# Eliminating Rank Reversal Problem Using a New Multi-Attribute Model—The RAFSI Method

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## Abstract

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## 1. Introduction

- Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) [3],
- Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [6],
- Analytical Hierarchy Process (AHP) [7],
- Elimination Et Choice Translating Reality (ELECTRE) [8],
- Multi-Attributive Border Approximation Area Comparison (MABAC) [9],
- Complex Proportional Assessment (COPRAS) [10],
- Combinative Distance-based Assessment (CODAS) [11],
- lattice MADM methods [12].

## 2. RAFSI Method

_{j}, j = 1, 2, …. n) meet the following condition $\sum _{j=1}^{n}{w}_{j}}=1$. Criteria ${C}_{1},{C}_{2},\dots ,{C}_{n}$ can be maximizing type (max) or minimizing type (min). Alternatives ${A}_{i}(i=1,2,\dots ,m)$ are defined by their respective values (a

_{ij}) on each criterion (c

_{j}). The initial decision matrix is shown as follows.

- (a)
- ${C}_{j}\in \left[{a}_{{N}_{j}},{a}_{{I}_{j}}\right]$, when ${C}_{j}$ belongs to max type criteria and
- (b)
- ${C}_{j}\in \left[{a}_{{I}_{j}},{a}_{{N}_{j}}\right]$, when ${C}_{j}$ belongs to min type criteria.

- (a)
- for max type criteria, if there is ${a}_{{x}_{j}}$ where ${a}_{{x}_{j}}>{a}_{{I}_{j}}$, then we have equality $f\left({a}_{{x}_{j}}\right)=f\left({a}_{{I}_{j}}\right)$
- (b)
- for min type criteria, if there is ${a}_{{x}_{j}}$ where ${a}_{{x}_{j}}<{a}_{{I}_{j}}$, then we have equality $f\left({a}_{{x}_{j}}\right)=f\left({a}_{{I}_{j}}\right)$

- (a)
- for the criteria C
_{j}($j=1,2,\dots ,n$) max type:$${\widehat{s}}_{ij}=\frac{{s}_{ij}}{2A}$$ - (b)
- for the criteria C
_{j}($j=1,2,\dots ,n$) min type:$${\widehat{s}}_{ij}=\frac{H}{2{s}_{ij}}$$

- (a)
- For max type criteria C
_{j}($j=1,2,\dots ,n$), we have the following condition.$$0<\frac{{n}_{1}}{2A}\le {\widehat{s}}_{ij}\le \frac{{n}_{2k}}{2A}<1$$Proof of (10):$$\frac{{n}_{2k}}{2A}=\frac{{n}_{2k}}{2\frac{{n}_{1}+{n}_{2k}}{2}}=\frac{{n}_{2k}}{{n}_{1}+{n}_{2k}}<\frac{{n}_{2k}+{n}_{1}}{{n}_{1}+{n}_{2k}}=1$$ - (b)
- for min type criteria C
_{j}($j=1,2,\dots ,n$), we have the following condition.$$0<\frac{H}{2{n}_{2k}}\le {\widehat{s}}_{ij}\le \frac{H}{2{n}_{1}}<1$$Proof of (11):$$\frac{H}{2{n}_{1}}=\frac{\frac{2}{\frac{1}{{n}_{2k}}+\frac{1}{{n}_{1}}}}{2{n}_{1}}=\frac{1}{{n}_{1}\left(\frac{1}{{n}_{2k}}+\frac{1}{{n}_{1}}\right)}=\frac{1}{1+\frac{{n}_{1}}{{n}_{2k}}}<1$$

_{i})) are calculated according to Equation (14) below. Alternatives are then ranked according to the descending order of the calculated (V(A

_{i})) values.

## 3. Case Study and Results

- (a)
- for max type criteria: ${C}_{1}\in \left[120,200\right]$; ${C}_{2}\in \left[6,12\right]$ i ${C}_{5}\in \left[2,8\right]$,
- (b)
- for min type criteria: ${C}_{3}\in \left[10,20\right]$ i ${C}_{4}\in \left[100,200\right]$.

_{i}-C

_{1}are obtained using the functions ${f}_{{A}_{i}}\left({C}_{1}\right)=0.06\cdot {C}_{1}-6.50$:

_{i}) of the alternatives are calculated, as exhibited in Table 1. Ranking pre-order of the alternatives is derived as per the descending order of V(A

_{i}) values, where the alternative with higher V(A

_{i}) values are always preferred.

## 4. Validation of the Results

#### 4.1. Comparing the Results with Other MADM Methods

- (1)
- In the first stage, the COPRAS and TOPSIS methods were slightly modified through the use of additive data normalization techniques in both methods. It was observed that both methods gave the same ranking order (Figure 2) for the considered alternatives under additive data normalization.
- (1)
- In the second stage, data normalization, as suggested in RAFSI method, was also used for TOPSIS, VIKOR, and COPRAS methods. After using the new normalization technique, identical rankings were obtained by all the methods. Based on these results, it can be concluded that the RAFSI method gives credible and reliable results.

