Optimal unit grouping in weighted voting systems

The voting system studied consists of n units that each provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise system output is 0. For a system consisting of voting units with given reliability characteristics one can maximize the entire system reliability by choosing proper unit weights and threshold values τ. In this paper it is shown that additional system reliability improvement can be achieved by grouping units in voting subsystems and tallying the weighted votes of these subsystems (groups) to make a final decision. An algorithm is suggested which finds the optimal element grouping as well as unit weights within each group, weights of the groups and corresponding system thresholds. The approach is based on using a universal generating function technique for evaluating system reliability. A genetic algorithm is used as the optimization tool. Examples are presented.