An exact branch-price-and-cut algorithm for the time-dependent cold chain pickup and delivery problem with incompatibility constraints

This paper addresses a cold chain transportation problem derived from a real-life situation, namely a time- dependent pickup and delivery problem with commodity incompatibility constraints (TDPDPI). In TDPDPI, the travel speed of these vehicles varies with the departure time, that is, the time-dependent travel speed. The quality delay cost of the perishable commodity also varies with the departure time, namely the time- dependent quality delay cost. The cost of this problem consists of two components: one is related to the total travel time, and the other is related to the quality delay cost of the perishable commodity. To solve TDPDPI, this paper develops an arc-based (mixed integer programming, MIP) model solved by CPLEX and a route-based (set-partitioning formulation, SPF) model. To address the SPF model, this paper proposes an exact branch-price-and-cut (BPC) algorithm. A specialized bidirectional labeling algorithm is developed to address the pricing problem. Additionally, subset row cuts (SRC) are employed as valid inequalities to enhance the quality of the lower bound in the SPF model. Extensive computational experiments are conducted to evaluate the efficacy of the proposed BPC algorithm. The results demonstrate that the BPC algorithm effectively solves the problem with 50 requests. Finally, this study conducts a sensitivity analysis of the key constraints and parameters of the model, providing valuable managerial insights.