# A Memetic Algorithm for the Cumulative Capacitated Vehicle Routing Problem Including Priority Indexes

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

**Parameters**

- N: Number of customers
- K: Number of vehicles available

- ${Q}_{max}$: Maximum capacity of any of the vehicles
- M: Maximum travel distance allowed (the same for all vehicles)

**Variables**

#### Reformulation Using Epsilon Constraint

## 3. Metaheuristic Algorithm

#### 3.1. Proposed Memetic Algorithm

#### 3.1.1. Constructive Procedure Based on Random Keys

Algorithm 1: Memetic algorithm with random keys. |

**Encoding mechanism:**To encode the solution, a real number drawn randomly from [0, 1) is assigned to every single position in ${R}_{a}$. Figure 3 depicts an example of this mechanism.

**Decoding mechanism:**The decoding mechanism is applied based on the information of the random key ${R}_{a}$. The ${R}_{a}$ chain is sorted in a non-decreasing order and their respective positions in ${S}_{a}$ are sorted correspondingly. As a result, a random ordered chain ${S}_{a}$ is obtained. Figure 4 exhibits the decoding mechanism.

**Assignment mechanism**: In every iteration, the algorithm selects the corresponding jth customer in ${S}_{a}$ and systematically tries to insert it into into a temporary set ${S}_{p}$ in the first available position (procuring to maintain feasibility in the capacity of the vehicles). For instance, if in the first potential route, two customers have been previously assigned, the next open position will be the third one. In the case that the customer cannot be inserted in the route due to the capacity constraints, the insertion will be evaluated in the next available route. It is important to emphasize that, since the number of routes is given in advance, the construction procedure considers a parallel routing mechanism. In other words, it performs the evaluation of feasible insertions over all of the routes. If so, the algorithm continues by selecting the next customer at ${S}_{a}$. Otherwise, the construction mechanism stops. If the algorithm reaches a feasible assignment, then $S\leftarrow {S}_{p}$ and the solution is inserted into the population ${Q}_{t}$. Otherwise, the algorithm destroys the partial constructed solution in ${S}_{p}$ and generates a new random key ${R}_{a}$ (to sort ${R}_{a}$).

Algorithm 2: Constructive procedure ($S,L,D$). |

#### 3.1.2. Crossover Procedure with Local Search Strategies

Algorithm 3: Crossover procedure. |

#### 3.1.3. Local Search (LS) Procedure

- Intra-route swap. The procedure exchanges the positions of two customers belonging to the same route. For instance, if the customers to exchange belong to positions h and i, then arcs $(h-1,h)$, $(h,h+1)$, $(i-1,i)$ and $(i,i+1)$ are removed and replaced by arcs $(h-1,i)$, $(i,h+1)$, $(i-1,h)$ and $(h,i+1)$. It is important to remark that these movements do not affect feasibility in terms of capacity.
- Intra-route reallocation. This mechanism deletes a customer from its current position and reinserts it into another position on the same route.
- Intra-route 2-opt. In this operator, two non-adjacent edges $(h,h+1)$ and $(i,i+1)$ in the path $0,1,2,\dots ,h,h+1,\dots ,i,i+1,\dots $ are deleted and replaced by $(i,h)$ and $(h+1,i+1)$, resulting in the new path $0,1,2,\dots ,h,i,\dots ,h+1,i+1,\dots $
- Inter-routes interchange. This strategy exchanges two customers belonging to different routes, as long as the move keeps feasibility (in terms of capacity).
- Inter-routes reallocation. For a given customer, the operator searches for the best position of the customer to move in any of the routes. If the best-identified position is different from the current one, the movement is performed.

Algorithm 4: Local search ($S,L,T$). |

## 4. Computational Results

#### 4.1. Test Instances

#### 4.2. Parameters Setting

#### Experimental Results

#### 4.3. Experimental Results for Larger Instances

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Solution routes minimizing the total latency: total latency = 1595.31; total tardiness = 4304.55.

