# A Model for Determining Weight Coefficients by Forming a Non-Decreasing Series at Criteria Significance Levels (NDSL)

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## Abstract

**:**

## 1. Introduction

## 2. Model for Determining Weight Coefficients by Forming a Non-Decreasing Series at Criteria Significance Levels

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 2.1. Forming a Non-Decreasing Series at Criteria Significance Levels

- Level ${L}_{1}$: At the ${L}_{1}$ level, the criteria from within the set S whose significance is equal to the significance of the criterion ${C}_{1}$ or up to two times as small as the significance of the criterion ${C}_{1}$ should be grouped. The criterion ${C}_{i}$ belonging to the ${L}_{1}$ level will be presented as ${C}_{i}\in [1,2),i\in \left\{1,2,\dots ,n\right\}$;
- Level ${L}_{2}$: At the ${L}_{2}$ level, the criteria from within the set S whose significance is exactly two times as small as the significance of the criterion ${C}_{1}$ or up to three times as small as the significance of the criterion ${C}_{1}$ should be grouped. The criterion ${C}_{i}$ belonging to the ${L}_{2}$ level will be presented as ${C}_{i}\in [2,3),i\in \left\{1,2,\dots ,n\right\}$;
- Level ${L}_{3}$: At the ${L}_{3}$ level, the criteria from within the set S whose significance is exactly three times as small as the significance of the criterion ${C}_{1}$ or up to four times as small as the significance of the criterion ${C}_{1}$ should be grouped. The criterion ${C}_{i}$ belonging to the ${L}_{3}$ level will be presented as ${C}_{i}\in [3,4),i\in \left\{1,2,\dots ,n\right\}$;
- Level ${L}_{k}$: At the ${L}_{k}$ level, the criteria from within the set S whose significance is exactly k times as small as the significance of the criterion ${C}_{1}$ or up to k + 1 times as small as the significance of the criterion ${C}_{1}$ should be grouped. The criterion ${C}_{i}$ belonging to the ${L}_{k}$ level will be presented as ${C}_{i}\in [k,k+1),i\in \left\{1,2,\dots ,n\right\}$.

- Level ${L}_{1}$: For ${C}_{i}\in [1,2)$, i.e., for ${t}_{1}=1$, it follows that ${\alpha}_{i}=0$, whereas for ${t}_{1}=2$, we obtain ${\alpha}_{i}=N/3$. Therefore, it follows that the values of the significance of the criteria (${\alpha}_{i}$) at the ${L}_{1}$ level range in the interval $0\le {\alpha}_{i}<N/3$, i.e., ${C}_{i}\in [1,2)\Rightarrow 0\le {\alpha}_{i}<N/3$;
- Level ${L}_{2}$: For ${C}_{i}\in [2,3)$, i.e., for ${t}_{2}=2$, it follows that ${\alpha}_{i}=N/3$, whereas for ${t}_{1}=3$, we obtain ${\alpha}_{i}=N/2$. The values of the significance of the criteria (${\alpha}_{i}$) at the ${L}_{2}$ level range in the interval $N/3\le {\alpha}_{i}<N/2$, i.e., ${C}_{i}\in [2,3)\Rightarrow N/3\le {\alpha}_{i}<N/2$;
- Level ${L}_{k}$: For ${C}_{i}\in [k,k+1)$, i.e., for ${t}_{k}=k$, it follows that ${\alpha}_{i}=N\cdot (k-1)/(k+1)$, whereas for ${t}_{k}=k+1$, we obtain ${\alpha}_{i}=N\cdot k/(k+2)$. The values of the significance of the criteria (${\alpha}_{i}$) at the ${L}_{k}$ level range in the interval $N\cdot (k-1)/(k+1)\le {\alpha}_{i}<N\cdot k/(k+2)$, i.e., ${C}_{i}\in [k,k+1)\Rightarrow N\cdot (k-1)/(k+1)\le {\alpha}_{x}<N\cdot k/(k+2)$.

**Example**

**1.**

- 1.
- Level ${L}_{1}$: ${C}_{i}\in [1,2)\Rightarrow 0\le {\alpha}_{i}<N/3$, then we have ${C}_{i}\in [1,2)\Rightarrow 0\le {\alpha}_{x}<50/3$;
- 2.
- Level ${L}_{2}$: ${C}_{i}\in [2,3)\Rightarrow N/3\le {\alpha}_{i}<N/2$, then we have ${C}_{i}\in [2,3)\Rightarrow 50/3\le {\alpha}_{x}<25$;
- 3.
- Level ${L}_{3}$: ${C}_{i}\in [3,4)\Rightarrow N\cdot 2/4\le {\alpha}_{x}<N\cdot 3/5$, then we have ${C}_{i}\in [3,4)\Rightarrow 25\le {\alpha}_{x}<30$.

