# Stochastic Drone Fleet Deployment and Planning Problem Considering Multiple-Type Delivery Service

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- (1)
- This paper studies a new stochastic DFDP with uncertain parcel demand, which determines (i) the drone fleet deployment, i.e., the numbers of different types of drones deployed, (ii) the drone service module, and (iii) the numbers of parcels delivered by drones and couriers under each scenario of demand.
- (2)
- For the problem, a novel two-stage stochastic programming formulation is proposed and a classic sample average approximation (SAA) method is first employed. Since SAA is very time-consuming, a hybrid genetic algorithm (GA) is further developed to achieve computational efficiency.
- (3)
- A case study based on the delivery service of Ele.me in Shanghai Jinshan industrial park in China demonstrates the applicability of the proposed methods. Computational results show that, under a given number of scenarios, the hybrid GA outperforms SAA in terms of total computational time with high solution quality.

## 2. Literature Review

#### 2.1. Operational-Level Drone Routing Problem

#### 2.2. Tactical-Level Drone Facility Location and Drone Fleet Deployment Problem

#### 2.3. Fleet Deployment and Planning Problem

## 3. Problem Description and Formulation

#### 3.1. Problem Description

#### 3.2. Formulation

**Variables:**

- -
- ${x}_{rl}$: Number of drones of type $l\in \mathcal{L}$ deployed on service route $r\in \mathcal{R}$, and $\mathit{x}={[{x}_{11},{x}_{12},\dots ,{x}_{\left|\mathcal{R}\right|,{N}_{\left|\mathcal{R}\right|}}]}^{\top}$.
- -
- ${\gamma}_{rl}$: A binary variable equal to 1 if there is a drone of type $l\in \mathcal{L}$ deployed on service route $r\in \mathcal{R}$, 0 otherwise, and $\mathit{\gamma}={[{\gamma}_{11},{\gamma}_{12},\dots ,{\gamma}_{\left|\mathcal{R}\right|,{N}_{\left|\mathcal{R}\right|}}]}^{\top}$.
- -
- ${\mu}_{rk}$: A binary variable equal to 1 if the service module $k\in \mathcal{K}$ is selected on route $r\in \mathcal{R}$, 0 otherwise, and $\mathit{\mu}={[{\mu}_{11},{\mu}_{12},\dots ,{\mu}_{\left|\mathcal{R}\right|,\left|\mathcal{K}\right|}]}^{\top}$.
- -
- ${y}_{ri}^{h}\left(\mathit{\xi}\right)$: Number of parcels of category $h\in \mathcal{H}$ transported by drones on leg $i\in {\mathcal{I}}_{r}$ of route $r\in \mathcal{R}$ under realized $\mathit{\xi}\in \mathsf{\Xi}$.
- -
- ${\nu}_{rlk}$: A nonnegative integral variable used to linearize ${x}_{rl}\xb7{\mu}_{rk}$, and ${\nu}_{rlk}={x}_{rl}\xb7{\mu}_{rk}$.
- -
- ${z}_{rki}^{h}\left(\mathit{\xi}\right)$: A nonnegative integral variable used to linearize ${\mu}_{rk}\xb7{y}_{ri}^{h}\left(\mathit{\xi}\right)$, and ${z}_{rki}^{h}\left(\mathit{\xi}\right)={\mu}_{rk}\xb7{y}_{ri}^{h}\left(\mathit{\xi}\right)$, $\forall \mathit{\xi}\in \mathsf{\Xi}$.
- -
- $Q(\mathit{x},\mathit{\gamma},\mathit{\mu},\mathit{\xi})$: The recourse function value, i.e., the labor cost, with given $\mathit{x}$, $\mathit{\gamma}$ and $\mathit{\mu}$ under realized $\mathit{\xi}\in \mathsf{\Xi}$ during the planning time period.

**P1**) for the problem can be expressed as follows:

**P1**):

## 4. Solution Approaches

**P1**by calling the off-the-shelf solvers. Therefore, in this section, based on the idea of SAA, we first propose an approximated SAA-based formulation, which can be solved by calling commercial solvers. As exactly solving the SAA-based model is rather time-consuming, a hybrid GA is further developed, to obtain feasible solutions in a reasonable time for the SAA-based formulation.

