# Developing a Stochastic Two-Tier Architecture for Modelling Last-Mile Delivery and Implementing in Discrete-Event Simulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Mathematical Modelling of Vehicle Routing Problems (VRP)

#### 2.2. Computer Simulation

#### 2.3. Geographic Information System

#### 2.4. Gaps in the Modelling of Freight Last-Mile Delivery

## 3. Methodology

#### 3.1. Research Objective

#### 3.2. Approach

^{®}(version 16.00, Rockwell Automation, Milwaukee, WI, USA). ArcGIS Pro

^{®}(version 2.6.3, Esri, Redlands, CA, USA) was applied to find cluster centres, calculate distances and run TSP simulation. TSP simulation is generally used to find the optimal route by organising the sequence of customer locations. ArcGIS was used for TSP simulation because it can provide more accurate routes and distances in terms of customer locations and road information with the existing database.

#### 3.2.1. Developing a Two-Tier Architecture

#### 3.2.2. Determining R

#### 3.2.3. Implementation Stages

## 4. Results

#### 4.1. Description of the Case Study

#### 4.2. Determining the Parameters for Two-Tier Architecture

#### 4.3. Results and Validation of Two-Tier Architecture

#### 4.4. Discrete-Event Simulation (DES) in Arena®

_{total}in the simulation is theoretically calculated by Equation (7).

#### 4.5. Simulation Results

^{3}. However, the actual volume limit is 70% of the ideal limit from the consideration of health and safety, which is 28 m

^{3.}

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**TSP results for 10 consignments. (

**a**) Route one; (

**b**) Route two; (

**c**) Route three; (

**d**) Route four; (

**e**) Route five; (

**f**) Route six; (

**g**) Route seven; (

**h**) Route eight; (

**i**) Route nine; (

**j**) Route ten.

**Figure A2.**TSP results for 15 consignments. (

**a**) Route one; (

**b**) Route two; (

**c**) Route three; (

**d**) Route four; (

**e**) Route five; (

**f**) Route six; (

**g**) Route seven; (

**h**) Route eight; (

**i**) Route nine; (

**j**) Route ten.

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**Figure 4.**Hotspots of consignment data with attributes (

**a**) Attribute of consignment number; (

**b**) Attribute of consignment weight; (

**c**) Attribute of consignment volume.

**Figure 6.**The Euclidean distance (D

_{e}) with a Gamma distribution fit with scale 107.3192 and shape 4.9074.

**Figure 7.**TSP results for n = 10 delivery routes. (

**a**) Example one; (

**b**) Example two; (

**c**) Example three; (

**d**) Example four.

**Figure 8.**TSP results for n = 15 delivery routes. (

**a**) Example one; (

**b**) Example two; (

**c**) Example three; (

**d**) Example four.

**Figure 10.**Mean values of route distance for different N cases for two-tier simulation (blue) compared to TSP results (orange). TSP results are only shown for n = 10 and 15.

**Figure 12.**DES delivery model: (

**a**) first-tier movement; (

**b**) second-tier movement; (

**c**) the return to the depot.

Methods | Description | Examples of Approaches That Have Been Attempted |
---|---|---|

Mathematical models (VRP) | Typical VPR models include NRP, TSP and ARP models. Mathematical models can be developed easily and rapidly. However, these models need to simplify the real system with little stochasticity. | [14,15,16,17,18,19,20,21] |

Computer simulation models | These models can simulate complex systems with a large number of uncertainties. The disadvantages are the long modelling period and the difficulty of representing specific routes. | [39,40] |

GIS models | GIS is able to find routes by implementing VRP algorithms, but it is limited to standalone cases. | [53,54,55,56] |

Integration of mathematical, simulation and GIS models | High daily variability in consignment/customer addresses, and number of consignments requires that the model be recomputed each time. The integration tends to be applied manually, rather than completely automated, so this takes time. For this reason, existing integration systems may not yet be ready for routine operational use in the industry. | [57,58] |

