# Computational Approaches for Grocery Home Delivery Services

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Attended Home Delivery

#### 2.2. Determining Feasible Time Slots

#### 2.3. Slotting and Pricing

## 3. Problem Description and Formal Model

#### 3.1. Computational Steps during the Ordering Phase

#### 3.1.1. Initialization Step

#### 3.1.2. Determination Step

#### 3.1.3. Insertion Step

#### 3.1.4. Improvement Step

#### 3.2. Vehicle Routing Problem with Time Windows

#### 3.3. Arrival Times and Feasibility

#### 3.3.1. Earliest and Latest Arrival Times

#### 3.3.2. Insertion Points

#### 3.3.3. Feasibility of an Insertion

#### 3.4. Slot Optimization Problem

## 4. Algorithms for the VRPTW

#### 4.1. Simple Insertion Heuristic

#### 4.2. Local Search Heuristic

#### 4.3. Adaptive Neighborhood Search Heuristic

#### 4.3.1. First/Last Customer

#### 4.3.2. Neighborhoods

- inside includes all operations with customers inside u:$in(u,\mathcal{A}):=\{i\in \mathcal{A}:i\in [f\left(u\right),l\left(u\right)\left]\right\};$
- outside represents operations with customers outside u:$out(u,\mathcal{A}):=\mathcal{A}\setminus \left(in\right(u,\mathcal{A})\cup \{0,n+1\left\}\right).$

#### 4.3.3. Loss and Free Time

#### 4.3.4. Feasibility and Infeasibility Conditions

#### 4.3.5. Algorithm

#### 4.4. Exact Approach for Solving a Sub-Problem

#### 4.5. Solution Approaches

- Simple insertion heuristic;
- ANS heuristic;
- A feasibility version of the suggested MILPs for solving the TSP(s)TW, which are applied for each time slot $u\in \mathcal{W}$ and for each $\mathcal{A}\in {\mathcal{T}}_{u}$.

- 1move + (1swap). The computationally cheap yet quite effective local search heuristic builds the foundation of the improvement step. We apply 1move (and 1swap) operations, where we focus on the 1move operations if possible, because they are computationally cheaper and, in general, more effective than 1swap operations. We stop our local search heuristic once we reach a local minimum of the objective function with respect to the selected neighborhood;
- 1move + 1swap + TSP(s)TW. After applying the local search heuristic, we additionally run our TSP(s)TW MILP on all tours that have changed since the last improvement step. We use the current tours of our delivery schedule as the initial solution for TSP(s)TW MILP. While 1move and 1swap exchange customers between different tours, the TSP(s)TW MILP re-orders them within the tours, which makes it a useful complement to the local search heuristics. In practice, optimizing the single tours of a schedule to optimality has proven to be critical to ensure driver satisfaction as it guarantees that drivers do not encounter any inefficiencies on their routes.

## 5. Computational Study and Analysis

#### 5.1. Design of the Instances

**${\mathcal{W}}_{\mathrm{NO}}$:**10 non-overlapping time windows with a length of 1 h each, e.g., 08:00–09:00, 09:00–10:00, etc. (b)

**${\mathcal{W}}_{\mathrm{OV}1.5}$:**10 overlapping time windows with a length of 1.5 h each (except for the last time window, which has a 1 h length), where each window overlaps the preceding time window by 30, e.g., 08:00–09:30, 09:00–10:30, etc. (c)

**${\mathcal{W}}_{\mathrm{OV}3}$:**12 overlapping time windows, consisting of nine windows with a length of 1 h each, 08:00–09:00, 10:00–11:00, …, 16:00–17:00 and (as used by Köhler et al. [21]) three time windows of 3 h length, morning: 08:00–11:00, noon: 11:00–14:00, afternoon: 14:00–17:00.

#### 5.2. Comparing Approaches for the Determination Step

#### 5.3. Comparing Approaches for the Improvement Step

- Average improvement over insertion step: the average reduction of the objective function when applying the optimization approaches to the schedule after inserting the new customer (given in percentage);
- Average improvement of the cost of insertion: the average reduction of the objective function relative to the increase of the objective function caused by inserting the new customer (given in percentage);
- Average number of TSPsTW MILPs solved;
- Average run time of each improvement strategy.

