bounded optimal coordination for human-robot teams g. ayorkor mills-tettey thesis proposal: 12 th...
DESCRIPTION
INTRODUCTIONRELATED WORKPROBLEM STATEMENTTECHNICAL APPROACHWORK TO DATECONCLUSION 3 Example Problem INTRODUCTION Compute: Allocation of tasks to agents Schedule for each agent Route for each agent Maximizing: (rewards for completed tasks) - (travel costs for agents’ routes) - (Cost for agents’ idle time)TRANSCRIPT
Bounded Optimal Coordination for
Human-Robot TeamsG. Ayorkor Mills-Tettey
Thesis Proposal: 12th December 2008
Committee: Anthony Stentz, M. Bernardine Dias,
Michael Trick, Nicola Muscettola
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Motivation
INTRODUCTION
[IMAGE CREDITS]
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem
INTRODUCTION
Compute:• Allocation of tasks to agents• Schedule for each agent• Route for each agent
Maximizing: (rewards for completed tasks)- (travel costs for agents’ routes)- (Cost for agents’ idle time)
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem
INTRODUCTION
Compute:• Allocation of tasks to agents• Schedule for each agent• Route for each agent
Maximizing: (rewards for completed tasks)- (travel costs for agents’ routes)- (Cost for agents’ idle time)
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Problem Features Tasks
Spatially distributed Multi-step Time constraints Location choice Precedence constraints Simultaneity constraints Proximity constraints
Agents Heterogeneous Capacity constraints
Locations Capacity constraints Mutual exclusion constraints
INTRODUCTION
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Thesis Goal: compute bounded optimal, anytime
solution to complex coordination problem Allocation of tasks to agents Schedule for each agent Route for each agent
Characteristics of application domains: Bounded optimality is important “Small” teams: 10s of agents, 10s-100s of tasks Planning time in minutes (or hours) acceptable Centralized planning acceptable
INTRODUCTION
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Related WorkProblem Features Vehicle
RoutingMulti-Robot Task Allocation
Spatial distribution
Multi-step
Time constraints
Location choice
Precedence constraints
Simultaneity constraints
Proximity constraints
Heterogeneity
Capacity constraints
Capacity constraints
Mutual exclusion constraints
Tasks
Agents
Locations
RELATED WORK
Optimal algorithms Approximate algorithms
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Vehicle Routing Problem Transportation of passengers / distribution of
goods between depots and final users
RELATED WORK
iD
j
l
k
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Vehicle Routing Problem Mathematical models – mixed integer linear
programming 3-index models (e.g. Cordeau, 2006) 2-index / set-partitioning models
(e.g. Savelsbergh & Sol, 1998)
RELATED WORK
iD
j
l
kaijx a
rxrIndicates if an agent a
traverses the edge from node i to node j
Indicates if an agent a performs route r
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Vehicle Routing Problem Exact approaches
Mathematical Programming Heuristic Approaches
Construction and improvement heuristics Tabu Search, Genetic Algorithms, Simulated
Annealing, etc
RELATED WORK
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Vehicle Routing Problem Problem features not addressed:
Precedence & simultaneity constraints with penalization of waiting/idle time
Location choice for tasks Capacity constraints on locations Proximity constraints Mutual exclusion constraints
RELATED WORK
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Multi-Robot Task Allocation In a multi-robot system, which robot should
execute which task?
