improving market-based task allocation with optimal seed schedules

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


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 Approachese.g. Mathematical Programming

Heuristic & Decentralized Approachese.g. Market-Based Approaches

iD

j

l

k

r

aijx

A task at (4, 2)

I can do it for $73It will cost me

$80arx


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


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


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


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


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


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!


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


Experiments In simulation & on robots

Tasks: Visit specified location

Objective function: Minimize total distance travelled by team


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


Experimental ProcedureUse 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


Results: Simulation2 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


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

Plan

ning

Tim

e (s

)

Terminated (timed-out) prior to proving optimality of solution

Observation: Combinatorial nature of the optimal planning problem


Results: Robots

Hindsight optimal

Seeded market-based

Pure market-based

0 0.5 1 1.5 2 2.5 3Suboptimality 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.


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)


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

Thank you! Questions?

Extra Slides


Branch-and-Price

AB E

CD

Start out with a subset of feasible routes


Branch-and-Price

AB E

CD

Start out with a subset of feasible routesSolve a relaxed version of the problem

AB E

CD


Branch-and-Price

AB E

CD

Start out with a subset of feasible routesSolve a relaxed version of the problemGenerate additional profitable routes

AB E

CD


Branch-and-Price

AB E

CD


AB E

CD


Branch-and-Price

AB E

CD


Repeat till no more profitable routes

AB E

CD


Branch-and-Price

AB E

CD



If constraints violated, branch

A B& together

AB E

CD


Branch-and-Price

AB E

CD




A B& together

AB E

CD

AB E

CD


Branch-and-Price

AB E

CD



A B& together

AB E

CD


AB E

CD


Branch-and-Price

AB E

CD


Repeat till no more profitable routesPrune nodes if possibleIf constraints violated, branch

A B& together

AB E

CD

AB E

CD


Branch-and-Price

AB E

CD



A B& together

AB E

CD

AB E

CD

A D& not together

Prune nodes if possibleIf constraints violated, branch

AB E

CD


Branch-and-Price

AB E

CD



A B& together

AB E

CD

Repeat till no more violated constraintsand no more nodes to process

AB E

CD

A D& not together


AB E

CD


Branch-and-Price

AB E

CD



A B& together

AB E

CD

Repeat till no more violated constraintsand no more nodes to process

AB E

CD

A D& not together


AB E

CD

Finds optimal solution!


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

improving market-based task allocation with optimal seed schedules

Documents

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