1. Example of Weighted Voting System Undersea target detection system

2. Weighted Voting System - system output (0,1,x) - voting units outputs (0,1,x) d 1 (I) d 2 (I) d 3 (I) d 4 (I) d 5 (I) d 6 (I)  I D(I)D(I) w 1 w 2 w 3 w 4 w 5 w 6 unit 1unit 2 unit 3unit 4 unit 5unit 6 - threshold - system input (0,1) - weights

3. Decision Making Rule Total weight of units voting for the proposition acceptance Total weight of units voting for the proposition rejection System output

4. Decision Making Rule if (1-  )W n 1 -  W n 0 <0 Wn0Wn0  W n 1 Accept Reject if W n 1 =W n 0 =0 otherwise

5. WVS as a Multi-state System Voting unit j: 3 states: 4 failure modes: d j (0)=1; d j (1)=0; d j (0)=x; d j (1)=x. (1-  )W n 1 -  W n 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. Input I

6. Asymmetric Weighted Voting System - system output (0,1,x) - voting units outputs (0,1,x) - acceptance weights d 1 (I) d 2 (I) d 3 (I) d 4 (I) d 5 (I) d 6 (I) w 1 1 w 1 2 w 1 3 w 1 4 w 1 5 w 1 6  I D(I)D(I) w 0 1 w 0 2 w 0 3 w 0 4 w 0 5 w 0 6 unit 1unit 2 unit 3unit 4 unit 5unit 6 - threshold - system input (0,1) - rejection weights

7. Decision Making Rule Total weight of units voting for the proposition acceptance Total weight of units voting for the proposition rejection System output

8. Types of Errors d j (0)=1 (unit fails stuck-at-1) too optimistic q 01 (j) d j (1)=0 (unit fails stuck-at-0) too pessimistic q 10 (j) d j (I)=x (unit fails stuck-at-x) too indecisive q 1x (j), q 0x (j) Voting unit parameters Decision making time t j Rejection weight w j 0 Acceptance weight w j 1 System threshold  System Parameters adjustable

9. Universal generating function technique Score distribution for m voters Score distribution for a single voter Composition operator

10. )w 01,w 11,…,w 0n,w 1n,  ) = arg{R(w 01,w 11,…,w 0n,w 1n,  )  max} System Success Probability Optimal adjustment problem Optimization problems

11. 145263 w 1 w 4 w 5 w 2 w 6 w 3  1  2  3 22 11 00 Optimal grouping R(w, ,  )  max

12. VU 1VU 2VU 3VU 4VU 5VU 6 w 1 w 2 w 3 w 4 w 5 w 6  P d 1 (P) d 2 (P) d 3 (P) d 4 (P) d 5 (P) d 6 (P) D(P) PG 3PG2PG1 v Optimal distribution among protected groups

13. Group vulnerability M-number of groups

14. Order of voting decisions Total weight of units with t j  t m voting for the proposition acceptance Total weight of units with t j  t m voting for the proposition rejection t1t1 t2t2 tmtm tntn … …

15. Wm0Wm0  W m 0 Reject  V 1 m+1 Wm0Wm0 Accept  V 0 m+1  W m 0 Accelerated Decision Making

16. Q ij m probability of making the decision D(i)=j at the time t m p 0, p 1 - input distribution System reliability and expected decision time

17. )w 01,w 11,…,w 0n,w 1n,  ) = arg{R(w 01,w 11,…,w 0n,w 1n,  )  max} subject to T (w 01,w 11,…,w 0n,w 1n,  )  T* R T Voting system optimization problem R  max T  min Two-objective problem: Constrained problem: R  max | T<T*

18. Numerical Example q1xq1x q 10 q0xq0x q 01 tjtj No of unit 0.120.290.310.22101 0.300.1030.070.35122 0.150.220.080.24383 0.010.200.050.10484 0.070.150.100.08555 0.050.100.010.08706 p 0 =0.3p 0 =0.5p 0 =0.7 0.450.500.76  0.8477 0.90050.9798 Q 00 0.1523 0.09950.0202 Q 01 0.02830.07190.1611Q 10 0.971660.92810.8389Q 11 0.93450.91430.9375R 34.99434.98734.994T Parameters of voting units Parameters of optimal system for T*=35 Reliability vs. expected decision time

19. References 1. 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).