# Parcel container capacity sizing optimization for light cargo logistics in last-mile delivery

Luis Marques, Yann Bouchery, Olivier Labarthe, Nafé Moradkhani

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Luis Marques, Yann Bouchery, Olivier Labarthe, Nafé Moradkhani. Parcel container capacity sizing optimization for light cargo logistics in last-mile delivery. ROADEF 2026, Feb 2025, Tours, France. hal-05537621

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Parcel container capacity sizing optimization for
light cargo logistics in last-mile delivery

Luis Marques

Yann Bouchery

Olivier Labarthe

Nafé Moradkhani

KEDGE Business School – CESIT

March 4, 2026
# Table of contents

Introduction

Current practices and our contribution

Problem statement

Modelization methods

Conclusion and perspectives
# Outline

## Introduction

Current practices and our contribution

Problem statement

Modelization methods

Conclusion and perspectives
# Urban logistics

Urban modern areas are very dense:

* 55% percent of the world population lives in urban areas

* Projected to 68% by 2050 by the United Nations

This poses considerable challenges for urban logistics:

* Air, noise, and light pollution

* Congestion and impact on commute time

* The World Economic Forum expects an increase of 36% for delivery vehicles from 2020 to 2030

<figure><img src="image_2.png" /></figure>
# Light cargo urban logistics

Light cargo vehicles (LCVs) are a suitable response to these challenges:

* Human-powered vehicles

* Electric assistance

Advantages:

* Close to zero pollution

* Less sensitive to congestion

* Easier to park

Disadvantages:

* Low capacity

* Low peak speed

* Low onboard energy

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>
# Light cargo logistics challenge: modal shift

Disadvantages of LCVs mean they are only suitable for low-distance deliveries (Melo and Baptista, 2017).

A modal shift occurs when shifting from one mode of transport (e.g., vans) to LCVs:

* Significant amount of labor and time for employees

* Significant land costs in the city centers

<figure><img src="image_6.png" /></figure>
Light cargo logistics challenge: modal shift

Containers are a solution to reduce the cost of the modal shift and to improve
standardization:

► group parcels into standardized units

► reduce handling time during modal shift

► allow the use of automated logistics
infrastructures

► inspired by containerization in maritime logistics

<figure><img src="image_7.png" /></figure>
Light cargo logistics challenge: modal shift

What is the optimal capacity of parcel containers?

► Too small → nearby parcels split across containers leading to an increased travel distance

► Too large → unused capacity

► Strategic decision!
# Outline

## Introduction

## Current practices and our contribution

## Problem statement

## Modelization methods

## Conclusion and perspectives
# Current practices: containerization in urban logistics

Companies have started using containers for urban logistics:

<figure><img src="image_8.png" /></figure>

Academic literature is parse:

* Few studies study parcel containers for urban logistics (Labarthe et al., 2024; Sina Mohri et al., 2024), but capacity sizing is not an important focus

* Many studies study containerization (especially in the Physical Internet literature), but focus on light cargo logistics is not an important focus
# Current practices: territory design in urban logistics

Territory design approaches are used strategically to partition the service area into *districts*:

* take as input a set of small geographical units named *basic units*

* form districts by aggregating basic units

* used for several objectives: politics, social facilities, waste collection, etc

We suppose:

* each container contains only parcels of a single basic unit

* each district is served only by a single LCV

<figure><img src="image_9.png" /></figure>
# Our contribution

Our contributions are to the literature are:

* Formalization of a strategic problem mixing capacity sizing and territory design

* Development of continuous approximations formulas to estimate tour lengths in this context (WIP)

* Efficient resolution methods for large-scale real-world instances (WIP)

Our contributions to current practices are:

* Managerial insights on when and how to use containers in light cargo logistics (WIP)

* Industrial case study with a French LSP (WIP)
# Outline

## Introduction

## Current practices and our contribution

## Problem statement

## Modelization methods

## Conclusion and perspectives
# Parcel Container Capacity Sizing (PCCS) problem

In practice:

* parcel demand is highly stochastic and difficult to fit
  We treat each day as a realization of this stochastic demand, i.e., a scenario, and optimize over a given number of days.