#### 5.3.1. Average-Sized Grocery Home Delivery Problems

#### 5.3.2. Dealing with Large-Problem Instances

#### 5.4. Interplay of Approaches for the Determination and the Improvement Step

- Average run time of the determination and improvement step;
- Average number of offered delivery time windows;
- Average total number of customer orders that have been inserted into the final schedules.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The first $f\left(u\right)$ and the last customer $l\left(u\right)$ of time window u in case of overlapping time windows. We indicate the positions of the insertion points ${\Theta}^{-}(j,\mathcal{A})$ and ${\Theta}^{+}(j,\mathcal{A})$ (vertical lines).

**Figure 4.**The loss time ${\chi}_{u}(j,\mathcal{A})$. Here, the case that the new customer j is inserted into a tour, with non-overlapping time windows, between two other customers assigned to time slot u, is considered.

**Table 1.**Summary of the computational experiments concerning the approaches for the determination step using ${\mathcal{W}}_{\mathrm{NO}}$ considering non-optimized and 1move-optimized schedules.

${\mathcal{W}}_{\mathbf{NO}}$ (10 Windows)—60 Vehicles | ||||||||
---|---|---|---|---|---|---|---|---|

Non-Optimized Schedules | Optimized Schedules | |||||||

Avg. $\widehat{p}$ | 408.0 | 1907.3 | ||||||

Fill level | 85% | 90% | 95% | 99% | 85% | 90% | 95% | 99% |

Avg. run time | ||||||||

(mm:ss.zzz) | ||||||||

simple insertion | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 |

TSP(s)TW insertion | 00:00.223 | 00:00.422 | 00:01.112 | 00:01.970 | 00:00.592 | 00:00.580 | 00:00.743 | 00:00.455 |

ANS | 00:00.024 | 00:00.089 | 00:00.468 | 00:04.031 | 00:00.049 | 00:00.052 | 00:00.077 | 00:01.377 |

Avg. number | ||||||||

of feasible slots | ||||||||

Simple insertion | 9.82 | 9.47 | 7.67 | 2.00 | 9.98 | 9.98 | 9.83 | 4.20 |

TSP(s)TW insertion | 9.85 | 9.62 | 8.60 | 4.30 | 9.98 | 9.98 | 9.83 | 4.29 |

ANS | 9.96 | 9.94 | 9.93 | 9.39 | 9.98 | 9.98 | 9.98 | 9.63 |

Combined | 9.96 | 9.94 | 9.93 | 9.43 | 9.98 | 9.98 | 9.98 | 9.63 |

**Table 2.**Summary of the computational experiments concerning the approaches for the determination step using ${\mathcal{W}}_{\mathrm{OV}1.5}$ considering non-optimized and 1move-optimized schedules.

${\mathcal{W}}_{\mathbf{OV}1.5}$ (10 Windows)–60 Vehicles | ||||||||
---|---|---|---|---|---|---|---|---|

Non-Optimized Schedules | Optimized Schedules | |||||||

Avg. $\widehat{p}$ | 628.0 | 1880.0 | ||||||

Fill level | 85% | 90% | 95% | 99% | 85% | 90% | 95% | 99% |

Avg. run time | ||||||||

(mm:ss.zzz) | ||||||||

Simple insertion | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 |

TSP(s)TW | 00:03.505 | 00:11.182 | 00:52.761 | 02:22.804 | 00:41.024 | 01:01.611 | 01:03.537 | 00:11.598 |

ANS | 00:00.032 | 00:00.138 | 00:00.683 | 00:07.837 | 00:00.036 | 00:00.040 | 00:00.066 | 00:08.343 |

Avg. number | ||||||||

of feasible slots | ||||||||

Simple insertion | 9.57 | 9.05 | 7.40 | 0.76 | 9.98 | 9.98 | 9.68 | 0.72 |

TSP(s)TW insertion | 9.84 | 9.68 | 8.96 | 5.04 | 10.00 | 10.00 | 9.84 | 0.70 |

ANS | 9.99 | 9.95 | 9.83 | 8.43 | 10.00 | 10.00 | 9.84 | 6.36 |

Combined | 9.99 | 9.95 | 9.87 | 8.76 | 10.00 | 10.00 | 10.00 | 6.36 |

**Table 3.**Summary of the computational experiments concerning the approaches for the determination step using ${\mathcal{W}}_{\mathrm{OV}3}$ considering non-optimized and 1move-optimized schedules.

