15-382 C OLLECTIVE I NTELLIGENCE – S19 L ECTURE 28: T ASK A LLOCATION 2 T EACHER : 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 rou outing g prob oblems 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-ha NP hard! d! 2
ST-SR-TA: G ENERALIZED A SSIGNMENT § If dependencies / constraints are included, problem is “more” NP-Hard § If the utility is related to traveling distances the problem falls in the class of m TSP, VRP problems Mul Multi-robot obot rout outing ng 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, … , &} § 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 P P z = d ij x ij min Check the additional slides in the file: i =1 j =1 tsp-formulations.pdf for a more n detailed discussion about different P s.t. x ij = 1 j = 1 , . . . n formulations of the Hamiltonian constraints i =1 n P x ij = 1 i = 1 , . . . n j =1 x ij ∈ { 0 , 1 } http://www.math.uwaterloo.ca/tsp/world/countries.html n P x ij ≤ | S | − 1 , 2 ≤ | S | ≤ n − 1 ∀ S ⊂ V, i,j ∈ S DFJ formulation, !(2 $ ) 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 col ollection on prob oblem: § 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 ! Assignment, not all places need a visit Time budget cannot be exceeded MTZ subtour elimination constraints 9
O RIENTEERING P ROBLEMS Age gent(s) à Tas Tasks (subs ubset t + path ath) 10
S ET C OVERING : F ORMULATION Are given a set ! of " “ activities” , and a set # of $ “ requirements ” § Each activity ! % can “cover” one or more requirements # & with cost ' % . § Select a subset of the activities such that all requirements are covered by at § least one activity and the total cost is minimized § Duplication on admitted : one requirement can be covered by multiple activities § * &% = 1 if ! % covers # & , 0 otherwise ( % variable corresponds to selection of activity ) § 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 x x R 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 AND T ASK A LLOCATION § Age gents à Tas Tasks § Do o all tasks at the minimum cos ost selecting g from om the age gent 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 : F ORMULATION Are given a set ! of " boxes, and a set # of $ items § Each box ! % can pack one or more items # & each delivering a profit ' & § Select a subset of the boxes such that a maximal number of items are § packed without duplicates and the total profit is maximized § No o 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 AND T ASK A LLOCATION § Age gents à Tas Tasks § Do o as as man any y tas tasks as pos ossible in or order to o maximize prof ofit selecting g with no o ov overlapping g from om the age gent 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 : F ORMULATION Are given a set ! of " boxes, and a set # of $ items § Each box ! % can pack one or more items # & delivering a profit/cost ' & § 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 o 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 AND T ASK A LLOCATION § Age gents à Tas Tasks § Do o al all tas tasks and maximize prof ofit (minimize cos osts) selecting g with no o ov overlapping g from om the age gent se 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 of custom omers / / tasks to o service: represented on a Euclidean graph Each client ! must be visited once and only once § Each client has associated a service demand " # § Each agent / vehicle has a maximum capacity $ (to deliver services) § Total length (time) of a closed path %&'() → ! + → ! , → … ! . → %&'() § cannot exceed a max value (e.g., energy, fuel, sleep, …) § Goa oal: Select the capacity-length feasible paths that service all customers at the minimum cost (no interfering paths, since client can’t be visited twice) 15781 Fall 2016: Lecture 13 17
V EHICLE ROUTING : S ET P ARTITIONING MODEL t ! of § Th The set of all paths feasible (for or capacity, lengt gth, …) is gi given ost " # if th # ∈ ! is § Th The cos if e each pa path is al also gi given (com omputed) % = number of tasks / customers § ' = number of available vehicles / agents (max number of paths) § ( )* = 1 if customer (node) , is included in path - , 0 otherwise § , * = cost of path - § 15781 Fall 2016: Lecture 13 / * = 1 if path - is selected in the solution, 0 otherwise § 18
T ASK A LLOCATION : S ELECT AND O RDER Finding an order on, sequencing the § Selecting subsets of agents § set of tasks assigned to each agents to cover all or a selection of / group of agents the tasks § Set (cover, partition, packing), Routing (TSP, VRP, TOP), Scheduling (including Joint problem: Selecting a § time) optimization formulations are models subsets of agents/tasks AND of task allocation and task sequencing finding an order on the § E.g., for moving between given tasks, for selected agents/tasks respecting priorities, for respecting time constraints, …. 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 partition § oning g prob oblem "# Cover (Partition) the elements in ! x x x 1 (Robots) using the elements in "# x x 2 (feasible coalition-task pairs) without ! S x x 3 duplicates (overlapping), and at the x x 4 min cost / max utility x x x 5 20
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