# Using Metaheuristics on the Multi-Depot Vehicle Routing Problem with Modified Optimization Criterion

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Modified Multi-Depot Vehicle Routing Problem

#### 3.1. Modification

#### 3.2. Impact of the Modified Criterion

## 4. Metaheuristic Solution

#### 4.1. Original Algorithm

#### 4.2. Additional Optimization Process

- Single route optimization (SRO),
- Mutual routes optimization (MRO).

Algorithms 1. Single route optimization in pseudocode. |

Algorithms 2. Mutual routes optimization in pseudocode. |

## 5. Experiments and Results

## 6. Practical Application

## 7. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Solutions for benchmark instance p03. (

**a**) MDVRP (best known solution) (

**b**) M-MDVRP (ACO algorithm).

**Figure 5.**Optimal logistics distribution on the battlefield [33].

Problem | BKS | ACO | Gap |
---|---|---|---|

MDVRP | (a) 641.19 | (c) 670.82 | (e) 4.62% |

M-MDVRP | (b) 207.23 | (d) 136.05 | (f) 52.32% |

Instance | Original ACO | Orig. ACO Runtime | ACO with AOP | ACO AOP Runtime | Solution Gap | Runtime Gap |
---|---|---|---|---|---|---|

p01 | 157.31 | 1.5 s | 152.54 | 1.7 s | 3.13% | 11.11% |

p02 | 129.00 | 1.6 s | 125.82 | 1.8 s | 2.53% | 14.58% |

p03 | 139.66 | 3.7 s | 136.05 | 4.3 s | 2.65% | 16.22% |

p04 | 518.37 | 23.1 s | 511.41 | 28.4 s | 1.36% | 22.94% |

p05 | 381.99 | 20.7 s | 378.17 | 25.6 s | 1.01% | 23.83% |

p06 | 314.87 | 25.7 s | 301.75 | 31.9 s | 4.35% | 24.25% |

p07 | 248.07 | 25.2 s | 228.95 | 30.9 s | 8.35% | 22.75% |

p08 | 2480.46 | 233.1 s | 2224.85 | 361.0 s | 11.49% | 54.85% |

p09 | 1468.86 | 267.3 s | 1364.70 | 418.3 s | 7.63% | 56.49% |

p10 | 1055.09 | 301.0 s | 960.76 | 468.5 s | 9.82% | 55.65% |

p11 | 795.28 | 278.5 s | 749.41 | 422.9 s | 6.12% | 51.85% |

p12 | 663.04 | 12.8 s | 659.48 | 15.4 s | 0.54% | 20.58% |

p13 | 659.48 | 13.4 s | 659.48 | 16.0 s | 0.00% | 19.59% |

p14 | 682.84 | 14.2 s | 682.84 | 16.9 s | 0.00% | 18.86% |

p15 | 730.20 | 124.6 s | 640.58 | 167.1 s | 13.99% | 34.08% |

p16 | 788.12 | 138.0 s | 651.55 | 188.1 s | 20.96% | 36.28% |

p17 | 817.65 | 111.4 s | 677.27 | 147.3 s | 20.73% | 32.26% |

p18 | 761.92 | 188.9 s | 652.24 | 279.2 s | 16.82% | 47.79% |

p19 | 870.90 | 175.3 s | 651.55 | 253.1 s | 33.67% | 44.40% |

p20 | 916.53 | 200.3 s | 681.13 | 292.8 s | 34.56% | 46.18% |

p21 | 862.00 | 466.0 s | 659.25 | 818.8 s | 30.75% | 75.70% |

p22 | 962.17 | 506.2 s | 657.90 | 952.4 s | 46.25% | 88.14% |

p23 | 1010.87 | 488.5 s | 686.69 | 895.2 s | 47.21% | 83.25% |

Instance (N/M) | Problem | BKS | ACO | Gap |
---|---|---|---|---|

p01 (50/4) | MDVRP | 576.87 | 607.66 | 5.34% |

M-MDVRP | 237.81 | 152.54 | 55.90% | |

p02 (50/4) | MDVRP | 473.53 | 495.34 | 4.61% |

M-MDVRP | 169.57 | 125.82 | 34.77% | |

p03 (75/5) | MDVRP | 641.19 | 670.82 | 4.62% |

M-MDVRP | 207.23 | 136.05 | 52.32% | |

p04 (100/2) | MDVRP | 1001.59 | 1021.36 | 1.97% |

M-MDVRP | 532.24 | 511.41 | 4.07% | |