RELATED WORK
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Multi-Robot Task Allocation Market-Based Approaches
RELATED WORK
A victim needs to be rescued at location (4, 2)
I can do it for $80
It will cost me $54 $73
$101
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Multi-Robot Task Allocation Related problem (Jones, 2007)
Planning, allocating and scheduling agent interactions for team tasks with precedence and simultaneity constraints
Solution approach: Market-based approach with combinatorial bidding and tiered auctions
Limitations for this thesis: Does not include all
problem features No optimality guarantees
RELATED WORK
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Multi-Robot Task Allocation Related problem (Koes, 2005)
Planning and Execution for Multirobot Coordination Solution approach: Mathematical programming (constraint
optimization) Limitations for this thesis:
Does not include all problem features Relies on off-the-shelve solvers, solution algorithm not
customized to problem domain
RELATED WORK
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Problem Statement Given
A set of tasks (could be multi-step) A team of heterogeneous agents A set of locations A set of constraints (compatibility, precedence,
simultaneity, proximity, mutual exclusion) Problem is to determine optimal:
Task allocation Schedule Routes
PROBLEM STATEMENT
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Technical Approach Mathematical programming
To guarantee bounded optimality Specifics:
Create a set-partitioning formulation of the problem, with side constraints
Design a branch-and-price algorithm to solve it
TECHNICAL APPROACH
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Technical Approach: Overview Background
Set-partitioning formulation Branch-and-bound Branch-and-price
Application to thesis problem Set-partitioning formulation with side constraints Formulation of pricing sub-problem
TECHNICAL APPROACH
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
ExampleSet-Partitioning Formulation
1P
B
3D
3P
1D
2D
2P
4D
A
4P
TECHNICAL APPROACH
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set-Partitioning Formulation Example
1P
B
3D
3P
1D
2D
2P
4D
A
4P
TECHNICAL APPROACH
Possible route
r0
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set-Partitioning Formulation Example
1P
B
3D
3P
1D
2D
2P
4D
A
4P
TECHNICAL APPROACH
Possible route
r1
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set-Partitioning Formulation Example
1P
B
3D
3P
1D
2D
2P
4D
A
4P
TECHNICAL APPROACH
Possible route
r2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set-Partitioning Formulation Example
1P
B
3D
3P
1D
4D
4P r0 r1 r2 r3 r4 r5
A: 0 0 1B: 0 0 0 0 1
Each agent assigned to at most 1 route:
r0 r1 r2 r3 r4 r5
1: 0 0 0 = 12: 0 0 = 1… … … … … … …
Each task assigned to exactly 1 route:
TECHNICAL APPROACH
Set Partitioning Formulation:
Arx1
Arx2
Arx3
Brx0
Brx4
Brx0
Arx3
Brx4
Arx1
Arx2
Arx3
r0
r1
r2
r3
r4
2DA
2P
Arx5
r1 Arx5
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set of all agents
Set-Partitioning Formulation Example
r0 r1 r2 r3 r4 r5
A: 0 0 1B: 0 0 0 0 1
Each agent assigned to at most 1 route:
r0 r1 r2 r3 r4 r5
1: 0 0 0 = 12: 0 0 = 1… … … … … … …
Each task assigned to exactly 1 route:
Arx1
Arx2
Arx3
Brx0
Brx4
Brx0
Arx3
Brx4
Arx1
Arx2
Arx3
TECHNICAL APPROACH
Set Partitioning Formulation:
kr
krx 1
1 Mk r
kr
kir
k
x
(for each agent, k)
(for each task, i)
Set of all feasible routes for agent k
Indicates if request i is on route r
Arx5
Arx5
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Branch-and-Bound
TECHNICAL APPROACH
P
P1 P2
20.67
22 21
min:Ar
Ar
Br
Br
Ar
Ar
Ar
Ar
Ar
Ar
Br
Br xcxcxcxcxcxc
554433221100
subject to: 15321 A
rAr
Ar
Ar xxxx
140 B
rBr xx
…
}1,0{,,,,,543210A
rBr
Ar
Ar
Ar
Br xxxxxx
)0,,,,,0( 32
31
31
31x
)0,0,0,0,1,1(x)1,1,0,0,0,0(x
Ar
Br
Ar
Ar
Ar
Br xxxxxx
54321071598615
route costs
01A
rx 11A
rx
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Solution
1P
B
3D
3P
1D
2D
2P
4D
A
4P
10B
rx
11A
rx
TECHNICAL APPROACH
r0
r1
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Branch-and-Price But, there might be too many routes to
enumerate explicitly. What to do? Column generation at each node of branch-and-
bound tree:
TECHNICAL APPROACH
P
P1 P2
P11 P12
At each node:• Compute the linear programming (LP)
relaxation with a subset of the routes• Solve a pricing sub-problem to find a
profitable route to include• Re-compute the LP relaxation• Repeat until no profitable routes
r0 r1 r2.
5.45.24.7r3.
r0 r1 r2. r3. r0 r1 r2. r3.
5.3 5.04.8r4.
4.9r5.
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Background:Branch-and-Price Pricing sub-problem for this example:
TECHNICAL APPROACH
kNi r
kikir
kr rMkvuc
k
,min
Cost of route r for agent k
Cost modification for each task (node) on the route
Cost modification for entire route
Dual variables
Shortest route problem in a graph with modified node costs!