#### 4.1. SAA

**P2**is proposed, in which the second-stage decisions depend on the first-stage decisions and on the realized demands under each scenario $\omega \in \mathsf{\Omega}$. SAA method is easy to implement and performs well under a sufficient number of scenarios (Wang and Ahmed [39]).

**P2**) with the purpose to apply the SAA method, where $\omega \in \mathsf{\Omega}$ denotes the index of scenarios.

**P2**):

**P2**can be exactly solved by calling commercial optimization softwares, such as CPLEX. The obtained solution includes the first-stage decisions and the second-stage decisions under given scenarios set $\mathsf{\Omega}$. The obtained first-stage solution, i.e., $\mathit{x},\mathit{\gamma}$ and $\mathit{\mu}$, can be considered as an approximately optimal first-stage drone fleet deployment, including the type, number, and service choice of drones deployed on each route, of the original problem. When the number of given scenarios $\left|\mathsf{\Omega}\right|$ is sufficiently large and $\left|\mathsf{\Omega}\right|\to \infty $, the optimal objective value of

**P2**converges to the true optimal objective value almost surely (Bertsimas et al. [40]).

#### 4.2. Hybrid GA

#### 4.2.1. Coding

#### 4.2.2. Crossover and Mutation

## 5. Computational Experiments

#### 5.1. Data Generation

#### 5.2. Computational Results

- (1)
- with the increase of the number of scenarios, the computational time of SAA increases dramatically;
- (2)
- given the same number of scenarios, the computational time of the hybrid GA is smaller than SAA with high solution quality;
- (3)
- with the increase of average drone speed and the total flight time, the number of drones deployed and the total cost decrease;
- (4)
- when the volume and weight capacities of drones increase, the number of drones deployed and the total cost decrease;
- (5)
- when the time intervals decrease, the number of drones deployed increases; and
- (6)
- the developed hybrid GA outperforms the SAA in terms of the computational time with high solution quality.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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i: | Index of legs on a service route. |

$\mathcal{R}$: | Set of service routes indexed by r. |

$\mathcal{L}$: | Set of all drone types indexed by l. |

$\mathcal{K}$: | Set of drone service modules indexed by k. |

$\mathcal{H}$: | Set of parcel categories indexed by h. |

${N}_{r}$: | Number of customers visited by service route $r\in \mathcal{R}$. |

${\mathcal{I}}_{r}$: | Set of customers on service route $r\in \mathcal{R}$, and ${\mathcal{I}}_{r}=\left(\right)open="\{"\; close="\}">1,2,\dots ,{N}_{r}$. |

${f}_{k}$: | Service frequency in the planning time period under service module $k\in \mathcal{K}$. |

${t}_{k}^{int}$: | Time interval between two consecutive drones under service module $k\in \mathcal{K}$. |

${V}_{l}$: | Volume capacity of a drone of type $l\in \mathcal{L}$. |

${W}_{l}$: | Weight capacity of a drone of type $l\in \mathcal{L}$. |

${T}_{rl}$: | Total travel time of a drone of type $l\in \mathcal{L}$ on service route $r\in \mathcal{R}$. |

${C}_{l}^{drone}$: | Fixed leasing (or amortized purchasing) cost in the planning time period of a drone of type $l\in \mathcal{L}$. |

${C}_{l}^{oper}$: | Variable cost for operating a drone of type $h\in \mathcal{H}$ per flight on each route. |

${C}_{ri}^{h}$: | Cost for a courier handling a parcel of category $h\in \mathcal{H}$ on the i-th leg of service route $r\in \mathcal{R}$. |

${\xi}_{ri}^{h}$: | Stochastic demand per time unit of category $h\in \mathcal{H}$ on leg i of route $r\in \mathcal{R}$. |

${v}_{h}$: | Volume of a parcel of category $h\in \mathcal{H}$. |

${w}_{h}$: | Weight of a parcel of category $h\in \mathcal{H}$. |

D: | Number of time units in the planning time period. |

M: | A large enough number. |

Parameter | Value (Hybrid GA) |
---|---|

Population size | 50 |

Generation number | 20 |

Crossover probability | 0.9 |

Mutation 1 probability | 0.2 |

Mutation 2 probability | 0.8 |

ID | Service Routes |
---|---|

1 | Pingan Square → Yunhe Village → Wujiazhai → Zhangjia → Ting South → Pingan Square |