p05 (100/2) | MDVRP | 750.03 | 750.72 | 0.09% |

M-MDVRP | 378.17 | 378.17 | 0.00% | |

p06 (100/3) | MDVRP | 876.50 | 902.91 | 3.01% |

M-MDVRP | 432.59 | 301.75 | 43.36% | |

p07 (100/4) | MDVRP | 885.80 | 907.55 | 2.46% |

M-MDVRP | 246.21 | 228.95 | 7.54% | |

p08 (249/2) | MDVRP | 4437.68 | 4449.65 | 0.27% |

M-MDVRP | 2474.82 | 2224.85 | 11.24% | |

p09 (249/3) | MDVRP | 3900.22 | 4085.51 | 4.75% |

M-MDVRP | 1836.00 | 1364.70 | 34.54% | |

p10 (249/4) | MDVRP | 3663.02 | 3825.73 | 4.44% |

M-MDVRP | 1049.12 | 960.76 | 9.20% | |

p11 (249/5) | MDVRP | 3554.18 | 3732.36 | 5.01% |

M-MDVRP | 808.21 | 749.41 | 7.85% | |

p12 (80/2) | MDVRP | 1318.95 | 1318.95 | 0.00% |

M-MDVRP | 659.48 | 659.48 | 0.00% | |

p13 (80/2) | MDVRP | 1318.95 | 1318.95 | 0.00% |

M-MDVRP | 659.48 | 659.48 | 0.00% | |

p14 (80/2) | MDVRP | 1360.12 | 1365.69 | 0.41% |

M-MDVRP | 686.69 | 682.84 | 0.56% | |

p15 (160/4) | MDVRP | 2505.42 | 2554.12 | 1.94% |

M-MDVRP | 646.92 | 640.58 | 0.99% | |

p16 (160/4) | MDVRP | 2572.23 | 2606.22 | 1.32% |

M-MDVRP | 694.27 | 651.55 | 6.56% | |

p17 (160/4) | MDVRP | 2709.09 | 2709.09 | 0.00% |

M-MDVRP | 690.55 | 677.27 | 1.96% | |

p18 (240/6) | MDVRP | 3702.85 | 3871.01 | 4.54% |

M-MDVRP | 895.50 | 652.24 | 37.30% | |

p19 (240/6) | MDVRP | 3827.06 | 3884.81 | 1.51% |

M-MDVRP | 694.27 | 651.55 | 6.56% | |

p20 (240/6) | MDVRP | 4058.07 | 4058.07 | 0.00% |

M-MDVRP | 690.55 | 681.13 | 1.38% | |

p21 (360/9) | MDVRP | 5474.84 | 5824.58 | 6.39% |

M-MDVRP | 936.62 | 659.25 | 42.07% | |

p22 (360/9) | MDVRP | 5702.16 | 5873.41 | 3.00% |

M-MDVRP | 719.64 | 657.90 | 9.38% | |

p23 (360/9) | MDVRP | 6095.46 | 6124.67 | 0.48% |

M-MDVRP | 690.55 | 686.69 | 0.56% |

Instance | Routes (Solution in Bold) | AvgDev (%) | StDev (%) |
---|---|---|---|

p01 | 152.24|151.86|151.02|152.54 | 0.477 (0.31%) | 0.661 (0.43%) |

p02 | 125.15|125.47|125.82|118.90 | 2.469 (1.96%) | 3.303 (2.63%) |

p03 | 133.27|133.52|134.14|133.84|136.05 | 0.754 (0.55%) | 1.104 (0.81%) |

p04 | 511.41|509.95 | 0.732 (0.14%) | 1.035 (0.20%) |

p05 | 378.17|372.55 | 2.808 (0.74%) | 3.971 (1.05%) |

p06 | 301.02|300.14|301.75 | 0.551 (0.18%) | 0.802 (0.27%) |

p07 | 228.95|224.81|226.79|227.00 | 1.088 (0.48%) | 1.695 (0.74%) |

p08 | 2224.81|2224.85 | 0.020 (0.00%) | 0.028 (0.00%) |

p09 | 1364.70|1362.17|1358.63 | 2.134 (0.16%) | 3.047 (0.22%) |

p10 | 957.14|960.76|949.74|958.09 | 3.347 (0.35%) | 4.720 (0.49%) |

p11 | 741.42|747.49|749.41|748.34|745.70 | 2.328 (0.31%) | 3.133 (0.42%) |

p12 | 659.48|659.48 | 0.000 (0.00%) | 0.000 (0.00%) |

p13 | 659.48|659.48 | 0.000 (0.00%) | 0.000 (0.00%) |

p14 | 682.84|682.84 | 0.000 (0.00%) | 0.000 (0.00%) |

p15 | 638.15|640.58|636.12|639.28 | 1.398 (0.22%) | 1.891 (0.30%) |

p16 | 651.55|651.55|651.55|651.55 | 0.000 (0.00%) | 0.000 (0.00%) |

p17 | 677.27|677.27|677.27|677.27 | 0.000 (0.00%) | 0.000 (0.00%) |

p18 | 646.11|641.43|647.94|652.24|650.51|632.78 | 5.375 (0.82%) | 7.135 (1.09%) |

