Wasserstein Distributionally Robust Equilibrium Optimization Under Random Fuzzy Environment for the Electric Vehicle Routing Problem

Abstract

This article innovatively combines the empirical distribution characteristics of random fuzzy variables to propose a new Wasserstein distributionally robust equilibrium optimization method and effectively applies to an electric vehicle routing problem with contactless delivery (EVRPCD). The proposed method characterizes customer demand and travel time in EVRPCD as random fuzzy variables with ambiguous probability distributions. Moreover, this studied EVRPCD integrates the location decision of the contactless delivery station and the routing decision of the electric vehicle, thus forming a bi-level optimization. Meanwhile, a tolerant load coefficient and an idle loss cost are introduced into the established bi-level optimization model to describe the safety and economic effects of vehicle overloading and underloading, respectively. Importantly, the proposed method generates Wasserstein ambiguity sets to effectively achieve the theoretical characterization of the ambiguous probability distribution of random fuzzy variables in EVRPCD. As the proposed method faces significant computational challenges, this article theoretically deduces its computable reformulation via utilizing the dual theory and the credibility measure method. The computable reformulation realizes the solvability of the model via transforming it into a mixed integer programming with a piecewise penalty function and multiple conditional constraints. An interactive iteration-based algorithm is then given to solve the reconstructed model numerically. The sensitivity analysis and comparative experimental results reveal the effectiveness of the proposed method. Experimental results show that the proportion of vehicle overweight may be reduced by appropriately increasing the penalty and the proposed method pays a small price of distributional robustness to resist the ambiguous probability distributions of random fuzzy variables.