Chli, Maria

Winsper, Michael

English

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Using the Max-Sum Algorithm for Supply Chain Formationin Dynamic Multi-Unit Environments(Extended Abstract)Michael Winsper and Maria ChliAston UniversityAston TriangleBirmingham, United Kingdom B4 7ET{winsperm, m.chli}@aston.ac.ukABSTRACTThe max-sum loopy belief propagation (LBP) algorithm wasshown in [4] to produce strong results in a simple decen-tralised supply chain formation (SCF) scenario where goodsare traded in single units. In this paper, we demonstrate theperformance of LBP in a multi-unit SCF scenario with ad-ditional constraints. We also provide experimental analysisof LBP’s performance in dynamic scenarios where the prop-erties and composition of participants are altered while thealgorithm is running. Our results suggest that LBP contin-ues to produce strong solutions in multi-unit scenarios, andthat performance remains solid in a dynamic setting.Categories and Subject DescriptorsI.2.11 [Distributed Artificial Intelligence]: MultiagentsystemsGeneral TermsEconomics, Algorithms, ExperimentationKeywordsSupply Chain Formation, Loopy Belief Propagation1. INTRODUCTIONComputational approaches to SCF model potential supplychain participants as individual computational agents whichexpress their capabilities and costs through a mechanismwhich determines the subset of agents capable of formingthe most efficient supply chain. Although centralised SCFtechniques [1] have allowed for multi-unit exchanges for sometime, the existing state of the art in decentralised [4, 3] SCFonly model simple scenarios where goods are exchanged insingle units. Additionally, [4] does not model the effect ofchanges to the properties or composition of participants oncethe process has begun. In this paper, we propose a frame-work for the representation of multi-unit supply chains andextend the LBP-based technique for decentralised SCF pre-sented in [4] to the multi-unit case. We also present a set ofexperiments analysing the performance of LBP in a dynamicenvironment, where changes to the properties and compo-sition of participants occur while the algorithm is running.Appears in: Proceedings of the 11th International Con-ference on Autonomous Agents and Multiagent Systems(AAMAS 2012), Conitzer, Winikoff, Padgham, and van der Hoek (eds.),4-8 June 2012, Valencia, Spain.Copyright c© 2012, International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved.Our results demonstrate that LBP is capable of producingefficient allocations over a range of network topologies inboth static and dynamic environments.2. MODELThe use of task dependency networks (TDNs) for the rep-resentation of supply chains was originally proposed in [3].For the first time, we extend this TDN representation to themulti-unit case by modelling input to output good ratios,production capacities and consumer desired good quanti-ties. An example of the extended representation is shown inFigure 1. Values below producers and consumers representreserve prices and production capacities, and consumptionvalues and desired consumable good quantities. Edges fromgoods to producers are labelled with the producer’s input tooutput ratio for that good. A producer with a single inputand an input ratio of 2 for that good requires two units ofthat good in order to produce one unit of its output.Figure 1: The Simple supply chain TDN from [3] extended tothe multi-unit case. Producers (P[x]) and a consumer (C1)are represented by rectangles, while goods are representedby circles. Edges indicate potential flows of goods.Producers and Consumers At initialisation, each pro-ducer is assigned a production capacity which specifies themaximum number of units each producer is able to produceof its output good, and an input to output ratio for each oftheir inputs. In order to produce one unit of their outputgood, producers are required to acquire a number of unitsof each of their input goods equal to their ratio for thatgood. A producer cannot produce its output good unless itacquires the necessary quantities of all of its input goods.Producers attach a reserve price Rp to their output good,which is linear with the number of units of its output goodthat it produces. Consumers seek to acquire a number ofunits of their consumable good no greater than their desiredconsumable good quantity. In each network, each consumeris assigned a static consumption value Vc representing thevaluation the consumer holds for obtaining a single unit ofits consumable good. The total value a consumer receives islinear with the number of goods it obtains.3. APPLYING THE MAX-SUM ALGORITHMThe max-sum algorithm is a variant of loopy belief propa-gation (LBP), a decentralised and distributed approximateinference scheme involving applying Pearl’s belief propaga-tion algorithm [2] to graphs containing cycles. It uses iter-ative stages of message passing as a means for estimatingthe maximum a posteriori assignment; in our case, this cor-responds to the network-wide state configuration that max-imises Eq. 3. Each state encodes a combination of purchasesand sales which may be made by an agent. States are associ-ated with costs - the unary cost, representing the cost to theallocation of the agent being in that state, and the pairwisecost, which encodes the compatibility of two states of neigh-bouring agents. At each iteration of the algorithm, everynode in the graph sends a message to each of its neighbours,representing the sender’s beliefs about the potential cost tothe total efficiency of the network of each of the recipient’sstates. This is calculated using Eq. 1, where xv is a state ofrecipient j, beli(xu) is sender i’s belief in its own state xu,mj→i(xu) is the message passed from j to i in the previousstep about i’s state xu, and gij(xu, xv) is the pairwise costof states xu and xv. Once values have been calculated forall of j’s states, i passes the message to j.mi→j(xv) = minxu(beli(xu)−mj→i(xu)+gij(xu, xv))(1)Once all nodes have sent a message to each of their neigh-bours, nodes then update their beliefs about their own statesbased upon the content of the messages they received usingEq. 