#### 4.2. Rank Reversal Problem

- (1)
- COPRAS method: The new candidate A7 was ranked second in the ranking order, so it is clear that all the candidates (except the first ranked) moved one place down in the ranking order. Furthermore, it is expected that the values of criteria functions f(Ai) for an old set of the alternatives f(A1), f(A2), …, f(A6) would not change, which signifies the function of an f(A7) of the new alternative would be ranked based on the old values of f(Ai). However, from Table 3, after introducing the new alternative, a change in f(Ai) values are observed for the COPRAS method. This fact can cause inconsistencies in ranking order of the alternatives.
- (2)
- TOPSIS method: The introduction of the new alternative resulted in a significant change in the ranking order as well as changes in f(Ai) values that are also observed. Alternative A7 is placed in the second position. Therefore, it is clear that the ranks of the other alternatives moved one place down. However, the same did not happen for alternative A3 as it remained in the fourth position in both new and old sets of alternatives. Additionally, alternative A2 was third in the old set, while, in the new set, it is in the fifth position instead of the fourth. These kinds of changes in alternatives’ ranking are observed with changes in f(Ai) values.
- (3)
- VIKOR method: In this method, similar changes happened as in the previous two methods. The new alternative A7 is placed in the third position. It is expected that, in the new set of alternatives, all the alternatives below the third rank would move one place down. However, some more drastic changes are noticed in the VIKOR method. For example, alternative A3 was in the fifth rank in the old set, but, in the new set, it is ranked last. Moreover, alternative A6 was last in the old set of alternatives, while it is in the second to last in the new set of alternatives. These changes in the ranking order also followed with the changes in f(Ai) values.
- (4)
- RAFSI method: This method showed stability in both sets of alternatives. All the alternatives kept the same f(Ai) values in both sets. Thus, it can be concluded that the RAFSI method has shown logical results following the new set of alternatives.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Alternative | V(A_{i}) | Rank |
---|---|---|

A1 | 0.5081 | 2 |

A2 | 0.4522 | 4 |

A3 | 0.4381 | 5 |

A4 | 0.4560 | 3 |

A5 | 0.5299 | 1 |

A6 | 0.4373 | 6 |

Alternative | S0 | S1 | S2 | S3 | S4 | S5 |
---|---|---|---|---|---|---|

A5 | 1 | 1 | 1 | 1 | 1 | 1 |

A1 | 2 | 2 | 2 | 2 | 2 | |

A4 | 3 | 3 | 3 | 3 | ||

A2 | 4 | 4 | 4 | |||

A3 | 5 | 5 | ||||

A6 | 6 |

f(Ai) | A1 | A2 | A3 | A4 | A5 | A6 | A7 | |
---|---|---|---|---|---|---|---|---|

RAFSI | Original f(Ai) | f(A1) = 0.508 | f(A2) = 0.452 | f(A3) = 0.438 | f(A4) = 0.456 | f(A5) = 0.530 | f(A6) = 0.437 | |

Original rank | 2 | 4 | 5 | 3 | 1 | 6 | ||

New f(Ai) | f(A1) = 0.508 | f(A2) = 0.452 | f(A3) = 0.438 | f(A4) = 0.456 | f(A5) = 0.530 | f(A6) = 0.437 | f(A7) = 0.495 | |

New rank | 2 | 5 | 6 | 4 | 1 | 7 | 3 | |

VIKOR | Original f(Ai) | f(A1) = 0.350 | f(A2) = 0.901 | f(A3) = 0.924 | f(A4) = 0.801 | f(A5) = 0.00 | f(A6) = 0.928 | |

Original rank | 2 | 4 | 5 | 3 | 1 | 6 | ||

New f(Ai) | f(A1) = 0.274 | f(A2) = 0.817 | f(A3) = 1.000 | f(A4) = 0.738 | f(A5) = 0.00 | f(A6) = 0.920 | f(A7) = 0.718 | |

New rank | 2 | 5 | 7 | 4 | 1 | 6 | 3 | |

TOPSIS | Original f(Ai) | f(A1) = 0.542 | f(A2) = 0.464 | f(A3) = 0.431 | f(A4) = 0.396 | f(A5) = 0.704 | f(A6) = 0.351 | |

Original rank | 2 | 3 | 4 | 5 | 1 | 6 | ||

New f(Ai) | f(A1) = 0.468 | f(A2) = 0.400 | f(A3) = 0.410 | f(A4) = 0.340 | f(A5) = 0.593 | f(A6) = 0.311 | f(A7) = 0.507 | |

New rank | 3 | 5 | 4 | 6 | 1 | 7 | 2 | |

COPRAS | Original f(Ai) | f(A1) = 0.964 | f(A2) = 0.950 | f(A3) = 0.951 | f(A4) = 0.932 | f(A5) = 1.00 | f(A6) = 0.930 | |

Original rank | 2 | 4 | 3 | 5 | 1 | 6 | ||

New f(Ai) | f(A1) = 0.962 | f(A2) = 0.952 | f(A3) = 0.957 | f(A4) = 0.933 | f(A5) = 1.00 | f(A6) = 0.933 | f(A7) = 0.998 | |

New rank | 3 | 5 | 4 | 6 | 1 | 7 | 2 |

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**MDPI and ACS Style**

Žižović, M.; Pamučar, D.; Albijanić, M.; Chatterjee, P.; Pribićević, I.
Eliminating Rank Reversal Problem Using a New Multi-Attribute Model—The RAFSI Method. *Mathematics* **2020**, *8*, 1015.
https://doi.org/10.3390/math8061015

**AMA Style**

Žižović M, Pamučar D, Albijanić M, Chatterjee P, Pribićević I.
Eliminating Rank Reversal Problem Using a New Multi-Attribute Model—The RAFSI Method. *Mathematics*. 2020; 8(6):1015.
https://doi.org/10.3390/math8061015

**Chicago/Turabian Style**

Žižović, Mališa, Dragan Pamučar, Miloljub Albijanić, Prasenjit Chatterjee, and Ivan Pribićević.
2020. "Eliminating Rank Reversal Problem Using a New Multi-Attribute Model—The RAFSI Method" *Mathematics* 8, no. 6: 1015.
https://doi.org/10.3390/math8061015