# Location-Routing Problem with Simultaneous Home Delivery and Customer’s Pickup for City Distribution of Online Shopping Purchases

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Formulation

#### 3.1. Problem Description

- (1)
- A set of pickup point locations;
- (2)
- Service options for customers;
- (3)
- Assignments between customers and selected pickup points;
- (4)
- Assignments between the selected pickup points and depots;
- (5)
- Vehicles´ scheduling and routing.

- (1)
- As online shoppers may be clustered in the same or nearby buildings, all the customers are divided into small groups according to their locations. We then take each small group as a customer with fixed demand;
- (2)
- Each customer can be served by either HD or CP service;
- (3)
- We do not consider the operation of the second delivery caused by uncertain factors in the current scheduling, and take the second delivery cost instead.

#### 3.2. Notions and the Proposed Model

## 4. Hybrid Approach

_{ac}is accepted. Additionally, a hybrid approach along with a two-phase solution generation heuristic is proposed to obtain high quality initial population; specific crossover and mutation operators are also proposed followed by a biased fitness function to evaluate the individuals. A high level overview of the presented HGALS is depicted below.

_{0}, maximum population M, crossover probability, mutation probability p

_{m}, infeasible solution accept probability p

_{ac}, solution improve probability p

_{ls}, terminate condition

_{0}

_{1}and X

_{2}from the population following probability p

_{c}, conduct crossover operation to generate offspring Y

_{1}and Y

_{2}, apply LS to improve Y

_{1}and Y

_{2}with probability p

_{ls}, Y

_{1}and Y

_{2}are improved to be Y

_{1}′ and Y

_{2}′, accept Y

_{1}′ and Y

_{2}′ by probability 1 or p

_{ac}

_{3}following probability p

_{m}to make new solution Y

_{3}by mutation operation, apply LS to improve Y

_{3}with probability p

_{ls}, Y

_{3}is improved to be Y

_{3}′, accept Y

_{3}′ by probability 1 or p

_{ac}

_{4}to be Y

_{4}′ by LS, accept Y

_{4}′ by probability 1 or p

_{ac}

_{0}solutions to make population P

_{0}, turn to step 3; otherwise, turn to step 8

#### 4.1. Solution Representation

_{c}. The third tier is customer tier, which stores the customers. The second tier is used to index the service options information: if the value of a gene belongs to {n

_{d}+ 1, …, n

_{d}+ n

_{p}}, it means that the corresponding customer in the third tier is served by CP and served by the pickup point. Otherwise, if the value of the gene is 0, it means that the customer is served by HD. The first tier is for depots and indicates the assignments between the depots and pickup points as well as depots and the customers with HD service.

_{d}depots, n

_{p}′ pickup points selected from N

_{P}, n

_{c}′ customers with HD service and a set of dummy zeros. Routes are started and ended at the same depots to serve the selected pickup points and customers with HD service one by one, zeros are used to terminate a route and start a new one in depots.

#### 4.2. Solution and Population Initialization

_{0}, turn to step 1; otherwise, put S into P

_{0}.

_{0}is reached, turn to step 5; otherwise, turn to step 1.

#### 4.3. Fitness Function

#### 4.4. Individual Evolution and Selection

_{close}closest neighbors be stored in set N

_{close}, the diversity contribution of individual I to the population in N

_{close}is defined as the average distance to the individuals in N

_{close}computed by Formula (21). A normalized Hamming distance based on pickup points’ status, customers’ service options and the assignments of two individuals is adopted to show the difference between solutions. The distance is computed according to Formula (22).

_{i}(I) stands for the assignment of customers with HD service, sp

_{i}(I) stands for the assignment of pickup facilities and pc

_{i}(I) stands for the assignment of customers with CP service. γ

_{1}, γ

_{2}and γ

_{3}are the coefficients of the three factors, γ

_{1}+ γ

_{2}+ γ

_{3}= 1.

_{close}customers, when solutions are selected by elitist strategy, excellent solutions can be protected as well as the diversity achieved.

#### 4.5. Crossover Operation

_{c}from the population.

_{c}], letting the smaller one be cp1 and the other cp2.