p19 | 645.53|643.08|650.47|649.13|651.55|645.04 | 2.917 (0.45%) | 3.387 (0.52%) |

p20 | 667.85|677.27|677.27|681.13|677.27|677.27 | 2.831 (0.42%) | 4.437 (0.65%) |

p21 | 659.25|638.74|635.47|637.61|644.01|653.75|644.89|655.23|655.63 | 7.814 (1.19%) | 8.948 (1.36%) |

p22 | 653.49|656.33|655.88|643.55|643.08|657.90|652.34|654.27|656.57 | 4.185 (0.64%) | 5.530 (0.84%) |

p23 | 675.58|683.13|677.27|679.28|673.42|686.69|681.15|683.13|685.00 | 3.671 (0.53%) | 4.460 (0.65%) |

Instance | BKS | Original ACO | ACO with AOP |
---|---|---|---|

StDev (%) | StDev (%) | StDev (%) | |

p01 | 75.912 (31.92%) | 3.547 (2.25%) | 0.661 (0.43%) |

p02 | 35.989 (21.22%) | 3.040 (2.36%) | 3.303 (2.63%) |

p03 | 50.421 (24.33%) | 0.667 (0.48%) | 1.104 (0.81%) |

p04 | 44.466 (8.35%) | 0.743 (0.14%) | 1.035 (0.20%) |

p05 | 4.457 (1.18%) | 0.605 (0.16%) | 3.971 (1.05%) |

p06 | 122.020 (28.21%) | 2.761 (0.88%) | 0.802 (0.27%) |

p07 | 21.018 (8.54%) | 9.725 (3.92%) | 1.695 (0.74%) |

p08 | 9.393 (0.42%) | 3.486 (0.14%) | 0.028 (0.00%) |

p09 | 480.583 (26.18%) | 8.838 (0.60%) | 3.047 (0.22%) |

p10 | 114.528 (10.92%) | 3.722 (0.35%) | 4.720 (0.49%) |

p11 | 96.537 (11.94%) | 24.424 (3.07%) | 3.133 (0.42%) |

p12 | 0.000 (0.00%) | 0.000 (0.00%) | 0.000 (0.00%) |

p13 | 0.000 (0.00%) | 0.000 (0.00%) | 0.000 (0.00%) |

p14 | 9.385 (1.37%) | 0.000 (0.00%) | 0.000 (0.00%) |

p15 | 23.746 (3.67%) | 20.894 (2.86%) | 1.891 (0.30%) |

p16 | 59.136 (8.52%) | 5.189 (0.66%) | 0.000 (0.00%) |

p17 | 10.837 (1.57%) | 24.148 (2.95%) | 0.000 (0.00%) |

p18 | 209.406 (23.38%) | 31.286 (4.11%) | 7.135 (1.09%) |

p19 | 46.958 (6.76%) | 22.779 (2.62%) | 3.387 (0.52%) |

p20 | 11.839 (1.71%) | 35.761 (3.90%) | 4.437 (0.65%) |

p21 | 199.647 (21.32%) | 41.186 (4.78%) | 8.948 (1.36%) |

p22 | 69.188 (9.61%) | 60.946 (6.33%) | 5.530 (0.84%) |

p23 | 9.598 (1.39%) | 80.569 (7.97%) | 4.460 (0.65%) |

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**MDPI and ACS Style**

Stodola, P.
Using Metaheuristics on the Multi-Depot Vehicle Routing Problem with Modified Optimization Criterion. *Algorithms* **2018**, *11*, 74.
https://doi.org/10.3390/a11050074

**AMA Style**

Stodola P.
Using Metaheuristics on the Multi-Depot Vehicle Routing Problem with Modified Optimization Criterion. *Algorithms*. 2018; 11(5):74.
https://doi.org/10.3390/a11050074

**Chicago/Turabian Style**

Stodola, Petr.
2018. "Using Metaheuristics on the Multi-Depot Vehicle Routing Problem with Modified Optimization Criterion" *Algorithms* 11, no. 5: 74.
https://doi.org/10.3390/a11050074