Distributionally robust equilibrious hybrid vehicle routing problem under twofold uncertainty

A novel nonlinear hybrid vehicle routing problem is examined, and its nonlinear component is linearised considering the transportation costs associated with electricity and traditional fuel-based driving. Due to the fact that the transportation cost and fuel consumption involve the twofold uncertainty of randomness and fuzziness, and only partial probability distribution information may be available. Therefore, these two parameters are considered as random fuzzy variables with ambiguous probability distributions. An ambiguous equilibrium risk value objective function and an ambiguous equilibrium chance constraint are formulated. Accordingly, a distributionally robust equilibrium approach is proposed, in which the ambiguity sets are used to characterize the ambiguous probability distributions of the random fuzzy variables. Specifically, this paper first applies the central limit theorem to construct the ambiguity sets. Subsequently, the inner ambiguous probability constraint and the outer credibility constraint are derived into their equivalent counterparts, respectively. In this manner, the proposed model is successfully converted into a mixed integer second-order cone programming model, where the conventional branch-and-cut algorithm is adopted to obtain the optimal routing. Finally, the performance of the proposed model and its price of distributional robustness are verified in the numerical experiments. Overall, the main achievements of this paper are summarized as (1) the proposal of a distributionally robust equilibrium optimization model for a nonlinear hybrid vehicle routing problem, (2) the definitions of an ambiguous equilibrium risk value objective function and an ambiguous equilibrium chance constraint under twofold uncertainty, and (3) the derivation of an equivalent second-order cone programming model for computationally solvable.