${\mathcal{W}}_{\mathbf{OV}3}$ (12 Windows)–60 Vehicles | ||||||||
---|---|---|---|---|---|---|---|---|

Non-Optimized Schedules | Optimized Schedules | |||||||

Avg. $\widehat{p}$ | 406.0 | 1897.9 | ||||||

Fill level | 85% | 90% | 95% | 99% | 85% | 90% | 95% | 99% |

Avg. run time | ||||||||

(mm:ss.zzz) | ||||||||

Simple insertion | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 | 00:00.001 |

TSP(s)TW insertion | 00:02.645 | 00:15.541 | 01:27.422 | 05:20.061 | 00:56.969 | 00:54.405 | 01:04.239 | 00:17.879 |

ANS | 00:00.012 | 00:00.112 | 00:00.847 | 00:08.412 | 00:00.052 | 00:00.057 | 00:00.073 | 00:08.981 |

Avg. number | ||||||||

of feasible slots | ||||||||

Simple insertion | 11.84 | 11.02 | 8.54 | 1.21 | 12.00 | 12.00 | 11.90 | 0.73 |

TSP(s)TW insertion | 11.93 | 11.73 | 10.78 | 6.65 | 12.00 | 12.00 | 11.92 | 0.69 |

ANS | 12.00 | 11.98 | 11.94 | 10.16 | 12.00 | 12.00 | 12.00 | 7.59 |

Combined | 12.00 | 11.99 | 11.97 | 10.73 | 12.00 | 12.00 | 12.00 | 7.59 |

**Table 4.**Summary of the computational experiments for the improvement approaches considering instances with 500 customers.

Average-Sized Grocery Home Delivery Problems | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathcal{W}}_{\mathrm{NO}}$ | ${\mathcal{W}}_{\mathrm{OV}1.5}$ | ${\mathcal{W}}_{\mathrm{OV}3}$ | |||||||

Vehicles | 16 | 18 | 20 | 16 | 18 | 20 | 16 | 18 | 20 |

Avg. $\widehat{p}$ | 450.80 | 491.60 | 498.00 | 459.60 | 495.80 | 499.20 | 453.00 | 496.60 | 499.00 |

Avg. number of | |||||||||

time windows offered | 9.01 | 9.84 | 9.97 | 9.21 | 9.93 | 9.98 | 10.83 | 11.92 | 12.00 |

Avg. run time | |||||||||

(mm:ss.zzz) | |||||||||

1move | 00:00.015 | 00:00.099 | 00:00.108 | 00:00.097 | 00:00.124 | 00:00.134 | 00:00.090 | 00:00.111 | 00:00.120 |

1move+1swap | 00:00.632 | 00:00.776 | 00:00.766 | 00:00.952 | 00:01.170 | 00:01.145 | 00:00.925 | 00:01.106 | 00:01.106 |

1move+1swap+TSP(s)TW | 00:00.678 | 00:00.833 | 00:00.821 | 00:03.481 | 00:03.757 | 00:03.655 | 00:01.314 | 00:01.478 | 00:01.485 |

Avg. improvement | |||||||||

over Insertion (%) | |||||||||

1move | 0.30 | 0.29 | 0.29 | 0.37 | 0.37 | 0.37 | 0.34 | 0.34 | 0.34 |

1move+1swap | 0.41 | 0.40 | 0.39 | 0.52 | 0.50 | 0.50 | 0.50 | 0.50 | 0.49 |

1move+1swap+TSP(s)TW | 0.45 | 0.43 | 0.42 | 0.60 | 0.57 | 0.56 | 0.55 | 0.54 | 0.53 |

