# Using Different Qualitative Scales in a Multi-Criteria Decision-Making Procedure

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**OQS**s). Likewise, in the framework of decision-making problems, some situations require that experts express their assessments taking into account their knowledge or experience by means of specific OQSs formed by linguistic terms (see Delgado et al. [1], Herrera and Herrera-Viedma [2], Herrera et al. [3] and de Andrés et al. [4], among others).

## 2. Preliminaries

**Definition**

**1**

**.**An ordinal proximity measure (

**OPM**) on $\mathcal{L}\phantom{\rule{0.166667em}{0ex}}$ with values in Δ is a mapping $\phantom{\rule{0.166667em}{0ex}}\pi :\mathcal{L}\times \mathcal{L}\u27f6\Delta $, where $\phantom{\rule{0.166667em}{0ex}}\pi ({l}_{r},{l}_{s})={\pi}_{rs}\phantom{\rule{0.166667em}{0ex}}$ represents the degree of proximity between ${l}_{r}$ and ${l}_{s}$, satisfying the following conditions:

- 1.
- Exhaustiveness: For every $\phantom{\rule{0.166667em}{0ex}}\delta \in \Delta $, there exist $\phantom{\rule{0.166667em}{0ex}}{l}_{r},{l}_{s}\in \mathcal{L}\phantom{\rule{0.166667em}{0ex}}$ such that $\phantom{\rule{0.166667em}{0ex}}\delta ={\pi}_{rs}$.
- 2.
- Symmetry: ${\pi}_{sr}={\pi}_{rs}$, for all $\phantom{\rule{0.166667em}{0ex}}r,s\in \{1,\dots ,g\}$.
- 3.
- Maximum proximity: ${\pi}_{rs}={\delta}_{1}\phantom{\rule{0.277778em}{0ex}}\iff \phantom{\rule{0.277778em}{0ex}}r=s$, for all $\phantom{\rule{0.166667em}{0ex}}r,s\in \{1,\dots ,g\}$.
- 4.
- Monotonicity: $\phantom{\rule{0.166667em}{0ex}}{\pi}_{rs}\succ {\pi}_{rt}\phantom{\rule{0.166667em}{0ex}}$ and $\phantom{\rule{0.166667em}{0ex}}{\pi}_{st}\succ {\pi}_{rt}$, for all $\phantom{\rule{0.166667em}{0ex}}r,s,t\in \{1,\dots ,g\}\phantom{\rule{0.166667em}{0ex}}$ such that $\phantom{\rule{0.166667em}{0ex}}r<s<t$.

**Definition**

**2**

**.**A linear metric on $\mathcal{L}\phantom{\rule{0.166667em}{0ex}}$ is a mapping $\phantom{\rule{0.166667em}{0ex}}d:\mathcal{L}\times \mathcal{L}\u27f6\mathbb{R}\phantom{\rule{0.166667em}{0ex}}$ satisfying the following conditions for all $\phantom{\rule{0.166667em}{0ex}}r,s,t\in \{1,\dots ,g\}$:

- 1.
- Positiveness: $\phantom{\rule{0.166667em}{0ex}}d({l}_{r},{l}_{s})\ge 0$.
- 2.
- Identity of indiscernibles: $d({l}_{r},{l}_{s})=0\phantom{\rule{0.277778em}{0ex}}\iff \phantom{\rule{0.277778em}{0ex}}{l}_{r}={l}_{s}$.
- 3.
- Symmetry: $d({l}_{s},{l}_{r})=d({l}_{r},{l}_{s})$.
- 4.
- Linearity: $\phantom{\rule{0.166667em}{0ex}}d({l}_{r},{l}_{t})=d({l}_{r},{l}_{s})+d({l}_{s},{l}_{t})\phantom{\rule{0.166667em}{0ex}}$ whenever $\phantom{\rule{0.166667em}{0ex}}r<s<t$.

