L ECTURE 28: T ASK A LLOCATION 2 T EACHER : G IANNI A. D I C ARO - - PowerPoint PPT Presentation
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)
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NP NP-ha hard! d!
Vehicle rou
g prob
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
http://www.math.uwaterloo.ca/tsp/world/countries.html
Check the additional slides in the file: tsp-formulations.pdf for a more detailed discussion about different formulations of the Hamiltonian constraints
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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 ! Assignment, not all places need a visit Time budget cannot be exceeded MTZ subtour elimination constraints
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Age gent(s) à Tas Tasks (subs ubset t + path ath)
15781 Fall 2016: Lecture 13
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§ 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
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
§ (% variable corresponds to selection of activity )
§ *&% = 1 if !% covers #&, 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}
15781 Fall 2016: Lecture 13
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§ Age gents à Tas Tasks § Do
g from
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
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% can pack one or more items #& each delivering a profit '&
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}
15781 Fall 2016: Lecture 13
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§ Age gents à Tas Tasks § Do
as man any y tas tasks as pos
g with no
g from
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 § Boxes (e.g., relocating) à Objects
15781 Fall 2016: Lecture 13
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% can pack one or more items #& delivering a profit/cost '&
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}
15781 Fall 2016: Lecture 13
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§ Age gents à Tas Tasks § Do
all tas tasks and maximize prof
g with no
g from
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
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§ A set of
/ tasks to
§ 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
the minimum cost (no interfering paths, since client can’t be visited twice)
15781 Fall 2016: Lecture 13
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§ Th The set t ! of
gth, …) is gi given § Th The cos
if e each pa path th # ∈ ! is is al also gi given (com
§ % = 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 - § /* = 1 if path - is selected in the solution, 0 otherwise
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§ Selecting subsets of agents to cover all or a selection of the tasks § Finding an order on, sequencing the set of tasks assigned to each agents / group of agents § Joint problem: Selecting a subsets of agents/tasks AND finding an order on the selected agents/tasks § Set (cover, partition, packing), Routing (TSP, VRP, TOP), Scheduling (including time) optimization formulations are models
§ E.g., for moving between given tasks, for respecting priorities, for respecting time constraints, ….
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g prob
"# !
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g prob
"#
1 2 3 4 5
!
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