# The Ordered Capacitated Multi-Objective Location-Allocation Problem for Fire Stations Using Spatial Optimization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}, W

_{2}, W

_{3}) in each repetition and the algorithm is produced by a random non-dominated method in order to produce all non-dominated solutions. In this research, the described method and the second strategy are chosen from among the various available weighting methods, for utilization in the problem under consideration.

#### 2.1. Minimizing the Distance between the Demand and the Fire Station

#### 2.2. Minimizing the Time for Reaching the Demand from the Fire Station

#### 2.3. Maximizing the Fire Station’s Coverage

## 3. Proposed Solution Approaches

#### 3.1. Genetic Optimization

#### 3.2. Simulated Annealing (SA) Optimization

## 4. Study Area and Data

#### 4.1. Study Area

^{2}. Figure 2a shows the populations of each census zone in this region. Figure 2b shows the distributed demand points (created and distributed randomly in each census zone in accordance with the population of each census zone and field observations). There are some areas in this figure 2b which are shown in white, indicating military areas where people cannot settle and live.

#### 4.2. Data

## 5. Results and Discussion

#### 5.1. Sensitivity Analysis and Tuning the Model Parameters

#### 5.2. Evaluation of the Algorithms

#### 5.3. Implementation of Genetic Model in Case Study

#### 5.3.1. Increasing the Number of Fire Stations While Retaining Existing Fire Stations

#### 5.3.2. Non-Dominated Solutions Retaining the Existing Fire Stations

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The populations of each census zone in region 11, Tehran; (

**b**) The distributed demand points.

**Figure 4.**Optimal location of fire stations, their capacity, and allocations in ArcGIS by means of best GA parameters with fixed weights $\left({W}_{1}={W}_{2}={W}_{3}=0.3333\right)$ for functions (

**a**) f1, (

**b**) f2, (

**c**), f3, and (

**d**) fitness output graph.

**Figure 5.**Optimal location of fire stations, their capacity, and allocations in ArcGIS by means of the best SA parameters with fixed weights $\left({W}_{1}={W}_{2}={W}_{3}=0.3333\right)$ for functions (

**a**) f1, (

**b**) f2, (

**c**) f3 and (

**d**) fitness output graph.

**Figure 6.**Optimal locations of fire stations, obtained from ordered capacitated multi-objective function with fixed weights (${W}_{1}={W}_{2}={W}_{3}=0.3333)$ and output fitness.

**Figure 7.**Optimal locations of the fire stations, obtained from capacitated multi-objective function with fixed weights $({W}_{1}={W}_{2}={W}_{3}=0.3333)$ and demand allocations of (

**a**) f1, (

**b**) f2, and (

**c**) f3; (

**d**) the service areas of the individual fire stations.

**Table 1.**Effect of crossover, mutation rate, and population size on objective function in genetic algorithms (GA) by sensitivity analysis.

