A Column Generation Tailored To Electric Vehicle Routing Problem With Nonlinear Battery Depreciation

Since the battery remains a significant cost component of electric vehicles (EVs), controlling the depreciation costs of EVs' batteries is of great importance, especially from the perspective of the electric vehicle routing problem (EVRP). However, most existing studies on the EVRP have not explicitly considered the battery depreciation in the cost function; instead, it has been treated as a linear function of an EV's travel distance or time. In fact, the depth-of-discharge (DOD) significantly affects the battery life and leads to a naturally nonlinear depreciation function, which can be used to calculate the battery life more accurately. In this paper, we adopt three battery depreciation methods to investigate and compare their influences on the EVRP with time windows (EVRPTW): (1) a nonlinear function of DOD; (2) a linear function of charge-discharge cycles; and (3) a linear function of total traveling distance. The first method is mainly studied, and the remaining two methods are verified as comparative analyses. Meanwhile, both full and partial charge policies are considered to formulate the problem. By combining the charging policies and battery depreciation methods, four mixed -integer programming models are formulated, aiming to minimize the total cost. To pursue exact solutions in acceptable computing time, a column generation algorithm (CG) that relies on four tailored labeling algorithms (LAs) is designed. The LAs are used to accelerate the calculation speed of the pricing problem. The most considerable difficulty of solving pricing problems lies in the complex endogenous relationships among recharging time, drivers' waiting time, battery state-of-charge (SOC), battery depreciation, battery rated capacity, and driving distance. The LAs need to find a tradeoff among them and generate the shortest paths. Given this, we design specific and elaborate resource extension functions (REFs) for the four models, respectively, which is an extension to Desaulniers et al. (2016). Although the REFs are intricate, they work well with the help of dominance rules. Benchmark instances are performed to verify the efficiencies of the CG and LAs, which shows that instances with 25-nodes are solvable. In computational studies, we find that models considering the DOD can increase the battery life and decrease the total cost by 9%-10%. Moreover, several insights for better management of EV fleets are obtained.