#### 2.2. Steps of the NDSL Model

- Level ${L}_{1}$: At the ${L}_{1}$ level, the criteria from within the set S whose significance is equal to the significance of the criterion ${C}_{1}$ or up to two times as small as the significance of the criterion ${C}_{1}$ should be grouped. The criterion ${C}_{i}$ belonging to the ${L}_{1}$ level will be presented as ${C}_{i}\in [1,2),i\in \left\{1,2,\dots ,n\right\}$;
- Level ${L}_{2}$: At the ${L}_{2}$ level, the criteria from within the set S whose significance is exactly two times as small as the significance of the criterion ${C}_{1}$ or up to three times as small as the significance of the criterion ${C}_{1}$ should be grouped. The criterion ${C}_{i}$ belonging to the ${L}_{2}$ level will be presented as ${C}_{i}\in [2,3),i\in \left\{1,2,\dots ,n\right\}$;
- Level ${L}_{k}$: At the ${L}_{k}$ level, the criteria from within the set S whose significance is exactly k times as small as the significance of the criterion ${C}_{1}$ or up to k + 1 times as small as the significance of the criterion ${C}_{1}$ should be grouped. The criterion ${C}_{i}$ belonging to the ${L}_{k}$ level will be presented as ${C}_{i}\in [k,k+1),i\in \left\{1,2,\dots ,n\right\}$.

- Level ${L}_{1}$: For ${C}_{i}\in [1,2)$, the values of the significance of criteria (${\alpha}_{i}$) range in the interval $0\le {\alpha}_{i}<N/3$, i.e., ${C}_{i}\in [1,2)\Rightarrow 0\le {\alpha}_{i}<N/3$;
- Level ${L}_{2}$: For ${C}_{i}\in [2,3)$, the values of the significance of criteria (${\alpha}_{i}$) range in the interval $N/3\le {\alpha}_{x}<N/2$, i.e., ${C}_{i}\in [2,3)\Rightarrow N/3\le {\alpha}_{i}<N/2$;
- Level ${L}_{k}$: For ${C}_{i}\in [k,k+1)$, the values of the significance of criteria (${\alpha}_{i}$) range in the interval $N\cdot (k-1)/(k+1)\le {\alpha}_{i}<N\cdot k/(k+2)$, i.e., ${C}_{i}\in [k,k+1)\Rightarrow N\cdot (k-1)/(k+1)\le {\alpha}_{x}<N\cdot k/(k+2)$.

_{l}(l = 1,2,..,m) alternative l can be obtained through the transformation of the NDSL model into a classical multi-criteria model by the application of the expression (14). By applying the simple additive weighted value function (14), which is the basic model for the majority of MCDM methods, the algorithm of the NDSL model transforms into a classical multi-criteria model, which can be used to evaluate m alternative solutions as per n optimization criteria.

## 3. Application of the NDSL Model

- Level ${L}_{1}$:$\left\{C2\right\}$;
- Level ${L}_{2}$:$\left\{C1,C4\right\}$;
- Level ${L}_{3}$:$\left\{C3\right\}$;
- Level ${L}_{4},{L}_{5},{L}_{6},{L}_{7}$: $\varnothing $;
- Level ${L}_{8}$:$\left\{C5\right\}$.

- (1)
- For Level One: Since the criterion C2 is the most significant criterion, it has been assigned the value ${\alpha}_{1}=0.$;
- (2)
- For Level Two: The criteria C1 and C4 have been estimated to have the same significance, which is exactly twice as small as the significance of the criterion C2, so they have been assigned the value ${\alpha}_{1}={\alpha}_{4}=8.33$;
- (3)
- For Level Three: The significance of the criterion C3 has been estimated to be slightly less than four times as small as the significance of the criterion C2, so it has been assigned the value ${\alpha}_{4}=14.9$;
- (4)
- For Level Eight: The significance of the criterion C5 has been estimated to be slightly more than eight times as small as the significance of the criterion C2, so it has been assigned the value ${\alpha}_{4}=19.5$;

## 4. Comparison and Discussion

_{AHP}= 0.029 and CR

_{BWM}= 0.000). A comparative presentation of the results of all three approaches is shown in Table 2.

