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Improving Market-Based Task Allocation with Optimal Seed Schedules
IAS-11, Ottawa.
September 1, 2010
G. Ayorkor Korsah1
Balajee Kannan1, Imran Fanaswala2, Bernardine Dias1,2
1 Robotics Institute, Carnegie Mellon University2 CS Department, Carnegie Mellon Qatar
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 2
Task Allocation
Key component of planning for team coordination
Example: disaster preparedness and response
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 3
Tradeoff: Optimality vs. Adaptivity
Optimality guarantees
Slow to compute not suitable for
dynamic problems
No optimality guarantees
Fast to compute suitable for dynamic
problems
Optimal & Centralized Approaches
e.g. Mathematical Programming
Heuristic & Decentralized Approaches
e.g. Market-Based Approaches
iD
j
l
k
r
aijx
A task at (4, 2)
I can do it for $73It will cost me
$80
arx
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 4
Real-World Problems
Many real-world problems have both static and dynamic components Some tasks known ahead of time, or some
likely scenarios known ahead of time New tasks arrive in real time and changed
information discovered in real time
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 5
Proposed Approach
Optimally pre-allocate static tasks then adapt plan (heuristically) as needed to handle dynamic situations
Can pre-compute several initial plans for various likely scenarios
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 6
Approach Overview
Mathematical Programming Approach• Used to compute optimal solution to the
static component of the problem• Use a branch-and-price approach
Market-Based Approach for Dynamism• Used to modify the initial optimal seed
schedule to handle dynamic component of the problem• Use TraderBots
Problem Decomposition• Identify static and dynamic components of
problem
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 7
Mathematical Model: Set-Partioning Integer Linear Program (ILP) Formulation
Objective Function (e.g. Total
Team Distance)
One route per agent
One agent per task
Minimize:
Subject to constraints:
agentsk routesr
kr
kr xd
1routesr
krx
1 agentsk routesr
kr
kjr x
agentsk
tasksj
“Route” = candidate time extended plan/task allocation for an agent
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 8
Branch-and-Price Approach Summary
Based on Branch-and-Bound Useful when variables cannot be
exhaustively enumerated (in our case, route variables) Allows progressive generation and inclusion
of profitable variables (in our case, routes) Enables computation of the optimal ILP
solution
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 9
Market-Based Approach Summary
Tasks are assigned via auctions Agents bid the marginal cost to perform the
new task Task is awarded to the lowest bidder
Centralized or decentralized Tasks auctioned by central operator or by
individual agentsMy bid: $280
My bid: $101
My bid: $73
Task at (3.5)Winner!
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 10
Proposed Seeded Market-Based Approach
Start out with the initial optimal plan Use market-based approach to modify
the optimal plan as changes occur Hold auctions for new tasks as they arrive Hold auctions for previously assigned tasks
if needed (environmental changes/ execution failure)
Task at (3.5)
My bid: $280
My bid: $101My bid:
$73
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 11
Experiments
In simulation & on robots
Tasks: Visit specified location
Objective function: Minimize total distance travelled by team
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 12
Experiments
Compare:
Post-execution evaluation: “Hindsight optimal” plan
(Optimal branch-and-price for static & dynamic tasks)
“Pure” Market-Based Plan (Auctions for static & dynamic tasks)
Seeded Market-Based Plan (Branch-and-price for static & auctions for dynamic tasks)
Team distance for (Seeded) Market-based planSuboptimality factor =
Team distance for “Hindsight Optimal” plan
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 13
Experimental Procedure
Use branch-and-price to compute initial optimal
plan for static tasks
Begin execution of computed plans
Continue execution, handling dynamism with market-based
approach
Compute “hindsight” optimal plan for static & dynamic
tasks
Compute “Sub-optimality factor”
Task at (4, 2)
$73
(Seeded) Market-based
=“Hindsight Optimal”
Complete Execution
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 14
Results: Simulation
2 agents, 12 tasks 2 agents, 16 tasks 5 agents, 20 tasks
(averaged over 5 random instances for each problem configuration)
Observation: With high % static tasks we see benefit of seeded market based approach
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 15
Median Planning Times for Branch-and-Price Planner (Simulation Experiments)
25% 50% 75% 100%0.1
1
10
100
1000
10000
100000
2 agents, 12 tasks
% Static Tasks
Pla
nn
ing
Tim
e (
s)
Terminated (timed-out) prior to proving optimality of solution
Observation: Combinatorial nature of the optimal planning problem
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 16
Results: Robots
Hindsight optimal
Seeded market-based
Pure market-based
0 0.5 1 1.5 2 2.5 3
Suboptimality factor
2 robots, 11 tasks (6 static)
(averaged over 5 runs for each approach)
Observation: more significant improvement of seeded market-based approach over pure market-based approach than in simulation.
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 17
Conclusion
Contributions: A seeded market-based approach for task
allocation Current & future directions:
Finer-grained characterization of seeded market-based approach
Handling inter-task order constraints (precedence, simultaneity, etc)
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 18
Acknowledgments
Sponsors: Qatar National Research Fund (QNRF) under contract NPRP 1-7-7-5
Collaborators: Anthony Stentz M. Freddie Dias Ameer Abdulsalam Wael Ghazzawi Victor Marmol Jaime Bourne
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 21
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routes
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 22
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problem
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 23
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 24
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 25
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 26
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
If constraints violated, branch
A B& together
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 27
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
If constraints violated, branch
A B& together
AB
E
C
D
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 28
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
A B& together
AB
E
C
D
If constraints violated, branch
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 29
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
A B& together
AB
E
C
D
If constraints violated, branch
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 30
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routesPrune nodes if possibleIf constraints violated, branch
A B& together
AB
E
C
D
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 31
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
A B& together
AB
E
C
D
AB
E
C
D
A D& not together
Prune nodes if possibleIf constraints violated, branch
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 32
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
A B& together
AB
E
C
D
Repeat till no more violated constraintsand no more nodes to process
AB
E
C
D
A D& not together
Prune nodes if possibleIf constraints violated, branch
AB
E
C
D
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 33
Branch-and-Price
AB
E
C
D
Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes
Repeat till no more profitable routes
A B& together
AB
E
C
D
Repeat till no more violated constraintsand no more nodes to process
AB
E
C
D
A D& not together
Prune nodes if possibleIf constraints violated, branch
AB
E
C
D
Finds optimal solution!
Korsah, Kannan, Fanaswala, Dias. “Improving Market-Based Task Allocation…” 34
Branch-and-price summary
Master Problem: Tries to assign known routes to agents by solving a mixed integer linear programming problem using branch-and-bound
Sub problem:At each node, generates additional useful routes to consider by solving a constrained shortest-route problem based on dual variables of master problem (column generation)
Start out with a subset of known routes
r0, r1, r2, r3, r4, r5…
Solve by searching a multi-dimensional space:• DD* Lite• Depth-1st search