#### 4.6. Mutation Operation

_{m.}

#### 4.7. Local Search

#### 4.8. Computational Complexity Analysis

_{max}is the maximum iteration. In the presented algorithm, the computational complexity for population initialization is $O({n}_{sp}({{n}_{p}}^{2}+{n}_{p}{n}_{c}+{n}_{d}{n}_{c}+{{n}_{c}}^{2}))$. In each iteration, the computational complexity for decoding, crossover operation, mutation, evaluation and local search are $O(2{n}_{s}+{n}_{c})$, $O(2{n}_{p}(2{n}_{s}+{n}_{c}))$, $O(2{n}_{s}+{n}_{c})$, $O(M({n}_{s}{n}_{c}+{n}_{s}{n}_{p}+{n}_{p}{n}_{c}))$ and $O({n}_{p}\cdot {(2{n}_{s}+{n}_{c})}^{2})$, respectively. The computational complexity of the whole algorithm can be calculated as:

_{p}and (2n

_{s}+n

_{c})

^{2}.

## 5. Computational Experiments

#### 5.1. Parameters Setting

_{0}, excellent solutions will be improved with much smaller probability, and it will take longer to find better solutions. Conversely, the population cannot sample the solution space which may lead to premature convergence. During the tests, since we found that the suitable initial population is closely related to n

_{p}and variable population strategy helps a lot, it was finally decided as M

_{0}= 4n

_{p}and M = 2M

_{0}. According to the decided M

_{0}and M, p

_{c}, p

_{m}, p

_{ls}and p

_{ac}are selected to be 0.45, 0.25, 1.0 and 0.25 respectively. Other parameters: h = 0.2, α

_{1}= 1000, α

_{2}= 200, α

_{3}= 100, α

_{4}= α

_{5}= 10, γ

_{1}= γ

_{3}= 0.3, γ

_{2}= 0.4.

#### 5.2. Tests Based on Benchmark Instance

#### 5.3. Real-World Instance

_{pc}= 600 m, u

_{H}= 1.5 min, u

_{P}= 10 min, p

_{H}= 10%, μ

_{sd}= 1.5. For the algorithm terminate conditions, to get a better solution, we take the maximum iterations I

_{max}and T

_{max}into consideration simultaneously; three terminate conditions are considered according to the instance scales: (I

_{max}, T

_{max}) = (1 × 10

^{4}, 600 s), (2 × 10

^{4}, 900 s), (3 × 10

^{4}, 1200 s). After running the algorithm, the obtained pickup point locations and customer assignments are reported in Table 4, and the vehicle routing information is given in Table 5.

_{ij}by d

_{ij}multiplying the unit routing cost. From the comparison of the total routing cost of the two scenarios, we can deduce that the travelling distance is reduced by 33%. As the carbon emission is closely related to the travelling distance, the proposed model can help to reduce the carbon emission effectively.

_{pc}influences the cost. When D

_{pc}ranges from 100 to 800 with step 100, the results are reported in Table 7 and Figure 6. It is worth mentioning that the scenario when D

_{pc}= 0 is with only HD service.

_{pc}≤ 100 m, the results obtained are the same with the only-HD scenario, which indicates that when D

_{pc}≤ 100 m, CP service shows no advantages when compared to HD and there is no need to provide CP service. When 100 < D

_{pc}≤ 600 m, there comes a continuous fall of the total cost, which means that with the distance increasing, better solutions are obtained due to the provided CP service. After that, the results stay unchanged which is constrained by the closest customers and the capacity of pickup points. From a cost optimization perspective, D

_{pc}= 600 m is the most acceptable distance when designing the Last Mile distribution system in this instance.

_{pc}, simultaneous HD and CP are effective at reducing the cost within a wide range of D

_{pc}, and a suitable distance exists for each Last Mile distribution system.

#### 5.4. Algorithm Components Performance Analysis

## 6. Conclusions

## 7. Future Research Directions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Sets | Description |
---|---|

$G$ | Network, $G=(N,A,C)$ |

$N$ | Node set, $N={N}_{D}\cup {N}_{P}\cup {N}_{C}$ |

${N}_{D}$ | Depot set, ${N}_{D}=\{1,\cdots ,{n}_{d}\}$ |

${N}_{P}$ | Pickup point set, ${N}_{P}=\{{n}_{d}+1,\cdots ,{n}_{d}+{n}_{p}\}$ |

${N}_{C}$ | Customer set, ${N}_{C}=\{{n}_{d}+{n}_{p}+1,\cdots ,{n}_{d}+{n}_{p}+{n}_{c}\}$ |

