Holistic reliability analysis of weighted voting systems from a multi-state perspective

Figures & data

Table 1 Notation

Variable	Definition
j	The jth decision-making unit, j = 1,…,n
I	Binary proposition for WVS
d j (I)	Decision from the jth decision-making unit
D(I)	Decision from the WVS
w j ^0	Weight associated with d j (I) = 0
w j ^1	Weight associated with d j (I) = 1
τ	WVS threshold
R	WVS reliability
q im ^j	P (d j (I) = m|I = i), i = 0,1; m = 0, x, 1
Q im	P (D (I) = m|I = i)
p i	P (D(I) = I)
x j	Current state of the jth decision-making unit
x	System state vector, x = (x 1, x 2,…,x n )
q i ^j	Probability vector of the jth decision-making unit associated with I = i
d i	Demand for case I = i.
b j	State vector for the jth decision-making unit
b jm	The mth state of the jth decision-making unit
ϕ	WVS structure function
Q i	Failure counter
Γ	Simulation runs
G	Index for optimization heuristic

Fig. 1 Multi-state parallel system model for a WVS.

Fig. 2 Transformation of WVS security system into a multi-state system.

Table 2 Data for security detection system

Detector	q 10 ^j	q 1x ^j	q 11 ^j	q 00 ^j	q 0x ^j	q 01 ^j	w j ^1	w j ^0
^1Visual	0.02	0.01	0.97	0.97	0.01	0.02	5	2
^2Heat	0.05	0.02	0.93	0.96	0.02	0.02	1	3

Table 3 Cut vectors for WVS under case d _i

Case d 0				Case d 1
x	x 1	x 2	ϕ (x)	X	x 1	x 2	ϕ (x)
1	2	−1.8	0.2	1	−1.2	0.4	−0.8
2	0	0	0	2	0	0	0
3	2	0	2	3	−1.2	0	−1.2
4	0	0.4	0.4	4	0	−1.8	−1.8
5	2	0.4	2.4	5	−1.2	−1.8	−3

Table 4 Data for WVS

Unit	q 10 ^j	q 1x ^j	q 11 ^j	q 00 ^j	q 0x ^j	q 01 ^j	w j ^1	w j ^0
System 1 p 0 = 0.5
1	0.020	0.010	0.970	0.970	0.010	0.020	5.000	2.000
2	0.050	0.020	0.930	0.960	0.020	0.020	1.000	3.000
System 2 p 0 = 0.5
1	0.287	0.025	0.688	0.567	0.209	0.224	0.402	0.372
2	0.219	0.106	0.675	0.680	0.077	0.243	1.796	0.124
3	0.103	0.197	0.700	0.719	0.073	0.208	2.848	0.031
4	0.197	0.014	0.789	0.751	0.249	0.000	0.526	2.693
5	0.133	0.067	0.800	0.628	0.168	0.204	0.557	0.65
System 3 p 0 = 0.5
1	0.150	0.200	0.650	0.900	0.080	0.020	2.486	2.486
2	0.180	0.000	0.820	0.940	0.000	0.060	1.878	1.878
3	0.070	0.050	0.880	0.880	0.050	0.070	2.403	2.403
4	0.120	0.160	0.720	0.760	0.160	0.080	1.685	1.685
5	0.160	0.120	0.720	0.700	0.120	0.180	1.547	1.547
System 4 p 0 = 0.5
1	0.35	0	0.65	0.65	0.35	0	2.759	2.759
2	0.23	0.1	0.67	0.56	0.1	0.34	0.172	0.172
3	0.36	0.06	0.58	0.82	0.07	0.11	3.060	3.060
4	0.07	0	0.93	0.58	0.12	0.3	4.009	4.009

Table 5 Results for WVS reliability simulation

	System 1 (p 0 = 0.5, τ = 0.60, R = 0.9795)			*System 2 (p 0 = 0.5, τ = 0.3, R = 0.7616)			System 3 (p 0 = 0.5, τ = 0.412, R = 0.9816)			System 4 (p 0 = 0.5, τ = 0.58, R = 0.891)
	Q 11 = 1	Q 00 = 0		Q 11 = 1	Q 00 = 0		Q 11 = 1	Q 00 = 0		Q 11 = 1	Q 00 = 0
Mean	0.9793	0.9796	0.9795	0.9806	0.5427	0.7617	0.9819	0.9813	0.9816	0.8345	0.9477	0.8911
MAPE	—	—	0.0001	—	—	0.0003	—	—	0.0001	—	—	0.0001
CPU time	3.5720	1.4150	4.9870	7.7610	5.0670	12.828	7.8810	4.0850	11.966	5.9800	4.3210	—

Fig. 3 MC CPU time as a function of number of units in the WVS.

Fig. 4 WVS reliability estimate variance as a function of number of units in the WVS.

Table 6 Quasi-optimal weights and threshold for symmetric WVS

System
	1	2	3	4
τ	0.4300	0.5270	0.4104	0.464
Q 00	0.9799	0.9553	0.9775	0.8444
Q 11	0.9794	0.9077	0.9837	0.9274
R	0.9796	0.9315	0.9806	0.8859
w	(1.75, 1)	(1, 1.106, 1.643, 2.594, 1.516)	(2, 1.417, 2.429, 1.70, 1)	(1.628, 1, 1.213, 1.541)
CPU (seconds)	1.994	4.940	4.750	4.3200
R GA	0.9796	0.9360	0.9815	0.8910
APE (%)	0.00	0.48	0.09	0.57

Table 7 Quasi-optimal weights and threshold for asymmetric WVS

	System
	1	2	3	4
τ	0.3740	0.552 80	0.5928	0.504
Q 00	0.9796	0.967 89	0.9785	0.8525
Q 11	0.9796	0.921 77	0.9751	0.9328
R	0.9796	0.945 00	0.9768	0.8927
w 1	(1, 1)	(1.085, 1, 1.168, 8.255, 1.191)	(9, 3, 2.571, 2.250, 1)	(8.708, 1, 3.091, 1.133)
w 2	(2.5, 1)	(1, 1.311, 2.786, 1.457, 2.158)	(1.2, 1, 2.571, 1.5, 1.125)	(1.029, 1.565, 1, 5.143)
CPU (seconds)	1.89	5.46	4.67	4.29
R GA	0.979 45	0.954 00	NA	NA
(APE%)	0.02	0.94	NA	NA

Levitin , G. 2002b . Asymmetric weighted voting systems . Reliability Engineering & System Safety , 76 : 205 – 212 .

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