**Definition**

**3**

**.**An OPM $\phantom{\rule{0.166667em}{0ex}}\pi :\mathcal{L}\times \mathcal{L}\u27f6\Delta \phantom{\rule{0.166667em}{0ex}}$ is metrizable if there exists a linear metric $\phantom{\rule{0.166667em}{0ex}}d:\mathcal{L}\times \mathcal{L}\u27f6\mathbb{R}\phantom{\rule{0.166667em}{0ex}}$ such that $\phantom{\rule{0.166667em}{0ex}}{\pi}_{rs}\succ {\pi}_{tu}\phantom{\rule{0.277778em}{0ex}}\iff \phantom{\rule{0.277778em}{0ex}}d({l}_{r},{l}_{s})<d({l}_{t},{l}_{u})$, for all $\phantom{\rule{0.166667em}{0ex}}r,s,t,u\in \{1,\dots ,g\}$.

## 3. The MCDM Procedure

#### 3.1. The Procedure

- Gather the agents’ assessments in the corresponding profiles $\phantom{\rule{0.166667em}{0ex}}{V}^{1},\dots ,{V}^{q}$.
- Calculate, for every pair of alternatives $\phantom{\rule{0.166667em}{0ex}}{x}_{i},{x}_{j}\in X$, the ordinal degrees of proximity between the assessments given by the agents, $\phantom{\rule{0.166667em}{0ex}}{\pi}^{k}\left({v}_{i}^{a,k},{v}_{j}^{a,k}\right)$, for all $\phantom{\rule{0.166667em}{0ex}}a\in A\phantom{\rule{0.166667em}{0ex}}$ and $\phantom{\rule{0.166667em}{0ex}}{c}_{k}\in C$.
- Homogenize the ordinal degrees of proximity coming from the metrizable OPMs considered in the different criteria by means of a mapping $\phantom{\rule{0.166667em}{0ex}}\rho :{\bigcup}_{k=1}^{q}{\Delta}^{k}\u27f6[0,1]$. Such mapping $\phantom{\rule{0.166667em}{0ex}}\rho \phantom{\rule{0.166667em}{0ex}}$ must satisfy the following conditions for every $\phantom{\rule{0.166667em}{0ex}}k\in \{1,\dots ,q\}$:
- Min-normalization: $\rho \left({\delta}_{1}^{k}\right)=0$.
- Max-normalization: $\rho \left({\delta}_{{h}_{k}}^{k}\right)=1$.
- Strict monotonicity: $\rho \left({\delta}_{r+1}^{k}\right)>\rho \left({\delta}_{r}^{k}\right)$, $\forall \phantom{\rule{0.166667em}{0ex}}r\in \{1,\dots ,{h}_{k}-1\}$.

- Assign a score to the alternatives, through the mapping $\phantom{\rule{0.166667em}{0ex}}S:X\u27f6\mathbb{R}\phantom{\rule{0.166667em}{0ex}}$ defined as$$\begin{array}{ccc}\hfill S\left({x}_{i}\right)& =& \sum _{a=1}^{m}\sum _{\begin{array}{c}k=1\\ {v}_{i}^{a,k}\succ {v}_{j}^{a,k}\end{array}}^{q}{w}_{k}\xb7\phantom{\rule{0.166667em}{0ex}}\rho \left({\pi}^{k}\left({v}_{i}^{a,k},{v}_{j}^{a,k}\right)\right)-\hfill \\ & & \sum _{a=1}^{m}\sum _{\begin{array}{c}k=1\\ {v}_{i}^{a,k}\prec {v}_{j}^{a,k}\end{array}}^{q}{w}_{k}\xb7\phantom{\rule{0.166667em}{0ex}}\rho \left({\pi}^{k}\left({v}_{i}^{a,k},{v}_{j}^{a,k}\right)\right).\hfill \end{array}$$In Equation (1), ${\pi}^{k}\left({v}_{i}^{a,k},{v}_{j}^{a,k}\right)$ measures for the criterion k and for the agent a, the proximity between the assessments ${v}_{i}^{a,k}$ and ${v}_{j}^{a,k}$. By means of these ordinal degrees of proximity, $S\left({x}_{i}\right)$ considers the scope of victories and defeats. The mapping $\rho $ converts the ordinal degrees of proximity to the interval [0,1]. Then, the final score is calculated multiplying the obtained results by the corresponding weights.
- Order the alternatives through the weak order ⪰ on X:$${x}_{i}\u2ab0{x}_{j}\phantom{\rule{0.277778em}{0ex}}\iff \phantom{\rule{0.277778em}{0ex}}S\left({x}_{i}\right)\ge S\left({x}_{j}\right).$$

