# Modification of the Logarithm Methodology of Additive Weights (LMAW) by a Triangular Fuzzy Number and Its Application in Multi-Criteria Decision Making

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Description of the Methodology Applied

#### 3.1. Model Description

#### 3.2. Fuzzy Number

- -
- X—universal set or a set of considerations based on which is defined the fuzzy set A;
- -
- μA(x) membership function of the element x (x ∈ X) to the set A; membership function can have any value between 0 and 1, and as the value of the function is closer to one, the membership of the element x to the set A is higher, and vice versa.

_{1}, t

_{2}, t

_{3}) is presented in Figure 2, where t

_{1}presents the left, and t

_{3}the right distribution of the confidence interval of the fuzzy number T and t

_{2}, the point in which the membership function of the fuzzy number has its maximum value.

_{1}, t

_{3}]. Therefore, a fuzzy variable can have the values only from the confidence interval. Defining confidence interval of every fuzzy variable is the task of the planner, and the most natural and often used solution is to adopt a confidence interval so that it matches the physical limits of the variable [49]. If the variable has no physical origin, some of the standard ones are adopted or it is defined as an abstract confidence interval [42]. The degree of membership is the value related to the confidence interval. In Figure 2, the confidence interval with the membership degree α can be observed, marked as $\left[{T}_{1}^{\alpha},{T}_{2}^{\alpha}\right]$.

#### 3.3. Fuzzy Logarithm Methodology of Additive Weights

## 4. Application of the FMLAW Method

#### 4.1. Defining of the Evaluation Criteria

#### 4.2. Selection of the Best Alternative Using the FLMAW Method

## 5. Validation of Results

#### 5.1. Comparison of the Results Obtained by the FLMAW with Other Methods

_{i}presents the difference between the rank of a given element in the vector w and the rank of the corresponding element in the reference vector and n presents the number of ranked elements.

#### 5.2. Change of Weight Coefficients of Criteria

#### 5.3. Validation of Results in a Simulation Software

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Criteria | Fuzzy Linguistic Descriptors | |||||
---|---|---|---|---|---|---|

Fuzzy number | All linguistic criteria | (1,1,2) | (1,2,3) | (2,3,4) | (3,4,5) | (4,4,5) |

Linguistic description | C2, C3, C4 | Very small (VS) | Small (S) | Middle (M) | High (H) | Very high (VH) |

C5, C6 | Very unfavorable (VU) | Unfavorable (U) | Middle (M) | Favorable (F) | Very favorable (VF) |

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | (8,9,9) | S | S | S | M | VU |

A2 | (5,6,7) | H | VS | VS | M | VU |

A3 | (4,5,6) | M | VS | S | VU | M |

A4 | (4,6,7) | M | H | M | VF | VF |

A5 | (3,4,6) | VH | VH | M | F | F |

A6 | (6,7,8) | VS | S | H | VU | U |

A7 | (5,6,8) | M | VS | VH | U | M |

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | (6,7,8) | S | VS | M | M | U |

A2 | (5,6,7) | VH | S | VS | M | U |

A3 | (5,5,6) | H | VS | S | M | M |

A4 | (4,5,6) | H | VH | M | F | VF |

A5 | (3,4,5) | H | VH | H | VF | VF |

A6 | (5,6,7) | S | H | M | U | U |

A7 | (7,7,8) | M | S | VH | VU | U |

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | (8,8,9) | S | S | M | M | M |

A2 | (4,5,6) | VH | S | S | U | M |

A3 | (4,5,6) | H | S | S | M | M |

A4 | (6,7,8) | M | VH | H | VF | F |

A5 | (3,3,6) | VH | VH | H | F | VF |

A6 | (5,7,8) | S | H | H | U | M |

A7 | (3,4,5) | H | S | H | VU | U |

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | (9,9,9) | M | VS | VS | M | U |

A2 | (6,6,7) | H | S | S | U | VU |

A3 | (3,4,5) | M | VS | S | U | M |

A4 | (5,6,7) | S | H | M | VF | F |

A5 | (2,5,6) | H | H | H | VF | F |

A6 | (4,5,7) | VS | M | H | VU | U |

A7 | (6,7,8) | H | VS | VH | VU | U |

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | (7.72,8.24,8.75) | (1.22,2.24,3.24) | (1,1.47,2.48) | (1.47,2.2,3.21) | (2,3,4) | (1.22,1.96,2.97) |

