Practical computational methods for aggregation of lower envelope

We consider the lower envelope model for characterizing uncertainty about a random phenomenon and the case where we are provided with several lower envelopes, each characterizing some aspect of the phenomenon. We wish to have a practical method for combining such characterizations into an aggregate characterization that reflects the collective knowledge, and we discuss several of the rules that have been proposed for aggregating lower envelopes as lower envelopes. While these rules have different properties, they all share the trait that, as the size of the underlying event algebra increases, the computational burden of aggregation quickly becomes insupportable. In our work, we investigate methods for reducing the computational burden.;We first present a technique called localization, which is a generalization of a procedure proposed by Kong for reducing the number of operations required for aggregating belief functions (a special case of lower envelopes) by Dempster's Rule of Combination. We develop criteria which can be used to determine if localization is applicable to a specific aggregation rule. Dempster's rule satisfies these criteria, as does another rule we introduce, the conjunction rule.;We then investigate the conjunction rule. While localization can be used for this rule, we develop arguments that suggest that it is not currently practical. We propose another procedure, the expansion technique, for reducing the computational load when aggregating by the conjunction rule.;We also consider the conjunction rule on a domain of lower envelopes that allows us to formulate the conjunction rule aggregation problem as a linear programming problem. We discuss traditional linear programming methods, as well as modification of the relaxation method based on the particular structure of our problem. Finally, we investigate the advantages and disadvantages of the various linear programming techniques when used in concert with the localization technique and the expansion technique.