# The Vehicle Routing Problem with Simultaneous Pick-Up and Delivery under Fuzziness Considering Fuel Consumption

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## Abstract

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## 1. Introduction

- The traditional VRPSDP aims at the minimization of the travel cost. In this work, the optimization criterion is the energy consumption linked with fuel consumption, which is a critical concern in transportation and distribution logistics.
- In contrast to the VRPSDP, this methodology treats payloads as fuzzy rather than fixed deterministic, aiming to encapsulate the inherent uncertainty present in real-world situations. The optimization solution method integrates fuzzy concepts based on the total integral value. The study investigated the impact of the uncertainty in customers’ quantities on the optimum route.

## 2. VRPSDP Formulation

#### 2.1. VRPSDP Description

- Each customer is served by exactly one vehicle;
- Each vehicle starts and returns back to the depot;
- The fuzzy payload for each tour is constrained not to surpass the vehicles’ available capacity.

#### 2.2. Fuzzy Numbers: Definition and Ranking

#### 2.3. Model Description

## 3. The Developed Genetic Algorithm

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**The total fuzzy pick-up payload for (

**a**) vehicle#1, (

**b**) vehicle#2, (

**c**) vehicle#3, and (

**d**) vehicle#4 for the test case of 21 customers.

**Figure 5.**The total fuzzy delivery payload for (

**a**) vehicle#1, (

**b**) vehicle#2, (

**c**) vehicle#3, and (

**d**) vehicle#4 for the test case of 21 customers.

Instance | Number of Routes | Fuel Consumption (Integral Value) |
---|---|---|

E-n22-k4 | 4 | 1,331,704 |

E-n23-k3 | 4 | 1,750,248 |

E-n30-k3 | 7 | 2,850,979 |

E-n33-k4 | 5 | 6,776,371 |

E-n51-k5 | 5 | 173,776 |

E-n76-k7 | 4 | 603,628 |

E-n76-k8 | 4 | 482,751 |

E-n76-k10 | 7 | 340,329 |

E-n76-k14 | 10 | 221,101 |

E-n101-k8 | 5 | 922,964 |

E-n101-k14 | 10 | 477,143 |

Vehicle# | Fuzzy Pick-Up Quantities | Fuzzy Delivery Quantities |
---|---|---|

1 | (3485, 4100, 5330) | (2380, 2800, 3640) |

2 | (5100, 6000, 7800) | (5100, 6000, 7800) |

3 | (5185, 6100, 7930) | (4590, 5400, 7020) |

4 | (5355, 6300, 8190) | (7055, 8300, 10,790) |

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**MDPI and ACS Style**

Zacharia, P.; Stavrinidis, S.
The Vehicle Routing Problem with Simultaneous Pick-Up and Delivery under Fuzziness Considering Fuel Consumption. *Vehicles* **2024**, *6*, 231-241.
https://doi.org/10.3390/vehicles6010009

**AMA Style**

Zacharia P, Stavrinidis S.
The Vehicle Routing Problem with Simultaneous Pick-Up and Delivery under Fuzziness Considering Fuel Consumption. *Vehicles*. 2024; 6(1):231-241.
https://doi.org/10.3390/vehicles6010009

**Chicago/Turabian Style**

Zacharia, Paraskevi, and Stavros Stavrinidis.
2024. "The Vehicle Routing Problem with Simultaneous Pick-Up and Delivery under Fuzziness Considering Fuel Consumption" *Vehicles* 6, no. 1: 231-241.
https://doi.org/10.3390/vehicles6010009