PMS 2012 April 1-4 2012 Leuven Introduction Scheduling Assigning - - PowerPoint PPT Presentation
The multiagent project scheduling problem: complexity of finding a minimum-makespan Nash equilibrium C. Briand A. Agnetis and J.-C Billaut ( briand@laas.fr, agnetis@dii.unisi.it, billaut@univ-tours.fr ) PMS 2012 April 1-4 2012 Leuven
(briand@laas.fr, agnetis@dii.unisi.it, billaut@univ-tours.fr)
April 1-4 2012 Leuven
Scheduling
– Assigning start times to a set of interdependent activities
satisfying a set of time and resource constraints
– A single objective function to be optimized – Scheduling = global decision function
Multiagent scheduling problem (MASP)
– Resources or/and activities are distributed among a set of
agents, each having his own decisional autonomy
– Agents must collaborate to carry out a common project or to
manage the sharing of common resources
– Ex : supply chain management, collaborative project
management, timetabling, grid computing
Agents’ knowledge can be imprecise ANR project no. 08-BLAN-0331-02 called “ROBOCOOP"
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Activities and resources are distributed among agents
Different kind of agents
activity-agent / resource-agent / mixed-agent
Agents have their own objective
A social (global) objective can be considered
Centralized vs. distributed scheduling approach
Full vs. Restricted agents’ knowledge
Activities Time constraint Resource Use
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Multiagent single-machine scheduling problem
algorithms for multi-agent scheduling to minimize total weighted completion time, Information Processing Letters, Volume 109, Issue 16, 31 July 2009.
scheduling problems with two competing agents, , J. of Sched., 12, pp 401-415.
max-form criteria", European Journal of Operational Research, 188(2), pp. 603-609.
minimize total weighted number of tardy jobs, Th. Computer Sci., 362, no 1-3, pp. 273-281.
two competing agents. Operations Research, 52(2), pp. 229-242.
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Resources are not shared between agents
–
The project is decomposed into time-dependent sub-projects
–
Unlimited resources
–
Social objective = Minimization of the project makespan
–
Centralized solving approach
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Introduction A multiagent project scheduling problem Complexity results A MIP for finding a minimum makespan NE Problem extensions Conclusion
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Introduction A multiagent project scheduling problem Complexity results Conclusion
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A project composed of n time-dependent activities
distributed amongst m agents (m ≤ n)
Agents can control the duration of their activities
– pi
=> pi = decision strategy
– The total compression cost pays by Au is
A customer pays at the completion of the project
– Rewards are offered in case of earliness (i.e., the project
ends before its worst completion time) = daily reward
– Rewards are shared among the agents in a predefined
way u = part of reward for Au
– Agent’s reward:
Agent’s objective
– Every agent want to maximize his profit – … so (possibly) shortening the project makespan ( SOF)
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Activity-on-arc graph with five activities : {a,b,c,d,e} Two agents : green (g) and red (r) Rewards and sharing: =120, wg=1/2, wr=1/2
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([px, px], Cx)
Multiobjective linear problem
– Note : single agent classical PERT-crashing (polynomial)
A collective strategy should be efficient
– There does not exist any other strategy that gives a better
profit for all the agents
– Pareto optimum
A collective strategy should be stable
– no agent can locally change his strategy, so improving his
profit
– Nash equilibrium
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Shortening the makespan
– Find a cut
for all Au, with :
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([px, px], Cx)
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([px, px], Cx)
t3-t0 Rewards Costg Costr Zg Zr 15
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([px, px], Cx)
t3-t0 Rewards Costg Costr Zg Zr 15
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([px, px], Cx)
t3-t0 Rewards Costg Costr Zg Zr 15 14 120 20 20 40 40
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([px, px], Cx)
t3-t0 Rewards Costg Costr Zg Zr 15 14 120 20 20 40 40
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([px, px], Cx)
t3-t0 Rewards Costg Costr Zg Zr 15 14 120 20 20 40 40
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([px, px], Cx)
t3-t0 Rewards Costg Costr Zg Zr 15 14 120 20 20 40 40 13 240 70 70 50 50
t3-t0 Rewards Costg Costr Zg Zr 15 14 120 20 20 40 40 13 240 70 70 50 50
S2 is a Pareto optimum … but not a Nash equilibrium
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t3-t0 Rewards Costg Costr Zg Zr 15 14 120 20 20 40 40 13 240 70 70 50 50
14 120 70 60
The green agent can improve his profit by modifying his strategy …
t0 t3 t1 t2 6 3 9 7 8 45
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t3-t0 Rewards Costg Costr Zg Zr 15 14 120 20 20 40 40 13 240 70 50 50 70
S2 is a Pareto optimum but not a Nash equilibrium S1 is not a Pareto optimum but a Nash equilibrium
GOAL: find a Nash equilibrium with minimum makespan
t0 t3 t1 t2 7 2 9 7 5
Introduction A multiagent project scheduling problem Complexity results Conclusion
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Is a strategy S a Pareto Optimum / Nash Equilibrium?
– Easy
Does there exist a Pareto Optimum / Nash Equilibrium?
– Easy
Does there exist a Nash equilibrium having a
makespan strictly lower than ?
– NP-complete in the strong sense – Reduction from 3-PARTITION
3-PARTITION
– Consider a set K of 3k integers such that
– Does there exist a partition in k subsets such that the sum of the
integer in every subset equals B ?
– Strongly NP-complete 22
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3-PARTITION
–
K=9, k=3, B=24 {7,7,7,7,8,8,9,9,10}
Reduction
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k agents in series
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Every agent manages K activities: pi[0, 1] and ci=ai
–
Makespan = k
–
Au reward: wu×=B+
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Q: Is there a NE such that makespan < k ?
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Find a cut such that the cost for each agent does not exceed the reward B+ Sol 3-PARTITION
Introduction A multiagent project scheduling problem Complexity results Conclusion
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Multiagent scheduling problem with controllable
processing times
–
Efficiency and stability notions
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Optimization problem = Find a Nash equilibrium that minimizes the project makespan
NP-hard in the strong sense
Other issues
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MIP formulation (wu = fixed parameter)
Finding the “best” project makespan
–
MIP formulation (wu = decision variables)
From the customer viewpoint
– What is the best reward sharing policy that offers the best (stable)
makespan ?
From one agent’s viewpoint
– What is the best profit (stable) the agent can expect given a project
makespan?
–
Distributed solving approach
–
Cooperative game theory
Agents can make coalitions
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