Avg. improvement | |||||||||

of cost of Insertion (%) | |||||||||

1move | 35.82 | 38.00 | 38.76 | 38.36 | 40.75 | 41.66 | 37.11 | 39.39 | 39.58 |

1move+1swap | 49.88 | 51.34 | 51.44 | 53.62 | 55.37 | 55.64 | 54.65 | 57.62 | 57.23 |

1move+1swap+TSP(s)TW | 53.83 | 55.34 | 55.26 | 61.80 | 63.33 | 63.48 | 59.53 | 62.37 | 61.90 |

Avg. number of | |||||||||

MILPs solved | |||||||||

1move+1swap+TSP(s)TW | 2.01 | 2.09 | 2.11 | 2.36 | 2.49 | 2.48 | 2.32 | 2.42 | 2.42 |

**Table 5.**Summary of the computational experiments for the improvement approaches considering instances with 2000 customers served by 80 vehicles. The improvement step is triggered after every ith ($i=10,20,30$) successful insertions.

Large-Scale Problem Instances—80 Vehicles | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathcal{W}}_{\mathrm{NO}}$ | ${\mathcal{W}}_{\mathrm{OV}1.5}$ | ${\mathcal{W}}_{\mathrm{OV}3}$ | |||||||

i | 10 | 20 | 30 | 10 | 20 | 30 | 10 | 20 | 30 |

Avg. $\widehat{p}$ | 1997.20 | 1998.50 | 1998.50 | 1999.40 | 1999.40 | 1999.40 | 1999.80 | 1999.80 | 1999.80 |

Avg. number of | |||||||||

time windows offered | 9.99 | 9.99 | 9.99 | 9.99 | 9.99 | 9.99 | 9.99 | 9.99 | 9.99 |

Avg. run time | |||||||||

(mm:ss.zzz) | |||||||||

1move | 00:04.303 | 00:06:907 | 00:10:578 | 00:05.360 | 00:08.930 | 00:12.587 | 00:04.939 | 00:08.126 | 00:11.208 |

1move+1swap | 00:59.003 | 01:28:161 | 01:57.892 | 01:25.017 | 02:02.721 | 02:31.010 | 01:13.358 | 01:40.780 | 02:10.395 |

1move+1swap+TSP(s)TW | 00:59.980 | 01:29.274 | 01:59.241 | 01:40.952 | 02:25.107 | 02:56.644 | 01:16.316 | 01:50.400 | 02:14.499 |

Avg. improvement | |||||||||

over Insertion (%) | |||||||||

1move | 1.62 | 3.27 | 4.93 | 1.69 | 3.54 | 5.12 | 1.60 | 3.29 | 4.82 |

1move+1swap | 2.29 | 4.16 | 5.94 | 2.46 | 4.57 | 6.28 | 2.33 | 4.26 | 5.92 |

1move+1swap+TSP(s)TW | 2.35 | 4.24 | 6.05 | 2.56 | 4.73 | 6.48 | 2.39 | 4.35 | 6.03 |

Avg. number of | |||||||||

MILPs solved | |||||||||

1move+1swap+TSP(s)TW | 23.43 | 30.64 | 33.58 | 24.65 | 31.41 | 34.48 | 24.09 | 31.09 | 34.22 |

**Table 6.**Summary of the computational experiments for different combinations of time window determination ($\mathrm{simple}\phantom{\rule{4.pt}{0ex}}\mathrm{insertion},\phantom{\rule{3.33333pt}{0ex}}\mathrm{ANS}$) and improvement approaches ($1\mathrm{move},\phantom{\rule{3.33333pt}{0ex}}1\mathrm{move}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1\mathrm{swap}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}\mathrm{TSP}\left(\mathrm{s}\right)\mathrm{TW}$) considering instances with 500 customers.

