# The Capacitated Location-Allocation Problem Using the VAOMP (Vector Assignment Ordered Median Problem) Unified Approach in GIS (Geospatial Information Systam)

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Definition

#### 1.2. Literature Review

#### 1.3. Developing the VAOMP Model Considering the Capacity Criterion (VAOCMP)

- I is the set of demandsJ is the set of facilities${a}_{i}$ the population or the number of demands in each building block or at location i
- k = 1, 2,…n is an index for the relative rank of service time, one assigned to each demand.${\lambda}_{k}$ the weight on the kth rank of service${d}_{ij}$ the distance or time (or generalized cost) between i and j.L the maximum number of levels of closeness being considered in the model for any demand (in this research is equal to 1)${\theta}_{il}$ the fraction of the time demand at i is served by its lth closest facility
- ${C}_{j}$ capacity of each facility or fire station that is equal to 50,000 people
- ${c}_{ij}^{l}=\{1\begin{array}{c}\mathrm{if}\mathrm{demandIi}\mathrm{assigns}\mathrm{to}\mathrm{facility}\mathrm{j}\mathrm{as}\mathrm{the}\mathrm{lth}\mathrm{closest}\mathrm{open}\mathrm{facility}\\ 0otherwise\end{array}$

- The required number of facilities is selected as the initial solution using spatial analysis. This solution will speed up the time required to reach the optimal solution.
- The OD Cost Matrix is created for all demands and facilities selected at the previous step.
- All the arrays in the OD Cost Matrix are arranged or ranked based on the minimum cost.
- Based on the order of matrix arrays, which was created in the previous step, and considering the capacity of each facility, the closest demand to each facility is allocated according to the type and the number of service levels.
- According to Equation (2), the partial sum for each demand is calculated based on the allocations of each demand to each facility and the possibility of allocating the demand to a facility according to the capacity criterion and the ranked level of each demand to any facility.
- Then, all the demands are sorted according to their minimum partial sum: the minimum value will be the lowest weight ${\lambda}_{k}$.
- In the last step by Equation (1), the minimum value of the function will be obtained for the selected facilities.

## 2. Material and Methods

## 3. Result

## 4. Discussion

#### 4.1. Scenario 1: Implementation of the VAOMP Model Aiming to Minimize Arrival Time for Existing Stations in the Study Area

#### 4.1.1. Implementation of the Tabu Search Algorithm

#### 4.1.2. Implementation of the Simulated Annealing

#### 4.1.3. Comparison of Algorithms and Validation of Models

#### 4.1.4. Relocation-Reallocation with the VAOMP Model to Service All Demands Using Both Algorithms

#### 4.1.5. Evaluating the VAOMP Model in a Larger Set

#### 4.2. Scenario 2: Implementation of the VAOCMP Model Aiming to Minimize Arrival Time for Existing Stations in the Study Area

