Formulations and branch-and-cut algorithms for the heterogeneous fleet vehicle routing problem with soft time deadlines

This paper investigates a variant of the heterogeneous fleet vehicle routing problem (HVRP) that incorporates soft time deadlines for customers and allows for tardiness at penalty costs. Distinct vehicle types feature varying fixed usage costs and utilize different road networks, resulting in differences in both travel times and travel costs. The objective is to optimize fleet assignment and vehicle service routes to minimize the total fixed vehicle usage costs, routing variable costs, and tardiness costs, while ensuring each customer is visited exactly once and respecting route duration limits. To address this problem, we introduce three compact formulations: the Miller-Tucker-Zemlin formulation (MTZF), single-commodity flow formulation (SCFF), and two-commodity flow formulation (TCFF), comparing their linear programming (LP) relaxations. Additionally, we propose two new families of valid inequalities, in conjunction with generalized subtour elimination constraints, to strengthen these LP relaxations, integrating them into branch-and-cut solution schemes. The theoretical results on the comparison of formulations and the validity of the proposed inequalities hold also for other HVRPs with limited route duration. Computational experiments demonstrate the superior performance of SCFF and TCFF over MTZF, the effectiveness of the proposed valid inequalities in tightening formulations, and the enhanced computational efficiency achieved by incorporating them. Finally, we explore the impact of depot relocation, varying degrees of urgency in customer requests, and varying fixed vehicle usage costs on optimal solutions.