Example of Weighted Voting System Undersea target detection system.

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Presentation transcript:

Example of Weighted Voting System Undersea target detection system

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

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

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

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

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

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

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

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

)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

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

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

Group vulnerability M-number of groups

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 … …

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

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

)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*

Numerical Example q1xq1x q 10 q0xq0x q 01 tjtj No of unit p 0 =0.3p 0 =0.5p 0 =  Q Q Q Q R T Parameters of voting units Parameters of optimal system for T*=35 Reliability vs. expected decision time

References 1. Weighted voting systems: reliability versus rapidity, G. Levitin, Reliability Engineering & System Safety, 89(2) pp (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 , (2003). 4. Asymmetric weighted voting systems, G. Levitin, Reliability Engineering & System Safety, vol. 76, pp , (2002). 5. Evaluating correct classification probability for weighted voting classifiers with plurality voting, G. Levitin, European Journal of Operational Research, vol. 141, pp , (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 , (2001). 7. Optimal unit grouping in weighted voting systems, G. Levitin, Reliability Engineering & System Safety vol. 72, pp , (2001). 8. Reliability optimization for weighted voting system, G. Levitin, A. Lisnianski, Reliability Engineering & System Safety, vol. 71, pp , (2001).