Approximation Algorithms for the Cumulative Vehicle Routing Problem with Stochastic Demands

TL;DR

This paper introduces improved randomized approximation algorithms for the stochastic vehicle routing problem, significantly reducing the approximation ratios compared to previous methods and extending to variants with multiple tours per customer.

Contribution

It presents the first randomized 3.456-approximation for Cu-VRPSD, improving previous ratios, and extends these results to VRPSD and Cu-VRP, also considering multiple tours per customer.

Findings

Achieved a 3.456-approximation for Cu-VRPSD.

Improved VRPSD approximation ratio to 3.25.

Enhanced Cu-VRP approximation ratio to 3.194.

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.