#### 4.2.1. Implementation of the Tabu Search Algorithm

#### 4.2.2. Implementation of the Simulated Annealing Algorithm

#### 4.2.3. Comparison of Algorithms and Validation of Models

#### 4.2.4. Relocation-Reallocation with the VAOVMP Model to Service All Demands Using Both Algorithms

#### 4.2.5. Evaluating the VAOCMP Model in a Larger Set

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lei, T.L.; Church, R.L.; Lei, Z. A unified approach for location-allocation analysis: Integrating GIS, distributed computing and spatial optimization. Int. J. Geogr. Inf. Sci.
**2016**, 30, 515–534. [Google Scholar] [CrossRef] - Arifin, S. Location Allocation Problem Using Genetic Algorithm and Simulated Annealing. A Case Study Based on School in Enschede. Master’s Thesis, Department of Geoinformation Science and Earth Observation, University of Twente, Enschede, The Netherlands, 2011. [Google Scholar]
- Brandeau, M.L.; Chiu, S.S. An overview of representative problems in location research. Manag. Sci.
**1989**, 35, 645–674. [Google Scholar] [CrossRef] - Farahani, R.Z.; Hekmatfar, M. Facility Location: Concepts, Models, Algorithms and Case Studies; Springer: New York, NY, USA, 2009. [Google Scholar]
- Church, R.; ReVelle, C. The maximal covering location problem. Pap. Reg. Sci. Assoc.
**1974**, 32, 101–118. [Google Scholar] [CrossRef] - Drezner, Z. On the conditional p-median problem. Comput. Oper. Res.
**1995**, 22, 525–530. [Google Scholar] [CrossRef] - Lei, T.L.; Church, R.L. Vector assignment ordered median problem: A unified median problem. Int. Reg. Sci. Rev.
**2014**, 37, 194–224. [Google Scholar] [CrossRef] - Yu, W.; Liu, Z. Better approximability results for min–max tree/cycle/path cover problems. J. Comb. Optim.
**2019**, 37, 563–578. [Google Scholar] [CrossRef] - Ding, W.; Qiu, K. A quadratic time exact algorithm for continuous connected 2-facility location problem in trees. J. Comb. Optim.
**2018**, 36, 1262–1298. [Google Scholar] [CrossRef] - Xu, Y.; Peng, J.; Xu, Y. The mixed center location problem. J. Comb. Optim.
**2018**, 36, 1128–1144. [Google Scholar] [CrossRef] - Yang, L.; Jones, B.F.; Yang, S.-H. A fuzzy multi-objective programming for optimization of fire station locations through genetic algorithms. Eur. J. Oper. Res.
**2007**, 181, 903–915. [Google Scholar] [CrossRef] [Green Version] - Hillsman, E.L. The p-median structure as a unified linear model for location—Allocation analysis. Environ. Plan. A
**1984**, 16, 305–318. [Google Scholar] [CrossRef] - Church, R.; Weaver, J. Theoretical links between median and coverage location problems. Ann. Oper. Res.
**1986**, 6, 1–19. [Google Scholar] [CrossRef] - Aghamohammadi, H.; Mesgari, M.S.; Molaei, D.; Aghamohammadi, H. Development a heuristic method to locate and allocate the medical centers to minimize the earthquake relief operation time. Iran. J. Public Health
**2013**, 42, 63. [Google Scholar] [PubMed] - Kovačević-Vujčić, V.; Čangalović, M.; Ašić, M.; Ivanović, L.; Dražić, M. Tabu search methodology in global optimization. Comput. Math. Appl.
**1999**, 37, 125–133. [Google Scholar] [CrossRef] [Green Version] - Notional Fire Protection Association. Standard for Developing Fire Protection Services for the Public (NFPA 1201); Notional Fire Protection Association: Quincy, MA, USA, 2000. [Google Scholar]
- Barr, R.C.; Caputo, A.P. Planning fire station locations. In Fire Protection Handbook, 18th ed.; Notional Fire Protection Association: Quincy, MA, USA, 1997; pp. 311–318. [Google Scholar]
- Cooper, L. Location-allocation problems. Oper. Res.
**1963**, 11, 331–343. [Google Scholar] [CrossRef] - Arostegui, M.A., Jr.; Kadipasaoglu, S.N.; Khumawala, B.M. An empirical comparison of tabu search, simulated annealing, and genetic algorithms for facilities location problems. Int. J. Prod. Econ.
**2006**, 103, 742–754. [Google Scholar] [CrossRef] - Yigit, V.; Aydin, M.E.; Turkbey, O. Solving large-scale uncapacitated facility location problems with evolutionary simulated annealing. Int. J. Prod. Res.
**2006**, 44, 4773–4791. [Google Scholar] [CrossRef] - Torrent-Fontbona, F.; Muñoz, V.; López, B. Solving large immobile location–allocation by affinity propagation and simulated annealing. Application to select which sporting event to watch. Expert Syst. Appl.
**2013**, 40, 4593–4599. [Google Scholar] [CrossRef] [Green Version] - Mahmoodpour, S.; Masihi, M.; Gholinejhad, S. Comparison of Simulated Annealing, Genetic, and Tabu Search Algorithms for Fracture Network Modeling. Iran. J. Oil Gas Sci. Technol.
**2015**, 4, 50–67. [Google Scholar] - 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. [Google Scholar] [CrossRef] [Green Version] - Bolouri, S.; Vafeainejad, A.; Alesheikh, A.; Aghamohammadi, H. Environmental sustainable development optimizing the location of urban facilities using vector assignment ordered median problem-integrated GIS. Int. J. Environ. Sci. Technol.
**2020**, 17, 3033–3054. [Google Scholar] [CrossRef] - Keil, J.; Mocnik, F.-B.; Edler, D.; Dickmann, F.; Kuchinke, L. Reduction of map information regulates visual attention without affecting route recognition performance. ISPRS Int. J. Geo-Inf.
**2018**, 7, 469. [Google Scholar] [CrossRef] [Green Version] - Novack, T.; Wang, Z.; Zipf, A. A system for generating customized pleasant pedestrian routes based on OpenStreetMap data. Sensors
**2018**, 18, 3794. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jokar, J.; Zipf, A.; Mooney, P.; Helbich, M. An Introduction to OpenStreetMap in Geographic Information Science: Experiences, Research, and Applications. In OpenStreetMap in GIScience. Experiences, Research, and Applications; Springer: New York, NY, USA, 2015; pp. 1–15. [Google Scholar]
- Amat, G.; Fernandez, J.; Arranz, A.; Ramos, A. Using Open Street Maps data and tools for indoor mapping in a Smart City scenario. In Proceedings of the 17th AGILE Conference on Geographic Information Science "Connecting a Digital Europe through Location and Place", Valencia, Spain, 6 June 2014. [Google Scholar]
- Rabadi, G.; Anagnostopoulos, G.; Mollaghasemi, M. A Simulated Annealing Algorithm for a Scheduling Problem with Setup Times. Available online: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.19.7117&rep=rep1&type=pdf (accessed on 10 November 2020).
- Wang, K.-J.; Makond, B.; Liu, S.-Y. Location and allocation decisions in a two-echelon supply chain with stochastic demand–A genetic-algorithm based solution. Expert Syst. Appl.
**2011**, 38, 6125–6131. [Google Scholar] [CrossRef] - Yang, K.; Cho, K. Simulated annealing algorithm for wind farm layout optimization: A benchmark study. Energies
**2019**, 12, 4403. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Implementation steps. VAOMP, Vector Assignment Ordered Median Problem; VAOCMP, Vector Assignment Ordered Capacitated Median Problem.