$A$ | Arc set, $A=\{(i,j):i,j\in N\}$ |

$C$ | Connection set, $C=\{(p,c):p\in {N}_{P},c\in {N}_{C}\}$ |

${V}_{D}$ | Available vehicle set |

Parameters | |

${n}_{d}$ | Number of depots |

${n}_{p}$ | Number of pickup points |

${n}_{c}$ | Number of customers |

${d}_{ij}$ | Distance of arc $(i,j)\in A$ |

${t}_{ij}$ | Travel time of arc $(i,j)\in A$ |

${c}_{ij}$ | Routing cost of arc $(i,j)\in A$ |

${q}_{i}$ | Demand of customer $i\in {N}_{C}$ |

${u}_{H}$ | Unit demand service time for HD service |

${u}_{P}$ | Fixed service time at a pickup point |

${\mathsf{\mu}}_{sd}$ | Probability of the second delivery |

${B}_{p}$ | Capacity of pickup point $p\in {N}_{P}$ |

${S}_{d}$ | Capacity of depot $d\in {N}_{D}$ |

${U}_{p}$ | Opening cost of pickup point $p\in {N}_{P}$ |

${Q}_{v}$ | Vehicle capacity |

${F}_{v}$ | Fixed vehicle cost |

${n}_{v}$ | Available vehicle number |

${D}_{pc}$ | Acceptable distance for CP service |

Decision variables | |

${x}_{ijk}^{d}$ | Equal to 1 if vehicle $k\in {V}_{D}$ departs from depot $d\in {N}_{D}$ that passes through arc $(i,j)\in A$; 0, otherwise |

${y}_{p}$ | Equal to 1 if pickup point $p\in {N}_{P}$ is selected to serve customers; 0, otherwise |

${z}_{ip}$ | Equal to 1 if customer $i\in {N}_{C}$ is served by pickup point $p\in {N}_{P}$; 0, otherwise |

${w}_{pd}$ | Equal to 1 if pickup point $p\in {N}_{P}$ is served by depot $d\in {N}_{D}$; 0, otherwise |

${\mathsf{\phi}}_{ijk}^{d}$ | Freight transported from depot $d\in {N}_{D}$by vehicle $k\in {V}_{D}$ that passes through arc $(i,j)\in A$ |

Instance | LB | BKR | CPU | Average | Best | |||
---|---|---|---|---|---|---|---|---|