* each container contains only parcels of a same basic unit and each district is served only by a single LCV, eventually in many trips to the depot due to low capacity

We model this and model fixed costs associated with each trip of LCV and each district designed.
# Parcel Container Capacity Sizing (PCCS) problem

## Inputs:

* $\mathcal{T}$: set of time periods, i.e., days

* $\mathcal{C}$: set of possible $(c, N_c)$ couples, where $c$ is the capacity and $N_c$ is the number of containers the LCV can carry

* $\mathcal{B}$: set of basic units (BUs), where each basic unit $b$ has, for each time period $t$, a demand $N_b^t$

## Decisions:

* Container capacity

* Territory design

## Objective:

* Minimize long-run costs:

* $\tilde{u} \in \mathbb{N}$: fixed cost per unit of distance traveled

* $\tilde{f}: \mathbb{N} \to \mathbb{N}$: fixed cost as function of number of trips done by an LCV, same for all LCVs

* $\tilde{g}: \mathbb{N} \to \mathbb{N}$: fixed cost as function of number of districts created
PCCS problem mathematical program

$$
\begin{array}{ll}
\text{minimize} & \alpha \\
\text{subject to} & \sum_{c \in C} x_c = 1, \\
& \sum_{d \in D} y_{b,d} = 1, \quad \forall b \in B, \\
& x \in \{0, 1\}^{|C|}, \\
& y \in \{0, 1\}^{|B| \times |D|}.
\end{array}
$$
# Outline

## Introduction

## Current practices and our contribution

## Problem statement

## Modelization methods

## Conclusion and perspectives
Method 1: Continuous approximations (CAs)

Continuous approximations based on Daganzo's analytical routing models:

► use district shapes and demand density to estimate tour length under assumption of uniform demand points.

$$ \underbrace{k}_{\text{routing constant}} \cdot \underbrace{\sqrt{A}}_{\text{area}} \cdot \underbrace{\sqrt{n}}_{\text{demand}} $$

$$ \underbrace{\left(k_1 + \frac{k_2 \cdot n}{c^2}\right)}_{\text{CVRP}} \cdot \underbrace{\sqrt{A}}_{\text{area}} \cdot \underbrace{\sqrt{n}}_{\text{demand}} $$

► We extend CAs to incorporate:

► Container capacity constraints

► Trip and district fixed costs (WIP)
Method 1: Continuous approximations (CAs)

$$
\alpha = \sum_{t \in T} \sum_{d \in D} \left( k_1 + k_2 \cdot \sum_{c \in C} \sum_{b \in B} \frac{\left[ \frac{N_b^t}{c} \cdot \underbrace{y_{b,d} \cdot x_c}_{\text{non-linear}} \right]}{N_c^2} \right) \cdot \sqrt{\sum_{b \in B} A(b) \cdot \underbrace{y_{b,d}}_{\text{non-linear}}} \cdot \sqrt{\sum_{c \in C} \sum_{b \in B} \left[ \frac{N_b^t}{c} \cdot \underbrace{y_{b,d} \cdot x_c}_{\text{non-linear}}} \cdot \sqrt{N_b^t \text{ mod } c}} + \underbrace{\sum_{b \in B} k_1 \cdot \sqrt{A(b)} \cdot \left( \sum_{c \in C} x_c \left( \left\lfloor \frac{N_b^t}{c} \right\rfloor \cdot \sqrt{c} + \sqrt{N_b^t \text{ mod } c} \right) \right)}_{\text{Continuous approximation in BUs}}
$$

Advantages:

► Model is very small

Disadvantages:

► Model is non-linear and non-convex / non-concave

Open questions:

► Is this approximation close to reality?

► How do we incorporate non-uniform demand points?
Method 2: Mix CAs and CVRP algorithms

Another method:

► models the number of parcels in each container as an integer variable

► uses CAs to estimate the cost of delivering the parcels of each container

► treats each container as a demand point in a CVRP

Objective function:

$$
\begin{align*}
\alpha = & \underbrace{\sum_{t \in T} \sum_{r \in \mathcal{R}} \sum_{b_1 \in \mathcal{B}} \sum_{i_1 \in S_{b_1,t}} \sum_{b_2 \in \mathcal{B}} \sum_{i_2 \in S_{b_2,t}}}_{\text{Explicit routing of containers}} c_{b_1, b_2} f_{s_1, s_2}^{r,t} + \underbrace{\sum_{t \in T} \sum_{r \in \mathcal{R}} \sum_{b \in \mathcal{B}} \sum_{i \in S_{b,t}} k_1 \cdot \sqrt{A(b)} \cdot \rho_{s_i}^{r,t}}_{\text{Continuous approximation of each container}} \\
& + \underbrace{\sum_{d \in \mathcal{D}} \sum_{k=1}^{U_d} \tilde{c}_k g_{k,t}^d}_{\text{Trip fixed costs}} + \underbrace{\sum_{d \in \mathcal{D}} \hat{c}_d Z_d}_{\text{District fixed costs}}
\end{align*}
$$
Method 2: Mix CAs and CVRP algorithms

