The distributionally robust CVRP over a marginalized moment set is a deterministic CVRP

Let $\mathcal P$ be a marginalized moment ambiguity set of the form (5) under the standing assumptions ($\underline{\boldsymbol q}\ge\mathbf 0$, $\boldsymbol\mu\in\operatorname{int}\mathcal Q$, componentwise convex $\boldsymbol\varphi_i$ with $\boldsymbol\varphi_i(\mu_i)<\boldsymbol\sigma_i$), let $\epsilon\in(0,1)$ and let the vehicle capacity be $Q\ge0$. Define the deterministic demands

Then for every assignment $\mathbf R=(R_1,\dots,R_m)$ of ordered customer lists to the $m$ vehicles,

Here RVRP($\mathcal P$)-feasibility means $\mathbf R\in\mathfrak P(V_C,m)$ and $\mathbb P[\sum_{i\in R_k}\tilde q_i\le Q]\ge1-\epsilon$ for all $\mathbb P\in\mathcal P$ and all $k$; deterministic feasibility means $\mathbf R\in\mathfrak P(V_C,m)$ and $\sum_{i\in R_k}q_i\le Q$ for all $k$.

Both problems minimize the same transportation cost $c(\mathbf R)$, so equality of their feasible sets is the paper's statement that the distributionally robust CVRP over (5) is equivalent to the deterministic CVRP with these demands. It allows any deterministic CVRP solver to be used for the robust problem.

Formalization Note Customers are Fin n (0-based), vehicles Fin m, routes List (Fin n); RVRPFeasible and CVRPFeasible both include the route-set condition $\mathbf R\in\mathfrak P(V_C,m)$. The demand of customer i is worstCaseVaR (marginalSet qlo qhi μ φ σ) ε {i}. The capacity is only assumed nonnegative ($Q\in\mathbb R_+$, p. 718).