_{BWM}= 0.00. Comparing criteria by applying a 9-degree scale (in the BWM), however, often leads to inconsistent results. Different from the BWM and AHP models, consistent results are always obtained when using the NDSL model because it applies an original methodology for grouping criteria as per significance, within which transitivity relations between criteria are retained. In the next part of the paper, a discussion is presented through a comparison of the NDSL model with the BWM and AHP models. The discussion aims to point to the limitations of the BWM and AHP models, which are eliminated by the application of the NDSL model. The discussion is organized through the following: (1) a comparative presentation of the number of criteria pairwise comparisons needed in the analyzed models; (2) the impact of the measuring scale on the results of the BWM, AHP, and NDSL models; (3) the consistency of the results of the analyzed models; (4) the problem of defining the best and worst criteria in the BWM and NDSL models; and (5) the problem of multi-optimality in the BWM.

## 5. Conclusions

_{B}) criterion with the C

_{x}criterion, an expert knows that the C

_{B}criterion is 2.5 times more significant than the C

_{x}criterion. In pairwise comparison methods that use the Saaty scale, such a relationship cannot be represented directly, since the Saaty scale involves only integer values. Through the formation of significance levels, the expert is given the opportunity to classify the C

_{x}criterion as belonging to another level in a logical manner, or based on their preferences, since they already know that the C

_{B}criterion is 2.5 times more important than the C

_{x}criterion. From this, we can conclude that the experts indirectly form the significance levels of the criteria. However, the mathematical formulation of existing models for pairwise comparisons requires experts to represent the significance of criteria by defining relationships over a numerical scale. In this way, criteria are indirectly grouped into levels of significance. However, such a procedure can lead to a misrepresentation of the significance of the criterion, which may be due to a misunderstanding of the mathematical apparatus of the method. Bearing all of the above in mind, the authors believe that this formulation of the interrelation between criteria enables the rational and logical expression of expert preferences, which further contributes to objective decision making.

## Author Contributions

## Funding

## Conflicts of Interest

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Criteria | C1 | C2 | C3 | C4 | C5 | Wj |
---|---|---|---|---|---|---|

C1 | 1.000 | 0.333 | 3.000 | 1.000 | 5.00 | 0.202 |

C2 | 3.000 | 1.000 | 5.000 | 3.000 | 7.00 | 0.464 |

C3 | 0.333 | 0.200 | 1.000 | 0.333 | 3.00 | 0.089 |

C4 | 1.000 | 0.333 | 3.000 | 1.000 | 5.00 | 0.202 |

C5 | 0.200 | 0.143 | 0.333 | 0.200 | 1.00 | 0.044 |

**Table 2.**A comparative presentation of the results obtained by applying the NDSL, Best Worst Method (BWM), and AHP methods.

Criteria | AHP (wi) | BWM (wi) | NDSL (wj) |
---|---|---|---|

C1 | 0.202 | 0.210 | 0.210 |

C2 | 0.464 | 0.421 | 0.421 |

C3 | 0.089 | 0.106 | 0.106 |

C4 | 0.202 | 0.210 | 0.210 |

C5 | 0.044 | 0.052 | 0.052 |

CR | 0.029 | 0.000 | - |

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**MDPI and ACS Style**

Žižović, M.; Pamučar, D.; Ćirović, G.; Žižović, M.M.; Miljković, B.D.
A Model for Determining Weight Coefficients by Forming a Non-Decreasing Series at Criteria Significance Levels (NDSL). *Mathematics* **2020**, *8*, 745.
https://doi.org/10.3390/math8050745

**AMA Style**

Žižović M, Pamučar D, Ćirović G, Žižović MM, Miljković BD.
A Model for Determining Weight Coefficients by Forming a Non-Decreasing Series at Criteria Significance Levels (NDSL). *Mathematics*. 2020; 8(5):745.
https://doi.org/10.3390/math8050745

**Chicago/Turabian Style**

Žižović, Mališa, Dragan Pamučar, Goran Ćirović, Miodrag M. Žižović, and Boža D. Miljković.
2020. "A Model for Determining Weight Coefficients by Forming a Non-Decreasing Series at Criteria Significance Levels (NDSL)" *Mathematics* 8, no. 5: 745.
https://doi.org/10.3390/math8050745