**Figure 2.**(

**a**,

**b**) Location of existing and potential fire stations and the main road network in Tehran’s 21st and 22nd districts (in UTM zone 39N map projection system).

**Figure 3.**(

**a**,

**b**) Population demand in the 21st and 22nd districts of Tehran and population distribution in statistical zones.

**Figure 10.**The allocation of each station using the Simulated Annealing algorithm in the VAOCMP model.

**Table 1.**The number of allocations for each station, runtime, and the optimal value of the function using the Tabu Search algorithm.

No. Stations | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Number of allocations | 25,760 | 18,200 | 35,840 | 33,240 | 21,960 | 17,520 | 51,240 | 17,760 | 9880 | 52,360 |

Runtime (sec) | 394.648 | |||||||||

Optimal value | 143,478,895.2 | |||||||||

Number of demands | 336,600 | |||||||||

Number of unallocated demands | 52,840 |

**Table 2.**The number of allocations for each station, runtime, and the optimal value of the function using the Simulated Annealing algorithm.

No. Stations | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Number of allocations | 25,760 | 18,200 | 35,840 | 33,240 | 21,960 | 17,520 | 51,240 | 17,760 | 9880 | 52,360 |

Runtime (sec) | 466.285 | |||||||||

Optimal value | 143,478,895.2 | |||||||||

Number of demands | 336,600 | |||||||||

Number of unallocated demands | 52,840 |

**Table 3.**The normalized standard deviation of solutions obtained using the VAOMP model with both algorithms.

Number of Fire Stations | Normalized Standard Deviation Solutions of Obtained Based on Tabu Search | Normalized Standard Deviation of Solutions Obtained Based on Simulated Annealing |
---|---|---|

11 | 0.0277 | 0.0288 |

12 | 0.0498 | 0.0618 |

13 | 0.0903 | 0.1041 |

Number of Fire Stations | Average of Optimal Values Based on Tabu Search | Average of Optimal Values Based on Simulated Annealing |
---|---|---|

11 | 110,336,112.9 | 110,905,133.8 |

12 | 106,997,363.5 | 107,456,433.3 |

13 | 104,312,426.7 | 104,614,621.7 |

Number of Fire Stations | Accuracy of Allocation Based on Tabu Search | Accuracy of Allocation Based on Simulated Annealing |
---|---|---|

