# Unconventional GVNS for Solving the Garbage Collection Problem with Time Windows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Research on TSP and TSP-TW

#### 2.2. Research on Algorithms for TSP and TSP-TW

## 3. A GPS Application for Garbage Trucks Expressed as a TSP with Time Windows

## 4. The Proposed Methodology

#### 4.1. Variable Neighborhood Search

- Cyclic neighbourhood shift phase: whether or not there is an enhancement with regard to the present alternative, the search proceeds to the next neighbourhood structure in the list.
- Pipe neighbourhood shift phase: if there is an enhancement of the present solution in some neighbourhood, the exploration will continue in that neighbourhood.
- Skewed neighbourhood shift step: recognize as new incumbent solutions not only enhancing solutions but also some of those that are worse than the present incumbent solution. The aim of such a neighborhood change step is to permit the exploration of valleys far from the current solution. The solution of the current trial must be assessed considering not only the objective values of the trial and the incumbent solution, but also the distance between them.

**Variable neighborhood search variants.**Many variants of VNS have been developed and applied for solving hard optimization problems [26]. The Basic VNS (BVNS), the Variable Neighborhood Descent (VND), the General VNS (GVNS), and the Reduced VNS (RVNS) are the most commonly used variants. In the BVNS a diversification method and a local search operator are alternated. VND consists of an improvement procedure in which the neighborhood structures are explored systematically, and a neighborhood change step. There are different VND variants according to their neighborhood change step. The pipe-VND, which uses the pipe neighborhood change step, seems to be the most efficient method for the solution of computationally challenging problems [26]. General Variable Neighborhood Search (GVNS) is a VNS variant where a VND method is used as the improvement procedure. GVNS has been successfully tested in many applications, as several recent works have demonstrated [27,28].

#### 4.2. Our Approach

#### 4.3. Perturbation, Neighborhood Structures

- Move backwards the violated nodes back,
- Move forward the non-violated nodes,
- Move the non-violated nodes backwards, and
- Move forward the violated nodes.

#### 4.4. Quantum Inspired Perturbation

## 5. Computational Results

#### 5.1. Implementation Technology and Instances Used

#### 5.2. Computational Results on Known Benchmarks

#### 5.3. A Practical Test Case with Real Data

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Pseudocode of the Meta-heuristic [4].

**Figure 4.**A simple visual example of TSP-TW with 5 nodes [4].

Name | N | Max WL | Best Sol. | Proposed by | RSC | Dif. from Best Sol. | QISC | Dif. from Best Sol |
---|---|---|---|---|---|---|---|---|

n20w120 | 20 | 120 | 265.6 | Ohlmann and Thomas (2007) | 357.49 | 34% | 354.03 | 33% |

n20w140 | 20 | 140 | 232.8 | Ohlmann and Thomas (2007) | 337.05 | 44% | 306.88 | 32% |

n20w160 | 20 | 160 | 218.2 | Ohlmann and Thomas (2007) | 331.00 | 51% | 390.72 | 79% |

n20w180 | 20 | 180 | 236.6 | Ohlmann and Thomas (2007) | 336.58 | 42% | 365.64 | 54% |

n20w200 | 20 | 200 | 241.0 | Ohlmann and Thomas (2007) | 365.87 | 51% | 365.36 | 51% |

n40w120 | 40 | 120 | 360.0 | Calvo (2000) | 521.18 | 44% | 551.84 | 53% |

n40w140 | 40 | 140 | 348.4 | Calvo (2000) | 493.35 | 41% | 518.36 | 49% |

n40w160 | 40 | 160 | 326.8 | Ohlmann and Thomas (2007) | 482.20 | 47% | 552.24 | 69% |

n40w180 | 40 | 180 | 326.8 | Calvo (2000) | 532.49 | 62% | 519.42 | 59% |

n40w200 | 40 | 200 | 313.8 | Ohlmann and Thomas (2007) | 493.65 | 57% | 468.01 | 49% |

