Routes (trains, buses, flights) § Personnel à Services (turns at hospitals/factories, cleaning areas) § Routes / Agents à Customers (trucks going through distribution centers, pick-up and delivery tasks) § Installation (plants, antennas, emergency hydrants) à Services 15781 Fall 2016: Lecture 13 12
S ET P ACKING Are given a set 𝐶 of 𝑙 “ boxes” , and a set 𝐽 of 𝑛 “ items ” § Each box 𝐶 1 can “pack” one or more items 𝐽 2 each delivering a profit 𝑑 2 § § Select a subset of the boxes such that a maximal number of items are packed without duplicates and the total profit is maximized § No duplicates admitted: the same item can’t go in two different boxes! B 1 2 3 4 5 max Z = p 1 x 1 + p 2 x 2 + p 3 x 3 + p 4 x 4 + p 5 x 5 s.t. x 1 + x 2 6 1 I x 1 + x 3 + x 5 6 1 x 2 + x 4 + x 5 6 1 x 3 6 1 x 1 6 1 15781 Fall 2016: Lecture 13 x 4 + x 5 6 1 13 x 1 , x 2 , x 3 , x 4 , x 5 ∈ { 0 , 1 }
S ET P ACKING Agents à Tasks § Do as many tasks as possible in order to maximize profit selecting with § no overlapping from the agent set § Agents do interfere / conflict: only one agent can be on a task at a time § Plants (e.g., incinerator) à Cities § SAR Agents (e.g., dogs) à Places to search 15781 Fall 2016: Lecture 13 § Boxes (e.g., relocating) à Objects 14
S ET P ARTITIONING Are given a set 𝐶 of 𝑙 “ boxes” , and a set 𝐽 of 𝑛 “ items ” § Each box 𝐶 1 can “pack” one or more items 𝐽 2 delivering a profit/cost 𝑑 2 § § Select a subset of the boxes such that the whole item set is partitioned (in the boxes) and total profit is maximized (or, total cost is minimized) § No duplicates admitted: the same item can’t go in two different boxes! B 1 2 3 4 5 6 min Z = d 1 x 1 + d 2 x 2 + d 3 x 3 + d 4 x 4 + d 5 x 5 + d 6 x 6 s.t. x 1 + x 2 + x 5 = 1 I x 1 + x 3 = 1 x 2 + x 4 = 1 x 3 + x 6 = 1 15781 Fall 2016: Lecture 13 x 2 + x 3 + x 6 = 1 x 1 , x 2 , x 3 , x 4 , x 5 x 6 ∈ { 0 , 1 } 15
S ET P ARTITIONING § Agents à Tasks § Do all tasks and maximize profit (minimize costs) selecting with no overlapping from the agent set § As in Set packing / Set covering but more strict (e.g., personnel can’t travel as passengers on a route to cover) § People at an event (some people can’t be together!) à Tables / rooms / buses 15781 Fall 2016: Lecture 13 16
V EHICLE ROUTING : S ET P ARTITIONING MODEL A set of customers / tasks to service: represented on a Euclidean graph § Each client 𝑑 must be visited once and only once § Each client has associated a service demand 𝑒 ; § Each agent / vehicle has a maximum capacity 𝑟 (to deliver services) § Total length (time) of a closed path 𝐸𝑓𝑞𝑝𝑢 → 𝑑 C → 𝑑 D → … 𝑑 E → 𝐸𝑓𝑞𝑝𝑢 § cannot exceed a max value (e.g., energy, fuel, sleep, …) Goal: Select the capacity-length feasible paths that service all customers at § 15781 Fall 2016: Lecture 13 the minimum cost (no interfering paths, since client can’t be visited twice) 17
V EHICLE ROUTING : S ET P ARTITIONING MODEL The set 𝑩 of all capacity / length feasible paths is given § 𝑛 = number of tasks / customers § 𝑤 = number of available vehicles / agents (max number of paths) § 𝑏 ;H = 1 if customer (node) 𝑑 is included in path 𝑞 , 0 otherwise § 𝑑 H = cost of path 𝑞 § 15781 Fall 2016: Lecture 13 𝑦 H = 1 if path 𝑞 is selected in the solution, 0 otherwise § 18
R OUGHLY , THREE MAIN CLASSES OF COP S § Finding an order on, sequencing the § Selecting a subset of solution set of solution components components (+ group subset of components) Modeling combinations of task § Joint problem: Selecting a allocation and task sequencing (e.g., for subset of solution components moving between given tasks, defining AND finding an order on the execution order) selected subset 19
ST-MR-IA: S ET P ARTITIONING - C OALITION F ORMATION § Model of the problem of dividing (partitioning) the set of robots into non-overlapping sub-teams (coalitions) to perform the given tasks instantaneously assigned This problem is mathematically equivalent to set partitioning problem § 𝐷𝑈 Cover (Partition) the elements in 𝑆 x x x 1 (Robots) using the elements in 𝐷𝑈 x x 2 (feasible coalition-task pairs) without 𝑆 S x x 3 duplicates (overlapping), and at the x x 4 min cost / max utility x x x 5 20
MT-MR-IA: S ET C OVERING - C OALITION F ORMATION § Model of the problem of dividing (partitioning) the set of robots into sub- teams (coalitions) to perform the given tasks instantaneously assigned § Overlap is admitted to model MT, a robot can be in multiple coalitions This problem is mathematically equivalent to set covering problem § 𝐷𝑈 Cover (Partition) the elements in 𝑆 x x x 1 (Robots) using the elements in 𝐷𝑈 x x 2 (feasible coalition-task pairs) admitting 𝑆 R x x 3 duplicates (overlapping) and at the min x x 4 cost / max utility x x x 5 21
O THER CASES § ST-MR-TA: Involves both coalition formation and scheduling, and it’s mathematically equivalent to MT-SR-TA § MT-MR-TA: Scheduling problem with multiprocessor tasks and multipurpose machines (quite complex) § Modeling of dependencies? → G. Ayorkor Korsah, Anthony Stentz, and M. Bernardine Dias. 2013. A comprehensive taxonomy for multi- robot task allocation. Int. J. Rob. Res. 32, 12 (October 2013), 1495- 1512. 22
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