# Optimization of the Weighted Multi-Facility Location Problem Using MS Excel

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Main Aim of This Work

#### 1.3. Problem Formulation

## 2. Materials and Methods

#### 2.1. Evolutionary Method Excel-Solver

#### 2.2. Metaheuristic Algorithm

- Generate random solution.
- Set temperature to its maximum limit.
- Repeat points 4 to 6 for a predefined number of times.
- Transform current solution.
- Replace the current solution by the transformed solution with the probability given by the Metropolis criterion.
- Save the best solution if found.
- Cool the temperature.
- When the temperature is not below its lower limit, repeat the whole process in point 3.
- Terminate the algorithm and return the best solution found.

## 3. Performance Evaluation

#### 3.1. Hardware and Software Configuration

**Hardware configurations.**Calculations were performed on three computers with different hardware configurations:

- HW 1-CPU: AMD A10-9620P RADEON R5 2.50 GHz; memory 8.00 GB RAM.
- HW 2-CPU: INTEL CORE i7-7700 3.60 GHz; memory 32.00 GB RAM.
- HW 3-CPU: INTEL CORE i9-9900K 3.60 GHz; memory 32.00 GB RAM.

**Software configuration.**A 64-bit operating system, Windows 10 Enterprise LSTC, MS Excel 2016, 64 bit, part of the Microsoft Office Suite, with the Solver add-in installed, which performs calculations. In the basic version it allows only one method of calculation using evolutionary algorithms. The Excel environment is used to enter data, calculate intermediate results, and display the result of the entire process.

#### 3.2. Benchmark Instances

- Each group is solved separately by an evolutionary method in Excel as a Single Facility Location Problem (SFLP) task. The sum of the resulting values is then the solution of the MFLP problem with this topology (best result 360,186.787);
- The lowest value of 300 calculations by the metaheuristic method on three hardware configurations (the best result 360,186.7863)—as this method provided the best solution, it serves as a reference value.

- by Excel-Solver evolutionary method on HW1;
- by Excel-Solver evolutionary method on HW2;
- by Excel-Solver evolutionary method on HW3;
- by metaheuristic algorithm on HW1;
- by metaheuristic algorithm on HW2;
- by metaheuristic algorithm on HW3.

- minimal solution;
- average solution;
- standard deviation;
- coefficient of variation;
- difference between the reference and the minimal solutions in percent;
- difference between the reference and the average solutions in percent;
- histogram, classes are from 1 percent to 10 percent deviation from reference;
- graph with coordinates for result for the reference and the minimal solution.

#### 3.3. Results

#### 3.3.1. Histograms and Solutions for Benchmark A

#### 3.3.2. Histograms and Solutions for Benchmark B

#### 3.3.3. Histograms and Solutions for Benchmark C

#### 3.3.4. Other statistical values for the Evolutionary and Metaheuristic method

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Parameter | Value |
---|---|

Max time | Unlimited |

Iterations | Unlimited |

Constraint precisions | 0.000001 |

Convergence | 0.0001 |

Population size | 100 |

Random seed | 0 |

Mutation rate | 0.075 |

Maximum time without improvement | 30 s |

Max subproblems | Unlimited |

Maximum feasible solutions | Unlimited |

Integer optimality | 1% |

Name of Term | Explanation |
---|---|

Reference | Optimal or reference solution for an instance. |

Minimal | Minimal value from set of 100 results |

Average | Average value from set of 100 results |

Dif. ref-min | Difference between reference and minimal value (%) |

Dif. ref-avg | Difference between reference and average value (%) |

Standard dev. | Standard deviation for set of 100 results |

Coefficient of variation | Relative standard deviation for set of 100 results |

Avg. time | Average time of computing one result in set of 100 results |

Name of Acronym | Explanation |
---|---|

EHW1 | Results of operations with evolutionary method on HW1 |

EHW2 | Results of operations with evolutionary method on HW2 |

EHW3 | Results of operations with evolutionary method on HW3 |

MHW1 | Results of operations with metaheuristic algorithm on HW1 |

MHW2 | Results of operations with metaheuristic algorithm on HW2 |

MHW3 | Results of operations with metaheuristic algorithm on HW3 |

Benchmark Instance | Number of Points | Number of Centers | Reference |
---|---|---|---|