2 | Pingan Square → Chenjiadai → Dongfeng → Wujiadai →Yuchi → Quanxin → Ting South → Pingan Square |

3 | Pingan Square → Dongfeng → Xupu Village → Tuanjie → Changlou → Pingan Square |

4 | Pingan Square → Changlou → Hexing Village → Qiaowan → Pingan Square |

5 | Pingan Square → Changlou → Fengjiazhai → Hexing Village → Taojiayuan → Qiaowan → Pingan Square |

6 | Pingan Square → Hexing Village → Wujiazhai → Shijiazhai → Jiangzhunag Village → Pingan Square |

7 | Pingan Square → Zhangjia → Wujiazhai → Yunhe Village → Fengjiazhai → Changlou → Dongfeng → Chenjiadai → Pingan Square |

8 | Pingan Square → Ting South → Wujiadai → Xupu Village → Tuanjie → Hexing Village → Qiaowan → Pingan Square |

9 | Pingan Square → Taojiayuan → Jiangzhuang Village → Yunhe village → Pingan Square |

10 | Pingan Square → Hexing Village → Wujiazhai → Shijiazhai → Tuanjie → Changlou → Pingan Square |

11 | Pingan Square → Xupu Village → Wujiadai → Yuchi → Quanxin → Chejiadai → Pingan Square |

Drone Type | Capacity | |||
---|---|---|---|---|

Volume (m${}^{3}$) | Weight (kg) | Operating Cost (Yuan) | Average Speed (km/h) | |

1 | 1 | 30 | 350 | 20 |

2 | 2 | 35 | 420 | 30 |

3 | 3 | 40 | 490 | 40 |

Parcel Category | Volume (m${}^{3}$) | Weight (kg) | Labor Cost Per Parcel Per Kilometer (Yuan) |
---|---|---|---|

1 | 0.01 | 1 | 0.2 |

2 | 0.02 | 2 | 0.4 |

3 | 0.01 | 2 | 0.3 |

4 | 0.02 | 1 | 0.3 |

Route | Deterministic Situation | SAA | HGA | |||
---|---|---|---|---|---|---|

${\mathit{N}}_{\mathit{drone}}$ | Service module | ${\mathit{N}}_{\mathit{drone}}$ | Service Module | ${\mathit{N}}_{\mathit{drone}}$ | Service Module | |

1 | 2 | 2 | 4 | 1 | 4 | 1 |

2 | 2 | 2 | 3 | 1 | 3 | 1 |

3 | 2 | 2 | 3 | 1 | 3 | 1 |

Obj | 5003 | 4900 | 4987 |

$\left|\mathsf{\Omega}\right|$ | SAA | HGA | ||||
---|---|---|---|---|---|---|

${\mathit{N}}_{\mathit{drone}}$ | Obj ($\times {10}^{5}$) | Time (s) | ${\mathit{N}}_{\mathit{drone}}$ | Obj ($\times {10}^{5}$) | Time (s) | |

3 | 50 | 7.3531 | 56.0 | 50 | 7.5165 | 48.2 |

5 | 50 | 6.3544 | 119.3 | 50 | 6.3795 | 66.9 |

10 | 50 | 6.8960 | 462.7 | 50 | 7.0713 | 79.8 |

15 | 50 | 6.9026 | 2883.2 | 50 | 6.9867 | 89.5 |

20 | 50 | 7.1125 | 3600.0 | 50 | 7.1602 | 112.4 |

25 | - | - | - | 50 | 7.0799 | 138.6 |

30 | - | - | - | 50 | 7.0001 | 164.2 |

35 | - | - | - | 50 | 6.9762 | 189.4 |

40 | - | - | - | 50 | 6.8648 | 249.8 |

50 | - | - | - | 50 | 6.8782 | 308.7 |

Average | 50 | 6.9237 | 1424.2 | 50 | 6.9913 | 144.8 |

Lower bound | 50 | 6.3544 | 56.0 | 50 | 6.3795 | 48.2 |

Upper bound | 50 | 7.3531 | 3600.0 | 50 | 7.5165 | 308.7 |

Increase of the | SAA with 10 Scenarios | HGA with 50 Scenarios | ||||
---|---|---|---|---|---|---|