Average-Sized Grocery Home Delivery Problems | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathcal{W}}_{\mathrm{NO}}$ | ${\mathcal{W}}_{\mathrm{OV}1.5}$ | ${\mathcal{W}}_{\mathrm{OV}3}$ | |||||||

Vehicles | 16 | 18 | 20 | 16 | 18 | 20 | 16 | 18 | 20 |

Determination: simple insertion, Improvement: 1move | |||||||||

Avg. $\widehat{p}$ | 453.35 | 491.80 | 498.65 | 455.75 | 495.10 | 498.65 | 451.00 | 495.50 | 499.60 |

Avg. number of | |||||||||

time windows offered | 9.07 | 9.84 | 9.97 | 9.15 | 9.90 | 9.98 | 10.81 | 11.88 | 11.99 |

Avg. run time | |||||||||

(mm:ss.zzz) | |||||||||

Determination | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 |

Improvement | 0:00.076 | 0:00.103 | 0:00.114 | 0:00.103 | 0:00.134 | 0:00.138 | 0:00.096 | 0:00.123 | 0:00.127 |

Determination: simple insertion, Improvement: 1move + 1swap + TSP(s)TW | |||||||||

Avg. $\widehat{p}$ | 454.25 | 494.00 | 499.05 | 456.95 | 495.95 | 499.00 | 452.80 | 496.00 | 499.65 |

Avg. number of | |||||||||

time windows offered | 9.09 | 9.87 | 9.98 | 9.15 | 9.92 | 9.98 | 10.86 | 11.91 | 12.00 |

Avg. run time | |||||||||

(mm:ss.zzz) | |||||||||

Determination | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 | 0:00.001 |

Improvement | 0:00.758 | 0:00.852 | 0:00.957 | 0:03.139 | 0:03.234 | 0:03.133 | 0:01.345 | 0:01.672 | 0:01.598 |

Determination: ANS, Improvement: 1move | |||||||||

Avg. $\widehat{p}$ | 455.35 | 498.50 | 499.55 | 458.00 | 499.05 | 499.75 | 453.10 | 499.44 | 499.80 |

Avg. number of | |||||||||

time windows offered | 9.11 | 9.97 | 9.99 | 9.16 | 9.98 | 9.99 | 10.88 | 11.99 | 12.00 |

Avg. run time | |||||||||

(mm:ss.zzz) | |||||||||

Determination | 0:00.098 | 0:00.022 | 0:00.017 | 0:00.090 | 0:00.014 | 0:00.012 | 0:00.120 | 0:00.020 | 0:00.017 |

Improvement | 0:00.060 | 0:00.077 | 0:00.086 | 0:00.073 | 0:00.091 | 0:00.100 | 0:00.073 | 0:00.092 | 0:00.098 |

Determination: ANS, Improvement: 1move + 1swap + TSP(s)TW | |||||||||

Avg. $\widehat{p}$ | 455.55 | 498.95 | 499.65 | 457.85 | 499.15 | 499.75 | 452.90 | 499.55 | 499.80 |

Avg. number of | |||||||||

time windows offered | 9.12 | 9.98 | 9.99 | 9.16 | 9.99 | 9.99 | 10.87 | 11.99 | 12.00 |

Avg. run time | |||||||||

(mm:ss.zzz) | |||||||||

Determination | 0:00.104 | 0:00.026 | 0:00.020 | 0:00.088 | 0:00.014 | 0:00.014 | 0:00.114 | 0:00.023 | 0:00.019 |

Improvement | 0:00.713 | 0:00.889 | 0:00.862 | 0:00.898 | 0:01.138 | 0:01.099 | 0:00.883 | 0:01.158 | 0:01.161 |

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## Share and Cite

**MDPI and ACS Style**

Truden, C.; Maier, K.; Jellen, A.; Hungerländer, P.
Computational Approaches for Grocery Home Delivery Services. *Algorithms* **2022**, *15*, 125.
https://doi.org/10.3390/a15040125

**AMA Style**

Truden C, Maier K, Jellen A, Hungerländer P.
Computational Approaches for Grocery Home Delivery Services. *Algorithms*. 2022; 15(4):125.
https://doi.org/10.3390/a15040125

**Chicago/Turabian Style**

Truden, Christian, Kerstin Maier, Anna Jellen, and Philipp Hungerländer.
2022. "Computational Approaches for Grocery Home Delivery Services" *Algorithms* 15, no. 4: 125.
https://doi.org/10.3390/a15040125