Name | N | Max WL | Best Sol. | Proposed by | SSC | Dif. from Best Sol. | QISC | Dif. from Best Sol |
---|---|---|---|---|---|---|---|---|

n20w120 | 20 | 120 | 265.6 | Ohlmann and Thomas (2007) | 371.78 | 39% | 354.03 | 33% |

n20w140 | 20 | 140 | 232.8 | Ohlmann and Thomas (2007) | 370.61 | 59% | 306.88 | 32% |

n20w160 | 20 | 160 | 218.2 | Ohlmann and Thomas (2007) | 398.08 | 82% | 390.72 | 79% |

n20w180 | 20 | 180 | 236.6 | Ohlmann and Thomas (2007) | 418.13 | 76% | 365.64 | 54% |

n20w200 | 20 | 200 | 241.0 | Ohlmann and Thomas (2007) | 414.70 | 72% | 365.36 | 51% |

n40w120 | 40 | 120 | 360.0 | Calvo (2000) | 549.97 | 52% | 551.84 | 53% |

n40w140 | 40 | 140 | 348.4 | Calvo (2000) | 563.83 | 61% | 518.36 | 49% |

n40w160 | 40 | 160 | 326.8 | Ohlmann and Thomas (2007) | 553.81 | 69% | 552.24 | 69% |

n40w180 | 40 | 180 | 326.8 | Calvo (2000) | 561.09 | 71% | 519.42 | 59% |

n40w200 | 40 | 200 | 313.8 | Ohlmann and Thomas (2007) | 558.65 | 78% | 468.01 | 49% |

Abbreviation | Full Name |
---|---|

N | Number of nodes |

Max WL | Max Window Length |

Best Sol | Best Solution |

RSC | Random Solution Cost |

RST | Random Solution Time |

dif. from best sol. | Difference from best solution |

SSC | Sorted Solution Cost |

SST | Sorted Solution Time |

QISC | Quantum Inspired Solution Cost |

Node | Longitude | Latitude | Comment |
---|---|---|---|

1 | 39.6422466 | 19.8223686 | the depot |

2 | 39.6404733 | 19.8229525 | a dumpster node |

3 | 39.6413691 | 19.8306238 | a dumpster node |

4 | 39.6407922 | 19.8370158 | a dumpster node |

5 | 39.6374761 | 19.8386927 | a dumpster node |

6 | 39.6329728 | 19.8526526 | a dumpster node |

7 | 39.6308208 | 19.8660169 | a dumpster node |

8 | 39.6246663 | 19.8778979 | a dumpster node |

9 | 39.6234445 | 19.8863486 | a dumpster node |

10 | 39.6247393 | 19.8899176 | a dumpster node |

11 | 39.6284544 | 19.8880917 | a dumpster node |

12 | 39.6297152 | 19.8887043 | a dumpster node |

13 | 39.6257976 | 19.9083606 | a dumpster node |

14 | 39.625156 | 19.9085661 | a dumpster node |

15 | 39.6243003 | 19.9219708 | a dumpster node |

16 | 39.6256647 | 19.9108012 | a dumpster node |

17 | 39.6262482 | 19.9179632 | a dumpster node |

18 | 39.6251191 | 19.9187309 | a dumpster node |

19 | 39.6254352 | 19.9195927 | a dumpster node |

20 | 39.6246342 | 19.9219791 | a dumpster node |

21 | 39.6243003 | 19.9219708 | a dumpster node |

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Papalitsas, C.; Andronikos, T.
Unconventional GVNS for Solving the Garbage Collection Problem with Time Windows. *Technologies* **2019**, *7*, 61.
https://doi.org/10.3390/technologies7030061

**AMA Style**

Papalitsas C, Andronikos T.
Unconventional GVNS for Solving the Garbage Collection Problem with Time Windows. *Technologies*. 2019; 7(3):61.
https://doi.org/10.3390/technologies7030061

**Chicago/Turabian Style**

Papalitsas, Christos, and Theodore Andronikos.
2019. "Unconventional GVNS for Solving the Garbage Collection Problem with Time Windows" *Technologies* 7, no. 3: 61.
https://doi.org/10.3390/technologies7030061