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A Scalable Message-PassingAlgorithm for Supply Chain Formation by Toni Penya-Alba, Meritxell Vinyals, Jesus Cerquides, and Juan A. Rodriguez-Aguilar for the 26th AAAI Conference
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THE PROB LEMAlice Dave-$2 $7 Carol -$5Bob Eve-$3 $8
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THE PROB L EM Alice Dave -$2 $7 Carol -$5 Bob Eve -$3 $8The Supply Chain Formation problem is that of ﬁndingthe feasible conﬁguration with maximum value.
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THE PROB L EMAlice-$2 Carol -$5 Eve $8 Supply Chain Value = $8 - $5 - $2 = $1
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THE GOALTo provide a scalable methodfor Supply Chain Formationin markets with highdegrees of competition.
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RI BUT IONCO NT Encoding in a novel graphical model. Optimized implementation of max-sum. BENEFReduced memory, communication and ITScomputation requirements.Higher quality solutions.
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factor graph is a graphical model torepresent functions. f g f(x,y) + g(y,z) x y zmax-sum is a message passing algorithm.max-sum can efficiently approximateoptimization problems.
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LoopyBeliefPropagation Michael Winsper Maria Chli
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LOOPY BELIEF PROPAGATIONmap the Supply Chain Formationproblem into a factor graph.apply max-sum over the factor graphto obtain a solution.
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MAPPING INTO A FACTOR GRAPH Alice Dave -$2 $7 Carol -$5 Bob Eve -$3 $8
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FACTOR GRAPH MAPPING Alice Carol Dave exchanges Bob Eve 0 do nothing 1 buy from Alice, sell to Dave 2 buy from Alice, sell to Eve 3 buy from Bob, sell to Dave 4 buy from Bob, sell to EveAgent variable encodes all of Carol’s possible exchanges.
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FACTOR GRAPH MAPPING Alice Carol Dave exchanges Bob Eve -5 inactive 0 otherwise -5Activation factor encodes Carol’s activation cost.
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FACTOR GRAPH MAPPING Alice Carol Dave-2 exchanges exchanges 7 exchanges Bob Eve-3 exchanges -5 exchanges 8
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FACTOR GRAPH MAPPING Alice Carol Dave -2 exchanges AC CD exchanges 7 exchanges Bob Eve -3 exchanges BC -5 CE exchanges 8Compatibility factors encode compatibility betweenCarol’s states and her neighbors’.
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LBP memory per O(G•A 2G+1) agentbandwidth per O(G•A G+1) agent computation O(G•A 2G+1)time per agent
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RB-LBP OUR APPROACHmap problem into a binary factorgraph containing binary variablesand logical constraints.optimized max-sum implementation.apply max-sum over the factor graphto obtain a solution.
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MAPPING INTO A BINARY FACTOR GRAPH Alice Dave -$2 $7 Carol -$5 Bob Eve -$3 $8
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BINARY FACTOR GRAPH MAPPINGAlice Carol DaveBob Eve
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BINARY FACTOR GRAPH MAPPING Alice Carol Dave activate Bob EveActivation variable encodes Carol’s decision to be active.
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BINARY FACTOR GRAPH MAPPING Alice Carol Dave activate Bob Eve -5Activation factor encodes Carol’s activation cost.
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BINARY FACTOR GRAPH MAPPING Alice Carol Dave buy sell from A to D activate Bob Eve buy from B -5 sell to EOption variables encode Carol’s decision to trade each ofher goods with each of her potential partners.
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BINARY FACTOR GRAPH MAPPING Alice Carol Dave buy sell from A to D S activate S Bob Eve buy from B -5 sell to ESelection factors guarantee that only one of theproviders is selected for each good.
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BINARY FACTOR GRAPH MAPPING Alice Carol Dave-2 activate S sell to C buy from A sell to D buy from C S activate 7 S activate S Bob Eve-3 activate S sell to C buy from B -5 sell to E buy from C S activate 8
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BINARY FACTOR GRAPH MAPPING Alice Carol Dave -2 activate S sell to C = buy from A sell to D = buy from C S activate 7 S activate S Bob Eve -3 activate S sell to C = buy from B -5 sell to E = buy from C S activate 8Equality factors guarantee that Carol takes coherentdecisions with her neighbors’.
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}Factors as logical constraintsThere is no need to store factors inmemory. SimplifiedSingle-valued messages messageContain agents’ willingness to calculationcollaborate with each other.
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LBP RB-LBP memory per O(G•A 2G+1) O(G•A) agentbandwidth per O(G•A G+1) O(G•A) agent computation O(G•A 2G+1) O(G•A2)time per agent
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FINAL RECA PEncoding in a novel graphical model.Optimized implementation of max-sum.Reduced memory, communication andcomputation requirements.Higher quality solutions.
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[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 ChainFormation : A Market Protocol and Competitive Equilibrium Analysis, Journal ofArtificial 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-unitcombinatorial auctions, Journal of Algorithms, vol. 63, no. 1-3, pp. 130-150,2008.
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