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
Introduction 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" C. Briand – PMS’2012 2
Multiagent scheduling problems 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 A 2 A 3 Time constraint A 1 Resource A 4 Use A 5 C. Briand – PMS’2012 3
Multiagent machine scheduling problems Multiagent single-machine scheduling problem A 3 A 2 A 1 A 4 Kangbok Lee, Byung-Cheon Choi, Joseph Y.-T. Leung, Michael L. Pinedo, Approximation algorithms for multi-agent scheduling to minimize total weighted completion time, Information Processing Letters, Volume 109, Issue 16, 31 July 2009. Agnetis A.,· de Pascale G., ·Pacciarelli D., 2009, A Lagrangian approach to single-machine scheduling problems with two competing agents, , J. of Sched., 12, pp 401-415. Cheng T. C. E., Ng C. T., Yuan J. J., 2008, Multiagent scheduling on a single machine with max-form criteria", European Journal of Operational Research, 188(2), pp. 603-609. Cheng T. C. E., Ng C. T., Yuan J. J., 2006, Multiagent scheduling on a single machine to minimize total weighted number of tardy jobs, Th. Computer Sci., 362, no 1-3, pp. 273-281. Agnetis A., Mirchandani P. B., Pacciarelli D., Pacifici, 2004, A. Scheduling problems with two competing agents. Operations Research, 52(2), pp. 229-242. C. Briand – PMS’2012 4
Multi-agent project scheduling 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 – A 2 A 3 A 1 A 4 A 5 C. Briand – ETFA'2011 5
Outline Introduction A multiagent project scheduling problem Complexity results A MIP for finding a minimum makespan NE Problem extensions Conclusion C. Briand – PMS’2012 6
Outlines Introduction A multiagent project scheduling problem Complexity results Conclusion C. Briand – PMS’2012 7
Problem definition A project composed of n time-dependent activities distributed amongst m agents ( m ≤ n ) Agents can control the duration of their activities – p i => p i = decision strategy – The total compression cost pays by A u 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 A u – Agent’s reward: Agent’s objective – Every agent want to maximize his profit – … so (possibly) shortening the project makespan ( SOF) C. Briand – PMS’2012 8
Example (inspired from Phillips and Dessouky, 1977) Activity-on-arc graph with five activities : { a,b,c,d,e } Two agents : green (g) and red (r) Rewards and sharing: =120, w g =1/2, w r =1/2 t 1 t 0 t 3 ([ p x , p x ], C x ) t i t j x t 2 C. Briand – PMS’2012 9
Strategy efficiency and stability 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 C. Briand – PMS’2012 10
Example (inspired from Phillips and Dessouky, 1977) Shortening the makespan – Find a cut on the critical graph such that for all A u , with : t 1 t 0 t 3 ([ p x , p x ], C x ) t i t j x t 2 C. Briand – PMS’2012 11
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 7 t 1 8 t 0 t 3 3 9 ([ p x , p x ], C x ) 5 t i t j x t 2 C. Briand – PMS’2012 12
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 7 t 1 8 t 0 t 3 3 9 ([ p x , p x ], C x ) 5 t i t j x t 2 C. Briand – PMS’2012 13
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 S 1 14 120 20 20 40 40 7 t 1 7 t 0 t 3 2 9 ([ p x , p x ], C x ) 5 t i t j x t 2 C. Briand – PMS’2012 14
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 S 1 14 120 20 20 40 40 7 t 1 7 t 0 t 3 2 9 ([ p x , p x ], C x ) 5 t i t j x t 2 C. Briand – PMS’2012 15
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 S 1 14 120 20 20 40 40 7 t 1 7 t 0 t 3 2 9 ([ p x , p x ], C x ) 5 t i t j x t 2 C. Briand – PMS’2012 16
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 S 1 14 120 20 20 40 40 S 2 13 240 70 70 50 50 6 t 1 7 t 0 t 3 3 9 ([ p x , p x ], C x ) 4 t i t j x t 2 C. Briand – PMS’2012 17
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 S 1 14 120 20 20 40 40 S 2 13 240 70 70 50 50 S 2 is a Pareto optimum … but not a Nash equilibrium 6 t 1 7 8 t 0 t 3 3 9 4 5 t 2 C. Briand – PMS’2012 18
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 S 1 14 120 20 20 40 40 S 2 13 240 70 70 50 50 The green agent can improve his profit by modifying his strategy … S’ 2 14 120 0 70 60 -10 6 t 1 7 8 t 0 t 3 3 9 4 5 t 2 C. Briand – PMS’2012 19
Example (inspired from Phillips and Dessouky, 1977) t 3 - t 0 Rewards Cost g Cost r Z g Z r S 0 15 0 0 0 0 0 S 1 14 120 20 20 40 40 S 2 13 240 70 50 50 70 S 2 is a Pareto optimum but not a Nash equilibrium S 1 is not a Pareto optimum but a Nash equilibrium 7 t 1 7 GOAL : find a Nash equilibrium with t 0 t 3 2 minimum makespan 9 5 t 2 C. Briand – PMS’2012 20
Outlines Introduction A multiagent project scheduling problem Complexity results Conclusion C. Briand – PMS’2012 21
Decision problems and complexity 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 3 k integers such that each integer a i ]B/4, B/2], for all i =1, …, K. • ∑ a i = kB • – Does there exist a partition in k subsets such that the sum of the integer in every subset equals B ? – Strongly NP-complete C. Briand – PMS’2012 22
Sketch of proof 3-PARTITION K =9, k =3, B =24 – {7,7,7,7,8,8,9,9,10} Reduction k agents in series – Every agent manages – K activities: p i [0, 1] and c i = a i Makespan = k – A u reward: w u × = B + – Q: Is there a NE such – that makespan < k ? Find a cut such that – the cost for each agent does not exceed the reward B + Sol 3-PARTITION C. Briand – PMS’2012 23
Outlines Introduction A multiagent project scheduling problem Complexity results Conclusion C. Briand – PMS’2012 24
Conclusion Multiagent scheduling problem with controllable processing times Efficiency and stability notions – Optimization problem = Find a Nash equilibrium that minimizes – the project makespan NP-hard in the strong sense Other issues MIP formulation ( w u = fixed parameter) – Finding the “best” project makespan MIP formulation ( w u = 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 C. Briand – PMS’2012 25
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