L ECTURE 23: T ASK A LLOCATION 2 I NSTRUCTOR : G IANNI A. D I C ARO - - PowerPoint PPT Presentation
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
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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
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15781 Fall 2016: Lecture 13
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min z =
n
P
i=1 n
P
j=1
dijxij s.t.
n
P
i=1
xij = 1 j = 1, . . . n entering city j once and only once
n
P
j=1
xij = 1 i = 1, . . . n exiting city i once and only once xij ∈ {0, 1}
{Solution is a Hamiltonian tour of n cities }
§ For n = 4, solution: (x13 = x24 = x31 = x42 = 1) and all other xij = 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) § Assignment is a relaxation of the TSP
15781 Fall 2016: Lecture 13
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min z =
n
P
i=1 n
P
j=1
dijxij s.t.
n
P
i=1
xij = 1 j = 1, . . . n
n
P
j=1
xij = 1 i = 1, . . . n xij ∈ {0, 1}
n
P
i,j∈S
xij ≤ |S| − 1, ∀S ⊂ V, 2 ≤ |S| ≤ n − 1 World TSP, ~2 ) 10+ cities
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§ One or more vehicles / agents with limited capacity § Customers / Task can have time windows § … Precedence relationships § … Conflicts § ….
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Start from 1 and returning in 𝑜 Assignment, not all places need a visit Time budget cannot be exceeded MTZ subtour elimination constraints
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Agent(s) à Tasks (subset + path)
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§ Are given a set 𝐵 of 𝑙 “activities”, and a set 𝑆 of 𝑛 “requirements” § Each activity 𝐵1 can “cover” one or more requirements 𝑆2 with cost 𝑑1. § 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 x x x x x x x x x x x x
1 2 3 4 5
R
A 1 2 3 4 5 6
One linear constraint per requirement
§ 𝑦1 variable corresponds to selection of activity 𝑘
§ 𝑏21 = 1 if 𝐵
1 covers 𝑆2, 0 otherwise
min Z = c1x1 + c2x2 + c3x3 + c4x4 + c5x5 + c6x6 s.t. x1 + x2 + x5 > 1 x1 + x3 > 1 x2 + x4 > 1 x3 + x6 > 1 x2 + x3 + x6 > 1 x1, x2, x3, x4, x5 x6 ∈ {0, 1}
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§ 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
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1 can “pack” one or more items 𝐽2 each delivering a profit 𝑑2
B 1 2 3 4 5 I
max Z = p1x1 + p2x2 + p3x3 + p4x4 + p5x5 s.t. x1 + x2 6 1 x1 + x3 + x5 6 1 x2 + x4 + x5 6 1 x3 6 1 x1 6 1 x4 + x5 6 1 x1, x2, x3, x4, x5 ∈ {0, 1}
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§ 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 § Boxes (e.g., relocating) à Objects
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1 can “pack” one or more items 𝐽2 delivering a profit/cost 𝑑2
1 2 3 4 5 6 B I
min Z = d1x1 + d2x2 + d3x3 + d4x4 + d5x5 + d6x6 s.t. x1 + x2 + x5 = 1 x1 + x3 = 1 x2 + x4 = 1 x3 + x6 = 1 x2 + x3 + x6 = 1 x1, x2, x3, x4, x5 x6 ∈ {0, 1}
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§ Agents à Tasks § Do all tasks and maximize profit (minimize costs) selecting with no
§ 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
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§ 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 → 𝑑C → 𝑑D → … 𝑑E → 𝐸𝑓𝑞𝑝𝑢 cannot exceed a max value (e.g., energy, fuel, sleep, …) § Goal: Select the capacity-length feasible paths that service all customers at the minimum cost (no interfering paths, since client can’t be visited twice)
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§ 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 § 𝑑H = cost of path 𝑞 § 𝑦H = 1 if path 𝑞 is selected in the solution, 0 otherwise
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§ Selecting a subset of solution components § Finding an order on, sequencing the set of solution components (+ group subset of components) § Joint problem: Selecting a subset of solution components AND finding an order on the selected subset Modeling combinations of task allocation and task sequencing (e.g., for moving between given tasks, defining execution order)
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𝐷𝑈 𝑆
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𝐷𝑈
1 2 3 4 5
𝑆
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