A2 | (4.98,5.74,6.74) | (3.49,4.49,5) | (1,1.73,2.74) | (1,1.47,2.48) | (1.47,2.48,3.49) | (1.22,1.68,2.71) |

A3 | (3.98,4.74,5.74) | (2.48,3.49,4.49) | (1,1.22,2.24) | (1,2,3) | (1.47,2.2,3.21) | (2,3,4) |

A4 | (4.73,5.98,6.99) | (1.96,2.97,3.98) | (3.49,4.49,5) | (2.24,3.24,4.24) | (3.74,4.74,5) | (3.49,4.49,5) |

A5 | (2.74,3.98,5.74) | (3.49,4.49,5) | (3.74,4.74,5) | (2.74,3.74,4.74) | (3.49,4.49,5) | (3.49,4.49,5) |

A6 | (4.98,6.23,7.49) | (1,1.47,2.48) | (2.2,3.21,4.22) | (2.74,3.74,4.74) | (1,1.47,2.48) | (1.22,2.24,3.24) |

A7 | (5.18,5.95,7.21) | (2.48,3.49,4.49) | (1,1.47,2.48) | (3.74,4.74,5) | (1,1.22,2.24) | (1.22,2.24,3.24) |

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | (1.88,1.94,2) | (1.24,1.45,1.65) | (1.2,1.29,1.5) | (1.31,1.45,1.68) | (1.4,1.6,1.8) | (1.24,1.39,1.59) |

A2 | (1.57,1.66,1.77) | (1.7,1.9,2) | (1.2,1.35,1.55) | (1.40,1.68,2) | (1.29,1.5,1.7) | (1.24,1.34,1.542) |

A3 | (1.45,1.54,1.66) | (1.5,1.7,1.9) | (1.2,1.24,1.45) | (1.33,1.5,2) | (1.29,1.44,1.64) | (1.4,1.6,1.8) |

A4 | (1.54,1.68,1.8) | (1.39,1.59,1.8) | (1.7,1.9,2) | (1.24,1.31,1.447) | (1.75,1.95,2) | (1.7,1.9,2) |

A5 | (1.31,1.45,1.65) | (1.7,1.9,2) | (1.75,1.95,2) | (1.21,1.27,1.36) | (1.7,1.9,2) | (1.7,1.9,2) |

A6 | (1.57,1.71,1.86) | (1.2,1.29,1.5) | (1.44,1.64,1.84) | (1.21,1.27,1.36) | (1.2,1.29,1.5) | (1.24,1.45,1.65) |

A7 | (1.59,1.68,1.82) | (1.5,1.7,1.9) | (1.2,1.29,1.5) | (1.2,1.21,1.27) | (1.2,1.25,1.45) | (1.24,1.45,1.65) |

The Name of the Fuzzy Linguistic Descriptor | Abbreviation | Fuzzy Number |
---|---|---|

Absolutely low | AL | (1,1,1) |

Very low | VL | (1,1.5,2) |

Low | L | (1.5,2,2.5) |

Medium | M | (2,2.5,3) |

Equal | E | (2.5,3,3.5) |

Medium high | MH | (3,3.5,4) |

High | H | (3.5,4,4.5) |

Very high | VH | (4,4.5,5) |

Absolutely high | AH | (4.5,5,5) |

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | (0.66,0.71,0.75) | (0.62,0.74,0.86) | (0.8,0.86,0.94) | (0.6,0.73,0.88) | (0.81,0.87,0.94) | (0.74,0.83,0.92) |