Result | Gap/BKR | Result | Gap/BKR | |||||

1 | 20-5-1 | 54,793 | 54,793 | 1.8 | 54,836 | 0.08 | 54,793 | 0 |

2 | 20-5-1b | 39,104 | 39,104 | 2.2 | 39,104 | 0 | 39,104 | 0 |

3 | 20-5-2a | 48,908 | 48,908 | 1.5 | 48,908 | 0 | 48,908 | 0 |

4 | 20-5-2b | 37,542 | 37,542 | 2.6 | 37,542 | 0 | 37,542 | 0 |

5 | 50-5-1 | 90,111 | 90,111 | 12.9 | 90,130 | 0.03 | 90,111 | 0 |

6 | 50-5-1b | 67,340 | 63,242 | 17.8 | 63,242 | 0 | 63,242 | 0 |

7 | 50-5-2 | 88,298 | 88,298 | 24.3 | 88,643 | 0.40 | 88,643 | 0.39 |

8 | 50-5-2 | 67,340 | 67,340 | 21.4 | 67,340 | 0 | 67,340 | 0 |

9 | 50-5-2bis | 84,055 | 84,055 | 20.8 | 84,139 | 0.10 | 84,055 | 0 |

10 | 50-5-2bbis | 51,822 | 51,822 | 16.2 | 51,958 | 0.26 | 51,822 | 0 |

11 | 50-5-3 | 86,203 | 86,203 | 19.3 | 86,456 | 0.29 | 86,456 | 0.29 |

12 | 50-5-3b | 61,830 | 61,830 | 18.1 | 61,830 | 0 | 61,830 | 0 |

Avg | 13.2 | 0.10 | 0.06 | |||||

13 | 100-5-1 | 275,993 | 274,814 | 92.1 | 277,135 | 0.84 | 277,035 | 0.81 |

14 | 100-5-1b | 214,392 | 213,615 | 122.4 | 2,145,034 | 0.42 | 214,313 | 0.33 |

15 | 100-5-2 | 194,598 | 193,671 | 91.8 | 194,366 | 0.36 | 194,124 | 0.23 |

16 | 100-5-2b | 157,173 | 157,095 | 132.5 | 157,560 | 0.3 | 157,095 | 0 |

17 | 100-5-3 | 200,246 | 200,079 | 124.2 | 201,844 | 0.88 | 201,628 | 0.77 |

18 | 100-5-3b | 152,586 | 152,441 | 107.7 | 154,484 | 1.34 | 152,992 | 0.36 |

Avg | 111.8 | 0.69 | 0.42 | |||||

19 | 100-10-1 | 290,429 | 287,695 | 113.3 | 291,339 | 1.27 | 290,243 | 0.89 |

20 | 100-10-1b | 234,641 | 230,989 | 127.5 | 235,057 | 1.76 | 233,512 | 1.09 |

21 | 100-10-2 | 244,265 | 243,590 | 95.6 | 249,180 | 2.29 | 245,123 | 0.63 |

22 | 100-10-2b | 203,988 | 203,988 | 130.9 | 205,511 | 0.74 | 204,667 | 0.33 |

23 | 100-10-3 | 253,344 | 250,882 | 133.5 | 257,059 | 2.46 | 253,865 | 1.19 |

24 | 100-10-3b | 204,597 | 204,317 | 122.7 | 205,449 | 0.55 | 205,232 | 0.48 |

Avg | 120.6 | 1.51 | 0.76 | |||||

25 | 200-10-1 | 479,425 | 475,294 | 877.1 | 484,748 | 1.99 | 479,926 | 0.97 |

26 | 200-10-1b | 378,773 | 377,043 | 922.3 | 382,595 | 1.47 | 380,613 | 0.94 |

27 | 200-10-2 | 450,468 | 449,006 | 823.6 | 451,835 | 0.63 | 450,312 | 0.29 |

28 | 200-10-2b | 374,435 | 374,280 | 789.4 | 380,667 | 1.70 | 375,674 | 0.37 |

29 | 200-10-3 | 472,898 | 469,433 | 865.7 | 476,230 | 1.45 | 473,875 | 0.94 |

30 | 200-10-3b | 364,178 | 362,653 | 900.5 | 364,762 | 0.58 | 364,201 | 0.43 |

Avg | 863.1 | 1.30 | 0.66 | |||||

Global Avg | 277.2 | 0.77 | 0.47 |

Algorithm | Performance | Algorithm | Performance | ||
---|---|---|---|---|---|

CPU | Best Gap/BKR | CPU | Best Gap/BKR | ||

GRASP | 96.5 | 3.57 | MACO | 176.4 | 0.40 |

MAPM | 76.7 | 1.35 | GRASP+ILP | 1163.0 | 0.12 |

LRGTS | 17.5 | 0.70 | GVTNS | 91.2 | 0.37 |

GRASP+ELS | 258.2 | 1.11 | Hybrid GA | 199.1 | 0.32 |

SALRP | 422.4 | 0.46 | HGALS | 277.2 | 0.47 |

ALNS | 4221.0 | 0.27 |

Pickup Point | Customer | Pickup Point | Customer | Pickup Point | Customer |
---|---|---|---|---|---|

5 | 54, 55, 56 | 6 | 60, 61, 62, 65 | 8 | 58, 59, 63, 64 |

10 | 69, 75, 83 | 11 | 73, 74, 76,77 | 12 | 71,72 |

15 | 85,86 | 17 | 91, 92, 93 | 22 | 137, 138 |

23 | 144, 145 | 24 | 141, 142, 148 | 25 | 146,147 |

26 | 151, 152, 153, 154 | 28 | 132, 149 | 29 | 157, 159 |

31 | 158, 160, 161, 162, 165 | 33 | 172, 173 | 34 | 167, 168, 170 |

35 | 101, 102, 103 | 37 | 104, 105, 106, 107 | 38 | 97, 98 |

39 | 95, 108 | 40 | 110, 111, 112 | 41 | 114, 115 |

44 | 119, 120 | 45 | 122, 123 | 46 | 128, 129 |

47 | 174, 175 | 48 | 177, 178, 179, 180 | 49 | 181, 812, 183 |

No. | Routing | No. | Routing |
---|---|---|---|

1 | 1-11-12-10-67-68-1 | 7 | 2-156-26-25-143-140-136-22-23-2 |

2 | 1-5-51-53-6-1 | 8 | 2-31-29-163-164-166-169-34-171-2 |

3 | 1-8-57-66-87-84-82-1 | 9 | 2-47-176-186-185-184-49-48-2 |

4 | 1-70-135-139-24-150-28-133-134-81-1 | 10 | 3-40-113-116-121-124-45-117-41-3 |