11 | 89 | 85 |

12 | 85 | 80 |

13 | 82 | 77 |

Number of Fire Stations | Average Percentage of All Demand Allocated Based on Tabu Search | Average Percentage of All Demand Allocated Based on Simulated Annealing |
---|---|---|

11 | 85.924 | 84.418 |

12 | 97.799 | 96.546 |

13 | 99.999 | 99.989 |

Number of Fire Stations | Solving Time Based on Tabu Search | Solving Time Based on Simulated Annealing |
---|---|---|

11 | 412.773 | 571.245 |

12 | 452.994 | 618.081 |

13 | 472.297 | 653.392 |

Algorithm | Tabu Search | Simulated Annealing |
---|---|---|

Runtime (sec) | 612.51 | 735.33 |

Optimal value | 425,884,271.54 | 425,884,271.54 |

Number of demands | 1,250,796 | 1,250,796 |

Number of unallocated demands | 104,200 | 104,200 |

**Table 9.**The normalized standard deviation of solutions obtained using the VAOMP model with both algorithms.

Number of Fire Stations | Normalized Standard Deviation Solutions of Obtained Based on Tabu Search | Normalized Standard Deviation of Solutions Obtained Based on Simulated Annealing |
---|---|---|

26 | 0.0691 | 0.0745 |

27 | 0.0913 | 0.1144 |

28 | 0.1327 | 0.1599 |

Number of Fire Stations | Average of Optimal Values Based on Tabu Search | Average of Optimal Values Based on Simulated Annealing |
---|---|---|

26 | 391,081,355.14 | 400,233,479.26 |

27 | 387,451,662.22 | 393,846,782.62 |

28 | 384,991,646.11 | 389,329,631.14 |

Number of Fire Stations | Accuracy of Allocation Based on Tabu Search | Accuracy of Allocation Based on Simulated Annealing |
---|---|---|

26 | 86 | 81 |

27 | 81 | 78 |

28 | 79 | 75 |

Number of Fire Stations | Average Percentage of All Demand Allocated Based on Tabu Search | Average Percentage of All Demand Allocated Based on Simulated Annealing |
---|---|---|

26 | 84.332 | 82.540 |

27 | 95.212 | 94.571 |

28 | 99.720 | 98.885 |

Number of Fire Stations | Solving Time Based on Tabu Search | Solving Time Based on Simulated Annealing |
---|---|---|

26 | 632.315 | 749.219 |

27 | 651.103 | 768.320 |

28 | 695.514 | 800.107 |

**Table 14.**The number of allocations for each station, runtime, and the optimal value of the function using the Tabu Search algorithm in the VAOCMP model.

No. Stations | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Number of allocations | 25,960 | 18,200 | 34,960 | 31,240 | 21,960 | 17,520 | 50,000 | 17,780 | 9880 | 50,000 |

Runtime (sec) | 525.36 | |||||||||

Optimal value | 142,474,495.67 | |||||||||

Number of demands | 336,600 | |||||||||

Number of unallocated demands | 59,080 |

**Table 15.**The number of allocations for each station, runtime, and the optimal value of the function using the Simulated Annealing algorithm in the VAOCMP model.

No. Stations | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Number of allocations | 25,960 | 18,200 | 35,960 | 33,240 | 21,960 | 17,520 | 50,000 | 17,800 | 9880 | 50,000 |

Runtime (sec) | 598.444 | |||||||||

Optimal value | 142,474,495.67 | |||||||||

Number of demands | 336,600 | |||||||||

Number of unallocated demands | 59,080 |

**Table 16.**The normalized standard deviation of solutions obtained using the VAOCMP model with both algorithms.

Number of Fire Stations | Normalized Standard deviation Solutions of Obtained Based on Tabu Search | Normalized Standard Deviation of Solutions Obtained Based on Simulated Annealing |
---|---|---|

11 | 0.0351 | 0.0398 |

12 | 0.0727 | 0.0841 |

13 | 0.1489 | 0.1855 |

Number of Fire Stations | Average of Optimal Values Based on Tabu Search | Average of Optimal Values Based on Simulated Annealing |
---|---|---|

