Distributed Group Analytical Hierarchical Process by Consensus

Introduction Related technologies GAHP by consensus Results Conclusions
Distributed Group Analytical Hierarchical Process
by Consensus
M. Rebollo, A. Palomares, C. Carrascosa
Universitat Polit`ecnica de Val`encia
DCAI, Toledo 2018
c b a
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Distributed Group Analytical Hierarchical Process by Consensus

Problem
Complex decisions in networks
Multicriteria optimization problem where criteria are organized into
a hierarchy and the decision is reduced to pairwise comparitions.
Problems to be addressed
local, private knowledge
agreement on the weights of the criteria
agreement on the values for the candidates
no central authority
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Outline
1 Analytical Hierarchical Process (AHP)
2 AHP by consensus
3 Comparative results
4 Conclusions
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Analytical Hierarchical Process (AHP)
Multi-objective optimization method for complex decisions
multi-objective
optimization
method
criteria structured
into a hierarchy
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Judgment Matrix
Tom Dick Harry Exp
Tom 1 1
4 4 0.217
Dick 4 1 9 0.717
Harry 1/4 1/9 1 0.066
candidates evaluated between 1 and 9 and compared with the rest
in a pairwise matrix
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Judgment Matrix
Exp Edu Char Age
Tom 0.217 0.188 0.743 0.265
Dick 0.171 0.081 0.194 0.672
Harry 0.066 0.731 0.063 0.063
This process is repeated for all the criteria
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Judgment Matrix
Exp Edu Char Age Goal
(0.547) (0.127) (0.270) (0.056)
Tom 0.119 0.024 0.201 0.015 0.358
Dick 0.392 0.010 0.052 0.038 0.492
Harry 0.036 0.093 0.017 0.004 0.149
values normalized depending on the relative relevance of the
criteria
higher value → best option
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Group Analytical Hierarchical Process (GAHP)
Multi-objective optimization method for complex decisions in a
group, with diﬀerent criteria
Experience
Education
Charisma
Age
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Consensus Process
1 each node has an initial
value xi (0)
2 exchanges xi (t) with its
neighbors
3 the new value is
calculated as
1 2
3 4
x1 = 0.4 x2 = 0.2
x3 = 0.3 x4 = 0.9
x2 = 0.2
x4 = 0.9
x3 = 0.3
xi (t + 1) = xi (t) + ε
j∈Ni
[xj(t) − xi (t)]
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Consensus Process
1 each node has an initial
value xi (0)
2 exchanges xi (t) with its
neighbors
3 the new value is
calculated as 0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x = 0.45
xi (t + 1) = xi (t) + ε
j∈Ni
[xj(t) − xi (t)]
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Multilayer Network
M = (G, C) where
G = {G1, . . . , Gp} is a set
of graphs
C = {Lαβ ⊆ Eα × Eβ
∀α, β ∈ [1, p], α = β} set
of connections between
Gα and Gβ
AHP criteria modeled as layers
Experience
Education
Charisma
Age
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Utility Function (local)
Gaussian local utility
functions
uα
i (xα
i ) = e
−1
2
xα
i
−lα
i
1−wα
i
2
n-Dimensional,
depending on the
number of criteria
ui (xi ) =
α
uα
i (xα
i )
Individual Utility Function for Each Agent
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Utility Function (global – unknown)
Global utility function as
the sum of the local ones
U =
i
ui (xi )
This function is never
calculated nor known by
the nodes
0
1
1
2
1
U
3
0.8
Initial Priorities of the Agents
4
0.5 0.6
5
0.4
0.2
0 0
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Combination of Consensus and Gradient
xi (0) at best ui (xi )
each layer executes a
consensus process
g(xα
1 , . . . , xα
n )
each node executes a
gradient ascent on its
private ui (xi )
h(x1
i , . . . , xp
i )
Individual Utility Function for Each Agent
xα
i (t+1) =
g(xα
i )
xα
i +
ε
wα
i j∈Nα
i
(xα
j (t) − xα
i (t)) + ϕ ui (x1
i (t), . . . , xp
i (t))
h(xα
i )
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Joint Solution
Agents begin in their
best decision
0
1
1
2
1
U
3
0.8
4
0.5 0.6
5
0.4
0.2
0 0
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Joint Solution
As consensus evolves,
agents agree with the
ﬁnal values for the
criteria
0 100 200 300 400 500
-1
0
1
2
x1
Consensus evolution on each priority
0 100 200 300 400 500
#epoch
-1
0
1
2
x2
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Joint Solution
If there are no local
optima, the process
converges to the best
solution
0
1
1
2
1
3
0.8
Final GAHP Priority by Consensus
4
0.5 0.6
5
0.4
0.2
0 0