- Anonymity: All agents are treated equally by the procedure.
- Neutrality: All alternatives are treated equally by the procedure.
- Monotonicity: If an agent improves the evaluation of an alternative on some criterion, all else remaining equal, then its score increases.
- Cancelation: In the uniform case, when two agents a and b increase and decrease at the same time their assessments, in such a way that agent a increases the assessment $\phantom{\rule{0.166667em}{0ex}}{v}_{i}^{a,k}={l}_{r}\phantom{\rule{0.166667em}{0ex}}$ to $\phantom{\rule{0.166667em}{0ex}}{l}_{r+1}\phantom{\rule{0.166667em}{0ex}}$ and agent b decreases the assessment $\phantom{\rule{0.166667em}{0ex}}{v}_{i}^{b,k}={l}_{r}\phantom{\rule{0.166667em}{0ex}}$ to $\phantom{\rule{0.166667em}{0ex}}{l}_{r-1}$, for some alternative ${x}_{i}$ and criterion ${c}_{k}$, then $S\left({x}_{i}\right)$ does not change.

**Definition**

**4.**

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. Linear Programming Formulation

**Linear Program:**

**Proposition**

**1.**

**Proof**

#### 3.3. Stochastic Analysis

- The rank acceptability indices ${r}_{i}^{p}$ (for each alternative ${x}_{i}$, for each ranking position p), which is the probability (in terms or relative frequency) that alternative ${x}_{i}$ is ranked in position p ($i,p=1,\dots ,n$).
- The pairwise winning indices $p(i,j)$ (for each pair of alternatives $({x}_{i},{x}_{j})$), which is the probability (in terms or relative frequency) that alternative ${x}_{i}$ is ranked better than ${x}_{j}$ ($i,j=1,\dots ,n$).

## 4. Practical Applications

#### 4.1. An Illustrative Example

#### 4.1.1. The Uniform Case

#### 4.1.2. A Non-Uniform Case 1

#### 4.1.3. A Non-Uniform Case 2

#### 4.1.4. Stochastic Analysis of the Non-Uniform Case 2

#### 4.2. Application of the Proposed MCDM Procedure to a Project Selection

## 5. Concluding Remarks

- The proposed procedure is developed in a multi-criteria setting where alternatives are assessed by means of a specific OQS for each criterion. This is an important difference regarding other MCDM procedures that use the same qualitative scale in all criteria.
- The procedure preserves and respects ordinal information of the OQSs by means of the concept of ordinal proximity measure that takes into account how agents perceive the proximities between the linguistic terms of the scales.
- The possibility of applying the procedure in real decision-making problems such as: clinical diagnosis, quality control, customer satisfaction measurement, etc.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**OPMs. Uniform case. (

**a**) Metrizable OPM associated with the proximity matrix ${A}_{22}$; (

**b**) Metrizable OPM associated with the proximity matrix ${A}_{2222}$; (

**c**) Metrizable OPM associated with the proximity matrix ${A}_{222}$.

**Figure 2.**OPMs. Non-uniform case 1. (

**a**) Metrizable OPM associated with the matrix ${A}_{23}$; (

**b**) Metrizable OPM associated with the matrix ${A}_{2332}$; (

**c**) Metrizable OPM associated with the matrix ${A}_{222}$.

**Figure 3.**OPMs. Non-uniform case 2. (

**a**) Metrizable OPM associated with the matrix ${A}_{32}$; (

**b**) Metrizable OPM associated with the matrix ${A}_{2332}$; (

**c**) Metrizable OPM associated with the matrix ${A}_{224}$.

**Figure 4.**OPMs corresponding to the R&D criteria. (

**a**) Metrizable OPM associated with the matrix ${A}_{23}$; (

**b**) Metrizable OPM associated with the matrix ${A}_{32}$; (

**c**) Metrizable OPM associated with the matrix ${A}_{32}$.

Location | Rooms | Service | |||||
---|---|---|---|---|---|---|---|

${l}_{1}^{1}$ | Poor | ${l}_{1}^{2}$ | Very uncomfortable | ${l}_{1}^{3}$ | Poor | ||