11 | 105,982,411.3 | 106,227,122.4 |

12 | 102,113,345.1 | 102,446,657.6 |

13 | 100,142,011.0 | 100,815,312.2 |

Number of Fire Stations | Accuracy of Allocation Based on Tabu Search | Accuracy of Allocation Based on Simulated Annealing |
---|---|---|

11 | 88 | 84 |

12 | 83.5 | 80 |

13 | 81 | 76 |

Number of Fire Stations | Average Percentage of All Demand Allocated Based on Tabu Search | Average Percentage of All Demand Allocated Based on Simulated Annealing |
---|---|---|

11 | 84.483 | 83.222 |

12 | 96.959 | 95.435 |

13 | 99.999 | 99.510 |

Number of Fire Stations | Solving Time Based on Tabu Search | Solving Time Based on Simulated Annealing |
---|---|---|

11 | 565.310 | 605.778 |

12 | 613.585 | 632.112 |

13 | 649.947 | 670.661 |

**Table 21.**Runtime and the optimal value of the function by Tabu Search and Simulated Annealing algorithm for existing stations.

Algorithm | Tabu Search | Simulated Annealing |
---|---|---|

Runtime (sec) | 653.18 | 791.25 |

Optimal value | 341,431,239.22 | 341,431,239.22 |

Number of demands | 1,250,796 | 1,250,796 |

Number of unallocated demands | 163,800 | 163,800 |

**Table 22.**The normalized standard deviation of solutions obtained using the VAOCMP model with both algorithms.

Number of Fire Stations | Normalized Standard Deviation Solutions of Obtained Based on Tabu Search | Normalized Standard Deviation of Solutions Obtained Based on Simulated Annealing |
---|---|---|

26 | 0.0812 | 0.1008 |

27 | 0.1258 | 0.1399 |

28 | 0.1739 | 0.2073 |

Number of Fire Stations | Average of Optimal Values Based on Tabu Search | Average of Optimal Values Based on Simulated Annealing |
---|---|---|

26 | 303,661,932.10 | 306,327,182.44 |

27 | 298,551,847.41 | 301,688,805.25 |

28 | 294,853,029.37 | 296,971,302.57 |

Number of Fire Stations | Accuracy of Allocation Based on Tabu Search | Accuracy of Allocation Based on Simulated Annealing |
---|---|---|

26 | 84 | 79 |

27 | 79 | 77 |

28 | 78 | 73 |

Number of Fire Stations | Average Percentage of All Demand Allocated Based on Tabu Search | Average Percentage of All Demand Allocated Based on Simulated Annealing |
---|---|---|

26 | 82.676 | 81.009 |

27 | 93.103 | 90.338 |

28 | 98.990 | 97.730 |

Number of Fire Stations | Solving Time Based on Tabu Search | Solving Time Based on Simulated Annealing |
---|---|---|

26 | 715.228 | 852.210 |

27 | 782.140 | 901.825 |

28 | 840.371 | 949.414 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vafaeinejad, A.; Bolouri, S.; Alesheikh, A.A.; Panahi, M.; Lee, C.-W.
The Capacitated Location-Allocation Problem Using the VAOMP (Vector Assignment Ordered Median Problem) Unified Approach in GIS (Geospatial Information Systam). *Appl. Sci.* **2020**, *10*, 8505.
https://doi.org/10.3390/app10238505

**AMA Style**

Vafaeinejad A, Bolouri S, Alesheikh AA, Panahi M, Lee C-W.
The Capacitated Location-Allocation Problem Using the VAOMP (Vector Assignment Ordered Median Problem) Unified Approach in GIS (Geospatial Information Systam). *Applied Sciences*. 2020; 10(23):8505.
https://doi.org/10.3390/app10238505

**Chicago/Turabian Style**

Vafaeinejad, Alireza, Samira Bolouri, Ali Asghar Alesheikh, Mahdi Panahi, and Chang-Wook Lee.
2020. "The Capacitated Location-Allocation Problem Using the VAOMP (Vector Assignment Ordered Median Problem) Unified Approach in GIS (Geospatial Information Systam)" *Applied Sciences* 10, no. 23: 8505.
https://doi.org/10.3390/app10238505