2, where fi(xu) is the unary cost of i’s state xu, andmj→i(xu) are the messages received from i’s set of neigh-bours Nu about state xu in the previous step. The pro-cess of message passing and belief update continues untilthe beliefs of each node stabilise. For more information onapplying LBP to SCF, we refer the reader to [4].beli(xu) = fi(xu) +∑j∈Numj→i(xu) (2)Allocation Before allocation is performed, each agent de-termines their final state - the state, when beliefs stabilise,which the agent believes holds the lowest cost. Once thefinal states of each of the agents have been determined, weclassify producers which successfully sell their output goodand consumers which acquire their consumable good as ac-tive. We calculate allocation values using Eq. 3, where Cis the set of active consumers C, Ac is the number of goodsacquired by consumer c, P is the set of active producers andMp is the number of goods manufactured by producer p.V al =∑c∈CVcAc −∑p∈PRpMp (3)4. RESULTSWe perform two sets of experiments, examining the perfor-mance of LBP in both static and dynamic environments. Inthe static environment, we compare LBP with a multi-unitimplementation of the SAMP-SB auction protocol from [3],extended by using multiple copies of each agent to representcapacities and desired good quantities. This representationdoes not allow for the use of input to output good ratios, sofor fair comparison we also test LBP with all ratios set to1, referred to as ratioless LBP. We test each technique over100 runs on each of the networks from [3]. We vary inputratios (drawn from [1 . . . 2]), consumer desired goods (from[2 . . . 3]), reserve prices (from U(0, 1)), and production ca-pacities ([4 . . . 5]) between each run, discarding runs in whichthe optimal allocation value, determined using mixed integerprogramming, is non-positive. Consumption values are fixedat the per-network values given in [3] over every run. In thedynamic environment, at a number (drawn from [6 . . . 10]) ofrandomly chosen steps during each run we randomly changeone of the aforementioned properties of a single agent, in-troduce a new producer or consumer, or remove a producer.We present our results in terms of average efficiency, cal-culated by dividing the total allocation values produced byeach method over 100 runs on each network by the maximumavailable value over the same 100 runs.Table 1: Average efficiency in each network produced byLBP and the SAMP-SB auction protocol from [3] on thenetworks from [3]. A result of 1.000 equates to the captureof 100% of available efficiency.Static Static Dynamic StaticLBP LBP LBP SAMP-SBNetwork ratioless with ratios with ratios ratiolessSimple 1.000 1.000 0.911 0.999Unbalanced 0.962 0.872 0.713 0.964Two-Cons 0.986 0.983 0.801 0.963Bigger 0.969 0.813 0.520 0.995Many-Cons 1.000 1.000 0.989 0.425Greedy-Bad 0.91 0.839 0.793 0.666Table 1 shows the average efficiency produced by each method.We see that both static LBP-based methods are able tomatch or outperform SAMP-SB on the majority of networks.As expected, LBP finds the optimal allocation 100% of thetime in static scenarios on acyclic networks, while still beingable to produce highly efficient allocations on more loopynetworks. We also see that LBP tended to perform betterwhen input to output good ratios are not present; this is tobe expected since the presence of ratios serves to constrainthe number of solutions available. In the dynamic setting,we see that for most networks, average efficiency is roughlycomparable to the results produced in a static environment.5. REFERENCES[1] J. Cerquides, U. Endriss, A. Giovannucci, andJ. Rodr´ıguez-Aguilar. Bidding Languages and WinnerDetermination for Mixed Multi-Unit CombinatorialAuctions. In Proceedings of the 20th International JointConference on Artificial Intelligence, pages 469–476,2007.[2] J. Pearl. Probabilistic Reasoning in Intelligent Systems:Networks of Plausible Inference, volume 1. MorganKaufmann, 1st edition, 1988.[3] W. Walsh and M. Wellman. Decentralized SupplyChain Formation: A Market Protocol and CompetitiveEquilibrium Analysis. Journal of Artificial IntelligenceResearch, 19:513–567, 2003.[4] M. Winsper and M. Chli. Decentralized Supply ChainFormation using Max-Sum Loopy Belief Propagation.Computational Intelligence, In Press, 2012.

Using the max-sum algorithm for supply chain formation in dynamic multi-unit environments

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Using the max-sum algorithm for supply chain formation in dynamic multi-unit environments

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