Online Companion for Robust Controls for Network Revenue Management

Proof of Proposition 1. With a slight abuse of notation, let us denote by D d the set of multivariate stochastic processes such that the aggregate demand equals d. Accordingly, Problem (5) can be decomposed as follows: ρ(y) = max d∈P max D∈D d max z∈F R(z, D) − R(y, D) = max d∈P max z∈F max D∈D d R(z, D) + max D∈D d −R(y, D). Let ξ j be the realized sales under a booking policy z when the demand is equal to d. Because the perfect hindsight booking limits z are chosen after observing the demand realization, one can assume without loss of generality that they are partitioned, For any d ∈ P, different demand arrival processes D ∈ D d are associated with different sales ξ, but the sales will always satisfy (A-1). Let us denote by ξ(D) the realized sales under the demand process D. Hence, for any d ∈ P, max D∈D d R(z, D) = R(z, D *) = r ξ(D *) ≤ max ξ:(A−1)holds r ξ. On the other hand, for any sales vector ξ, there always exists a demand arrival process that gives rise to those sales. Let us denote by D d (ξ) the set of demand processes that lead to the sales ξ. Hence, for any d ∈ P, max ξ:(A−1)holds r ξ = r ξ * = R(z, D) ∀D ∈ D d (ξ *) ≤ max D∈D d R(z, D).