**Figure 2.**Solution routes minimizing the total tardiness: total latency = 2972.84; total tardiness = 0.

Instance Name | n | k | Gurobi | MA-RK v1 | MA-RK v2 |
---|---|---|---|---|---|

FNO1 | 12 | 5 | 17 | 9 | 15 |

FNO2 | 15 | 8 | 16 | 15 | 4 |

Instance Name | Gurobi | MA-RK v1 | MA-RK v2 |
---|---|---|---|

FNO1 | 3,641.179 | 0.177 | 0.125 |

FNO2 | 13,742.185 | 0.256 | 0.149 |

Instance Name | Exact | MA-RK v1 | MA-RK v2 | |||
---|---|---|---|---|---|---|

Max | Avg | Max | Avg | Max | Avg | |

FNO1 | 0.41624 | 0.117976 | 0.469738 | 0.211416 | 0.225514 | 0.106864 |

FNO2 | 0.180034 | 0.121934 | 0.167452 | 0.0844987 | 0.410824 | 0.320028 |

Instance Name | Exact | MA-RK v1 | MA-RK v2 |
---|---|---|---|

FNO1 | 0.821569 | 0.596382 | 0.64612 |

FNO2 | 0.799919 | 0.454889 | 0.432928 |

X’/X” | Exact | MA-RK v1 | MA-RK v2 |
---|---|---|---|

Exact | 0 | 1 | 1 |

MA-RK v1 | 0 | 0 | 0.066 |

MA-RK v2 | 0 | 0.778 | 0 |

X’/X” | Exact | MA-RK v1 | MA-RK v2 |
---|---|---|---|

Exact | 0 | 1 | 1 |

MA-RK v1 | 0 | 0 | 0.25 |

MA-RK v2 | 0 | 0.333 | 0 |

Instance Name | n | k | MA-RK v1 | MA-RK v2 |
---|---|---|---|---|

FNO3 | 20 | 3 | 8 | 5 |

FNO4 | 22 | 8 | 8 | 6 |

FNO5 | 75 | 4 | 9 | 11 |

FNO6 | 75 | 7 | 5 | 18 |

FNO7 | 75 | 10 | 7 | 13 |

FNO8 | 75 | 7 | 10 | 5 |

FNO9 | 75 | 8 | 13 | 11 |

FNO10 | 100 | 5 | 8 | 12 |

Instance Name | Type of Objective | MA-RK v1 | MA-RK v2 | ||
---|---|---|---|---|---|

