The document proposes a scalable message-passing algorithm called RB-LBP for solving supply chain formation problems. RB-LBP maps the problem into a novel binary factor graph and implements an optimized max-sum algorithm. This reduces memory, communication, and computation requirements compared to existing loopy belief propagation approaches. Experimental results on small and large networks show RB-LBP uses up to 105 times less memory and 787 times less bandwidth while obtaining higher quality solutions in some cases.

1. A Scalable Message-Passing
Algorithm for Supply Chain Formation
by Toni Penya-Alba, Meritxell Vinyals,
Jesus Cerquides, and Juan A. Rodriguez-Aguilar
for the 26th AAAI Conference

2. THE PROB
LEM
Alice Dave
-$2 $7
Carol
-$5
Bob Eve
-$3 $8

3. THE PROB
L EM
Alice Dave
-$2 $7
Carol
-$5
Bob Eve
-$3 $8
The Supply Chain Formation problem is that of finding
the feasible configuration with maximum value.

4. THE PROB
L EM
Alice
-$2
Carol
-$5
Eve
$8
Supply Chain Value = $8 - $5 - $2 = $1

5. THE GOAL
To provide a scalable method
for Supply Chain Formation
in markets with high
degrees of competition.

6. RI BUT ION
CO NT
Encoding in a novel graphical model.
Optimized implementation of max-sum.
BENEF
Reduced memory, communication and ITS
computation requirements.
Higher quality solutions.

7. factor graph is a graphical model to
represent functions.
f g
f(x,y) + g(y,z)
x y z
max-sum is a message passing algorithm.
max-sum can efficiently approximate
optimization problems.

8. Loopy
Belief
Propagation
Michael Winsper
Maria Chli

9. LOOPY BELIEF PROPAGATION
map the Supply Chain Formation
problem into a factor graph.
apply max-sum over the factor graph
to obtain a solution.

10. MAPPING INTO A FACTOR GRAPH
Alice Dave
-$2 $7
Carol
-$5
Bob Eve
-$3 $8

11. FACTOR GRAPH MAPPING
Alice Carol Dave
Bob Eve

12. 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 Eve
Agent variable encodes all of Carol’s possible exchanges.

13. FACTOR GRAPH MAPPING
Alice Carol Dave
exchanges
Bob Eve
-5
inactive 0
otherwise -5
Activation factor encodes Carol’s activation cost.

14. FACTOR GRAPH MAPPING
Alice Carol Dave
-2 exchanges exchanges
7
exchanges
Bob Eve
-3 exchanges
-5 exchanges 8

15. FACTOR GRAPH MAPPING
Alice Carol Dave
-2 exchanges
AC CD exchanges
7
exchanges
Bob Eve
-3 exchanges
BC -5 CE exchanges 8
Compatibility factors encode compatibility between
Carol’s states and her neighbors’.

16. LBP
memory per O(G•A 2G+1)
agent
bandwidth per O(G•A G+1)
agent
computation O(G•A 2G+1)
time per agent

17. RB-LBP
Our Approach

18. RB-LBP OUR APPROACH
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.

19. MAPPING INTO A BINARY FACTOR GRAPH
Alice Dave
-$2 $7
Carol
-$5
Bob Eve
-$3 $8

20. BINARY FACTOR GRAPH MAPPING
Alice Carol Dave
Bob Eve

21. BINARY FACTOR GRAPH MAPPING
Alice Carol Dave
activate
Bob Eve
Activation variable encodes Carol’s decision to be active.

22. BINARY FACTOR GRAPH MAPPING
Alice Carol Dave
activate
Bob Eve
-5
Activation factor encodes Carol’s activation cost.

23. BINARY FACTOR GRAPH MAPPING
Alice Carol Dave
buy sell
from A to D
activate
Bob Eve
buy
from B -5 sell
to E
Option variables encode Carol’s decision to trade each of
her goods with each of her potential partners.

24. BINARY FACTOR GRAPH MAPPING
Alice Carol Dave
buy sell
from A to D
S activate
S
Bob Eve
buy
from B -5 sell
to E
Selection factors guarantee that only one of the
providers is selected for each good.

25. 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

26. 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
Equality factors guarantee that Carol takes coherent
decisions with her neighbors’.

27. }
Factors as logical constraints
There is no need to store factors in
memory. Simplified
Single-valued messages message
Contain agents’ willingness to calculation
collaborate with each other.

28. LBP RB-LBP
memory per O(G•A 2G+1) O(G•A)
agent
bandwidth per O(G•A G+1) O(G•A)
agent
computation O(G•A 2G+1) O(G•A2)
time per agent

29. Experimental
Analysis

30. EXPERIMENTAL SETUP
Small networks [Walsh2003].
Large networks [Vinyals2008].
Measure memory, communication,
time, and quality.

31. small networks
<20 goods
<40 agents

32. SMALL NETWORKS
up to 13 times less memory
up to 5 times less bandwidth
same solution quality
negligible time differences

33. large networks
50 goods
40-500 agents

34. LARGE NETWORKS
up to 105 times less memory
up to 787 times less bandwidth
up to 20 times faster
up to twice bettersolutions

35. LARGE NETWORKS
up to 105 timesless memory

36. FINAL
RECA
P
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.