RB-LBP

Presentation as given at AAAI2012

Supply Chain Formation (SCF) is the process of determining the participants in a supply chain, who will exchange what with whom, and the terms of the exchanges. Decentralized SCF appears as a highly intricate task because agents only possess local information and have limited knowledge about the capabilities of other agents. The decentralized SCF problem has been recently cast as an optimization problem that can be efficiently approximated using max-sum loopy belief propagation. Along this direction, in this paper we propose a novel encoding of the problem into a binary factor graph (containing only binary variables) as well as an alternative algorithm. We empirically show that our approach allows to significantly increase scalability, hence allowing to form supply chains in market scenarios with a large number of participants and high competition.

Transcript

1. by Toni Penya-Alba, Meritxell Vinyals, Jesus Cerquides, and Juan A. Rodriguez-Aguilar A Scalable Message-Passing Algorithm for Supply Chain Formation for the 26th AAAI Conference   
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2. Alice Carol Bob Dave Eve $8 $7 -$5 -$3 -$2 THE PROBLEM
3. The Supply Chain Formation problem is that of finding the feasible configuration with maximum value. Alice Carol Bob Dave Eve $8 $7 -$5 -$3 -$2 THE PROBLEM
4. THE PROBLEM Alice Carol Eve $8 -$5 -$2 Supply Chain Value = $8 - $5 - $2 = $1
5. To provide a scalable method for Supply Chain Formation in markets with high degrees of competition. THE GOAL
6. CONTRIBUTION Encoding in a novel graphical model. Optimized implementation of max-sum. BENEFITS Reduced memory, communication and computation requirements. Higher quality solutions.
7. factor graph is a graphical model to represent functions. max-sum is a message passing algorithm. max-sum can efficiently approximate optimization problems. x f g y z f(x,y) + g(y,z)
8. Loopy Belief Propagation Michael Winsper Maria Chli
9. map the Supply Chain Formation problem into a factor graph. apply max-sum over the factor graph to obtain a solution. LOOPY BELIEF PROPAGATION
10. MAPPING INTO A FACTOR GRAPH Alice Carol Bob Dave Eve $8 $7 -$5 -$3 -$2
11. Carol Bob Alice Eve Dave FACTOR GRAPH MAPPING
12. exchanges Carol Bob Alice Eve Dave Agent variable encodes all of Carol’s possible exchanges. do nothing buy from Alice, sell to Dave buy from Alice, sell to Eve buy from Bob, sell to Dave buy from Bob, sell to Eve 0 1 2 3 4 FACTOR GRAPH MAPPING
13. exchanges -5 Carol Bob Alice Eve Dave Activation factor encodes Carol’s activation cost. 0 -5 inactive otherwise FACTOR GRAPH MAPPING
14. exchanges -5 Carol exchanges 7 exchanges 8 exchanges exchanges Bob Alice Eve Dave -2 -3 FACTOR GRAPH MAPPING
15. exchanges CE CD BC AC -5 Carol exchanges 7 exchanges 8 exchanges exchanges Bob Alice Eve Dave -2 -3 Compatibility factors encode compatibility between Carol’s states and her neighbors’. FACTOR GRAPH MAPPING
16. LBP memory per agent O(G•A2G+1) bandwidth per agent O(G•AG+1) computation time per agent O(G•A2G+1)
17. RB-LBP Our Approach
18. map problem into a binary factor graph containing binary variables and logical constraints. optimized max-sum implementation. apply max-sum over the factor graph to obtain a solution. RB-LBP OUR APPROACH
19. MAPPING INTO A BINARY FACTOR GRAPH Alice Carol Bob Dave Eve $8 $7 -$5 -$3 -$2
20. Carol Bob Alice Eve Dave BINARY FACTOR GRAPH MAPPING
21. activate Carol Bob Alice Eve Dave Activation variable encodes Carol’s decision to be active. BINARY FACTOR GRAPH MAPPING
22. activate -5 Carol Bob Alice Eve Dave Activation factor encodes Carol’s activation cost. BINARY FACTOR GRAPH MAPPING
23. sell to E sell to D buy from B buy from A activate -5 Carol Bob Alice Eve Dave Option variables encode Carol’s decision to trade each of her goods with each of her potential partners. BINARY FACTOR GRAPH MAPPING
24. sell to E sell to D buy from B buy from A activate S S -5 Carol Bob Alice Eve Dave Selection factors guarantee that only one of the providers is selected for each good. BINARY FACTOR GRAPH MAPPING
25. buy from C buy from C sell to E sell to D buy from B buy from A activate S S -5 Carol activate 7 S activate 8 S sell to C sell to C activate -2 S activate -3 S Bob Alice Eve Dave BINARY FACTOR GRAPH MAPPING
26. buy from C buy from C sell to E sell to D buy from B buy from A activate S = = = = S -5 Carol activate 7 S activate 8 S sell to C sell to C activate -2 S activate -3 S Bob Alice Eve Dave Equality factors guarantee that Carol takes coherent decisions with her neighbors’. BINARY FACTOR GRAPH MAPPING
27. There is no need to store factors in memory. Factors as logical constraints Simplified message calculation Contain agents’ willingness to collaborate with each other. Single-valued messages }
28. LBP RB-LBP memory per agent O(G•A2G+1) O(G•A) bandwidth per agent O(G•AG+1) O(G•A) computation time per agent O(G•A2G+1) O(G•A2)
29. Experimental Analysis
30. EXPERIMENTAL SETUP Small networks [Walsh2003]. Large networks [Vinyals2008]. Measure memory, communication, time, and quality.
31. networks goods agents small <20 <40
32. less memory less bandwidth solution quality time differences up to 13 times up to 5 times same negligible SMALL NETWORKS
33. networks goods agents large 50 40-500
34. up to 105 times up to 787 times up to 20 times up to twice less memory less bandwidth faster better solutions LARGE NETWORKS
35. up to 105 timesless memory LARGE NETWORKS
36. FINAL RECAP Encoding in a novel graphical model. Optimized implementation of max-sum. Reduced memory, communication and computation requirements. Higher quality solutions.
37. Coming soon ...
38. Coming soon ... ... Better valued solutions
39. ThankYou
40. Questions?
41. [Winsper2003] M. Winsper and M. Chli, Decentralised Supply Chain Formation: A Belief Propagation-based Approach, Agent-Mediated Electronic Commerce, 2010. [Walsh2003] W. E. Walsh and M. P. Wellman, Decentralized Supply Chain Formation : A Market Protocol and Competitive Equilibrium Analysis, Journal of Artificial intelligence Research (JAIR), vol. 19, pp. 513-567, 2003. [Vinyals2008] M. Vinyals, A. Giovannucci, J. Cerquides, P. Meseguer, and J. A. Rodriguez-Aguilar, A test suite for the evaluation of mixed multi-unit combinatorial auctions, Journal of Algorithms, vol. 63, no. 1-3, pp. 130-150, 2008.