Example of Weighted Voting System

Undersea target detection system

Weighted Voting SystemWeighted Voting System

-system output (0,1,x)

-voting units outputs (0,1,x)

d1(I) d2(I) d3(I) d4(I) d5(I) d6(I)

I

D(I)

w1 w2 w3 w4 w5 w6

unit 1 unit 2 unit 3 unit 4 unit 5 unit 6

-threshold

-system input (0,1)

-weights

.0if,

0),(if,0

0),(if,1

)(01

01011

01011

nn

nnnnn

nnnnn

WWx

WWWWW

WWWWW

ID

Decision Making Rule

)1)((11

1

n

jjjn IdwW Total weight of units voting for

the proposition acceptance

)0)((11

0

n

jjjn IdwW Total weight of units voting for

the proposition rejection

System output

Decision Making Rule

0)( ID if (1-)Wn1-Wn

0<0

Wn0

Wn1

Accept

Reject

xID )( if Wn1=Wn

0=0

1)( ID otherwise

WVS as a Multi-state System

Voting unit j:

3 states:

4 failure modes:

dj(0)=1; dj(1)=0;

dj(0)=x; dj(1)=x.

x

d j 10

(1-)Wn1-Wn

0

Entire WVS:

Multiple states characterized by different scores

3 possible outputs:

4 failure modes:

D(0)=1; D(1)=0;

D(0)=x; D(1)=x.

x

D 10

Input I

10I

Asymmetric Weighted Voting System

-system output (0,1,x)

-voting units outputs (0,1,x)

-acceptance weights

d1(I) d2(I) d3(I) d4(I) d5(I) d6(I)

w11 w1

2 w13 w1

4 w15 w1

6

I

D(I)

w01 w0

2 w03 w0

4 w05 w0

6

unit 1 unit 2 unit 3 unit 4 unit 5 unit 6

-threshold

-system input (0,1)

-rejection weights

.0if,

0),(if,0

0),(if,1

)(01

01011

01011

nn

nnnnn

nnnnn

WWx

WWWWW

WWWWW

ID

Decision Making Rule

)1)((11

11

n

jjjn IdwW Total weight of units voting for

the proposition acceptance

)0)((11

00

n

jjjn IdwW Total weight of units voting for

the proposition rejection

System output

Types of Errors

dj(0)=1 (unit fails stuck-at-1) too optimistic q01(j)

dj(1)=0 (unit fails stuck-at-0) too pessimistic q10(j)

dj(I)=x (unit fails stuck-at-x) too indecisive q1x(j), q0x(j)

Voting unit parameters

Decision making time tj

Rejection weight wj0

Acceptance weight wj1

System threshold

System Parameters

adjustable

Universal generating function technique

Score distribution for m voters

01

)()()()( 000

0)1(

01jj w

xw

j zjqzjqzjqzu

)()()(

)...()()(

)()(

1

112

11

zuzUzU

zuzUzU

zuzU

mmm

K

k

H

h

K

k

jkGihGjkih

jkGjk

H

h

ihGihji zsszszszUzU

1 1 11)()()(

Score distribution for a single voter

Composition operator

)w01,w11,…,w0n,w1n, ) = arg{R(w01,w11,…,w0n,w1n, )max}

))(Pr( IIDR System Success Probability

Optimal adjustment problem

Optimization problemsOptimization problems

1 4 5 2 6 3

w1 w4 w5 w2 w6 w3

1 2 3

21

0

Optimal grouping

R(w,,)max

VU 1 VU 2 VU 3 VU 4 VU 5 VU 6

w1 w2 w3 w4 w5 w6

P

d1(P) d2(P) d3(P) d4(P) d5(P) d6(P)

D(P)

PG 3PG2PG1

v

Optimal distribution among protected groups

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3

v

S

M=1

M=2

M=5 M=4

Optimal distribution among protected groups

Group vulnerability

M-number of groups

Order of voting decisions

)1)((11

11

m

jjjm IdwW Total weight of units with tjtm voting

for the proposition acceptance

)0)((11

00

m

jjjm IdwW Total weight of units with tjtm voting

for the proposition rejection

t1

t2tm

tn

……

n

mjm

jwV

1

111

n

mjm

jwV

1

001

Wm0

Wm0

Reject

V1m+1

Wm0

Accept

V0m+1

Wm0

Accelerated Decision Making

Qijm probability of making the decision

D(i)=j at the time tm

p0, p1 - input distribution

System reliability and expected decision time

mmm

n

mm

mmn

mtQQptQQpT )()( 1110

110100

10

mk

m

mk

mk QpQptR 11

1100

10)(

)w01,w11,…,w0n,w1n, ) = arg{R(w01,w11,…,w0n,w1n, )max}subject to T(w01,w11,…,w0n,w1n, )T*

RT

Voting system optimization problem

R max

T minTwo-objective problem:

Constrained problem: R max | T<T*

Numerical Example

No of unittjq01q0xq10q1x

1100.220.310.290.12

2120.350.070.1030.30

3380.240.080.220.15

4480.100.050.200.01

5550.080.100.150.07

6700.080.010.100.05

p0=0.7 p0=0.5p0=0.3

0.760.500.45

Q000.97980.90050.8477

Q010.02020.09950.1523

Q100.16110.07190.0283

Q110.83890.92810.97166

R0.93750.91430.9345

T34.99434.98734.994

Parameters of voting units

Parameters of optimal systemfor T*=35

0.77

0.82

0.87

0.92

0.97

15 20 25 30 35 40 45 50 55 60T

R

Po=0.7 Po=0.5 Po=0.3

Reliability vs. expected decision time

References1. Weighted voting systems: reliability versus rapidity, G. Levitin, Reliability Engineering &

System Safety, 89(2) pp.177-184 (2005).

2. Maximizing survivability of vulnerable weighted voting systems, G. Levitin, Reliability Engineering & System Safety, vol. 83, pp.17-26, (2003).

3. Threshold optimization for weighted voting classifiers, G. Levitin, Naval Research Logistics, vol. 50 (4), pp.322-344, (2003).

4. Asymmetric weighted voting systems, G. Levitin, Reliability Engineering & System Safety, vol. 76, pp.199-206, (2002).

5. Evaluating correct classification probability for weighted voting classifiers with plurality voting, G. Levitin, European Journal of Operational Research, vol. 141, pp.596-607, (2002).

6. Analysis and optimization of weighted voting systems consisting of voting units with limited availability, G. Levitin, Reliability Engineering & System Safety, vol. 73, pp. 91-100, (2001).

7. Optimal unit grouping in weighted voting systems, G. Levitin, Reliability Engineering & System Safety vol. 72, pp. 179-191, (2001).

8. Reliability optimization for weighted voting system, G. Levitin, A. Lisnianski, Reliability Engineering & System Safety, vol. 71, pp. 131-138, (2001).