Crossover | Mutation | Pop Size | Fitness |
---|---|---|---|

Data set 1 | |||

0.4 | 0.1 | 10 | 0.2009 |

0.2 | 0.1921 | ||

0.3 | 0.1925 | ||

0.4 | 0.1889 | ||

0.5 | 0.1 | 10 | 0.2099 |

0.2 | 0.2108 | ||

0.3 | 0.2217 | ||

0.4 | 0.2011 | ||

0.6 | 0.1 | 10 | 0.2099 |

0.2 | 0.2094 | ||

0.3 | 0.2092 | ||

0.4 | 0.2001 | ||

0.7 | 0.1 | 10 | 0.2148 |

0.2 | 0.2155 | ||

0.3 | 0.2287 | ||

0.4 | 0.1999 | ||

0.8 | 0.1 | 10 | 0.2148 |

0.2 | 0.2196 | ||

0.3 | 0.2199 | ||

0.4 | 0.2019 | ||

0.9 | 0.1 | 10 | 0.2542 |

0.2 | 0.2012 | ||

0.3 | 0.2227 | ||

0.4 | 0.2007 | ||

Data set 2 | |||

0.4 | 0.1 | 20 | 0.2109 |

0.2 | 0.1995 | ||

0.3 | 0.1984 | ||

0.4 | 0.1971 | ||

0.5 | 0.1 | 20 | 0.2198 |

0.2 | 0.2199 | ||

0.3 | 0.2200 | ||

0.4 | 0.1993 | ||

0.6 | 0.1 | 20 | 0.2195 |

0.2 | 0.2191 | ||

0.3 | 0.2201 | ||

0.4 | 0.2001 | ||

0.7 | 0.1 | 20 | 0.2455 |

0.2 | 0.2222 | ||

0.3 | 0.2280 | ||

0.4 | 0.2021 | ||

0.8 | 0.1 | 20 | 0.2317 |

0.2 | 0.2264 | ||

0.3 | 0.2266 | ||

0.4 | 0.2017 | ||

0.9 | 0.1 | 20 | 0.2431 |

0.2 | 0.2132 | ||

0.3 | 0.2241 | ||

0.4 | 0.2019 |

Initial Temperature | Cooling Rate | Fitness |
---|---|---|

Data set 1 | ||

50 | 0.8 | 0.3222 |

0.9 | 0.3015 | |

0.95 | 0.3110 | |

100 | 0.8 | 0.2914 |

0.9 | 0.2999 | |

0.95 | 0.2814 | |

Data set 2 | ||

200 | 0.8 | 0.2805 |

0.9 | 0.2807 | |

0.95 | 0.2799 | |

300 | 0.8 | 0.2791 |

0.9 | 0.2774 | |

0.95 | 0.2764 |

Model | Time (min) | Fitness |
---|---|---|

SA | 10.46 | 0.2771 |

GA | 5.41 | 0.1892 |

Data Set | ${\mathbf{\chi}}_{1}$ | ${\mathit{\delta}}_{2}^{2}$ | ${\mathbf{\chi}}_{2}$ | ${\mathit{\delta}}_{2}^{2}$ |
---|---|---|---|---|

1 | 0.210017 | 0.000188 | 0.301233 | 0.012312 |

2 | 0.216258 | 0.000187 | 0.279000 | 0.019462 |

Run | ${\mathit{w}}_{1}{\mathit{f}}_{1}^{\prime}$ | ${\mathit{w}}_{2}{\mathit{f}}_{2}^{\prime}$ | ${\mathit{w}}_{3}{\mathit{f}}_{3}^{\prime}$ | Min Dist (m) | Max Dist (m) | Sum | Demand Allocations of f_{3} | Number of Solutions |
---|---|---|---|---|---|---|---|---|