5 | 1-15-88-89-90-96-38-37-99-79-1 | 11 | 3-118-44-99-125-109-39-94-17-3 |

6 | 1-80-35-127-126-130-156-131-46-100-78-1 |

Item | Simultaneous HD and CP | Only HD | Difference |
---|---|---|---|

Vehicle number | 11 | 26 | −15 |

Pickup point number | 30 | - | +30 |

Fixed vehicle cost | 2200 | 5200 | −3000 |

Routing cost | 156 | 233 | −77 |

Pickup point opening cost | 2370 | - | +2370 |

Second delivery cost | 186 | 529 | −343 |

Total cost | 4912 | 5962 | −1050 |

Item | D_{pc} | |||||||
---|---|---|---|---|---|---|---|---|

100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | |

Vehicle number | 26 | 21 | 13 | 11 | 11 | 11 | 11 | 11 |

Pickup point number | - | 16 | 36 | 33 | 30 | 30 | 30 | 30 |

Fixed vehicle cost | 5200 | 4000 | 2600 | 2200 | 2200 | 2200 | 2200 | 2200 |

Routing cost | 233 | 225 | 109 | 157 | 152 | 156 | 156 | 156 |

Pickup point opening cost | - | 1120 | 2640 | 2480 | 2400 | 2370 | 2370 | 2370 |

Second delivery cost | 529 | 417 | 140 | 188 | 209 | 186 | 186 | 186 |

Total cost | 5962 | 5762 | 5489 | 5025 | 4961 | 4912 | 4912 | 4912 |

Instance | SGALS | HGALS | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

CPU | Average | Best | CPU | Average | Best | |||||

Result | Gap | Result | Gap | Result | Gap | Result | Gap | |||

I2-10-50 | 272.9 | 1833.2 | 1.69 | 1806.2 | 0.19 | 143.2 | 1804.5 | 0.99 | 1802.7 | 0 |

I2-15-60 | 533.7 | 2473.2 | 4.56 | 2382.2 | 0.71 | 413.8 | 2388.6 | 0.98 | 2365.3 | 0 |

I2-18-80 | 600.1 | 3315.7 | 6.37 | 3117.1 | 0 | 435.6 | 3202.7 | 2.75 | 3117.1 | 0 |

I2-20-100 | 900 | 4404 | 14.46 | 4098.6 | 6.5 | 795.1 | 3958.6 | 2.88 | 3847.8 | 0 |

I2-22-120 | 862.3 | 4471.4 | 6.81 | 4220.7 | 0.82 | 852.4 | 4276.1 | 2.15 | 4186.3 | 0 |

I2-25-150 | 900 | 4967.8 | 6.13 | 4787.1 | 2.27 | 900 | 4769.3 | 1.89 | 4680.8 | 0 |

I2-28-180 | 1183 | 7166 | 9.21 | 7074.4 | 7.82 | 1200 | 6710.6 | 2.28 | 6561.3 | 0 |

I2-30-200 | 1200 | 6748.7 | 6.74 | 6543 | 3.49 | 1200 | 6426.9 | 1.65 | 6322.6 | 0 |

Agv | 807 | 7.0 | 2.73 | 742 | 1.83 | 0 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, L.; Wang, X.; Ni, L.; Lin, Y. Location-Routing Problem with Simultaneous Home Delivery and Customer’s Pickup for City Distribution of Online Shopping Purchases. *Sustainability* **2016**, *8*, 828.
https://doi.org/10.3390/su8080828

**AMA Style**

Zhou L, Wang X, Ni L, Lin Y. Location-Routing Problem with Simultaneous Home Delivery and Customer’s Pickup for City Distribution of Online Shopping Purchases. *Sustainability*. 2016; 8(8):828.
https://doi.org/10.3390/su8080828

**Chicago/Turabian Style**

Zhou, Lin, Xu Wang, Lin Ni, and Yun Lin. 2016. "Location-Routing Problem with Simultaneous Home Delivery and Customer’s Pickup for City Distribution of Online Shopping Purchases" *Sustainability* 8, no. 8: 828.
https://doi.org/10.3390/su8080828