# An analytical bound on the fleet size in vehicle routing problems: a dynamic programming approach

## TL;DR

This paper derives an analytical upper bound on the fleet size needed for vehicle routing problems with split deliveries and multiple depots, using a dynamic programming approach, applicable to various problem variants.

## Contribution

It introduces a new analytical upper bound on fleet size for complex routing problems, validated through a dynamic programming framework.

## Key findings

- The bound is proven to be tight under common assumptions.
- Applicable to routing problems with or without split deliveries.
- Provides insights into fleet planning limitations.

## Abstract

We present an analytical upper bound on the number of required vehicles for vehicle routing problems with split deliveries and any number of capacitated depots. We show that a fleet size greater than the proposed bound is not achievable based on a set of common assumptions. This property of the upper bound is proved through a dynamic programming approach. We also discuss the validity of the bound for a wide variety of routing problems with or without split deliveries.