Average Drone Speed | ${\mathit{N}}_{\mathit{drone}}$ | Time (s) | Obj($\times {10}^{5}$) | ${\mathit{N}}_{\mathit{drone}}$ | Time (s) | Obj($\times {10}^{5}$) |

0 | 50 | 463.2 | 6.8960 | 50 | 312.7 | 6.8782 |

5 | 44 | 465.3 | 6.2485 | 44 | 306.3 | 6.8488 |

10 | 42 | 468.9 | 6.1849 | 42 | 317.9 | 6.8390 |

15 | 39 | 472.1 | 6.2458 | 39 | 317.2 | 6.8243 |

20 | 35 | 471.6 | 6.2346 | 36 | 318.3 | 6.8047 |

25 | 34 | 465.8 | 6.1793 | 34 | 306.6 | 6.7998 |

30 | 30 | 466.9 | 6.1345 | 31 | 309.9 | 6.7802 |

Average | 39.1 | 467.7 | 6.3034 | 39.4 | 312.7 | 6.8250 |

Increase of Volume | SAA with 10 Scenarios | HGA with 50 Scenarios | ||||
---|---|---|---|---|---|---|

and Weight Capacities | ${\mathit{N}}_{\mathit{drone}}$ | Time (s) | Obj($\times {10}^{5}$) | ${\mathit{N}}_{\mathit{drone}}$ | Time (s) | Obj($\times {10}^{5}$) |

(0, 0) | 50 | 467.5 | 6.8960 | 50 | 313.3 | 6.8782 |

(0.5, 5) | 37 | 472.6 | 6.7510 | 37 | 311.0 | 6.8145 |

(0.5, 10) | 37 | 468.9 | 6.2560 | 37 | 309.8 | 6.8145 |

(1, 5) | 37 | 469.8 | 6.2346 | 36 | 320.4 | 6.8047 |

(1, 10) | 37 | 470.6 | 6.1793 | 37 | 322.7 | 6.7998 |

Average | 39.6 | 469.9 | 6.4634 | 39.4 | 315.4 | 6.8223 |

Decrease of | SAA with 10 Scenarios | HGA with 50 Scenarios | ||||
---|---|---|---|---|---|---|

Time Intervals | ${\mathit{N}}_{\mathit{drone}}$ | Time (s) | Obj($\times {10}^{5}$) | ${\mathit{N}}_{\mathit{drone}}$ | Time (s) | Obj($\times {10}^{5}$) |

0 | 50 | 466.1 | 6.8960 | 50 | 314.6 | 6.8782 |

1 | 62 | 464.2 | 2.4349 | 62 | 310.4 | 2.9725 |

2 | 81 | 478.1 | 0.9162 | 81 | 313.4 | 1.2973 |

3 | 117 | 472.9 | 0.5733 | 117 | 319.5 | 0.5769 |

4 | 229 | 469.8 | 1.1221 | 228 | 320.3 | 1.1221 |

Average | 107.8 | 470.22 | 2.3885 | 107.6 | 315.6 | 2.5694 |

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## Share and Cite

**MDPI and ACS Style**

Liu, M.; Liu, X.; Zhu, M.; Zheng, F.
Stochastic Drone Fleet Deployment and Planning Problem Considering Multiple-Type Delivery Service. *Sustainability* **2019**, *11*, 3871.
https://doi.org/10.3390/su11143871

**AMA Style**

Liu M, Liu X, Zhu M, Zheng F.
Stochastic Drone Fleet Deployment and Planning Problem Considering Multiple-Type Delivery Service. *Sustainability*. 2019; 11(14):3871.
https://doi.org/10.3390/su11143871

**Chicago/Turabian Style**

Liu, Ming, Xin Liu, Maoran Zhu, and Feifeng Zheng.
2019. "Stochastic Drone Fleet Deployment and Planning Problem Considering Multiple-Type Delivery Service" *Sustainability* 11, no. 14: 3871.
https://doi.org/10.3390/su11143871