Approximation Algorithms for the Cumulative Vehicle Routing Problem with Stochastic Demands

2025-07-11

AI Summary

This paper studies the Cumulative Vehicle Routing Problem with Stochastic Demands (Cu-VRPSD), where customer demands are unknown upon vehicle departure from the depot and revealed only upon visit; routes must respect capacity constraints while minimizing cumulative cost—defined as the sum of distances weighted by real-time vehicle load. As an NP-hard problem, we propose an iterative algorithm integrating a metric TSP 1.5-approximation, randomized routing strategies, and probabilistic analysis. Theoretically, we improve the best-known approximation ratio for Cu-VRPSD from 6 to 3.456. Moreover, our analysis tightens bounds for related problems: achieving 3.25 for the classical VRPSD and 3.194 for the deterministic Cumulative VRP—both current state-of-the-art. The algorithm naturally extends to allow multiple visits per customer, enhancing robustness and practical applicability.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

In the Cumulative Vehicle Routing Problem (Cu-VRP), we need to find a feasible itinerary for a capacitated vehicle located at the depot to satisfy customers' demand, as in the well-known Vehicle Routing Problem (VRP), but the goal is to minimize the cumulative cost of the vehicle, which is based on the vehicle's load throughout the itinerary. If the demand of each customer is unknown until the vehicle visits it, the problem is called Cu-VRP with Stochastic Demands (Cu-VRPSD). Assume that the approximation ratio of metric TSP is $1.5$. In this paper, we propose a randomized $3.456$-approximation algorithm for Cu-VRPSD, improving the best-known approximation ratio of $6$ (Discret. Appl. Math. 2020). Since VRP with Stochastic Demands (VRPSD) is a special case of Cu-VRPSD, as a corollary, we also obtain a randomized $3.25$-approximation algorithm for VRPSD, improving the best-known approximation ratio of $3.5$ (Oper. Res. 2012). For Cu-VRP, we give a randomized $3.194$-approximation algorithm, improving the best-known approximation ratio of $4$ (Oper. Res. Lett. 2013). Moreover, if each customer is allowed to be satisfied by using multiple tours, we obtain further improvements for Cu-VRPSD and Cu-VRP.

Problem

• Develops approximation algorithms for Cumulative Vehicle Routing Problem with Stochastic Demands
• Improves best-known approximation ratios for Cu-VRPSD and VRPSD
• Enhances solutions for Cu-VRP with multiple tours per customer

Innovation

• Randomized 3.456-approximation for Cu-VRPSD
• Improved 3.25-approximation for VRPSD
• Randomized 3.194-approximation for Cu-VRP