1 iff task i is on route rk
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Recap Thesis Problem Problem Features:
Tasks:
Agents: Locations:
Technical Approach: Branch-and-price algorithm on set-partitioning
formulation with side constraints
TECHNICAL APPROACH
Spatially distributed, Multi-step, Location choice, Time windows,Precedence, Simultaneity, Mutual exclusion, Proximity constraints
Heterogeneous, Capacity constraints
Capacity constraints, Mutual exclusion constraints
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem
1
A
2
3a
4a
3b
4b
B
C
Silo 1
Silo 2
Field 1
Field 2
5
3b
4b 5
6
6D
TECHNICAL APPROACH
Harvest field 1
Harvest field 2
Move field 1 grain to silo
Move field 2 grain to silo
Monitor field 1 grain unloading
Monitor field 2 grain unloading
1
2
3a 3b
4a 4b
5
6
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem
Task Subtasks Compatible locations
Compatible agents
Harvest field 1 Field 1
Harvest field 2 Field 2
3. Transport field 1 grain Load Unload
Field 1Silo 1, Silo 2
4. Transport field 2 grain Load Unload
Field 2Silo 1, Silo 2
Monitor field 1 grain unloading
Monitor field 2 grain unloading
A
A
B C
B C
3a3b
4a4b
1
2
5
6
D
D
TECHNICAL APPROACH
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example ProblemPrecedence constraints: Harvest field 1 before Load grain from field 1 Harvest field 2 before Load grain from field 2
Simultaneity constraints: Monitor field 1 grain unloading concurrent with Unload field 1 grain Monitor field 2 grain unloading concurrent with Unload field 2 grain
Proximity constraints: Monitor field 1 grain unloading collated with Unload field 1 grain Monitor field 2 grain unloading collated with Unload field 2 grain
1
2
3a
3b
3a
4b
5
6
5
6
3b
4b
TECHNICAL APPROACH
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
1
2
3.1
4.1
B
C
A
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
1
2
3.1
4.1
B
C
A
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
1
2
3.1
4.1
B
C
A
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
1
2
3.1
4.1
B
C
A
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set Partitioning Formulation with Side Constraints
Objective Function
1 route per agent 1 route per task
Valid start time for taskValid arrival time for agent for task
Valid idle time for agent for task
Synchronization constraints
Proximity constraints
Precedence constraints
Non-overlapping constraints
Mutual exclusion constraints
Location capacity constraints
Maximize:
Subject to:
TECHNICAL APPROACH
Sid
e co
nstra
ints
Sta
ndar
d se
t-par
titio
ning
fo
rmul
atio
n
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Objective Function
Maximize
Rewards for completed tasks
Travel costs for selected routes
Costs for agents’ idle/waiting time
TECHNICAL APPROACH
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Objective Function
Maximize
Mj Kk r
kr
kjrj
k
xr
Kk r
kr
Rkr
k
xc
Ni l Kk
lki
Wk
i
wc
Rewards for completed tasks
Travel costs for selected routes
Costs for agents’ idle/waiting time
Route selection variable: 1 if agent k is assigned route r
Idle/waiting time of agent k for subtask i at location l
TECHNICAL APPROACH
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set Partitioning Formulation with Side Constraints
Objective Function
1 route per agent 1 route per task
Valid start time for taskValid arrival time for agent for task
Valid idle time for agent for task
Synchronization constraints
Proximity constraints
Precedence constraints
Non-overlapping constraints
Mutual exclusion constraints
Location capacity constraints
Maximize:
Subject to:
TECHNICAL APPROACH
Sid
e co
nstra
ints
Sta
ndar
d se
t-par
titio
ning
fo
rmul
atio
n
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Pricing Sub-problemTECHNICAL APPROACH
Reward for each completed task
∆cost for entire route
∆cost for each subtask
linear ∆cost for each subtask
∆cost for each subtask in a precedence/simultaneity constraint
∆cost for each subtask at a capacity-constrained location
∆cost for each subtask in a non-overlapping constraint
∆cost for each subtask at a location with mutual-exclusion
Find route (sequence of subtask/location pairs) that maximizes:
+
route
-Travel cost-
-
-
--
--
Subject to constraints:
Each route includes only 1 location per subtask
All subtasks of a given task are on the same route
∆reward for each completed task -
∆cost for each subtask in a proximity constraint
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Work to Date Implemented branch-and-bound algorithm for
simplified version of thesis problem Includes precedence constraints Does not include other side constraints Explicitly enumerates all routes – does not involve
column generation
WORK TO DATE
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem: 12 single-step tasks, 5 agents, max 3 tasks/agent
WORK TO DATE
Tasks