${l}_{2}^{1}$ | Acceptable | ${l}_{2}^{2}$ | Somewhat uncomfortable | ${l}_{2}^{3}$ | Fair | ||

${l}_{3}^{1}$ | Good | ${l}_{3}^{2}$ | Neither comfortable nor uncomfortable | ${l}_{3}^{3}$ | Good | ||

${l}_{4}^{2}$ | Somewhat comfortable | ${l}_{4}^{3}$ | Excellent | ||||

${l}_{5}^{2}$ | Very comfortable |

Weights | |
---|---|

${w}_{1}$ | 0.20 |

${w}_{2}$ | 0.50 |

${w}_{3}$ | 0.30 |

Hotel 1 | Hotel 2 | Hotel 3 | Hotel 1 | Hotel 2 | Hotel 3 | |||
---|---|---|---|---|---|---|---|---|

Location | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | ${l}_{2}^{1}$ | Location | ${l}_{3}^{1}$ | ${l}_{2}^{1}$ | ${l}_{2}^{1}$ | |

Rooms | ${l}_{4}^{2}$ | ${l}_{5}^{2}$ | ${l}_{3}^{2}$ | Rooms | ${l}_{5}^{2}$ | ${l}_{4}^{2}$ | ${l}_{4}^{2}$ | |

Service | ${l}_{2}^{3}$ | ${l}_{1}^{3}$ | ${l}_{3}^{3}$ | Service | ${l}_{4}^{3}$ | ${l}_{3}^{3}$ | ${l}_{3}^{3}$ | |

Assessments given by the agent 1. | Assessments given by the agent 2. | |||||||

Hotel 1 | Hotel 2 | Hotel 3 | Hotel 1 | Hotel 2 | Hotel 3 | |||

Location | ${l}_{1}^{1}$ | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | Location | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | ${l}_{3}^{1}$ | |

Rooms | ${l}_{5}^{2}$ | ${l}_{3}^{2}$ | ${l}_{5}^{2}$ | Rooms | ${l}_{3}^{2}$ | ${l}_{3}^{2}$ | ${l}_{4}^{2}$ | |

Service | ${l}_{1}^{3}$ | ${l}_{2}^{3}$ | ${l}_{3}^{3}$ | Service | ${l}_{2}^{3}$ | ${l}_{3}^{3}$ | ${l}_{3}^{3}$ | |

Assessments given by the agent 3. | Assessments given by the agent 4. | |||||||

Hotel 1 | Hotel 2 | Hotel 3 | Hotel 1 | Hotel 2 | Hotel 3 | |||

Location | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | ${l}_{2}^{1}$ | Location | ${l}_{3}^{1}$ | ${l}_{3}^{1}$ | ${l}_{3}^{1}$ | |

Rooms | ${l}_{3}^{2}$ | ${l}_{4}^{2}$ | ${l}_{2}^{2}$ | Rooms | ${l}_{4}^{2}$ | ${l}_{5}^{2}$ | ${l}_{3}^{2}$ | |

Service | ${l}_{4}^{3}$ | ${l}_{3}^{3}$ | ${l}_{3}^{2}$ | Service | ${l}_{3}^{3}$ | ${l}_{1}^{3}$ | ${l}_{3}^{3}$ | |

Assessments given by the agent 5. | Assessments given by the agent 6. |

Agent 1 | Agent 2 | Agent 3 | ||||||||

${\mathit{v}}_{\mathit{i}}$ | ${\mathit{v}}_{\mathit{j}}$ | $\mathit{\pi}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})$ | ${\mathit{v}}_{\mathit{i}}$ | ${\mathit{v}}_{\mathit{j}}$ | $\mathit{\pi}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})$ | ${\mathit{v}}_{\mathit{i}}$ | ${\mathit{v}}_{\mathit{j}}$ | $\mathit{\pi}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})$ | ||

H1 vs. H2 | Location | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | ${\delta}_{2}$ | ${l}_{3}^{1}$ | ${l}_{2}^{1}$ | ${\delta}_{2}$ | ${l}_{1}^{1}$ | ${l}_{2}^{1}$ | ${\delta}_{2}$ |

