Multi-criteria group consensus under linear cost opinion elasticity

Consensus is a pivotal concept in group decision making. Many times, such a consensus is achieved by the experts shifting their opinion towards a point of mutual consent. Such a shift in many cases is the result of laborious negotiations, which escalates the cost of reaching the consensus. Moreover, many times the group decision is multi-criteria oriented in which the experts need to agree on each criterion separately. This paper describes three problems where experts of unequal importance and with a linear cost of changing their opinion (opinion elasticity) consider a single and a multi-criteria decision consensus. These problems achieve a minimum cost consensus without a budget limit. It turns out that the optimal consensus point is at the median opinion for rectilinear cost function and at the weighted average opinion for squared geometric distance calculations. Linear-time algorithms are presented for all cost consensus problems with no budget limits. Proofs, computational complexity and examples are provided for these algorithms.