Decentralized Supply Chain Formation Using Max‐sum Loopy Belief Propagation

Abstract

Supply chain formation is the process by which a set of producers within a network determine the subset of these producers able to form a chain to supply goods to one or more consumers at the lowest cost. This problem has been tackled in a number of ways, including auctions, negotiations, and argumentation-based approaches. In this paper we show how this problem can be cast as an optimization of a pairwise cost function. Optimizing this class of energy functions is NP-hard (Boykov et al., 2001) but efficient approximations to the global minimum can be obtained using loopy belief propagation (LBP). Here we detail a max-sum LBP-based approach to the supply chain formation problem, involving decentralized message-passing between supply chain participants. Our approach is evaluated against a well-known decentralized double-auction method and an optimal centralized technique, showing several improvements on the auction method: it obtains better solutions for most network instances which allow for competitive equilibrium † while also optimally solving problems where no competitive equilibrium exists, for which the double-auction method frequently produces inefficient solutions.

Using the Max-Sum Algorithm for Supply Chain Emergence in Dynamic Multiunit Environments

“…In [2], an LBP-based approach was applied to the SCF problem, noting that the passing of messages in LBP is comparable to the placing of bids in standard auction-based approaches. The decentralized and distributed nature of LBP also allows for the avoidance of the scalability issues present in centralized approaches such as combinatorial auctions.…”

“…Reference [20] has established worst-case bounds on the quality of solutions produced by max-sum LBP, although these guarantees hold only when all unary and pairwise potentials are nonnegative, which is not the case in our model. Despite these limitations, LBP has seen great success in a number of areas, including turbo codes, low density parity check codes and stereo vision, as well as in communication in sensor networks [15], [24], and more recently, in supply chain emergence [2], [25]- [27].…”

“…In [2], it was shown that when the SCF problem is cast as an optimization of a pairwise cost function, max-sum loopy belief propagation (LBP) is capable of quickly finding consistently optimal allocations in a decentralized manner over a wide range of network structures. However, this paper examines a very simple scenario in which goods are only traded one unit at a time, production capacities are not accounted for, input to output ratio is always assumed to be 1:1 and grants agents little autonomy-agents are forced to actively participate for the entirety of the process and are unable to change any of their properties, such as the price they wish to charge for the goods they sell or their capacity, once the process has begun.…”

“…The max-sum LBP-based technique for decentralized SCF presented in [2] is used as a starting point and it is extended to accommodate the multiunit, variable capacity and input-output ratio scenario. We also present a set of experiments demonstrating the performance 2168-2216 c 2015 IEEE.…”

A Tutorial on Optimization for Multi-Agent Systems

“…Diminishing the negative eect of these two issues is subject of current research. Thus, approximate distributed solvers, based in parallel auctions [102] or max-sum [104,105], have been proposed for certain types of MMUCAs. Despite their expressiveness, one of the main drawbacks of MMUCAs is that they are not able to manage time information, although there is ongoing work to extend the bidding language and WDP in that direction [106].…”