Rooms | ${l}_{4}^{2}$ | ${l}_{5}^{2}$ | ${\delta}_{2}$ | ${l}_{5}^{2}$ | ${l}_{4}^{2}$ | ${\delta}_{2}$ | ${l}_{5}^{2}$ | ${l}_{3}^{2}$ | ${\delta}_{3}$ | |

Service | ${l}_{2}^{3}$ | ${l}_{1}^{3}$ | ${\delta}_{2}$ | ${l}_{4}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{2}$ | ${l}_{1}^{3}$ | ${l}_{2}^{3}$ | ${\delta}_{2}$ | |

H1 vs. H3 | Location | ${l}_{2}^{1}$ | ${l}_{2}^{1}$ | ${\delta}_{1}$ | ${l}_{3}^{1}$ | ${l}_{2}^{1}$ | ${\delta}_{2}$ | ${l}_{1}^{1}$ | ${l}_{3}^{1}$ | ${\delta}_{3}$ |

Rooms | ${l}_{4}^{2}$ | ${l}_{3}^{2}$ | ${\delta}_{2}$ | ${l}_{5}^{2}$ | ${l}_{4}^{2}$ | ${\delta}_{2}$ | ${l}_{5}^{2}$ | ${l}_{5}^{2}$ | ${\delta}_{1}$ | |

Service | ${l}_{2}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{2}$ | ${l}_{4}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{2}$ | ${l}_{1}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{3}$ | |

Agent 4 | Agent 5 | Agent 6 | ||||||||

${\mathit{v}}_{\mathit{i}}$ | ${\mathit{v}}_{\mathit{j}}$ | $\mathit{\pi}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})$ | ${\mathit{v}}_{\mathit{i}}$ | ${\mathit{v}}_{\mathit{j}}$ | $\mathit{\pi}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})$ | ${\mathit{v}}_{\mathit{i}}$ | ${\mathit{v}}_{\mathit{j}}$ | $\mathit{\pi}({\mathit{v}}_{\mathit{i}},{\mathit{v}}_{\mathit{j}})$ | ||

H1 vs. H2 | Location | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | ${\delta}_{2}$ | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | ${\delta}_{2}$ | ${l}_{3}^{1}$ | ${l}_{3}^{1}$ | ${\delta}_{1}$ |

Rooms | ${l}_{3}^{2}$ | ${l}_{3}^{2}$ | ${\delta}_{1}$ | ${l}_{3}^{2}$ | ${l}_{4}^{2}$ | ${\delta}_{2}$ | ${l}_{4}^{2}$ | ${l}_{5}^{2}$ | ${\delta}_{2}$ | |

Service | ${l}_{2}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{2}$ | ${l}_{4}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{2}$ | ${l}_{3}^{3}$ | ${l}_{1}^{3}$ | ${\delta}_{3}$ | |

H1 vs. H3 | Location | ${l}_{2}^{1}$ | ${l}_{3}^{1}$ | ${\delta}_{2}$ | ${l}_{2}^{1}$ | ${l}_{2}^{1}$ | ${\delta}_{1}$ | ${l}_{3}^{1}$ | ${l}_{3}^{1}$ | ${\delta}_{1}$ |

Rooms | ${l}_{3}^{2}$ | ${l}_{4}^{2}$ | ${\delta}_{2}$ | ${l}_{3}^{2}$ | ${l}_{2}^{2}$ | ${\delta}_{2}$ | ${l}_{4}^{2}$ | ${l}_{3}^{2}$ | ${\delta}_{2}$ | |

Service | ${l}_{2}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{2}$ | ${l}_{4}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{2}$ | ${l}_{3}^{3}$ | ${l}_{3}^{3}$ | ${\delta}_{1}$ |