A | 100 | 20 | 310,048.77 * |

B | 100 | 20 | 360,186.79 |

C | 100 | 20 | 3,399,273.00 |

Benchmark Instance | A | B | C |
---|---|---|---|

Reference value | 310,048.77 | 360,186.77 | 3,399,273.00 |

Minimal | 319,256.49 | 364,155.78 | 3,496,336.34 |

Average | 369,489.23 | 422,268.28 | 3,666,323.69 |

Dif. ref-min (%) | 2.97 | 1.10 | 2.86 |

Dif. ref-avg (%) | 19.17 | 17.24 | 7.86 |

Standard dev. | 24,062.23 | 42,651.21 | 91,422.78 |

Coefficient of variation | 0.0651 | 0.1010 | 0.0249 |

Benchmark Instance | A | B | C |
---|---|---|---|

Reference value | 310,048.77 | 360,186.77 | 3,399,273.00 |

Minimal | 314,559.20 | 360,530.87 | 3,407,806.82 |

Average | 343,051.52 | 397,906.86 | 3,513,082.48 |

Dif. ref-min (%) | 1.45 | 0.10 | 0.25 |

Dif. ref-avg (%) | 10.64 | 10.47 | 3.35 |

Standard dev. | 14,906.14 | 37,515.08 | 56,379.08 |

Coefficient of variation | 0.0435 | 0.0943 | 0.0160 |

Benchmark Instance | A | B | C |
---|---|---|---|

Reference value | 310,048.77 | 360,186.79 | 3,399,273.00 |

Minimal | 314,772.06 | 360,344.72 | 3,417,691.86 |

Average | 341,912.54 | 388,651.12 | 3,524,354.39 |

Dif. ref-min (%) | 1.52 | 0.04 | 0.54 |

Dif. ref-avg (%) | 10.28 | 7.90 | 3.68 |

Standard dev. | 12,411.75 | 37,700.39 | 59,072.10 |

Coefficient of variation | 0.0363 | 0.0970 | 0.0168 |

Benchmark Instance | A | B | C |
---|---|---|---|

Reference value | 310,048.77 | 360,186.79 | 3,399,273.00 |

Minimal | 310,048.77 | 360,186.79 | 3,399,273.00 |

Average | 312,878.10 | 360,214.37 | 3,420,905.56 |

Dif. ref-min (%) | $2\times {10}^{-7}$ | $2\times {10}^{-7}$ | 0 |

Dif. ref-avg (%) | 0.91 | 0.01 | 0.64 |

Standard dev. | 5747.78 | 58.32 | 17,009.00 |

Coefficient of variation | 0.0184 | 0.0002 | 0.0050 |

Benchmark Instance | A | B | C |
---|---|---|---|

Reference value | 310,048.77 | 360,186.79 | 3,399,273.00 |

Minimal | 310,048.77 | 360,186.79 | 3,399,823.32 |

Average | 312,993.06 | 360,484.42 | 3,419,586.89 |

Dif. ref-min (%) | $2\times {10}^{-7}$ | 0 | 0.02 |

Dif. ref-avg (%) | 0.95 | 0.08 | 0.60 |

Standard dev. | 5639.52 | 2671.73 | 13,667.40 |

Coefficient of variation | 0.0180 | 0.0074 | 0.0040 |

Benchmark Instance | A | B | C |
---|---|---|---|

Reference value | 310,048.77 | 360,186.79 | 3,399,273.00 |

Minimal | 310,048.77 | 360,186.79 | 3,400,034.97 |

Average | 313,718.26 | 360,207.91 | 3,422,185.23 |

Dif. ref-min (%) | $2\times {10}^{-7}$ | $2\times {10}^{-7}$ | $6.22\times {10}^{-3}$ |

Dif. ref-avg (%) | 1.18 | $5.87\xb7{10}^{-3}$ | 0.66 |

Standard dev. | 6288.90 | 41.25 | 15,981.47 |

Coefficient of variation | 0.0200 | 0.0001 | 0.0047 |

Benchmark Instance | EHW1 | MHW1 |
---|---|---|

Dif. ref-min (%) | 2.86 | 0 |

Dif. ref-avg (%) | 7.86 | 0.64 |

Coefficient of variation | 0.0249 | 0.0050 |

Benchmark Instance | A | B | C |
---|---|---|---|

EHW1 | 2.97 | 1.10 | 2.86 |

EHW2 | 1.45 | 0.10 | 0.25 |

EHW3 | 1.52 | 0.04 | 0.54 |

Benchmark Instance | A | B | C |
---|---|---|---|

EHW1 | 19.17 | 17.24 | 7.86 |

EHW2 | 10.64 | 10.47 | 3.35 |

EHW3 | 10.28 | 7.90 | 3.68 |

Benchmark Instance | A | B | C |
---|---|---|---|

EHW1 | 0.0651 | 0.1010 | 0.0249 |

EHW2 | 0.0435 | 0.0944 | 0.0160 |

EHW3 | 0.0363 | 0.0970 | 0.0168 |

Benchmark Instance | A | B | C |
---|---|---|---|

EHW1 | 125 | 108 | 79 |

EHW2 | 55 | 54 | 50 |

EHW3 | 51 | 47 | 44 |

MHW1 | 22 | 25 | 27 |

MHW2 | 8 | 8 | 8 |

MHW3 | 6 | 7 | 7 |

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## Share and Cite

**MDPI and ACS Style**

Němec, P.; Stodola, P.; Pecina, M.; Neubauer, J.; Blaha, M.
Optimization of the Weighted Multi-Facility Location Problem Using MS Excel. *Algorithms* **2021**, *14*, 191.
https://doi.org/10.3390/a14070191

**AMA Style**

Němec P, Stodola P, Pecina M, Neubauer J, Blaha M.
Optimization of the Weighted Multi-Facility Location Problem Using MS Excel. *Algorithms*. 2021; 14(7):191.
https://doi.org/10.3390/a14070191

**Chicago/Turabian Style**

Němec, Petr, Petr Stodola, Miroslav Pecina, Jiří Neubauer, and Martin Blaha.
2021. "Optimization of the Weighted Multi-Facility Location Problem Using MS Excel" *Algorithms* 14, no. 7: 191.
https://doi.org/10.3390/a14070191