# Determining Optimal Locations of Postal Access Points Based on Simulated Annealing

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Research Background

#### 3.2. Data Preparation and Methodology for Calculating the Postal Service Accessibility

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#### 3.3. Brute-Force Search Algorithm

#### 3.4. Simulated Annealing

For given searchable network with known mutual road distances and corresponding house numbers, find a solution of placing P postal offices in such way that sum of all distances for all house numbers to nearest postal office is minimal.

Algorithm 1: Pseudo code for the SA algorithm. |

$x:=generateRandominitialSolution()$ |

$f\left(x\right)$ := $costFunction\left(x\right)$ |

$t$ := $setInitialTemperature()$ |

for$i$ := 0 to $k$ do |

${x}_{new}$ := $generateNeighbor\left(x\right)$ |

$f\left({x}_{new}\right)$ := $costFunction\left({x}_{new}\right)$ |

$\mathsf{\Delta}f\left(x\right):=f\left(x\right)-f\left({x}_{new}\right)$ |

if $\mathsf{\Delta}f\left(x\right)<0$ then |

$x:={x}_{new}$ |

$\Delta f\left(x\right):=f\left({x}_{new}\right)$ |

else |

if $random\left(0,1\right)<\mathrm{exp}\left(\frac{-\Delta f\left(x\right)}{t}\right)$ then |

$x:={x}_{new}$ |

end if |

end if |

$t:=decreaseTemperature\left(t\right)$ |

end for |

return$x$ |

**costFunction1**(earlier described). Starting temperature is set to the initial value. At each iteration, new neighbor ${x}_{new}$ is generated, and if it has a lower evaluated value, it becomes $x$. If ${x}_{new}$ does not become $x$ due to the evaluation, the change can appear with probability calculated by $\mathrm{exp}(-\frac{\Delta f\left(x\right)}{t})$. After iteration, the temperature is decreased by multiplying the current temperature by $\alpha $. This process is known as cooling. Parameter $\alpha $ can be calculated as:

## 4. Results

## 5. Discussion

#### Alternative Access Point (AAP) Scenario

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Letter and parcel ratio volumes by 2025 (compound annual growth rate, %) Authors according to [5].

**Figure 2.**Reasons for sending or receiving mail over the period of the next 5–10 years (% of respondents) Authors according to [2].

**Figure 4.**The optimal distances between two road nodes presented as the sum of partial distances over the optimal path.

**Figure 6.**An example catchment area for one possible solution for positioning $P=3$ postal offices and $N=13$ road nodes; the borders of the catchment areas are presented with dashed lines.

**Table 1.**The number sof combinations without repetitions for different $HN$ (house numbers) and $N$ (number of road nodes).

No of Post Offices, P | $\mathbf{Number}\mathbf{of}\mathbf{Combinations},{\mathit{C}}_{\mathit{P}}\left(\mathit{H}\mathit{N}\right)$ | $\mathbf{Number}\mathbf{of}\mathbf{Combinations},{\mathit{C}}_{\mathit{P}}\left(\mathit{N}\right)$ |
---|---|---|

1 | 3408 | 365 |

2 | 5,805,528 | 66,430 |

3 | 6,591,209,456 | 8,038,030 |

4 | 5,610,767,049,420 | 727,441,715 |

5 | 3,819,810,207,245,136 | 52,521,291,823 |

No of Post Offices, P | Solutions | Sum of All Distances |
---|---|---|

1 | 114 | 304.63 |

2 | 19, 103 | 197.80 |

3 | 15, 64, 89 | 143.00 |

4 | 15, 37, 65, 89 | 111.00 |

5 | 15, 37, 57, 89, 308 | 93.87 |

Solution [Nodes] | Sum of All Distance | Percentage % |
---|---|---|

[37, 58, 90] | 156.12 | 9.17 |

[16, 37, 58] | 183.72 | 28.47 |

[16, 37, 90] | 206.75 | 44.58 |

[16, 58, 90] | 261.56 | 82.91 |

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**MDPI and ACS Style**

Mostarac, K.; Mostarac, P.; Kavran, Z.; Šarac, D.
Determining Optimal Locations of Postal Access Points Based on Simulated Annealing. *Sustainability* **2022**, *14*, 8635.
https://doi.org/10.3390/su14148635

**AMA Style**

Mostarac K, Mostarac P, Kavran Z, Šarac D.
Determining Optimal Locations of Postal Access Points Based on Simulated Annealing. *Sustainability*. 2022; 14(14):8635.
https://doi.org/10.3390/su14148635

**Chicago/Turabian Style**

Mostarac, Katarina, Petar Mostarac, Zvonko Kavran, and Dragana Šarac.
2022. "Determining Optimal Locations of Postal Access Points Based on Simulated Annealing" *Sustainability* 14, no. 14: 8635.
https://doi.org/10.3390/su14148635