Min | Max | Min | Max | ||

FNO3 | Latency | 5327.35 | 7987.42 | 4852.72 | 7335.90 |

Tardiness | 4853.59 | 19,782.90 | 3296.29 | 11,060.70 | |

FNO4 | Latency | 660.85 | 1102.98 | 759.182 | 975.53 |

Tardiness | 233.65 | 1159.62 | 290.55 | 2072.82 | |

FNO5 | Latency | 8275.60 | 10,376.60 | 6788.84 | 10,460.50 |

Tardiness | 27,515.3 | 66,396.30 | 23,903.30 | 51,934.70 | |

FNO6 | Latency | 7366.98 | 9251.54 | 6933.74 | 16,841.5 |

Tardiness | 20,883.80 | 31,319.70 | 11,558.3 | 16,063.70 | |

FNO7 | Latency | 12,511.70 | 15,026.20 | 10,027.10 | 11,253.50 |

Tardiness | 82,739.9 | 124,013.00 | 67,365.4 | 144,892 | |

FNO8 | Latency | 13,683.50 | 16,203.90 | 12,833.5 | 14,208.1 |

Tardiness | 116,976.00 | 199,977.00 | 101,302 | 154,659 | |

FNO9 | Latency | 9617.94 | 14,949.70 | 9150.29 | 11,955.2 |

Tardiness | 69,126.70 | 130,754.00 | 58,969.5 | 123,515.00 | |

FNO10 | Latency | 31,092.50 | 40,164.90 | 26,484.2 | 35,107.6 |

Tardiness | 343,722.00 | 530,071.00 | 272,748.00 | 335,684.00 |

Instance Name | MA-RK v1 | MA-RK v2 |
---|---|---|

FNO3 | 0.262 | 0.817 |

FNO4 | 0.768 | 1.087 |

FNO5 | 6.415 | 6.586 |

FNO6 | 9.919 | 11.837 |

FNO7 | 62.262 | 70.162 |

FNO8 | 47.039 | 47.819 |

FNO9 | 48.816 | 54.017 |

FNO10 | 96.541 | 97.29 |

Instance Name | MA-RK v1 | MA-RK v2 |
---|---|---|

FNO3 | 0.680387 | 0.836718 |

FNO4 | 0.752206 | 0.648306 |

FNO5 | 0.397265 | 0.823822 |

FNO6 | 0.642281 | 0.929850 |

FNO7 | 0.472473 | 0.646997 |

FNO8 | 0.433841 | 0.847476 |

FNO9 | 0.527335 | 0.836342 |

FNO10 | 0.390825 | 0.959196 |

Instance Name | MA-RK v1 | MA-RK v2 | ||
---|---|---|---|---|

Max | Avg | Max | Avg | |

FNO3 | 0.579687 | 0.308544 | 0.503147 | 0.389821 |

FNO4 | 0.418041 | 0.256448 | 0.643338 | 0.344496 |

FNO5 | 0.483586 | 0.166288 | 0.742506 | 0.168127 |

FNO6 | 0.516558 | 0.298917 | 0.483949 | 0.107271 |

FNO7 | 0.342773 | 0.179116 | 0.628917 | 0.176940 |

FNO8 | 0.371050 | 0.212086 | 0.386249 | 0.248088 |

FNO9 | 0.606253 | 0.161301 | 0.574856 | 0.179261 |

FNO10 | 0.387176 | 0.187895 | 0.221685 | 0.0603717 |

Instance Name | X’/X” | Exact | |
---|---|---|---|

MA-RK v1 | MA-RK v2 | ||

FNO3 | MA-RK v1 | 0 | 0 |

MA-RK v2 | 0.875 | 0 | |

FNO4 | MA-RK v1 | 0 | 0.333 |

MA-RK v2 | 0.500 | 0 | |

FNO5 | MA-RK v1 | 0 | 0 |

MA-RK v2 | 0.889 | 0 | |

FNO6 | MA-RK v1 | 0 | 0 |

MA-RK v2 | 0.800 | 0 | |

FNO7 | MA-RK v1 | 0 | 0 |

MA-RK v2 | 1 | 0 | |

FNO8 | MA-RK v1 | 0 | 0 |

MA-RK v2 | 1 | 0 | |

FNO9 | MA-RK v1 | 0 | 0 |

MA-RK v2 | 1 | 0 | |

FNO10 | MA-RK v1 | 0 | 0 |

MA-RK v2 | 1 | 0 |

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**MDPI and ACS Style**

Nucamendi-Guillén, S.; Flores-Díaz, D.; Olivares-Benitez, E.; Mendoza, A.
A Memetic Algorithm for the Cumulative Capacitated Vehicle Routing Problem Including Priority Indexes. *Appl. Sci.* **2020**, *10*, 3943.
https://doi.org/10.3390/app10113943

**AMA Style**

Nucamendi-Guillén S, Flores-Díaz D, Olivares-Benitez E, Mendoza A.
A Memetic Algorithm for the Cumulative Capacitated Vehicle Routing Problem Including Priority Indexes. *Applied Sciences*. 2020; 10(11):3943.
https://doi.org/10.3390/app10113943

**Chicago/Turabian Style**

Nucamendi-Guillén, Samuel, Diego Flores-Díaz, Elias Olivares-Benitez, and Abraham Mendoza.
2020. "A Memetic Algorithm for the Cumulative Capacitated Vehicle Routing Problem Including Priority Indexes" *Applied Sciences* 10, no. 11: 3943.
https://doi.org/10.3390/app10113943