The Uniform Case | Non-Uniform Case 1 | Non-Uniform Case 2 |
---|---|---|

Hotel 3 | Hotel 2 | Hotel 1 |

Hotel 1 Hotel 2 | Hotel 1 | Hotel 2 |

Hotel 3 | Hotel 3 |

Alternative | Rank 1 (Best) | Rank 2 | Rank 3 |
---|---|---|---|

Hotel 1 | 0.892 | 0.108 | 0 |

Hotel 2 | 0 | 0.543 | 0.457 |

Hotel 3 | 0.108 | 0.349 | 0.543 |

**Table 7.**Pairwise winning indices: probability of row alternative being better than column alternative.

Hotel 1 | Hotel 2 | Hotel 3 | |
---|---|---|---|

Hotel 1 | - | 1.00 | 0.892 |

Hotel 2 | 0.00 | - | 0.543 |

Hotel 3 | 0.108 | 0.457 | - |

Alternative | Rank 1 (Best) | Rank 2 | Rank 3 |
---|---|---|---|

Hotel 1 | 0.340 | 0.486 | 0.174 |

Hotel 2 | 0.321 | 0.263 | 0.416 |

Hotel 3 | 0.339 | 0.251 | 0.410 |

Hotel 1 | Hotel 2 | Hotel 3 | |
---|---|---|---|

Hotel 1 | - | 0.569 | 0.517 |

Hotel 2 | 0.431 | - | 0.501 |

Hotel 3 | 0.483 | 0.499 | - |

Originality | Prospects | Qualification of the Applicant | |||||
---|---|---|---|---|---|---|---|

${l}_{1}^{1}$ | Further development of previous ideas | ${l}_{1}^{2}$ | Success is hardly probable | ${l}_{1}^{3}$ | Unknown | ||

${l}_{2}^{1}$ | There are new elements in the proposal | ${l}_{2}^{2}$ | Success is rather probable | ${l}_{2}^{3}$ | Normal | ||

${l}_{3}^{1}$ | Absolutely new idea and/or approach | ${l}_{3}^{2}$ | High probability of success | ${l}_{3}^{3}$ | High |

Originality | Prospects | Qualification | |
---|---|---|---|

${x}_{1}$ | ${l}_{3}^{1}$ | ${l}_{2}^{2}$ | ${l}_{2}^{3}$ |

${x}_{2}$ | ${l}_{2}^{1}$ | ${l}_{2}^{2}$ | ${l}_{3}^{3}$ |

${x}_{3}$ | ${l}_{1}^{1}$ | ${l}_{3}^{2}$ | ${l}_{2}^{3}$ |

${x}_{4}$ | ${l}_{3}^{1}$ | ${l}_{1}^{2}$ | ${l}_{3}^{3}$ |

${x}_{5}$ | ${l}_{2}^{1}$ | ${l}_{3}^{2}$ | ${l}_{1}^{3}$ |

${x}_{6}$ | ${l}_{1}^{1}$ | ${l}_{2}^{2}$ | ${l}_{2}^{3}$ |

${x}_{7}$ | ${l}_{1}^{1}$ | ${l}_{2}^{2}$ | ${l}_{3}^{3}$ |

${x}_{8}$ | ${l}_{1}^{1}$ | ${l}_{1}^{2}$ | ${l}_{3}^{3}$ |

${x}_{9}$ | ${l}_{2}^{1}$ | ${l}_{3}^{2}$ | ${l}_{2}^{3}$ |

Proposed Procedure |
---|

${x}_{1}$ |

${x}_{9}$ |

${x}_{2}$${x}_{4}$ |

${x}_{3}$${x}_{7}$ |

${x}_{5}$ |

${x}_{6}$ |

${x}_{8}$ |

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## Share and Cite

**MDPI and ACS Style**

González del Pozo, R.; Dias, L.C.; García-Lapresta, J.L.
Using Different Qualitative Scales in a Multi-Criteria Decision-Making Procedure. *Mathematics* **2020**, *8*, 458.
https://doi.org/10.3390/math8030458

**AMA Style**

González del Pozo R, Dias LC, García-Lapresta JL.
Using Different Qualitative Scales in a Multi-Criteria Decision-Making Procedure. *Mathematics*. 2020; 8(3):458.
https://doi.org/10.3390/math8030458

**Chicago/Turabian Style**

González del Pozo, Raquel, Luis C. Dias, and José Luis García-Lapresta.
2020. "Using Different Qualitative Scales in a Multi-Criteria Decision-Making Procedure" *Mathematics* 8, no. 3: 458.
https://doi.org/10.3390/math8030458