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.
A Scalable Message-Passing Algorithm for Supply Chain Formation (AAAI2012)
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.
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
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
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
36. FINAL
RECA
P
Encoding in a novel graphical model.
Optimized implementation of max-sum.
Reduced memory, communication and
computation requirements.
Higher quality solutions.
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.