Tail mean and related robust solution concepts

Robust optimisation might be viewed as a multicriteria optimisation problem where objectives correspond to the scenarios although their probabilities are unknown or imprecise. The simplest robust solution concept represents a conservative approach focused on the worst-case scenario results optimisation. A softer concept allows one to optimise the tail mean thus combining performances under multiple worst scenarios. We show that while considering robust models allowing the probabilities to vary only within given intervals, the tail mean represents the robust solution for only upper bounded probabilities. For any arbitrary intervals of probabilities the corresponding robust solution may be expressed by the optimisation of appropriately combined mean and tail mean criteria thus remaining easily implementable with auxiliary linear inequalities. Moreover, we use the tail mean concept to develope linear programming implementable robust solution concepts related to risk averse optimisation criteria.