A2 | (0.61,0.67,0.73) | (0.72,0.79,0.87) | (0.79,0.87,0.95) | (0.62,0.77,0.93) | (0.79,0.87,0.94) | (0.75,0.82,0.91) |

A3 | (0.59,0.65,0.72) | (0.69,0.77,0.87) | (0.8,0.85,0.93) | (0.59,0.74,0.94) | (0.79,0.86,0.93) | (0.78,0.85,0.93) |

A4 | (0.61,0.67,0.74) | (0.67,0.76,0.86) | (0.86,0.91,0.95) | (0.58,0.69,0.83) | (0.85,0.89,0.94) | (0.81,0.87,0.93) |

A5 | (0.54,0.63,0.74) | (0.72,0.79,0.87) | (0.87,0.91,0.95) | (0.57,0.68,0.8) | (0.84,0.89,0.94) | (0.81,0.87,0.93) |

A6 | (0.61,0.68,0.75) | (0.61,0.71,0.84) | (0.84,0.89,0.95) | (0.57,0.68,0.8) | (0.77,0.84,0.93) | (0.74,0.83,0.93) |

A7 | (0.62,0.67,0.74) | (0.69,0.77,0.87) | (0.8,0.86,0.94) | (0.58,0.65,0.75) | (0.78,0.83,0.92) | (0.74,0.83,0.93) |

${\tilde{\mathit{Q}}}_{\mathit{i}}$ | ${\mathit{Q}}_{\mathit{i}}$ | Rank | |
---|---|---|---|

A1 | (4.235,4.738,5.296) | 4.747 | 4 |

A2 | (4.285,4.784,5.329) | 4.792 | 2 |

A3 | (4.242,4.726,5.323) | 4.745 | 5 |

A4 | (4.374,4.796,5.249) | 4.801 | 1 |

A5 | (4.357,4.77,5.219) | 4.776 | 3 |

A6 | (4.143,4.632,5.195) | 4.644 | 7 |

A7 | (4.195,4.63,5.153) | 4.645 | 6 |

FLMAW | FMAIRCA | FVIKOR | FMABAC | FCOPRAS | FSAW | |
---|---|---|---|---|---|---|

FLMAW | 1 | 0.964 | 0.964 | 0.964 | 0.964 | 1 |

FMAIRCA | 1.000 | 1.000 | 1.000 | 0.929 | 0.964 | |

FVIKOR | 1 | 1 | 0.929 | 0.964 | ||

FMABAC | 1 | 0.929 | 0.964 | |||

FCOPRAS | 1 | 0.964 | ||||

FSAW | 1 |

Initial Scenario | Scenarios 1–7 | Scenarios 8–12 | Scenarios 13–20 | |
---|---|---|---|---|

A1 | 4 | 5 | 5 | 5 |

A2 | 2 | 2 | 3 | 3 |

A3 | 5 | 4 | 4 | 4 |

A4 | 1 | 1 | 1 | 2 |

A5 | 3 | 3 | 2 | 1 |

A6 | 7 | 7 | 7 | 7 |

A7 | 6 | 6 | 6 | 6 |

v (%) | Rank | t (%) | Rank | |
---|---|---|---|---|

A1 | 6.477 | 4 | 7.12 | 5 |

A2 | 3.001 | 2 | 3.015 | 2 |

A3 | 9.055 | 6 | 6.893 | 4 |

A4 | 1.613 | 1 | 2.97 | 1 |

A5 | 4.462 | 3 | 4.103 | 3 |

A6 | 12.442 | 7 | 9.123 | 6 |

A7 | 8.041 | 5 | 9.498 | 7 |

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## Share and Cite

**MDPI and ACS Style**

Božanić, D.; Pamučar, D.; Milić, A.; Marinković, D.; Komazec, N.
Modification of the Logarithm Methodology of Additive Weights (LMAW) by a Triangular Fuzzy Number and Its Application in Multi-Criteria Decision Making. *Axioms* **2022**, *11*, 89.
https://doi.org/10.3390/axioms11030089

**AMA Style**

Božanić D, Pamučar D, Milić A, Marinković D, Komazec N.
Modification of the Logarithm Methodology of Additive Weights (LMAW) by a Triangular Fuzzy Number and Its Application in Multi-Criteria Decision Making. *Axioms*. 2022; 11(3):89.
https://doi.org/10.3390/axioms11030089

**Chicago/Turabian Style**

Božanić, Darko, Dragan Pamučar, Aleksandar Milić, Dragan Marinković, and Nenad Komazec.
2022. "Modification of the Logarithm Methodology of Additive Weights (LMAW) by a Triangular Fuzzy Number and Its Application in Multi-Criteria Decision Making" *Axioms* 11, no. 3: 89.
https://doi.org/10.3390/axioms11030089