1 | 0.1537 | 0.0081 | 0.5576 | 1019.776 | 2625.404 | −0.3024 | 274,253 | 10 |

0.0145 | 0.0238 | 0.4922 | 993.009 | 1738.385 | 254,578 | |||

0.2622 | 0.0006 | 0.3068 | 1061.513 | 2309.535 | 258,854 | |||

0.0005 | 0.2376 | 0.3635 | 973.342 | 2625.404 | 264,658 | |||

0.0934 | 0.0202 | 0.6232 | 1022.032 | 2625.404 | 274,589 | |||

0.1756 | 0.0027 | 0.3636 | 1002.435 | 1839.701 | 249,858 | |||

0.0033 | 0.0849 | 0.6478 | 1047.78 | 2309.535 | 274,548 | |||

0.0045 | 0.0244 | 0.6163 | 1047.78 | 1794.085 | 270,001 | |||

0.5262 | 0.0005 | 0.0993 | 782.465 | 2025.198 | 252,157 | |||

0.0492 | 0.0419 | 0.6825 | 635.148 | 2913.485 | 276,671 | |||

2 | ------- | ------- | ------- | ------- | ------- | −0.2852 | ------- | 7 |

3 | ------- | ------- | ------- | ------- | ------- | −0.2820 | ------- | 9 |

4 | ------- | ------- | ------- | ------- | ------- | −0.2334 | ------- | 17 |

5 | 0.0140 | 0.0771 | 0.6456 | 983.523 | 2875.103 | −0.3437 | 278,621 | 17 |

0.0248 | 0.0606 | 0.5994 | 990.272 | 1941.282 | 271,894 | |||

0.0005 | 0.1846 | 0.4826 | 1098.580 | 1941.282 | 269,995 | |||

0.1343 | 0.0001 | 0.3958 | 661.675 | 2625.404 | 248,444 | |||

0.1000 | 0.0061 | 0.4351 | 661.675 | 2625.404 | 256,841 | |||

0.0418 | 0.0482 | 0.4703 | 661.675 | 2309.535 | 259,654 | |||

0.1133 | 0.0098 | 0.4802 | 661.675 | 2309.535 | 269,415 | |||

0.0711 | 0.0447 | 0.4355 | 661.675 | 2309.535 | 280,000 | |||

0.0503 | 0.0474 | 0.4514 | 635.148 | 2309.535 | 259,587 | |||

0.0002 | 0.2447 | 0.2358 | 1022.032 | 2625.404 | 262,356 | |||

0.0857 | 0.0136 | 0.4546 | 928.856 | 2625.404 | 262,221 | |||

0.1086 | 0.0354 | 0.4566 | 585.251 | 2937.662 | 262,549 | |||

0.0841 | 0.0373 | 0.4587 | 585.251 | 2937.662 | 266,425 | |||

0.0131 | 0.1456 | 0.5803 | 585.251 | 2625.404 | 275,527 | |||

0.0000 | 0.2508 | 0.3526 | 585.251 | 2625.404 | 277,458 | |||

0.0120 | 0.0780 | 0.5754 | 585.251 | 2625.404 | 276,457 | |||

0.0995 | 0.0502 | 0.6200 | 585.251 | 2937.662 | 275,248 | |||

6 | ------- | ------- | ------- | ------- | ------- | −0.1496 | ------- | 10 |

7 | ------- | ------- | ------- | ------- | ------- | −0.3346 | ------- | 8 |

8 | ------- | ------- | ------- | ------- | ------- | −0.1648 | ------- | 16 |

9 | ------- | ------- | ------- | ------- | ------- | −0.3434 | ------- | 7 |

10 | ------- | ------- | ------- | ------- | ------- | −0.1996 | ------- | 10 |

Min | # of Execution | Max | # of Execution | |
---|---|---|---|---|

${w}_{1}{f}_{1}^{\prime}$ | 0 | 5 | 0.5262 | 1 |

${w}_{2}{f}_{2}^{\prime}$ | 0 | 10 | 0.4808 | 6 |

${w}_{3}{f}_{3}^{\prime}$ | −0.6825 | 1 | −0.0477 | 4 |

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**MDPI and ACS Style**

Bolouri, S.; Vafaeinejad, A.; Alesheikh, A.A.; Aghamohammadi, H. The Ordered Capacitated Multi-Objective Location-Allocation Problem for Fire Stations Using Spatial Optimization. *ISPRS Int. J. Geo-Inf.* **2018**, *7*, 44.
https://doi.org/10.3390/ijgi7020044

**AMA Style**

Bolouri S, Vafaeinejad A, Alesheikh AA, Aghamohammadi H. The Ordered Capacitated Multi-Objective Location-Allocation Problem for Fire Stations Using Spatial Optimization. *ISPRS International Journal of Geo-Information*. 2018; 7(2):44.
https://doi.org/10.3390/ijgi7020044

**Chicago/Turabian Style**

Bolouri, Samira, Alireza Vafaeinejad, Ali Asghar Alesheikh, and Hossein Aghamohammadi. 2018. "The Ordered Capacitated Multi-Objective Location-Allocation Problem for Fire Stations Using Spatial Optimization" *ISPRS International Journal of Geo-Information* 7, no. 2: 44.
https://doi.org/10.3390/ijgi7020044