Agents
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem: Optimal Solution 12 single-step tasks, 5 agents, max 3 tasks/agentNo precedence constraints
WORK TO DATE
0 5 10 15 time
Timeline:
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem: Optimal Solution12 single-step tasks, 5 agents, max 3 tasks/agent2 precedence constraints: t4<t8, t9<t8
WORK TO DATE
0 5 10 15 time
Timeline:
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem: Optimal Solution12 single-step tasks, 5 agents, max 3 tasks/agent4 precedence constraints: t4<t8, t9<t8, t8<t5, t5<t3
WORK TO DATE
0 5 10 15 time
Timeline:
Waiting/idle time
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Quantitative Results
WORK TO DATE
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Bounded Optimality:Example with 4 precedence constraints
# Iterations0 100 200 300 400 500
30,000
25,000
20,000
15,000
10,000
5,000
0
Solu
tion
Valu
e
Legend
Best solution found
Upper bound
Found optimal solution
Proved optimality of solution
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Overview of Remaining Work Develop branch-and-price algorithm
Extend implementation to include all side constraints Develop algorithms to solve pricing sub-problem Implement column generation
Develop heuristic algorithms for comparison Evaluate branch-and-price against heuristic
algorithms Solution quality Solution time
WORK TO DATE
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Schedule
CONCLUSION
Dec ’08 – Jan ‘09 Formulate benchmark problems
Jan ‘09 Implement omitted side constraints
Feb – Apr ’09 Design and implement algorithms for pricing sub-problem
Apr – May ’09 Implement column generation
Jun – Jul ’09 Evaluate branch-and-price algorithm, develop comparison heuristic approaches
Aug – Sep ’09 Refine algorithms and continue evaluation
Oct – Nov ’09 Write thesis
Dec ’09 Defend Thesis
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Contributions First branch-and-price approach to optimal
robotic team coordination Set-partitioning formulation of novel complex
coordination problem First optimal task allocation approach
enabling cooperation in human-robot teams
CONCLUSION
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Acknowledgements Sponsors:
Qatar National Research Fund (QNRF) NASA Jet Propulsion Laboratory (JPL)
Thesis Committee: Anthony Stentz, M. Bernardine Dias,
Michael Trick, Nicola Muscettola rCommerce Lab Members Friends and Family
CONCLUSION
Questions?
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Image Credits “Motivation” Slide:
Construction site: http://www.freefoto.com/images/13/12/13_12_2_prev.jpg
Combine harvester: http://www.freefoto.com/images/07/37/07_37_65_prev.jpg
Tractor-trailer: http://www.freefoto.com/images/07/37/07_37_70_prev.jpg
Disaster response (with people & cranes): http://anitokid.blogspot.com/2008/05/china-earthquake-leaves-students.html
Earthquake: http://www.cemeteryspot.com/blog/?p=180
Extra slides
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Vehicle Routing Problem Transportation of passengers / distribution of
goods between depots and final users Some variants:
Capacitated vehicle routing with time-windows Multi-depot vehicle routing problems Pickup and delivery / dial-a-ride
RELATED WORK
iD
j
l
k
57
INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Multi-Robot Task Allocation Gerkey and Matarić’s (2004) categorization of
MRTA problems
RELATED WORK
ST
MT
SR
MR
IA TA
Single versus multi task robots
Instantaneous versus time-extended assignment
Single versus multi robot tasks
Our problem:
(ST-SR-TA or ST-MR-TA) + inter-task constraints+ location choice
58
INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set-Partitioning Formulation Example
1P
B
3D
3P
1D
2D
2P
4D
A
4P
TECHNICAL APPROACH
Possible route
r3
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set-Partitioning Formulation Example
1P
B
3D
3P
1D
2D
2P
4D
A
4P
TECHNICAL APPROACH
Possible route
r4
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Set-Partitioning Formulation Example
1P
B
3D
3P
1D
2D
2P
4D
A
4P
TECHNICAL APPROACH
Possible route
r5
61
INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
2
3.1
4.1
B
C
A
1
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
2
3.1
4.1
B
C
A
1
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
1
2
3.1
4.1
B
C
A
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
1
2
3.1
4.1
B
C
A
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Example Problem:Feasible Route
1
2
3.1
4.1
B
C
A
3.2
4.2
Silo 1
Silo 2
5
3.2
4.2 5
6
6D
TECHNICAL APPROACH
Field 1
Field 2
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Dual Variables
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INTRODUCTION RELATED WORK PROBLEM STATEMENT TECHNICAL APPROACH WORK TO DATE CONCLUSION
Details of Pricing Sub-problem