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Break in Groups
When the global utility
function presents several
local maxima. . .
0
1
0.5
1
1
U
1.5
2
0.5
0.5
2.5
0 0
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Break in Groups
. . . the method allows
node i to break the links
with those neighbors in
which position the utility
of i is near zero
ui (xj) ≈ 0 → (i, j) = 0
0 100 200 300 400 500
-1
0
1
2
x1
Consensus evolution on each priority
0 100 200 300 400 500
#epoch
-1
0
1
2
x2
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Break in Groups
. . . and the network
splits apart into two (or
more) groups
0
1
0.5
1
1
1.5
0.8
Final GAHP Priority by Consensus
2
0.5 0.6
2.5
0.4
0.2
0 0
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Scenarios
Methods
aggregation of individual judgments (AIJ)
aggregation of individual priorities (AIP)
loss function approach (LFA)
GAHP with preferential diﬀerences and ranking (PDR)
Data aggregation
arithmetic mean
geometric mean
Measures
Pareto Optimality (Scenarios 1 and 2)
Homogeneity condition (Scenarios 3 and 4)
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Pareto Optimality
Agent Scenario A1 A2 A3 A4 Pareto Opt.
DM1 1 0.236 0.418 0.164 0.181 A1 > A3
2 0.161 0.421 0.159 0.259 A1 > A3
DM2 1 0.490 0.127 0.173 0.210 A1 > A3
2 0.485 0.140 0.204 0.174 A1 > A3
DM3 1 0.238 0.262 0.063 0.437 A1 > A3
2 0.190 0.248 0.104 0.458 A1 > A3
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Pareto Optimality
Method Scenario A1 A2 A3 A4 Pareto Opt.
AIJ(WAMM) 1 0.290 0.321 0.133 0.254 A1 > A3 Yes
2 0.181 0.353 0.184 0.281 A1 < A3 No
AIJ(WGMM) 1 0.279 0.344 0.130 0.246 A1 > A3 Yes
2 0.164 0.370 0.190 0.276 A1 < A3 No
AIP(WAMM) 1 0.312 0.300 0.146 0.241 A1 > A3 Yes
2 0.264 0.302 0.161 0.272 A1 > A3 Yes
AIP(WGMM) 1 0.318 0.288 0.149 0.244 A1 > A3 Yes
2 0.252 0.297 0.172 0.279 A1 > A3 Yes
LFA(WAMM) 1 0.253 0.343 0.128 0.275 A1 > A3 Yes
2 0.177 0.370 0.165 0.288 A1 > A3 Yes
LFA(WGMM) 1 0.289 0.343 0.133 0.234 A1 > A3 Yes
2 0.165 0.375 0.190 0.269 A1 < A3 No
PDR 1 0.370 0.264 0.041 0.325 A1 > A3 Yes
2 0.329 0.276 0.063 0.331 A1 > A3 Yes
COJ 1 0.238 0.262 0.063 0.436 A1 > A3 Yes
2 0.288 0.3651 0.152 0.1934 A1 > A3 Yes
COP 1 0.313 0.299 0.146 0.241 A1 > A3 Yes
2 0.505 0.048 0.312 0.174 A1 > A3 Yes
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Homogeneity
Agent Scenario A1 A2 A3 A4 Homogeneity
DM1 3 0.332 0.338 0.166 0.164 µ1,3 = 2.00
4 0.264 0.364 0.143 0.229 µ4,3 = 1.60
DM2 3 0.458 0.119 0.229 0.194 µ1,3 = 2.00
4 0.445 0.139 0.160 0.257 µ4,3 = 1.60
DM3 3 0.205 0.257 0.102 0.436 µ1,3 = 2.00
4 0.232 0.290 0.183 0.295 µ4,3 = 1.60
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Homogeneity
Method Scenario A1 A2 A3 A4 Homogeneity
AIJ(WAMM) 3 0.290 0.271 0.212 0.228 µ1,3 = 1.37 No
4 0.268 0.304 0.149 0.279 µ4,3 = 1.88 No
AIJ(WGMM) 3 0.302 0.311 0.171 0.216 µ1,3 = 1.77 No
4 0.268 0.320 0.145 0.267 µ4,3 = 1.84 No
AIP(WAMM) 3 0.344 0.256 0.172 0.228 µ1,3 = 2.00 Yes
4 0.312 0.282 0.156 0.250 µ4,3 = 1.60 Yes
AIP(WGMM) 3 0.352 0.248 0.176 0.223 µ1,3 = 2.00 Yes
4 0.312 0.270 0.161 0.258 µ4,3 = 1.60 Yes
LFA(WAMM) 3 0.273 0.293 0.163 0.271 µ1,3 = 1.68 No
4 0.243 0.331 0.139 0.286 µ4,3 = 2.05 No
LFA(WGMM) 3 0.287 0.322 0.171 0.220 µ1,3 = 1.68 No
4 0.266 0.330 0.146 0.259 µ4,3 = 1.77 No
PDR 3 0.352 0.238 0.073 0.337 µ1,3 = 4.83 No
4 0.371 0.267 0.074 0.288 µ4,3 = 3.91 No
COJ 3 0.331 0.338 0.165 0.166 µ1,3 = 2.00 Yes
4 0.199 0.330 0.183 0.287 µ4,3 = 1.57 (*)
COP 3 0.362 0.423 0.181 0.032 µ1,3 = 2.00 Yes
4 0.208 0.260 0.203 0.328 µ4,3 = 1.60 Yes
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Consensus vs. Geometric Mean-Based Methods
Solution generated
by consensus
optimizes the
utility function of
AHP, whereas AIP
methods agrees on
suboptimal values
based on the
geometric mean
on each dimension
of the problem
0 0.5 1
0
0.5
1
x1x1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
x2x2
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
x3x3
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
0 0.5 1
0
0.5
1
x4x4
Consensus vs Geom. mean solutionConsensus vs Geom. mean solution
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Conclusions
method based on a combination of consensus and gradient
ascent to solve GAHP
fully distributed, with local information
private decision criteria
Pareto optimality and homogeneity condition achieved
converges to the optimal of the global utility function (unique)
network automatically divided into groups → coalitions
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Distributed Group Analytical Hierarchical Process by Consensus

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