Parameters | Assumptions |
---|---|

Number of consignments | 10 and 15 |

Addresses | Randomly selected |

Truck capacity | One truck is sufficient for assumed consignment numbers |

Truck speed | Average speed including truck stop and start times |

Backhaul | None |

Area | Count of Consignments | Sum of Weight | Sum of Volume |
---|---|---|---|

Wigram | 69.26% | 70.16% | 74.74% |

Sockburn | 26.23% | 25.55% | 21.63% |

Central city | 1.03% | 2.23% | 1.05% |

Addington | 0.94% | 0.32% | 0.78% |

Hornby | 0.57% | 0.42% | 0.42% |

Halswell | 0.54% | 0.30% | 0.47% |

Upper Riccarton | 0.26% | 0.18% | 0.10% |

Sydenham | 0.11% | 0.05% | 0.06% |

Harewood | 0.11% | 0.10% | 0.08% |

Hillmorton | 0.09% | 0.06% | 0.09% |

Middleton | 0.09% | 0.06% | 0.05% |

Waltham | 0.06% | 0.02% | 0.01% |

Riccarton | 0.06% | 0.21% | 0.22% |

Templeton | 0.03% | 0.00% | 0.01% |

Islington | 0.03% | 0.00% | 0.01% |

Wainoni | 0.03% | 0.01% | 0.02% |

Bishopdale | 0.03% | 0.01% | 0.03% |

Airport | 0.03% | 0.00% | 0.00% |

Hoon Hay | 0.03% | 0.01% | 0.04% |

Hornby South | 0.03% | 0.07% | 0.02% |

Unknown data | 0.46% | 0.23% | 0.20% |

Number of Consignments | Delivery Case | Total Second-Tier Distance $\sum}{\mathit{D}}_{\mathit{S}$ (m) | Total Distance for Two-Tier Architecture ${\mathit{D}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ (m) | Total Distance for TSP ${\mathit{D}}_{\mathit{t}}$ (m) | R-Value |
---|---|---|---|---|---|

n = 10 | 1 | 5265 | 6790 | 7645 | 1.163 |

2 | 6176 | 7701 | 6066 | 0.735 | |

3 | 6152 | 7677 | 4906 | 0.550 | |

4 | 4557 | 6082 | 5417 | 0.854 | |

5 | 5248 | 6773 | 7459 | 1.131 | |

6 | 4191 | 5716 | 5175 | 0.871 | |

7 | 5840 | 7365 | 5241 | 0.636 | |

8 | 3712 | 5237 | 4115 | 0.698 | |

9 | 5439 | 6964 | 6397 | 0.896 | |

10 | 5905 | 7430 | 6906 | 0.911 | |

n = 15 | 1 | 8337 | 9861 | 7956 | 0.771 |

2 | 8126 | 9651 | 7242 | 0.704 | |

3 | 7999 | 9524 | 7416 | 0.736 | |

4 | 8385 | 9910 | 7311 | 0.690 | |

5 | 6694 | 8219 | 7131 | 0.837 | |

6 | 6988 | 8513 | 6426 | 0.701 | |

7 | 6951 | 8476 | 7599 | 0.874 | |

8 | 7750 | 9275 | 6584 | 0.653 | |

9 | 8099 | 9624 | 8674 | 0.883 | |

10 | 7701 | 9226 | 7912 | 0.829 |

Two-Tier Parameters | |
---|---|

${D}_{f}$ (m) | 762.5 |

${D}_{s}$ (m) | ${D}_{e}\ast R$ = GAMM(107.319, 4.907) ∗ NORM(0.806, 0.153) |

Parameter | Variable | Value |
---|---|---|

Consignment number | N | 10 and 15 |

Truck speed (km/h) | S | 30.893 |

Consignment weight (kg) | W | 0.999 + EXPO(409) |

Consignment volume (m^{3}) | V | −0.001 + EXPO(1.38) |

Freight loading time (s) | T_{l} | 26 + EXPO(21.6) |

Freight unloading time (min) | T_{u} | 0.5 + GAMM(7.81, 1.15) |

First-tier distance (m) | D_{f} | 762.5 |

Second-tier distance (m) | D_{s} | GAMM(107.319, 4.907) ∗ NORM(0.806, 0.153) |

n = 10 | n = 15 | ||
---|---|---|---|

Truckload weight (t) | mean | 4.125 | 6.261 |

std dvn | 1.207 | 1.511 | |

Truckload volume (m^{3}) | mean | 13.506 | 20.513 |

std dvn | 3.769 | 5.374 | |

Capacity utilisation | mean | 48% | 73% |

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**MDPI and ACS Style**

Lyu, Z.; Pons, D.; Chen, J.; Zhang, Y.
Developing a Stochastic Two-Tier Architecture for Modelling Last-Mile Delivery and Implementing in Discrete-Event Simulation. *Systems* **2022**, *10*, 214.
https://doi.org/10.3390/systems10060214

**AMA Style**

Lyu Z, Pons D, Chen J, Zhang Y.
Developing a Stochastic Two-Tier Architecture for Modelling Last-Mile Delivery and Implementing in Discrete-Event Simulation. *Systems*. 2022; 10(6):214.
https://doi.org/10.3390/systems10060214

**Chicago/Turabian Style**

Lyu, Zichong, Dirk Pons, Jiasen Chen, and Yilei Zhang.
2022. "Developing a Stochastic Two-Tier Architecture for Modelling Last-Mile Delivery and Implementing in Discrete-Event Simulation" *Systems* 10, no. 6: 214.
https://doi.org/10.3390/systems10060214