$$
\alpha = \underbrace{\sum_{t \in T} \sum_{r \in R} \sum_{b_1 \in B} \sum_{i_1 \in S_{b_1,t}} \sum_{b_2 \in B} \sum_{i_2 \in S_{b_2,t}}}_{\text{Explicit routing of containers}} c_{b_1, b_2} f_{s_1, s_2}^{r,t} + \underbrace{\sum_{t \in T} \sum_{r \in R} \sum_{b \in B} \sum_{i \in S_{b,t}} k_1 \cdot \sqrt{A(b)} \cdot \rho_{s_i}^{r,t}}_{\text{Continuous approximation of each container}} \\
+ \underbrace{\sum_{d \in D} \sum_{k=1}^{U_d} c_k g_{k,t}^d}_{\text{Trip fixed costs}} + \underbrace{\sum_{d \in D} c_d Z_d}_{\text{District fixed costs}}
$$

Advantages:

► Captures operational constraints and fixed costs explicitly

► Contains only products of binary variables which can be easily linearized

Disadvantages:

► Weak continuous relaxation due to the many linearizations and many symmetries

► Too many variables: preliminary experiments show out-of-memory errors for 1-day instances
# Outline

Introduction

Current practices and our contribution

Problem statement

Modelization methods

Conclusion and perspectives
Conclusion and perspectives

Contributions:

► First study linking container capacity sizing and territory design

► Extension of continuous approximation models

► Development of MIP and MILP formulations to solve the PCCS problem

Perspectives:

► Implement non-uniform demand points in continuous approximation models.

► Develop the case study with French LSP:

► use sample average approximation methods to compute the number of time periods necessary for a good estimation of mean long-run costs
References

Labarthe, O., Ahmadi, G., Klibi, W., Deschamps, J.-C., and Montreuil, B. (2024). A sustainable on-demand urban delivery service enabled by synchromodality and synergy in passenger and freight mobility. *Transportation Research Part C: Emerging Technologies*, 161:104544.

Melo, S. and Baptista, P. (2017). Evaluating the impacts of using cargo cycles on urban logistics: Integrating traffic, environmental and operational boundaries. *European Transport Research Review*, 9(2):30.

Sina Mohri, S., Mohammadi, M., and Van Woensel, T. (2024). Designing zero-emissions containerized last-mile delivery systems: A case study for melbourne. *Transportation Research Part C: Emerging Technologies*, 159:104492.
Parcel container capacity sizing optimization for
light cargo logistics in last-mile delivery

Luis Marques

Yann Bouchery

Olivier Labarthe

Nafé Moradkhani

KEDGE Business School – CESIT

March 4, 2026
Method 1: Continuous approximations (CAs)

$$
\alpha = \underbrace{\sum_{t \in T} \sum_{d \in D} \sum_{c \in C} x_c \cdot \left( k_1 + k_2 \cdot \sum_{c \in C} \sum_{b \in B} \frac{\lfloor \frac{N_b^t}{c} \rfloor \cdot y_{b,d}}{N_c^2} \right)}_{\text{Continuous approximation in districts}} \cdot \sqrt{\sum_{b \in B} A(b) \cdot y_{b,d}} \cdot \underbrace{\sqrt{\sum_{c \in C} \sum_{b \in B} \frac{\lfloor \frac{N_b^t}{c} \rfloor \cdot y_{b,d}}{N_c^2}}}_{\text{Continuous approximation in BUs}} \\
+ \sum_{b \in B} k_1 \cdot \sqrt{A(b)} \cdot \left( \sum_{c \in C} x_c \cdot \left( \lfloor \frac{N_b^t}{c} \rfloor \cdot \sqrt{c} + \sqrt{N_b^t \mod c} \right) \right)
$$

Observations:

► Model is non-convex / non-concave even for fixed capacity

► objective function is of form $\ell(y) \cdot \sqrt{f(y) \cdot g(y)}$

► for concavity, we need $\ell(y)$ positive and non-increasing